Thermal and moisture transport in fibrous materials (Woodhead Publishing in Textiles)

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Thermal and moisture transport in fibrous materials

Related titles: Innovation and technology of women’s intimate apparel (ISBN-13: 978-1-84569-046-5; ISBN-10: 1-84569-046-X) Ladies’ intimate apparel holds the majority of the intimate apparel market and therefore leads the research in the area. Women’s intimate apparel is more complex than men’s when it comes to research and technology and the knowledge developed can easily be transferred to men’s intimate apparel where appropriate. Innovation and technology of women’s intimate apparel is the first monograph to offer a thorough review and critical appreciation of the technological and innovative developments of ladies’ intimate apparel. It describes the research principles and scientific understandings of size, materials, pattern and fit to achieve the functional and technical design of various types of lingerie.

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Clothing biosensory engineering (ISBN-13: 978-1-85573-925 3; ISBN-10: 1-85573-925-9) Clothing biosensory engineering is a systematic and integrative way of translating consumers’ biological and sensory responses and psychological feelings and preferences about clothing into perceptual elements of design. It is a link between scientific experimentation and commercial application to develop economic solutions to practical technical problems. This book quantifies the decision-making processes through which physics, mathematics, neurophysiological and engineering techniques are applied to optimally convert resources to meet various sensory requirements. It includes theoretical and experimental observations, computer simulations, test methods and examples of actual product development.

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Thermal and moisture transport in fibrous materials Edited by N. Pan and P. Gibson

CRC Press Boca Raton Boston New York Washington, DC

WOODHEAD

PUBLISHING LIMITED Cambridge, England

Published by Woodhead Publishing Limited in association with The Textile Institute Woodhead Publishing Limited, Abington Hall, Abington Cambridge CB1 6AH, England www.woodheadpublishing.com Published in North America by CRC Press LLC, 6000 Broken Sound Parkway, NW, Suite 300, Boca Raton FL 33487, USA First published 2006, Woodhead Publishing Limited and CRC Press LLC © 2006, Woodhead Publishing Limited The authors have asserted their moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the authors and the publishers cannot assume responsibility for the validity of all materials. Neither the authors nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress. Woodhead Publishing ISBN-13: 978-1-84569-057-1 (book) Woodhead Publishing ISBN-10: 1-84569-057-5 (book) Woodhead Publishing ISBN-13: 978-1-84569-226-1 (e-book) Woodhead Publishing ISBN-10: 1-84569-226-8 (e-book) CRC Press ISBN-13: 978-0-8493-9103-3 CRC Press ISBN-10: 0-8493-9103-2 CRC Press order number: WP9103 The publishers’ policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elementary chlorine-free practices. Furthermore, the publishers ensure that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by Replika Press Pvt Ltd, India Printed by T J International Limited, Padstow, Cornwall, England

Contents

Contributor contact details Introduction Part I

Textile structure and moisture transport

1

Characterizing the structure and geometry of fibrous materials

xi xiv

3

N. PAN and Z. SUN, University of California, USA

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 2

Geometrical characterization of single fibers Basic parameters for porous media Characterization of fibrous materials Mathematical descriptions of the anisotropy of a fibrous material Pore distribution in a fibrous material Tortuosity distributions in a fibrous material Structural analysis of fibrous materials with special fiber orientations Determination of the fiber orientation The packing problem References

3 4 6

19 33 37 38

Understanding the three-dimensional structure of fibrous materials using stereology

42

11 14 17

D. LUKAS and J. CHALOUPEK, Technical University of Liberec, Czech Republic

2.1 2.2 2.3 2.4 2.5 2.6

Introduction Basic stereological principles Stereology of a two-dimensional fibrous mass Stereology of a three-dimensional fibrous mass Sources of further information and advice References

42 54 64 82 98 98

vi

Contents

3

Essentials of psychrometry and capillary hydrostatics

102


3.1 3.2 3.3 3.4 3.5 3.6

Introduction Essentials of psychrometry Moisture in a medium and the moisture sorption isotherm Wettability of different material types Mathematical description of moisture sorption isotherms References

102 103 106 115 119 132

4

Surface tension, wetting and wicking

136

W. ZHONG, University of Manitoba, Canada

4.1 4.2 4.3 4.4

136 136 138

4.5 4.6

Introduction Wetting and wicking Adhesive forces and interactions across interfaces Surface tension, curvature, roughness and their effects on wetting phenomena Summary References

5

Wetting phenomena in fibrous materials

156

143 152 153

R. S. RENGASAMY, Indian Institute of Technology, India

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

156 156 158 160 167 176 178 180

5.10 5.11

Introduction Surface tension Curvature effect of surfaces Capillarity Surface roughness of solids Hysteresis effects Meniscus Instability of liquid flow Morphological transitions of liquid bodies in parallel fiber bundles Sources of further information and advice References

6

Interactions between liquid and fibrous materials

188

183 184 184


6.1 6.2 6.3 6.4 6.5

Introduction Fundamentals Complete wetting of curved surfaces Liquid spreading dynamics on a solid surface Rayleigh instability

188 188 193 195 199

Contents

6.6 6.7 6.8

Lucas–Washburn theory and wetting of fibrous media Understanding wetting and liquid spreading References

vii

203 214 219

Part II Heat–moisture interactions in textile materials 7

Thermal conduction and moisture diffusion in fibrous materials

225

Z. SUN and N. PAN, University of California, USA

7.1 7.2 7.3 7.4 7.5

225 226 233 237

7.6 7.7 7.8 7.9 7.10 7.11

Introduction Thermal conduction analysis Effective thermal conductivity for fibrous materials Prediction of ETC by thermal resistance networks Structure of plain weave woven fabric composites and the corresponding unit cell Prediction of ETC by the volume averaging method The homogenization method Moisture diffusion Sensory contact thermal conduction of porous materials Future research References

8

Convection and ventilation in fabric layers

271

241 249 259 262 265 266 266

N. GHADDAR, American University of Beirut, Lebanon; K. GHALI, Beirut Arab University, Lebanon; and B. JONES, Kansas State University, USA.

8.1 8.2 8.3

8.5 8.6 8.7

Introduction Estimation of ventilation rates Heat and moisture transport modelling in clothing by ventilation Heat and moisture transport results of the periodic ventilation model Extension of model to real limb motion Nomenclature References

298 301 302 305

9

Multiphase flow through porous media

308

8.4

271 275 283

P. GIBSON, U.S. Army Soldier Systems Center, USA

9.1 9.2 9.3 9.4

Introduction Mass and energy transport equations Total thermal energy equation Thermodynamic relations

308 308 328 336

viii

Contents

9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12

Mass transport in the gas phase Gas phase convective transport Liquid phase convective transport Summary of modified transport equations Comparison with previously derived equations Conclusions Nomenclature References

338 340 341 344 347 351 352 355

10

The cellular automata lattice gas approach for fluid flows in porous media

357

D. LUKAS and L. OCHERETNA, Technical University of Liberec, Czech Republic

10.1 10.2 10.3 10.4

357 364 378

10.5 10.6

Introduction Discrete molecular dynamics Typical lattice gas automata Computer simulation of fluid flows through porous materials Sources of further information and advice References

11

Phase change in fabrics

402

381 395 399

K. GHALI, Beirut Arab University, Lebanon; N. GHADDAR, American University of Beirut, Lebanon; and B. JONES, Kansas State University, USA

11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 12

Introduction Modelling condensation/evaporation in thin clothing layers Modelling condensation/evaporation in a fibrous medium Effect of fabric physical properties on the condensation/ evaporation process Modelling heating and moisture transfer in PCM fabrics Conclusions Nomenclature References Heat–moisture interactions and phase change in fibrous material

402 407 411 416 418 420 421 422 424

B. JONES, Kansas State University, USA; K. GHALI, Beirut Arab University, Lebanon; and N. GHADDAR, American University of Beirut, Lebanon

12.1 12.2 12.3

Introduction Moisture regain and equilibrium relationships Sorption and condensation

424 426 427

Contents

12.4 12.5 12.6 12.7

Mass and heat transport processes Modeling of coupled heat and moisture transport Consequences of interactions between heat and moisture References

ix

428 431 434 436

Part III Textile–body interactions and modelling issues 13

Heat and moisture transfer in fibrous clothing insulation

439

Y.B. LI and J. FAN, The Hong Kong Polytechnic University, Hong Kong

13.1 13.2 13.3 13.4 13.5 13.6 13.7

Introduction Experimental investigations Theoretical models Numerical simulation Conclusions Nomenclature References

439 439 448 456 463 465 466

14

Computer simulation of moisture transport in fibrous materials

469

D. LUKAS, E. KOSTAKOVA and A. SARKAR, Technical University of Liberec, Czech Republic

14.1 14.2 14.3 14.4 14.5 15

Introduction Auto-models Computer simulation Sources of further information and advice References

470 478 509 536 538

Computational modeling of clothing performance

542

P. GIBSON, U.S. Army Soldier Systems Center, USA; J. BARRY and R. HILL, Creare Inc, USA; P. BRASSER, TNO Prins Maurits Laboratory, The Netherlands; and M. SOBERA and C. KLEIJN, Delft University of Technology, The Netherlands

15.1 15.2 15.3 15.4 15.5 15.6 15.7

Introduction Material modeling Material modeling example Modeling of fabric-covered cylinders Full-body modeling Conclusions References

542 543 545 546 554 558 558

x

Contents

16

The skin’s role in human thermoregulation and comfort

560

E. ARENS and H. ZHANG, University of California, Berkeley, USA

16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9

Introduction Body–environment exchange Skin Heat exchange at the skin surface Moisture exchange at the skin surface Typical skin temperatures Sensation and comfort Modeling human thermal regulation and comfort References

560 561 564 578 584 585 589 596 597

Index

603

Contributor contact details

(* = main contact)

Editors

Chapter 2

Dr Ning Pan Division of Textiles and Clothing University of California One Shields Ave Davis CA 95616-8722 USA

Professor David Lukas* Katedra netkanych textilii Technicka univerzita v Liberci Halkova 6 Liberec 1 Czech Republic 461 17

E-mail: [email protected]

E-mail: [email protected] [email protected]

Phil Gibson Macromolecular Science Team AMSRD-NSC-SS-MS Building 3 (Research) Room 321 U.S. Army Soldier Systems Center Natick MA 01760-5020 USA

Ing. Jiri Chaloupek Katedra netkanych textilii Technicka univerzita v Liberci Halkova 6 Liberec 1 Czech Republic 461 17


Chapter 4

Chapters 1, 3, 6 and 7 Dr Ning Pan Division of Textiles and Clothing University of California One Shields Ave Davis CA 95616-8722 USA E-mail: [email protected]


Wen Zhong Department of Textile Sciences University of Manitoba Winnipeg MB, R3T 2N2 Canada E-mail: [email protected]

xii


Chapter 5

Chapter 10

R. S. Rengasamy Department of Textile Technology Indian Institute of Technology Hauz Khas New Delhi – 110 016 India

Professor David Lukas* Katedra netkanych textilii Technicka univerzita v Liberci Halkova 6 Liberec 1 Czech Republic 461 17


E-mail: [email protected] [email protected]

Chapter 8 Professor N. Ghaddar Department of Mechanical Engineering Faculty of Engineering and Architecture American University of Beirut P.O. Box 11-236 - Riad El Solh Beirut 1107 2020 Lebanon E-mail: [email protected]

Chapter 9 and 15 Phil Gibson Macromolecular Science Team AMSRD-NSC-SS-MS Building 3 (Research) Room 321 U.S. Army Soldier Systems Center Natick MA 01760-5020 USA E-mail: [email protected]

Ing. Larisa Ocheretna Doktorand Katedra netkanych textilii Technicka univerzita v Liberci Halkova 6 Liberec 1 Czech Republic 461 17 E-mail: [email protected]

Chapters 11 Professor Kamel Ghali Department of Mechanical Engineering Beirut Arab University New Road Beirut Lebanon E-mail: [email protected]

Chapter 12 B. Jones Engineering Experiment Station Kansas State University 1048 Rathbone Hall Manhattan KS 66506-5202 USA E-mail: [email protected]


xiii

Chapter 13

Chapter 16

Jintu Fan ST606 Institute of Textiles and Clothing The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

Edward Arens* Center for the Built Environment University of California Berkeley CA 94720 USA


Center for the Built Environment University of California Berkeley CA 94720 USA

Chapter 14 Professor David Lukas* Technicka univerzita v Liberci Halkova 6 Katedra netkanych textilii Liberec 1 Czech Republic 461 17 E-mail: [email protected] [email protected] Ing. Eva Kostakova Doktorand/PhD Katedra netkanych textilii Technicka univerzita v Liberci Halkova 6 Liberec 1 Czech Republic 461 17 E-mail: [email protected] Arindam Sarkar, MTech. (Indian Institute of Technology, Delhi) Doktorand Technicka univerzita v Liberci Halkova 6 Katedra netkanych textilii Liberec, 1 Czech Republic 461 17 E-mail: [email protected]

E-mail: [email protected] Zhang Hui


Introduction

In recent years, there has been a resurgence in the opportunities and challenges facing engineers, chemists, and textile scientists responsible for developing and applying new fiber-based materials. The explosive growth of manufactured nonwoven fibrous products, the continued development of textile processing technology, and the increasing applications of nanotechnology in the form of nanoparticles and nanofibers incorporated into fibrous materials have all led to the need for new approaches to characterize the behavior of these materials. The role of heat and mass transfer is often critical to the manufacture or function of devices, structures, and engineered items incorporating fibrous materials. Two aspects of thermal and moisture transport in fibrous materials are examined in this book: the basic nature of the transport process itself and the engineering factors important to the performance of manufactured articles incorporating fibrous materials. The purpose of this book is to survey the present state of the art with respect to the engineering and scientific aspects of heat and mass transfer through fibrous materials. Research on these materials is driven by the needs of industry to develop functional materials that perform well in their intended application. A welcome trend in recent years is to look outside of the textile research community to other engineering fields for insights into the properties of this unique class of engineering materials. The general treatment of fiberand textile-based materials as a soft, porous, deformable, multi-component matrix has equivalent applications in materials and fields such as particulates (soil and sand), composites, food, and biomaterials. The general approach of this book is to treat fibrous materials as ‘soft condensed matter’ in the jargon of physics, even though for many the term fibrous material is a little exotic already. However, this classification has many positive implications for the manufacture, use, and performance of these materials in that they will be just as rigorously studied as any other counterparts of engineering materials under the blessing of being a member of the group; something that has never happened systematically before. Many of the chapters in this book treat fibrous materials as a porous media – a solid material phase permeated by an interconnected network of pores (voids) xiv

Introduction

xv

filled with fluids (liquid or gas). The solid matrix and the pore spaces are assumed to form two interpenetrating phases, which may be either continuous or discontinuous. The coverage of chapters emphasizes heat and mass transfer, with mass transfer referring primarily to fluids such as gases, vapors, and liquids in continuous phases. A significant area of mass transfer that is missing from this book is particle filtration, a topic, albeit very interesting and technologically important, outside the scope of this book. The chapters in this book comprise an eclectic mix of applied, theoretical, and engineering-oriented approaches to the problem of heat and mass transfer in fibrous materials. The varied perspectives are valuable. Some of the physicsbased approaches provide a fundamental framework for understanding the interactions between the various elements of a fibrous ‘system’ of polymer fibers, pore spaces, liquids, and gases. Other chapters provide excellent applied examples that illuminate the factors contributing to the performance of a fiber-based product (such as a clothing system). More specifically, the first major issue is the description or characterization of fibrous materials, and this is covered by the first couple of chapters in the book. Nearly all the new challenges in dealing with transport phenomena in fibrous materials can be traced back to the complexities of the fibrous structures: we cannot even define such a simple physical quantity as the density of fibrous materials without running into the problem of the state of the material (fiber packing, volume fraction, etc.) How can we apply PDEs to a material system for which we even cannot define where the material boundary is? How to conveniently specify the fibrous materials and consequently incorporate the information into various governing equations would be, arguably, the most challenging problem. Another most interesting characteristic of fibrous materials is that the solid matrix is often a participating media in the overall transport process. Many treatments of transport phenomena in porous media (such as geology) treat the solid matrix as an inert space-filling substance that does not participate in mass transport other than defining the geometry of the pores. Even in heat transfer analyses, the solid matrix is often given a defined thermal conductivity, and is neglected or ignored in favor of the transport phenomena taking place in the fluid contained in the pore spaces. A deformable solid matrix composed of polymeric fibers requires that more attention be paid to the solid portion of the porous material model. Fibers can absorb the vapor or liquid phase, causing the fibers to swell or shrink. The entire porous material can be easily deformed under mechanical stress, changing such characteristics as porosity, and perhaps expelling or taking up more liquid or gas into the porous structure. Coupling phenomena between heat and mass transfer, or between mechanical stress and heat/mass transfer, can make a full analysis and simulation of the behavior of a fibrous polymeric material extremely complicated, particularly

xvi

Introduction

when various liquid and vapor components are involved. Some bold and insightful attempts in tackling the problem are reported in several chapters with more detailed deliberations. The intriguing nature of the issues in fibrous materials calls for powerful tools, and computer modeling is the most robust available: it is even able to take the structural irregularities into account. There are various ways computers can help unravel the mysteries surrounding the material, such as numerical solutions of otherwise intractable governing equations, discrete simulations using lattice or cellular automate approaches or stochastic algorithms based on thermodynamics and statistical mechanics. There are two chapters solely devoted to this topic. A few of the chapters delve into approaches and disciplines that have not yet been applied widely to the science and technology of fibrous materials, but which may provide inspiration for extending the formalism of techniques such as stereology and lattice methods to further application in this field. Another difficult area is dealing with the interfacing between fibrous materials (clothing in this case) and the human body, which is the main incentive in studying such a heat-moisture–swelling fiber complex. Our exploration in this area should not end at the manikins and we have to examine more intimately the interactions between clothing and the human body to make more sense out of this complex system. We have one whole chapter, as well as several sections spread over other individual chapters, introducing the human physiology relating to or determining skin–clothing interactions. We are in fact very pleased to have included such a component in this book, for it is very likely that nano-science, computer modeling and human physiology may revitalize textile science as a whole. N. Pan and P. Gibson

1 Characterizing the structure and geometry of fibrous materials N . P A N and Z . S U N, University of California, USA

The textile manufacturing process is remarkably flexible, allowing the manufacture of fibrous materials with widely diverse physical properties. All textiles are discontinuous materials in that they are produced from macroscopic sub-elements (finite length fibers or continuous filaments). The discrete nature of textile materials means that they have void spaces or pores that contribute directly to some of the key properties of the textiles, for example, thermal insulating characteristics, liquid absorption properties, and softness and other tactile characteristics. Fibrous materials can be defined as bulk materials made of large numbers of individual fibers, so to understand the behaviors of fibrous materials, we have to discuss issues related to single fibers. However, it should be noted that the behavior of fibrous materials is remarkably different from that of their constituent individual fibers. For instance, the same wool fiber can be used to make a summer T-shirt or a winter coat; structural factors have to be included to explain the differences.

1.1

Geometrical characterization of single fibers

1.1.1

The fiber aspect ratio

A fiber is, in essence, merely a concept associated with the shape or geometry of an object, i.e. a slender form characterized by a high aspect ratio of fiber length lf to diameter Df

s=

lf Df

[1.1]

with a small transverse dimension (or diameter) at usually 10–6 m scale.

1.1.2

The specific surface

For a given volume (or material mass) Vo, different geometric shapes generate different amounts of surface area by which to interact with the environment. 3

4


For heat and moisture transport, a shape with higher specific surface Sv value is more efficient. For a sphere

s vs =

( )

3 4π 3 1 Vo3

1 3

≈ 4.836 1 Vo3

[1.2]

For a cube

s vc = 61 Vo3 For a fiber (cylinder)

[1.3]

2π r 2 + 2 [1.4] Vo r That is, for a given volume Vo, a cubic shape will generate more surface area than a spherical shape. However, since the fiber radius r can be an independent variable as long as s vf =

l π r 2 = Vo remains constant, so r=

Vo πl

[1.5]

reduces as the fiber length increases. In other words, theoretically, the specific surface area for fiber s vf could approach infinite if r → 0 so l → ∞. This is one of the advantages of nano fibers; also why the capillary effect is most significant in fibrous materials. It may be argued that a cuboid with sides a, b and c such that the volume Vo = abc remains constant would have the same advantage, i.e. 2( ab + bc + ca ) s ve = =2 1 + 1 + 1 [1.6] Vo a b c V where we have used c = o so that c and ab cannot change independently; ab if we choose c → ∞ then ab → 0, in other words, the cuboid becomes a fiber with non-circular (rectangular) cross sectional shape.

(

)

1.2

Basic parameters for porous media

1.2.1

Total fiber amount – the fiber volume fraction Vf

For any mixture, the relative proportion of each constituent is obviously the most desirable parameter to know. There are several ways to specify the proportions, including fractions or percentages by weight or by volume.


5

For practical purpose, weight fraction is most straightforward. For a mixture of n components, the weight fraction Wi for component i (= 1, 2, …, n) is defined as Mi Mt

Wi =

[1.7]

where Mi is the net weight of the component i, and Mt is the total weight of the mixture. However, it is the volume fraction that is most often used in analysis; this can be readily calculated once the corresponding weight fractions Mi and Mt and the densities ρi and ρt are known:

Vi =

( M i / ρi ) ρ M ρ = i t = Wi t ρi ( M t / ρt ) M t ρi

[1.8]

For a fibrous material formed of fibers and air, it should be noted that, although the weight fraction of the air is small, its volume fraction is not due to its low density.

1.2.2

Porosity ε

The porosity of a material is defined as the ratio of the total void spaces volume Vv to the total body volume V:

ε=

Vv V

[1.9]

Obviously, the porosity ε is dependent on the definition of the pore sizes, for at the molecular level everything is porous. So, in the case of circular pore shape, the porosity is a function of the range of the pore size distribution from rmax to rmin

ε=

∫

rmax

rmin

dε = dr

∫

rmax

f ( r ) dr

[1.10]

rmin

where

f ( r ) = dε dr

[1.11]

is the so-called pore size probability density function (pdf) and satisfies the normalization function.

∫

∞

0

f ( r ) dr = 1

6

1.2.3


Tortuosity ξ

The tortuosity is the ratio of the body dimension l in a given direction to the length of the path lt traversed by the fluid in the transport process,

ξ=

1.2.4

lt l

[1.12]

Pore shape factor δ

The pore shape factor reflects the deviation of the pore shape from an ideal circle. In the case of an oval shape with longer axis a and shorter axis b;

δ= b a

[1.13]

Apparently, δ < 1.

1.3

Characterization of fibrous materials

Even for a fibrous material made of identical fibers, i.e. the same geometrical shapes and dimensions and physical properties, the pores formed inside the material will exhibit huge complexities in terms of sizes and shapes so as to form the capillary geometry for transporting functions. The pores will even change as the material interacts with fluids or heat during the transport process; fibers swell and the material deforms due to the weight of the liquid absorbed. Such a tremendous complexity inevitably calls for statistical or probabilistic approaches in describing internal structural characteristics such as the pore size distribution as a prerequisite for studying the transport phenomenon of the material.

1.3.1

Description of the internal structures of fibrous materials

Fibrous materials are essentially collections of individual fibers assembled via frictions into more or less integrated structures (Fig. 1.1). Any external stimulus on such a system has to be transmitted between fibers through either the fiber contacts and/or the medium filling the pores formed by the fibers. As a result, a thorough understanding and description of the internal structure becomes indispensable in attempts to study any behavior of the system. In other words, the issue of structure and property remains just as critical as in other materials such as polymers: with similar internal structures, except for the difference in scales.


1.3.2

7

Fiber arrangement – the orientation probability density function

Various analytic attempts have already been made to characterize the internal structures of the fibrous materials. There are three groups of slightly different approaches owing to the specific materials dealt with. The first group aimed at paper sheets. The generally acknowledged pioneer in this area is Cox. In his report (Cox, 1952), he tried to predict the elastic behavior of paper (a bonded planar fiber network) based on the distribution and mechanical properties of the constituent fibers. Kallmes (Kallmes and Corte, 1960; Corte and Kallmes, 1962; Kallmes and Bernier, 1963; Kallmes et al., 1963; Kallmes 1972) and Page (Seth and Page, 1975, 1996; Page et al., 1979; Page and Seth, 1980 a, b, c, 1988 Michell, Seth et al., 1983; Schulgasser and Page, 1988; Page and Howard, 1992; Gurnagul, Howard et al., 1993; Page, 1993, 2002) have contributed a great deal to this field through their research work on properties of paper. They extended Cox’s analysis by using probability theory to study fiber bonding points, the free fiber lengths between the contacts, and their distributions. Perkins (Perkins and Mark, 1976, 1983a, b; Castagnede, Ramasubramanian et al., 1988; Ramasubramanian and Perkins, 1988; Perkins and Ramasubramanian, 1989) applied micromechanics to paper sheet analysis. Dodson (Dodson and Fekih, 1991; Dodson, 1992, 1996; Dodson and Schaffnit, 1992; Deng and Dodson, 1994a, b; Schaffnit and Dodson 1994; Scharcanski and Dodson, 1997, 2000; Dodson and Sampson, 1999; Dodson, Oba et al., 2001; Scharcanski, Dodson et al., 2002) tackled the problems along a more theoretical statistics route. Another group focused on general fiber assemblies, mainly textiles and other fibrous products. Van Wyk (van Wyk, 1946) was among the first who studied the mechanical properties of a textile fiber mass by looking into the microstructural units in the system, and established the widely applied compression formula. A more complete work in this aspect, however, was carried out by Komori and his colleagues (Komori and Makishima, 1977, 1978; Komori and Itoh, 1991, 1994, 1997; Komori, Itoh et al., 1992). Through a series of papers, they predicted the mean number of fiber contact points and the mean fiber lengths between contacts (Komori and Makishima, 1977, 1978; Komori and Itoh, 1994), the fiber orientations (Komori and Itoh, 1997) and the pore size distributions (Komori and Makishima, 1978) of the fiber assemblies. Their results have broadened our understanding of the fibrous system and provided new means for further research work on the properties of fibrous assemblies. Several papers have since followed, more or less based on their results, to deal with the mechanics of fiber assemblies. Lee and Lee (Lee and Lee, 1985), Duckett and Chen (Duckett and Cheng, 1978; Chen and Duckett, 1979) further developed the theories on the compressional properties (Duckett and Cheng, 1978; Beil and Roberts, 2002). Carnaby and

8


Pan studied fiber slippage and compressional hysteresis (Carnaby and Pan, 1989), and shear properties (Pan and Carnaby, 1989). Pan also discussed the effects of the so called ‘steric hinge’ (Pan, 1993b), the fiber blend (Pan et al., 1997) and co-authored a review monograph on the theoretical characterization of internal structures of fibrous materials (Pan and Zhong, 2006). The third group is mainly concerned with fiber-reinforced composite materials. Depending on the specific cases, they chose either of the two approaches listed above with modification to better fit the problems (Pan, 1993c, 1994; Parkhouse and Kelly, 1995; Gates and Westcott, 1999 Narter and Batra et al., 1999). Although Komori and Makishima’s results are adopted hereafter, we have to caution that their results valid only for very loose structures, for if the fiber contact density increases, the effects of the steric hinge have to be accounted to reflect the fact that the contact probability changes with the number of fibers involved (Pan, 1993b, 1995).

1.3.3

Characterization of the internal structure of a fibrous material (Pan,1994)

A general fibrous structure is illustrated in Fig. 1.1. As mentioned earlier, we assume that all the properties of such a system are determined collectively by the bonded areas and the free fiber segments between the contact points as well as by the volume ratios of fibers and voids in the structure. Therefore, attention has to be focused first on the characterization of this microstructure, or more specifically, on the investigation of the density and distribution of the contact points, the relative proportions of bonded portions and the free fiber segment between two contact points on a fiber in the system of given volume V. According to the approach explored by Komori and Makishima (1977, 1978), let us first set a Cartesian coordinate system X1, X2, X3 in a fibrous Free length b

Volume V

1.1 A general fibrous structure.


9

structure, and let the angle between the X3-axis and the axis of an arbitrary fiber be θ, and that between the X1-axis and the normal projection of the fiber axis onto the X1X2 plane be φ. Then the orientation of any fiber can be defined uniquely by a pair (θ, φ), provided that 0 ≤ θ ≤ π and 0 ≤ φ ≤ π as shown in Fig. 1.2. Suppose the probability of finding the orientation of a fiber in the infinitesimal range of angles θ ~ θ + dθ and φ ~ φ + dφ is Ω(θ, φ) sin θdθdφ where Ω(θ, φ) is the still unknown density function of fiber orientation and θ is the Jacobian of the vector of the direction cosines corresponding to θ and φ. The following normalization condition must be satisfied:

∫

π

dθ

0

∫

π

dφ Ω (θ , φ ) sin θ = 1

[1.14]

0

Assume there are N fibers of straight cylinders of diameter D = 2rf and length lf in the fibrous system of volume V. According to the analysis by Komori and Makishima (1977), the average number of contacts on an arbitrary fiber, n , can be expressed as n=

2 DNl 2f l V

[1.15]

where l is a factor reflecting the fiber orientation and is defined as

Ι=

∫

π

dθ

0

∫

π

dφ J (θ , φ ) Ω (θ , φ ) sin θ

[1.16]

0

where J (θ , φ ) =

∫

π

dθ ′

0

∫

π

dφ ′ Ω (θ ′ , φ ′ ) sin χ (θ , φ , θ ′ , φ ′ )sin θ ′ [1.17]

0

(l , θ , φ )

θ

φ

1.2 The coordinates of a fiber in the system.

10


and

sin χ = [1 – (cos θ cos θ ′ + sin θ sin θ ′ cos (φ – φ ′ )) 2 ] 2

1

[1.18]

χ is the angle between two arbitrary fibers. The mean number of fiber contact points per unit fiber length has been derived by them as 2 DNl f nl = n = I = 2 DL I V V lf

[1.19]

where L = Nlf is the total fiber length within the volume V. This equation can be further reduced to nl =

2 Vf 2 DL I = πD L 8l = 8 l πD V 4V π D

[1.20]

2 where V f = πD L is the fiber volume fraction and is usually a given parameter. 4V It is seen from the result that the parameter I can be considered as an indicator of the density of contact points. The reciprocal of n l is the mean length, b , between the centers of two neighboring contact points on the fiber, as illustrated in Fig. 1.3, i.e.

b = πD 8 IV f

[1.21]

The total number of contacts in a fiber assembly containing N fibers is then given by 2 n = N n = DL I V 2

[1.22]

The factor 12 was introduced to avoid the double counting of one contact. Clearly these predicted results are the basic microstructural parameters and the indispensable variables for studies of any macrostructural properties of a fibrous system.

Contact points

Mean free length b

1.3 A representative micro-structural unit.


1.4

11

Mathematical descriptions of the anisotropy of a fibrous material

As demonstrated previously, the fiber contacts and pores in a fibrous material are entirely dependent on the way that the fibers are put together. Let us take a representative element of unit volume from a general fibrous material in such a way that a simple repetitive packing of such elements will restore the original whole material. Consider on the representative element a cross-section, as shown in Fig. 1.4, of unit area whose normal is defined by direction (Θ, Φ), just as we defined a fiber orientation previously. Here we assume all fibers are identical, with length lf and radius rf. If we ignore the contribution of air in the pores, the properties of the system in any given direction are determined completely by the amount of fiber involved in that particular direction. Since, for an isotropic system, the number of fibers at any direction should be the same, the anisotropy of the system structure is reflected by the fact that, at different directions of the system, the number of fibers involved is a function of the direction and possesses different values. Let us designate the number of fibers traveling through a cross-section of direction (Θ, Φ) as Ψ(Θ, Φ). This variable, by definition, has to be proportionally related to the fiber orientation pdf in the same direction (Pan, 1994), i.e. Ψ(Θ, Φ) = NΩ(Θ, Φ)

[1.23]

where N is a coefficient. This equation, in fact, establishes the connection between the properties and the fiber orientation for a given cross-section. The total number of fibers contained in the unit volume can be obtained by integrating the above equation over the possible directions of all the crosssections of the volume to give Fiber cut ends

Apex circle r (Θ, Φ)

r Cross-section C (Θ, Φ)

1.4 The concept of the ‘aperture circle’ of various radii on a crosssection. Adapted from Komori, T. and K. Makishima (1979). ‘Geometrical Expressions of Spaces in Anisotropic Fiber Assemblies.’ Textile Res. J., 49: 550–555.

12


∫

Ψ

Ψ ( Θ, Φ ) d Ψ ( Θ, Φ ) =

∫

π

dΘ

0

∫

π

d Φ N Ω ( Θ , Φ )sin Θ = N [1.24]

0

That is, the constant N actually represents the total number of fibers contained in the unit volume, and is related to the system fiber volume fraction Vf by the expression N=

Vf π r f2 l f

[1.25]

Then, on the given cross-section (Θ, Φ) of unit area, the average number of cut ends of the fibers having their orientations in the range of θ ~ θ + dθ and φ ~ φ + dφ is given, following Komori and Makishima (1978), as dΨ = Ψ(Θ, Φ)lf | cos χ | Ω(θ, φ) sin θdθdφ

[1.26]

where, according to analytic geometry, cos χ = cos Θ cos θ + sin Θ sin θ cos (φ – Φ)

[1.27]

with χ being the angle between the directions (Θ, Φ) and θ, φ). Since the area of a cut-fiber end at the cross-section (Θ, Φ), ∇S, can be derived as

∇S =

π r f2 , |cos χ |

[1.28]

the total area S of the cut-fiber ends of all possible orientations on the crosssection can be calculated as S ( Θ, Φ ) =

∫

π

dθ

0

= Ψ ( Θ, Φ )

∫

π

d φ × ∇S × dΨ × Ω (θ , φ )sin θ

0

∫

π

0

dθ

∫

π

d φπr f2 l f Ω (θ , φ )sin θ = Ω ( Θ , Φ ) Nπr f2 l f

0

[1.29] As S(Θ, Φ) is in fact equal to the fiber area fraction on this cross-section of unit area, i.e. S(Θ, Φ) = Af(Θ, Φ),

[1.30]

we can therefore find the relationship in a given direction (Θ, Φ) between the fiber area fraction and the fiber orientation pdf from Equations [1.29] and [1.30] A f (Θ, Φ) = Ω(Θ, Φ) Nπr f2 l f = Ω(Θ, Φ)Vf

[1.31]

This relationship has two practical yet important implications. First, it can provide a means to derive the fiber orientation pdf; at each system cross-


13

section (Θ, Φ), once we obtain through experimental measurement the fiber area fraction Af(α, Φ), we can calculate the corresponding fiber orientation pdf Ω(Θ, Φ) for a given constant Vf. So a complete relationship of Ω(Θ, Φ) versus (Θ, Φ) can be established from which the overall fiber orientation pdf can be deduced. Note that a fiber orientation pdf is by definition the function of direction only. Secondly, it shows in Equation [1.31] that the only case where Af = Vf is when the density function Ω(Θ, Φ) = 1; this happens only in the systems made of fibers unidirectionally oriented at direction (Θ, Φ). In other words, the difference between the fiber area and volume fractions is caused by fiber misorientation. The pore area fraction Aa(Θ, Φ), on the other hand, can be calculated as Aa(Θ, Φ) = 1 – Af(Θ, Φ) = 1 – Ω(Θ, Φ)Vf

[1.32]

In addition, the average number of fiber cut-ends on the plane, ν(Θ, Φ), is given as

ν ( Θ, Φ ) =

∫

π

dθ

0

∫

π

d φ × dΨ × Ω (θ , φ )sin θ

0

= N Ω( Θ, Φ ) l f

∫

π

dθ

0

=

∫

π

d φ | cos χ | Ω (θ , φ )sin θ

0

Vf Ω(Θ, Φ)ϒ(Θ, Φ) π r f2

[1.33]

where ϒ( Θ , Φ ) =

∫

π

0

dθ

∫

π

d φ | cos χ | Ω (θ , φ )sin θ

[1.34]

0

is the statistical mean value of | cos χ |. Hence, the average radius of the fiber cut-ends, ρ(Θ, Φ), can be defined as

ρ ( Θ, Φ ) =

S Θ, Φ ) = rf πν Θ , Φ )

1 ϒ ( Θ, Φ )

[1.35]

Since ϒ(Θ, Φ) ≤ 1 there is always ρ(Θ, Φ) ≥ rf. All these variables (S, ν and ρ) are important indicators of the anisotropic nature of the short-fiber system structure, and can be calculated once the fiber orientation pdf is given. Of course, the fiber area fraction can also be calculated using the mean number of the fiber cut-ends and the mean radius from Equations [1.30] and [1.35], i.e. A f (Θ, Φ) = ν(Θ, Φ)πρ2(Θ, Φ)

[1.36]

14


It should be pointed out that all the parameters derived here are the statistical mean values at a given cross-section. These parameters are useful, therefore, in calculating some system properties, such as the system elastic modulus in the direction whose values are based on averaging rules of the elastic moduli of the constituents at this cross-section. As to the study of the local heterogeneity and prediction of other system properties such as the strength and fracture behavior, which are determined by the local extreme values of the properties of the constituents, more detailed information on the local distributions of the properties of the constituents, as deduced below, is indispensable.

1.5

Pore distribution in a fibrous material

In all the previous studies on fibrous system behavior, the system is assumed, explicitly or implicitly, to be quasi-homogeneous such that the relative proportion of the fiber and air (the volume fractions) is constant throughout the system. This is to assume that fibers are uniformly spaced at every location in the system, and the distance between fibers, and hence the space occupied by air between fibers, is treated as identical. Obviously, this is a highly unrealistic situation. In practice, because of the limit of processing techniques, the fibers even at the same orientation are rarely uniformly spaced. Consequently, the local fiber/air concentration will vary from point to point in the system, even though the total fiber and air volume fractions remain constant. As mentioned above, if we need only to calculate the elastic properties such as the modulus at various directions, a knowledge of A f (Θ, Φ) alone will be adequate, as the system modulus is a statistical average quantity. However, in order to investigate the local heterogeneity and to realistically predict other system properties such as strength, fracture behavior, and impact resistance, we have to look into the local variation of the fiber fraction or the distribution of the air between fibers. In general, the distribution of air in a fibrous system is not uniform, nor is it continuous, due to the interference of fibers. If we cut a cross-section of the system, the areas occupied by the air may vary from location to location. According to Ogston (1958) and Komori and Makishima (1979), we can use the concept of the ‘aperture circle’ of various radius r, the maximum circle enclosed by fibers or the area occupied by the air in between fibers, to describe the distribution of the air in a cross-section, as seen in Fig. 1.4. In order to derive the distribution of the variable r, let us examine Fig. 1.5 where an aperture circle of radius r is placed on the cross-section (Θ, Φ) of unit area along with a fiber cut-end of radius ρ(Θ, Φ). According to Komori and Makishima (1979), these two circles will contact each other when the center of the latter is brought into the inside of the circle of radius r + ρ, concentric with the former. The probability f (r)dr, that the aperture circle


15

ΘΦ

ρ + r + dr

ρ+r

dr r

a fiber

ρ

1.5 An aperture circle of radius r is placed on the cross-section (Θ, Φ) of unit area along with a fiber cut end of radius ρ (Θ, Φ). Adapted from Komori, T. and K. Makishima (1979). ‘Geometrical Expressions of Spaces in Anisotropic Fiber Assemblies.’ Textile Res. J., 49: 550– 555.

and the fiber do not touch each other, but that the slightly larger circle of radius r + dr, does touch the fiber, is approximately equal to the probability when ν(Θ, Φ) points (the fiber cut-ends) are scattered on the plane, no point enters the circle of radius r + ρ, and at least one point enters the annular region, two radii of which are r + ρ and r + ρ + dr. When x fiber cut-ends are randomly distributed in a unit area, taking into account the area occupied by the fiber, the probability that at least one point enters the annular region 2π (r + ρ)dr is 2 xπ ( r + ρ ) dr 1 – [ π ( r + ρ ) 2 – πρ 2 ]

(x = 0, 1, 2, …)

[1.37]

and the probability of no point existing in the area π (r + ρ)2 – πρ2 is {1 –[π (r + ρ)2 – πρ 2]}x

[1.38]

Then the joint probability, fx(r)dr, that no point is contained in the circle of radius r + ρ but at least one point is contained in the circle of radius r + ρ + dr, is given by the product of the two expressions as fx(r)dr = 2xπ (r + ρ)dr{1 – [π (r + ρ)2 – πρ2]}x–1

[1.39]

Because the number of fiber cut-ends is large and they are distributed randomly, their distribution can be approximated by the Poisson’s function

16


ν x e –ν x!

[1.40]

Therefore, the distribution function of the radii of the aperture circles f(r) is given by ∞ x f ( r ) dr = Σ ν e – ν f x ( r ) dr x =0 x !

∞

= 2 πν ( r + ρ )e – ν dr Σ

x =0

= 2 πν ( r + ρ )e – ν e ν + πνρ

[ν + πνρ 2 – πν ( r + ρ ) 2 ] x –1 ( x – 1)! 2 –πν ( r + ρ ) 2

2

dr

2

= 2 πν ( r + ρ ) e πνρ e – πν ( r+ρ ) dr

[1.41]

It can be readily proved that

∫

∞

∫

f ( r ) dr =

0

∞

2

2

2

2

2 πν ( r + ρ ) e πνρ e –πν ( r+ρ ) dr = e πνρ e –πνρ = 1

0

So this function is valid as the pdf of distribution of the aperture circles filled with air, or it provides the distribution of the air at the given cross-section. The result in Equation [1.41] is different from that of Komori and Makishima (1979), which ignores the area of fiber cut-ends and hence does not satisfy the normalization condition. The average value of the radius, r ( Θ , Φ ) , can then be calculated as

r ( Θ, Φ ) =

∞

∫

rf ( r ) dr =

0

∫

∞

2

2

2 rπν ( r + ρ ) e πνρ e –πν ( r+ρ ) dr

0

= e πνρ

2

∫

∞

2 πν t ( t – ρ ) e –πνt dt ≈ e πνρ 2

2

0

=

∫

∞

2

πνρ 2 2 πν t 2 e –πνt dt = e 2 ν

0

[1.42]

where t = (r + ρ) has been used in the integration. Similarly, the variance Ξr(Θ, Φ) of the radius can be calculated as

Ξ r ( Θ, Φ ) =

∫

∞

r 2 f ( r ) dr

0

=

∫

∞

0

πνρ 2 2 2 r 2 πν ( r + ρ ) e πνρ e –πν ( r + ρ ) dr = e πν

2

=

2 r ( Θ, Φ ) π ν [1.43]


17

Note that for a given structure, the solution of the equation d Ξ r ( Θ, Φ ) =0 d ( Θ, Φ )

[1.44]

gives us the cross-sections in which the pore distribution variation reaches the extreme values, or the cross-sections with the extreme distribution nonuniformity of the air material.

1.6

Tortuosity distributions in a fibrous material

The variable r specifies only the areas of the spaces occupied by the air material. The actual volumes of the spaces are also related to the depth or length of the pores. The tortuosity is thus defined as the ratio of the length of a true flow path for a fluid and the straight-line distance between inflow and outflow in Fig. 1.6. This is, in effect, a kinematical quantity as the flow itself may alter the path. In a fibrous system, the space occupied by air material is often interrupted because of the existence or interference of fibers. If we examine a line of unit length in the direction (Θ, Φ), the average number of fiber intersections on this line is provided by Komori and Makishima (1979) and Pan (1994) as n(Θ, Φ) = 2rf Nl f J (Θ, Φ) = 2

Vf J ( Θ, Φ ) π rj

[1.45]

where J(Θ, Φ) is the mean value of | sin χ |, J ( Θ, Φ ) =

∫

π

0

dθ

∫

π

dφ |sin χ | Ω (θ , φ ) sin θ

0

a parameter reflecting the fiber misorientation. Free apex circle r (Θ, Φ)

Tortuosity lt (Θ, Φ)

1.6 Tortuosity in a fibrous material.

[1.46]

18


Following Komori and Makishima (1979) at a given direction, we define the free distance as the distance along which the air travels without interruption by the constituent fibers, or the distance occupied by the air between two interruptions by the fibers. Here we assume the interruptions occur independently. Suppose that n(Θ, Φ) segments of the free distance are randomly scattered along this line of unit length. The average length of the free distance, lm, is given as

lm ( Θ, Φ ) =

1 – πr f2 Nl f 1 – Vf = n ( Θ, Φ ) n ( Θ, Φ )

[1.47]

According to Kendall and Moran’s analysis (1963) on non-overlapping intervals on a line, the distribution of the free distance l is given as l

– f ( l ) dl = 1 e lm dl lm

[1.48]

It is easy, as well, to prove that

∫

∞

f ( l ) dl =

0

∫

∞

0

1

1 e – lm dl = 1 lm

[1.49]

This is also a better result than the one given by Komori and Makishima (1979), for their result again does not satisfy the normalization condition. We already have lm in Equation [1.47] as the mean of l, and the variance of l is given by

Ξ l ( Θ, Φ ) =

∫

∞

0

l 2 f ( l ) dl =

∫

∞

0

1

– l 2 1 e lm dl = 2 l m2 lm

[1.50]

These statistical variables can be used to specify the local variations of the fiber and air distributions or the local heterogeneity of a system. Also, because of the association of the local concentration of the constituents and system properties, these variables can be utilized to identify the irregular or abnormal features caused by the local heterogeneity in a system. However, when dealing with a system with local heterogeneity, the system properties are location dependent. Consequently, using the system or overall volume fractions will not be valid, and the concept of local fiber volume fraction is more relevant. Locations where the radius of the aperture circles and the free distance possess the highest or lowest values will likely be the most irregular spots in the system.


1.7

19

Structural analysis of fibrous materials with special fiber orientations

Since we have all the results of the parameters defining the distributions of constituents in a fibrous system, it becomes possible to predict the irregularities of the system properties. To demonstrate the application of the theoretical results obtained, we will employ the two simple and hypothetical cases below.

1.7.1

A random distribution case

For simplicity, let us first consider an ideal case where all fibers in a system are oriented in a totally random manner with no preferential direction; the randomness of fiber orientation implies that the density function is independent of both coordinates θ and φ. Therefore, this density function would have the form of [1.51] Ω(θ, φ) = Ω0 where Ω0 is a constant whose value is determined from the normalization condition as [1.52] Ω0 = 1 2π Using this fiber orientation pdf, we can calculate the system parameters by replacing (Θ, Φ) with (0, 0). The results are provided below to reveal the internal structure of the material: • cos χ = cos (Θ, θ, Φ, φ) = cos (0, θ, 0, φ) = cos θ; • sin χ = sin θ; • ϒ(Θ, Φ) = 1 ; 2 π • J(Θ, Φ) = ; 4 • A ( Θ, Φ ) = 1 V f 2π Vf • ν ( Θ, Φ ) = ; 4 ππr f2 • ρ ( Θ, Φ ) =

2r f

• n ( Θ, Φ ) =

Vf ; 2rf

• r ( Θ, Φ ) =

πr f V2 πf e ; Vf

20

Thermal and moisture transport in fibrous materials V

f 4 πr f2 • Ξ r ( Θ, Φ ) = e 2π 2 ; Vf

  • l m ( Θ , Φ ) = 2 r f  1 – 1  Vf  2

• Ξ r ( Θ, Φ ) =

8 r j2

 1   V – 1 ;  f 

The following discussion of several other system parameters provides detailed information on the distributions of both the fibers and air in this isotropic fibrous system. As seen from the above calculated results, for this given fiber orientation pdf, all of the distribution parameters are dependent on the system fiber volume fraction Vf and fiber radius rf, regardless of the fiber length lf. Therefore, we will examine the relationships between the distribution parameters and these two factors. Figure 1.7 depicts the effects of these two factors on the number of fiber cut-ends ν per unit area on an arbitrary cross-section using the calculated results. As expected, for a given system fiber volume fraction Vf, the thinner the fiber, the more fiber cut-ends per unit area, whereas for a given fiber ν (Cut ends/mm2)

rf = 5 × 10–3 mm

800

600

400

rf = 10 × 10–3 mm

200

rf = 10 × 10–3 mm Vf

0 0.2

0.4

0.6

0.8

1.7 Effects of fiber volume fraction Vf and fiber radius rf on the number of fiber cut ends ν per unit area. Adapted from Pan, N. (1994). ‘Analytical Characterization of the Anisotropy and Local Heterogeneity of Short-fiber Composites – Fiber Fraction as a Variable.’ Journal of Composite Materials 28(16): 1500–1531.


21

radius rf, increasing the system fiber volume fraction will lead to more fiber cut-ends. The distribution density function f (r) of the radius r of the aperture circles is constructed based on Equation [1.41], and the illustrated results are produced accordingly. Figure 1.8(a) shows the distribution of f (r) at three fiber radius f (r ) rf = 15 × 10–3 mm

40

vf = 0.6

30 vLf1

20

rf = 10 × 10–3 mm

vLf 2

rf = 5 × 10–3 mm

10

r (mm) 0.02 0.04 0.06 0.08 0.1 0.12 0.14

(a)

f (r ) 40

vLf 1 rf = 5 × 10–3 mm

30

vf = 0.4 20

vf = 0.6

vLf 2

10

vf = 0.2 0 0.02

0.04

0.06 (b)

0.08

r (mm) 0.1

1.8 Distribution of the aperture circles radius r in random case. Adapted from Pan, N. (1994). ‘Analytical Characterization of the Anisotropy and Local Heterogeneity of Short-fiber Composites – Fiber Fraction as a Variable.’ Journal of Composite Materials 28(16): 1500–1531. (a) at three fiber radius rf levels while Vf = 0.6. (b) at three Vf levels while rf = 5.0 × 10–3 mm (c) variance Ξr of r against Vf (d) the mean radius r against Vf .

22

Thermal and moisture transport in fibrous materials Ξr

0.025

0.02

0.015

rf = 15 × 10–3 mm 0.01 Ξro 0.01

0.005

rf = 1 0 × 1 –3 0 m rf = 5 m × 1 0 –3 mm

Vf 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (c)

r (mm)

0.14 0.12

rf = 15 × 10–3 mm

0.1 0.08

rf = 1 0 × 1 –3 0

0.06

mm

0.04 0.02

rf = 5 × 1 0 –3 mm

Vf 0.2

0.4

0.6

0.8

(d)

1.8 Continued

rf levels when the overall fiber volume fraction Vf = 0.6, whereas Fig. 1.8(b) is the result at three Vf levels when the fiber radius is fixed at rf = 5.0 × 10–3 mm. It is seen in Fig. 1.8(a) when the fiber becomes thicker, there are more aperture circles with smaller radius values. The pore sizes become less spread out. Decreasing the overall fiber volume fraction Vf has a similar effect, as seen in Fig. 1.8(b). To verify the conclusions, the variance Ξr of the aperture circle radius distribution is calculated using Equation 1.43 as shown in Fig. 1.8(c). Again, a finer fiber or a greater Vf will lower the variation of the aperture circle radius r. Moreover, since the extreme fiber volume fractions are related to


23

high variation of r values, we can define the allowable local fiber volume fraction vLf1 and vL/2 to bound the allowable variance level Ξro represented by the dotted line in the figure, and the condition Ξr ≤ Ξro will in turn determine the corresponding allowable fiber size rf and the system fiber volume fraction Vf to avoid a massive number of large aperture circles. Finally, Fig. 1.8(d) is plotted based on Equation [1.42], showing the average radius r of the aperture circles as a function of the system fiber volume fraction at three fiber size levels. The average radius of the aperture circles will decrease when either the fiber radius reduces (meaning more fibers for the given fiber volume fraction Vf), or the system fiber volume fraction increases. The distribution function f (l) of the free distance l is formed from Equation 1.48, and the results are illustrated in Fig. 1.9(a) and (b). When increasing either the fiber size rf or the system fiber volume fraction Vf , the number of free distances with shorter length will increase and those with longer length f (l ) 100

rf = 15 × 10–3 mm vLf 1 80

vf = 0.5

60

rf = 10 × 10–3 mm vLf 2

40

20

rf = 5 × 10–3 mm

l (mm)

0 0.02

0.04 0.06

0.08 (a)

0.1

0.12

0.14

1.9 Distribution of the tortuosity length l in a random case. Adapted from Pan, N. (1994). ‘Analytical Characterization of the Anisotropy and Local Heterogeneity of Short-fiber Composites – Fiber Fraction as a Variable.’ Journal of Composite Materials 28(16): 1500–1531. (a) at three fiber radius rf levels; (b) at three Vf levels; (c) variance Ξr of the tortuosity length l; (d) the mean value lm against Vf.

24

Thermal and moisture transport in fibrous materials f (l ) 50

vf = 0.6

40

rf = 15 × 10–3 mm

30

vLf 1

VLf2

20

10

vf = 0.4 vf = 0.2

0.05

0.1

0.15

0.2

0.25

0.3

l (mm)

(b) Ξr 17.5

rf = 15 × 10–3 mm

15

12.5

10

7.5

rf = 10 × 10–3 mm

5 Ξlo 2.5

rf = 5 × 10–3 mm

0.02

0.04 (c)

1.9 Continued

0.06

0.08

0.1

Vf


25

Im (mm) 0.25

rf = 15 × 10–3 mm

0.2

0.15

0.1

rf = 10 × 10–3 mm 0.05 rf = 5 × 10–3 mm 0.2

0.4 (d)

0.6

Vf 0.8

1.9 Continued

will decrease. Again, the allowable range of the free distances is defined by the two local fiber volume fractions vL f1 and vL f 2. The variance Ξi of the free distance distribution as well as the critical value Ξlo is provided in Fig. 1.9(c), and the effects of rf and Vf on Ξl are similar but less significant compared to the case in Fig. 1.8(c). Furthermore, it is interesting to see that, although the system dealt with here is an isotropic one in which all fibers are oriented in a totally random manner with no preferential direction, there still exist variations or irregularities in both r and l, leading to a variable local fiber volume fraction vLf value from location to location. In other words, the system is still a quasi-heterogeneous one. Figure 1.9(d) shows the effects of the two factors on the average free distance lm of the air material using Equation 1.49. It follows the same trend as the average radius of the aperture circles, i.e. for a given fiber volume fraction Vf, thinner fibers (more fibers contained) will lead to a shorter lm value. A reduction of lm value can also be achieved when we increase the system fiber volume fraction, while keeping the same fiber radius.

1.7.2

A planar and harmonic distribution

The planar 2-D random fiber orientation is of practical significance since planar cases are independent of the polar angle. We can hence set in the following analysis θ = Θ = π . To illustrate the effect of the structural 2 anisotropy, let us assume a harmonic pdf as the function of the base angle φ, i.e.

26


Ω(φ) = Ω0 sin φ

[1.53]

where Ω0 again is a constant whose value is determined using the normalization condition as

Ω0 = 1 2

[1.54]

Using this fiber orientation pdf, we can calculate the system parameters to illustrate the internal structure of the material. Because of the randomness of fiber orientation, all the related parameters are calculated below: • cos χ = cos (φ – Φ); • sin χ =

1 – cos 2 ( Θ , φ ) = sin (φ – Θ ) ;

• ϒ ( Φ ) = 1 cos Φ + π sin Φ ; 2 4 • J ( Φ ) = π cos Φ – 1 sin Φ ; 4 2 • A ( Φ ) = 1 V f sin Φ ; 2 Vf sin Φ cos Φ + π sin Φ • ν (Φ) = 2 4 πr j2

(

• ρ (Φ) = rf

1 ; 1 cos Φ + π sin Φ 2 4

• n (Φ) =

2Vf J ( Φ ); πr f

• r (Φ) =

sin Φ 1 e 2π ; 2 ν (Φ)

• Ξr (Φ) = • lm ( Φ ) = • Ξl (Φ) =

)

Vf

Vf

sin Φ 1 e 2π ; πν ( Φ )

(1 – V f ) πr f ; 2Vf J (Φ) (1 – V f ) 2 π 2 r f2 2 V f2 J ( Φ ) 2

;

The system parameters as the functions of direction Φ are illustrated in Fig. 1.10(a) through Fig. 1.13. The fiber orientation pdf in Equation [1.53] indicates a non-uniform fiber concentration at different directions, with lowest value


27

at Φ = 0° and the highest at Φ = 90°. This is clearly reflected in the characteristics of the aperture circle radius r shown in Fig. 1.7. Figure 1.10(a) illustrates the distribution of r at three selected directions, and Fig. 1.10(b) provides the corresponding variance of r. In Fig. 1.10(a), r value ranges with the widest span from 0 to infinity at direction Φ = 0°, but covers narrowest range at direction Φ = 90°. Consequently, the mean radius r of the aperture circles shown in Fig. 1.10(c) reaches its maximum value (approaching infinity) at direction Φ = 0° and descends to the minimum at Φ = 90°, whereas the variance in Fig. 1.10(b) is the highest at Φ = 0° and lowest at direction Φ = 90° correspondingly. (For easy comparison, the variance value at Φ = 18° direction is used in Fig. 1.10(b) to replace the infinity value at Φ = 0°. Moreover, the average number of fiber cut-ends ν(Φ) in Fig. 1.11 possesses the minimum values at Φ = 0° but the maximum values at around 70° to 80°, and becomes slightly lower at the direction Φ = 90° due to the more severe fiber-obliquity effect at high Φ levels. f (r ) 35

30

vLf 1 Φ = 90°

rf = 15 × 103 mm Vf = 0.6

25

20 Φ = 30° 15

vLf2

10

5 Φ = 0° 0

0.02

0.04

0.06

0.08

r (mm) 0.1

(a)

1.10 Distribution of the aperture circles radius r in an anisotropic case. Adapted from Pan, N. (1994). ‘Analytical Characterization of the Anisotropy and Local Heterogeneity of Short-fiber Composites – Fiber Fraction as a Variable.’ Journal of Composite Materials 28(16): 1500–1531. (a) r at three cross-sections; (b) variance Ξr of r at three cross-sections; (c) the mean value r versus the direction Φ.

28

Thermal and moisture transport in fibrous materials Ξr 0.02 Φ = 18°

0.015

rf = 15 × 10–3 mm

0.01

Ξro 0.005 Φ = 30° Φ = 90°

0.1

0.2

0.3

0.4 0.5 (b)

0.6

0.7

Vf 0.8

r (Φ) (mm)

0.3

0.25

rf = 10 × 10–3 mm

0.20

0.15

0.1

vf = 0.2 vf = 0.4

0.05

vf = 0.6 20

40

60 (c)

1.10 Continued

80

Φ (degree)

Characterizing the structure and geometry of fibrous materials v (Φ)

29

vf = 0.6

800

rf = 10 × 10–3 mm 600

vf = 0.4

400

vf = 0.2 200

20

40

60

80

Φ (degree)

1.11 Mean fiber cut ends ν(Φ) versus the direction Φ. Adapted from Pan, N. (1994). ‘Analytical Characterization of the Anisotropy and Local Heterogeneity of Short-fiber Composites – Fiber Fraction as a Variable.’ Journal of Composite Materials 28(16): 1500–1531.

The effect of the system fiber volume fraction Vf on r distribution is depicted in Figs 1.10(b) and (c). It is easy to understand that increasing Vf value will reduce the number of the aperture circles with larger r values, causing lower variance of the r values in Fig. 1.10(b), and resulting in a smaller value of the mean radius r of the aperture circles in Fig. 1.10(c). Furthermore, as specified above, either extreme of the r value will lead to violation of the boundaries defined by the allowable local fiber volume fractions vL f1 and vL f 2 . It can hence be concluded from Fig. 1.10(a) that direction Φ = 0° with most extreme r values is the weakest direction in the system, while direction Φ = 90° with least extreme r values is the strongest direction; a reflection of the anisotropic nature of this system. Additionally, the vL f 1 and vL f 2 restraints can be translated into the allowable variance value Ξro in Fig. 1.10(b) which in turn determines the minimum allowable system fiber volume fraction Vf so as to eliminate the excessive number of large r aperture circles. There is one more direction, Φ = 30°, provided in Figs 1.10(a) and (b) for comparison. It is deduced from the results that when Φ value decrease from Φ = 90° to Φ = 30°, the r distribution will shift towards the region of greater values, leading to more larger aperture circles and fewer smaller ones. Overall, reduction of Φ value in the present case results in greater variance or more

30


diverse r distribution as seen in Fig. 1.10(b). On the other hand, there are two other parameters related to the fiber cut-ends and the air free length in the list of calculated results: ϒ( Φ ) = 1 cos Φ + π sin Φ 2 4

and

J ( Φ ) = π cos Φ – 1 sin Φ 4 2

Both expressions reach their extremes at the direction Φ = 57.518°. Correspondingly, our predictions indicate that the average radius of the fiber cut-ends, ρ(Φ), becomes the minimum in Fig. 1.12, while the average free length lm(Φ) of the air material in Fig. 1.10(c) approaches its maximum at this direction, because of the fact of too few fibers oriented in this direction. Further evidence is provided in Figs 1.13(a) and (b). Figure 1.13(a) shows the distribution of the free distance l at three directions at given fiber size rf and total fiber quantity Vf. It is seen that l value is distributed over the full spectrum from 0 to the infinity at the cross-section Φ = 57.518°, again because of the extremely small number of fibers associated with this direction, leading to an excessively great range of l, and high variance value at this direction as seen in Fig. 1.13(b). (For the same reason as above, the variance at Φ = 72° instead of the infinity value at Φ = 57.518° is shown here.) Likewise, the allowable range of the l value is indicated by the vL f1 and vL f 2 boundary in Fig. 1.13(a), and the minimum system fiber volume fraction Vf is given in Fig. 1.13(b) according to the condition Ξl ≤ Ξlo. It can be ρ( Φ ) rf 1.2 1.175 1.15 1.125 1.1 1.075 1.05 20

40

60

80

Φ (degree)

1.12 Relative cut fiber ends ρ(Φ) versus the direction Φ. Adapted from Pan, N. (1994). ‘Analytical Characterization of the Anisotropy and Local Heterogeneity of Short-fiber Composites – Fiber Fraction as a Variable.’ Journal of Composite Materials, 28(16): 1500–1531.


31

speculated, based overall on Fig. 1.13, that when Φ < 57.518°, contrary to the case of r distribution in Fig 1.10, increasing Φ value will shift the l distribution in the direction of greater value, and at the same time result in greater variance. The trend will reverse once Φ > 57.518°. In addition, it can be concluded from Figures 1.10 to 1.13 that even at a given cross-section Φ in the system, the parameters such as r and l are still variables at different locations on the cross-section. In other words, this system is both anisotropic and quasi-heterogeneous. It may suggest, based on the above two general distribution cases, the spatial random and planar harmonic, that quasi-heterogeneity is an inherent feature of fiber systems, and it exists in all fiber systems regardless of the fiber distributions. Even for a unidirectional fiber orientation, although it is possible to achieve a quasihomogenity at individual cross-sections, irregularities of local fiber volume fraction between cross-sections still exist.

f (l ) 50

Φ = 0° 40

vLf 1 rf = 15 × 10–3 mm vf = 0.6

30

20

vLf 2

10 Φ = 90° Φ = 57.518° 0.02 0.04 0.06 0.08 (a)

0.1 0.12 0.14

l (mm)

1.13 Distribution of the tortuosity length l in an anisotropic case. Adopted from Pan, N. (1994). ‘Analytical Characterization of the Anisotropy and Local Heterogeneity of Short-fiber Composites - Fiber Fraction as a Variable.’ Journal of Composite Materials 28(16): 1500– 1531. (a) l at three cross-sections; (b) the variance of l at three crosssections; (c) the mean value lm versus the direction Φ.

32

Thermal and moisture transport in fibrous materials Ξr 200

Φ = 72° 150

rf = 15 × 10–3 mm

100

50

Φ = 90° Φ = 0°

Ξro 0.02

0.04

0.06

0.08

0.1

0.12

Vf

(b)

Im (Φ) (mm)

4

rf = 10 × 10–3 mm 3

2

1

vf = 0.4

vf = 0.6 vf = 0.2 20

40

60 (c)

1.13 Continued

80

Φ (degree)


1.8

33

Determination of the fiber orientation

It has to be admitted that, although the statistical treatment using the fiber orientation pdf is a powerful tool in dealing with the structural variations, the major difficulty comes from the determination of the probability density function for a specific case. Cox (1952) proposed for a fiber network that such a density function can be assumed to be in the form of Fourier series. The constants in the series are dependent on specific structures. For simple and symmetrical orientations, the coefficients are either eliminated or determined without much difficulty. However, it becomes more problematic for complex cases where asymmetrical terms exist. Pourdeyhimi et al. have published a series of papers on determination of fiber orientation pdf for nonwovens (Pourdeyhimi and Ramanathan, 1995; Pourdeyhimi, Ramanathan et al., 1996; Pourdeyhimi and Kim, 2002). Because of the central limit theorem, the present author has proposed (Pan, 1993a) to apply the Gaussian function, or its equivalence in periodic case, the von Mises function (Mardia, 1972) to approximate the distribution in question, provided that the coefficients in the functions can be determined through, most probably, experimental approaches. Sayers (1992) suggested that the coefficients of the fiber orientation function of any form be determined by expanding the orientation function into the generalized Legendre functions. Recent work by Tournier, Calamante et al. (2004) proposed a method to directly determine the fiber orientation density function from diffusion-weighted MRI data using a spherical deconvolution technique.

1.8.1

BET–Kelvin method for pore distribution

Litvinova (1982) proposed a method of determining some of these parameters on the basis of the BET equation for a given sorption isotherm. In the beginning, the sorption isotherm curve is almost linear (usually for 0.01 < M < 0.35). When the capillary walls are covered by a monomolecular layer of liquid, the BET equation can be written as follows: aw = 1 – c–1 cVA M (1 – a w ) cVA

[1.55]

where M is the moisture content at sorbed air humidity aw, VA is the volume of monomolecular layer and c is the constant resulting from thermal effect of sorption. By plotting M vs. aw using given data, the above equation gives a straight line on the graph with slope (c – 1)/cVA and intercept 1/cVA. It thus allows calculation of the ‘volume’ of a monomolecular layer of water and then the specific surface of porous body a (m2/g) a = sVAN

[1.56]

where s is the surface occupied by molecules and N is the Avogadro’s number.

34


Strumillo and Kudra proposed another method by which we can calculate the corresponding pore radius r and pore volume V (Strumillo and Kudra 1986). From the Kelvin–Thomson equation, r=

2σVm cos γ RT ln (1/ RH )

[1.57]

where Vm is the molar volume. For a given relative humidity RH and the corresponding value of the moisture content M on the desorption isotherm, the radius of the pore can be calculated from above equation. Hence the volume of pores of radius r filled with water can be expressed as (m3/kg of dry material) V=M 1 ρ

[1.58]

Repeating these calculations for a range of RH, the function V = f (r) can be obtained. By means of graphical differentiation, the pore size distribution can be easily acquired. For example, the sorption isotherm of a fiber mass is given in Fig. 1.14(a), and the data is also listed in Table 1.1. We can then determine the integral and differential curves of the pore size distribution for the fiber mass, given the parameters in Equation [1.57] as σ = 71.97 × 10–3 N/M, Vm = 0.018 m3/ mole P = 0.101 MPa, T = 293 K, cos γ = 0.928, R = 8314 J/mol. K), ρ = 998.2 kg/m3. For each RH value we can find the corresponding moisture content M from Fig. 1.14(a) Table 1.1. Then by using Equations [1.57] and [1.58], we can calculate the pore radius r and the corresponding pore volume V as in Table 1.1. Table1.1 Results of calculations RH

r ∗ 10–10 m

M

V ∗ 105 m3

0.04 0.06 0.08 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

3.07 3.51 3.91 4.29 6.13 8.20 10.77 14.24 19.32 27.67 44.23 93.68

0.0390 0.0445 0.0505 0.0565 0.0750 0.0925 0.108 0.122 0.135 0.149 0.165 0.185

3.91 4.46 5.06 5.66 7.51 9.27 10.82 12.22 13.52 14.93 16.52 18.53

Adapted from Strumillo, C. and T. Kudra (1986)


35

By plotting the data, we obtain the pore volume distribution V = f(r) curve shown in Fig. 1.14(b) and differentiating the figure yields the differential pore volume distribution curve in Fig. 1.14(c). RH 1.0

0.5

M

0 0.04

0.10

0.15

0.20

(a)

V × 105m3

15

10

5

ln (r × 1010), m 0 1.0

2.0

3.0

4.0

5.0

(b)

1.14 BET–Kelvin method for pore distribution. Adapted from Strumillo, C. and T. Kudra (1986). Drying: Principles, Applications and Design. New York, Gordon and Breach Publishers. (a) the sorption isotherm curve RH/M of a fiber mass; (b) the pore volume V/pore radius r ; (c) the differential pore volume distribution curve dV /r. dr

36

Thermal and moisture transport in fibrous materials dV × 105m3 dr 1.5

1.0

0.5

ln (r × 1010), m 0

1.0

2.0

3.0

4.0

(c)

1.14 Continued

1.8.2

The Fourier transformation method for fiber orientation

The fiber orientation function (ODF) can also be determined using the Fourier transformation method. An image of a fibrous structure shows the special arrangement of fibers in the form of brightness transitions from light to dark and vice versa. Thus, if the fibers are predominantly oriented in a given direction, the change in frequencies in that direction will be low, whereas the change in frequencies in the perpendicular direction will be high. We use this characteristic of the Fourier transformation to obtain information on the fiber orientation distribution in the fibrous structure. Fourier transformation decomposes an image of the spatial distribution of fibers into the frequency domain with appropriate magnitude and phase values. The frequency form of the image is also depicted using another image in which the gray scale intensities represent the magnitude of the various frequency components. In two dimensions, the direct Fourier transformation is given as

F ( u, v ) =

+∞

+∞

–∞

–∞

∫ ∫

f ( x , y ) exp [– j 2 π ( ux + vy )] dxdy

[1.59]

where f (x, y) is the image and F(u, v) is its transformation, u refers to the frequency along x-direction and v represents the frequency along the y-axis. Since the Fourier transformation has its reference in the center, orientations may be directly computed from the transformed image by scanning the image radially. An average value of the transform intensity is found for each


37

of the angular cells. Subsequently, the fiber orientation distribution function (ODF) is determined by normalizing the average values with the total transform intensity at a given annulus. A full description of this Fourier transformation method can be found in Kim (2004).

1.9

The packing problem

Research on the internal structure and geometry of fibrous materials is still very primitive. In order to understand the behavior of fibrous structures, we have to better examine the micro-structure or the discrete nature of the structure. Yet a thorough study of a structure formed by individual fibers is an extremely challenging problem. It is worth mentioning that the problem of the micro-geometry in a fiber assembly can be categorized into a branch of complex problems in mathematics called ‘packing problems’. Taking, for example, the sphere packing problem, also known as the Kepler problem, based on the conjecture put forth in 1611 by the astronomer Johannes Kepler (Peterson, 1998; Chang, 2004), who speculated that the densest way to pack spheres is to place them in a pyramid arrangement known as face centred cubic packing (Fig. 1.15). This statement has become known as ‘Kepler’s conjecture’ or simply the sphere packing problem. To mathematically solve the sphere packing problem has been an active area of research for mathematicians ever since, and its solution remains disputable (Stewart, 1992; Li and Ng, 2003; Weitz, 2004). Yet, it seems that sphere packing would be the simplest packing case, for one only needs to consider one characteristic size, i.e. the diameter of perfect spheres, and ignore the deformation due to packing. Therefore it does not seem to be the case that

1.15 The Kepler conjecture – The sphere packing problem. Adapted from Kenneth Chang, ‘In Math, Computers Don’t Lie. Or Do They?’, The New York Times, April 6, 2004.

38


the fiber packing problem, which obviously is much more of a complex topic, can be solved completely anytime soon.

1.10

References

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39

Kallmes, O. and H. Corte (1960). ‘The Structure of Paper: I. The Statistical Geometry of an Ideal Two-dimensional Fiber Network.’ Tappi 43: 737. Kallmes, O., H. Corte, and G. Bernier (1963). ‘The Structure of Paper: V. The Free Fiber Length of a Multiplanar Sheet.’ Tappi 46: 108. Kendall, M. G. and P. A. P. Moran (1963). Geometrical Probability. London, Charles Griffin and Co. Ltd. Kim, H. S. (2004). ‘Relationship Between Fiber Orientation Distribution Function and Mechanical Anisotropy of Thermally Point-Bonded Nonwovens.’ Fibers And Polymers 5(3): 177–181. Komori, T. and M. Itoh (1991). ‘Theory of the General Deformation of Fiber Assemblies.’ Textile Research Journal 61(10): 588–594. Komori, T. and M. Itoh (1994). ‘A Modified Theory of Fiber Contact in General Fiber Assemblies.’ Textile Research Journal 64(9): 519–528. Komori, T. and M. Itoh (1997). ‘Analyzing the Compressibility of a Random Fiber Mass Based on the Modified Theory of Fiber Contact.’ Textile Research Journal 67(3): 204– 210. Komori, T., M. Itoh, et al. (1992). ‘A Model Analysis of the Compressibility of Fiber Assemblies.’ Textile Research Journal 62(10): 567–574. Komori, T. and K. Makishima (1977). ‘Numbers of Fiber to Fiber Contacts in General Fiber Assemblies.’ Textile Research Journal 47(1): 13–17. Komori, T. and K. Makishima (1978). ‘Estimation of Fiber Orientation and Length in Fiber Assemblies.’ Textile Research Journal 48(6): 309–314. Komori, T. and K. Makishima (1979). ‘Geometrical Expressions of Spaces in Anisotropic Fiber Assemblies.’ Textile Res. J., 49: 550–555. Lee, D. H. and J. K. Lee (1985). Initial Compressional Behavior of Fiber Assembly. Objective Measurement: Applications to Product Design and Process Control. S. Kawabata, R Postle, and M. Niwa, Osaka, The Textile Machinery Society of Japan: 613. Li, S. P. and K. L. Ng (2003). ‘Monte Carlo study of the sphere packing problem.’ Physica a-Statistical Mechanics and Its Applications 321(1–2): 359–363. Litvinova, T. A. (1982). Calculation of Sorption-structural Characteristics of Textile Materials. Moscow, Moscow Textile Institute. Mardia, K. V. (1972). Statistics of Directional Data. New York, Academic Press. Michell, A. J., R. S. Seth, and D. H. Page (1983). ‘The Effect of Press Drying on Paper Structure.’ Paperi Ja Puu-Paper and Timber 65(12): 798–804. Narter, M. A., S. K. Batra and D. R. Buchanan (1999). ‘Micromechanics of three-dimensional fibrewebs: constitutive equations.’ Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences 455(1989): 3543–3563. Ogston, A. G. (1958). ‘The Spaces in a Uniform Random Suspension of Fibers.’ Trans. Faraday Soc. 54: 1754–1757. Page, D. H. (1993). ‘A Quantitative Theory of the Strength of Wet Webs.’ Journal of Pulp and Paper Science 19(4): J175–J176. Page, D. H. (2002). ‘The Meaning of Nordman Bond Strength.’ Nordic Pulp & Paper Research Journal 17(1): 39–44. Page, D. H. and R. C. Howard (1992). ‘The Influence of Machine Speed on the Machinedirection Stretch of Newsprint.’ Tappi Journal 75(12): 53–54. Page, D. H., R. S. Seth, et al. (1979). ‘Elastic Modulus of Paper. 1. Controlling Mechanisms.’ Tappi 62(9): 99–102. Page, D. H. and R. S. Seth (1980a). ‘The Elastic Modulus of Paper. 2. The Importance of Fiber Modulus, Bonding, and Fiber Length.’ Tappi 63(6): 113–116.

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Page, D. H. and R. S. Seth (1980b). ‘The Elastic Modulus of Paper 3. The Effects of Dislocations, Microcompressions, Curl, Crimps, and Kinks.’ Tappi 63(10): 99–102. Page, D. H. and R. S. Seth (1980c). ‘Structure and the Elastic Modulus of Paper.’ Abstracts of Papers of the American Chemical Society 179(MAR): 27–CELL. Page, D. H. and R. S. Seth (1988). ‘A Note on the Effect of Fiber Strength on the Tensile Strength of Paper.’ Tappi Journal 71(10): 182–183. Pan, N. (1993a). ‘Development of a Constitutive Theory for Short-fiber Yarns, Part III: Effects of Fiber Orientation and Fiber Bending Deformation.’ Textile Research Journal 63: 565–572. Pan, N. (1993b). ‘A Modified Analysis of the Microstructural Characteristics of General Fiber Assemblies.’ Textile Research Journal 63(6): 336–345. Pan, N. (1993c). ‘Theoretical Determination of the Optimal Fiber Volume Fraction and Fiber–Matrix Property Compatibility of Short-fiber Composites.’ Polymer Composites 14(2): 85–93. Pan, N. (1994). ‘Analytical Characterization of the Anisotropy and Local Heterogeneity of Short-fiber Composites – Fiber Fraction as a Variable.’ Journal of Composite Materials 28(16): 1500–1531. Pan, N. (1995). ‘Fiber Contact in Fiber Assemblies.’ Textile Research Journal 65(10): 618–618. Pan, N. and G. A. Carnaby (1989). ‘Theory of the Shear Deformation of Fibrous Assemblies.’ Textile Research Journal 59(5): 285–292. Pan, N., J. Chen, M., Seo, and S. Backer (1997). ‘Micromechanics of a Planar Hybrid Fibrous Network.’ Textile Research Journal 67(12): 907–925. Pan, N. and W. Zhong (2006). ‘Fluid Transport Phenomena in Fibrous Materials.’ Textile Progress: in press. Parkhouse J. and A. Kelly (1995). ‘The Random Packing of Fibers In Three Dimensions.’ Proc: Math. and Phy. Sci. Roy. Soc. A 451: 737. Perkins, R. W. and R. E. Mark (1976). ‘Structural Theory of Elastic Behavior of Paper.’ Tappi 59(12): 118–120. Perkins, R. W. and R. E. Mark (1983a). ‘Effects of Fiber Orientation Distribution on the Mechanical Properties of Paper.’ Paperi Ja Puu-Paper and Timber 65(12): 797–797. Perkins, R. W. and R. E. Mark (1983b). ‘A Study of the Inelastic Behavior of Paper.’ Paperi Ja Puu-Paper and Timber 65(12): 797–798. Perkins, R. W. and M. K. Ramasubramanian (1989). Concerning Micromechanics Models for the Elastic Behavior of Paper. New York, The American Society of Mechanical Engineering. Peterson, I. (1998). ‘Cracking Kepler’s Sphere-packing Problem.’ Science News 154(7): 103. Pourdeyhimi, B. and H. S. Kim (2002). ‘Measuring Fiber Orientation in Nonwovens: The Hough Transform.’ Textile Research Journal 72(9): 803–809. Pourdeyhimi, B. and R. Ramanathan (1995). ‘Image-analysis Method for Estimating 2D Fiber Orientation and Fiber Length in Discontinuous Fiber-reinforced Composites.’ Polymers and Polymer Composites 3(4): 277–287. Pourdeyhimi, B., R. Ramanathan, et al. (1996). ‘Measuring fIber Orientation in Nonwovens.1. Simulation.’ Textile Research Journal 66(11): 713–722. Ramasubramanian, M. K. and R. W. Perkins (1988). ‘Computer Simulation of the Uniaxial Elastic–Plastic Behavior of Paper.’ Journal of Engineering Materials and Technology– Transactions of the ASME 110(2): 117–123. Sayers, C. M. (1992). ‘Elastic Anisotropy of Short-fiber Reinforced Composites.’ Int. J. Solids Structures 29: 2933–2944.


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Schaffnit, C. and C. T. J. Dodson (1994). ‘A New Analysis of Fiber Orientation Effects on Paper Formation.’ Paperi Ja Puu-Paper and Timber 76(5): 340–346. Scharcanski, J. and C. T. J. Dodson (1997). ‘Neural Network Model for Paper-forming Process.’ IEEE Transactions on Industry Applications 33(3): 826–839. Scharcanski, J. and C. T. J. Dodson (2000). ‘Simulating Colloidal Thickening: Virtual Papermaking.’ Simulation 74(4): 200–206. Scharcanski, J., C. T. J. Dodson, et al. (2002). ‘Simulating Effects of Fiber Crimp, Flocculation, Density, and Orientation on Structure Statistics of Stochastic Fiber Networks.’ Simulation – Transactions of the Society for Modeling and Simulation International 78(6): 389–395. Schulgasser, K. and D. H. Page (1988). ‘The Influence of Transverse Fiber Properties on the Inplane Elastic Behavior of Paper.’ Composites Science and Technology 32(4): 279–292. Seth, R. S. and D. H. Page (1975). ‘Fracture Resistance – Failure Criterion for Paper.’ Tappi 58(9): 112–117. Seth, R. S. and D. H. Page (1996). ‘The Problem of Using Page’s Equation to Determine Loss in Shear Strength of Fiber–fiber Bonds upon Pulp Drying.’ Tappi Journal 79(9): 206–210. Stewart, I. (1992). ‘Has the Sphere Packing Problem Been Solved.?’ New Scientist 134: 16. Strumillo, C. and T. Kudra (1986). Drying: Principles, Applications and Design. New York, Gordon and Breach Publishers. Tournier, J. D., F. Calamante, et al. (2004). ‘Direct Estimation of the Fiber Orientation Density Function from Diffusion-weighted MRI Data using Spherical Deconvolution.’ Neuroimage 23(3): 1176–1185. van Wyk, C. M. (1946). ‘Note on the Compressibility of Wool.’ Journal of Textile Institute 37: 282. Weitz, D. A. (2004). ‘Packing in the Spheres.’ Science 303: 968–969.

2 Understanding the three-dimensional structure of fibrous materials using stereology D. L U K A S and J. C H A L O U P E K, Technical University of Liberec, Czech Republic

Stereology is a unique mathematical discipline used to describe the structural parameters of fibrous materials found in textiles, geology, biology, fibrous composites, and in corn-grained solids, where fibre-like structures are created by the edges of grains in contact with each other. This chapter is compiled from lectures delivered to post-graduate students taking ‘Stereology of Textile Materials’ at the Technical University of Liberec (Lukas, 1999), and is relevant to students and researchers involved in interpreting flat images of fibrous materials in order to explain their behaviour, or to design new fibrous materials with enhanced properties. There are a number of excellent monographs on stereology, ranging from the basic to the expert. This chapter outlines an elementary technique for deriving most of the stereological formulae, avoiding those demanding either lengthy explanations or a specialised mathematical background. The chapter concentrates on the set of tools needed for a geometrical description of fibrous mass, and provides comprehensive references for further information on this relatively new field.

2.1

Introduction

Stereology was developed to solve various problems in understanding the internal structure of three-dimensional objects, such as fibrous materials, and especially textiles. The relevant geometrical features are mainly expressed in terms of volume, length, surface area, etc. (detailed in Section 2.1.1), and there are three main obstacles facing efforts to quantify these features. The first two difficulties are practical in nature and the third theoretical. (i) The internal structure of an opaque object can only be examined in thin sections, comprising projections of its fibres. Sections of textile materials may be cut using sharp tools, or created virtually by applying the principles of tomography, confocal microscopy, etc. (ii) The dimensions of an object under investigation are usually proportionately much greater than the characteristic dimensions of its 42

Understanding the three-dimensional structure

43

internal structure; for instance, fibre diameters will be orders of magnitude smaller that the width and the length of the fabric they form. Hence, it is not practicable to study an entire object in detail. (iii) Occasionally, investigators must determine an appropriate set of geometrical parameters to describe real structures and their properties. Specific parameters will be associated with either mechanical or adsorption properties of fibrous materials. Various disciplines require information on the internal structures of objects, including biology, medicine, geology, material engineering and mathematics itself. The evolution of methods to quantify structural features laid the foundation for what is now known as stereology, and the concept has continued to evolve since it was proposed in 1961 by a small group of scientists at Feldberg in Germany, under the leadership of Hans Elias (Elias, 1963). For the purposes of this chapter, the following definition of stereology is used: Stereology is a mathematical method of statistical selection and processing of geometrical data to estimate geometrical quantities of an n-dimensional object through measurements of its sections and projections, which have dimensions less than n. The relationship between the geometrical quantities of an n-dimensional object and measurements of its sections and projections is quite logical and familiar. Figure 2.1 reminds us of the procedure for ascertaining the volume of a three-dimensional body. The volume of a three-dimensional body K, say V(K), may be expressed by a definite integral, laid out as:

V (K) =

∫

H

a ( z ) dz

[2.1]

0

where a(z) is the area of a planar cross-section of the body K and is perpendicular to the z-axis. H is the longitudinal length of the projection of the body on the z-axis. The left-hand side of the formula, i.e. the volume V(K), represents a parameter of the three-dimensional object. The right-hand side reveals another parameter of the body in question, a(z), which results from an analysis of its flat cross-section. The two-dimensional parameter a(z) symbolises the area of the flat section cut in the body K by a plane, normal to the z-axis, thus, expressing its cross-sectional area as a function of z. Thus the relationship between three- and two-dimensional parameters is established through integration. The above relationship may also be demonstrated through Cavalieri’s principle. The conceptualisation was framed by Cavalieri, a student of Galileo in the 17th century (Naas and Schmidt, 1962; Russ and Dehoff, 2000), for two- and three-dimensional objects. For two dimensions, the principle states

44

Thermal and moisture transport in fibrous materials z

K dZ

a (z )

H y

x

2.1 Volume, V(K), of a three-dimensional body, K, being expressed as a sum of the volumes of its elementary thin sections of thickness, dz, that are parallel to the x–y plane. H is the length of the body K, perceived as its upright projection on the z-axis.

that the areas of two figures included between parallel lines are equal if the linear cross-sections parallel to and at the same distance from a given base line have equal lengths. For three dimensions, the principle states that the volumes of two solids included between parallel planes are equal if the planar cross-sections parallel to and at the same distance from a given plane have equal areas. This is illustrated in Fig. 2.2. Cavalieri’s principle thus provides further evidence of the relationship between the parameters of threeand two-dimensional objects and their sections. Cauchy’s formula for surface area also supports the existence of the relationship between objects and their lower-dimension projections. According to this formula, the surface area S(K) of a three-dimensional convex body K is four times the mean area of its planar projection. This can easily be verified by considering a sphere of radius R, whose surface area S is 4π R2, and each of its planar projections has an area of π R2. These quantities are proportional to each other, being related by a factor of 4. A similar relationship for two-dimensional convex bodies will be established in Section 2.3.4. The definition of a convex body will be specified in Section 2.1.1. However, these attempts to colligate the dimensional aspects of objects with their sections and projections are based only on geometry. Stereology involves statistical methodology in combination with geometry and gives us the ability to model geometrical relations where measurement is impractical or even impossible. To understand the effectiveness of this method, it is necessary to review an interesting experiment carried out in the 18th century.


2

45

3

3

1

2.2 Illustration of Cavalieri’s principle: Volumes of the two solid bodies included between parallel planes are equal if the corresponding planar cross-sections (shown as 3) at any position are equal and parallel to a given plane (shown as 1). 2

d

L (j )

1

2.3 Buffon’s needle (shown as 1) of length L( j ) is located on a warp of parallel lines (2), which are separated by a distance d.

In 1777, the French naturalist Buffon was attracted by the probability, P, that a randomly thrown needle, j, of length L( j) will hit a line among a given set of parallel lines in a plane with each of the neighbouring lines separated by a distance d, so as to conform to a precondition of d > L( j). The situation is depicted in Fig. 2.3. Buffon (1777) deduced P as 2L( j)/(π ∗ d). The estimated value [P] of probability P, from a large number of throws, N, could

46


be estimated through a relative frequency of hits. Precisely, the value of P equalled the limiting value of [P], while N tended to infinity, i.e. P = lim [ P ] . N →∞ The relative frequency, [P], was defined as: [ P] = n N

[2.2]

where n is the number of positive trials, and N the total number of throws. From this relationship, an unbiased estimation of the distance between parallel lines, [d] can be obtained. The concept of ‘estimators’ will be detailed in sub-section 2.3.1.

[d ] =

2 L ( j ) 2 L( j ) N = π [ P] πn

[2.3]

The above relation [2.3] will be used in Section 3.3.2 to estimate the lengths of curves or fibrous materials in a plane. Equation [2.3] can be verified by imagining a series of random needle throws. The needle has to be thrown in such a way as to ensure equal probabilities of its landing at various locations on the parallel lines in all possible orientations. This can be done by throwing the needle repeatedly in the same way, while rotating the parallel lines by an angle kπ/M. For each particular orientation of the lines, groups of equal number of trials are carried out. Here, k is the sequence number of a particular group of trials and M the total number of groups of trials. Equation [2.3] shows that the one-dimensional geometrical parameter d may be estimated from the number of times Buffon’s needle intersects a line. Since the intersection points are zero-dimensional, the connection between dimensions of an object with those of its sections is confirmed. The next point noteworthy in the context of the Buffon’s needle problem, concerns the Ludolf number π, which may be estimated statistically after rearranging Equation [2.3] to obtain an expression of [π] as 2L(j) ∗ N/(d ∗ n) and, subsequently, using known values of the other parameters. The value of d has to be known exactly to estimate π. The Ludolf number is therefore estimated using a known set of parameters of L( j), N, d, and n. There are three different classes of analysis for investigating the internal structures of a material, and the most appropriate method or combination of methods is chosen for the particular problem at hand. (i) The first class of analysis comprises estimations of the global geometrical parameters of a structure or the total values of its individual components, such as total volume, total length, and total numbers of particles. The geometrical parameters do not depend on the shape or distribution of the structure or its components in space. Accordingly, the corresponding stereological methods are independent as far as shapes and spatial


47

distribution of the structural features are concerned. This class of study is characterised by estimations of total volumes, areas, lengths and densities. (ii) The second class of study involves estimating the properties of individual parts and elements of a structure; for instance, estimating the distribution function of a chosen particle parameter. Sizes of particles and their projections are the most commonly measured parameters in this case. (iii) The last class covers analyses of mutual spatial positions of structural features. The above two classes of study are not influenced in any way by the scatter of features in space. An analysis typical of this third class involves evaluating the planar anisotropy of fibrous systems, and this is described in more detail in Section 2.3.5. The importance of this area of study was highlighted by Pourdeyhimi and Koehl (2000a), who dealt with methods to examine the uniformity of a non-woven web. An understanding of the mutual spatial location of fibres and yarns is vital for the automatic recognition of fabric patterns, as described by Jeon (2003). Inter-fibre distances in paper and non-wovens have been studied by Dent (2001).

2.1.1

Structural features and their models

Textile engineering began with a classification of the various types of textiles, either according to their corresponding technologies or according to their most meaningful structural attributes, as described by Jirsak and Wadsworth (1999). Stereology provides the scientific basis for describing structures and their features, and these structural features are described below. The notion of a ‘feature’ may be explained with reference to a complex structure, such as that of a non-woven textile. Figure 2.4 shows a pointbonded non-woven fabric made of thermoplastic fibres. The figure shows rectangular regions where many fibres adhere together. These regions are generally formed by the impacts of the rollers when the surface screen reaches the temperature at which the thermoplastic fibres melt and bond to form the fabric. These types of non-wovens are referred to as ‘point-bonded’. Although the fibres are apparently randomly oriented, a deeper investigation reveals their preferred orientation. Some of the fibres have more crimp than others, and the distribution of the intra-rectangular bonded areas is nearly regular. Inside the squares, however, there are holes, or pores. Pores are found among the fibres as well, and the spatial distribution of the pores is irregular, as is the distribution of fibres. The internal components of fibrous materials show morphological and dimensional variations along with a wide range of mutual spatial organisations, and a reasonable simplification of the complexity of such a structure appears unattainable. From a purely practical standpoint, a perfect description of the

48


2.4 A point bonded nonwoven fabric, reinforced with thermoplastic fibers. The square-form spots were created by a regular grid of projections on one of the calender rollers. The projections, when they reached the melting temperature of fibers, bonded the nonwoven material in the predetermined pattern of spots with the thermoplastic fibres.

entire structure is unlikely to be helpful; it is more useful to examine the components that are responsible for the property under examination. The elements of the structure to be studied have to be spatially limited and experimentally distinguishable, otherwise quantitative measurements are not possible. The components that satisfy these conditions are called ‘structural features’, or simply ‘features’, and the combination of these features makes up the ‘internal structure’ of an object (Saxl, 1989). The property(ies) of an object depend on its structure, which is studied or explained in terms of measurements of structural parameter. Properties such


49

as textile permeability are studied without reference to the types of materials involved, as all materials trap gases and liquids, thus hindering their movement. On the other hand, the tensile strength of a fibrous material is related to the types of fibres and how they are bound. The classification of structural elements as features also depends on how a sample is processed for stereological measurements. Figure 2.5 shows two different situations. Both diagrams display the same part of a blended fibrous mass, the right-hand image (b) differing with respect to shades. While it is not easy to distinguish between the two kinds of features in image a, it is possible in b. An exact recognition of structural features is important for processing images digitally. Koehl et al. (1998) developed a method for extracting geometrical features from digitised cross-sectional images of yarns. Researchers must select the features that will enable them to investigate effectively the property of the object that is of interest. Features are mostly three-dimensional formations, distributed in three-dimensional space, but sometimes lower dimensions are more appropriate. One example of a lowerdimension investigation is for extremely thin textiles, where the investigation is restricted to their planar projections. Fibres may even be considered as one-dimensional features, and thus zero-dimensional points, such as centres of tiny dust particles in a fibrous filter, are features pertinent to the study of their distribution. Cross-sections or projections of three-dimensional objects may also be regarded as objects with their own intrinsic structure. In this chapter, such cross-sections and projections will be regarded as ‘induced structures’. Mathematical descriptions of internal structures are necessary to create a model of the feature that is both powerful enough to describe real objects

(a)

(b)

2.5 Images of a fibrous object can have different kinds of features with different colour combinations. As the fibrous structure in (a) cannot be differentiated with respect to colour, it has only one type of feature. On the other hand, different shades of colours of the fibres in image (b) characterize it as a two-featured one.

50


and simple enough to model based on standard rules and regulations. The necessary attributes for this rarely coexist. The more generalised the model, the fewer the regulations, and therefore a rather careful choice of the feature model has to be carried out. Three models of features are described below, including some of the conceivable mathematical complications related to their usage, and their pertinence for describing fibrous materials. (i) Compact sets A compact set is a generalised model of structural features, and is discussed here in the context of fibrous features. Fibrous features are limited in space due to their well-defined boundaries, which therefore allow the existence of an n-dimensional cube with finite edge lengths that contains the chosen feature entirely. Therefore, fibrous features may also be referred to as closed sets, and thus a general model of structural features consists of limited and closed sets. Sets of points in Euclidian space obeying these properties are called ‘compact sets’. Some compact sets have rather curious properties, as demonstrated by their characteristic finite volumes or areas, where determination of surface areas or boundary lengths causes a range of problems. An example of such a peculiar set is the von Koch flake. The base for its construction is an abscissa, < 0, 1 >, known as the ‘initiator’. It is divided into three equal sections, with the mid-section substituted by two line segments of equal lengths. Each of the segments has a length identical to that of the removed middle section. As shown in Fig. 2.6 (b), the segments meet together at an angle to form the vertex of an equilateral triangle. Subsequent repetitions of these steps produce the results shown in Fig. 2.6 (c). The basic unit, comprising a buckled line with four sections of equal lengths, is called the ‘generator’. Each of the four parts of the generator is replaced with a unit that is a diminished version of the generator in the ratio of 1:3. The resulting pattern has 16 sections of equal length. If the same procedure is repeated infinitely and each successive step ensures a reduction of the generator unit with respect to the previous step by the same ratio, a von Koch’s curve is obtained in the interval < 0,1 >. Using three initiators, joined together in a triangle form, a similar process will result in the von Koch’s flake. One of its construction stages is shown in Fig. 2.6 (d). If the flake’s boundary is observed with a gradual increment of magnification, newer details will start emerging in stages. This unique feature, common to both von Koch’s curve and flake, is why determining their length and area is problematic. Similarly, three-dimensional sets can be constructed with very complex boundaries whose surface areas and volumes are not easily determined. These unique objects are called ‘fractals’, as described by Mandelbrot (1997). To exclude sets with


51

(a)

(b)

(c)

(d)

2.6 Von Koch curve and von Koch flake: Shows the initiator (a), and the generator (b) to enable constructions of the curve (c) and the flake (d). The parts (c) and (d) represent early stages of both the constructions.

complex boundaries, the ‘convex body’, a more specific class of model of feature, is used (see below). Kang et al. (2002) investigated fibrous mass from the point of view of fractals, to model fabric wrinkle. Summerscales et al. (2001) explored Voronoi tessellation and fractal dimensions for the quantification of microstructures of woven fibre-reinforced composites. (ii) Convex bodies Convex bodies are characterised by the shortest link connecting two

52


arbitrary points. If the straight line linking the points is enclosed fully inside the body, the body is then considered to be convex. Figure 2.7 illustrates three-dimensional convex and two-dimensional non-convex bodies. This model of convex bodies is inadequate to describe fibrous materials because the loops in the fibrous structure violate the model. The concept of convex bodies, nevertheless, is significant in stereology because simple rules govern their properties. Figure 2.8 shows intersections of convex (a) and non-convex (b) two-dimensional bodies

(a)

(b)

2.7 A three-dimensional convex body (a), and a two-dimensional non-convex body (b), obey the mutual relationship of the body and the shortest line connecting two of its arbitrary points. The straight link in-between the points has to lie fully inside the body to make it a convex one.

(a)

(b)

2.8 Intersections of a convex (a) and a non-convex two-dimensional body (b) with straight lines. Number of intersections for a nonconvex body with such a line depends on the mutual position of the body and the line.


53

with straight lines. A convex body can be intersected by a straight line only once, and the intersection is itself convex. For a non-convex body, the number of intersections depends on the mutual orientation and position of the body and the straight line. For a non-convex body, an intersection may not be convex, but may be composed of several isolated parts. In other words, it is impossible to correlate the numbers of nonconvex bodies and intersections from only knowing the number of intersections. The shortcomings of using convex bodies in describing fibrous structures must be overcome by an additional model, the ‘convex ring’, as described below. (iii) Convex rings A convex ring is defined as the union of a finite number of convex bodies. Figure 2.9 shows some two-dimensional bodies that demonstrate this concept, illustrating that not all convex rings are suitable for describing real fibrous structures. For our purposes, features pertinent to fibrous structures will be visualised in the context of convex rings.

2.9 Two-dimensional bodies, so-called figures, belong to the set of the convex ring. Using more and more appropriately chosen convex bodies, one can create fibre-like objects either in two- or threedimensional space.

54

2.2


Basic stereological principles

This section examines the geometrical characteristics of the volume of threedimensional bodies, in the context of mapping the volume of a threedimensional geometrical object with a set, R, of real numbers. A characterization theorem demonstrates how many groups there are of geometrical characteristics with the same set of attributes as the volume. Finally, we cite a generalised notion of section, which will be used as a tool to open opaque three-dimensional structures.

2.2.1

Content of convex ring sets and characterization theorem

One of the most frequently used parameters of features is their n-dimensional content, which generally refers to volume, surface area, and length. Accordingly, volume is regarded as a three-content, area two-content, and length as onecontent in the parlance of stereology. Now let us examine the generic properties of contents, along with the parameters that define n-dimensional objects of a convex ring and have the characteristic set of properties of content, using the example of the volume, or the three-content, of a three-dimensional prism h. The volume, V(h), for any element, h, of the set of all possible prisms, H, is defined simply by a product of a, b and c, which exactly represent the lengths of the prism’s perpendicular edges. As the set, H, of all prisms is connected to the set of real numbers, representing volume, V(h), by means of ‘onto mapping’, the volume may be deemed as a functional. Generally, a functional is defined as the mapping of any set to a set of numbers. The well-known properties of functional, V(h), are listed below: (i) The functional, V(h), does not depend on the location and the orientation of the prism, h, in space. This property is known as translational invariance. (ii) If splitting the original prism, h, gives rise to two non-intersecting prisms, A and B, with at the most one common edge or side, then their corresponding volumes, V(A) and V(B), fulfil the relation V(h) = V(A ∩ B) = V(A) + V(B), where the functional V(h) has its usual significance. The relation expresses a simple additivity of the volume functional. (iii) The functional, V(h), is positively defined, i.e., V(h) ≥ 0 for each prism, h, from the set of prisms, H. (iv) The functional, V(h), is normalised. Thus for each V(h), the properties (i), (ii), and (iii) satisfy a functional V′(h) = α · a · b · c, where α is the normalisation factor and α > 0. When the value of α attains unity, then V(h) has a unit value for a unit cube, with each of its edges, a, b and c


55

having unit length. Commonly, the normalization parameter α is taken to be one. A fascinating theorem pertaining to convex bodies and convex ring sets gives a solution for the total number of linearly independent functionals of n-dimensional bodies in n-dimensional spaces that fulfil the same list of properties (i)–(iv) as does the volume V. A detailed explanation of the solution was given by Hadwiger (1967). Sera (1982) brought in a characterization theorem to state that every such body had just (n + 1) linearly independent invariant characteristics, so-called ‘invariant measures’. Saxl (1989) listed a specially chosen set of such measures for convex bodies and bodies from convex rings. An edited version of the list for convex ring bodies is provided in Table 2.1. The only characteristic that will not be discussed in this chapter is the integral of the mean curvature of the surface of three-dimensional bodies. In Table 2.1, this characteristic is highlighted in italics. Euler–Poincaré characteristics will be described in Section 2.3.3.

2.2.2

Sections and ground sections

Usually, the terms ‘section’ and ‘ground section’ refer to a two-dimensional section of a three-dimensional body. This concept can, however, be generalised. Using different kinds of sections to investigate various materials is highly advantageous, because they help us to analyse the internal structure of objects that are otherwise imperceptible. Taking care to prepare the sections appropriately preserves the original mutual positions of the features in different materials. The notion of a section can be generalised as the intersection of a threedimensional object with a two-dimensional space, i.e. the plane of a section made by a cutting tool or by the movement of a grindstone in the case of a Table 2.1 List of linearly independent and invariant structural characteristics, also known as invariant measures, for objects of various dimensions from the convex ring

Dimension of object

Linearly independent invariant structural characteristics (invariant measures) n -content (n -1)-content (n -2)-content (n -3)-content 3

Volume

Surface area

Integral of the mean curvature of the surface

2

Area

Perimeter length

Euler–Poincaré characteristics

1

Length

Euler–Poincaré

0


characteristics


56


ground section. Two-dimensional sections may also be generated on the focal plane of a confocal microscope (Lukas, 1997). Therefore, the general definition of a section may be based on the intersection of an object under investigation, with another body having dimensions equal to or less than that of the investigated object. By choosing the dimensions of the different bodies, various types of sections can be obtained. Sections obtained from the intersection of two three-dimensional bodies are called three-dimensional sections or, more frequently, thin sections. Normally, this kind of section has the shape of a layer between two parallel planes, as shown in Fig. 2.10 (a). Section 2.4.4 uses thin sections to evaluate the average values of curvature and torsions of linear features. Block-like three-dimensional sections, which will be described in detail in Section 2.4.5, are used as dissectors for counting isolated parts of internal structures. A two-dimensional section is obtained by intersecting a three-dimensional body with a plane, as shown in Fig. 2.10 (b). The intersection of a threedimensional body with a straight line results in a one-dimensional section, as is depicted in Fig. 2.11 (a). The intersection of a three-dimensional body with a point located on a line, as shown in Fig. 2.11 (b), produces a section of zero dimensions. Figures can have two-, one- and zero-dimensional sections, while curves can have only one and zero-dimensional sections. Figure 2.12, where a part of fibrous structure is embedded in a block of region Ω, demonstrates the kinds of information about three-dimensional structures that is available from various sections. According to the

(a)

(b)

2.10 Three-dimensional (a) and two-dimensional (b) cross-sections of a three-dimensional object.

(a)

(b)

2.11 One-dimensional (a) and zero-dimensional cross-sections (b) of a three-dimensional body.


57

A

Ω

(a)

(b)

(c)

(d)

2.12 A three-dimensional object A in a region Ω and its various cross-sections: (a) shows a three-dimensional cross-section with the induced structure embedded in it; (b) shows a two-dimensional cross-section with the induced structure; (c) shows a onedimensional cross-section to which belongs the induced structure composed of a piece of a line; (d) shows a zero-dimensional crosssection represented by a point.

58


characterization theorem and Table 2.1, four common characteristics can be assigned to the structure in this figure. The characteristics are the volume, surface area, the integral of the mean curvature of the surface expressed in terms of length, and the Euler–Poincaré characteristic taken here for convenience as the number of isolated convex parts of the object. The three-dimensional section is also a three-dimensional structure and contains information about all four aforementioned characteristics of the structure. The fewer the dimensions of the section, the less information it contains. The correspondence between information conveyed by a section and the geometrical parameters of the original structure can be shown using a three-dimensional fibrous object enclosed in a three-dimensional region Ω such as that in Fig. 2.12(a). The set of independent parameters of the structure A comprises the volume, surface area, length of the very thin fibres (because thin fibres have only length as their physical dimension, they play a role very similar to the integral of the mean curvature of the surface. More information about integrals of curvature can be found in Saxl (1989)), and the number of isolated parts of the structure. The independence of the parameters implies that none of them can be expressed using linear combinations of the remaining ones. This independency can be explained using their dissimilar physical dimensions. Denoting the physical dimension of the length as L1, dimensions of the volume, the surface area, and the number of isolated structural parts take the form of L3, L2 and L0, respectively. The three-dimensional section of A, as depicted in Fig. 2.12(a), contains the induced structure of the threedimensional object, and so contains information about all four independent parameters. The two-dimensional section carries information about only three parameters, because its induced structure is described using only the three independent measures of surface area, boundary length and number of isolated parts. Since the number of isolated non-convex bodies of a convex ring cannot be estimated from their sections of lower dimensions, it is impossible to determine the number of isolated parts of an original structure with this type of section. This is because the number of intersections in a convex ring is manifold and does not depend solely on the total number of bodies there. It also depends on the position and orientation of the section, as was indicated in Fig. 2.8 for two-dimensional convex bodies. One-dimensional sections contain information about length and the number of isolated line segments described on them as induced structures. As before, it is not possible to estimate the number of isolated bodies of the original structure from this type of section. Zero probability of an intersection of a straight line with a line or a curve cannot be used also to estimate the feature length of an original structure. The one-dimensional fibre here represents all parameters with a physical dimension L1 including integrals of the surface main curvature.


59

Finally, the zero-dimensional section, or point, contains information only about the volume of an original structure because the probability of a point section intersecting with points, curves and surfaces embedded in a threedimensional space, is zero. The above statements concerning the information contained in various sections of a three-dimensional object are summarised in Table 2.2. Guidelines for interpreting Table 2.2 are given below: (i) The second row in the table shows the three-dimensional structural parameter, which is volume with a physical dimension of L3. The row expresses that each type of section can be used to estimate this parameter. (ii) The fourth row includes sections specifically used to measure onedimensional parameters, such as length with the physical dimension L1. As has been observed before, the probability of a one-dimensional body intersecting with a line or a point section in three-dimensional space is nil. Hence, such a parameter can only be estimated from three and two-dimensional sections. (iii) The fourth column corresponds to sections of one dimension. This type of section provides intersections among two- and three-dimensional features with a non-zero probability. Thus, one-dimensional sections are useful for estimating the volumes and surface areas of threedimensional bodies. The above analysis may be extended to any object of arbitrary dimensions through Equation [2.4]. This equation associates the dimension of an induced structure, the dimension of a structural feature, and the dimension of a body used to create sections, with that of an investigated body. Here, the dimension of an investigated body is the same as that of the space occupied by it. The term d(α) stands for the dimension of a structural feature, α , of an investigated body, A, having dimension d(A). The structural feature, α, under consideration occasionally stands for the surface of a three-dimensional body. Therefore, d(α) and d(A) have values of 2 and 3, respectively. Structural features and Table 2.2 Dimensions of structural parameters of a three-dimensional object and dimensions of bodies from which their sections are created to determine the dimensions of induced structures on sections. For three-dimensional objects, the dimensions of the body used to carry out sectioning are equal to dimensions of the corresponding sections Structural parameters of threedimensional objects (and their dimensions)

Dimension of the sectioning bodies —————————————————— 3 2 1 0

Volume (3) Surface area (2) Length (1) Euler–Poincaré characteristics (0)

3 2 1 0

2 1 0

1 0

0

60


their structural parameters have the same dimensions. This principle has already been used in Table 2.2. The term d(β ) denotes the dimension of a body, β, from which sections of the investigated body, A, are created. For instance, β may be considered as a straight line with a dimension, d(β ), of 1. Finally, d(α ∩ β ) describes the dimension of the induced structure created out of an intersection of the structural feature, α, and the surface of the body, β. The dimensional terms d(α), d(A), d(β ) and d(α ∩ β ) are related by the following formula, as introduced by Wiebel (1979). All the data in Table 2.2 can be derived from it. d(α ∩ β ) = d(α ) + d(β ) – d (A)

[2.4]

The above relation is easily verified in the context of this chapter. The threedimensional body, A, with the dimension, d(A), of 3, has the dimension of the surface area of its relevant feature, α, as d(α), having a value of 2. If the feature is examined with the one-dimensional body, β, having the dimension, d( β ), ascribed with a value of 1, the induced structure α ∩ b, which is created by the intersection of the surface, α, of the three-dimensional body, A, and the straight-line, β, takes the form of a point. Consequently, its dimension is d(α ∩ b) = 0. By assigning the above-mentioned values for the corresponding terms on the right-hand side, the relationship is verified. The above relationship may be extended to objects having fewer than three dimensions. If A is any two-dimensional area embedded with a fibrous system (material) α, then the relevant term d(A), has a value of 2. Accordingly, the internal structure consists of a one-dimensional fibrous system, α, characterised by a value of d(α) as 1. A body, β, with a dimension, d( β ), of 0, may be used, hopefully, to estimate the length as a geometrical parameter of the internal structure, which consists of one-dimensional fibrous material α. Fitting the values into the equation gives the value of d(α ∩ β ) as –1, which is ignored due to its physical insignificance. A similar argument explains the empty box of the row for the surface area in Table 2.2.

2.2.3

Lattices and test systems

Measuring part of an object, X, can be facilitated by incorporating a test system that is composed of a regular lattice of fundamental regions along with a regular distribution of probes. A lattice of fundamental regions consists of regions α0, α1, α2, . . . , αn with the following attributes: (i) Each of the fundamental regions, αi, contains at the most one point of an n-dimensional space. (ii) All fundamental regions are distributed regularly in space with respect to translational symmetry. Thus each fundamental region, αi, can be exactly displaced to any other region, αj, and the displacement vector consists of a linear combination of basic lattice vectors. Multiplication constants in this linear combination are integers.


61

The most common lattices of fundamental regions consist of squares, oblongs, triangles, hexagons, etc. According to the attribute (i), tightly packed lattices in a plane comprise fundamental regions whose boundaries are partly open, to preclude overlapping of the boundary points. Lattices of fundamental regions are illustrated in Fig. 2.13. Test systems are constructed so that set B is encompassed by each fundamental region, where B is distributed in the lattice with the same translational symmetry as that of the spatial distribution of the fundamental regions in the lattice. This means that the local view for each fundamental region is identical with the others. The set B is known as a probe, and is generally represented as points, curves or figures. As described in Section

α0

α1

α2

Fundamental region αi (a) α0

α1

α2

Fundamental region

αi (b) α0

α1

α2

αi Fundamental region

(c)

2.13 Three examples of two-dimensional lattices of fundamental regions: a square lattice (a), a lattice with the fundamental region of parallelogram type (b), and a hexagonal lattice (c).

62


3.4.5, dissectors are the only systems that are based on the use of threedimensional probes. To model a probe on a transparent sheet or a foil, marking tools leave behind trails and spots whose respective widths and diameters are significantly thick; therefore, these spots and trails do not correspond to points or onedimensional lines. The same argument is valid for grids, lines, and points created by graphics software on monitor screens. An uncertainty therefore persists about the precision of the presumed intersections of the probes used to study structural features. Figure 2.14 demonstrates this imprecision. To counter this problem, a pointed probe in a testing system is expressed as an intersection of the edges of two mutually perpendicular trails. A onedimensional curvilinear probe is represented by the chosen edge of a trail. The positions of the probes must be in a uniform random distribution with respect to the object under examination, in order to arrive at an unbiased estimation of the selected structural parametric value. In other words, stereological measurements are carried out in a series of uniform random and isotropic sections. Pertaining to a body, A, and a test section, T, there are uniform random sections A ∩ T corresponding to a point, X ∈ T, randomly located in A with the same probability of appearing at each region of A, provided that the isotropic orientation of T in three-dimensional space remains unaffected by the position of X in A. An analogous definition may be framed for two-dimensional space, whereas for one-dimensional space the only condition is the uniform randomness. Two of uniform random and isotropic cross-sections of three-dimensional object are portrayed in Fig. 2.15(a). Uniform random and isotropic sections are, in fact, obtained by microphotographs or micro-images. These are subsequently used to measure the chosen parameters of the internal structures, using testing systems as sketched in Fig. 2.15(b). The position of the testing system in the section has to be

1

2 (a)

(b)

2.14 Inaccuracy of a point and a curve probe using a pencil trail, where thickness of trails hinder clarity of intersection of a point or a line with an object or its boundary, is depicted (a). The point (1) and the line probe (2) can be more sharply represented by edges of trails, as is highlighted in (b) using bold lines.


63

T2

A X T1 Y X

(a)

αo

(b)

2.15 Application of uniform random and isotropic sections for measurements of geometrical parameters of internal structure of the object A. Two such sections, A ∩ T1 and A ∩ T2, created by two plains T1 and T2, are depicted in (a). Cross-sections A ∩ Ti are used for measurements with test systems that are uniform random and isotropic in their locations on these sections. One such instance with respect to a cross-section of object A is shown in the part (b). Dark gray objects in (b) represent cross-sections of the inner structure of A.

uniform random and isotropic. Due to the translational symmetry of the test system, a point Y, chosen from the object section, can be displaced in a uniform random manner on a selected fundamental area, α0, of the test system. For each new position of the point Y, a rotation of the testing system, with respect to the section, may be carried out simultaneously. The angular positions of the testing system must be isotropic. In some cases, the efficacy of a stereological measurement is enhanced by integral testing systems (Jensen and Gundersen, 1982). In this chapter, integral testing systems will be used for estimating the surface areas of threedimensional objects and the lengths of curves in three-dimensional space in Sections 2.4.2 and 2.4.3. The word ‘integral’ here implies the simultaneous usage of several types of probes (points, lines, figures) in a test system. An example of a fundamental region of an integral test system is shown in Fig. 2.16. A synopsis of various kinds of testing systems and their notations is included in Wiebel (1979).

64


a

b c

1

2.16 Fundamental region of an integral test system containing three point probes (arrows are pointing towards them): A curve probe of the length c, an excluding line (1), see Section 2.3.3, and a twodimensional probe of oblong shape with edge lengths a and b.

2.3

Stereology of a two-dimensional fibrous mass

Here we describe selected methods for stereological measurements of twodimensional fibrous materials. In particular, we estimate the geometrical parameters of an entire structure according to the area, length, and count of selected structural features, and define the Euler–Poincaré characteristic. Circular granulometry is introduced as a typical example of the second class of tasks for structural analysis. We will focus on one property of the individual parts of the structure, namely the distribution of particles using a typical length scale. The last example introduced in this section concerns the planar anisotropy of plain fibrous systems, which is a typical example of the third group of structure analysis problems, describing the mutual space distribution of structural features. This distribution will be represented by mutual fibre orientation, not taking into account the distances between them.

2.3.1

Point counting method for area and area density measurement

Volumes and volume densities of fibrous masses determine several of their properties, including air permeability, tensile strength and filtration efficiency. Glagolev (1933) and Thompson (1930) demonstrated that the cross-sectional area of a three-dimensional object is related to a random point counting procedure conducted on its two-dimensional section. Glagolev and Thompson


65

worked in the field of geology. A similar method was independently introduced in biology by Chalkley (1943). Two-dimensional parameters of twodimensional objects, the areas of figures, can be estimated using their zerodimensional sections. The probe has to be a point or a finite set of points in a test system. For such sections, the following example shows a point counting method. Using a two-dimensional reference region Ω and a two-dimensional object B that is embedded fully or partly in Ω, we will solve the question of how to estimate the area of B inside Ω using uniform random zero-dimensional sections. As a reference region, we can consider a microphotograph or a part of it. The situation is shown in Fig. 2.17. We start with the probability p that a uniform random point in Ω intersects the object B.

p=

S ( B) S (Ω)

[2.5]

The area of the region Ω is here denoted as S(Ω) and the particular area of the object B that is embedded in the region Ω is S(B). The probability p is expressed as the ratio of two surfaces and hence it is called a geometrical probability. Carrying out n measurements with the point probe we derive from Equation [2.5] np = nS(B)/S(Ω). The number of non-empty intersections, denoted as I, is equal to np. Then we obtain I ≅ S ( B) n S (Ω)

[2.6]

Due to the finite number of trials, we only estimate the probability p as I/n, so the left-hand side of Equation [2.6] does not represent the exact value of p but a very good estimation of the fraction S(B)/S(Ω) that is equal to p. From Equation [2.6] we can draw two conclusions. Knowing the area S(Ω) B

Ω

2.17 A two-dimensional space containing a region Ω, within which parts of an object B are embedded.

66


exactly, we can estimate the area S(B), or we can estimate the area density S(B)/S(Ω) of the object B inside Ω directly. But first we will state the notation for estimating a feature parameter value. We have mentioned that, for instance, p is estimated using the fraction I/n. This can be expressed as p ≅ I/n. Without additional explanation, however, this does not tell the reader which quantities in the relationship are measured with complete or high accuracy, and which of them are estimated. Here the estimated quantity is p while I and n are measured accurately. To underline these facts, we write [p] = I/n which we understand as – p is estimated as the fraction of known values I and n, and the symbol [p] denotes the estimator of p. Let us return to Equation [2.6]. If we know exactly the area of the reference region Ω, which is, as a rule, the area of our micro-photograph or monitor screen, we can express from Equation [2.6] the estimator [S(B)] of the area S(B) inside Ω as: [ S ( B )] = I S{Ω ) n

[2.7]

When our interest is focused on the B area density S(B)/S(Ω) inside region Ω, we can state from Equation [2.6] its estimator in the following form: [ S ( B )] = I [ S ( Ω )] n

[2.8]

We have written the left-hand side of Equation [2.8] in the form [S(B)]/ [S(Ω)] rather than [S(B)/S(Ω)] because S(B) is in fact estimated with I and S(Ω) is estimated using n. The measurement procedure can be improved using a test system with zero-dimensional probes. When we wish to estimate the total area of B inside the region Ω, or the area density of B within Ω, we have to cover Ω with the test system as sketched in Fig. 2.18. The number of hits I on the figure B by test system point probes is equal to 4 in this example. These hits are denoted using empty squares. The total number n of point probes falling into Ω is 14 in this case, and these hits are marked either by empty squares or by black circles. The situation in Fig. 2.18 leads to the approximate value of the area density [S(B)]/[S(Ω)] = 0.286, a poor estimate from only one particular position of the test system. We can enhance the accuracy of our measurement significantly by increasing the number of uniform random and isotropic trials. The point counting method for estimating area and area density of figures in the reference region is in fact a direct extension of the well-known method based on square grids and counting the number of squares that are fully contained within B, as shown in Fig. 2.19. However, this method is less accurate than the point counting method. Increasing the accuracy of the grid method by measuring squares that are only partly contained in B is more laborious than using the Glagolev and Thompson point counting method.


67

B

Ω

2.18 A test system having one point probe at left bottom corner of each of its fundamental regions (indicated by arrows), covering the region Ω completely with embedded objects, B. Hits of probes with B are denoted with squares and residual probes in Ω are encircled in black. A very rough estimation of area density of B in Ω may be calculated here as [S(B)]/[S(Ω)] = 4/14 = 0.286. B

A

2.19 A simple estimation of area covered by B using a square grid and counting the areas of fundamental squares fully embedded in B. If the area of a grid cell is A, then S(B) stands for the particular case shown in the figure, having an estimated value of 8A.

2.3.2

Buffon’s needle and curve length estimation

A thorough investigation of a fibrous mass often requires information about total fibre length or fibre length density. The influence of fibre length and

68


fibre distribution on the strength of fibres in yarns, and the relation between cross-sectional counts of fibres and their length, have been investigated by Zeidman and Sawhney (2002). This subsection examines how to estimate the length L(C) of one-dimensional linear features, i.e. curves, embedded into two-dimensional space, or into a plane. It will be shown, based on Buffon’s needle problem, that [L(C)] = (π /2)dI, where d is the distance between equidistantly spaced parallels and I is the number of intersections between the curve and the system of parallels. Buffon’s needle, as described in Section 2.1, identifies the probability p with which a uniform random and isotropic abscissa j, the so-called Buffon’s needle, of length L ( j ), touches the warp of equidistant parallels under the condition that the needle falls on it and nowhere else. The relation L ( j ) < d ensures, at maximum, one hit for each trial. Figure 2.20 shows this in more detail. If the mutual orientation of the needle and the warp is fixed, this suggests that the needle is uniformly random, but anisotropic. We initially select its orientation perpendicular to the warp lines. The probability P of the anisotropic needle hitting one of the parallel lines is given as the fraction L( j )/d, which follows from the concept of geometrical probability given as the ratio of the areas of two point sets. The first set is composed of the locations of a chosen fixed point on the needle for all possible trials when the needle hits the warp, and the second consists of the area of the point set created by all locations of the same selected point on the needle for all possible trials. Thanks to the warp periodicity, we can restrict our attention to the region between two pairs of neighbouring parallel warp lines, so we consider nothing outside such bands. Both bands are parallel with the warp lines. The first has width equal to the needle length L( j) and the latter has the width d that fills the entire space between neighbouring parallel lines. The lengths of both bands can be taken as equal, so we only need to take the widths into account. With the aid of Fig. 2.20 we can conclude that P = L( j)/d. d y L(j )

Φ

j

2.20 Buffon’s needles are anisotropically distributed and are all perpendicular to the parallel warp lines. The distance between parallels is d and the Buffon’s needle length is L( j ). A hit of a needle with one of the warp lines is denoted by a small circle. The probability of the hit is evaluated from the ratio L( j )/d. On the right part of the figure is depicted a declined needle that makes an angle θ with parallels. Its projection on the normal to the warp lines is y.


69

Repeating the process for a needle with some chosen fixed angle θ to the parallel warp lines, as shown in Fig. 2.20, instead of the needle’s length, we will be concerned with its parallel projection y on the direction perpendicular to the straight lines that make up the warp. For y = L( j ) sin (θ ) we can write the probability Pθ of hits by the angled needle as:

Pθ =

y L ( j ) sin (θ ) = d d

[2.9]

The last step in solving Buffon’s needle problem is to consider isotropic orientations of a uniform random needle. Here we need to calculate the average value for L ( j ) sin (θ )/d, where L( j ) and d are constants. Using the well-known formula for the mean value f of a function f (x) on an interval < a, b > written as f =

1 b–a

∫

b

f ( x ) dx, we obtain the final

a

relation for the probability of hits of a uniform random and parallel needle as: p=

L( j) πd

∫

π

0

sin θ dθ =

L( j ) 2 L( j ) [– cos θ ]π0 = πd πd

[2.10]

where b – a = π – 0 = π is the length of the interval in question. Before we extend Buffon’s problem to the estimation of curve length in a plane, let us look at the formula for the mean value of a function on an interval. We have introduced geometrical probability as a generally accepted approach and now we can describe the geometrical interpretation of the mean value of a function. Imagine a very thin aquarium containing fine sand. We will arrange the sand into the shape of sin θ on the interval < 0, π >. The volume of the sand pile is proportional to

∫

π

sin θdθ (see Fig. 2.21). Tapping

0

the aquarium gently will produce a flat block of sand from the previously sinusoidal heap. We have destroyed our original curve but the height of the sand in the aquarium is now equal to the average value of the function in question and, moreover, the volume of sand (which is conserved) is now easily expressed as π f . Equilibrating both formulae for the volume of the sand, we obtain a formula from which the average function value f on the interval < 0, π > can be easily derived as π f =

∫

π

sin θdθ .

0

We will now investigate a curve in a plane of total length L(C). Imagine the curve is divided into very short straight segments of equal length; these segments can be taken as Buffon’s needles with uniform lengths L ( j ). Unlike previous discussions of the Buffon’s needle problem, here j denotes the j-th piece from the total number of n linear pieces composing the curve. Length

70


f (Θ) = sin(Θ)

2 1

π

0

f 〈0,π〉 π

0

2.21 A sinusoidal sand pile in a narrow aquarium is streamed using a gentle percussion to create a flat block. The sand volume is conserved during the motion. d = 0.01m C

n 2

1

2.22 A curve C is approximated using a set of straight pieces. Their lengths are assumed nearly equal. The number of hits I of the curve C and the warp lines in this case is 10. Hence a rough estimation of the curve length L(C ) from a single trial is πd10/2, as given by Equation [2.11].

L( j) is shorter than the distance d between parallel straight lines. The warp now represents our test system. We overlap the curve with this test system as shown in Fig. 2.22. As the curve is composed of n Buffon’s needles, it means nL ( j ) = L(C ), and the number of hits I of the curve in the uniform random and isotropic system of parallels will be equal to the n-multiple of the probability p in Equation [2.10]: I = n[ p ] = n

2[ L ( j )] 2[ L ( C )] = πd πd

[2.11]

The only measurement done with the curve overlapping the test system provides us with a very pure estimation of L(C) from a single trial. The derivation of the formula [2.10] for p was based on uniform random and isotropic needles, so we have to carry out a lot of measurements to ensure this condition by rotating and shifting the warp and by counting and averaging all the hits. These experiments provide us with a more exact estimation of


71

[L(C)]. From Equation [2.11], the final formula for curve length estimation in two-dimensional space can be easily derived as: [ L ( C )] =

π dI 2

[2.12]

where I is the average number of hits per single measurement calculated from numerous uniform random and isotropic trials of test system position with respect to a curve. The equidistant and parallel system of lines represents a lattice of fundamental regions, each of them being oblong in shape as the lines are restricted to a plane. The area of a particular oblong between neighbouring parallels represents a fundamental region. In each fundamental region, there is only one piece of a line as a probe, represented by one of the parallels.

2.3.3

Feature count in two-dimensional space and the Euler–Poincaré characteristic

Feature count is useful for instance in identifying an economic wool fibre where scale frequency plays an important role, as shown by Wortham et al. (2003). The count of fuzz and pill formation on knitted samples as a function of enzyme dose for treatment has been investigated by Jensen and Carstensen (2002) and is another example of the importance of feature count techniques for fibrous materials. Before we discuss the stereological method for estimating feature count in two-dimensional space, we will describe the Euler–Poincaré characteristic ν (A). This characteristic is the functional that evaluates the connectivity of compact sets, which is why it can also be used for convex ring sets. The connectivity of a set A can be defined in various ways that reflect an intuitive view. We will use an approach similar to that described by DeHoff and Rhines (1968), Wiebel (1979) and Saxl (1989), aiming at a visual and rigorous introduction of the Euler–Poincaré characteristic. We take the position that a set composed of two disjoined cubes has the same value of connectivity as another set consisting of two disjoined spheres. In addition, we note that connectivity does not depend on the size of the bodies involved. On the contrary, it depends on the numbers of holes and cavities in the bodies and on their nature, which is consistent with the number of isolated parts of the body boundaries. We distinguish between open holes, for example a hole created by a perforation of a sphere, and a closed cavity, which results in the sphere having a boundary composed of two isolated parts. The degree of connectivity depends on the behaviour of a body with respect to a section. If we draw a curve that lies in a plane on the body’s surface, then we can extract the part of the body that lies within this plane and is restricted by the curve on the boundary. A sphere without holes or a sphere with a closed cavity are both broken up by such a section. A sphere

72


with an open hole only breaks in some cases, and is hence the more connected set. These situations are sketched schematically in Fig. 2.23. The numerical value of the Euler–Poincaré characteristic ν depends also on the dimension of the set. Generalising the above leads us to the following rules for the determination of Euler–Poincaré characteristic values: (i) For a one-dimensional set A composed of N isolated curves, ν (A) = N. (ii) The two-dimensional set B consisting of N isolated parts with total number of N′ cavities (in two-dimensional space cavities are always closed) has the Euler–Poincaré value ν(B) = N – N′. (iii) For the three-dimensional set C of N isolated parts with the total number of N″ open holes and N′ closed cavities, ν (C) = N + N′ – N″.

(a)

(b)

2.23 The sphere without an open hole detaches into two parts after each cutting, followed by withdrawal of the sphere part lying within this section and restricted by its planar curve on the spherical surface (a). The sphere with an open hole does not disintegrate after such cutting (b).


73

This gives us the result that, for a circle with a cavity and for a sphere with an open hole, ν = 0. An arbitrary single body or figure without holes or cavities has a Euler–Poincaré characteristic equal to one, which is why the Euler–Poincaré characteristic is identical to the feature count for objects without any holes. The connectivity of a sphere with a closed cavity is evaluated as ν = 2. Saxl (1989) introduces the Euler–Poincaré characteristic by having the boundary of a half space in such a position that the origin of the coordinate system lies within it, and the investigated structure A lies fully in the righthand of the half space as drawn in Fig. 2.24. The boundary is swept from the left to the right side along the perpendicular axis. The boundary is plane in three-dimensional space, a line in two-dimensional space and a point in onedimensional space. We count the values of the left-hand side limit Sweeping boundary y

A

0 X1 X2 X3 X4

X5 X6

X7

X8 X

X1 X2 X3 X4 lim (υ–υ(X–ε)) ε→0+

X5 X6

X7

X8 X

υ(X) 2

1

0

1

–1

X1 X2 X3 X4

X5 X6

X7

X8 X υ=3–2=1

2.24 Euler–Poincaré characteristics as introduced by Serra (1976) and Saxl (1989). The Euler–Poincaré characteristic value of the object A is equal to 1.

74


lim (ν ( x ) – ν ( x – ε )) composed of the subtraction of the Euler–Poincaré ε → 0+ characteristic values of the induced structure in sections of the moving boundary with the investigated structure, in cases where the limit values are non-zero. The subsequent stages of this method are shown in Fig. 2.24. Now to discuss the problems of counting features notwithstanding the number and the nature of the holes they contain. To estimate features in a selected area of a two-dimensional structure, we use a test system with the so-called excluding line introduced by Gundersen et al. (1988). A probe A in this system is two-dimensional and as a rule it has an oblong shape. Its area will be denoted here as S(A). The excluding line is an infinite straight line running along a portion of the boundary of probe A which changes direction twice. The excluding line falls particularly on two neighbouring sides of the oblong A. The mutual position of probe A and the excluding line is shown in Fig. 2.25. This probe is inserted into a lattice of fundamental regions to create a test system. The estimation of the feature count NA in a certain area of the object is conducted according to the following procedure: (i) Count all figures (i.e. all isolated parts of the object) that have nonempty intersections with a chosen probe A and at the same time have no hits with the excluding line. Their count is denoted as Q. (ii) Repeat this measurement for each probe in the test system and for all its uniform random and isotropic positions with respect to the fixed object. The estimation of the total count of features N in the reference region Ω is then: [N] =

QS ( Ω ) S( A)

[2.13]

where Q is the feature count per probe of area S(A) and S(Ω) is the area of the reference region. The estimator of the feature count area density [N]/[S(Ω)] in the object is simply:

Q [N] = [ S( Ω )] S ( A )

[2.14]

The estimation accuracy increases with the number of uniform random and isotropic trials conducted in different test system positions.

2.3.4

Linear characteristics of convex ring sets and circular granulometry

For many practical applications, it is valuable to introduce a numerical linear parameter that estimates the representative size of structural features. For


75

1

10 mm

A2

A1

2

A3

A4

2.25 Test system for estimation of particle numbers: The grey particles are counted exclusively. The residual ones either hit the excluding lines or have no intersection with fundamental regions and are not counted, according to the counting procedure. Excluding lines are in bold (1). Two-dimensional probes (2) are arranged in a lattice of fundamental regions. The rough estimation of the feature count density [N]/[S(Ω)] from this particular trial comes out to be Q/S = 15/(4S(A)), where S = ∑ S(Ai) = 4S(A) is the area of all the oblong probes used for the purpose.

example, Neckar and Sayed (2003) described pores between fibres in general fibre assemblies with particular focus on their linear characteristics, such as pore dimension, perimeter and length. Pore length and radius were used by Miller and Schwartz (2001) as critical parameters for a forced flow percolation model of liquid penetration into samples of fibrous materials. Lukas et al.

76


(1993a) compared the breadth and diameter of maximal pores in thin nonwoven fabric with their radius values measured using the bubble counting method. Cotton fibre width and its distribution using image analysis was measured by Huang and Xu (2002). Farer et al. (2002) studied fibre diameter distribution in melt-blown non-woven webs. Brenton and Hallos (1998) investigated the size distribution, morphology, and composition of dust particles gathered from the vicinity of various commonly performed processes in industrial wool fibre preparation. Here we discuss the estimation of breadth w, diameter d and width t of structural features, and then introduce the effective method for estimating diameter known as circular granulometry. Consider an n-dimensional body A, part of a convex ring, and an arbitrary r direction u as is sketched in Fig. 2.26. The support plane is taken as that which creates the boundary of the ‘smallest’ half space that contains the r body A in the direction u , hence it touches A. Since this half-space unfolds r from the support plane in the direction u , there is no part of A in the residual r half-space. For each support plane perpendicular to the chosen direction u , r there is a parallel twin for the opposite direction – u . We will denote the r distance between the two support planes as the breadth w(A, u ) of a body A r r in the direction u , and consequently also in the direction – u . The isotropic r r r average of breadths w(A, u ) is denoted as w ( A, u ) , where all u directions have the same weight. r [2.15] w ( A ) = w ( A, u ) The maximum breadth value is diameter d(A) and the minimum is width t(A). Extending this to n < 3 dimensions is straightforward. As an example, we will calculate the average breadth w ( S ) of the square S with side length a (see Fig. 2.27). For the breadth w of square S we have: u

–u

d (A ) A A

t (A )

w (A )

2.26 Linear characteristics of a set A of a convex ring having breadth w (A), width t (A), and diameter d (A). Supporting planes are r r perpendicular to u and –u .


77

S α

a

r u

a

r w (u )

r 2.27 The breadth w (S, u ) of a square S.

r w ( S , u ) = a 2 cos α

[2.16]

Taking periodicity into account, we will consider only π /2 rotations of the twin support lines with respect to the square. The isotropic average value of r r w(S, u ) in the interval of u directions < 0, π /2 > is: w (S) = 2 π

=

∫

π /4

– π /4

a 2 cos α d α =

2 2a 2 = 4a π π 2

2 2a [sin α ]π– π/4/4 π

[2.17]

Noting the above relationship between square S perimeter O(S) = 4a and its average breadth w(S), then for a square, O(S) = π w(S). The same relation holds for a circle C with perimeter O(C) = 2π r and with average breadth w(C) = 2r. The general relation: O(B2) = π w(B2)

[2.18]

is valid for all two-dimensional convex sets B2; hence their average breadths are commonly calculated from their perimeters. Circular granulometry is a simple method for estimating diameter d in the distribution of two-dimensional particles or projections of three-dimensional ones. The method is based on a special type of test system consisting of circles of various diameters. Stereotypes of circles are commonly used, with diameters expanding equidistantly in steps of one millimetre. Then we select at random a particle from the magnified image and assign to it the smallest circle that can fully contain that particle. We count the numbers of particles assigned to circles of various diameters and we plot their total relative counts

78


pd as a histogram that estimates the probability densities or probability distribution function. The probability density histogram expresses the appearance of the particle diameter in the interval (di – ∆ d, di), where ∆d is the incremental step used for the construction of the circle stereotypes. An example of circular granulometry analysis is indicated in Fig. 2.28.

2.3.5

Analysis of planar anisotropy of two-dimensional fibrous structures

Fibrous materials often present as thin, nearly planar fibrous systems; for instance thin webs, sheets, some yarn tangles, woven and knitted textiles,

d4

d1 d2 d3 d4 Test system of circles

d1 d2 d2

d3

d1 d2

d4 d2

d2

d1

d4

d3

p 5/12

1/4

1/4 1/6

d2 d4 d1 d3 Probability density histogram

d

2.28 Circular granulometry: To randomly chosen particles of an investigated structure are assigned the smallest circles from the test system that can circumscribe the chosen particles completely. The special test system, represented here by stereotypes of circles with various diameters, is shown in the right upper corner. The histogram relates the frequencies of estimated particle diameters with the diameters.


79

and vessels in bladders. Planar fibre systems can also be projections of threedimensional fibrous materials. The intensity of light scattering and its distribution in non-woven fabric as a function of fibre mass arrangement in space has been studied by Zhou et al. (2003). Pourdeyhimi and Kim (2002) outlined the theory and application of the Hough transform (Hough, 1962) in determining fibre orientation distribution in a series of simulated and real non-woven fabrics. Farer and colleagues (2002) studied fibre orientation in melt-blown non-woven webs. A general model of directional probability in homogeneous, anisotropic non-woven structures was presented by Mao and Russel (2000), in which fibre diameter, porosity and particularly fibreorientation distribution were considered as structural parameters. A method for non-destructive fibre tracing in a three-dimensional fibre mass using Xray microphotography was developed by Eberhardt and Clarke (2002). Karkkainen and colleagues (2002) developed stereological formulae based on the scaled variation of grey shades in digital images of fibrous materials to estimate the rose of directions. Thin fibrous systems can also be modelled and analysed using established theory of fibre processes, as is described thoroughly in Stoyan et al. (1995). In this section, we describe the simple graphical method for evaluating planar fibre mass anisotropy introduced by Rataj and Saxl (1988), beginning with a discussion of planar anisotropy. Imagine a curve or a thread of total length L fully embedded in a plane thanks to its negligibly small diameter. It is understood for anisotropy that equal angle intervals (βι , β ι + ∆β ) do not contain equal lengths of thread elements pointing to the corresponding directions (see Fig. 2.29). A parameter of anisotropy is therefore the angular density of the thread f (β ) governing 90°

∆β

180°

0°

270°

2.29 The left-hand side of the figure represents a thread of a total length L. The broken segments of the thread have an orientation within an angular interval (– ∆β, + ∆β ) of an equidistant net of angles as shown on the right-hand side.

80


the particular length of the thread L(β, β + ∆β) leading up to the interval (β, β + ∆β): L ( β , β + ∆β ) = L

∫

β + ∆β

β

f ( β ) dβ

[2.19]

where L is the total thread length. The density function f ( β ) is known as the rose of directions or the texture function. There are additional experimental methods that enable us to estimate the rose of directions f ( β ). The direct method, as indicated in Fig. 2.29, was described by Sodomka (1981). According to this procedure, we first identify the part of the thread with the highest curvature. Inside this part, we demarcate a piece of the thread for which tangent directions vary within the interval ± 12 ∆β. The remaining part of the thread is then divided into elements of equal length, this length being determined by the length of the section in the most curved part of the thread. Each such thread element will be counted to a corresponding angular interval (β ι , β ι + ∆β ). The fractions Nβ /N, where Nβ are the counts inside the interval (β ι , β ι + ∆β ) and N is the total number of counts, give the estimations for values of the rose of directions. It is clear that experimental implementation of this procedure would be laborious and time-consuming. Its advantage, however, is the clarity with which it helps us introduce the notion of the rose of directions. More effective methods for estimating fibrous planar anisotropy are based on measuring a rose of intersections (Rataj and Saxl, 1988). The rose of intersections is obtained using the method shown in Fig. 2.30(a, b) and in Table 2.3. The rose of directions is constructed from rose of intersection data by a simple graphical construction using a Steiner compact in the following five steps: (i) Place a net of angles drawn on a transparent foil over the structure being studied, or a computer-aided net over the image on a monitor screen. An example of such a net is shown in Fig. 2.30(a). The net consists of arms of equal length that intersect each other at their central points, and the number of arms has to be equal to or smaller than 18, otherwise the method does not produce sufficiently stable, or reproducible, results. The angular distance among all arms is equal to π divided by the number of arms. The example in Fig. 2.30(a) has four arms with the angular distance π /4. (ii) Count the intersections of the fibrous features with each arm separately, as shown in Fig. 2.30(b). Repeat this measurement in uniform randomly chosen parts of the fibrous structure, keeping the orientation of the angular net strictly fixed. Take the direction of a line in the object and its images and denote it as direction 0°. One of the arms of the net of angles must then be parallel with this line for each measurement. Put together the total number of intersections for each arm into a table such


81

β3

β4

β2

β 1 = 0°

0°

(a)

(b)

β3

d

β4

c

β2

b c a

b

d

β1

a

(c)

(d)

2.30 Construction of a rose of directions using a simple graphical method: A net of angles is composed of equal arm lengths (a); intersections of a net of angles with a planar fibrous structure and a chosen direction in it (b); Steiner compact of side lengths a, b, c, d with arrow pointing towards βi, belonging to the side c (c); a rose of directions (d). Table 2.3 The values of the rose of intersections for the fibrous system as depicted in Fig. 2.30(b). The last column of the figure contains values of this rose after rotation by π /2, used for construction of the Steiner compact in Fig. 2.30(c) Angle

Rose of intersections values

π /2 rotated values

0° 45° 90° 135°

3 4 3 3

3 3 3 4

as Table 2.3. The intersection count data can then be expressed graphically in a polar diagram, known as the rose of intersections. Rotate the rose of intersections by the angle π /2 clockwise or anticlockwise or shift values in the table. The fibres are not orientated up or down, so it is not

82


important to distinguish between those fibre segments that point in direction β or β + π. Hence the angular density f (β ) is the periodic function with the period π. Clockwise and anticlockwise rotations of the rose of intersections differ by π /2 + π/2 = π, and this periodicity provides us with the same information about f (β ). (iii) Plot the count number from the rotated rose of intersection data into a polar diagram, using an appropriate scale, to obtain the π /2 rotated geometrical interpretation of a rose of intersections. (iv) Raise verticals from each point of the π /2 rotated rose of intersections to obtain a polygon restricted to containing the origin of the polar diagram. This polygon must be convex and centrally symmetric, and is known as the Steiner compact (see Fig. 2.30(c)). The distance between neighbouring vertices, i.e. the Steiner compact side length, is the estimation of the angular density f (β i ) of the rose of directions value for a direction identical to the direction of the side in question. Hence, using the length of the side pointing in the direction β i we can estimate the angular density f (β i ) within the interval β i ± 1/2 · ∆β. (v) Construct arcs with their centres in the polar graph to finish the rose of directions. Each arm of these arcs is proportional to the length of the corresponding side of the Steiner compact. Similarly, like the Steiner compact, the rose of directions must also be centrally symmetric. The resultant rose of directions for our example is depicted in Fig. 2.30(d). To normalise the construction, we have used a scale where the total length value of the arms of the rose of directions is equal to 1. Figure 2.31 shows us various simple planar curve systems, a regular square grid, a grid of rectangles and a system of circles. Each grid is shown with its rose of directions. The reader is invited to estimate them using the simple graphical method described above. We should point out that, for each measurement, the net of angles must be fully embedded into the fibrous system. The reader will probably observe some nearly negligible angular density values estimated for directions that are not present in the system. That is the cost paid for the method’s simplicity.

2.4

Stereology of a three-dimensional fibrous mass

Adding a dimension helps us to fully appreciate the power of using stereological methods to estimate three-dimensional parameters of features from measurements of their two- and lower dimensional sections. Here, we introduce methods for estimating volumes, surfaces, lengths and their densities in three-dimensional reference regions. We then describe methods for estimating average curvature and torsion of fibrous materials in three-dimensional space. Finally we discuss feature counts.


83

(a)

(b)

(c)

2.31 Roses of directions belonging to various fibre structures: a regular square grid (a); a rectangular frame (b); a system of circles (c).

2.4.1

Estimation of volume and volume density

To illustrate the importance of volume estimations, we refer to the fact that pore volume or pore volume density are critical parameters in the air permeability of fibrous materials, as has been shown for instance by Mohammadi et al. (2002). Fibre bulk density heavily influences the compressibility of fibrous materials, as shown by Taylor and Pollet (2002) or in classic work on this topic by Van Wyck (1964). The porosity of a fabric and the volume fraction of fibres were considered critical parameters for coupled heat and liquid moisture transfer in porous textiles by Li and colleagues (2002). The point counting method introduced by Glagolev and Thompson to estimate the areas of figures was actually aimed at ultimately estimating volumes, and we will now extend the results discussed above to threedimensional space in order to estimate volumes and volume densities of real fibrous objects.

84


Ω

A Y

2.32 In a reference region is embedded a three-dimensional object Y. A point A represents a zero-dimensional section in the region Ω that does not strike the object Y.

Imagine a three-dimensional reference region Ω and an object Y embedded in it, as shown in Fig. 2.32. The conditional probability p with which a uniform random point A in Ω has a non-empty intersection with the object Y is given by the relation:

p=

V (Y ) V (Ω)

[2.20]

where V(Ω) is the volume of the reference region Ω and V(Y) is the volume of the object Y to be estimated. The geometrical probability p of the hit is equal to the fraction of the aforementioned volumes V(Y)/V(Ω). When we carry out n measurements with the uniform random point in the threedimensional region Ω, it will hit the object I times, where I is close to the product pn; in other words I/n estimates p. Hence, by knowing the volume V(Ω) with sufficient accuracy, we can express the estimation of the volume [V(Y)] of the object Y as: [ V ( Y )] = I V ( Ω ) n

[2.21]

The volume fraction is hence estimated by: [ V ( Y )] = I [ V ( Ω )] n

[2.22]

To improve the efficiency of three-dimensional volume and volume fraction measurements, as a rule we use uniform random two-dimensional sections on which we carry out zero-dimensional sections, i.e. point hit trials, using test systems. This procedure is indicated in Fig. 2.33 for a single measurement. To enhance the accuracy of our measurements, we have to take further two-


85

Y Ω

Ω

Y (a)

(b)

2.33 A two-dimensional section of a reference region Ω with embedded objects, Y (a), consists of two parts of the twodimensional section (b). The cross-section is overlapped with a test system containing zero-dimensional probes at the bottom right-hand corner of the fundamental zones, as highlighted by arrows. The total number, N, of probes in the test system is 44 and the number of hits with Y as I = 16. The volume density can be roughly estimated as [V(Y)/V(Ω)] to be I/n = 16/44 = 0.364 from this measurement.

dimensional sections of the body and apply more trials on them using the test system. All trials must be uniform, random and isotropic. Measurements of volume density correspond with a fundamental principle of stereology that was proved long before this mathematical discipline was established in 1961. The French geologist Delesse (1847) showed that the volume densities of various components making up rocks can be estimated from random ground sections by measuring the relative areas of their profiles. The same statement is contained in Equation [2.22] because the right-hand side is identical with the right-hand side of Equation [2.8] for area density estimation, and we claimed that with the test system of point probes we made our measurements on planar, i.e. two-dimensional, sections. That is why:

[ V ( Y )] [ S ( Y2 )] = = I [ V ( Ω )] [ S ( Ω 2 )] n

[2.23]

where [S(Y2)]/[S(Ω2)] is taken as the average value from a series of measurements carried out on a sufficient number of uniform random twodimensional sections of Ω and Y. Quantities Y2 and Ω2 represent induced structures of Y and Ω on the two-dimensional sections. Symbols I and n have the same meanings as before. Another approach for deriving the Delesse principle is based on integration, as introduced in the integral relation [2.1] commenting on the definition of stereology. Having a function of both cross-sectional areas SY (z) and SΩ(z) for Y and Ω using the same incremental step ∆z, we obtain:

86


V (Y ) = V (Ω)

∫ ∫

H

≅

0 H

0

H / ∆z

SY ( z ) dz SΩ ( z ) dz

Σ SY ( z i ) ∆z

i =1 H / ∆z

Σ SΩ ( z i ) ∆z

H / ∆z

= Σ

i =1

SY ( z i ) S ( Y2 ) = SΩ ( z i ) S ( Ω 2 )

i =1

[2.24] The term of the left of Equation [2.24] is the mean value of the fraction of areas. Integrals in Equation [2.24] have been estimated using a finite number H/∆z of sections, where H is the total height of the reference region Ω and ∆z is the step, i.e. the constant distance between parallel and neighbouring sections. For more details see Fig. 2.34. The relation [2.24] is independent of the choice of z-axis direction and hence estimation of volumes and volume densities can be carried out on one series of parallel sections, which is unusual in stereology since sections must normally be isotropic.

2.4.2

Surface area and surface area density of threedimensional features

Surface and surface area density estimations are critical for explaining the sorption characteristics of a fibrous mass. Kim and colleagues (2003) carried out research on fibre structure and pore size in wiping cloths. The filtration properties of fibrous materials with respect to their surface areas were investigated by Lukas (1991). Z

Ω

Y

Sz ( Y )

H

∆Z

Sz (Ω)

2.34 Delesse’s principle: The volume density V(Y)/V(Ω) is estimated through the average value of area densities Sz(Y )/Sz(Ω).


87

To describe a stereological method for surface area estimation, we start with the three-dimensional reference region Ω of volume V(Ω). In this volume is placed a three-dimensional body with surface Y and of surface area S(Y). We will use a test needle T of length L(T) to estimate the surface area S(Y). Assume the needle is uniform, random and isotropic in Ω, being the region to which the appearance of the needle is restricted. The isotropy of the needle means that if we moved its lower point to the origin of a coordinate system, the upper point hits the small area dµ on a sphere with radius r = L(T) with the probability p1 for which: p1 =

dµ 2 πr 2

[2.25]

Taking one small, flat piece y on the surface Y with the area S(y), then the whole area S(Y) is built of n such elementary surface pieces y. The probability p2 with which the uniform random and completely anisotropic needle T, i.e. a needle with fixed orientation, hits y in a region Ω will be expressed as a geometrical probability. The probability of the hit p2 is now given by the fraction of two volumes. The first is the volume of a point set composed of locations of the needle fixed point (let it be located on its lower edge) for all cases when the needle hits the small area y. The second volume is that of Ω. This volume is proportional to all possible locations of the fixed point of the needle. The first volume, the small area y, and the needle, are depicted in Fig. 2.35. For p2, the following is true:

µy

θ

µy

µt

µt

y

T

2.35 The probability of intersection of the needle T (having a fixed orientation in space) and a small r surface piece y is proportional r to the volume V = S(y)L(T)/cos Θ. u y is perpendicular to y and u t of unitary length lies in the needle direction.

88


p2 =

S( y ) L ( T )/cos θ / V (Ω)

[2.26]

where S(y)L(T)/cos θ/ is the volume of the first point set when T hits y. Straight brackets denote the absolute value of the cosine of angle θ that contains the needle and the normal perpendicular to the small area y. All we have to do now is to express the average value of p for uniform random positions and isotropic orientations of the needle, which means we have to express the average value of the function /cos θ/, where all needle directions will have equal weight. To do that we return to the area dµ on the sphere and consider the sphere radius r = 1. Envisage the situation depicted in Fig. 2.36, which helps us to obtain the relation dµ = sin θ dΦ dθ. The total area covered by all dµ’s for various needle orientations is one half of the unit sphere surface area 2π. One half of the sphere surface is used here because we do not wish to distinguish between up and down orientation of the needles. The dµ elements are of various areas for various θ as can be seen in Fig. 2.36. The area dµ is much smaller near the sphere’s apex than in the vicinity of the sphere’s equator. Considering the geometrical interpretation of a function average value on a chosen interval; here we have a two-dimensional interval 2π which has the shape of one half of the unitary sphere surface. In the interval value, r2 is implicit because r = 1. This interval is determined using angles Φ and θ in Fig. 2.36. Τhe function for which the average is sought is /cos θ/. The factor sin θ in the relation dµ = sin θ d Φ dθ tells us how the area dµ varies with various values of the angle θ, representing various attitudes on the sphere. The interval for θ is and Φ is from . The

sin θ d Φ sin θ

dµ r=1 θ

Φ

dθ

dΦ

2.36 A small piece of surface dµ on a sphere has its area expressed in terms of angles Φ and θ.


89

average value of /cos θ/ for an isotropic needle on the interval 2π is then given by:

∫

/ cos θ / = 1 2π = 1 2π

=

∫

∫

π /2

0

/cos θ / dµ

A 2π

0

dΦ

∫

π /2

/cos θ / sin θd Φ dθ

0

π /2

1 /cos θ / sin θ dθ =  sin 2θ  2 0

= 1 2

[2.27]

The first integral in the relation is taken over one half of the unitary sphere surface denoted here as A. Now we substitute this result into Equation [2.26] to obtain p as the average value of p2 with respect to the isotropic orientation of the needle: p=

S ( y ) L ( T ) /cos θ / S ( y ) L ( T ) = V (Ω) 2V (Ω)

[2.28]

The relation [2.28] is valid for each surface piece y. All y’s cover the whole surface Y. To sum the probability p for the total number n of y’s we obtain: np = I =

L (T ) n L(T ) S (Y ) Σ S ( yi ) = i =1 2V (Ω) 2V (Ω)

[2.29]

where the product of the probability p and the number N of elementary areas y is expressed as the number of hits I between the needle of length L(T) and surface Y. We estimate p from a finite number of measurements. That is why the estimation of the surface S(Y) for known volume V(Ω) has the shape: [ S ( Y )] =

2 IV ( Ω ) L(T )

[2.30]

For the surface density [S(Y)]/[V(Ω)] of Y in the three-dimensional reference region Ω we can write: [ S ( Y )] 2I = [ V ( Ω )] L ( T )

[2.31]

For measurements using a test system containing the total length L of all needles, I is the total number of all hits belonging to the total length L of all needles. Hence the formula [2.31] is valid for test systems after the substitution of L for L(T) according to this new meaning of I. We will now demonstrate the use of an integral test system to estimate the surface area S(Y) of a surface Y that is embedded in a three-dimensional region Ω. We prepare uniform random and isotropic sections from our specimen and then we place over them the integrated test system as shown in Fig. 2.37.

90


Ω

Y

X Y Y

X

Y

X A

X

2.37 A three-dimensional object X, e.g. a fibrous mass, has its surface denoted here as Y (right side of the figure). The number of intersections between test needles and the surface Y of the object X to estimate the surface area S(Y ) can be realised through twodimensional sections (left-hand side of the figure). The number of intersections of point probes with cross-section of the region Ω is denoted by Q (Q = 9 in this case). Q estimates the total needle length L in a particular section as L = Q · L(T). The number of hits of test needles with Y is I = 2. Hence, a rough estimation of the surface density from the measurement is [S(Y )]/[V(Y ) = 2I /(QL(T ) = 4 /(9L(T )).

The total length L of the needles inside Ω is estimated by the count Q of needle reference points that fall inside Ω rather than by time-consuming measurements of the needle length if they are only partly involved in Ω. The estimation of the total needle length in Ω is then L = L(T )Q. We can also estimate S(Y ) from two-dimensional sections of Y, denoted Y2, using Buffon’s needle. We simply substitute the distance d between parallels in Equation [2.12] with S(Ω2)/L, where Ω2 is now the reference area of the two-dimensional section of Ω and L is the total length of parallel lines lying in Ω2. The substitution into [2.12] gives us [L (Y2)] = π S(Ω2)I/(2L). Here, L(Y2) is the perimeter length of Y2. The surface area S(Y) is then estimated according to [2.30], using the following formula for known V(Ω): [ S ( Y )] =

4[ L ( Y2 )] V (Ω) π [ S ( Ω 2 )]

[2.32]

or the surface density S(Y)/V(Ω) in the reference region Ω can be estimated as:

[ S ( Y )] 4[ L ( Y2 )] = [ V ( Ω )] π [ S ( Ω 2 )]

[2.33]

The surface area S(Ω2) can be measured by point counting methods using zero-dimensional probes in an appropriate test system.

2.4.3

Length and length density in three-dimensional space

Linear, fibre-like structures in biological tissues support a wide variety of physiological functions, including membrane stabilisation, vascular perfusion,


91

and cell-to-cell communication; thus stereological estimations of the parameters of fibre-like three-dimensional structures are of primary interest. Smith and Guttman (1953) demonstrated a stereological method to estimate the total length density of linear objects based on random intersections with a twodimensional sampling probe. The method presented by Mouton (2002) uses spherical probes that are inherently isotropic to measure the total length of thin nerve fibres in the dorsal hippocampus of the mouse brain. Hlavickova et al. (2001) studied bias in the estimator of length density for fibrous features in a three-dimensional space using projections of vertical slices. Cassidy (2001) estimated the total length of fibres in a fibrous mass simultaneously with the count of fibres, providing an estimation of average fibre length that was used to investigate fatigue breaks in wool carpets. Consider a fibre mass composed of negligibly thin fibres. We treat this fibrous system as a curve C of a total length L(C ) in the three-dimensional reference region Ω, having the volume V(Ω). To estimate the curve length, we will use a test tablet T of known area S(T ). This test tablet T will sit inside Ω uniform random and isotropic positions. For T this means, accordingly with remarks in Section 2.2.3: (i) (ii)

the chosen fixed point X of T is uniform random in the reference region Ω; and the orientation of the testing surface T is isotropic independent of the r position of T in Ω, which means in this case that the normal vector u T perpendicular to T is isotropic in three-dimensional space. This situation is shown in Fig. 2.38.

ut

Y

Ω

T

X ut T X

2.38 A reference region Ω of volume, V (Ω), contains a fibrous system Y of a total length, L (Y ). The length, L(Y ), is estimated from the number of intersections, I, between Y and a test piece of a plane, whose surface area is S(T ). Two uniform random and isotropic positions of T are indicated in the figure.

92


Imagine an element c of the curve C of length L(c) which is so short that it can be considered to be straight, appearing together with the uniform random and isotropic tablet T in Ω. The probability p that T will be hit by c is the same as in the subsection below dealing with the estimation of a threedimensional object and its surface area. For the current example, we exchange S(T ) for S(Y ), and L(c) for L(T ) in Equation [2.28]. In other words, the testing probe becomes the measured object and vice versa. Using these substitutions, we obtain the following formula for the hit probability: p=

S(T ) L(c) 2V (Ω)

[2.34]

From this relation we derive the formula for estimating the length L(C) of the curve C using the sum over all its elements ci. We suppose that there are n such elements constituting C, thus: I=

S(T ) n Σ L( c i ) 2 V ( Ω ) i =1

[2.35] n

where I is the number of hits represented by the product np and Σ L( c i ) is i =1

equal to the total curve length L(C ). We estimate L(C ) from a finite number of measurements, and hence we can write from Equation [2.35] the relation: [ L ( C )] =

2 IV ( Ω ) S(T )

[2.36]

This relation is desirable for known volumes V(Ω) of the reference region. The length density of the curve C in the reference region Ω is then estimated as: [ L ( C )] = 2I [ V ( Ω )] S ( T )

[2.37]

To estimate the curve length or the curve length density in a three-dimensional reference region Ω using testing systems, we first have to prepare uniform random and isotropic sections of a three-dimensional sample, as suggested in Fig. 2.39. We then use test systems with two-dimensional probes and the excluding line. For these measurements, I is the total number of crosssections of the curve in all two-dimensional probes, and S(T ) is estimated as count Q of the fixed points in each probe that hits the section of Ω under investigation. The area of the two-dimensional probe is denoted a. We can then write: [ L ( C )] 2 I = [ V ( Ω )] aQ

[2.38]


93

Ω (a)

a

4 mm

(b)

a

c

c

c

c c c

c

(c)

2.39 An isotropic and uniform random two-dimensional section of a reference region Ω is sketched (a) while the used test system with excluding lines is given by (b). The area of each two-dimensional probe of the test system is a. The total number of objects, counted by the test system is denoted as I while p is the total number of twodimensional probes with area a used to count particles, as shown in (c). A rough estimation of the length density from only one measurement in a particular case is [L(C)] /[V (Ω)] = 2I /S (T ) = 4 / (2S (T )).

94

2.4.4


Average curvature and average torsion of linear features in three-dimensional space

Understanding torsion and curvature values in three-dimensional space is important where the compression behaviour of fibrous materials is critical, for instance in some furniture and automotive applications. The method described here was first introduced by DeHoff (1975). Wool fibre curvatures were calculated by Munro and Carnaby (1999) from their internal geometry and shrinkage. We introduce the notions of curvature and torsion of fibres in three-dimensional space, and then describe the count method for estimating average values, without deriving the respective formulae. Curvature is usually considered in studies of the compression behaviour of a fibrous mass (Beil et al., 2002), while torsion is generally ignored. Changes in both these values during the compression of a very small fibrous mass were estimated in Lukas et al. (1993). Curvature and torsion are local characteristics of curves in three-dimensional space. The latter vanishes when the curve is fully embedded in a plane. Our definitions of the curve and its torsion are based on the osculation plane, the osculation circle, the tangent, the normal and the binormal. We start by investigating the vicinity of a point A on a curve in three-dimensional space as shown in Fig. 2.40. As well as point A, two points B and C are located on the same curve so that A is between them. These three points determine the circle going through all of them. The limit circle for B → A and C → A is the z b

t C

A B

r

n S x

y

2.40 A curve in three-dimensional space with three points A, B, C r that determines the osculation circle with centre at S. The tangent t r n lie in the osculation plane whilst the and the normal vector r binormal vector b is perpendicular to it.


95

osculation circle to the curve in the point A. This osculation circle determines r the osculation plane. The normal vector n to the curve is embedded in this plane, which is unitary, has its origin in the point A and points in the direction A to S, where S is the centre of the osculation circle. The unitary vector lying in the osculation plane that is perpendicular to r r these the normal n is the tangent r r vector t of the curve in point A. Both r orthogonal vectors n and t determine the next unitary vector b which is perpendicular to them. This vector is denoted as binormal of the curve in point A. By shifting point A along the rcurve by distance d λ , the orientation r r of all these three vectors n , t and b can be changed. The new vectors between the shifted point and the original one generally contain non-zero angles. We will denote the angle between tangents as dθ and the angle that contains binormals as dγ. The curvature k at point A is defined as k = dθ dλ

[2.39]

and is equal to 1/r where r is the radius of the osculation circle belonging to point A on the curve. The torsion τ at point A has the defining relation:

τ=

dγ dλ

[2.40]

From the definitions, it is clear that the curvature relates to orientation changes of the tangent while torsion is related to orientation changes of the binormal. The average values of curvature k and torsion τ along a curve of total length L are then expressed as average values of functions on the interval (0, L) in the following manner: k = 1 L

∫

L

0

k ( λ ) dλ

τ= 1 L

∫

L

τ ( λ ) dλ

[2.41]

0

Stereological estimations and measurements of these average values are based on the investigation of projections of thin sections of a fibrous mass as depicted in Fig. 2.41. The average value of torsion τ is estimated from the relation:

[τ ] =

π IA 2NL

[2.42]

where IA is the number of inflex points in a unit area of the projection. The inflex points are marked as squares in Fig. 2.41 and they represent those points on the curve where the centre of the osculation circle belonging to the planar projection of the curve jumps from one side of the curve to the other. For instance, the letter ‘S’ has one such point in its centre while ‘C’ and ‘O’ have no inflection points. The quantity NL is the average number of intersections between the testing line and the curve per unit length of the testing line, as

96


a = 3 cm

b = 5 cm

Thin section of a reference region Ω

Projection of a thin section

Test line

2.41 Projection of a thin section containing linear features, i.e. fibres. The tangential positions of a test line, moved along the fibres, are denoted by triangles. Inflection points are marked with small squares and hits of the fibres with the test line are denoted using empty circles. A rough estimation of the average torsion from one particular r measurement is [τ ] = π IA /(2 NL) = (π 9 /(ab))/(4 /a), while that of the average curvature is [k ] = πTA /(2 N L) = (π 7/(ab ))/(4 /a).

shown in Fig. 2.41. These testing lines have to be uniform random and isotropic. The average value of curvature k is estimated using the formula: [k ] =

π TA 2NL

[2.43]

The symbol TA denotes the average count of the tangential positions of a sweeping testing line per projection unit area. We refer to a tangential position as that where the sweeping line first touches the curve. The sweeping line is moved slowly across the projection, perpendicular to a previously chosen direction. The average number of counts is then calculated from all isotropic orientations and directions along which the sweeping line has moved. The count of tangential positions for each orientation of the sweeping line is then divided by the area of the sample projection (across which the line has swept), to obtain TA. Some tangential positions of the sweeping line are shown in Fig. 2.41.

2.4.5

Feature count and feature count density: dissectors

The introduction of dissectors into stereology represents a major turning point for this discipline. Dissectors, described by Gundersen (1988b), can, without exaggeration, be considered a methodological conception as significant as the contributions of Delesse, and Glagolev and Thompson.


97

Table 2.1 (Section 2.2.2) shows that only three-dimensional probes can measure the feature count in three-dimensional space. Unlike the methods described above, dissectors consist of three-dimensional probes and hence they cannot be expressed using two-dimensional test systems. The use of dissectors is demonstrated in Fig. 2.42. The dissector can be envisaged as a prism-shaped three-dimensional probe. The base of this prism A has surface area S(A), and it has height h. The volume of the dissector D is then V(D) = S(A)h. Critical parts of the dissector are the so-called excluding walls. In Fig. 2.42, parts of these excluding walls are shown using different shades. The excluding walls are infinite plains that involve three mutually perpendicular walls of the prism. Using the dissector consists of determining an object count NV belonging to the dissector’s volume V(D). The decision procedure for counting concrete features is similar to the feature count method in two-dimensional space, viz. that given in Section 2.3.3, where we used test systems with the excluding line. Here, we count only features that fulfil the following requirements: (i) The object has a non-empty intersection with the dissector’s prism. (ii) The object does not touch any of the three excluding walls. The unbiased estimation of object count volume density NV is then: [ NV ] =

I V ( D)

[2.44]

where I is the number of counted objects in the dissector D that respect the conditions (i) and (ii). In the example in Fig. 2.42, we count only particles 1, 2, 3 and 4 because the others have either an empty intersection with the dissector’s prism or

h = 2 cm

D

6

2

3 4

1

5

A S ( A ) = 12 cm

2.42 The dissector D on the figure of volume V(D) = hS (A) has height h and base A of area S(A). Parts of three excluding walls are shaded grey. Only particles No. 1, 2, 3 and 4 are counted in D as the rest hit the excluding walls. The volume density of the object count for this particular case may be estimated as [NV] = I /V (D) =4/V(D).

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they touch at least one of the excluding walls. Measurements have to be repeated using a number of uniform random dissectors. Counting long fibrous features is extremely arduous as we have to follow an entire fibre outside the dissector prism to make sure that the fibre does not hit one of the excluding walls. The best way to count fibres is to count their origins and divide the final count by two, because each fibre has two ends.

2.5

Sources of further information and advice

We have introduced a number of stereological methods useful for investigating fibrous materials, focusing mostly on explaining the basic stereological tools. We have not covered the statistical side of processing experimental data, which is broadly described in Russ (2000), Saxl (1989) and Elias and Hyde (1983). Recent information about stereology and its application regarding fibrous materials can be found in the Journal of Microscopy, the official journal of the International Society for Stereology, and in the Textile Research Journal and The Journal of The Textile Institute. We refer the reader to the following recent works for a greater understanding of stereology: Baddeley (2005), Coleman (1979), Ambartzumian (1982), Russ (1986), Hilliard (2003), Mouton (2002), Underwood (1981), Vedel Jensen (1998) and DeHoff (1968). Stereological methods could be also useful for identifying fabric defects in a dynamic inspection process. A dynamic inspection system for fast image acquisition with a linear scan digital camera is described by Kuo (2003). Changes in appearance due to mechanical abrasion may be evaluated with respect to changes in image texture properties, as has been shown by Berkalp et al. (2003).

2.6

References

Ambartzumian R V (1982), Combinatorial Integral Geometry: with Applications To Mathematical Stereology, New York, Chichester, Wiley. Baddeley A (2005), Stereology for Statisticians, Boca Raton, Chapman & Hall/CRC. Beil N B, William W and Roberts J (2002), ‘Modeling and computer simulation of the compressional behavior of fibre assemblies’, Textile Res. J., 72(5), 375–382. Berkalp O B, Pourdehimi B, Seyam A and Holmes R (2003), ‘Texture retention of the fabric-to-fabric abrasion’, Textile Res. J., 73(4), 316–321. Brenton J R and Hallos R S (1998), ‘Investigation into the composition, size, and morphology of dust generated during wool processing’, J. Text. Inst., 89(2), 337–353. Buffon G L L (1777), ‘Essai d’arithmetique morale’, Suppl. A l’Histoire Naturale, Paris, 4. Cassidy B D (2001), ‘Type and location of fatigue breaks in wool carpets, Part II: Quantitative examination’, J. Text. Inst., 92(1), 88–102. Chalkley H W (1943), ‘Methods for quantitative morphological analysis of tissue’ J. Nat. Cancer Inst., 4, 47.


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Coleman R (1979), An Introduction to Mathematical Stereology, Aarhus, University of Aarhus. DeHoff R T (1975), ‘Quantitative microscopy of linear features in three dimensions’, 4th Int. Congress of Stereology, Goithersburg, p. 29. DeHoff R T and Rhines F N (1968), Quantitative Microscopy, New York, McGraw Hill. Delesse M A (1847), ‘Procede mecanique pour determiner la composition des roches’, C.R. Acad. Sci., Paris, 25, 544. Dent R W (2001), ‘Inter-fibre distances in paper and non-wovens’, J. Text. Inst., 92(1), 63–74. Eberhardt C N and Clarke A R (2002), ‘Automated reconstruction of curvilinear fibres from 3D datasets acquired by X-ray microphotography’, J. Microsc., 206(1), 41–53. Elias H (1963), ‘Address of the President’, 1st Int. Congress for Stereology, Wien, Congressprint, p. 2. Elias H and Hyde D M (1983), A Guide to Practical Stereology, New York, Krager Continuing Education Series, Switzerland. Farer R et al. (2002), ‘Meltblown structures formed by robotic and meltblowing integrated systems: impact of process parameters on fibre orientation and diameter distribution’, Textile. Res. J., 72(12), 1033–1040. Glagolev A A (1933), ‘On the geometrical methods of quantitative mineralogic analysis of rocks’, Trans. Ins. Econ. Min., Moscow, 59, 1. Gundersen H J G et al. (1988a), ‘Some new single and efficient stereological methods and their use in pathological research and diagnostics’, APMIS, 96, 379–394. Gundersen H J G (1988b), ‘The new stereological tools: dissector, fractionator, nucleator and point sampled intercepts, and their use in pathological research and diagnostics’, APMIS, 96, 857–881. Hadwiger H (1967), Vorlesungen uber Inhalt, Oberflache und Isoperimetrie, Berlin, Heidelberg, New York, Springer Verlag. Hilliard J E (2003), Stereology and Stochastic Geometry, Boston, Kluwer Academic Publishers. Hlavickova M, Gokhale A M and Benes V (2001), ‘Bias of a length density estimator based on vertical projections’, J. Microsc., 204(3), 226–231. Hough R V (1962), Method and means for recognizing complex patterns, U.S. Patent 306954. Huang Y and Xu B (2002), ‘Image analysis for cotton fibres; Part I: longitudinal measurements’, Textile Res. J., 72(8), 713–720. Jensen E B and Gundersen H J G (1982), ‘Sterological ratio estimation based on counts from integral test systems’, J. Microscopy, 125, 51–66. Jensen K L and Carstensen J M (2002), ‘Fuzz and piles evaluated on knitted textiles by image analysis, Textile Res. J., 72(1), 34–38. Jeon B.S. (2003), ‘Automatic recognition of woven fabric patterns by a neural network’, Textile Res. J., 73(7), 645–650. Jirsak O and Wadsworth L C (1999), Non-woven Textiles, Durham, North Carolina, Carolina Academic Press. Kang T J, Cho D H and Kim S M (2002), ‘Geometric modeling of cyber replica system for fabric surface property grading’, Textile Res. J., 72(1), 44–50. Karkkainen S, Jensen E B V and Jeulin D (2002), ‘On the orientational analysis of planar fibre system’, J. Microsc., 207(1), 69–77. Kim S H, Lee J H and Lim D Y (2003), ‘Dependence of sorption properties of fibrous assemblies on their fabrication and material characteristics’ Textile Res. J., 73(5), 455–460.

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Koehl L, Zeng X, Ghenaim A and Vasseur C (1998), ‘Extracting geometrical features from a continuous-filament yarn by image-processing techniques’, J. Text. Inst., 89(1), 106–116. Kuo C H J, Lee C J and Tsai C C (2003), ‘Using a neural network to identify fabric defects in dynamic cloth inspection’, Textile Res. J., 73(3), 238–244. Lukas D (1991), ‘Hodnocení filtračních vlastností vlákenných materiálů pomocí stereologických metod’, 1st Conf. Filtračné a Absorbčné Materiály, Starý Smokovec, 25–33. Lukas D, Hanus J and Plocarova M (1993), ‘Quantitative microscopy of non-woven material STRUTO’, 6th European Conference of Stereology, Prague, p. iv–13. Lukas D, Jirsak O and Kilianova M (1993a), ‘Stanoveni Maximální Velikosti Pórů Textilních Filtračních Materiálů Omocí Přístroje Makropulos 5’, Textil, 7, 123–125. Lukas D (1997), ‘Konfokální mikroskop TSCM’, 3rd Conf. STRUTEX’97, Liberec, Nakladatelství Technická Univerzita v Liberci, 18–19. Lukas D (1999), Stereologie Textilnich Materialu, Liberec, Technicka Univerzita v Liberci. Mandelbrot B B (1997) Fractals, Form, Chance and Dimensions, San Francisco, W.H. Freeman and Co. Mao N and Russel S J (2000), ‘Directional probability in homogeneous non-woven structures; Part I: The relationship between directional permeability and fibre orientation’, Textile Res. J., 91(2), 235–243. Miller A and Schwartz P (2001), ‘Forced flow percolation for modeling of liquid penetration of barrier materials’, J. Text. Inst., 92(1), 53–62. Mohammadi M and Banks-Lee P (2002), ‘Air permeability of multilayered non-woven fabrics: comparison of experimental and theoretical results’, Textile Res. J., 72(7), 613–617. Mouton P R (2002), Principles and Practices of Unbiased Stereology :An Introduction for Bioscientists, Baltimore, Johns Hopkins University Press. Mouton P R, Gokhale A M, Ward N L and West M J (2002a), ‘Stereological length estimation using spherical probes’, J. Microsc., 206(1), 54–64. Munro W A and Carnaby G A (1999), ‘Wool-fibre crimp; Part I: The effects of microfibrillar geometry’, J. Text. Inst., 90(2), 123–136. Naas J and Schmidt H L (1962), Mathematics Worterbuch, Band I A-K, Berlin, Akademie Verlag Gmbh. Neckar B and Sayed I (2003), ‘Theoretical approach for determining pore characteristics between Fibres’, Textile Res. J., 73(7), 611–619. Pourdeyhimi B and Kim H S (2002), ‘Measuring fibre orientation in non-wovens: The Hough transform’, Textile Res. J., 72(9), 803–809. Pourdeyhimi B and Kohel L (2002a), ‘Area based strategy for determining web uniformity’, Textile Res. J., 72(12), 1065–1072. Rataj J and Saxl I (1988), ‘Analysis of planar anisotropy by means of Steiner compact: a simple graphical method’, Acta Stereologica, 7(2), 107–112. Russ J C (1986), Practical Stereology, New York, Plenum Press. Russ J C and Dehoff R T (2000), Practical Stereology, New York, Kluwer Academic/ Plenum Publishers. Saxl I (1989), Stereology of Objects with Internal Structure, Amsterdam, New York, Elsevier. Sera J (1982), Image Analysis and Mathematical Morphology, London, Academic Press. Smith C S and Guttman L (1953), ‘Measurement of internal boundaries in three-dimensional structures by random sectioning’, Trans AIME, 197, 81–92.


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Sodomka L (1981), ‘Studium textury pavucin, rouna a netkane textilie’, Textil, 36, 129. Stoyan D, Kendall W S and Mecke K R (1995), Stochastic Geometry and its Applications, Chichester, J. Wiley. Summer scales J, Peare N R L, Russell P and Guld F J (2001). ‘Vornonoi cells, fractal dimensions and fibre composites, Journal of Microscopy, 201(2), 153–162. Taylor P M and Pollet D M (2002), ‘Static load lateral compression of fabrics’, Textile Res. J., 72(11), 983–990. Thompson E (1930), ‘Quantitative microscopic analysis’, J. Geol., 38, 193. Underwood E E (1981), Quantitative Stereology, Addison-Wesley Pub. Co. Van Wyck C M (1964), ‘A study of the compressibility of wool with special reference to South African Merino wool’, Ondersteoort J. Vet. Sci. Anim. Ind., 21(1), 99–226. Vedel Jensen E B (1998), Local Stereology, Singapore, World Scientific. Wiebel E R (1979), Stereological Methods; Vol. 1 Practical Methods for Biological Morphometry, New York, Academic Press. Wortham F J, Phan K H and Augustin P (2003), ‘Quantitative fibre mixture analysis by scanning electron microscopy, Textile Res. J., 73(8), 727–732. Yil, Zhu Q and Yeung K W (2002), ‘Influence of thickness and porosity on coupled heat and liquid moisture transfer in porous textiles’, Textile Res. J., 72(5), 435–446. Zhou S, Chu C and Yan H (2003), ‘Backscattering of light in determining fibre orientation distribution and area density of non-woven fabrics’, Textile Res. J., 73(2), 131–138. Ziedman M and Sawhney P S (2002), ‘Influence of fibre length distribution on strength efficiency of fibres in yarn’, Textile Res. J., 72(3), 216–220.

3 Essentials of psychrometry and capillary hydrostatics N. P A N and Z. S U N, University of California, USA

3.1

Introduction

From the general engineering approach, water flow in solid porous media should be treated as a problem of hydromechanics. Thus the fundamental laws, such as the continuity or conservation equations, the rheological conditions and the Navier–Stokes equations supposedly govern the phenomena. However, several unique characteristics of fluids transport in fibrous materials render these tools nearly irrelevant or powerless. For instance, except during the wet processing period where higher speed flow may be encountered, low speed, low viscosity and small influx of the fluids make such issues as the interactions between fluids and solid media much more prevalent over the fluids flow problem itself; the pore size, often so tiny as to be on the same scale level as the free molecular path length in the fluid, highlights the need for consideration of the so-called molecular flow, where problems such as absorption and capillary action dominate. In other words, a more microscopic view and associated approaches become indispensable. Further, if our focus is mainly on fluid transport in porous media during static or quasi-static conditions, it raises another question related to the phase change. The solid fibrous media may cause some of the fluids (e.g. moist air) to condense back to liquid phase, which in turn brings out other issues such as capillary condensation, moisture absorption, associated change of the properties and behaviors of the fibrous materials, and generation of sorption heat. The above issues and discussions in fact dictate the content and focus of this chapter.

102


3.2

Essentials of psychrometry (Skaar, 1988; Siau, 1995; Morton and Hearle, 1997)

3.2.1

Atmosphere and partial pressures

103

Our unique ambient environment conditions provide a proper combination of such factors as air, moisture, temperature, and pressure indispensably suitable for life on earth. The whole system is a dynamic one in which every physical entity constantly interacts with others, yet maintains the equilibrium most of the time for our survival and prosperity. Moisture is one of the three states in which water manifests itself and its existence and behavior in the atmosphere is one of the fundamental issues in our discussion. It is common knowledge that the dry air surrounding us comprises a mixture of gases, the approximate percentages of which are shown in Table 3.1; these are known as the dry gases of the atmosphere. Based on this composition, the molecular mass of dry air is calculated as 28.9645. For a given atmospheric conditions, the dry gases will inevitably absorb water moisture and become a humid mixture termed the moist air. Psychrometrics deals with the thermodynamic properties of moist air and uses these properties to analyze conditions and processes involving moist air. In dealing with the connection of behaviors between the system and its constituents, our problem here is rare where the Rule of Mixtures is actually valid – that is, the water vapor is completely independent of the dry atmospheric gases in that its behavior is not affected by their presence or absence. For instance, in moist air, the dry gases and the water vapor behave according to Dalton’s law of Partial Pressures, i.e. they act independently of one another and the pressure each exerts combines to produce an overall ‘atmospheric pressure’ patm. patm = pg + pv where pg and pv are termed the partial pressures of the dry gases and of the water vapor, respectively. From the ideal gas laws, the partial pressures are

Table 3.1 The approximate percentage (composition) of dry air Nitrogen Oxygen Argon Carbon dioxide Neon Helium Methane Sulfur dioxide Other

78.0840% 20.9476% 0.9340% 0.0314% 0.001818% 0.000524% 0.0002% 0 to 0.0001% 0.0002%

104


related to other thermodynamic variables such as the volume V and temperature T of the constituent i as piVi = niRTi = NikTi

[3.1]

where the subscript i = atm, g or v, respectively. • • • •

n = number of moles R = universal gas constant N = number of molecules k = Boltzmann constant = 1.38066 × 10–23 J/K = R/NA, NA – Avogadro’s number = 6.0221 × 1023/mol

Since the mole fraction (xi) of a given component in a mixture is equal to the number of moles (ni) of that component divided by the total number of moles (n) of all components in the mixture, then the mole fractions of dry air and water vapor are, respectively: xg =

ng pg pg = = ng + nv pg + pv patm

[3.2]

xv =

pv p nv = = v ng + nv pg + pv patm

[3.3]

and

By definition, xa + xv = 1. However, upon the changing of environment conditions, the mass of water vapor will change due to condensation or evaporation (also known as dehumidification and humidification respectively), but the mass of dry air will remain constant. It is therefore convenient to relate all properties of the mixture to the mass of the dry gases rather than to the combined mass of dry air and water vapor. The evaporation of water is a temperature-activated process and, as such, the saturated vapor pressure psv (the maximum of pv) may be calculated with relatively good precision using an Arrhenius-type (Skaar, 1988; Siau, 1995) equation:

(

p sv = A exp – E RT

)

[3.4]

where psv = saturated water vapor pressure, A = constant; E = escape energy. The equation in fact offers the relationship between vapor saturation and the ambient temperature, and increasing temperature will lead to a greater saturated vapor pressure psv. For instance, with increasing temperature there is an increase in molecular activity and thus more water molecules can escape from the liquid water and be absorbed into the gas. After a while, however, even at this increased


105

temperature, the air will become fully saturated with water vapor so that no more water can evaporate unless we again increase the temperature. The pressure produced by the water vapor in this fully saturated condition is known as the saturated vapor pressure (psv) and, since at a given temperature the air cannot absorb more water than its saturated condition, the saturated vapor pressure is the maximum pressure of water vapor that can occur at any given temperature.

3.2.2

Percentage saturation and relative humidity

To describe the water vapor concentration in the atmosphere, the most natural way is to determine its volume or weight in a given volume of the air. However, the obvious difficulties in actually handling the vapor volume or weight prompt other more feasible measures for the purpose. The first one is the Percentage Saturation PS PS (%) =

hv × 100 hsv

[3.5]

where hv is the actual mass of vapor in a unit volume of the air and hsv is the saturated vapor mass. So the PS value indicates the degree of saturation of the atmosphere at a given temperature. Another more frequently used measure is the relative humidity (RH), defined based on the ratio of the partial vapor pressures RH (%) =

pv × 100 p sv

[3.6]

For most practical purposes, the ratio of the partial vapor pressures is very close to the ratio of the humidities, i.e. p hv ≈ v p sv hsv

[3.7]

RH ≈ PS

[3.8]

or

Although the relative humidity and the percentage saturation have been treated as interchangeable in many applications, it is often useful to remember their differences.

3.2.3

Dew-point temperature (Tdp )

Since the molecular kinetic energy is greater at higher temperature, more molecules can escape the surface and the saturated vapor pressure is correspondingly higher. Besides the two characteristic temperatures which

106


Moisture absorbed

P = constant

RH = 100%

RH = 50%

RH = 25% B A

Vapor

Liquid

Tdp

Temperature

3.1 Dew temperature and relative humidity.

affect the state of water, namely, the ice point and boiling point, the dew point temperature is yet another one. This is the temperature at which the saturation state (RH = 100%) of the mixture of air and water vapor during a cooling process, at constant pressure and without any contact with the liquid phase, is reached. If the temperature drops lower than this point, water vapor will begin to condense back into liquid water as indicated by the arrow A in Fig. 3.1.

3.3

Moisture in a medium and the moisture sorption isotherm

3.3.1

Moisture regain and moisture content

Similar to the case of vapor in the atmosphere, we need to find a way to specify the amount of total moisture in a material. If we can determine the weight D of dry material and weight W of moisture in the material, there are two definitions commonly used in the textile and fiber industries (Morton and Hearle, 1997). Moisture regain (R) R (%) = W × 100 D Moisture content (M) M (%) =

W × 100 ( W + D)

It is obvious that R > M and relation between R and M:

[3.9]

[3.10]


R (%) =

107

M (%)  1 – M (%)   100 

and M (%) =

R (%)  1 + R (%)   100 

[3.11]

Note that in literature, as well as in our discussion hereafter, the terms of both moisture regain and moisture content are often treated as interchangeable. Equilibrium moisture content (EMC) is the moisture content at which the water in a medium is in balance with the water in the surrounding atmosphere. Although the temperature and relative humidity of the surrounding air are the principal factors controlling EMC, it is also affected by species, specific gravity, extractives content, mechanical stress, and previous moisture history. The curve relating the equilibrium moisture content of a material with the relative humidity at constant temperature is called the sorption isotherm. A collection of moisture sorption isotherms of several fibers is provided in Fig. 3.2 (Morton and Hearle, 1997). At a given set of standard atmospheric 30

25

Regain (%)

20

15

oo W

l

Vi

10

s

s co

e

sil

k

n tto e at et Ac on Nyl

Co

5

Orlo 0

20

x) ppro n (a Terylene

40 60 80 Relative humidity (%)

100

3.2 Sorption isothermals for various fibers. From Morton, W. E. and J. W. S. Hearle (1997). Physical Properties of Textile Fibers. UK, The Textile Institute.

108


conditions, the EMC for each fiber type is a constant, and hence is termed as ‘official’ or ‘commercial’ regain for trading purpose (Morton and Hearle, 1997). Table 3.2 shows the data including the ‘commercial’ regains for some common textile fibers (Morton and Hearle, 1997).

3.3.2

Moisture sorption isotherm

The relationship between the moisture content in a material and the ambient relative humidity at a constant temperature yields a moisture sorption isotherm when expressed graphically. Determination of a moisture sorption isotherm is the general approach for characterizing the interactions between water and solids. This isotherm curve can be obtained experimentally in one of two ways (see Fig. 3.3). (i) An adsorption isotherm is obtained by placing a completely dry material into various atmospheres of increasing relative humidity and measuring the weight gain due to water uptake; Table 3.2 Moisture sorption data for major fibers Moisture absorption of fibres Material

Recommended allowance or commercial regain or conventional allowance* (%)

Absorption regain (%) (65% R.H., 20°C)†

Difference in desorption and absorption regains (65% R.H., 20°C)†

Cotton Mercerized cotton Hemp Flax Jute Viscose rayon Secondary acetate Triacetate Silk Wool Casein Nylon 6.6, nylon 6 Polyester fibre Acrylic fibre Modacrylic fibre Poly(vinyl chloride) Poly(vinyl alcohol) Glass, polyethylene

8.5 – 12 12 13.75 13 9 – 11 14–19 – 53/4 or 61/4 1.5 or 3 – – – – –

7–8 Up to 12 8 7 12 12–14 6, 6.9 4.5 10 14, 16–18 4.1 4.1 0.4 1–2 0.5–1 0 4.5–5.0 0

0.9 1.5 – – 1.5 1.8 2.6 – 1.2 2.0 1.0 0.25 – – – – – –

Adapted from Morton and Hearle (1997) * As given in B.S. 4784:1973; other standardizing organizations may quote different values. † The earlier measurements were at 70 °F (21.1°C).

Essentials of psychrometry and capillary hydrostatics Moisture regain R (%)

109

Moisture regain R (%)

Desorption Desorption

Hysteresis Time

Absorption 0

Absorption

RH (%) T = constant

3.3 Two ways of depicting the sorption isotherms and hysteresis. From Morton, W. E. and J. W. S. Hearle (1997). Physical Properties of Textile Fibers. UK, The Textile Institute.

(ii) A desorption isotherm is found by placing an initially wet material under the same relative humidities, and measuring the loss in weight. The adsorption and desorption processes are both referred to as the sorption behavior of a material, and are not fully reversible; a distinction can be made between the adsorption and desorption isotherms by determining whether the moisture levels within the material are increasing, indicating wetting, or whether the moisture is gradually lowering to reach equilibrium with its surroundings, implying that the product is being dried. On the basis of the van der Waals adsorption of gases on various solid substrates, Brunauer et al. (1938) classified adsorption isotherms into five general types (see Fig. 3.4). Type I is termed the Langmuir, and Type II the sigmoid-shaped adsorption isotherm; however, no special names have been attached to the other three types. Types II and III are closely related to Types V and IV, respectively. For the same adsorption mechanisms, if they occurred in ordinary solids, Types II and III depict two typical isotherms. If, however, the solid is porous so that it has an internal surface, then the thickness of the adsorbed layer on the walls of the pores is necessarily limited by the width of the pores. The form of the isotherm is altered correspondingly; Type II turns into Type V and Type III corresponds to Type IV (Gregg and Sing, 1967). Moisture sorption isotherms of most porous media are nonlinear, generally sigmoidal in shape, and have been classified as Type II isotherms. Caurie (1970), Rowland (in Brown, 1980), Rao and Rizvi (1995) and Chinachoti and Steinberg (1984) explained the mechanisms and material types (mainly foods) leading to different shapes of the adsorption isotherms. Morton and Hearle have collected most comprehensive experimental results regarding the moisture sorption behaviors (e.g. Fig. 3.3) of fibrous materials including moisture sorption isotherms for various fibers. Al-Muhtaseb et al. published

110

Thermal and moisture transport in fibrous materials Moisture regain R (%)

IV V

I

II

III

RH (%)

3.4 Different moisture sorption behaviors. Reprinted from Brunauer, S., P. H. Emmett, et al. (1938). ‘Adsorption of gases in multimolecular layers.’ Journal of the American Chemical Society 60: 309. With permission from American Chemical Society.

a comprehensive review on moisture sorption isotherm characteristics (AlMuhtaseb et al. 2002). For interpretation purposes, a generalized moisture sorption isotherm for a hypothetical material system may be divided into three main regions, as detailed in Fig. 3.5 (Al-Muhtaseb, et al. 2002). Region A represents strongly bound water with an enthalpy of vaporization considerably higher than that of pure water. A typical case is sorption of water onto highly hydrophilic biopolymers such as proteins and polysaccharides. The moisture content theoretically represents the adsorption of the first layer of water molecules. Usually, water molecules in this region are un-freezable and are not available for chemical reactions or as plasticizers. Region B represents water molecules that are less firmly bound, initially as multi-layers above the monolayer. In this region, water is held in the solid matrix by capillary condensation. This water is available as a solvent for low-molecular weight solutes and for some biochemical reactions. The quantity of water present in the material that does not freeze at the normal freezing point usually is within this region. In region C or above, excess water is present in macro-capillaries or as part of the liquid phase in high moisture materials. It exhibits nearly all the properties of bulk water, and thus is capable of acting as a solvent. The variation in sorption properties of materials reported in the literature is caused

Moisture content


111

Desorption

A Adsorption

B

20

C


100

3.5 Three main regions in a generalized moisture sorption isotherm. Reproduced with permission from Al-Muhtaseb, A. H., McMinn, W. A. M. and Magee, T.R.A. (2002). ‘Moisture sorption isotherm characteristics of food products: a review.’ Trans. IChemE, Part C, 80: 118–128.

by property variations, pretreatment, and differences in experimental techniques adopted (gravimetric, manometric or hygrometric) (Saravacos, et al. 1986). The mechanisms of moisture sorption, especially in hydrophilic fiber materials, are further complicated by a continuous change of the structure of the fibers owing to swelling (Preston and Nimkar, 1949). High internal temperature change caused by heat of sorption with large amounts of moisture also introduces more difficulties to the kinetics of the moisture sorption (Urquhart and Williams, 1924).

3.3.3

Water activity and capillary condensation

In describing the state of any medium, the free energy (∆G) of the system is one of the most important parameters along with temperature T, volume V, concentration c and pressure p. On a molar basis, the free energy becomes the chemical potential F (cal/mole), and is defined as F = Fo + RT ln a

[3.12]

where R = gas constant and T = absolute temperature in °K. The dimensionless variable a is termed the thermodynamic activity of the medium, which as reflected clearly in the equation, determines the system energy at a given

112


temperature. A substance with a greater a value is thermodynamically more active. The water activity aw is a measure of the energy status of the water in a specific system such as in the air or in a fiber mass. Different materials systems will generate different aw values. As a potential energy measurement aw is a driving force for water movement from regions of high water activity to regions of low water activity. In other words, the water activity is the cause for water (liquid or moisture) transport in porous media (Berlin, 1981; Luck, 1981; Van den Berg and Bruin, 1981). There are several factors that control water activity in a system, and they have been summarized mathematically in the well known Kelvin equation (McMinn and Magee, 1999) as

aw =

pv =e p sv

–2 γ M rρRT

[3.13]

where M = molecular weight of water, γ = surface tension; ρ = density of water, T the absolute temperature and r the capillary radius. Although there have been questions on the validity of the Kelvin equation, it has been proven (Powles, 1985) that the equation is valid to a few per cent even for temperatures approaching the critical temperature and for microscopic drops insofar as homogeneous thermodynamics is valid. One word of caution is that according to Equation [3.13], aw → 0 when r → 0, i.e. an adequately low aw would require a capillary radius too small to be practical; a lower boundary should thus be observed in specific cases. On open surfaces, moisture condensation sets in when saturation vapor pressure has been reached. However, it follows from the Kelvin equation that the vapor saturation pressure reduces inside capillaries of narrower sizes. As a result, for the same vapor pressure, the saturation point becomes lower in smaller pores so that water condenses inside the pores. This means that the tightest pores will be filled first with condensed liquid water. This ‘prematured’ condensation in pores is termed the capillary condensation. This is an extremely important phenomenon widely observable in our daily life. The process of such condensation continues until vapor pressure equilibrium is reached, i.e. up to the point at which the vapor pressure of the water in the surrounding gaseous phase is equal to the vapor pressure inside the pores. Further, from Equation [3.13], several major factors which can lower the water activity aw value are identified. Temperature is an obvious one and there is a special section later in this chapter on its influence. Next, the nature of the material system the water is in; including the impurities or dissolved species (e.g. salt or dyestuff) in liquid water which interact in three dimensions with water through dipole–dipole, ionic, and hydrogen bonds, leading to the associated colligative effects which will alter


113

such properties as the boiling or freezing point and vapor pressure. Raoult’s Law (Labuza, 1984) sometimes is used to account for these factors. In a solution of Nw moles water as the solvent and Ns moles of dissolved solute, aw = α

Nw Nw + Ns

[3.14]

where α is termed the activity coefficient and α = 1 for an ideal solute. The presence of the solute Ns reduces the water activity aw and thus leads to the colligative effects. Also the surface interactions in which water interacts directly with chemical groups on un-dissolved solid ingredients (e.g. fibers and proteins) through dipole–dipole forces, ionic bonds (H3O+ or OH–), van der Waals forces and hydrogen bonds, as reflected by the change of the surface tension (Taunton, Toprakcioglu et al., 1990; Duran, Ontiveros et al., 1998). Finally, the structural influences, which are reflected through the capillary size r where water activity is less than that of pure water because of changes in the hydrogen bonding between water molecules. It is a combination of all these factors in a material that reduces the energy of the water and thus reduces the water activity as compared to pure water (Al-Fossail and Handy, 1990; Hirasaki, 1996; Reeves and Celia, 1996; Tas, Haneveld et al., 2004).

3.3.4

Water activity and sorption types

As described in the Kelvin equation, moisture trapped in the small pores exerts a vapor pressure less than that of pure water at the given temperature. In other words, water has a lower activity once trapped inside a material system. The solids in which this effect can be observed exhibit so-called hygroscopic properties. The phenomenon of hygroscopicity can be interpreted by a sorption model such as the Brunauer, Emmett and Teller (BET) Equation (Brunauer, Emmett et al., 1938) which proposes a multi-molecular sorption process as shown in Fig. 3.6, based on the different levels of the water activity aw. • aw ≤ 0.2, formation of a monomolecular layer of water molecules on the pore walls • 0.2 < aw < 0.6, formation of a multi-molecular layers of water molecules building up successively on the monolayer; • aw ≥ 0.6, the process of capillary condensation takes place as described by the Kelvin equation.

3.3.5

Pore size effects

Just as indicated in the Kelvin equation, the wetting mechanisms change with the pore sizes r.

114


100

Unbound moisture

Water activity aw (%)

Bound moisture

aweq

Free moisture Equilibrium moisture

0

Req

Rmax Moisture regain R (%)

3.6 Various kinds of moisture in a material. Reprinted from Brunauer, S, P. H. Emmett et al. (1938). ‘Adsorption of gases in multimolecular layers.’ Journal of the American Chemical Society 60: 309. With permission from American Chemical Society.

• Pore size < 10–7m – capillary-porous bodies within which the moisture is maintained mainly through surface tension • Pore size >10–7m – porous bodies within which the gravitational forces have to be considered, apart from the capillary forces On the other hand, taking into consideration the mechanism of liquid and gaseous phase motion, the assumed value of 10–7m is of the same order as the mean free path of water vapor under atmospheric pressure. Luikov (Luikov, 1968; Strumillo and Kudra, 1986) divided capillaries into micro-capillaries with radii less than 10–7m. Therefore, in the micro-capillaries in which the free path is larger than the capillary radius, gas is transported by means of ordinary diffusion, i.e. chaotic particle motion. In micro-capillaries, the capillary tubes filled up with liquid due to capillary condensation on capillary walls, with a mono-molecular liquid layer of about 10–7m thick formed. In the case of polymer adsorption, the layers formed on the opposite capillary walls can be joined and the whole capillary volume is filled with a liquid phase. Macro-capillaries with radii bigger than 10–7m are, on the other hand, filled up with liquid phase only when they are in a direct contact with liquid – no more capillary condensation. Such a division into macro- and microcapillaries has been confirmed by Kavkazov (Kavkazov, 1952; Luikov, 1968; Strumillo and Kudra, 1986) who observed that capillary-porous bodies of


115

r > 10–7 m did not absorb moisture from humid air, but on the contrary released the moisture into atmosphere. It is worth mentioning that when vapor and temperature equilibrium are obtained, the water activity in the atmosphere is now equal to the relative humidity of surrounding air, i.e.

aw =

RH (%) pv h ≈ v = p sv 100 hsv

[3.15]

This equation connects a material property, the water activity, with the ambient condition. The more tightly water is bound with the material, the lower its activity aw becomes. Equation [3.15] has wide implications and applications. For instance, the moisture sorption isotherm can be expressed in two ways; moisture regain ~ relative humidity presents how the ambient condition affects the moisture in the material, as in many fiber related cases (Morton and Hearle, 1997); whereas moisture regain ~ water activity reveals the interconnection between the two material properties.

3.4

Wettability of different material types

Leger and Joanny (1992), Zisman (1964) and de Gennes (1985) have each written an extensive review on the liquid wetting subject. The following is just a brief summary of what been dealt with by them. Based on the cohesive energy or surface tension, there are two types of solids (de Gennes, 1998). • Hard solids – covalent, ionic or metallic bonded, high-energy surfaces with surface tension γSO ~ 500 to 5000 erg/cm2; • Weak molecular crystals – van der Waals (VW) forces, or in some special cases, hydrogen bonds bonded, low-energy surfaces, with γSO ~ 50 erg/ cm2.

3.4.1

Typical behaviors of high-energy surfaces

Most molecular liquids achieve complete wetting with high-energy surfaces. Assuming that chemical bonds control the value of γSO, while physical ones control the liquid/solid interfacial energies, when there is no contact between the solid and liquid, the total energy of the system is γSO + γ where γ is the surface tension of the liquid. However, once the solid and liquid are in contact, the interfacial energy becomes

γSL = γSO + γ – VSL

[3.16]

Here the term –VSL describes the attractive van der Waals interactions at the S/L interface. Similarly, when bringing two portions of the same liquid together, the system energy changes from 2γ to

116


γLL = 2γ – VLL = 0

[3.17]

where –VLL represents the L/L interfacial attractions. Thus the spreading parameter S, which measures the energy difference between the bare solid and the solid covered with the liquid, is defined as (de Gennes, 1985) S = γSO – (γSL + γ) = –2γ + VSL = VSL – VLL

[3.18]

and the complete wetting (S > 0) occurs when VSL > VLL

[3.19]

That is, the high energy surfaces are wetted by molecular liquids, not because γSO is high, but rather because the interfacial attraction between the solid and liquid VSL is higher than the attraction between the liquid and liquid VLL.

3.4.2

Low-energy surfaces and critical surface tensions

For solids of low-energy surface, wetting is not complete. A useful way of representing these results is to plot the contact angle cos θ versus the liquid surface tension γ (See Fig. 3.7 for example). Although in many cases we never reach complete wetting so that cos θ = 1, we can extrapolate the plot down to a value γ = γc when cos θ = 1; γ > γc indicates a partial wetting and γ < γc a total wetting (de Gennes et al., 2003). In general, we expect γc to be dependent on both the solid and liquid.

cosθ 1

CH3

0.9

(CH2)n Si Cl ClH Cl Si

0.8

20 γc

22

+++++ Glass

24

26

γ (dyn/cm)

3.7 The contact angle versus the liquid surface tension; From Brochard-Wyart, F., ‘Droplets: Capillarity and Wetting’, in Soft Matter Physics, M. Daoud, C.E. Williams, Editors. 1999, Springer: New York. p. 1–45. With kind permission of Springer Science and Business Media.


117

However, when dealing with simple molecular liquids (where VW forces are dominant), Zisman (1964) observed that γc is essentially independent of the nature of the liquid, and is a characteristic of the solid alone. Typical values of γc are listed in Table 3.3. So if we want to find a molecular liquid that wets completely a given low energy surface, we must choose a liquid with surface tension γ ≤ γc. This critical surface tension γc is clearly an essential parameter for many practical applications. In general, the chemical constitutions of both the solid S and liquid L affect the wetting behavior of the S/L system (Zisman, 1964), and some concluding remarks are listed below. (i) Wettability is proportional to the polarity of a solid; (ii) The systems of high γc (Nylon, PVC) are those wettable by organic liquids. (iii) Among systems controlled by VW interactions, we note that CF2 groups are less wettable (less polar) than CH2 groups. In practice, many protective coatings (antistain, waterproofing etc.) are based on fluorinated systems. Usually, glassy polymers, when exposed to a range of relative humidities, show differing absorption behavior at low and high relative humidities (i.e. low or high activities of the penetrant species) (Karad and Jones, 2005). At low activities, sorption of gases and vapors into glassy polymers is successfully described by a dual mode sorption theory, which assumes a combination of Langmuir-type trapping within pre-existing holes and Henry’s Law type dissolution of penetrant into the glassy matrix. At high activities, strong positive deviations from Henry’s Law are observed, which indicated that the sorbed molecules diffuse through the macromolecular array according to a different mechanism (Jacobs and Jones, 1990). In fact, the high cohesive energy of water leads to a phenomenon of cluster forming in nonpolar polymers. The water molecule is relatively small and is strongly associated through hydrogen bond formation. This combination of features distinguishes it from the majority of organic penetrants. As a result, strong localized interactions may develop between the water molecules and suitable polar groups in the polymer. On the other hand, in relatively nonpolar materials, clustering or association of the sorbed water is encouraged. Rodriquez et al. (2003) confirmed that polymers having strong interactions Table 3.3 The critical surface tension γc for some polymers

γc (mN/m)

Nylon

PVC

PE

PVF2

PTFE

46

39

31

28

18

Reprinted from Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. pp. 21, de Gennes, P. G., Brochard-Wyart, F. and Quere, D. Copyright (2003), with kind permission of Springer Science and Business Media

118


with water have negligible degrees of water clustering, while the more hydrophobic polymers exhibit a higher degree of clustering. The quantitative description of penetrant diffusion into micro-heterogeneous media has evolved over the last three decades and has become known as the dual mode sorption theory. Based on Meares’ (1954) concept of microvoids in the glassy state, Barrer et al. (1958) suggested two concurrent mechanisms of sorption – ordinary dissolution and ‘hole-filling’. Brown (1980) concluded through an extensive study that, at low partial pressures or relative humidities, water is distributed uniformly throughout the polymers, but probably preferentially where hydrogen bonding is possible. At higher pressures, chains of water molecules form at hydrogen bonding sites. The initial sorption process can be described by a conventional solution theory and the enhancement process can be viewed as one of occupancy of sites.

3.4.3

Retention of water inside a sorbent

All the natural fibers have groups in their molecules that attract water, referred to as the hydrophilic groups (Morton and Hearle, 1997). However, after all the hydrophilic groups have absorbed water molecules directly, the newly arrived water molecules may form further layers on top of the water molecules already absorbed. These two groups of water molecules are termed the directly and indirectly attached water, as shown in Fig. 3.8. The former is firmly bonded with the sorbent and hence is limited in movement and exhibits physical properties significantly different from those of free, or bulk, water (Berlin, 1981). According to Luck (1981), bound water has a reduced solubility for other compounds, causing a reduction in the diffusion of water-soluble solutes in Polymer

H2O

H2O

H 2O

H2O

H2O

Direct water

H2O

H2O

H 2O

H2O

H2O

Indirect water

H2O

H 2O

H 2O

3.8 Direct and indirect water. Adapted from Morton, W. E. and J. W. S. Hearle (1997). Physical Properties of Textile Fibers. UK, The Textile Institute.


119

the sorbent, and a decrease in diffusion coefficient with decreasing moisture content. The decreased diffusion velocity impedes drying processes because of slower diffusion of water to the surface. Some of the characteristics of bound water include a lower vapor pressure, high binding energy as measured during dehydration, reduced mobility as seen by nuclear magnetic resonance (NMR), non-freezability at low temperature, and unavailability as a solvent (Labuza and Busk, 1979). Although each of these characteristics has been used to define bound water, each gives a different value for the amount of water which is bound. As a result of this, as well as the complexities and interactions of the binding forces involved, no universal definition of bound water has been adopted. Indirectly attached water groups whose activity is in between those of the directly attached water and the free liquid water are held relatively loosely. In fact, this division of two water groups inside a sorbent forms the basis on which the first theory on moisture sorption was constructed in 1929 by Peirce (1929).

3.5

Mathematical description of moisture sorption isotherms

Water transport in porous material systems can be classified into three categories (Rizvi and Benado, 1984). (i) Structural aspects: to describe the mechanism of hydrogen bonding and molecular positioning by spectroscopic techniques; (ii) Dynamic aspects: to study molecular motions of water and their contribution to the hydrodynamic properties of the system; The use of these two approaches is restricted by the limited information on the theory of water solid interactions. (iii) Thermodynamic aspects: to understand the water equilibrium with its surroundings at a certain relative humidity and temperature. Since thermodynamic functions are readily calculated from sorption isotherms, this approach allows the interpretation of experimental results in accordance with a statement of theory (Iglesias et al. 1976). Various theories have been proposed and modified in the past centennial to describe the sorption mechanisms of individual fiber materials (Barrer, 1947; Hill, 1950; Taylor, 1954; Al-Muhtaseb et al., 2002). Langmuir (1918) developed the classical model for adsorption isotherms which is applicable for gases adsorbed in a monolayer on material surfaces. Largely based on Langmuir’s work, Brunauer et al. (1938) derived a widely used model for multi-layer adsorption. Independently, Peirce introduced in 1929 a model which is based on the assumption of direct and indirect sorption of water molecules on attractive groups of the fibrous materials (Peirce, 1929); and a theory also dealing with fibrous materials, in which the interaction between water and

120


the binding sites was classified into three types of water with different associating strengths, was later proposed by Speakman (1944). Young and Nelson (1967) developed a complete sorption–desorption theory, starting from the assumption of a distinct behavior of bound and condensed water. The moisture uptake leads to swelling of hygroscopic cellular fibers, a dimensional change due to breaking of inter- and intra-molecular hydrogen bonds between the cellular molecules (Gruber, Schneider et al., 2001). Also the equilibrium moisture isotherms show a distinct hysteresis between the sorption and desorption cycle, indicating structural changes of the fiber caused by the interaction with water (Hermans, 1949). Labuza (1984) noted that no single sorption isotherm model could account for the data over the entire range of relative humidity, because water is associated with the material by different mechanisms in different water activity regions. Of the large number of models available in the literature (Van den Berg and Bruin, 1981), some of those more commonly used are discussed below, a most recent account referring to Sánchez-Montero et al. (2005).

3.5.1

Selected theories on sorption isotherm

Amongst several brilliant pieces of work, Peirce proposed in 1929 one of the earliest mathematical models to describe the absorption process. Given the simplicity of his treatment, the model is surprisingly robust in comparison with the more sophisticated models that followed. Peirce first divided the absorbed water molecules into two parts, directly and indirectly attached water molecules: C = C a + Cb

[3.20]

where C, Ca and Cb are the total, direct and indirect water molecules absorbed per available absorption site. The value C in fact is related to the moisture regain R by

R=

CM w κ Mo

[3.21]

where Mw, Mo are the molecular weights of water and of per absorption site, W respectively, and κ = t , and Wt, Wo are the total masses of the material Wo and of all absorption sites. Peirce then derived the expressions for both Ca and Cb Ca = 1 – e–C

[3.22]

Cb = C – Ca = C – 1 + e–C

[3.23]

and


121

So that

C=

3κR 100

[1.24]

By replacing the moisture regain with the ratio of pressures, and working out the result for the coefficient κ for a case of soda-boiled cotton, Equation [3.20] was turned into 1–

pv = (1 – 0.4 Ca ) e –5.4 Cb p sv

[3.25]

A comparison between the experiments and predictions is shown in Fig. 3.9 (Peirce 1929). The Brunauer–Emmett–Teller (BET) model (Brunauer, Emmett et al., 1938) has been the most widely used method for predicting moisture sorption by solids. An important application of the BET isotherm is the surface area evaluation for solid materials. In general, the BET model describes the isotherms well up to a relative humidity of 50%, depending on the material and the type of sorption isotherm. The range is limited because the model cannot describe properly the water sorption in multilayers due to its three rather crude assumptions (Al-Muhtaseb et al. 2002): (i) the rate of condensation on the first layer is equal to the rate of evaporation from the second layer; 20

Regain (%)

16

12

8

4

0

20


100

3.9 Comparison of Peirce’s theory with experiment. Soda-boiled cotton at 110°C. From Peirce, F. T. (1929). ‘A two-phase theory of the absorption of water vapor by cotton cellulose.’ Journal of Textile Institute 20: 133T.

122


(ii) the binding energy of all of the adsorbate on the first layer is equal; (iii) the binding energy of the other layers is equal to those of the pure adsorbate. However, the equation has been useful in determining an optimal moisture content for drying and storage stability of materials, and in the estimation of the surface area of a material (Van den Berg, 1991). The BET equation is generally expressed in the form: aw = 1 + α – 1 aw α Ro R (1 – a w ) α Ro

[3.26]

where R is the moisture regain, Ro is the monolayer moisture regain, aw is the water activity, and α is approximately equal to the net heat of sorption. The advantage of this expression is that the RHS of the equation is a linear function of aw or the relative humidity. A plot of the equation in comparison with experimental data of various fibers is seen in Fig. 3.10 (Morton and Hearle, 1997). Dent in 1977 proposed a revised theory in which he improved the BET model by lifting the assumption that the binding energy of the other layers is equal to those of the pure adsorbate: this led to a better prediction (Dent, 1977). Hailwood and Horrobin (1946) developed a model in which the first vapor layer of water molecules was treated as being chemically bonded with the polymer groups and the successively absorbed water was viewed as solution inside the polymer. Their final result yielded:

RM = HK + HKK1 1800 1 – HK 1 + HKK1

[3.27]

where R is the moisture regain of the polymer; M the molecular weight of the polymer group; K1 is the equilibrium constant; and K is the ratio of the masses of the water solution and water vapor. By choosing the last three constants for best fitting with experimental data, they achieved a close agreement between the theoretical predictions and the testing data for both wool and cotton fibers, as illustrated in Fig. 3.11 (Hailwood and Horrobin, 1946). In order to analyze the sorption isotherm over a wider range of relative humidities, a model, known as the Guggenheim–Anderson–de Boer (GAB) theory, was also proposed by Guggenheim (1966), Anderson (1946) and de Boer (1968), based on some modified assumptions of the BET model, including the presence of an intermediate adsorbed layer having different adsorption and liquefaction heats and also the presence of a finite number of adsorption layers. The GAB equation provides the monolayer sorption values and could also be used for solid surface area determinations. At the same time, the equation covers a broader range of humidity conditions (Timmermann, 2003).

Essentials of psychrometry and capillary hydrostatics 0.40

0.35

0.30

Experimental points Nylon Acetate Cotton Silk Viscose Wool Full lines follow B.E.T. equation

123

Nylon

Acetate

  p r   (p – p 0) 

0.25

0.20

Cotton

0.15

Silk

Visco

0.10

se

Wool

0.05

0 0.1

0.2

0.3

0.4 p /p o

0.5

0.6

0.7

3.10 Comparison between the experiments and BET model. From Morton, W. E. and J. W. S. Hearle (1997). Physical Properties of Textile Fibers. UK, The Textile Institute.

Both BET and GAB methods have become very popular in food science, where the theory of mono and multilayer adsorption is applied to the sorption of water by a wide variety of dehydrated foods. The two theories are often expressed in the same format; the BET equation W=

Wm cp / p o (1 – p / p o )/(1 – p / p o + cp / p o )

[3.28]

and the GAB equation W=

Wm ckp / p o (1 – kp / p o )/(1 – kp / p o + ckp / p o )

[3.29]

where W is the weight of adsorbed water, Wm the weight of water forming a monolayer, c the sorption constant, p/po the relative humidity and k the additional constant for the GAB equation. Using gravimetrically obtained data, the constants in the two equations were obtained by an iterative technique,

124

Thermal and moisture transport in fibrous materials Calculated curves Experimental results, wool 30 Experimental results, cotton Wool 25

Regain (%)

20

15 Cotton

10

Dissolved water Cotton

5

Water in hydrate 0

20


100

3.11 Comparison between the experiments and predictions. Reproduced by permission of the Royal Society of Chemistry from Hailwood, A. J. and S. Horrobin (1946). ‘Absorption of water by polymers: analysis in terms of a simple model.’ Trans. Faraday Soc. 42B: 84.

so that both methods were applied in Roskar and Kmetec (2005) to evaluate the sorption characteristics of several excipients. Microcalorimetric analysis was also performed in order to evaluate the interaction between water and the substances. As shown in Fig. 3.12 from (Roskar and Kmetec, 2005), the experiments showed excellent agreement between data and the BET model up to 55% RH, confirming the previous conclusion and the GAB model over the entire humidity range, indicated also by high values of the statistical correlation coefficients in Roskar and Kmetec (2005). Furthermore, microcalorimetric measurements suggested that the hygroscopicity of solid materials could be estimated approximately using these approaches. A kinetic study of moisture sorption and desorption on lyocell fibers was recently conducted by Okubayashi et al. (2004). The authors summarized the various moisture sorption modes as shown in Fig. 3.13 and discussed the


125

60

Moisture content (%)

50 40 30 20 10 0 0

0.2

0.4 0.6 Relative humidity

0.8

1

3.12 Moisture sorption isotherms of Kollidone CL fitted by the BET (dotted line) and GAB (solid line) models to the experimental data (Roskar and Kmetec, 2005). With kind permission from the Pharmaceutical Society of Japan.

H2O H 2O

2

1 4

3

OH

OH

H 2O

H 2O

OH

OH

OH a

OH OH H2O

c OH b

H2O H 2O OH

OH

d

OH

a: Crystallites 1: External sorption b: Amorphous regions 2: Sorption onto amorphous c: Interfibrillartie molecules region d: Void 3: Sorption onto inner surface 4: Sorption onto crystallites

H2O : Direct water molecule H2O : Indirect water molecule

3.13 A schematic diagram of direct and indirect moisture sorption onto external surface (1), amorphous regions (2), inner surface of voids (3), and crystallites (4). Reprinted from Carbohydrate Polymers, 58, Okubayashi, S., U.J. Griesser, and T. Bechtold, ‘A kinetic study of moisture sorption’, 293–299, Copyright (2004), with permission from Elsevier.

results of quantitative and kinetic investigations of moisture adsorption in a man-made cellulose lyocell fiber by using a parallel exponential kinetics (PEK) model proposed by Kohler, Duck et al. (2003). A mechanism of water adsorption into lyocell is applied by considering the BET surface area, water retention capacity and hysteresis between the moisture regain isotherms and

126


is compared to those of cotton fibers. The simulation curves showed good fits with the experimental data of moisture regain in both sorption isotherms (Fig. 3.14) and sorption hysteresis (Fig. 3.15). The enthalpy change (∆H) provides a measure of the energy variations 20

Minf(total) (%)

15

10

5

0 0

20

40 60 Relative humidity (%)

80

100

3.14 Equilibrium moisture sorption and desorption isotherms of lyocell (black) and cotton (white) at 20 °C. Reprinted from Carbohydrate Polymers, 58, Okubayashi, S., U.J. Griesser, and T. Bechtold, ‘A kinetic study of moisture sorption’, 293–299, Copyright (2004), with permission from Elsevier. 50

Hysteresis (%)

40

30

20

10

0 0

10

20

30 40 50 60 Relative humidity (%)

70

80

90

3.15 Effects of relative humidity on hysteresis between sorption and desorption isotherms for lyocell (black) and cotton (white) at 20 °C. Reprinted from Carbohydrate Polymers, 58, Okubayashi, S., U.J. Griesser, and T. Bechtold, ‘A kinetic study of moisture sorption, 293 – 299, Copyright (2004), with permission from Elsevier.


127

occurring on mixing water molecules with sorbent during sorption processes, whereas the entropy change (∆S) may be associated with the binding, or repulsive, forces in the system and is associated with the spatial arrangements at the water–sorbent interface. Thus, entropy characterizes the degree of order or randomness existing in the water-sorbent system and aids interpretation of processes such as dissolution, crystallization and swelling. Free energy (∆G), based on its sign, is indicative of the affinity of the sorbent for water, and provides a criterion as to whether water sorption is a spontaneous (–∆G) or non-spontaneous process (+∆G) (Apostolopoulos and Gilbert, 1990). The relation between differential enthalpy (∆H) and differential entropy (∆S) of sorption is given by the equation (Everett, 1950): ln a w = – ∆H + ∆S RT R

[3.30]

where aw is water activity; R is universal gas constant (8.314 J mol–1K–1) and T is temperature (K). From a plot of ln (aw) versus 1/T using the equilibrium data, ∆H and ∆S values were determined from the slope and intercept, respectively. Applying this at different moisture contents (X) allowed the dependence of ∆H and ∆S on moisture content to be determined (Aguerre, et al. 1986).

3.5.2

Moisture sorption hysteresis

As in many nonlinear complex phenomena, there is hysteresis in the moisture sorption process, typically depicted by the different paths on a regain–time curve between absorption and desorption isotherm processes. Taylor (1952, 1954) has shown that hysteresis occurs even in cycles at low relative humidities. The interpretations proposed for sorption hysteresis can be classified into one, or a combination, of the following categories (Arnell, 1957; Kapsalis, 1987): (i) Hysteresis in porous solids: for instance in polymers, the uneven breaking and reforming of the cross-links due to capillary pressure during the absorption and desorption processes causes the hysteresis (Urquhart and Eckersall, 1930; Hermans, 1949; Morton and Hearle, 1997). (ii) Hysteresis in non-porous solids: this is observed in materials such as protein, where the theory is based on partial chemisorption, surface impurities, or phase changes (Berlin, 1981); (iii) Hysteresis in non-rigid solids: this is observed in materials such as in single fibers, where the theory is based on changes in structure due to swellings which hinder the further penetration of the moisture (Meredith, 1953; Ibbett and Hsieh, 2001). Given the complexity of the issue, a more effective way to analyze the

128


sorption hysteresis is to investigate the hysteresis in the contact angle during sorption processes. Any wetting process is extremely sensitive to heterogeneities or chemical contamination and one of the most spectacular manifestations of the inhomogeneity is the contact angle hysteresis (Leger and Joanny, 1992). On a real solid surface one almost never measures the equilibrium contact angle given by Young’s law, but a static contact angle that depends on the history of the sample. If the liquid–vapour interface has been obtained by advancing the liquid, (after spreading of a drop, for example) the contact angle has a value θA larger than the equilibrium value; if, on the contrary, the liquid–vapour interface has been obtained by receding the liquid (by retraction or aspiration of a drop), the measured contact angle θR is smaller than the equilibrium contact angle in Fig. 3.16. Even when the solid surface is only slightly heterogeneous, the difference θA – θR can be as large as a few degrees; in more extreme situations, when the spreading liquid is not a simple liquid but a solution, differences of the order of 100 degrees have been observed (Leger and Joanny, 1992). Contact angle hysteresis explains many phenomena observed in everyday life. A raindrop attached to a vertical window should flow down under the action of its weight; on a perfect window the capillary force exactly vanishes. On a real window, in the upper parts of the drop the liquid has a tendency to recede and the contact angle is the receding contact angle; in the lower parts of the drop, the liquid has a tendency to advance and the contact angle is the advancing contact angle; the difference in contact angles creates a capillary force directed upwards that can balance the weight (Leger and Joanny, 1992). The most common heterogeneities that are invoked to explain contact angle hysteresis are roughness and chemical heterogeneities due to contamination that we discuss in more detail below. Any kind of heterogeneity of the solid may, however, create contact angle hysteresis: examples are the porosity of the solid or the existence of amorphous and crystalline regions at the surface of a polymeric solid. Another source of contact angle hysteresis may come from the liquid itself; when it is not a simple liquid but a solution, the irreversible adsorption of solutes leads to strong hysteretsis effects. The following are just two examples of various models proposed for specific surfaces. Advancing

θA

θR

3.16 Advancing and receding contact angles.


129

(i) Contact angle hysteresis on a rough surface The early models to describe contact angle hysteresis considered surfaces with parallel or concentric groves (Mason, 1978). The simplest example is that of a surface with a periodic roughness in one direction u = uo sin qx when the contact line is parallel to the groves in the y direction. In this geometry, Young’s law can be applied locally and leads to a contact angle between the liquid–vapour interface and the local slope of the solid θo. The apparent contact angle θ is, however, the angle between the liquid–vapour interface and the average solid surface. If quo < 1:

θ = θ o – du [3.31] dx For stability reasons, in an advancing experiment, the contact angle must increase; θ thus reaches its maximum value θ = θo + quo and the contact line must then jump one period towards the next position where this value can be attained; the advancing contact angle is thus θA = θo + quo

[3.32]

Similarly, in a receding experiment, the contact angle must decrease and the receding contact angle is the lowest possible contact angle

θR = θo – quo

[3.33]

This very simple model thus leads to contact angle hysteresis ∆θ = θA – θR = 2quo and predicts jumps of the contact line between equilibrium positions. It, however, contains some unrealistic features. (ii) Surface with a single defect Far away from the contact line, the liquid–vapour interface is flat and shows a contact angle θA. Following Young’s arguments, the extra force due to the defects on the contact line is γLV (cos θo – cos θA). The dissipated energy for one defect is D = UγLV (cos θo – cos θA)

[3.34]

where U is the advancing speed. This dissipated energy is due to the jump of the contact line on the defects and may be thus calculated directly. The number of defects swept per unit time and unit length of the contact line is nU and WA is the surface energy D = nUWA

[3.35]

Comparing these two expressions we obtain the advancing angle as

γLV (cos θo – cos θA) = nWA

[3.36]

Similarly in a receding experiment,

γLV (cos θo – cos θR) = nWR The contact angle hysteresis is then

[3.37]

130


γLV (cos θR – cos θA) = n(WA + WR)

[3.38]

For a smooth defect we thus predict a contact angle hysteresis γLV (cos θR – cos θA) This dilute defect model has several important limitations; it is restricted to small contact angles, to small distortions of the contact lines (that we have assumed approximately flat) and to extremely dilute defects.

3.5.3

Heat and temperature effects on sorption isotherm

When a material absorbs water, heat is released, depending on the state of the water. For liquid water, this heat is denoted as Ql, or Qv for vapor. The two differ by the condensation heat Qc at constant temperature, i.e. Qv = Ql + Qc

[3.39]

There are two ways to describe or calculate the heat released (Watt and McMahon, 1966; Morton and Hearle, 1997; Mohamed, Kouhila et al., 2005). (i) The differential heat of sorption Q(J per gram of water absorbed): Heat evolved for l gram water to be completely absorbed by a material of infinite mass at a given moisture regain level R. Data for some fibers are shown in Table 3.4 (Morton and Hearle, 1997). (ii) The integral heat of sorption W (J per gram of dry material): Heat evolved for l gram dry mass to be completely wet (absorption from the liquid state) at a given moisture regain level R as shown in Fig. 3.17 for several fibers (Morton and Hearle, 1997). W × 100(%) =

∫

Rs

Ql dR

[3.40]

R

where RS is the saturation moisture regain at the constant temperature; Table 3.4 The differential heat of sorption for some fibers Differential heats of sorption (kJ/g) Relative humidity (%) Material

0

15

30

45

60

75

Cotton Viscose rayon Acetate Mercerized cotton Wool Nylon*

1.24 1.17 1.24 1.17 1.34 1.05

0.50 0.55 0.56 0.61 0.75 0.75

0.39 0.46 0.38 0.44 0.55 0.55

0.32 0.39 0.31 0.33 0.42 0.42

0.29 0.32 0.24 0.23 – –

– 0.24 – – – –

Adapted from Morton and Hearle (1997) *From sorption isotherms.


131

Integral heat of sorption (J/g)

100

80

Wool

60

40

20

Viscose Cotton Acetate

0

5 10 Regain (%)

15

Heat evolved 0 to 65% r.h. (J/g of fibre)

3.17 The integral heat of sorption for some fibers. From Morton, W. E. and J. W. S. Hearle (1997). Physical Properties of Textile Fibers. UK, The Textile Institute.

Ardil 80

Mercerized cotton

Wool Tenasco Fortisan

60 Silk 40

20

0

Cotton Acetate Nylon

10 Regain (%) at 65% r.h.

20

3.18 Heat evolved from 0 to 65% RH for major fibers. From Meredith, R. (1953). From Fiber Science. J. M. Preston. Manchester, The Textile Institute: p. 246.

or [3.41] Ql = –100 dW dR Heat evolved from 0 to 65% RH for major fibers is provided in Fig. 3.18 (Meredith, 1953).

132


The differential heat of sorption is the amount of energy above the heat of water vaporization associated with the sorption process. This parameter is used to indicate the state of absorbed water by the solid particles. Free energy and differential heat of sorption are commonly estimated by applying the Clausius–Clapeyron equation to sorption isotherms (Kapsalis, 1987; Yang and Cenkowski, 1993): ln

Q a2 = s  1 – 1  a1 R  T1 R2 

[3.42]

where ai is the water activity at temperature Ti °K, Qs the heat of sorption in cal/mole, a function of the moisture content. There is no analytical way to determined Qs other than to conduct tests at two temperature levels to determine the moisture sorption isotherms, from which Qs can be derived (Labuza, 1984). R the gas constant = 1.987 cal/mole °K, aw value increases as T increases at a constant moisture content. In describing a moisture sorption isotherm, one must specify the temperature and hold it constant. Morton and Hearle (1997) have shown by using the equation that an increase in moisture regain ∆a of 0.6 causes the temperature to increase by 10.3 °C. Although, in theory, this sorption heat can serve as a thermal buffer for clothing materials (for evaporation of sweat from a hot body absorbs the heat to more or less chill the body), in practice, sweat often blocks the air flow channels in the clothing, and causes fiber swelling which in turn reduces the free pores in the clothing. Both hinder the ‘breath-ability’ of the clothing. Furthermore, the sorption heat can be a safety hazard for materials storage. The collective sorption heat can raise the temperature to the burning point!

3.6

References

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4 Surface tension, wetting and wicking W. ZHONG, University of Manitoba, Canada

4.1

Introduction

Surface tension, wicking and wetting are among the most frequently observed phenomena in the processing and use of fibrous materials, when water or any other liquid chemical comes into contact with and is transported through the fibrous structures. The physical bases of surface tension, wetting and wicking are molecular interactions within a solid or liquid, or across the interface between a liquid and a solid. Wetting/wicking behaviors are determined by surface tensions (of solid and liquid) and liquid/solid interfacial tensions. Curvature and roughness of contact surface are two other critical factors for wetting phenomena, especially in the case of wetting in fibrous materials. These factors and their effects on wetting phenomena in fibrous materials will also be discussed.

4.2

Wetting and wicking

4.2.1

Wetting

The term ‘wetting’ is usually used to describe the displacement of a solid–air interface with a solid–liquid interface. When a small liquid droplet is put in contact with a flat solid surface, two distinct equilibrium regimes may be found: partial wetting with a finite contact angle θ, or complete wetting with a zero contact angle (de Gennes, 1985), as shown in Fig. 4.1. The forces in equilibrium at a solid–liquid boundary are commonly described by the Young’s equation:

γSV – γSL – γLV cos θ = 0

[4.1]

where γSV, γSL, and γLV denotes interfacial tensions between solid/vapor, solid/liquid and liquid/vapor, respectively, and θ is the equilibrium contact angle. 136


137

Vapor Liquid

θ

θ

Solid (a)

(b)

(c)

4.1 A small liquid droplet in equilibrium over a horizontal surface: (a) partial wetting, mostly non-wetting, (b) partial wetting, mostly wetting, (c) complete wetting.

The parameter that distinguishes partial wetting and complete wetting is the so-called spreading parameter S, which measures the difference between the surface energy (per unit area) of the substrate when dry and wet:

or

S = [Esubstrate]dry – [Esubstrate]wet

[4.2]

S = γSo – (γSL + γLV)

[4.3]

where γSo is surface tension of a vapor-free or ‘dry’ solid surface. If the parameter S is positive, the liquid spreads completely in order to lower its surface energy (θ = 0). The final outcome is a film of nano-scale thickness resulting from competition between molecular and capillary forces. If the parameter S is negative, the drop does not spread out, but forms at equilibrium a spherical cap resting on the substrate with a contact angle θ. A liquid is said to be ‘mostly wetting’ when θ ≤ π /2, and ‘mostly non-wetting’ when θ > π /2 (de Gennes et al., 2004). When contacted with water, a surface is usually called ‘hydrophilic’ when θ ≤ π /2, and ‘hydrophobic’ when θ > π /2.

4.2.2

Wicking

Wicking is the spontaneous flow of a liquid in a porous substrate, driven by capillary forces. As capillary forces are caused by wetting, wicking is a result of spontaneous wetting in a capillary system (Kissa, 1996). In the simplest case of wicking in a single capillary tube, as shown in Fig. 4.2, a meniscus is formed. The surface tension of the liquid causes a pressure difference across the curved liquid/vapor interface. The value for the pressure difference of a spherical surface was deduced in 1805 independently by Thomas Young and Pierre Simon de Laplace, and is represented with the socalled Young–Laplace equation (Adamson and Gast, 1997):

∆P = γ LV  1 + 1   R1 R2 

[4.4]

For a capillary with a circular cross-section, the radii of the curved interface R1 and R2 are equal. Thus:

138


r R θ

h

4.2 Wicking in a capillary.

∆P = 2γLV/R where R = r/cos θ

[4.5] [4.6]

and r is the capillary radius. As the capillary spaces in a fibrous assembly are not uniform, usually an indirectly determined parameter, effective capillary radius re is used instead.

4.3

Adhesive forces and interactions across interfaces

The above discussions show that both wicking and wetting behaviors are determined by surface tensions (of solid and liquid) and liquid/solid interfacial tensions. These surface/interfacial tensions, in macroscopic concepts, can be defined as the energy that must be supplied to increase the surface/interface area by one unit. In microscopic concepts, however, they originate from such intra-molecular bonds as covalent, ionic or metallic bonds, and such longrange intermolecular forces as van der Waals forces and short range acid– base interactions. Therefore, the physical bases of wetting and wicking are those molecular interactions or adhesive forces within a solid or liquid, or across the interface between a liquid and a solid. These adhesive forces include Lifshitz–van de Waals interactions and acid–base interactions.

4.3.1

Lifshitz–van der Waals forces

Molecules can attract each other at a moderate distances and repel each other at a close range, as denoted by the Lennard–Jones potential:


w (r ) = A – C6 12 r r

139

[4.7]

where w(r) is the interactive potential between two molecules at distance r, and A and C are intensities of the repellency and attraction, respectively. The attractive forces, represented by the second term at the right-hand side of Equation [4.7], are collectively called ‘van der Waals forces’. They are some of the most important long-range forces between macroscopic particles and surfaces. They are general forces which always operate in all materials and across phase boundaries. Van der Waals forces are much weaker than chemical bonds. Random thermal agitation, even around room temperature, can usually overcome or disrupt them. However, they play a central role in all phenomena involving intermolecular forces, including those interactions between electrically neutral molecules (Israelachvili, 1991; Good and Chaudhury, 1991). When those intermolecular forces are between like molecules, they are referred to as cohesive forces. For example, the molecules of a water droplet are held together by cohesive forces. The cohesive forces between molecules inside a liquid are shared with all neighboring atoms. Those on the surface have no neighboring atoms beyond the surface, and exhibit stronger attractive forces upon their nearest neighbors on the surface. This enhancement of the intermolecular attractive forces at the surface is called surface tension, as shown in Fig. 4.3. Intermolecular forces between different molecules are known as adhesive forces. They are responsible for wetting and capillary phenomena. For example, if the adhesive forces between a liquid and a glass tube inner surface are larger than the cohesive forces within the liquid, the liquid will rise upwards along the glass tube to show a capillary phenomenon, as shown in Fig. 4.2. To derive the van der Waals interaction energy between two bodies/surfaces from the pair potential w(r) = –C/r6, Hamaker (1937) introduced an additivity assumption that the total interaction can be seen as the sum over all pair

Gas

Surface tension

Liquid

4.3 Liquid surface tension caused by cohesive forces among liquid molecules.

140


interactions between any atom in one body and any atom in the other, thus obtaining the ‘two-body’ interaction energy, such as that for two spheres (Fig. 4.4(a)), for a sphere near a surface (Fig. 4.4(b)), and for two flat surfaces (Fig. 4.4(c)) (Israelachvili, 1991). And the Hamaker constant A is given as a function of the densities of the two bodies: A = π 2 Cρ1ρ2

[4.8]

Hamaker’s theory has ever since been used widely in studies of surface– interface interactions and wetting phenomena, although there have been concerns about its additivity assumption and ignorance of the influence of neighboring atoms on the interaction between any atom pairs (Israelachvili, 1991; Wennerstrom, 2003). The problem of additivity is completely avoided in Lifshitz’s theory (Garbassi et al., 1998; Wu, 1982; Wennerstrom, 2003; Israelachvili, 1991). The atomic structure is ignored, and interactive bodies are regarded as dielectric continuous media. Then the van der Waals interaction free energies W between large bodies can be derived in terms of such bulk properties as their dielectric constants and refractive indices. And the net result of a rather complicated calculation is that Lifshitz regained the Hamaker expressions in Fig. 4.4, but with a different interpretation of the Hamaker constant A. An approximate expression for the Hamaker constant of two bodies (1 and 2) interacting across a medium 3, none of them being a conductor (Israelachvili, 1991; Wennerstrom, 2003), is

A1,2 =

3 hv e ( n12 – n32 )( n 22 – n32 ) 8 2 ( n12 + n32 )1/2 ( n 22 + n32 )1/2 [( n12 + n32 ) + ( n 22 + n32 )1/2 ]

ε – ε3 ε2 – ε3 + 3 kT 1 ε1 + ε 3 ε 2 + ε 3 4

[4.9]

where h is the Planck’s constant, ve is the main electronic adsorption frequency in the UV (assumed to be the same for the three bodies, and typically around 3 × 1015 s–1), and ni is the refractive index of phase i, εi is the static dielectric constant of phase i, k is the Boltzmann constant, and T the absolute temperature.

D

R1

R2

ρ1

ρ2

R

D

(a) Two spheres R1R 2 W= – A 6D R1 + R 2

(b) Sphere–surface W = – AR 6D

D (c) Two surfaces A W= – 12πD 2 per unit area

4.4 Van der Waals interaction free energies between selected bodies.


141

Alternatively, from a macroscopic view, the creation of an interface with interfacial free energy γ12 by bringing together two different phases from their infinitely separately states, characterized by their surface energies γ1 and γ2, results in a molecular reorganization in the surface layers of each phase, as well as in interphase molecular interactions. These effects can be expressed thermodynamically as the work of adhesion, Wa: Wa = γ1 + γ2 – γ12

[4.10]

It was suggested by Fowkes that the equilibrium work of adhesion between two surfaces for a system involving only apolar interactions (Fowkes, 1962) is: Wa = 2(γ1γ2)1/2

[4.11]

Combining Equations [4.10] and [4.11], we obtain:

γ12 = γ1 + γ2 – 2(γ1γ2)1/2, i and j apolar = ( γ1 –

γ 2 )2

[4.12]

For greater generality, polar components should be taken into consideration. This will be examined in the following section.

4.3.2

Acid–base interactions

While Lifshitz–van der Waals (LW) interactions (γ LW) represent the apolar component of interfacial forces, acid–base (AB) interactions (γ AB) account for the polar component. Hydrogen bonds constitute the most important subclass of acid–base interactions. The Lifshitz–van der Waals/acid–base approach, or acid–base approach for short claimed that, for any liquid or solid, the total surface tension γ can be uniquely characterized by these two surface tension components (van Oss, 1993; Good, 1992; Good et al., 1991):

γ = γ LW + γ AB

[4.13]

This approach came into existence when the thermodynamic nature of the interface was re-examined by van Oss et al. (1987a) in the light of Lifshitz theory. The apolar interaction between a protein and a low energy surface solid is repulsive and hence solely the apolar interaction cannot explain the strong attachment of biopolymer on the low energy solid. A polar term, short-range interaction, later called Lewis acid–base (AB) interaction, was introduced to explain the attraction. The LW component in Equation [4.13] can be derived by Equation [4.12]. AB interactions, on the other hand, are not ubiquitous as are LW interactions. They occur when an acid (electron acceptor) and a base (electron donor) are brought close together. Accordingly, the acid–base surface tension component

142


comprises two non-additive parameters: acid surface tension parameter γ + and base surface tension parameter γ –:

γ

AB

= 2 γ +γ

–

[4.14]

The AB interactions across an interface may be expressed in the form

γ 12AB = (2 γ 1+ –

γ 2+ )( γ 1– –

γ 2– )

[4.15]

The existence of acid–base interactions can substantially improve wetting and adhesion. The high energy associated with acid–base interactions is due to their short range (2–3A) Coulombic forces. The interfacial tension for solid/liquid systems, therefore, can be obtained through a combination of Equations [4.12]–[4.15] (van Oss, 1993; Kwok et al., 1994):

γ SL = γ S + γ L – 2 (γ SLW γ LLW )1/2 – 2 (γ S+ γ L– )1/2 – 2(γ S– γ L+ )1/2 [4.16] It is well known that surface tensions of liquids may readily be measured directly by force methods such as the Wilhelmy plate or the du Nouy ring. However, there is no well-accepted direct method to measure the surface tensions of solid polymers. When using the Young’s equation [4.1] to derive the solid surface tension from liquid surface tension and contact angle, a valid approach to determine interfacial tensions between liquid and solid is very important. The acid–base approach has therefore been used frequently to estimate the solid surface tensions (Kwok et al., 1994, 1998; van Oss et al., 1990) or interfacial adhesion (Greiveldinger and Shanahan, 1999; Chehimi et al., 2002). In order to calculate the solid surface tension components from the acid– base approach, Equation [4.16] combined with Young’s Equation [4.1] yields (Lee, 1993):

γ L (1 + cos θ ) = 2 (γ SLW γ LLW )1/2 + 2 (γ S+ γ L– )1/2 + 2(γ S– γ L+ )1/2 [4.17] under the assumption that vapor adsorption is negligible. From Equation [4.17], the solid surface tension components, γ SLW , γ S+ and γ S– can be calculated by simultaneous solution of three equations if the measurement of contact angles with respect to three different liquids is known on the solid substrates. Three liquids of known surface tension components (γ LLW , γ L+ and γ L– ) are also required. Usually, the van der Waals component γ SLW can be first determined by using an apolar liquid. Then two other polar liquids can be used to determine the acid–base components of the solid, γ L+ and γ L– (Kwok et al., 1994; van Oss et al., 1990).


4.4

143

Surface tension, curvature, roughness and their effects on wetting phenomena

There has been numerous research work published on the wetting process on solid surface, including several comprehensive reviews (Good, 1992; de Gennes, 1985), which cover topics from contact angle, contact line, liquid– solid adhesion, wetting transition (from partial wetting to complete wetting) and dynamics of spreading. However, wetting of fibrous materials becomes an even more complex process as it involves interaction between a liquid and a porous medium of curved, intricate and tortuous structure, yet with a soft and rough surface, instead of a simple solid, flat and smooth surface.

4.4.1

Surface tension and wettability

From studies on the bulk cohesive energy, we learn that there are two main types of solids: hard solids (bound by covalent, ionic or metallic) with socalled ‘high energy surfaces’, and weak molecular crystals (bound by van der Waals forces, or in some cases by acid–base interactions) with ‘low energy surfaces’. The surface tension, γsv, is in the range of 500 to 5000 mN/ m for high energy surfaces, and 10 to 50 mN/m for low energy surfaces (Fowkes and Zisman, 1964). Most organic fibers belong to the ‘low energy surfaces’ category. Most molecular liquids achieve complete wetting with high-energy surfaces (de Gennes, 1985). In the idealized case where liquid–liquid and liquid– solid interactions are purely of the van der Waals type (no chemical bonding nor polar interactions), solid–liquid energy could be deducted as follows: If a semi-infinite solid and a semi-infinite liquid are brought together, they start with an energy γLV + γSo, and end in γSL, as the van der Waals interaction energy VSL between solid and liquid is consumed. This process can be expressed as:

γSL = γSo + γLV – VSL

[4.18]

To a first approximation, the van der Waals couplings between two species are simply proportional to the product of the corresponding polarizabilities α (de Gennes, 1985): VSL = kαSαL

[4.19]

Similarly, if two liquid portions are brought together, they start with energy 2γLV, and end up with zero interfacial energy: 2γLV – VLL = 0

[4.20]

The same applies to solids: 2γSo – VSS = 0

[4.21]

144


Equation [4.3] combined with [4.18]–[4.21] gives: S = γSo – (γSL + γLV) = VSL – VL L = k(αS – αL)αL

[4.22]

Therefore, a liquid spreads completely if αS > αL so as to make S positive. Low-energy surfaces can give rise to partial or complete wetting, depending on the liquid chosen (de Gennes, 1985). The empirical criterion of Zisman (Zisman, 1964; de Gennes et al., 2004) is that each solid substrate has a critical surface tension γC, and there is partial wetting when the liquid surface tension γ > γC and total wetting when γ < γC. The critical surface tension can be determined by the so-called Zisman plot. A series of homologous liquid (usually n-alkanes, with n the variable) is chosen for the study. Cos θ as a function of γ is plotted to give the critical surface tension, as shown in Fig. 4.5 (de Gennes et al., 2004; de Gennes, 1985). Equation [4.22] is an interpretation of spreading coefficient S in terms of van der Waals forces only. To extend the wetting criteria for liquid/solid interfaces to include both long-range and short-range interactions, two key parameters are used: the effective Hamaker constant Aeff and the spreading coefficient S (Brochard-Wyart et al., 1991; Lee, 1993). The effective Hamaker constant describes the long-range interactions: Aeff = ASL – ALL

[4.23]

And the spreading coefficient S contains contributions from short-range interactions in its original expression [4.3]. It is also important to note that both S and Aeff are independent variables, and both can have positive or negatives values. Using two parameters, S and Aeff, as wetting criteria, results in four possibilities of wetting behaviors: (i) S > 0 and Aeff > 0, complete wetting. A small droplet put in contact with a flat solid surface spreads out and forms a thin ‘pancake’ film, as shown in Fig. 4.6(a). cos θ 1

0

γC

γ of n-alkanes (mN/m)

4.5 A typical Zisman plot to determine critical surface tension γ C.


145

Droplet

Thin pancake Precursor film

(a) S > 0 and Aeff > 0

Dry

Drop

(c) S < 0 and Aeff > 0

(b) S > 0 and Aeff < 0

Dry

Drop

(d) S < 0 and Aeff < 0

4.6 Various kinds of wetting.

(ii) S > 0 and Aeff < 0, pseudo partial wetting. The final equilibrium state of the liquid drop is a spherical cap with a precursor film, as shown in Fig. 4.6(b). (iii) S < 0 and Aeff > 0, partial wetting. The solid around the droplet is dry. The liquid profile is hyperbolic and curved downward, as shown in Fig. 4.6(c). (iv) S < 0 and Aeff < 0, partial wetting. The solid around the droplet is dry. The liquid profile is hyperbolic and curved upward, as shown in Fig. 4.6(d). Over the past two decades, considerable interest has developed in the field of acid–base, or electron acceptor/donor theory and their applications in evaluating surface and interfacial tensions, as described in the previous section. One of the appealing features of the concept based on acid–base theory is that it introduces the possibility of negative interfacial tensions, as exist in spontaneous emulsification or dispersion phenomena. Negative interfacial tensions were impossible within the confines of van der Waals bonding (van Oss et al., 1987b; Leon, 2000).

4.4.2

Curvature and wetting

Wetting of fibrous materials is dramatically different from the wetting process on a flat surface, due to the geometry of the cylindrical shape. A liquid that fully wets a material in the form of a smooth planar surface may not wet the same material when presented as a smooth fiber surface. Brochard (1986) discussed the spreading of liquids on thin cylinders, and stated that, for nonvolatile liquids, a liquid drop cannot spread out over the cylinder if the spreading coefficient S is smaller than a critical value Sc,

146


instead of 0. At the critical value Sc, there is a first-order transition from a droplet to a sheath structure (‘manchon’). The critical value was derived as

a 2/3 Sc = 3 γ   2  b

[4.24]

where a is the molecular size, b is the radius of the cylinder. There was also plenty of research work on the equilibrium shapes of liquid drops on fibers (Neimark, 1999; McHale et al., 1997, 1999, 2001; Quere, 1999; Bauer et al., 2000; Bieker and Dietrich, 1998; McHale and Newton, 2002). It was reported that two distinctly different geometric shapes of droplet are possible: a barrel and a clam shell, as shown in Fig. [4.7]. In the absence of gravity, the equilibrium shape of a drop surface is such that the Laplace excess pressure, across the drop surface is everywhere constant, as shown in Equation [4.4]. (McHale et al. (2001) solved this equation for the axially symmetric barrel shape subject to the boundary condition that the profile of the fluid surface meets the solid at an angle given by the equilibrium contact angle θ:

∆P =

2γ LV ( n – cos θ ) x1 ( n 2 – 1)

[4.25]

where n = x2/x1, is the reduced radius as shown in Fig. 4.7(a). Their (McHale et al., 1999) solution for the barrel shape droplet was subsequently used to compute the surface free energy, defined as F = γLVALV + (γSL – γSV)ASL

[4.26]

where ALV and ASV are the liquid/vapor and solid/liquid interfacial areas, respectively. In contrast to the barrel-shape droplet problem, no solution to Laplace’s equation for the asymmetric clam-shell shape is reported except for such numerical approaches as finite element methods (McHale and Newton, 2002). There are, however, papers discussing the roll-up (barrel to clam-shell) transition (McHale et al., 2001, McHale and Newton, 2002) in the wetting process on a fiber.

x1

x2 Fiber

(a) Barrel shape

(c) Clam-shell shape

4.7 Equilibrium liquid droplet shapes on a fiber.


147

In addition, there is work with respect to gravitational distortion of barrelshape droplets on vertical fibers (Kumar and Hartland, 1990). To represent the heterogeneous nature of fibrous materials in the wetting process, Mullins et al. (2004) incorporated a microscopic study of the effect of fiber orientation on the fiber-wetting process when subjected to gravity, trying to account for the asymmetry of wetting behavior due to fiber orientation and gravity. The theory concerning the droplet motion and flow on fibers is based on the balance between drag force, gravitational force and the change in surface tension induced by the change in droplet profile as the fiber is angled. As a result, there comes out an angle where droplet flow will be maximized. In reality, fibrous materials are porous media with intricate, tortuous and yet soft surfaces, further complicating the situation. As a result, a precise description of the structure of a fibrous material can be tedious. Therefore, much research work has adopted Darcy’s law, an empirical formula that describes laminar and steady flow through a porous medium in terms of the pressure gradient and the intrinsic permeability of the medium (Yoshikawa et al., 1992; Ghali et al., 1994; Mao and Russell, 2003):

u = – K ∇p µ

[4.27]

where u is the average velocity of liquid permeation into the fibrous material, µ the Newtonian viscosity of the liquid, K the permeability, and ∇p the pressure gradient. In the case of wetting, the driving pressure is usually the capillary pressure as calculated by the Laplace equation. The permeability K is either determined by experiments or by the empirical Kozeny–Carman relations as a function of fiber volume fraction (Mao and Russell, 2003). Darcy’s law reflects the relationship of pressure gradient and average velocity only on a macroscopic scale. To reach the microscopic details of the liquid wetting behavior in fibrous media, various computer simulation techniques have been applied in this field to accommodate more complexity so as to investigate more realistic systems, and to better understand and explain experimental results. Molecular Dynamics (MD) and Monte Carlo (MC) are best-known, standard simulation formulae emerging from the last decades (Hoffmann and Schreiber, 1996) and, accordingly, most of the simulation for clarifying liquid wetting behaviors falls into these two categories. Fundamentally, wetting behaviors of liquids in fibrous materials stem from interactions between liquid/solid and within the liquid at the microscopic level. The most important task for the various models and simulations is, therefore, to define and treat these interactions. In Molecular Dynamics, all potentials between atoms, solid as well as liquid, are described with the standard pairwise Lennard–Jones interactions:

148


  σ ij  12  σ ij  6  Vij ( r ) = 4 ε ij    –  r       r 

[4.28]

where r is the distance between any pair of atoms i and j, εij is an energy scale (actually the minimum of the potential), and σij is a length scale (the distance at which the potential diminishes to zero). Large-scale MD simulations have been adopted to study the spreading of liquid drops on top of flat solid substrates (Semal et al., 1999; van Remoortere et al., 1999). If the system contains enough liquid molecules, the macroscopic parameters, such as the density, surface tension, viscosity, flow patterns and dynamic contact angle, can be ‘measured’ in the simulation. However, the computational cost for MD simulations is huge, as they are dealing with the individual behaviors of a great number of single molecules. And, the application of MD simulations for liquid spreading on a fiber or transport in intricate fibrous structures is still pending, although there are already reports on microscopic understanding of wetting phenomena on cylindrical substrates for simple fluids whose particles are governed by dispersion forces and are exposed to long-ranged substrate potentials (Bieker and Dietrich, 1998). Based on a microscopic density functional theory, the effective interface potential for a liquid on a cylinder has been derived. To solve the problem of huge computation, simulation techniques have been invented to cope with the so called ‘cell’, or small unit of the system, instead of single molecules. The statistical genesis of the process of liquid penetration in fibrous media can be regarded as the interactions and the resulting balance among the media and liquid cells that comprise the ensemble. This process is driven by the difference of internal energy of the system after and before a liquid moves from one cell to the other. In the 1990s, Manna et al. (1992) presented a 2D stochastic simulation of the shape of a liquid drop on a wall due to gravity. The simulation was based on the so called Ising model and Kawasaki dynamics. Lukkarinen et al. (1995) studied the mechanisms of fluid droplets spreading on flat solids using a similar model. However, their studies dealt only with flow problems on a flat surface instead of a real heterogeneous structure. Only recently has the Ising model been used in the simulation of wetting dynamics in heterogeneous fibrous structures (Lukas et al., 1997; Lukas and Pan, 2003; Zhong et al., 2001a, 2001b). As a ‘meso-scale’ approach, stochastic models and simulations deal with discrete and digitalized cells or subsystems instead of individual molecules. They lead to considerable reduction of computational cost, naturally.

4.4.3

Surface roughness and wetting

The Young’s Equation [4.1] describes the mechanical balance at the triple line of the three-phase solid–liquid–vapor system. However, the equilibrium


149

contact angle θ in the equation can be obtained only experimentally on a perfectly smooth and homogeneous surface. In the real world, the roughness and heterogeneity of the solid surface produces the contact angle hysteresis (de Gennes, 1985): ∆θ = θa – θr ≥ 0

[4.29]

The advancing angle θa is measured when the solid–liquid contact area increases, while the receding angle θr is measured when the contact area shrinks, as shown in Fig. 4.8. The equilibrium contact angle lies between them:

θr < θ < θa

[4.30]

The most important source of contact angle hysteresis is the surface roughness. Early studies on the effect of surface roughness concentrated on periodic surfaces, such as a surface with parallel grooves (Cox, 1983; Oliver et al., 1977). In the simplest case where the triple line is parallel to the grooves, as shown in Fig. 4.9, the energy barrier for liquid spreading over a ridge of the rough surface can be computed numerically. When the grooves are rather deep, vapor bubbles may be trapped at the bottom of the grooves, as shown in Fig. 4.9(b). These vapor bubbles would lead to much smaller barriers, which was also observed in experimental work. With the increase of roughness, that is, with the increase of the depth of the grooves, there is first a corresponding decrease of receding angle θr; but when the grooves become deep enough, θr increases as the entrapped vapor bubbles reduce the barriers (de Gennes, 1985).

Advancing

Liquid

θa

Receding

θr

Solid

4.8 Advancing and receding contact angles for a liquid on a solid surface. Triple line Liquid

Triple line Liquid

Vapor bubble Solid Solid (a)

(b)

4.9 Wetting of rough surfaces without and with vapor bubbles.

150


A more realistic representation of a rough surface is a random surface (Joanny and de Gennes, 1984). The irregularities of the surface can be defined in a random function h(x, y). Consider a single ‘defect’, which is defined as a perturbation h(x, y) localized near a particular point (xd, yd) and with finite linear dimension d, as shown in Fig. 4.10. A triple line becomes anchored to the defect. Far from the defect, the line coincides with y = yL. An approximation of the total force f exerted by the defect on the line is: f ( ym ) =

∫

∞

–∞

[4.31]

dxh ( x , y m )

Assuming a Gaussian defect,  ( x – x d ) 2 + ( y – yd ) 2  h ( x , y ) = h0 exp  –  2d 2  

[4.32]

The force f(ym) is also Gaussian:  ( y – y )2  f ( y m ) = (2 π )1/2 h0 d exp  – m 2 d  2d  

[4.33]

In equilibrium, the force expressed in Equation [4.33] is balanced by an elastic restoring force fe, which tend to bring ym back to the unperturbed line position yL. Assume that this has the simple Hooke form: fe = k(yL – ym)

[4.34]

Therefore: k(yL – ym) = f (ym)

[4.35]

The equation can be solved graphically in Fig. 4.11. When the magnitude of the defect h0 is small, there is only one root ym for any specified yL, and no y d

ym yd Triple line

yL

yd

4.10 A triple line anchored in a defect.

X


151

f k (y m – y L )

f ( ym)

yL ym1

ym2

ym3

ym

4.11 Equilibrium positions of a triple line in the presence of a local defect.

hysteresis. If h0 reaches a certain threshold, there are three roots for a specified yL, and hysteresis occurs. This means that weak perturbation create no hysteresis. Accordingly, for a good determination of equilibrium contact angle, a surface with irregularities below a certain threshold would be enough if an ideal surface is not available. The above arguments can be further extended to a dilute system of defects. However, it only applies to defects with diffuse edges. The case of shaped edge defects, where the function h(x, y) has step discontinuities, is a completely different story. Hysteresis can happen for very small h0. An alternative approach to study the influence of surface roughness on the contact angle hysteresis is to examine the Wenzel’s roughness factor rW, defined as (Wenzel, 1936):

rW =

Areal A = real ≥ 1 Ageom bl

[4.36]

where Areal is the real area of the rough solid surface of width b and length l. And the measured contact angle, or Wenzel angle θW, is given by cos θW = rW cos θ

[4.37]

Introducing equation [4.37] into the Young’s equation [4.1]: rW(γSV – γSL) = γLV cos θW

[4.38]

An empirical ‘friction force’ F was used by good (1952) to explain the contact angle hysteresis: rW(γSV – γSL) = γLV cos θa + F

[4.39]

rW(γSV – γSL) = γLV cos θr – F

[4.40]

F reflects the influence of the surface roughness on the triple line. If F is assumed to be the same for both wetting and de-wetting processes, it is obtained that

152


2rW(γSV – γSL) = γLV (cos θa + cos θr)

[4.41]

Combining Equations [4.37], [4.38] and [4.41] gives an expression to derive the equilibrium contact angle from the advancing and receding angles: cos θ =

cos θ a + cos θ r 2r W

[4.42]

According to Equation [4.36], the Wenzel roughness factor rW can be determined by appropriate scanning force microscopy (SFM) (Kamusewitz and Possart, 2003) or atomic force microscopy (AFM) (Semal et al., 1999) measurement of the surface topography of the solid. In general, it is agreed that contact angle hysteresis increases steadily with the microroughness of the solid surface.

4.5

Summary

Surface tensions, wicking and/or wetting are among the most frequently encountered phenomena when processing and using fibrous materials. Wetting is a process of displacing a solid–air interface with a solid–liquid interface, while wicking is a result of spontaneous wetting in a capillary system. The physical bases of surface tension, wetting and wicking are those molecular interactions within a solid or liquid, or across the interface between liquid and solid. These adhesive forces include the Lifshitz–Van de Waals interactions and acid–base interactions. The Lifshitz–Van de Waals (LW) interactions are general, long-range forces which always operate in all materials and across phase boundaries. The Lewis acid–base (AB) interactions are polar, short-range interactions that occur only when an acid (electron acceptor) and a base (electron donor) are brought close together. Existence of acid– base interactions can substantially improve wetting and adhesion between two surfaces. Wetting and wicking behaviors are determined by surface tensions (of solid and liquid) and liquid/solid interfacial tensions. Curvature and roughness of contact surfaces are two critical factors for wetting phenomena, especially in the case of wetting in fibrous materials, which are porous media of intricate, tortuous and yet soft, rough structure. A liquid that fully wets a material in the form of a smooth planar surface may not wet the same material when presented as a smooth fiber surface, let alone a real fibrous structure. To reach the microscopic details of the liquid wetting behaviors in fibrous media, various computer simulation techniques have been applied in this field to accommodate more complexity so as to investigate more realistic systems, and to better understand and explain experiment results. On the other hand, the surface roughness is the most important source of contact angle hysteresis. In general, it is agreed that contact angle hysteresis increases steadily with microroughness of solid surface.


4.6

153

References

Adamson, A. W. and Gast, A. P. (1997) Physical Chemistry of Surfaces, New York, Wiley. Bauer, C., Bieker, T. and Dietrich, S. (2000) Wetting-induced effective interaction potential between spherical particles. Physical Review E, 62, 5324–5338. Bieker, T. and Dietrich, S. (1998) Wetting of curved surfaces. Physica A – Statistical Mechanics and Its Applications, 252, 85–137. Brochard-wyart, F., Dimeglio, J. M., Quere, D. and de Gennes, P. G. (1991) Spreading of nonvolatile liquids in a continuum picture. Langmuir, 7, 335–338. Brochard, F. (1986) Spreading of liquid-drops on thin cylinders – the Manchon–Droplet transition. Journal of Chemical Physics, 84, 4664–4672. Chehimi, M. M., Cabet-Deliry, E., Azioune, A. and Abel, M. L. (2002) Characterization of acid–base properties of polymers and other materials: relevance to adhesion science and technology. Macromolecular Symposia, 178, 169–181. Cox, R. G. (1983) The spreading of a liquid on a rough solid-surface. Journal of Fluid Mechanics, 131, 1–26. de Gennes, P.-G., Brochard-Wyart, F. and Quere, D. (2004) Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves New York, Springer. de Gennes, P. G. (1985) Wetting – statics and dynamics. Reviews of Modern Physics, 57, 827–863. Fowkes, F. M. (1962) Determination of interfacial tensions, contact angles, and dispersion forces in surfaces by assuming additivity of intermolecular interactions in surfaces. Journal of Physical Chemistry, 66, 382. Fowkes, F. M. and Zisman, W. A. (1964) Contact Angle, Wettability and Adhesion: The Kendall Award Symposium Honoring William A. Zisman, Washington, American Chemical Society. Garbassi, F., Morra, M. and Occhiello, E. (1998) Polymer Surfaces: From Physics to Technology, Chichester, England; New York, Wiley. Ghali, K., Jones, B. and Tracy, J. (1994) Experimental techniques for measuring parameters describing wetting and wicking in fabrics. Textile Research Journal, 64, 106–111. Good, R. J. (1952) A thermodynamic derivation of Wenzel’s modification of Young’s equation for contact angle, together with a theory of hysteresis. Journal of the American Chemistry Society 74, 5041–5042. Good, R. J. (1992) Contact angle, wetting, and adhesion – a critical review. Journal of Adhesion Science and Technology, 6, 1269–1302. Good, R. J. and Chaudhury, M. K. (1991) Theory of adhesive forces across interfaces: 1. The Lifshitz – van der Waals component of interaction and adhesion, in Lee, L. H. (Ed.) Fundamentals of Adhesion Plenum Press. Good, R. J., Chaudhury, M. K. and Van OSS, C. J. (1991) Theory of adhesive forces across interfaces: 2. Interfacial hydrogen bonds as acid–base phenomena and as factors enhancing adhesion, in Lee, L. H. (Ed.) Fundamentals of Adhesion, Plenum Press. Greiveldinger, M. and Shanahan, M. E. R. (1999) A critique of the mathematical coherence of acid base interfacial free energy theory. Journal of Colloid and Interface Science, 215, 170–178. Hamaker, H. C. (1937) The London van der Waals attraction between spherical particles. Physica, 4, 1058–1072. Hoffmann, K. H. and Schreiber, M. (1996) Computational Physics: Selected Methods, Simple Exercises, Serious Applications, Berlin; New York, Springer.

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Israelachvili, J. N. (1991) Intermolecular and Surface Forces, London; San Diego, CA, Academic Press. Joanny, J. F. and de Gennes, P. G. (1984) A model for contact-angle hysteresis. Journal of Chemical Physics, 81, 552–562. Kamusewitz, H. and Possart, W. (2003) Wetting and scanning force microscopy on rough polymer surfaces: Wenzel’s roughness factor and the thermodynamic contact angle. Applied Physics A – Materials Science and Processing, 76, 899–902. KISSA, E. (1996) Wetting and wicking. Textile Research Journal, 66, 660–668. Kumar, A. and Hartland, S. (1990) Measurement of contact angles from the shape of a drop on a vertical fiber. Journal of Colloid and Interface Science, 136, 455–469. Kwok, D. Y., Lee, Y. and Neumann, A. W. (1998) Evaluation of the Lifshitz–van der Waals acid–base approach to determine interfacial tensions. 2. Interfacial tensions of liquid–liquid systems. Langmuir, 14, 2548–2553. Kwok, D. Y., Li, D. and Neumann, A. W. (1994) Evaluation of the Lifshitz–van der Waals acid–base approach to determine interfacial tensions. Langmuir, 10, 1323–1328. Lee, L. H. (1993) Roles of molecular interactions in adhesion, adsorption, contact-angle and wettability. Journal of Adhesion Science and Technology, 7, 583–634. Leon, V. (2000) The mechanical view of surface tension is false. Journal of Dispersion Science and Technology, 21, 803–813. Lukas, D., Glazyrina, E. and Pan, N. (1997) Computer simulation of liquid wetting dynamics in fiber structures using the Ising model. Journal of the Textile Institute, 88, 149–161. Lukas, D. and Pan, N. (2003) Wetting of a fiber bundle in fibrous structures. Polymer Composites, 24, 314–322. Lukkarinen, A., Kaski, K. and Abraham, D. B. (1995) Mechanisms of fluid spreading – Ising model simulations. Physical Review E, 51, 2199–2202. Manna, S. S., Herrmann, H. J. and Landau, D. P. (1992) A stochastic method to determine the shape of a drop on a wall. Journal of Statistical Physics, 66, 1155–1163. Mao, N. and Russell, S. J. (2003) Anisotropic liquid absorption in homogeneous twodimensional nonwoven structures. Journal of Applied Physics, 94, 4135–4138. McHale, G., Kab, N. A., Newton, M. I. and Rowan, S. M. (1997) Wetting of a high-energy fiber surface. Journal of Colloid and Interface Science, 186, 453–461. McHale, G. and Newton, M. I. (2002) Global geometry and the equilibrium shapes of liquid drops on fibers. Colloids and Surfaces a-Physicochemical and Engineering Aspects, 206, 79–86. McHale, G., Newton, M. I. and Carroll, B. J. (2001) The shape and stability of small liquid drops on fibers. Oil and Gas Science and Technology – Revue de L’ Institut Français du Petrole, 56, 47–54. McHale, G., Rowan, S. M., Newton, M. I. and Kab, N. A. (1999) Estimation of contact angles on fibers. Journal of Adhesion Science and Technology, 13, 1457–1469. Mullins, B. J., Agranovski, I. E., Braddock, R. D. and Ho, C. M. (2004) Effect of fiber orientation on fiber wetting processes. Journal of Colloid and Interface Science, 269, 449–458. Neimark, A. V. (1999) Thermodynamic equilibrium and stability of liquid films and droplets on fibers. Journal of Adhesion Science and Technology, 13, 1137–1154. Oliver, J. F., Huh, C. and Mason, S. G. (1977) The apparent contact angle of liquids on finely-grooved solid surfaces – a SEM study. Journal of Adhesion, 8, 223–234 Quere, D. (1999) Fluid coating on a fiber. Annual Review of Fluid Mechanics, 31, 347– 384.


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Semal, S., Blake, T. D., Geskin, V., de Ruijter, M. J., Castelein, G. and de Coninck, J. (1999) Influence of surface roughness on wetting dynamics. Langmuir, 15, 8765– 8770. van OSS, C. J. (1993) Acid–base interfacial interactions in aqueous media. Colloids and Surfaces A – Physicochemical and Engineering Aspects, 78, 1–49. van OSS, C. J., Chaudhury, M. K. and Good, R. J. (1987a) Monopolar Surfaces. Advances in Colloid and Interface Science, 28, 35–64. van OSS, C. J., Giese, R. F. and Good, R. J. (1990) Reevaluation of the surface tension components and parameters of polyacetylene from contact angles of liquids. Langmuir, 6, 1711–1713. van OSS, C. J., Ju, L. K., Good, R. J. and Chaudhury, M. K. (1987b) Negative interfacial tensions between polar liquids and some polar surfaces 2. Liquid surfaces. Abstracts of Papers of the American Chemical Society, 193, 172–COLL. van Remoortere, P., Mertz, J. E., Scriven, L. E. and Davis, H. T. (1999) Wetting behavior of a Lennard–Jones system. Journal of Chemical Physics, 110, 2621–2628. Wennerstrom, H. (2003) The van der Waals interaction between colloidal particles and its molecular interpretation. Colloids and Surfaces A – Physicochemical and Engineering Aspects, 228, 189–195. Wenzel, R. N. (1936) Resistance of solid surface to wetting by water. Industrial and Engineering Chemistry 28, 988. WU, S. (1982) Polymer Interface and Adhesion, New York, Marcel Dekker. Yoshikawa, S., Ogawa, K., Minegishi, S., Eguchi, T., Nakatani, Y. and Tani, N. (1992) Experimental study of flow mechanics in a hollow-fiber membrane module for plasma separation. Journal of Chemical Engineering of Japan, 25, 515–521. Zhong, W., Ding, X. and Tang, Z. L. (2001a) Modeling and analyzing liquid wetting in fibrous assemblies. Textile Research Journal, 71, 762–766. Zhong, W., Ding, X. and Tang, Z. L. (2001b) Statistical modeling of liquid wetting in fibrous assemblies. Acta Physico-Chimica Sinica, 17, 682–686. Zhong, W., Ding, X. and Tang, Z. L. (2002) Analysis of fluid flow through fibrous structures. Textile Research Journal, 72, 751–755. Zisman, W. A. (1964) Contact angle, wettability and adhesion, in Fowkes, F. M. (Ed.) Advances in Chemistry Series. American Chemical Society, Washington, D. C.

5 Wetting phenomena in fibrous materials R . S . R E N G A S A M Y, Indian Institute of Technology, India

5.1

Introduction

Wetting of fibrous materials is important in a diverse range of applications in textile manufacture such as desizing, scouring, bleaching, dyeing and spinfinish application, cleaning, coating and composite manufacture. Clothing comfort also depends on wetting behavior of fibrous structure. In fibre composites, the adhesion between the fibers and resin is influenced by the initial wetting of the fibers by resin, which governs the resin penetration into the voids between the fibers and subsequently the performance of the composites. On the other hand, surgical fabrics should not let liquid and solid particles pass through easily. Wetting processes are considered extremely important in the application of fibrous filters, where wetting of the fibre surface is the key mechanism for the separation of two different liquids from their mixture; for instance, in separating oil from sea-water during a cleaning process after an oil spillage. Wetting and wicking behavior of the fibrous structures is a critical aspect of the performance of products such as sports clothes, hygiene disposable materials, and medical items. Wetting is a complex process complicated further by the structure of the fibrous assembly. Fibrous assemblies do not meet the criteria of ideal solids. Most practical surfaces are rough and heterogeneous to some extent. Fibers are no exception to this. In addition, curvature of fibers, crimps on fibers, and orientation and packing of fibers in fibrous materials make evaluation of wetting phenomena of fibrous assemblies more complicated.

5.2

Surface tension

A molecule on the surface of a liquid experiences an imbalance of forces due to the presence of free energy at the surface of the liquid which tends to keep the surface area of the liquid to a minimum and restrict the advancement of the liquid over the solid surface. This can be conceived as if the surface of a liquid has some kind of contractable skin. The surface energy is expressed 156


157

per unit area. Precise measurement of surface energy is not generally possible; the term surface tension refers to surface energy quantified as force per length (mN/m or dynes/cm). For a liquid to wet a solid completely or for the solid to be submerged in a liquid, the solid surfaces must have sufficient surface energy to overcome the free surface energy of the liquid. When a liquid drop is placed on an ideal flat solid surface (i.e. smooth, homogeneous, impermeable and non-deformable), the liquid drop comes to an equilibrium state corresponds to minimization of interfacial free energy of the system. The forces involved in the equilibrium are given by the wellknown Young’s equation:

γSV – γSL = γLV cos θ

[5.1]

The terms γSV, γSL, and γLV represent the interfacial tensions that exists between the solid and vapor, solid and liquid and liquid and vapor respectively. The last term is also commonly referred as the surface tension of the liquid. θ is the equilibrium contact angle. The term ‘γLV cos θ ’, is the ‘adhesion tension’ or ‘specific wettability’. Young’s equation has been widely used to explain wetting and wicking phenomena. Contact angle is the consequences of wetting, not the cause of it, and is determined by the net effect of three interfacial tensions. For a hydrophilic regime, γSV is larger than γSL and the contact angle θ lies between 0 and 90°, i.e. cos θ is positive. For a hydrophobic regime, γSV is smaller than γSL, and the contact angle lies between 90° and 180°. With increasing wettability, the contact angle decreases and cos θ increases. Complete wetting implies a zero contact angle, but equating θ = 0 may lead to incorrect conclusions and it is better to visualize that, when the contact angle approaches zero, wettability has its maximum limit.1 A lower contact angle for water wets the surface and at high contact angle water run off the surface. According to Adam,2 equilibrium condition cannot exist when the contact angle is zero, and Equation [5.1] does not apply. The equilibrium contact angle is the single valued intrinsic contact angle described by the Young equation for an ideal system. An experimentally observed contact angle is an apparent contact angle, measured on a macroscopic scale, for example, through a low-power microscope. On rough surfaces, the difference between the apparent and intrinsic contact angles can be considerable.3 Immersion, capillary sorption, adhesion, and spreading are the primary processes involved in wetting of fibrous materials. A solid–liquid interface replaces the solid–vapor interface during immersion and capillary penetration/ sorption. For spontaneous penetration, the work of penetration has to be positive. Work of adhesion, WA, is equal to the change of surface free energy of the system when the contacting liquid and the solid are separated: WA = γSV + γLV – γSL = γLV (1 + cos θ )

[5.2]

158


During spreading, the solid–liquid and liquid–vapor interfaces increase, whereas the solid–vapor interface decreases. For spreading to be spontaneous, the work of spreading or the spreading coefficient, WS, has to be positive, which is related as: WS = γSV – γLV – γSL

5.3

Curvature effect of surfaces

5.3.1

Wetting of planar surfaces

[5.3]

For a sufficiently small drop of a partial wetting or non-wetting liquid placed on a planar surface, gravity effects can be neglected. For such a drop, hydrostatic pressure inside the drop equilibrates and the drop adopts a shape to conform to the Laplace law:

∆P = γLV(1/R1 + 1/R2)

[5.4]

where ∆P is the pressure difference between two sides of a curved interface characterized by the principal radii of curvature R1 and R2. The drop shape would be spherical. For complete wetting of a flat surface, this pressure can be reduced towards zero by simultaneously increasing both R1 and R2 conserving the volume of the liquid.

5.3.2

Wetting of curved surfaces

A fluid that fully wets a material in the form of smooth planar surface may not wet the same material if it is presented as a smooth fiber form. On a flat surface, vanishing contact angle is a sufficient condition for the formation of a wetting film. On a chemically identical fiber surface, the indefinite spreading is inhibited and the equilibrium is not necessarily a thin sheathing film about the fiber, but can have a microscopic profile. This shows that vanishing contact angle is not a sufficient condition for the formation of a wetting film on a fiber. The Laplace excess pressure inside a liquid drop resting on a fiber is: 1 + 1 = ∆P R⊥ RII γ

[5.5]

The two radii of curvature R⊥ and RII of a drop, are measured normal to and along the fiber axis respectively. For a droplet on a fiber, the radii of curvature cannot both be increased while maintaining the volume of liquid. It is necessary to reduce one radius of curvature as the other is increased. Nevertheless, the excess pressure given by the Laplace law can still be reduced toward zero, although not to zero, by making RII negative. The other radius R⊥ cannot be reduced below the radius of curvature of the fiber; a minimization of the


159

excess pressure can be obtained while maintaining finite values for the radii of curvature.

5.3.3

Wetting of fiber surfaces

In the case of fiber, three distinct droplet configurations are observed, as shown in Fig. 5.1: a series of axisymmetrical ‘barrel’ shaped (unduloid) droplets around the fiber, commonly connected by a film in the order of a nanometer (Configuration I); axially asymmetric ‘clam-shell’ shaped droplets around the fiber (Configuration II), the flow usually being broken into distinct droplets by Rayleigh instability; and a sphere for a non-wetting liquid (Configuration III). The droplet-on-fiber system becomes a droplet-on-aplane-surface in the limiting case of an extremely large fiber radius (very low fiber curvature). It has been shown that barrel-shaped droplets, even under vertical fibers, becomes axially asymmetric under the influence of drag forces.4 On a fiber, the equilibrium shape of a barreling droplet is only approximately a spherical cap rotated about the axis of the fiber. Under certain conditions, the curvature goes through a point of inflexion as it approaches the solid surface at the three-phase interface, before then changing the sign of curvature as shown in Fig. 5.2. For a high-energy fiber, when the diameter of the fiber reduces, the inflection angle increases and the transition to the lower value of contact angle occurs very rapidly as the drop profile nears the fiber surface. This makes the measurement of contact angle difficult. An improved estimation of the equilibrium contact angle can be obtained by measuring the inflection angle, and the reduced length and thickness of the droplets. Transition or roll-up from one conformation to other can occur. It is reported that for large drops with contact angle < 90°, barrel shapes will be stable for any fiber radius.7 The parameters that influence the roll-up process have been investigated by Briscoe et al.5 Increasing the parameters of contact

Configuration III (Nonwetting droplets)

Configuration I (Barrel) Configuration II (Clamshell)

5.1 Droplets shapes on fiber. Reprinted from Colloids and Surfaces, Vol. 56, B. J. Briscoe, K. P. Galvin, P. F. Luckham, and A. M. Saeid, pp. 301–312, Copyright (1991), with permission from Elsevier.

160

Thermal and moisture transport in fibrous materials x A

X1

θ

B

θ1 Fiber

0

z

Liquid

X2

L

5.2 Geometrical parameters for the description of a drop on a single fiber. X1 is the fiber radius; X2, the maximum drop height; θ, the contact angle; θ1 the inflection angle; and L the drop length ‘Reprinted from International Journal of Adhesion and Adhesives, Vol. 19, S. Rebouillat, B. Letellier, and B. Steffenino, pp. 303–314, Copyright (1999), with permission from Elsevier’.

angle, surface tension of liquid and diameter of fibers, or reducing the volume of the droplets favors change of confirmation of droplets from Configuration I to III. Local surface anomalies due to chemical or physical heterogeneity can lead to two completely different droplet profiles on the same fiber.

5.3.4

Role of droplet shapes in wet fiber filtration

The formation of droplets of different shapes has a significant role in influencing the efficiency of wet-fiber filters in removing sticky and viscous particles. During wet filtration of solid or liquid aerosols, droplets attached to the fibers are observed to rotate under the influence of induced airflow. Barrelshaped droplets, being smaller in size, rotate as a rigid body and the droplets laden with particles frequently flow down the fiber under gravity. The larger droplets, i.e. clamshells, have significant capacity to contain particulates, but rotate like less rigid bodies and can flow-off the fiber rather than flowing down with entrained particles. This is not advantageous in self-cleaning as it is likely to lead to re-entrainment of the particles back into the air stream.8

5.4

Capillarity

Transport of a liquid into a fibrous assembly may be caused by external forces or by capillary forces only. In most of the wet processing of fibrous materials, uniform spreading and penetration of liquids into pores are essential for the better performance of resulting products.9 Capillarity falls under the general framework of thermodynamics that deals with the macroscopic and statistical behavior of interfaces rather than with the details of their molecular structure.10 The interfaces are in the range of a few molecular diameters.


161

Wicking is one example of the more general set of phenomena termed ‘capillarity’. For wicking to be significant, the ratio of solid–liquid (SL) interfacial area to liquid volume must be large. Wicking can only occur when a liquid wets fibers assembled with capillary spaces between them. The resulting capillary forces drive the liquid into the capillary spaces, increasing the solid–liquid interface and decreasing the solid–air interface. For the process to be spontaneous, free energy has to be gained and the work of penetration has to be positive, i.e. γSV must exceed γSL .

5.4.1

Capillary flow

When a liquid in a capillary wets the walls of the capillary, a meniscus is formed. The pressure difference ∆P across the curved liquid–vapor interface driving the liquid in a small circular capillary of radius r, is related as:

∆P = 2γLV cos θ /r

[5.6]

For a positive capillary pressure, the values of θ have to be between 0° and 90°. Accordingly, the smaller the pore size, the greater is the pressure within the capillary, and so the smallest fill first. During draining of the capillary under external pressure, the smaller pores drain last. For most systems, wicking does not occur when the contact angle is between 90 to 180°. According to Marmur,11 partial penetration of the capillary can occur even if the contact angle is 90°, provided the pressure within the bulk of the liquid is substantial enough to force the liquid into the capillary. This occurs only when the liquid reservoir is small, i.e. a drop of liquid. In a drop of liquid, the radius of curvature of the drop can be high enough such that the pressure directly outside of the capillary is increased, and thus the pressure difference, leading to penetration of liquid into the capillary. The flow in a porous medium is considered as flow through a network of interconnected capillaries. The Lucas–Washburn equation12 is widely used to describe this flow,

γ LV r cos θ – r 2 ρL g /8 η [5.7] 4ηh The first term on the right side of the equation accounts for the spontaneous uptake of liquid into the material while the second term accounts for the gravitational resistance. The second term in the above equation is negligible if either the flow is horizontal or r is very small (r 2 = 0). The term h is the distance that the liquid has traveled at time t; and ρL and η are the density and viscosity of the liquid, respectively. When the capillary forces are balanced by the gravitational forces, liquid rise stops and equilibrium is reached as given by: dh / dt =

γLV cos θ 2π r = π r 2 ρL gh

[5.8]

162


Hence, equilibrium wicking height is: heq = 2γLV cos θ /rρLg

[5.9]

The linear rate of liquid flow (u) is: u = dh/dt = rγLV cos θ /4η h

5.4.2

[5.10]

Wicking in fibrous materials

In the case of capillarity in fibrous assemblies, the term ‘wicking’ is used in a broader practical sense to describe two kinetically different processes: a spontaneous flow of a liquid within the capillary spaces accompanied by a simultaneous diffusion of the liquid into the interior of the fibers or a film on the fibers.13 If the penetration of liquid is limited to the capillary spaces and the fibers do not imbibe the liquid, the wicking process is termed ‘capillary penetration’ or ‘capillary sorption’. Swelling of the fibers caused by the sorption of the liquid into the fibers can reduce capillary spaces between fibers and change the kinetics of wicking. The interpretation of wetting results can be misleading if the effects of sorption in fibers or finishes on fibers are overlooked.14 For a theoretical treatment of capillary flow in fabrics, the fibrous assemblies are usually considered to have a number of parallel capillaries. The advancement of the liquid front in a capillary can be visualized as occurring in small jumps. The fibrous assembly is a non-homogeneous capillary system due to irregular capillary spaces having various dimensions and discontinuities of the capillaries leading to small jumps in the wetting front. The capillary spaces in yarns and fabrics are not uniform, and an indirectly determined effective capillary radius has to be used instead of the radius r.15 Fibers in textile assemblies form capillaries of effective radius re so the horizontal liquid transport rate becomes:

h2 =

γ LV cos θ a′ re t = ks t 2η

[5.11]

where ks is the capillary liquid transport constant for the penetration of a liquid into a definite fiber assembly.16 Equation [5.11] applies only to a system where the free surface of the liquid reservoir feeding the capillary tube is substantially flat, i.e. the capillary pressure on the reservoir surface is zero.17 According to Lucas–Washburn, neglecting gravitational forces, the wicking height h is directly proportional to the square root of time t :18 h = (r γ LV cos θ /2τ 2 η)1/2t 1/2 = k t1/2 where hτ is the actual distance traveled, τ is the tortuosity factor.

[5.12]


163

The Lucas–Washburn equation is used primarily to describe flow into vertically hung materials and it has been shown to be a good estimate of the flow rate within many textile materials. According to the Lucas–Washburn equation, the liquid uptake into the material is in direct correlation with the product of γLV and cos θ. If the contact angle is very large, use of surfactants will improve liquid uptake; on the other hand, if the initial contact angle before addition of surfactant is very low, adding surfactant only reduces the value γLV to a greater extent than it increases the cos θ value. As a result, the product γLV cos θ reduces, lowering the wicking rate. The Lucas–Washburn equation has been extended for the case of radial expansion of a wicking liquid originating at the centre of a flat sample, relating liquid mass uptake mA, the distance traveled by the liquid L and a constant K as19: dmA/dt = (K/mA) – L

[5.13]

Most textile processes are time limited, and often the rate of wicking is therefore very critical. However, the wicking rate is not solely governed by interfacial tension and the wettability of the fibers, but by other factors as well. The mechanisms of water transport for an isolated single fiber differs from water sorption in a fiber bundle or assembled fibers where capillary spaces exist.20 Ito and Muraoka21 have reported that water transport is suppressed as the number of fibers in the yarn decreases. When the number of fibers is greater, water moves along even untwisted fibers. But when the number of fibers is reduced, wicking occurs only for twisted fibers and, if reduced further, wicking may not occur at all. This indicates that sufficient number and continuity of pores are important for wicking.

5.4.3

Wicking in yarns

A yarn may be assumed to have oriented cylindrical fibers. Lord22 has discussed a theory for yarn wicking. Using hydraulic radius theory for an assembly of parallel cylindrical fibers, the value of the hydraulic mean radius rm is: rm =

Af rf = ( K p Kc ) 2π rf n 2

[5.14]

A correction factor Kc is applied in the above equation for cases when the fibers are not cylindrical or inclined to the axis of the yarn. In the above equation: Af is area of fluid between fibers of a yarn, Kp is packing factor, rf = radius of fiber, and n = number of fibers in the yarn cross-section. Equivalent wicking height is given by: h′ =

2γ LV cos θ ρL gK c K p r f

[5.15]

164


For a given fiber and liquid, 2γLV cos θ /ρL grf is constant, and hence, hα (KcKp)–1 Both Kc and Kp are functions of twist multiple, fiber type, and packing and migration of fibers in the yarn, which are related to yarns produced by different technologies. The presence of smaller pores at the core of the open-end yarn wicks dye solution to a greater height. The wicking rate and equilibrium height observed for ring yarn is higher than that of compact yarns. This indicates that the number of pores, pore size and continuity are important factors in yarn wicking.23 The orientation of fibers in a yarn influences wicking. In air-jet textured yarns, the presence of long, drawn-out loops such as floats and arcs offers a less tortuous path for the liquid to travel; as a result, a greater percentage of floats and arcs leads to a higher wicking height. The equilibrium wicking height and wicking rate are higher for air-jet textured yarn than for the corresponding feeder yarn. Equilibrium wicking height initially increases and then decreases with increasing tension on the yarns during wicking. The initial increase in height is due to partial alignment of the filaments; further increase in tension may bring the filaments closer to each other, reducing the capillary radii and possibly discontinuity in the capillaries.24 The packing density of the filaments influences more greatly the wicking in crenulated viscose filaments than in circular nylon filaments. Viscose filaments under loose condition show abnormally high wicking; when the packing of filaments increases, the crenulations mesh like gear teeth, the open space reduces greatly without any corresponding reduction in the yarn diameter, and thus the wicking rate diminishes.17

5.4.4

Wicking in fabrics

When a liquid drop is placed on a fabric, it will spread under capillary forces. The spreading process may be split conveniently into two phases: I liquid remains on the surface, and II liquid is completely contained within the substrate, as suggested by Gillespie.25 For two-dimensional circular spreading in textiles during phase II, Kissa26 developed Gillespie’s equation to propose the following exponential sorption: A = K(γLV /η) u V m t n

[5.16]

where A is the area covered by the spreading liquid, K is the capillary sorption coefficient, η is the viscosity of the liquid, V is the volume of the liquid, t is the spreading time. Wicking occurs when a fabric is completely or partially immersed in a liquid or in contact with a limited amount of liquid, such as a drop placed on the fabric. Capillary penetration of a liquid can therefore occur from an infinite (unlimited) or limited (finite) reservoir. Wicking processes from an


165

infinite reservoir are immersion, transplanar wicking, and longitudinal wicking. Wicking from a limited reservoir is exemplified by a drop placed onto the fabric surface.

5.4.5

Porosity of fabrics and spreading of liquids

The porosity φ of material is defined as the fraction of void space within the material.27

φ = 1 – (ρF /ρf)

[5.17]

where ρf is the density of the fiber and ρF is the density of the fabric; the latter is the ratio of fabric weight to thickness. The maximum liquid absorption capacity Cm is: Cm = [( ρl φ )/ρf (1 – φ)]

[5.18]

where, ρl is the liquid density. The pores within the structure are responsible for the liquid flow through a material and the size and connectivity of the pores in the fabric influence how fast and how much liquid is transported through the material. Hsieh et al., 28,29 reported that, in the case of woven, non-woven and knitted fabrics, a distribution of pore sizes along any planar direction is expected. Hsieh27 has also shown that with poor wetting, many pores in fabrics are not filled by water due the effect of reduced cos θ in driving the water into the pores, e.g. with polyester fabric. When liquid moves into a fiber assembly, the smaller pores are completely filled and the liquid then moves to the larger pores. The sizes and shapes of fibers as well as their alignment will influence the geometric configurations and topology of the pores, which are channels with widely varying shape and size distribution and may or may not be interconnected.29–31 The shape of fibers in an assembly affects the size and geometry of the capillary spaces between fibers and consequently the wicking rates. The flow in capillary spaces may stop when geometric irregularities allow the meniscus to reach an edge and flatten.15 The distance of liquid advancement is greater in a smaller pore because of the higher capillary pressure, but the mass of liquid retained in such a pore is small. A larger amount of liquid mass can be retained in larger pores but the distance of liquid advancement is limited. Therefore, fast liquid spreading in fibrous materials is facilitated by small, uniformly distributed and interconnected pores, whereas high liquid retention can be achieved by having a greater number of large pores or a high total pore volume.27 Wicking is affected by the morphology of the fiber surface, and may be affected by the shape of the fibers as well. Fiber shape does not affect the wetting of single fibers. However, the shape of the fibers in a yarn and fabric

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affects the size and geometry of the capillary spaces between the fibers, and consequently the rate of wicking.3 Randomness of the arrangement of the fibers in the yarns considerably influences the amount of water and transport rate of the fabrics. The same factor also seems to control the ease of wetting of the surface of fabrics. Non-woven fabrics are highly anisotropic in terms of fiber orientation, which depends greatly on the way in which the fibers are laid (random, cross-laid and parallel-laid) during web formation and any further processing. The in-plane liquid distribution is important in spreading the liquid over a large area of the fabric for faster evaporation of perspiration in clothing or maximum liquid drawing capacity of the secondary layer of baby diapers. Classical capillary theory, based on equivalent capillary tubes applied for yarns and woven fabrics, is inadequate to study the liquid absorption in nonwovens.32 The former structures are compact with a porosity in the range of 0.6–0.8 and have better defined fiber alignment, whereas non-wovens have porosity generally above 0.8 and as high as 0.99 in some high-loft structures. Further, wicking in woven fabrics is mainly concerned with liquid movement in between the fibers in the yarn33 and the larger pores that exist between the yarns are therefore less important34. The structure of non-wovens is markedly different from the traditional structures in that they have larger spaces between fibers, and high variation of size, shape and length of capillary channels.32 Orientation of fibers in non-wovens is found to influence the in-plane liquid transportation in different directions. To characterize the capillary pressure during liquid transportation in nonwoven fabrics, instead of using the pore size, an alternative theory was developed by Mao and Russell35,36 based on hydraulic radius theories proposed by Kozeny37 and Carman.38 In hydraulic radius theories the channels usually have a non-circular shape and the hydraulic radius is defined by the surface area of the porous medium. Mao and Russell employed Darcy’s law39 to quantify the rate of liquid absorption in non-woven fabrics. Based on Darcy’s law, they related specific or directional permeability of sample k(θ) in m2 in the direction θ from reference and angle of fiber with respect to reference α as:

  k (θ ) = – 1 δ  32 φ    2

∫

π

0

  ST   2 2 Ω (α ){T cos (θ – α ) + S sin (θ – α )} dα   [5.19]

where δ is the fiber diameter, φ is the volume fraction of solid material, Ω is the fiber orientation distribution probability function that defines the arrangement of fibers within the fabric. S and T are functions in terms of φ.


167

By assuming that the capillary pressure in the fabric plane is hydraulically equivalent to a capillary tube assembly in which there are a number of cylindrical capillary tubes of the same hydraulic diameter, the equivalent hydraulic diameter DH (θ) was formulated. Using the equivalent hydraulic diameter DH (θ) in the Laplace equation, the capillary pressure in the direction θ in the fabric was calculated. For a given contact angle β, wicking rate V (θ) was shown as:   V (θ ) = – 1 δ  32 φ    2

4φ ×

∫

∫

π

0

π

0

  ST   2 2 Ω (α ){T cos (θ – α ) + S sin (θ – α )} dα  

Ω (α ) |cos (θ – α )| dα

δ (1 – φ )

γ LV cos β 1 ηL

[5.20]

Fiber diameter, fiber orientation distribution and fabric porosity are the important structural parameters that influence the spreading rate of liquid in non-wovens. The anisotropy of liquid absorption in non-woven fabric largely depends on a combination of the fiber orientation distribution and the fabric porosity. Konopka and Pourdeyhimi40 carried out experiments on non-woven fabrics to study in-plane liquid distribution using a modified GATS apparatus and found that fiber orientation factor is the dominant factor in determining where the liquid will spread in the material. Kim and Pourdeyhimi41 simulated in-plane liquid distribution in non-wovens using the above equation and found reasonable agreement between the simulated and experimentally observed results. Fiber orientation factor influences the rate of spreading in different directions as well as the mass of liquid transported in the dynamic state. The spreading of liquid in a thermally bonded non-woven is more elliptical than that in the woven, which is closer to isotropic.

5.5

Surface roughness of solids

The wetting of surfaces involves both chemistry and geometry. Geometry can be either local, in the form of rough or patterned surfaces, or it can be global, in the form of spheres, cylinders/fibers, etc. Amplification of hydrophobicity due to surface roughness is frequently seen in nature. Water droplets are almost spherical on some plant leaves and can easily roll off (lotus effect or super hydrophobic effect), cleaning the surface in the process. There are many applications of artificially prepared ‘self-cleaning’ surfaces. A drop placed on a rough surface can sit either on the peaks or wet the

168


grooves, depending on how it is formed, determined by the geometry of the surface roughness. One that sits on the peaks will have a larger contact angle with higher energy. It has ‘air pockets’ along its contact with the substrate; hence it is termed a ‘composite contact’. It is this type of surface that is desirable in applications such as ‘self-cleaning’ surfaces. Wenzel42 studied the wetting behavior of a rough substrate. The apparent contact angle of a rough surface θ * depends on the intrinsic contact angle (Young’s contact angle) θ, and the roughness ratio, r (called ‘Wenzel’s roughness ratio); the latter is the ratio of rough to planar surface areas. cos θ * = r cos θ

[5.21]

The underlying assumption of the above relationship is that hydrophilic surfaces that wet (θ < 90°) if smooth will wet even better if rough. According to this relationship, if roughness is increased, the apparent contact angle will decrease. This much-quoted equation immediately suggests that: if

θs < π /2 then θro < θs; but if θs > π /2 then θro > θs

θs is the contact angle for a smooth or ideal surface and θro the contact angle for a rough surface.

5.5.1

Heterogeneity of surfaces

In the case of chemically heterogeneous smooth surface consisting of two kinds of small patches, occupying fractions f1 and f2 of the surfaces, then the apparent contact angle is:10

γLV cos θ * = f1(γS1V – γS1L) + f2(γS2V – γS2L)

[5.22]

Alternatively, cos θ * = f1 cos θ1 + f2 cos θ2

[5.23]

In the case of microscopically heterogeneous surfaces, forces rather than surface tensions are averaged,10 hence: (1 + cos θ *)2 = f1(1 + cos θ1)2 + f2(1 + cos θ2)2

[5.24]

In the case of a rough surface or a composite surface, such as a fabric, incompletely wetted by a liquid, if f w is the area fraction of substrate that is wetted and fu is the fraction of unwetted (open area of fabric) surface (i.e. 1 – fw), then in Equation [5.22] γS2V is zero (due to air entrapment) and γS2L is simply γLV; wettability of such surfaces is then expressed by Equation [5.37]43: cos θ * = fw cos θ – fu

[5.25]

If the contact angle is large and the surface is sufficiently rough, the liquid


169

may trap air so as to give a composite surface with the relation as given by Cassie:44 cos θ * = rfw cos θ – fu Alternatively, Cassie and Baxter

[5.26] 45

have shown that:

cos θ* = φs (1 + cos θ) – 1

[5.27]

where φs is the surface fraction and, 1 – φs is the air fraction. θ * > θ unless the roughness factor is relatively large. Several workers have found that the apparent contact angle for water drops on paraffin metal screens, textile fabrics, and embossed polymer surfaces does vary with fu in approximately the same manner predicted by Equation [5.25]. The Wenzel and Cassie states for a drop on a hydrophobic textured surface are shown in Fig. 5.3. Shuttleworth and Bailey47 have pointed out that a rough surface causes the contact line to distort locally, which give rise to a spectrum of microcontact angles near the solid surface. Consequently, θ*, will be less than or greater than θ according to the expression:

θ* = θ ± α

[5.28]

where α is the maximum angle (±) of the local surface, representing the roughness. In contrast to Wenzel’s relationship, the above equation predicts that the apparent contact angle will increase as roughness increases. This discrepancy in the predicted effects of roughness on wetting has been investigated experimentally by Hitchcock et al. They approximated Wenzel’s roughness ratio and α as: r = 1 + c1(R/λ)2 and α = tan–1 (c2R/λ)

[5.29]

where c1 and c2 are constants, and R and λ are RMS surface height and average distance between surface asperities, respectively. For several liquids and a variety of solid substrates, they found agreement with the predictions of Shuttleworth and Bailey47 in that wetting decreased

θ*

θ*

(a)

(b)

5.3 Two possible states for a drop on a hydrophobic textured surface: (a) Wenzel state; and (b) Cassie’s state ‘Reprinted from Microelectronic Engineering, Vol. 78–79, M. Callies, Y. Chen, F. Marty, A. Pépin, and D Quéré, pp. 100–105, Copyright (2005), with permission from Elsevier’.

170


with increased roughness ratio with the exception of a few examples (improved wetting with increased roughness, i.e. Wenzel’s behavior). However, Johnson and Dettre49 and Nicholas and Crispin50 working on ‘very well-wetting systems’ found Wenzel’s behavior.

5.5.2

Global geometry of surfaces

Nakae et al.51 studied water wetting a paraffin surface made of hemispherical and hemi round-rod close-packed solids. The Wenzel’s roughness factors were 1.6 and 1.9 for these surfaces, respectively, and were found to be independent of the radii of the spheres and cylinders. When the height roughness of the hemi-cylindrical surfaces was increased, the contact angle increased initially and then decreased when the roughness was increased beyond 50 µm.

5.5.3

Chemically textured surfaces

Shibuichi et al.52 carried out experiments on the effect of chemical texturing of a surface on contact angle as a function of wettability of the solid. They plotted the measured cos θ * as a function of cos θ determined on a flat surface of the same material and varied using different liquids. Their results are shown in Fig. 5.4. As soon as the substrate becomes hydrophobic (θ > 90°), cos θ * sharply decreases, corresponding to a jump of contact angle θ * to a value of the order of 160°. On the hydrophilic side, the behavior is quite different: in a first regime, cos θ* increases linearly with cos θ, with a slope larger than 1,

cos θ*

1

0

–1 –1

0 cos θ

1

5.4 Experimental results of the Kao group (from Shibuichi et al. [52]. The cosine of the effective contact angle θ* of a water drop is measured as a function of the cosine of Young’s angle θ (determined on a flat surface of the same material and varied using different liquids). ‘Reprinted from Colloids Surfaces A: Physicochemical and Engineering Aspects, Vol. 206, J. Bico, U. Thiele, and D. Quere, pp. 41–46, Copyright (2002), with permission from Elsevier’.


171

indicating improved wetting with a rough surfaces in agreement with Wenzel’s relation. In a second regime (small contact angles), cos θ* again increases linearly with cos θ, with a much smaller slope. Complete wetting of rough surfaces (θ* = 0°) is only reached if the substrate itself becomes wettable (θ = 0°). These successive behaviors have been modeled and explained by Bico et al.53 In the super-hydrophobic regimes, when a liquid is deposited on a model surface, air is trapped below the liquid, inducing a composite interface between the solid and liquid as Cassie’s state. The condition for stability for this state is: cos θ < ( f – 1)/(r – f )

[5.30]

where f is the fraction of the solid–liquid interface below the drop (dry surface). For a very rough surface, r is very large, and cos θ < 0° expresses the usual condition for hydrophobicity. For a Young’s contact angle θ between 90° and the threshold value given by the Equation [5.30], air pockets should be metastable. For hydrophilic solids, the solid–liquid interface is likely to follow the roughness of the solid as γSV > γSL, which leads to a Wenzel contact angle as in Equation [5.21]. As r > 1 and θ < 90°, Equation [5.21] implies θ* < θ: the surface roughness makes the solid more wettable. The linear relation found in Equation [5.21] is in good agreement with the first part of the hydrophilic side.

5.5.4

Roughness and surface-wicking

A textured solid can be considered as a 2D porous material in which the liquid can be absorbed by hemi-wicking (surface wicking), which is intermediate between spreading and imbibitions (0° < θ < 90°). When the contact angle is smaller than a critical value θcr, a film propagates from a deposited drop, a small amount of liquid is sucked into the texture, and the remaining drop sits on a patchwork of solid and liquid – a case very similar to the super-hydrophobic one, except that here the vapor phase below the drop is replaced by the liquid phase. In a partial wetting, as shown in Fig. 5.5, the top of the spikes remain dry as the imbibition front progresses. If f is the solid fraction in dry state, then θ < θcr with: cos θcr = (1 – f )/(r – f )

[5.31]

For a flat surface, r = 1 and θcr = 0, indicating spreading at vanishing of the contact angle. For a rough surface, r > 1 and f < 1, so that condition in the above equation defines the critical contact angle θcr in between 0° and 90°. The nature of the texture determined by r and f decides if condition in Equation [5.31] is satisfied or not. If the surface composition is such that 90° > θ > θcr, the solid remains dry beyond the drop, and Wenzel’s relation

172

Thermal and moisture transport in fibrous materials Front

dx

Air Liquid Solid

5.5 Liquid film invading the texture of a solid decorated with spikes (or micro channels). The front is marked with an arrow. In the case of partial wetting, the tops of the spikes remain dry. ‘Reprinted from Colloids Surfaces A: Physicochemical and Engineering Aspects, Vol. 206, J. Bico, U. Thiele, and D. Quere, pp. 41–46, Copyright (2002), with permission from Elsevier’. θ*

5.6 A film invades solid texture; a drop lies on a solid/liquid composite surface. The apparent contact angle θ* lies between 0° and θ (contact angle on a flat homogeneous solid). ‘Reprinted from Colloids Surfaces A: Physicochemical and Engineering Aspects, Vol. 206, J. Bico, U. Thiele, and D. Quere, pp. 41–46, Copyright (2002), with permission from Elsevier’.

applies. If the contact angle is smaller than θcr, a film develops in the texture and the drop sits upon a mixture of solid and liquid, as shown in Fig. 5.6. For the hemi-wicking case: cos θ* = f (cos θ – 1) + 1

[5.32]

This shows that the film beyond the drop has improved wetting (θ* < θ), but it does so less efficiently than with the Wenzel scenario. The angle deduced from Equation [5.32] is significantly larger than the one derived from Eq [5.21]. When the film advances, it smooths out the roughness, thus preventing the Wenzel effect from taking place. Roughness of a surface can influence wicking on that surfaces. It is very common that fibrous materials encounter roughness on surfaces and walls of pores. The driving force for such surface wicking depends on the geometry of the grooves, the surface tension of the liquid, and the free energies of the solid–gas and solid–liquid interfaces.54

5.5.5

Hemi-wicking in fabrics

In a fabric, the distance between the most advanced and less advanced liquid front gets larger with time in most imbibition processes. In fabrics, the distances between the yarns are larger than the ones between the fibers. The liquid in between fibers propagates much faster than that between the yarns.


173

Fabric as a porous material can be modeled as a tube decorated with spikes, as shown in Fig. 5.7. The observed phenomena in fabrics can be explained based on this model. Considering the length scales being much smaller than the capillary rise, the wetting liquid should invade both the tube itself (between yarns) and the texture (between fibers) if the condition of Equation [5.31] is satisfied. The texture acts as a reservoir for the film and hence the film propagates faster along the decorations than in the tube. Different capillary rises are likely to take place in such a tube. The film in between the fibers propagates faster than the main meniscus, which leads to a broadening of the front as times goes on. The main meniscus moves along a composite surface and the apparent contact angle for it is given by Equation [5.32]. The dynamics of the rise of the main meniscus are influenced by this contact angle. As the texture affects the value of the apparent contact angle, the value deduced from the dynamics of the rise can be different, and sometimes anomalously lower, than the one measured on the flat surface of the same material. Pezron et al. 55 performed experiments on wicking in cotton woven fabrics to see the relationship between the mass of the liquid absorbed and square root of time, to test the validity of the Lucas–Washburn equation. The graph for m vs. t1/2 displayed a non-linear relationship. The m vs. t1/2 could be represented by two straight lines; one that wicks the liquid inside the fabric structure and the other, surface wicking due to alveoli which could not absorb liquid to a great height because of their large capillary size. When the fabric surface was coated with a gel to eliminate the alveoli, the m vs. t1/2 displayed a linear relationship.

5.7 Tube decorated with spikes, as an example of the modeling of a porous material ‘Reprinted from Colloids Surfaces A: Physicochemical and Engineering Aspects, Vol. 206, J. Bico, U. Thiele, and D. Quere, pp. 41–46, Copyright (2002), with permission from Elsevier’.

174


Liquid spreading rate in a non-woven is influenced by surface wicking during the in-plane wicking test using GATS (Grammetric Absorbency Test System), when plates are placed below, or on top of, or at both faces of, the fabric. The added capillaries increase the wicking rate due to surface wicking. The shape of the liquid spreading in a non-woven is not affected by the extra capillaries when material distribution is uniform throughout the non-woven fabric. However, non-uniform distribution of material influences the shape of the liquid spread.40

5.5.6

Roughness anisotropy and grooves

If the roughness geometry is isotropic, then the drop shape is almost spherical and the apparent contact angle of the drop is nearly uniform along the contact line. If the roughness geometry is anisotropic, e.g. parallel grooves, then the apparent contact angle and the shape of the drop is no longer uniform along the contact line56. For the case of a composite contact of a hydrophobic drop, the apparent contact angle in a plane normal to the grooves is larger than the one along the grooves. This is a consequence of the squeezing and pinning of the drop in the former and the stretching of the drop in the latter planes, respectively. Both these apparent contact angles are usually larger than the intrinsic values of the substrate material (i.e. the one for the smooth surface). Wenzel and Cassie’s equations are insufficient to understand this anisotropy in the wetting of rough surfaces. Yost et al.57 demonstrated that, in extensive wetting, the arc length of wetting has a fractal character which is shown to arise from rapid flow into groove-like channels in the rough surface. This behavior is due to the additional driving force for wetting exerted by channel capillaries, resulting in flow into and along the valleys of the nodular structure. Several workers have shown that continuous paths of internodular grooves having α > θ would explain the profuse wetting on rough surfaces. It has been shown that rough substrates having α < θ do not show Wenzel behavior. Flow in a straight V-shaped groove has been modeled. When the straight walls of the groove are oriented at an angle of α to the surface and the liquid fills the groove to a depth y, the curvature of the liquid surface (1/R) becomes: 1/R = sin (α – (θ) tan α /y

[5.33]

This shows that flow into the groove can only occur if α > θ. This clearly emphasizes that fluid is drawn only into grooves satisfying this inequality and provides an alternative path to its derivation originally provided by Shuttleworth and Bailey. Further, it was shown that the area of spreading of the liquid, A(t) is related as: A(t) = βDt

[5.34]


175

where β is a proportionality coefficient including a tortuosity factor; the diffusion coefficient D is found to increase with α. This lends support to the notion that extensive wetting and spreading is driven by capillary flow into the valleys of rough surfaces.

5.5.7

Roughness of fibrous materials

Fabrics constructed from hydrophobic microfilament yarns have higher contact angles than others. Aseptic fabrics (sterilized) have mostly higher contact angles than non-aseptic fabrics. Rough surfaces may facilitate fast spreading of liquid along troughs offered by the surface roughness. Alkaline hydrolysis causes pitting of the surface of polyester fibers and improves their wettability, as indicated by contact angle measurements.58 The enhanced wettability is due to an increase in either the number or the accessibility of polymer hydrophilic groups to water and/or an increase in the roughness of the sample surfaces. Hollies et al.,16 reported that differences in yarn surface roughness give rise to differences in wicking of yarns and fabrics made from the yarns. Increase in yarn roughness due to random arrangement of its fibers gives rise to a decrease in the rate of water transport, and this is seen to depend on two factors directly related to water transfer by a capillary process: (i) the effective advancing contact angle of water on the yarn is increased as yarn roughness is increased; (ii) the continuity of capillaries formed by the fibers of the yarn is seen to decrease as the fiber arrangement becomes more random. The measurement of water transport rates in yarns is thus seen to be a sensitive measure of fiber arrangement and yarn roughness in textiles assemblies.16 Plasma-treated polypropylene melt-blown webs develop surface roughness as a result of chemical reactions and micro etchings on fiber surfaces. However, it has been pointed out that the improved water wettability after plasma treatment is due to the increased polarity of the surface; surface roughness is not a primary reason for improved wettability, but may increase it.59

5.5.8

Wetting of textured fabrics

The natural hydrophobicity of surfaces can be enhanced by creating texture on them, especially if the surfaces are microtextured. Surfaces that are rough on a nanoscale tend to be more hydrophobic than smooth surfaces because of the extremely reduced contact area between the liquid and solid, analogous to so-called ‘lotus-effect’ (repellency of lotus leaves).60 This gives a selfcleaning effect to surgical fabrics, i.e. particles adhering to the fabric surface are captured by rolling water due to the very small interfacial area between the particle and the rough fabric surface.61 Super hydrophobic surfaces can be created using a nanofiber web made from hydrophobic materials. In this

176


kind of structure, the apparent contact angle θ * will be very high since the fraction of the surface in contact with the liquid φs may be very low, coupled with a high intrinsic contact angle θ as evident from Equation [5.27]; a drop placed on them easily rolls-off without wetting the surface and subsequently hindering wicking in the material. Electrospun nanofibrous webs have potential application as barriers to liquid penetration in protective clothing systems for agricultural workers. Research work is in progress to create microporous web made from nanofibers such as cellulose acetate and polypropylene laminated with conventional fabric for this application.62 It is envisaged that the microporous web with small pore sizes will prevent liquid penetration, and the laminate will provide a selective membrane system that prevents penetration of pesticide challenged liquids while allowing the release of moisture vapor to provide thermal comfort.

5.6

Hysteresis effects

For an ideal surface wet by a pure liquid, the contact angle theory predicts only one thermodynamically stable contact angle. For many solid–liquid interactions, there is no unique contact angle and an interval of contact angles is observed. The largest contact angle is called ‘advancing’ and the smallest contact angle is called ‘receding’. The work of adhesion during receding is larger. Liquid droplets placed on a surface may produce an advancing angle if the drop is placed gently enough on the surface, or a receding angle if the deposition energy forces the drop to spread further than it would in the advancing case. Hysteresis occurs due to a wide range of metastable states as the liquid meniscus scans the surface of a solid at the solid–liquid–vapor interface. The true equilibrium contact angle is impossible to measure as there are free energy barriers between the metastable states. It is essential to measure both the contact angles and report the contact angle hysteresis to fully characterize a surface. The hysteresis effect can be classified in thermodynamic and kinetic terms. Roughness and heterogeneity of the surface are the sources of thermodynamic hysteresis. Kinetic hysteresis is characterized by the time-dependent changes in contact angle which depend on deformation, reorientation and mobility of the surface, and liquid penetration. Difference in hysteresis among fibers sheds light on the differences that exist in their chemical and physical structures.63

5.6.1

Characterization of hysteresis

Wetting hysteresis can be characterized in three different ways: the arithmetic difference between the values of the advancing and receding contact angles 䉭θ = θa – θ r ; the difference between the cosines of the receding and advancing


177

contact angles ∆cos θ = cos θr – cos θa; and a dimensionless form,64 referred to as ‘reduced hysteresis’ H, H = (θa – θr)/θa

[5.35]

Wetting hysteresis is also characterized as the ratio of the work of adhesion in the receding mode to that in the advancing.

5.6.2

Hysteresis on micro-textured surfaces

On micro-textured surfaces, the contact angle hysteresis is affected by the state of the drop. The Wenzel state is characterized by a huge hysteresis in the range of 50° to 100° which makes it very sticky compared to the Cassie state, which is very slippery because of its low hysteresis (in the range of 5° to 20°). This is due to the fact the drop interacts with many defects on the surface in the first case, whereas it hardly feels the surface and can easily roll off in the second case.46

5.6.3

Hysteresis on fibrous materials

Since fibrous materials are complicated by surface roughness and heterogeneity, the measured (apparent) contact angle exhibits hysteresis and the advancing contact angle is usually employed in discussions of wicking.65 Surface contamination, roughness, and molecular structure of fibers are the factors responsible for wetting hysteresis.66 The wetting index while receding is governed mostly by the chemical make-up of the fiber; the index during advancing is affected additionally by the physical and morphological structures which include molecular orientation, crystallinity, roughness, and surface texture. Whang and Gupta67 tested wetting characteristics of chemically similar cellulosic fibers, viz. cotton, regular rayon (roughly round but crenulated shape), and trilobal-shaped rayon, using the Wilhelmy technique. The contact angles during receding for these fibers are similar due to their similar chemical structures. The wetting hysteresis for cotton, regular rayon and trilobal rayon were 1.06, 1.25 and 1.01, respectively. Very little or no hysteresis values for the trilobal rayon fiber and high values for regular rayon fiber may be explained on the basis of chemical purity, cross-sectional morphologies, and orientation of molecules in the fibers. The trilobal rayon fibers had high purity, were smoother and had more homogeneous surfaces than regular rayon fibers. These differences are partly responsible for the difference in the hysteresis values of the two rayon fibers. Pre-wetting and absorption can also influence hysteresis for some fibers.63,68 Surface contamination of fibers can also cause hysteresis.69

178

5.7


Meniscus

When a fiber is dipped in a fluid, a meniscus is formed on it. When it is withdrawn, the meniscus is deformed, and a layer of fluid covers the fiber and is entrained with it. Two regions of meniscus can be described, as shown in Fig. 5.8. The dynamic region is high above the meniscus where the fluid layer is nearly constant and the hydrodynamic equations can be simplified and solved; and the static meniscus region is near the surface of the fluid bath, where the capillary equation of Laplace is integrated. The Landau–Lavich–Derjaguin (LLD) theory forecasts the limit film thickness h0, present on an inclined plate withdrawn from a liquid bath, by matching the curvature between the apex of the static meniscus and the bottom of the steady-state region of the dynamic regime using the expression: h0 = (0.945/(1 – cos α0)1/2)(ηv0 /γLV)2/3(γLV /ρg)1/2

[5.36]

where α0 is the inclination angle in degrees of the plate with the horizontal; v0 is the plate velocity, and g is the gravity constant. The second term represents the capillary number and the final term is related to the inverse of the bond number.

5.7.1

Meniscus on single fiber

Rebouillat et al.70 extended their work to a meniscus on an inclined fiber and showed that z r g

v0

R ho

Constant thickness region

Dynamic meniscus

so

L

s

Fiber

Static meniscus

5.8 Withdrawal of a fiber from a bath of wetting liquid: the static meniscus is deformed and strained for a length L and a layer of constant thickness ho covers the fiber above the meniscus. Reprinted from Chemical Engineering Science, Vol. 57, S. Rebouillat, B. Steffenino, and B. Salvador, pp. 3953–3966, Copyright (2002), with permission from Elsevier’.


h0 /( R + h0 ) = 1.34 Ca2/3 / 1 – cos α 0

179

[5.37]

where R is the fiber radius and Ca is the capillary number expressed as (ηv0/γLV) The fluid radii in the dynamic meniscus region S and in the constant thickness region S0 are related by the expression: S = S0 + B exp (–z/E)

[5.38]

where B and E are the parameters of the model and z is the distance along the fiber from the level of the liquid bath. It is shown that, for a monofilament withdrawn from a bath of liquid, with increasing meniscus height, fluid radius decreases and for a given rise of liquid on the withdrawing fiber, the larger the withdrawal speed of the fiber, the larger is the fluid radii in the dynamic region.

5.7.2

Meniscus on multifilament

In the case of a multifilament, the complexity comes essentially from the influence of the porosity existing inside the structure between the filaments, which increases the surface of contacts as compared with a monofilament of the same size. Using images, it was shown that, at low velocity, the fluid seems to be dragged inside the fibers; that is to say, the structure seems to be swollen under the capillary suction effect. Nevertheless, at high speeds, the porous structure may become saturated and fluid is dragged around the cylinder composed of the multifilaments, internal fluid filling the porosity formed by the filament structure. The ratio of fluid thickness on the fiber to radius of the fiber is found to be similar for monofilament and multifilament when the fiber is withdrawn at highspeeds, as if the multifilament fibers behave like a cylinder of apparent radius encompassing the majority of the filaments. The height of the dynamic meniscus L for velocities 20–120 m/min is expressed as:

L=

( ho ( h 0 + R )

[5.39]

Wiener and Dejiová30 modeled the curvature of the meniscus during wicking in multifilament yarns. The curvature of the liquid along the fibers is infinite and the radius of the curved meniscus between the fibers R, by simplifying the Laplace equation, yields ∆P = γLV/R. When the capillary pressure driving the liquid front is balanced by the hydrostatic pressure, ρLgh, then R is: R = γLV /ρLgh

[5.40]

According to the above equation, the curvature of the liquid surface increases (or radius decreases) as the liquid rises to a greater height between the fibers. This is shown in Fig. 5.9.

180


(A)

(B) Liquor

Fiber

Surface of liquor

(C)

5.9 Influence of hydrostatic negative pressure or liquid height on the curvature of meniscus in a parallel fiber bundle. Height increases from C to B and then to a maximum height A (From Wiener and Dejiova, Autex Research Journal.30

5.8

Instability of liquid flow

Flow of liquid under certain conditions experiences instability. Instability of liquid flow influences the uniformity of coating of fibrous materials, including spin–finishes on synthetic filament yarns and filling of voids between fibers during fiber–composite production. Droplet formation occurs on fibers due to flow instability. During wet filtration, aerosol particles are captured by the liquid drops formed on the fibers rather than being directly captured by the fibers, and by providing sufficient liquid, the filter is self-cleaning and filtration efficiency is greatly increased.

5.8.1

Curvature of cylindrical surfaces

A uniform cylindrical bubble possesses a critical length beyond which it is unstable toward necking in at one end and bulging at the other. This length equals the circumference of the cylinder. A cylinder of length greater than this critical value thus promptly collapses into a smaller and a larger bubble. The same is true of a cylinder of liquid, i.e. a stream of liquid emerging from a circular nozzle.10 A fluid film layer flowing either on the outside or inside of a vertical cylinder is more unstable than on a vertical plane wall. The stability of the flow on the cylindrical wall is characterized by the curvature of the free surface rather than that of the cylinder.71 As the radius of the cylinder decreases, flow becomes more unstable. Even when the liquid is at rest, the layer of fluid is unstable because of the disturbance of the wave number beyond a certain critical value. With increasing curvature of the film, the range of unstable wave numbers and the wave number of the most amplified wave increase. For low curvature, the wave number of the most amplified


181

wave decreases with the Reynolds number or Weber number, while for high curvatures it increases.

5.8.2

Fluid jets

A slow-moving, thin, cylindrical stream of water undergoes necking-in, becomes non-uniform in diameter and eventually breaks up into alternately smaller and larger droplets. This is an example of capillary break-up (a column of liquid in a capillary) and it is commonly known as Rayleigh instability. A stream or jet of fluid emerging from a circular nozzle undergoes a process of necking-in, leading to break-up of the jet into alternate smaller and larger drops72,73. Weber73 considered the break-up of a jet of fluid and, according to his theory, the most rapidly growing mode is given by: 2π a /λ = 0.707 [1 + (9η2/2 ργ LV a)1/2] –1/2

[5.41]

where a is the initial radius of the liquid cylinder, η is the viscosity of the fluid, and λ is the wave length of the disturbance. For a cylindrical jet, Rayleigh calculated that the most unstable disturbance wavelength, λ , is about nine times the radius of the jet. In the case of a thin annular coating of liquid on the inside of a capillary, the disturbance is much faster than the case where liquid completely fills in the capillary. The liquid film breaks up into droplets of equal length more quickly. A standing wave develops, which grows in amplitude until droplets are producted.74 Ponstein75 studied jets of rotating fluids and observed that an increasing angular velocity decreases the stability of a solid jet and increases the stability of a ‘hollow infinitely thick’ jet. Investigations of annular jets with both surfaces free, showed that, in some cases, non-axially symmetric disturbances are more stable than axially symmetric ones, whereas in non-rotating jets, only axially symmetric disturbances are unstable. Tomotika76 considered a cylinder of bi-component fluids (one liquid surrounded by the other). The most rapidly growing mode of disturbance is given by: 2π a/λ = 0 if the ratio of viscosities is either zero or infinite and 2π a/λ ≠ 0 for finite values of the ratio.

5.8.3

Marangoni effect

Surface tension gradient on a liquid, known as the ‘Marangoni effect’, leads to an erratic and slow wicking rate of the liquid. Spin finishes are applied to synthetic fibers to control friction during downstream processes. Spin finishes are multicomponent liquid systems containing surfactant and are applied to yarns moving at high speeds. For uniform spreading of the finish within the yarn structure, it is important that the rate of wicking be high and the finish film not retract due to lack of adhesion as the carrier evaporates during the

182


storage of yarn packages. It is observed that the absorption of surfactant molecules on the fiber surface at the wicking front results in a decrease in the surface energy of the fiber and an increase in the surface tension of the liquid, with a concomitant decrease in the cosine of the contact angle and capillary forces. Equilibrium conditions are re-established when the surfactant molecules diffuse from the more concentrated regions into less concentrated region (leading edge of the meniscus).77 These effects, often termed transient effects, arise due to depletion and replenishment of surfactants at the liquid surface. The overall results of adsorption of surfactant molecules and surface tension gradient of liquid is erratic wicking behavior and a lower wicking rate. This depletion effect is more pronounced in dilute solutions and decreases as concentration of surfactant increases. The concentration of surfactant needed to overwhelm the depletion is equal to, or in excess of, critical micellar concentration.

5.8.4

Dewetting process

The rupture of a thin film on the substrate (liquid or solid) and formation of droplets, can be understood as dewetting: it is the opposite process of spreading of a liquid on a substrate, i.e. S < 0. Dewetting is one of the processes that can occur at a solid–liquid or liquid–liquid interface. Dewetting is an unwanted process in applications such as lubrication, protective coating and printing, because it destroys the applied thin film. Even in the case of S < 0, the film does not dewet immediately if it is in a metastable state, e.g. if the temperature of the film is below the Tg of the polymer forming the film. Annealing of the film above its Tg increases the mobility of the polymer chain molecules; dewetting starts from randomly formed holes (dry patches) in the film. These dry patches grow and the material is accumulated in the rim surrounding the growing hole, a polygon network of connected strings of material forming. These strings then can break-up into droplets by the process of ‘Rayleighinstability’. Dewetting of resin on glass fiber has to be controlled during composite manufacture. It has been shown that the presence of high surface energy components on the glass surface (treated with finishes) tends to resist dewetting of the receding fluid front, lowering the receding angle.78

5.8.5

Fiber coating

Droplet formation can occur in the case of coating of synthetic fibers with water for lubrication. Droplets can be formed on the inside of fiber assemblies from the thin liquid coating left behind either when the liquid drains from the tube or larger air bubbles pass through the tube.79 In order to give cohesion between multifilaments to prevent them from being damaged in further


183

operations or imparting lubrication and specific surface properties (hydrophilic, hydrophobic, functional etc.) in textile applications, fiber impregnation process is used. This process is usually done by passing the material through a liquid bath. The impregnation speed is of the order of 10 m s–1. Rebouillat et al.70 studied the high-speed fiber impregnation fluid layer formation on monoand multifilaments. During high-speed impregnation, the predominant phenomenon is inertia followed by surface tension, viscosity and then gravity. At high speeds, the inertia effect tends to drag more quantities of fluid on the substrate and the meniscus takes a critical size; the capillary forces perpendicular to the fiber are no longer negligible and drops are formed as various forces tend to minimize the fluid surfaces. These formed drops are dragged under the effect of inertia.

5.9

Morphological transitions of liquid bodies in parallel fiber bundles

The fundamentals of non-homogeneous liquid flow dealing with thin films on flat surfaces, capillary instability and surface gradient effects have been well researched. A few attempts have been made to exploit non-homogeneous flow for practical applications involving fibrous materials in the areas of fiber coating, wet filters and development of liquid-barrier fabrics. Wetting phenomena occurring between two or three equidistant, parallel cylinders have been studied.62 By changing the ratio of spacing d between the cylinders and radii r of the cylinders, different morphologies can be observed for liquid shapes between the cylinders. As the ratio d/r is increased, one can observe that the morphology of the liquid changes from ‘disintegrated column’ to ‘unduloid shape’ through ‘channel-filling column’ (Fig. 5.10). (d / r )

4

3

3

‘Unduloid’

2 1 Channel-filling column

0.2

0.1

θ

Disintegrated column 0

0

10

20

30

40

5.10 Morphology of liquid for three-cylinder system.

50 62

60

184


This observation has a far-reaching impact on designing liquid-barrier fabrics by manipulating the pore size of nanofiber webs.

5.10


Many works of a significant nature have been published on the wetting of solids rather in the fiber wetting field. Little work has so far been done dedicated to the gas filtration of liquid aerosols using fiber filters. Most of the studies reported on wet filtration are on a macroscopic level investigating the efficiency of the wetted fiber filter without examining the actual processes occurring inside the filter. Microscopic works in the area of wet filters to enhance the understanding of the physical phenomena have been carried out by Mullins et al.8 in developing the model for the oscillation of clamshell droplets in the Reynolds transition flow region; Mullins et al.80 on dynamic effects of water build-up on the fiber, flow down the fiber leading to a selfcleaning effect, fiber rewetting and cake removal after evaporative drying; and Contal et al.81 on a qualitative description of clogging of fiber filters by liquid droplets in terms of the change in the mass of deposit on fibers vs. pressure drop. Formation of barrel-shaped droplets is preferable to clamshell to improve the efficiency of the wet filter. Fine wettable fibers favor the barrel configuration for droplets. Future investigations should be directed towards selection of fibers and their fineness, surface modification of fibers by finishes/plasma treatment and the arrangement of these fibers in terms of angle and spacing to design efficient wet fiber filters. Another promising area of research involving fibers is the development of liquid-barrier fabrics using nanofibers. Here again, little has been done except a work wherein the theory of liquid-instability is applied to develop a model for liquid instability between cylinder analogs to fibers.62 Methods of quantifying wetting of fibers, yarns and fabrics, effects of various parameters influencing wetting phenomena and modeling of wetting phenomena on fibrous materials including simulation are also very important and these are reviewed in a monograph ‘Wetting and wicking in fibrous materials’.82

5.11

References

1. E. Kissa (1984) in Handbook of Fiber Science and Technology Part II (edited by B. M. Lewin and S. B. Sello), Marcel Dekker, NY, p. 144. 2. N. K. Adam (1968) The Physics and Chemistry of Surfaces, Dover, New York, p. 179. 3. A. Marmur (1992) in Modern Approaches to Wettability, (edited by M. E. Schrader and G. I. Loeb), Plenum Press, New York. 4. B. J. Mullins, I. E. Agranovski and R. D. Braddock (2004) J. Colloid Interface Sci., 269 (2), 449–458.


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5. B. J. Briscoe, K. P. Galvin, P. F. Luckham and A. M. Saeid (1991) Colloids and Surfaces, 56, 301–312. 6. S. Rebouillat, B. Letellier and B. Steffenino (1999) Int. J. Adh. Adhes., 19 (4), 303– 314. 7. G. McHale, M. I. Newton and B. Carrol (2001) Oil Gas Sci. Technol., 26 (1), 47– 54. 8. B. J. Mullins, R. D. Braddock, I. E. Agranovski, R. A. Cropp and R. A. O’ Leary, (2005) J. Colloid Interface Sci., 284, 245–254. 9. N. R. Bertoniere and S. P. Rowland (1985) Text. Res. J., 55 (1), 57–64. 10. A. W. Adamson (1990) Physical Chemistry of Surfaces, Wiley-Inter Science, New York. 11. A. Marmur (1992) Advances in Colloid and Interface Science, 39, 13–33. 12. R. Lucas (1918) Kolloid Z., 23, 15. 13. J. J. De Boer (1980) Text. Res. J., 50 (10), 624–631. 14. C. Heinrichs, S. Dugal, G. Heidemann and E. Schollmeyer (1982) Text. Prax. Int., 37 (5), 515–518. 15. E. Kissa (1996) Text. Res. J., (1996) 66 (10), 660–668. 16. N. R. S. Hollies, M. M. Kaessinger and H. Bogaty, (1956) Text. Res. J., 26, 829– 835. 17. F. W. Minor, A. M. Schwartz, E. A. Wulkow and L. C. Buckles (1959) Text. Res. J., 29 (12), 931–939. 18. K. T. Hodgson and J. C. Berg (1988), J. Coll. Interface Sci., 121, (1), 22–31. 19. http:trc.ucdavis.edu/textiles/ntc%20projects/M02-CD03-04panbrief.htm. 20. B. Miller (1977) The Wetting of Fibers in Surface Characteristics of Fibers and Textiles, Part II (edited by M. J. Schick), Marcel Dekker, NY, USA, p. 417. 21. H. Ito and Y. Muraoka (1993) Text. Res. J., 63 (7), 414–420. 22. P. R. Lord (1974) Text. Res. J., 44, 516–522. 23. K. K. Wong, X. M. Tao, C. W. M. Yuen and K. W. Yeung (2001) Text. Res. J., 71 (1), 49–56. 24. A. K. Sengupta, V. K. Kothari and R. S. Rengasamy, (1991) Indian J. Fiber Text. Res., 16 (2), (1991) 123–127. 25. T. Gillespie 1958 J. Coll. Interface Sci., 13, 32–50. 26. E. Kissa (1981) J. Coll. Interface Sci., 83 (1), 265–272. 27. Y. -L. Hsieh (1995) Text. Res. J., 65 (5), 299–307. 28. Y. -L. Hsieh, J. Thompson and A. Miller (1996) Text. Res. J., 66 (7), 456–464. 29. Y. -L. Hsieh, A. Miller and J. Thompson (1996) Text. Res. J., 66 (1), 1–10. 30. J. Wiener and P. Dejiová (2003) AUTEX Res. J., 3 (2), 64–71. 31. Y. -L. Hsieh, B. Yu, and M. M. Hartzell (1992) Text. Res. J., 62 (12), 697–704. 32. N. Mao and S. J. Russell, J. Appl. Phy., 94 (6), 4135–4138. 33. N. R. Hollies M. M. Kaessinger, B. S. Watson, and H. Bogaty (1957) Text. Res. J., 27 (1), 8–13. 34. D. Rajagopalan, A. P. Aneja and J. M. Marchal (2001) Text. Res. J., 71 (9), 813–821. 35. M. Mao and S. J. Russell (2000) J. Text. Inst., 91 Part 1(2), 235–243. 36. M. Mao and S. J. Russell (2000) J. Text. Inst., 91 Part 1 (2), 244–258. 37. J. Kozeny (1997) Proc. Royal Academy of Sci., Vienna, Class 1, 136, p. 271. 38. P. C. Carman (1956) Flow of Gases through Porous Media, Academic Press, New York. 39. H. D’Arcy (1856) Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris. 40. A. Konopka and B. Pourdeyhimi (2002) Int. Non-wovens J., 11 (2), 22–27.

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Thermal and moisture transport in fibrous materials H. S. Kim and B. Pourdeyhimi (2003) Int. Non-wovens J., 12 (2), 29–33. R. N. Wenzel (1936) Ind. Eng. Chem., 28, 988. S. A. Kulinich and M. Farzaneh (2005) Vacuum, 79, 255–264. A. B. D. Cassie (1948) Discuss. Faraday Soc., 3, 11. A. B. D. Cassie and S. Baxter (1944) Trans. Faraday Soc. 40, 546. M. Callies, Y. Chen, F. Marty, A. Pépin and D Quéré (2005) Microelectronic Engg., 78–79, 100–105. R. Shuttleworth and G. L. J. Bailey (1948) Disc. Faraday Soc., 3, 16. S. J. Hitchcock, N. T. Carrol and M. G. Nicholas (1981) J. Mater. Sci., 16, 714. R. E. Johnson and R. H. Dettre (1964) Contact Angle, Wettability and Adhesion (edited by R. F. Gould), Advances in Chemistry Series 43, ACS, Washington DC. M. G. Nicholas and R. M. Crispin (1986) J. Mater. Sci., 21, 522. H. Nakae, R. Inui, Y. Hirata and H. Saito (1998) Acta Mater., 46 (7), 2313–2318. S. Shibuichi, T. Onda, N. Satoh and K. Tsujii (1996) J. Phys. Chem., 100, 19512– 19517. J. Bico, U. Thiele and D. Quere (2002) Colloids Surfaces A: Phy. Eng. Aspects, 206, 41–46. J. C. Berg (1985) The Role of Surfactants in Absorbency (edited by P. K. Chatterjee), Elsevier, New York, p. 179. I. Pezron, G. Bourgain and D. Quere (1995) J. Colloid and Interface Sci., 173, 319– 327. Y. Chen, B. He, J. Lee, and N. A. Patankar (2005) J. Colloid Interface Sci., 281, 458– 464. F. G. Yost, J. R. Michael and E. T. Eisenmann (1995) Acta Metall. Mater., 43 (1), 299–305. E. M. Sanders and S. H. Zeronian (1982) J. App. Poly. Sci., 27 (11), 4477–4491. P. P. Tsai, L. C. Wadsworth, and J. R. Roth (1997) Text. Res. J., 67 (5), 359–369. W. Barthlott and C. Neinhuis (1995) Plant, 202, 1–8. ˙ B. Lehmann, and A. Vitakauskas (2003) Mat. Sci., 9 (4), 410–413. M. Pociute, N. Pan, Y. L. Hsieh, K. Obendorf and S. Witaker, http://trc.ucdavis.edu/textiles/ ntc%20projects/M02–CD03 B. Miller and R. A. Young (1975) Text. Res. J., 45 (5), 359–365. C. W. Extrand and Y. Kumagai (1995) J. Colloid Interface Sci. 170, 515. R. H. Dettre and R. E. Johnson (1964) in ‘Contact Angle, Wettability and Adhesion’ (edited by R. F. Gould), Advances in Chemistry Series, Vol. 43, ACS, Washington, D. C., p. 136. P. Luner and M. Sandell (1969) J. Poly. Sci. Part C, 28, 115–142. H. S. Whang and B. S. Gupta (2000) Text. Res. J., 70 (4), 351–358. G. Giannotta, M. Morra, E. Occhiello, F. Garbassi, L. Nicolais, and A. D’Amore (1992) J. Coll. Interface Sci., 148 (2), 571–578. H. J. Barraza, M. J. Hwa, K. Blakley, E. A. O’Rear and B. P. Grady (2001) Langmuir, 17 (17), 5288–5296. S. Rebouillat, B. Steffenino and B. Salvador, (2002) Chemical Engg. Sci., 57, 3953– 3966. E. Hasegawa and C. Nakaya (1970) J. Phy. Soc. Japan, 29 (6), 1634–1639. L. Rayleigh (1879) Appendix I Proc Roy. Soc. A, 29, 71. C. Weber Z. (1931) Angew Math. Mech., 11, 136. K. D. Bartle, C. L. Wooley, K. E. Markides, M. L. Lee and R. S. Hansen (1987) J. High Resol. Chromatog. Chromatogr. Comm., 10, 128.


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75. J. Ponstein (1959) Appl. Sci. Res. A, 8, 425. 76. S. Tomotika (1935) Proc. Roy. Soc. A, 150, 322. 77. Y. K. Kamath, S. B. Hornby, H. D. Weigmann and M. F. Wilde (1994) Text. Res. J., 64 (1), 33–40. 78. K. Van de Velde and P. Kiekens (2000) Indian J. Fiber Text. Res., 25 (1), 8–13. 79. S. L. Goren (1962) J. Fluid Mech., 12, 309–319. 80. B. J. Mullins, R.D. Braddock and I. E. Agranovski (2004) J. Colloid Interface Sci., 279, 213–227. 81. P. Contal, J. Simao, D. Thomas, T. Frising, S. Calle, J. C. Appert-Collin and D. Bémer (2004) Aerosol Sci., 35, 263–278. 82. A Patnaik, R. S. Rengasamy, V. K. Kothari and A Ghosh (2006) ‘Wetting and Wicking in Fibrous Materials’, Text Prog., 38 (1).

6 Interactions between liquid and fibrous materials N. P A N and Z. S U N, University of California, USA

6.1

Introduction

The interaction of liquids with fibrous materials may involve one or several physical phenomena (Skelton, 1976; Leger and Joanny, 1992; Keey, 1995; Batch, Chen et al., 1996; Kissa, 1996). On the basis of the relative amount of liquid involved and the mode of the liquid–fabric contact, the wicking processes can be divided into two groups: wicking from an infinite liquid reservoir (immersion, trans-planar wicking, and longitudinal wicking), and wicking from a finite (limited) liquid reservoir (a single drop wicking into a fabric). According to fiber–liquid interactions, the wicking processes can also be divided into four categories: capillary penetration only; simultaneous capillary penetration and imbibition by the fibers (diffusion of the liquid into the interior of the fibers); capillary penetration and adsorption of a surfactant on fibers; and simultaneous capillary penetration, imbibition by the fibers, and adsorption of a surfactant on fibers. When designing tests to simulate liquid–textile interactions of a practical process, it is essential to understand the primary processes involved and their kinetics (Batch, Chen et al., 1996; Perwuelz, Mondon et al., 2000; Baumbach, Dreyer et al., 2001). One of the fundamental parameters which dictates the liquid–solid interactions is the geometry of the solid, including the shape and relative positions of the structural components in the system, as explicitly reflected in the Laplace pressure law showing that the pressure drop is proportional to the characteristic curvatures. Consequently, for the same material, its wetting behavior will be different, in some cases drastically, when made into a film, a fiber, a fiber bundle or a fibrous material, as demonstrated in this chapter.

6.2

Fundamentals

Surface tension only occurs at the interface, and is therefore determined by both the media at the interface. Surface tension between two media (e.g. two non-miscible liquids) A and B is termed as γAB, except in the case of a water/ 188


189

air interface where the surface tension is often denoted simply as γ. The following are some of the liquid/solid interfacial relationships fundamental to understanding the interactions between liquid and fibrous media. We will restrict our discussion to the case of non-volatile liquids.

6.2.1

A liquid drop on a fiber – in shape or not in shape

There has been much research work on the equilibrium shapes of liquid drops on fibers (Carroll 1976, 1984, 1992; McHale, Kab et al., 1997; Bieker and Dietrich, 1998; McHale, Rowan et al., 1999; Neimark, 1999; Quere, 1999; Bauer, Bieker et al., 2000; McHale and Newton, 2002). In a complete wetting case, a liquid drop will form a barrel shape covering the fiber as shown in Fig. 6.1. Such a wetting liquid drop on a fiber of radius b has a profile z(x) described by de Gennes et al. (2003) as ∆p 2 z – (z – b2) = b 2 γ 2 ˙ 1+z

[6.1]

The maximum radius of the drop zmax = Rm when z˙ = dz = 0 . The above dx equation gives ∆p 2 ∆γ ( Rm – b 2 ) = Rm – b or Rm = –b 2γ ∆p

[6.2]

∆p is the so-called over-pressure and roughly equals the Laplace capillary 2γ for complete wetting (de Gennes et al., 2003). pressure Rm + b

6.2.2

Meniscus on a fiber – what if the fiber is standing in water?

If a fiber is vertically inserted into a liquid bath, assuming the rise is very low so that the effect of gravity on the liquid is negligible and there is a complete wetting between fiber and the liquid, the liquid in the meniscus is in equilibrium with the liquid bath so that ∆p = 0. Equation 6.1 hence becomes Z (x ) 2b

Rm x Fiber

6.1 A liquid drop forming a barrel shape covering a fiber.

190


z =b 1 + z˙ 2

[6.3]

At a height x, the vertical projections of forces is balanced 2π zyγ cos θ = 2π bγ and tan θ = ż. Bringing both conditions into the above equation yields the solution of the profile of the meniscus x z = b cosh    b

[6.4]

This is a hanging chain equation known as a centenary curve (see Fig. 6.2).

6.2.3

The capillary number and the spreading speed – dimensionless and dimensional

When a fiber is pulled out from a liquid at a speed V, the capillary force causes some liquid to move with the fiber, yet the liquid viscosity η resists any such movement. A dimensionless ratio of the two forces is called the capillary number Ca:

Ca =

ηV γ

[6.5]

A characteristic number with a dimension of speed Vs =

η = Ca V γ

[6.6]

is called the spreading speed.

b Z 0

z (x )

θ

X

6.2 Liquid meniscus as a hanging chain or a centenary curve.


6.2.4

191

Capillary adhesion – water serving as glue

Two glass surfaces can adhere to each other if there is a liquid drop in between, as shown in Fig. 6.3. The Laplace pressure within the drop requires

(

)

∆p = p o – p w = γ 1 + 1 = γ  1 – cos θ  R R′ R H /2

[6.7]

where po and pw are the pressures in the air and water, respectively; γ is the liquid–air surface tension and θ < π /2 to assure an attractive pressure ∆p < 0. R and H are the radius of the liquid drop and the gap between the two surfaces, respectively. The capillary adhesion ∆p reduces into ∆p ≈

2γ cos θ H

[6.8]

if H 0; the liquid is then referred to as a wetting liquid. Based on Young’s relation

γsa – γsl = γ cos θ

[6.13]

I > 0 is equivalent to θ < π /2 as mentioned above. The factor I is closely related to the spreading parameter S (Brochard, 1986) by, I=S+γ

[6.14]

Therefore, the capillary rising criterion is more restrictive than that of the spreading; if all other conditions remain the same, it is easier for a liquid to rise in a capillary tube than to spread. When H >> R, a very thin tube, the total energy E of the liquid column due to the capillary rise can be calculated as E = 1 π R 2 H 2 ρg – 2 π RHI 2

[6.15]

where the first term is the cost in terms of gravitational potential energy, and the second term is the surface energy. Minimizing the total energy (and note that I = γ cos θ ) yields the equilibrium (or Jurin’s) height:

H=

2 γ cos θ ρgR

[6.16]

(i) H is the height a liquid of density ρ can climb in a small tube due to the capillary effect. This value agrees with the experiments of Francis Hauskbee (1666–1713). H is inversely proportional to R, and is independent of the outer pressure and thickness of the tube wall. (ii) I = γ cos θ and H share the same sign, I > 0, H > 0 capillary rises, otherwise capillary descends. (iii) H reaches maximum when θ = 0. Further increase in I > γ will lead to S > 0; a microscopic film forms ahead of the meniscus. (iv) When the condition H >> R is not true, corrections must be made in the equation (de Gennes et al, 2003). Equation [6.16] is often referred to as Jurin’s Law. (v) If θ ≥ π /2, H < 0, i.e. the liquid will not penetrate – a non-wetting situation; the secret for Gore-Tex and other waterproof finishes.


6.3

193

Complete wetting of curved surfaces

According to Brochard (1986) we define the complete wetting of a single fiber of radius b as the state when the fiber is covered by a liquid ‘manchon’ or barrel, as this liquid geometry is less energy demanding than the nearly spherical droplet sessile on the fiber. Let us denote by γSa, γSL and γ the surface tensions of the solid fiber, the solid/liquid interface, and the liquid (or liquid/air), respectively. The liquid film thickness in the manchon is represented by a parameter e. This liquid manchon formation occurs when the so-called Harkinson spreading parameter S (Brochard, 1986), defined as S = γSa – γSL – γ

[6.17]

reaches the critical value SCF derived in Brochard (1986).

SCF =

eγ b

[6.18]

That is, the fiber will be covered by the liquid manchon in the case of the following inequality

S > SCF =

eγ b

[6.19]

Compared to the wetting of planes, the wetting of individual fibers is a more energy-consuming process according to the Young Equation (Young, 1805), as for complete wetting of a flat solid it only requires S>0

[6.20]

In other words, for a plane, the critical spreading parameter SCP holds SCP = 0

[6.21]

From Equations [6.19] and [6.21] we see that it is obvious that liquids will wet a solid plane more promptly than wet a fiber. Next, let us examine the case of a fiber bundle formed by n parallel fibers as seen in Fig. 6.4, each with a radius b. Let us focus on the less energydemanding case and assume that the manchon is a cylindrically symmetric liquid body with an equivalent radius R, as shown in Fig. 6.5. The equilibrium configurations of limited amounts of liquid in horizontal assemblies of parallel cylinders have been introduced and described in detail by Princen (Princen 1969, 1992; Princen, Aronson et al. 1980). The criterion of complete wetting of a vertical fiber bundle dipped partially in a liquid will be derived here by the comparison of the surface energy Wm of such a manchon liquid geometry with the surface energy Wb of a dry fiber bundle. For a length L of the dry fiber bundle, Wb = 2π bnLγSa

[6.22]

194

Thermal and moisture transport in fibrous materials n=7 R = 3b

Liquid

b

R Fiber

6.4 A fiber bundle formed by n parallel fibers; From Lukas, D. and N. Pan (2003). ‘Wetting of a fiber bundle in fibrous structures.’ Polymer Composites 24(3): 314–322 with kind permission of the Society of Plastics Engineers.

R

L

6.5 A liquid body with an equivalent radius R covering the fiber bundle; From Lukas, D. and N. Pan (2003). ‘Wetting of a fiber bundle in fibrous structures.’ Polymer Composites 24(3): 314–322 with kind permission of the Society of Plastics Engineers.

whereas the same length of liquid formed manchon on the fiber bundle has the surface energy Wm = 2π bnLγ SL + 2π RLγ

[6.23]

That is, the energy Wm is composed of both terms of the solid/liquid interface and the liquid/air interface. The complete wetting sets in when the wet state of the system is energetically more favorable compared with the dry one, i.e. Wb > Wm. Or from previous equations

R⋅γ [6.24] >0 n⋅b Inserting Harkinson spreading coefficient from Equation [6.17] into Equation [6.24] yields

γ Sa – γ SL –

S = R – n ⋅ bγ [6.25a] n⋅b So the critical value SCb for the complete wetting of the bundle system is SCb = R – n ⋅ b γ n⋅b

[6.25b]


195

The radius of the manchon R could be smaller than the total sum of fibers radii nb. Figure 6.4 shows us such an example when the cross-section of the seven-fiber bundle is covered by a liquid cylinder. The value of SCb is clearly only –4/7 γ. The above results show that it is highly probable to have a solid/liquid system in which, on one hand, the liquid will wet a solid plane but not a single fiber, and on the other hand, the liquid will wet a fiber bundle, even before it does the solid plane. This, of course, is attributable to the familiar capillary mechanism. However, the above simple analysis also explains the excellent wetting properties of a fiber mass in terms of energy changes: the consequence of the collective behavior of fibers in the bundle allows the manchon energy Wm to decrease more rapidly with the fiber number n in the bundle than the dry bundle energy Wb.

6.4

Liquid spreading dynamics on a solid surface

6.4.1

Fiber pulling out of a liquid – the Landau–Levich– Derjaguin (LLD) law

When a fiber is pulled out of a liquid pool, it drags a liquid film of thickness e along with it; a phenomenon resulting from several competing factors including the interfacial surface tensions, liquid viscosity and density. According to Derjaguin’s law (Derjaguin and Levi, 1943), when Ca 2π ro. We can thus conclude that a drop with shorter wavelength than 2π ro cannot be formed since the surface energy of the drop should always be lower than that of the original smooth cylinder. We have to stress that the exact value for the wavelength of the Rayleigh instability cannot be derived based merely on the conservation of free energy, for the transformation of a liquid body shape is coupled with the mutation of its surface area, causing change of both energy and entropy at the liquid–gas interface as discussed in Grigorev and Shiraeva (1990).

6.5.2

A more dynamic approach

The Rayleigh instability of liquid jets is the consequence of a temporal development and magnification of the originally tiny perturbations, also known as the capillary waves (de Gennes et al., 2003). We assume the perturbations to be harmonic with an exponentially growing amplitude. While such a perturbation is developing along a liquid jet, some of the liquid surface energy turns into the kinetic energy associated with a liquid flow, thus causing the cylindrical liquid column to be transformed into a chain of individual droplets. We anticipate the perturbations to develop with various speeds, depending on their wavelengths, and the perturbation that grows the most will quickly prevail so as to determine the wavelength, or distance between the neighboring droplets (Brochard, 1986). For practical purposes, we further assume that the resulting wavelength is entirely determined by the earliest state of the perturbations. We will develop more details of this idea below. The perturbation wave propagates on the liquid column of the originally cylindrical shape. By coinciding the column axis with the axis z of the Cartesian coordinate system, the radius of the liquid body changes according to our assumption above in space and time as r = ro + aeqt cos (kz)

[6.40]

where ro is a constant, and a denotes the initial amplitude of the perturbation. The growing parameter for the surface wave is q, and k is the wave vector (k = 2πλ–1). For convenience, we will use in the following text a parameter α(t) = aeqt. Given the assumption that the whole process is determined by the early state of the perturbation, we take into account only the first non-zero term in the expansions of surface and kinetic energies of the developing perturbation.

202


Our further procedure will be a qualitative one, working with the following previously obtained findings. (i) The relevant parameters for the surface and kinetic energy changes are those involved in Equation [6.38], which is our first estimation of the Rayleigh wavelength. The relevant parameters include the radius of the original cylinder ro, the wave vector k, and the amplitude α (t) = a exp(–qt). (ii) According to the Plateau inequality [6.39] written in the form ro k < 0.996~1.012, or approximately ro k < 1, the dimensionless parameter ro k, will play a critical role in changes of surface and kinetic energies. (iii) When the surface energy change is positive, there must be rok > 1, and the change is negative when ro k < 1 as dictated by the Plateau inequality. The surface energy change ∆W(t) per unit length of the liquid jet after some mathematical manipulations in de Gennes et al. (2003) can be written as

∆W ( t ) ≈ α 2 ( t )(1 – k 2 ro2 )

[6.41]

The kinetic energy per unit length, ∆T = T1 – T2, has to contain a relevant parameter proportional to the velocity squared. The only time-dependent relevant parameter is α (t) = ae qt whose physical unit is length, and its time derivation

d α ( t ) = α˙ ( t ) = aqe qt = qα ( t ) [6.42] dt has the meaning of velocity. Therefore, we have ∆T proportional to α˙ 2 ( t ) . The dependence of ∆T with the remaining parameters k and ro has to be estimated based on the kinetic energy required to transport an equal volume of liquid an equal distance in the same time. The flux in a tube is proportional to r2v and its energy to r2Lv2 where r is the radius of the tube and L is the distance on which the liquid is transported through at average velocity v. From the equality of fluxes in tubes with various radii r1 and r2 follows r2 v1 = 22 and so the ratio of the kinetic energies T1 and T2 for different radii v2 r1 r1 and r2 is r2 T1 = 22 [6.43] T2 r1 So, the kinetic energy estimated here has to be inversely proportional to ro2 . The only way to incorporate the wave vector k into the equation so as to comply with the constraint on the kinetic energy by the condition (ii), the dependence on dimensional parameter rok, is to assume ∆Τ to be inversely proportional to the square of k as well. The resultant estimation of the kinetic energy of the perturbation is thus ∆T ≈ πρ ro2 α˙ 2 ( t ) 1 2 [6.44] ( kro )


203

where ρ is the liquid mass density. The law of energy conservation, ∆T + ∆W = 0, leads to the relation

q2

1 + [1 – ( kr ) 2 ] = 0 o ( kro ) 2

[6.45]

with its extreme value for the growing parameter q given as the function of the dimensionless product kro by the equation

d dx

( kro ) 2 ( k 2 ro2 – 1) = 0

[6.46]

Equation [6.46] has the solution kro 1/2 = 0.707, where k = 2πλ–1, which provides us with the estimation of the Rayleigh wavelength λest = 2 2 πro = 2.83πro. The exact result for the Rayleigh wavelength is achieved by means of Navier–Stokes such that Equation [6.46] can be expressed in terms of (ikro) ( J o′ ( ikro ))/ J o ( ikro ) , where Jo(ikro) is the Bessel function of zero order, and J o′ is its first-order derivative, r is the cylindrical coordinate, and i denotes the imaginary term. The maximum growing coefficient q then has the value of 0.69 and the Rayleigh wavelength thus obtained is 2.88πro, in good agreement both with results from Equation [6.46] and from (Rayleigh, 1878).

6.6

Lucas–Washburn theory and wetting of fibrous media

6.6.1

Liquid climbing along a fiber bundle

Study of fiber wetting behavior is critical in prediction of properties and performance of fibrous structures such as fiber reinforced composites and textiles. On the other hand, the most often studied cases in physics for wetting phenomena are the wetting of solid planes. Compared to the plane wetting situation, the wetting of a fiber exhibits some unique features due to the inherent fiber curvature (Brochard, 1986; Bacri, Frenois et al., 1988). Brochard, for instance, derived the critical spreading parameter SCF for complete fiber wetting transition and proved that this parameter is greater than that for a plane of the same liquid/solid system. It means that liquids are more willing to wet planes than individual fibers of the same material, due to fiber curvature. However, in spite of this higher inertia of wetting process of individual fibers, one of the best known and most frequently used materials for liquid absorption is fiber assemblies. Their excellent behavior during wetting processes could be intuitively explained by the capillary effect due to their collectively large inner surface area, but a more quantitative theory of fiber assembly wetting at the microscopic level has yet to be fully developed.

204


We attempt here to extend the approach presented by Brochard (Brochard, 1986; Brochard-Wyart and Dimeglio, 1987) and Bacri (Bacri, Frenois et al., 1988; Bacri and Brochard-Wyart, 2000) obtained for single fiber wetting, to the spreading of a liquid along a fiber bundle. We then develop a theory to predict the ascension profile of a liquid along a vertical fiber bundle. The non-linear relationship between the liquid profile and the bundle properties observed experimentally will be predicted by the theoretical tool. Brochard’s deduction of a liquid body profile in a wetting regime for a single fiber is easily extendable to a small bundle of parallel fibers, with the assumption of axial symmetry of the sessile liquid body. Our goal here is to obtain the relationship between the liquid body profile Φ(x) measured from the bundle to the liquid/air interface. The equivalent radius of the fiber bundle is denoted above as R, and the bundle is vertically dipped into the liquid as shown in Fig. 6.10.

γ sa

0

R

Φ(x )

γ γ sl θ

x

Liquid

6.10 A fiber bundle vertically dipped into the liquid; From Lukas, D. and N. Pan (2003). ‘Wetting of a fiber bundle in fibrous structures.’ Polymer Composites 24(3): 314–322 with kind permission of the Society of Plastics Engineers.


205

The base for the derivation is the equilibrium of the projections onto the bundle axis x of the capillary forces (Brochard, 1986). The particular force projections taking part in the equilibrium include the one spreading the liquid on a fiber caused by γSO, parallel with the bundle axis, the force due to the fiber/liquid surface tension γSL, parallel with but opposite to γSO, and the third one in the direction with an angle θ from the x axis representing the liquid surface tension γ as illustrated in Fig. 6.10. In Laplace force regime, the equilibrium of the capillary forces acting on the liquid spread on the fiber bundle is 2π n · bγSO = 2π n · bγ SL + 2π (Φ(x) + R) cos θ.

[6.47]

In our consideration, we neglect the gravity effects, since addition of a gravitational term into Equation [6.47] will make it mathematically unsolvable. Yet it has been indicated (Manna et al., 1992) that, for relatively short fibers (≤ 10 cm), the effects of the gravitational force are negligible. Using the following relations cos θ =

1 1 + tan 2θ

[6.48a]

and

tan θ =

dΦ( x) = Φ′ ( x ) dx

[6.48b]

Equation [6.47] can be rewritten in the form of a differential equation,

R + Φ(x) 1 + Φ′ 2 ( x )

[6.49]

= np

where p is a system constant p = b  S + 1 γ 

[6.50]

The solution of Equation [6.49] is the function Φ(x) that represents the equilibrium profile of the liquid mass clinging onto the fiber bundle Φ ( x ) = np cosh  

x – xo  –R np 

[6.51a]

where xo specifies the peak point of the macroscopic meniscus. We can set xo = 0 so that Φ ( x ) = np cosh  x  – R  np 

0≤x 0, which translates into

S > – 1 or γ > γ SO SL γ

[6.52]

Φ 0 (x ) 0.6 0.5 0.4

n = 10 b = 20 s =1 γ

0.3 0.2 0.1

n = 50 10

20

30 (a)

n = 100 40

50

60

x (mm)

Φ0 (x) 0.6

b = 30 mm 0.5

n = 50

0.4

s =1 γ

b = 20 mm 0.3 0.2

b = 10 mm 0.1

10

20

30 (b)

40

50

60

x (mm)

6.11 The liquid profile Φ(x) over a fiber bundle. From Lukas, D. and N. Pan (2003). ‘Wetting of a fiber bundle in fibrous structures.’ Polymer Composites 24(3): 314–322 with kind permission of the Society of Plastics Engineers. (a) Φ (x) distribution at different fiber number n; (b) Φ(x) distribution at different fiber radius b; (c) Φ(x) distribution at different spreading ratio S/γ.


207

Φ0 (x) 0.6 0.5

n = 50 b = 20 mm

0.4

s = 10 γ

s =1 γ

0.3 0.2 0.1

s =0 γ

10

20

30 (c)

40

50

60

x (mm)

6.11 Continued

The physical implication of this inequity is obvious – a necessary condition for wetting a fiber bundle is that the surface tension of the fiber γSO has to be greater than the surface tension of the fiber/liquid, γSL. Furthermore, from Equation [6.51b], we can see that there is a criterion for determining the equivalent fiber bundle radius R, since Φ(x) ≥ 0 so that R ≤ np cosh  x   np 

[6.53]

As cosh(x) achieves the minimum when x = 0, and cosh(0) = 1, we have the limit for R R ≤ np = nb  S + 1 γ 

[6.54]

In the case R > np, the mathematical solution of Φ(x) no longer has physical meaning. Shown in Equation [6.54], the spacing between fibers in the bundle is limited by the spreading ratio S/γ . By using Equation [6.52], i.e. S/γ > –1, the minimum value of the bundle radius R = Rmin > 0. Furthermore, when x = 0, and cosh (0) = 1, then Equation [6.51b] gives Φ(0) = np – R. It means that, according to Equation [6.54], beneath the liquid meniscus with the hyperbolic cosine shape, there exists a microscopic liquid film on the fiber bundle, whose thickness is Φ(0) = np – R > 0

[6.55]

This may indicate that, at the point where the liquid mass profile starts, i.e. x = 0, the liquid first coats the fiber bundle with a thin layer of thickness np – R, a phenomenon similar to what is reported in Brochard (Brochard,

208


1986) for the single-fiber wetting case. However, it is recommended that this conclusion be verified in view of the omission of the gravitational effects in the analysis. Considering the upper limit for the bundle radius R = np in Equation [6.55], the lower limit Φo(x) of the liquid profile Φ(x) in Equation [6.51b] can be expressed in terms of the Harkinson spreading parameter S and the liquid surface tension γ:     S  x Φ o ( x ) = nb  S + 1 cosh   – nb  γ + 1 0 ≤ x ≤ ∞ γ    S  nb +1    γ [6.56] That is, Φo(x) is a function of the height x, the spreading ratio S/γ reflecting the surface properties of liquid, the fiber, and the liquid/fiber interfacial property, as well as the fiber parameters nb, as plotted in Fig. 6.11 based on Equation [6.56] (which may be regarded as [6.51c] in the series). In general, Φo(x) increases with x when other parameters are given. The effect of the number of fibers in a bundle is seen in Fig. 6.11(a) where a small bundle (small n value) will have a greater amplitude of Φo(x) at a given position x. The fiber radius b has the similar influence on Φo(x), i.e. Φo(x) increasing with b for a given x, except that it also determines the maximum value, Φm(x) and the maximum height xm as seen in Fig. 6.11(b); when b is smaller, the Φm(x) value as well as xm will be accordingly smaller. Figure 6.11(c) shows that the same thing can be said about the effect of the spreading ratio S/γ ; a smaller ratio S/γ results in a smaller Φm(x) and xm. Once again, the solution to Equation [6.51] has a shortcoming resulting from the exclusion of gravity in the analysis. The consequence is an asymptotical behavior of Φ(x) that does not converge to the flat horizontal surface of the liquid source perpendicular to the fiber bundle.

6.6.2

Lucas–Washburn theory

The first attempt to understand the capillary driven non-homogeneous flows for practical applications was made by Lucas (1918) and Washburn (1921). Good (1964) and Sorbie et al. (1995) have successively derived more generalized expressions of the theory. The theory aroused public excitement in England in 1999 about what is called dunking, or dipping a biscuit into a hot drink such as tea or coffee to enhance flavor release by up to ten times (Fisher, 1999). Lucas–Washburn theory has been used in, and further developed for, the


209

textile area by a few authors. Chatterjee (1985) dealt with these kinds of flow in dyeing. Pillai and Advani (1996) conducted an experimental study of the capillarity-driven flow of viscous liquids across a bank of aligned fibers. Hsieh (1995) has discussed wetting and capillary theories, and applications of these principles to the analysis of liquid wetting and transport in fibrous materials. Several techniques employing fluid flow to characterize the structure of fibrous materials were also presented in Hirt et al. (1987). Lukas and Soukupova (1999) carried out a data analysis to test the validity of the Lucas–Washburn approach for some fibrous materials and obtained a solution for the Lucas–Washburn equation including the gravity term. Non-homogeneous flows have also been studied using stochastic simulation since the beginning of the 1990s. Manna, Herrmann and Landau (1992) presented a stochastic simulation that generates the shape of a two-dimensional liquid drop, subject to gravity, on a wall. The system was based on the socalled Ising model, with Kawasaki dynamics. They located a phase transition between a hanging and a sliding droplet. Then Lukkarinen (1995) studied the mechanisms of fluid droplet spreading on flat solids, and found that in the early stages the spreading is of nearly linear behavior with time, and the liquid precursor film spreading is dominated by the surface flow of the bulk droplet on the solid; whereas in the later stages, the dynamics of liquid spreading is governed by the square root of time. A similar study of fluid droplet spreading on a porous surface was also recently reported (Starov, et al. 2003). First attempts to simulate liquid wetting dynamics in fiber structures using the Ising Model have been done by Lukas et al. (Lukas, Glazyrina et al., 1997; Lukas and Pan, 2003; Lukas et al., 2004), also by Zhong et al. (Zhong, Ding et al., 2001, 2001a, though the simulation was restricted to 2-D systems only. For both scientific and practical purposes, the so-called wicking (or absorbency) rate is of great interest. EDANA and INDA recommended tests (EDANA, 1972; INDA, 1992) to determine the vertical speed at which the liquid is moving upward in a fabric, as a measure of the capillarity of the test material. The vertical rate of absorption is measured from the edges of the test specimen strips suspended in a given liquid source. The resultant report of the test contains a record of capillary rising heights after a time 10 s, 30 s, 60 s (and even 300 s if required). Gupta defined absorbency rate as the quantity that is characterized based on a modification of the Lucas–Washburn equation, and he then modified it to apply to a flat, thin, circular fabric on which fluid diffuses radially outward (Gupta, 1997). Miller and Friedman (Miller et al., 1991; Miller and Friedman, 1992) introduced a technique for monitoring absorption rates for materials under compression. Their Liquid/Air Displacement Analyser (LADA) measures the rate of absorption by recording changes of the liquid weight when liquid is sucked into a flat textile specimen connected to a liquid source.

210


A more scientific definition of the wicking rate is based on the Lucas– Washburn theory. This simple theory deals with the rate at which a liquid is drawn into a circular tube via capillary action. Such a capillary is a grossly simplified model of a pore in a real fibrous medium with a highly complex structure (Berg, 1989). The theory is actually a special form of the Hagen– Poiseuille law (Landau, 1988) for laminar viscous flows. According to this law, the volume dV of a Newtonian liquid with viscosity µ that wets through a tube of radius r, and length h during time dt is given by the relation 4 dV = πr ( p1 – p 2 ) dt 8hµ

[6.57]

where p1 – p2 is the pressure difference between the tube ends. The pressure difference here is generated by capillarity force and gravitation. The contact angle of the liquid against the tube wall is denoted as θ, and β is the angle between the tube axis and the vertical direction shown in Fig. 6.12. The capillary pressure p1 has the value p1 =

2γ cos θ r

[6.58]

while the hydrostatic pressure p2 is p2 = hζg cos β

[6.59]

where γ denotes the liquid surface tension, ζ is the liquid density, g is gravitational acceleration and h, in this case, is the distance traveled by the liquid, measured from the reservoir along the tube axis. This distance obviously is the function of time, h = h(t), for a given system. When we substitute the 1

3

r

Θ β

h

2

6.12 A single fiber in a liquid pool; From Lukas, D. and N. Pan (2003). ‘Wetting of a fiber bundle in fibrous structures.’ Polymer Composites 24(3): 314–322 with kind permission of the Society of Plastics Engineers.


211

quantities p1, p2, and h(t) into Equation 6.57, expressing the liquid volume in the capillary V as π r2h, we obtain the Lucas–Washburn equation 2 dh = rγ cos θ – r ζg cos β 8µ dt 4 µh

[6.60]

For a given system as shown in Fig. 6.13(a), parameters such as r, γ, θ, µ, ζ, g, and β remain constant. We can then reduce the Lucas–Washburn Equation [6.60] by introducing two constants K′ =

rγ cos θ , and 4µ

L′ =

rζg cos β 8µ

[6.61]

into a simplified version

dh = K ′ – L ′ dt h

[6.62]

The above relation is a non-linear ordinary differential equation that is solvable only after ignoring the parameter L′; this has a physical interpretation, when either the liquid penetration is horizontal (β = 90°), or r is small, or the rising liquid height h is low so that K ′ >> L ′ or L′ → 0, the effects of the gravitation h field are negligible and the acceleration g vanishes. The Lucas–Washburn Equation [6.62] could thus be solved with ease

h=

6.6.3

2 K ′t ,

[6.62]

Radial spreading of liquid on a fibrous material

Now we turn our attention back to Gupta’s (1997) approach to wicking rate where a fluid from a point source in the centre of a substrate is spreading T w

1

h

2 T

h 3

(a)

(b)

6.13 Two liquid spreading routes in fibrous materials. (a) liquid spreading in radial directions; (b) liquid ascending vertically. Adapted from Lukas, D. Soukupova, V., Pan, N. and Parikh, D. V. (2004). ‘Computer simulation of 3-D liquid transport in fibrous materials.’ Simulation-Transactions of The Society For Modeling and Simulation International 80(11): 547–557.

212


radially outward, instead of the ascending liquid front in a fibrous substrate partially dipped into a liquid, as illustrated in Fig. 6.13(a) and (b), respectively. It is useful now to transfer the Lucas–Washburn equation into a modified version by replacing the distance h with liquid mass uptake m. Such a transition is described in detail in Ford (1933) and Hsieh (1995). This manipulation does not influence the fundamental shape of Equation [6.63], because the relationship between h and m is linear for a circular tube of fixed crosssection. Furthermore, for the radial spreading, the liquid mass mR = πh2TζVL and for the ascending liquid front mA = whTζVL, denoting T as the thickness of the substrate, and VL as the liquid volume fraction inside the substrate of width w. For the radial liquid spreading in a flat textile specimen, as in Fig. 6.13(b), we can then write, using Equation [6.63]

Q=

mR = 2 π K ′TζVL t

[6.64]

where Q is the liquid wicking (absorbency) rate used by Gupta (1997), which is independent of time during the spreading process. Equation [6.64] can be used to predict a drop radial spreading as shown in Fig. 6.14. Let us now substitute liquid mass uptake mA into the original Lucas– Wasbhurn Equation [6.62], with the result as

R (t )

6.14 Radial spreading of a liquid drop. From Brochard-Wyart, F., ‘Droplets: Capillarity and Wetting’, in Soft Matter Physics, M. Daoud, C.E. Williams, Editors. 1999, Springer: New York. pp. 1–45. With kind permission of Springer Science and Business Media.


dm A = K –L mA dt

213

[6.65]

The new constants K and L are K = (wTζVL)2 K′,

L = wTζVLL′

[6.66]

It is obvious that the constant K in the modified Lucas–Washburn Equation Equation [6.65] is proportional to the wicking (absorbency) rate Q which is defined in Equation [6.64], and from Equations [6.64] and [6.66] it follows that

Q=

2π K w 2 TζVL

[6.67]

Hence, the parameter K can be used as a measure of the spreading wicking rate Q in the experiments when a fabric is hung vertically into a liquid. The values of K and L can be derived from the slope and intercept of the dmA/dt versus 1/mA. On the other hand, Equations [6.62] and [6.65] can be solved in terms of the functions t(h) or t(mA) without dropping the gravity term g, as shown by Lukas and Soukupova (1999). For the liquid mass uptake Lucas–Washburn Equation 6.65, one obtains for the ascending liquid front the relationship

(

)

mA [6.68] – K2 ln 1 – L m A K K L Conversely, however, we are unable to acquire the inverse solution mA(t) using the common functions. The Lucas–Washburn approach presents an approximate but effective tool to investigate the wicking and wetting behaviour of textiles despite the complicated, non-circular, non-uniform, and non-parallel structure of their pore spaces. It has been shown that Equations [6.62] and [6.65] hold for a variety of fibrous media, including paper and textile materials [(Berg, 1989; Everet et al., 1978) and 3-D pads (Miller and Jansen, 1982). Nevertheless, this theory is unable to deal with issues such as the influence of structure, e.g. fiber orientation and deformation, on wetting and wicking behavior of fibrous media. t (mA) = –

6.6.4

Capillary rise in a fibrous material

Wetting a fiber assembly is very different from wetting a single fiber, for the specific surface areas in the two cases are very different. Instead of a single dimension of fiber radius r, we have to deal with a medium of complex surface structural geometry made of fibers and irregular pores. For a medium with regular pore diameter dp, the specific surface area As (m2/kg) is defined as the total surface area per kg of the medium, and can be approximately calculated as

214


As ≅

1 ρs d p

[6.69]

where ρs is the solid density of the medium without any pores. For a pore diameter dp = 10 µm and ρs = 1 g/cm3, As is in the order of 100 m2/kg. This value will be only 6 m2/kg if no pores exist. If we know the volume fraction of the fibers Vf, then density of the fibrous material is ρsVf and the total specific surface area is A f = As ρs V f ≅

Vf dp

[6.70]

Now consider a column made of this medium, with cross-sectional area S and thus a wet volume Sh. When the height increases from h to h + dh, there is a corresponding change in capillary energy dEcap = AfSdh (γSL – γSa)

[6.71]

and in liquid volume (assuming that all the pores are accessible by the liquid of density ρl) dM = ρl (1 – Vf )Sdh

[6.72]

Associated with dM is a change in gravitational energy dEg dEg = ghdM

[6.73]

At equilibrium, the total energy change vanishes so that dEcap + dEg = 0, so that the new height

h=

Vf A f (γ Sa – γ SL ) V f (γ Sa – γ SL ) γ cos θ ≅ = ρl (1 – V f ) g d p ρl (1 – V f ) g d p ρl (1 – V f ) g

[6.74]

Although this result is completely analogous to Jurin’s law, it is expressed in explicit macroscopic parameters of the fibrous materials. When pore diameter dp = 10 µm, Vf = 0.5 and water is the liquid, this results in h = 10 cm!

6.7

Understanding wetting and liquid spreading

Leger and Joanny (1992), de Gennes (1985) and Joanny (1986) have each written a comprehensive and excellent review on the liquid spreading phenomena. Some of the relatively new developments and discoveries in these reviews are summarized below.

6.7.1

The long-range force effects and disjoining pressure

In a situation of partial wetting, the liquid does not spread completely and shows a finite contact angle on a solid surface. Partial wetting behavior on


215

perfect solid surfaces is well described by classical capillarity. Heterogeneities of the solid surface lead to contact angle hysteresis. Experimentally, it is very easy to tell the difference between partial wetting and complete wetting. In the latter case, there exists a microscopic liquid film underneath the water droplet covering the fiber, so that the contact angle θ = 0 (as mentioned in Fig. 6.7) (Brochard–Wyart 1999). In a complete wetting situation, the liquid forms a film on a solid surface with a thickness in the mesoscopic range. The direct long-range interaction between liquid and solid described by the so-called disjoining pressure governs the physics of these films. Films of mesoscopic thickness also appear in the spreading kinetics of liquids. These precursor films form ahead of macroscopic advancing liquid fronts. The spreading kinetics is extremely slow. In fact, it is only recently that it has been fully recognized that an essential aspect of the physics of thin films, i.e. long range force effects, has to be added to classical capillarity (Leger and Joanny, 1992). When a liquid spreads on a solid or on another immiscible liquid, thin liquid zones always appear close to the triple line. There, as soon as the thickness becomes smaller than the range of molecular interactions, the interfacial tensions are not sufficient to describe the free energy of the system: a new energy term has to be included, which takes into account the interactions between the two interfaces (solid– liquid and liquid–gas for a liquid spreading on a solid). This new free energy contribution has a pressure counterpart which is the disjoining pressure introduced by Derjaguin (1955) to describe the physics of thin liquid films. It may dominate the spreading behavior, especially in situations of total wetting in the late stages of spreading where thin films are likely to appear. A recent paper by Rafai et al. (2005) has pointed out that wetting transition proceeds in two schemes: the first-order process and the critical process, depending on the thermal fluctuations, i.e. the competition between the shortrange interactions and the long-range van der Waals interactions. The sign of the system’s Hamaker constant determines the outcome of the competition. First-order implies a discontinuity in the first derivative of the surface free energy. This discontinuity then suggests a jump in the liquid layer thickness. Thus, at a first-order wetting transition, a discontinuous change in film thickness occurs, such as in the case of Rayleigh’s instability. However, the critical wetting is a continuous transition between a thin and a thick adsorbed film at bulk two-phase coexistence.

6.7.2

Experimental investigation of the liquid wetting and spreading processes

The macroscopic scales are the easiest to investigate and have been most widely studied for a long time, either by observation through an optical microscope or by contact angle measurements. With the development of

216


computer image analysis, this can now be performed in an automated way (Cheng, 1989), either for an advancing or a receding liquid front. A less classical method, based on the use of the whole drop as a convex mirror reflecting a parallel beam of light into a cone of aperture angle 2θ has been proposed by Allain et al. (1985), as it allows one to test simultaneously the whole periphery of the drop. Sizes and thicknesses can be deduced from direct observations through a microscope. If monochromatic light is used, equal thickness fringes are quite a convenient way of investigating drop profiles, with a vertical resolution of A/2n (A is the wavelength of the radiation used and n the index of refraction of the liquid), the first black fringes being located at a thickness of A/4n, i.e. typically 800 Å for visible light (Tanner, 1976). In order to investigate thinner parts of the drop, typical thin film methods have to be used. As a liquid is present, methods requiring high vacuum are inadequate. Teletzke, Davis et al. (1988) have settled on a description of spreading, including the long-range force contributions, which has stimulated a strong activity in the field, both theoretically and experimentally. Decisive progress has thus recently been achieved in the understanding of spreading and wetting phenomena. This progress has only been possible because of the parallel development of very refined experimental techniques that allow the detailed investigation of the properties of thin liquid films (Cazabat, 1990). As a spreading drop may develop characteristic features at various thicknesses, ranging from microscopic (a few Å) to macroscopic (larger than 0.1 mm), complementary techniques have to be used in order to completely probe the spreading behavior.

6.7.3

The scale effects

One of the most interesting features is the variety of length scales involved in these problems: macroscopic scales for liquid thicknesses larger than a few thousand angstroms, mesoscopic scales for liquid thicknesses between 10 and 1000 Å, and even microscopic scales at the molecular level (de Gennes et al., 2003). At the macroscopic level the liquid is characterized by thermodynamic quantities and the spreading kinetics have been described as a hydrodynamic process. For simple liquids on ideal solid surfaces, the agreement between theory and experiment seems rather good both for static and dynamic properties. This is particularly true for Tanner’s (1976) law, giving the variation of the dynamic contact angle with the advancing velocity that has been extensively verified experimentally. The extension of this law to more complex situations where the spreading is driven by other than the capillary forces, or to situations where the spreading is unstable, also gives good quantitative descriptions of the experimental results.


217

On the other hand, the viscous effects predominate over inertial effects when length scale becomes sufficiently small. Therefore, the dissipative mechanism in destabilizing a liquid cylinder becomes dominant and has to be considered (Schultz and Davis, 1982; Eggers, 1997). Another major additional difficulty comes from the fact that the thickness may vary rapidly with distance from the center of the drop, especially at mesoscopic scales. High spatial resolution is then required and the number of available techniques is not very large. Ellipsocontrast, i.e. observation under a microscope in reflected polarized light, has proven to be very useful to probe thicknesses in the range 100 Å and up, with a spatial resolution of 1 nm (Ausserre et al., 1986); it is not however, up to now, fully quantitative. Ellipsometry (Azzam, 1977) appears to be the technique of choice, and tricks have been developed to increase the spatial resolution (Leger, Erman et al., 1988; Heslot, Cazabat et al., 1989). One has to notice, however, that it only gives access to the product ne (n is the index of refraction of the liquid, e its thickness). X-ray reflectivity has proven to be a unique tool to study spreading processes (Daillant et al., 1988, 1990). The spatial resolution is poorer than in ellipsometry, as grazing incidence is used, and the dimensions of the illuminated area of the sample cannot be decreased below 100 nm × 1 or 2 mm. It is, however, a unique tool, because it gives access independently to three important characteristics of the liquid film: its thickness, its density and its roughness. It is thus valuable for microscopic scales and for studying the late stages of spreading. Many other techniques have been used to visualize the presence of thin liquid films, such as dust particle motion, vapor blowing patterns (Hardy, 1919) and the use of fluorescent or absorbing dyes, but they can hardly lead to quantitative profiles determination.

6.7.4

Heterogeneity

As in all surface phenomena, heterogeneities of the solid surface play an important role which is only partially understood. There are several models for contact angle hysteresis but very few quantitative experiments on this matter. In the case of partial wetting, the spreading kinetics of a liquid on a heterogeneous surface have been studied only in very artificial geometries and the spreading law (relation between the contact angle and the advancing velocity) on a strongly heterogeneous surface is not known either experimentally or theoretically (Joanny, 1986). In a case of complete wetting, the dynamic contact angle only depends very weakly on the nature of the solid surface and heterogeneities play a less important role. At the mesoscopic level, the properties of thin liquid films are described by continuum theories that ignore the molecular nature of the liquid and by macroscopic hydrodynamics; the long-range character of the molecular interactions is then taken into account through the disjoining pressure.

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For liquids for which the interactions are well known, the calculated static properties of the film are in very good agreement with the experimental measurements: this agreement is extremely good for superfluid helium but it is also satisfactory for van der Waals liquids such as silicone oils. The most spectacular recent progress in this field is the important development of surface scattering techniques such as X-ray reflection ellipsometry, which now allows measurements of thicknesses with a precision of the order of 1Å or less; one should note, however, that the lateral resolution of these techniques is in the micron range and that the measured thicknesses are averaged over this size, thus eliminating heterogeneities of the film at small sizes. For many liquids, however, and in particular for water, the disjoining pressure is only poorly known and this is a strong limitation of the theory. Recent studies start to consider cases where the disjoining pressure is nonmonotonic. A qualitatively different spreading behavior is observed that is not entirely understood. These very refined techniques have also been applied to the study of precursor films that form ahead of spreading drops. Detailed determinations of the precursor film profile have been made experimentally; they are in qualitative agreement with the semi-microscopic theory but no quantitative agreement has been obtained, the reason for that being unclear. For liquids spreading on high energy surfaces, the continuum description of the liquid breaks down in the last stages of the spreading where the beautiful experiments of Heslot et al. (1998, a, b) have shown that the liquid shows well-defined layers of molecular thickness. Some phenomenological theories have been proposed to describe this layered spreading but a systematic description of these experiments is far from being available. This looks like a very promising subject for future studies.

6.7.5

For liquids other than water

Other extensions of the hydrodynamic theory than the one discussed here have been made; for instance, to the spreading on a liquid substrate or to the case where the external phase is not a vapor, or systems of immiscible viscous liquids (Pumir, 1984; Joanny and Andelman, 1987). For more complex liquids such as polymeric liquids or surfactant solutions, our understanding of the spreading dynamic is poorer and further theoretical work is certainly needed to understand in more detail the role of surface tension gradients and the spreading hydrodynamics of polymer melts. Finally, most of the theoretical studies of liquids spreading describe the spreading as a purely hydrodynamic process and use classical hydrodynamics down to liquid thicknesses of a few molecular diameters. In certain cases this works surprisingly well (as is known from helium physics) but should certainly be questioned for more complex liquids such as polymeric liquids or liquid


219

crystals. Even for simple liquids, the spreading may involve non-hydrodynamic processes such as the evaporation and re-condensation of the liquid (which we have avoided, focusing on non-volatile liquids). This has received very little theoretical attention but experimentally, volatile liquids often show an instability when they spread (Williams, 1977). We would like to finish this chapter using a paragraph by Herminghaus (2005) in his preface for a recent special edition of J. Phys.: Condens. Matter entirely devoted to the topic – ‘By the mid-nineties, the physics of wetting had made its way into the canon of physical science topics in its full breadth. The number of fruitful aspects addressed by that time is far too widespread to be covered here with any ambition to completeness. The number of researchers turning to this field was continuously growing, and many problems had already been successfully resolved, and many questions answered. However, quite a number of fundamental problems remained, which obstinately resisted solution.’

6.8

References

Allain, C., Ausserre, D. and Rondelez, F. (1985). ‘A new method for contact angle measurement of sessile drops’ J. Colloid Interface Sci. 107: 5. Ausserre, D., Picard, A. M. and Leger, L. (1986). Existence and role of the precursor film in the spreading of polymer liquids’ Phys. Rev. Lett. 57: 2671–2674. Azzam, R. M. and Bashara, A. (1977). Ellipsometry and Polarized Light. Amsterdam, North-Holland. Bacri, J. C., Frenois, C. et al. (1988). ‘Magnetic Drop-sheath Wetting Transition of a Ferrofluid on a Wire.’ Revue De Physique Appliquee 23(6): 1017–1022. Bacri, L. and F. Brochard-Wyart (2000). ‘Droplet suction on porous media.’ European Physical Journal E 3(1): 87–97. Batch, G. L., Chen, Y. T. et al. (1996). ‘Capillary impregnation of aligned fibrous beds: Experiments and model.’ Journal of Reinforced Plastics and Composites 15(10): 1027– 1051. Bauer, C., Bieker, T. et al. (2000). ‘Wetting-induced effective interaction potential between spherical particles.’ Physical Review E 62(4): 5324–5338. Baumbach, V., Dreyer, M. et al. (2001). ‘Coating by capillary transport through porous media.’ Zeitschrift Fur Angewandte Mathematik Und Mechanik 81: S517–S518. Berg, J. C. (1989). The Use of Single-fibre Wetting Measurements in the Assessment of Absorbency. Nonwovens Advanced. Tutorial. F. T. Allin and L. V. Tyrone (eds), Atlanta, TAPPI Press: 313. Bieker, T. and Dietrich S. (1998). ‘Wetting of curved surfaces’ Physica A 252(1–2), 85– 137. Brochard-Wyart, F. (1999). Droplets: Capillarity and Wetting. Soft Matter Physics. Daoud, M. and Williams C.E. (eds) New York, Springer: 1–45. Brochard, F. (1986). ‘Spreading of Liquid-drops on Thin Cylinders – the Manchon– Droplet Transition.’ Journal of Chemical Physics 84(8): 4664–4672. Brochard-Wyart, F. and Dimeglio, J. M. (1987). ‘Spreading of Liquid-drops on Fibers.’ Annali Di Chimica 77(3–4): 275–283.

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Carroll, B. J. (1976). ‘Accurate Measurement of Contact Angle, Phase Contact Areas, Drop Volume, and Laplace Excess Pressure in Drop-on-Fiber Systems.’ Journal of Colloid and Interface Science 57(3): 488–495. Carroll, B. J. (1984). ‘The Equilibrium of Liquid-drops on Smooth and Rough Circular Cylinders.’ Journal of Colloid and Interface Science 97(1): 195–200. Carroll, B. J. (1992). ‘Direct Measurement of the Contact Angle on Plates and on Thin Fibers – Some Theoretical Aspects.’ Journal of Adhesion Science and Technology 6(9): 983–994. Cazabat, A. M. (1990). ‘Experimental Aspects of Wetting’ Liquids at Interfaces. J. Charvolin. Amsterdam, North-Holland, 371. Cazabat, A. M., Gerdes, S. et al. (1997). ‘Dynamics of wetting: From theory to experiment.’ Interface Science 5(2–3): 129–139. Chatterjee, P. K. (1985). Absorbency. New York, Elsevier. Cheng, P., Li, D., Boruvka, L., Rotenberg, Y. and Neuman, A.W. (1989). ‘Colloids and Surfaces.’ 43: 151. Daillant, J., Benattar, J. J. and Leger, L. (1990). Phys. Rev. Lett., A 41: 1963. Daillant, J., Bennattar, J. J, Bosio, L. and Leger, L. (1988). Europhys. Lett. 6: 431. de Gennes, P. G. (1985). ‘Wetting: Statics and dynamics.’ Rev Mod Physics 57: 827–863. de Gennes, P. G., Brochard-Wyart, F. and Quere, D. (2003). Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. New York, Springer. Derjaguin, B. V. (1955). ‘The definition and magnitude of disjoining pressure and its role in the statics and dynamics of thin fluid films.’ Kolloid Zhurnal 17: 204. Derjaguin, B. V. and Levi S. M. (1943). Film Coating Theory. London, The Focal Press. EDANA (1972). Recommended Test: Absorption. 10. Eggers, J. (1997). ‘Nonlinear dynamics and breakup of free-surface flows.’ Reviews Of Modern Physics 69(3): 865–929. Everet, D. H., Haynes, J. M. and Miller, R. J. (1978). ‘Kinetics of capillary imbibition by fibrous materials. In Fibre-water interactions in papermaking, edited by the Fundamental Research Committee. London: Clowes.’ Fisher, L. (1999). Physics takes the biscuit, Nature, V. 397, 11 Feb., p. 469. Ford, L. R. (1933). Differential Equations. New York: McGraw-Hill. Good, R. J. (1964). Contact Angle, Wettability, and Adhesion, Washington DC, ACS, p. 74. Grigorev, A. I. and Shiraeva S. O. (1990). ‘Mechanism of electrostatic polydispersion of liquid.’ J. Phys. D: Appl. Phys. 23: 1361–1370. Gupta, B. S. (1997). Some Recent Studies of Absorbency in Fibrous Nonwovens. XXV. International Nonwovens Colloquium, Brno, Czech. Hardy, H. (1919). Phil. Mag. 38: 49. Herminghaus, S. (2005). ‘Wetting: Introductory note.’ Journal of Physics: Condensed Matter 17(9): S261. Heslot, F., Cazabat, A. M. et al. (1989). ‘Diffusion-controlled Wetting Films.’ Journal of Physics – Condensed Matter 1(33): 5793–5798. Heslot, F., Cazabat, A. M. et al. (1989). ‘Dynamics of Wetting of Tiny Drops – Ellipsometric Study of the Late Stages of Spreading.’ Phys. Rev. Lett. 62: 1286. Heslot, F., Fraysse, N. and Cazabat, A. M. (1989b). ‘Molecular layering in the spreading of wetting liquid drops.’ Nature 338: 640. Hirt, D. G., Adams, K. L., Hommer, R. K. and Rebenfeld, L. (1987). ‘In-plane radial fluid flow characterization of fibrous materials.’ J. Thermal Insulation 10: 153. Hsieh, Y. L. (1995). ‘Liquid Transport in Fibrous Assemblies.’ Textile Res. J. 65: 299– 307.


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7 Thermal conduction and moisture diffusion in fibrous materials Z . S U N and N . P A N, University of California, USA

7.1

Introduction

7.1.1

Thermal conduction

Thermal transfer is a subject analyzing the energy change of a system. Of the three main physical mechanisms for heat transfer, i.e. conduction, convection and radiation, thermal convection refers to heat passing through the movement of substances and, if occurring, it occurs only at the surface of a normal solid material. The situation changes when we come to a fibrous material; as a multiphase system, all the thermal transfer processes become possible, depending on the construction and environmental conditions. Theoretically, thermal conduction always happens as long as a temperature gradient is present between a material system and the environment. When that temperature gradient is small, heat transfer via radiation can be ignored. Furthermore, if the fiber volume fraction is high enough, convection is suppressed by the tiny pores between fibers. Consequently, thermal conduction turns out to be the only or the most dominant heat transfer mechanism. Unlike many other porous media, since the pores in a fibrous material are virtually all interconnected, at low fiber volume fraction, heat loss due to convection can become dominant, as in the case of wearing a loosely knitted sweater on a windy day. In the engineering field, because of such complexities, effective thermal resistance is usually adopted to characterize thermal properties of fibrous material systems by approximating a complex thermal process to an equivalent thermal conduction process in normal solids (Martin and Lamb, 1987; Satsumoto, Ishikawa et al., 1997; Jirsak, Gok et al., 1998). The other advantage in dealing with the thermal conduction problem is that the mathematical formulation of thermal conduction is better documented. The equations governing different initial and boundary conditions have been more widely explored and more analytical and numerical tools are thus made available for ready applications. 225

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7.1.2


Similarity and difference between thermal conduction and moisture diffusion

There are many similarities between thermal conduction and moisture diffusion. Governing equations for both thermal conduction and moisture diffusion are in the same form. Thus, analysis methods and results would be analogous for both processes when system scale, material properties, and initial and boundary conditions are similar. A more detailed comparison of conduction and diffusion processes is available in the literatures (Crank, 1979; Bird, Stewart et al., 2002). Macroscopic similarity between these two processes results from microscopic physical mechanisms. Both of the processes are governed by statistical behaviors of micro-particles’ (atoms, molecules, electrons) random movement in the system. Thermal conduction deals with changes in system internal energy; heat flow is a result of a change of system internal energy due to spatial and temporal temperature differences. In this process, the change of system energy is achieved by changing vibration, collision and migration energy of the micro-particles. Moisture diffusion describes the migration of water molecules and/or the assembly of water molecules in the system. Thus, mass diffusivity of moisture in air is much larger than it is in fibers, whereas the thermal conductivity of fibers is larger than that of air. Furthermore, for most fibers, which are composed of polymers, anomalous mass diffusion processes are observed due to the effects of water molecules on large macromolecules. Although the governing equations for both processes are built on a requirement for balance, thermal conduction is based on energy conservation, and moisture diffusion requires mass conservation. In this chapter, we focus mainly on continuum approaches to thermal conduction and moisture diffusion. This means the micro-level interactions will not be present in the formulations. The fibrous system will therefore be treated as a continuum or several continua, characterized by macroscopic material properties. Most analysis methods will be illustrated for thermal conduction; analogies, to moisture diffusion condition whenever they exist, will be mentioned. More detailed treatment of moisture diffusion, however, such as anomalous diffusion in polymers, are briefly reviewed in Section 7.8.

7.2

Thermal conduction analysis

Generally, the goal of thermal transfer analysis is to determine the temporal and spatial distributions of the scalar temperature field in a given system. To achieve this, the governing equation, and the initial and boundary conditions need to be formulated. Conceptually, detailed information about temperature and derived variables of the system, such as heat flow rate and heat flux through a given surface, will all be available from solutions of the governing


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equations with auxiliary conditions. Although formulation of the governing equation for pure thermal conduction in a homogeneous system is rather simple, a good understanding of the procedure not only illustrates the basic idea about transport processes in general, but builds up fundamentals to extend the analysis of heterogeneous systems such as fibrous systems. When dealing with a physical process in homogeneous and isotropic materials, it is implied that every differential part inside the system will contribute the same response to the process. Thus, the governing equation and bulk material properties can be derived based on one differential unit of the material. Consider an arbitrary volume V of a homogenous and isotropic material bounded by the surface A. The heat flow rate across the surface A, is given by –

∫

q ⋅ ndA

[7.1]

A( t )

where n denotes the unit outward directed normal to A. Assuming no bulk movement of the material, the transfer rate of thermal energy can be related to the change rate of the internal energy in the volume V, ∂ ∂t

∫

V

ρ edV = –

∫

q ⋅ nd A +

A

∫

Φ dV

[7.2]

V

where Φ is the heat generating rate inside volume V, including the adsorption heat, condensation latent heat and so on. Applying the divergence theorem, the surface integral can be changed into a volume integral, and Equation [7.2] becomes

∫

V

 ∂e   ρ ∂t + ∇ ⋅ q + Φ  = 0

[7.3]

Since the volume V is chosen arbitrarily, the governing equation is thus given as

ρ

∂e +∇⋅q+Φ=0 ∂t

[7.4]

However, as we have four unknown variables, e and qi (i = 1, 2, 3), with only one equation now, additional equations have to be established. First, the specific heat, i.e. heat capacity per unit mass, is introduced to describe the relationship between the system’s internal energy and temperature change. The specific heat of a material at constant volume is defined as  ∂e  Cv ( T ) =    ∂T  ρ

[7.5]

The specific heat has dimensions of [energy][temperature]–1[mass]–1. Specific

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heats for general fibers are listed in Table 7.1 (Morton and Hearle, 1993). The constitutive equation for heat flux is the well-known Fourier’s Law. When in differential form, q = – k∇T

[7.6]

where another material property is introduced, the thermal conductivity, k, with dimensions [energy][time] –1[temperature] –1 [length] –1 . Thermal conductivities of some polymer materials that are used as textile fibers are listed in Table 7.2 (Morton and Hearle, 1993; Warner, 1995). Strictly, Fourier’s Law is not a law of nature but an approximation, and potentially it may lead to the problem that heat excitations would be transferred with infinite speed (Ali and Zhang, 2005). However, Equation [7.6] does have some theoretical basis, and has been widely and successfully used in many science and engineering applications (Bird, Stewart et al., 2002). Table 7.1 Specific heats of general fibers Fiber

Specific heat (J g–1 K–1)

Cotton Rayon Wool Silk Nylon 6 Polyester Terylene Asbestos Glass

1.21 1.26 1.36 1.38 1.43 1.34 1.05 0.80

Adapted from Morton and Hearle (1997)

Table 7.2 Thermal conductivity of polymer materials used in textile fibers Material

Thermal conductivity (mW m–1 K–1)

Poly(vinyl chloride) Cellulose acetate Nylon Polyester Polyethylene Polypropylene Polytetrafluoroethylene PET Glycerol Cotton (cellulose) Cotton bats Wool bats Silk bats

160 230 250 140 340 120 350 140 290 70 60 54 50

Adapted from Morton and Hearle (1997)


229

With the relationships shown above, the governing equation for thermal transfer with temperature as the field variable is given by

 ∂T  ρ c v   = ∇ ⋅ ( k∇T ) + Φ  ∂t 

[7.7]

This equation is valid for constant volume processes. For constant pressure cases, however, a corresponding constant pressure specific heat, cp, should be substituted. The difference between the two values is negligible for solids yet relatively larger for liquids and gases (Carslaw and Jaeger, 1986; Bird, Stewart et al., 2002). Considering the processes in fibrous systems in which we are interested, the constant pressure form is obviously more appropriate. For given material properties, the classical three-dimensional conduction equation for constant pressure processes is obtained as ∂T = α∇ 2 T + Φ ρc p ∂t

[7.8]

where α, called the thermal diffusivity, is a combined material property with the dimensions [length]2[time]–1. It is clear that thermal diffusivity has the same dimensions as mass diffusivity D. The dimensionless ratio between these two properties, called the Lewis number, indicates the relative ease of thermal conduction versus mass diffusion transport in a material. This governing partial differential equation shares the same form as the time-dependent diffusion equation when Φ = 0. The corresponding steady-state equation is in the elliptical form. The properties of these equations have been well explored and can be found in books dealing with partial differential equations (Haberman,1987; Arfken and Weber, 2005). In order to obtain the distribution of temperature field, the boundary conditions and initial condition are needed to determine the constants resulting from integration of the governing differential equations. The initial condition for transient thermal conduction is a given temperature distribution in the form of T(x, 0) = f (x)

[7.9]

where f (x) is a known function whose domain coincides with the region the material occupied. A solution of the governing equation, T(x, t) with t > 0, has to satisfy the initial condition lim T ( x , t ) → f ( x ). t →0 The boundary conditions describe the physical behavior at the surface of the material. They are determined from experiments at a given operation environment. Three kinds of boundary condition are often used to approximate real-world situations. (i) Prescribed temperature The prescribed temperature could be constant or a function of time,

230


position or both of them. This boundary condition is mostly well explored and is applicable to model conditions where material boundaries are in contact with a well-controlled thermal environment, such as a thermal guard plate. (ii) Prescribed thermal flux across the boundary surface ∂T This boundary condition implies k = g at the boundary surface, for ∂n t > 0. When the prescribed function g is equal to zero, it represents an insulated condition which is particularly important when fibrous materials are used for thermal insulation. (iii) Linear thermal transfer at the boundary surface This boundary condition assumes that thermal flux varies linearly with temperature difference between the boundary and the environment, given by k

∂T + h( T – Tenv ) = 0 for t > 0 ∂n

[7.10]

in which h is a positive measured variable called the surface heat transfer coefficient. This boundary condition is generally referred to as the ‘Newton’s law of cooling’ and describes a material cooled by an external, well-stirred fluid. Also, it is applicable to black-body or near black-body radiation at boundaries where the temperature difference between the material and the environment is not too large. There are still many other boundary conditions, including both linear and non-linear forms. Some of them are listed in Carslaw and Jaeger (1986). Choosing, or setting up, appropriate boundary conditions depends on one’s understanding of the process and is critical for further analysis. The thermal conduction governing equation with certain initial and boundary conditions can be solved by both analytical and numerical methods. General discussions about analytical methods and their results, such as separation variables, integral transformation and Green functions methods, are available in both applied mathematics and transport phenomena books (Carslaw and Jaeger, 1986; Haberman, 1987; Bird, Stewart et al., 2002; Arfken and Weber, 2005). Numerical methods for thermal conduction problems, such as finite difference and finite elements analysis, are also well developed (Shih, 1984; Minkowycz, 1988). These results are critical not only for thermal analysis but are also important for measurement of thermal conductivity. By carefully setting up experiments, a one-dimensional steady-state heat transfer solution has been applied widely to guide the static hot-plate thermal conductivity measurement (Satsumoto, Ishikawa et al., 1997; Jirsak, Gok et al., 1998; Mohammadi, Banks-Lee et al., 2003). Transient thermal conduction results have also found their application in dynamic measurement of fabric thermal conductivities (Martin and Lamb, 1987; Jirsak, Gok et al., 1998). In order to


231

improve experimental design and data analysis, however, a deeper understanding of these theoretical results and their limitations are required. In fibrous materials, anisotropic characteristics are of predominant importance. It is known that the longitudinal and lateral thermal conductivities of a single fiber are significantly different owing to its anisotropic nature (Woo, Shalev et al., 1994a,b; Fu and Mai, 2003). Furthermore, this directional dependence of thermal conductivity is magnified in fiber assemblies due to asymmetry packing of fibers. In this context, we would like to review some fundamental characteristics of anisotropic thermal conductivity and its effects on the conduction process. The generalization of Fourier’s Law for anisotropic materials is given by q = K · ∇T

[7.11]

where k is the thermal conductivity tensor. In the Cartesian coordinate system, it is written in matrix form as

 k xx  K =  k yx   k zx

kxy k yy k zy

k xz   k yz   k zz 

[7.12]

Depending on the system symmetry, the conductivity matrix can be simplified. It has been proved that the thermal conductivity matrix is symmetrical, based on Onsager’s principle of microscopic reversibility, i.e. krs = ksr for all r and s. The other important aspect for the thermal conductivity tensor is the transformation of the coordinate system. Assume that we try to consider a new Cartesian system x′, y′ and z′, whose directional cosines relative to the old coordinate x, y, z system are (c11, c21, c31), (c12, c22, c32), (c13, c23, c33) respectively. The components of conductivity tensor k ik′ in the new system are given by 3

3

k ik′ = Σ Σ c ri c sk k rs r =1 s =1

[7.13]

These are just the transformation laws for a second-order tensor. With the introduction of the thermal conductivity tensor, the governing equation for homogenous anisotropic materials without heat generation is given by

ρc p

2 2 2 2 ∂T = k xx ∂ T2 + k yy ∂ T2 + k zz ∂ T2 + ( k x y + k yx ) ∂ T ∂t ∂ x∂ y ∂x ∂y ∂z

2 2 ( k xz + k zx ) ∂ T + ( k yz + k zy ) ∂ T ∂ x ∂z ∂ y∂z

[7.14]

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It can be shown that a transformation to a particular Cartesian system ξ, η, ζ leads to the simplified representation

ρc p

2 2 2 ∂T = k1 ∂ T2 + k 2 ∂ T2 + k 3 ∂ T2 ∂t ∂ξ ∂η ∂ζ

[7.15]

These new axes are called the principal axes of thermal conductivity and k1, k2 and k3 are the principal conductivities. The directions of the principal axes depend on the symmetry of the system in question. For an orthotropic system, which has different conductivities k1, k 2 and k3 in three mutually perpendicular directions, these directions coincide with the principal axes. Different from isotropic materials, an important characteristic for heat conduction in anisotropic media is that the heat flux vector does not locate in the same direction as the temperature gradient. Thus, two thermal conductivities at a given point P in an anisotropic material are defined. km is defined as the conductivity in the direction of the flux vector at P, and satisfies qm = – km

∂T ∂m

[7.16]

∂T are the flux and rate of change of temperature along the ∂m direction of flux vector at point P. Similarly, the conductivity normal to the isothermals at P, kn is defined by relating the heat flux and rates of temperature change in the direction normal to the isothermal at P,

where qm and

fn = – K n

∂T ∂n

[7.17]

Relationships between these conductivities with principal conductivities are also found. Assuming the flux vector has directional cosines (l, m, n) relative to the principal axes of the conductivity, the conductivity in direction m, km, is given by 1 = l2 + m2 + n2 km k1 k2 k3

[7.18]

whereas the conductivity normal to isothermal kn, whose normal has direction cosines (l′, m′, n′) relative to the principal axes, is given by kn = l′2k1 + m′2k2 + n′2k3

[7.19]

Depending on the measurement method, km or kn will be measured (Carslaw and Jaeger, 1986).


233

For more discussion about the geometrical properties of thermal conductivities and their effects on the thermal conduction process, one can refer to the classic treatise by Carslaw and Jaeger (1986).

7.3

Effective thermal conductivity for fibrous materials

7.3.1

Introduction

Fibrous materials are widely used in various engineering fields, such as textile fabrics as reinforcements in fiber-reinforced composites, fibrous thermal insulators, and fibrous scaffold in tissue engineering, to just name a few (Tong and Tien, 1983; Tong, Yang et al., 1983; Christensen, 1991; Freed, Vunjaknovakovic et al., 1994). Also, most biological tissues, e.g. tendons, muscles, are intrinsically fibrous materials (Skalak and Chien, 1987). In these applications, fibrous materials are often referred to as assemblies of fibers. The behaviors of these fiber assembles are significantly different from those of single fibers. Systems with fibers are generally heterogeneous. For example, textile fabrics are a mixture of fibers and air, and become a mixture of fibers and water when fully wetted. Fiber-reinforced composite materials are composed of a fiber assembly and matrix materials between fibers. Generally, we treat these mixtures as a whole, heterogeneous material system and analysis of the responses of these heterogeneous materials to external disturbances is our objective in research for engineering applications. Clearly, internal structure, properties of each component, and interactions among components, will determine the behaviors of the whole heterogeneous material. Ideally, a fully discrete analysis based on characterization of each fiber, interstitial materials and interface conditions will provide the most detailed information for the system. But the large number of fibers, often intricate internal structure, and complex interactions of components render the discrete analysis very expensive, if not impossible. One way to overcome the difficulties in analysis of heterogeneous materials is to try to find a hypothetical homogeneous material equivalent to the original heterogeneous one (Bear and Bachmat, 1990; Christensen, 1991; Whitaker, 1999); the same external disturbances will lead to the same macro-responses. The properties of this equivalent homogeneous material are denoted as ‘effective material properties’. As soon as the effective material properties are determined, the analysis of a heterogeneous material can be reduced to that of a homogeneous one, a much easier case to tackle. As in all mixed systems, some of the properties, such as the effective density and specific heat in the thermal conduction case, can easily be obtained by some form of averaging over the corresponding properties of each

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component. However, there are other system properties, including the effective thermal conductivity, that depend not only on the properties of each component, but also on the way those components are assembled into the whole system, i.e. the internal structure and the interactions among the individual components. Sometimes, the effective thermal conductivity can be measured directly. But, there are often many difficulties and practical limitations in the experimental approach. For example, when testing a fibrous material, many issues have to be settled before the test can proceed, such as the time to reach a steady state, influence of other thermal transfer processes, effect of applied pressure, and so on. Also, the results only can be applied in certain environment ranges, and costs are often expensive. Thus, prediction effective thermal conductivity by setting up constitutive laws from component properties and structure is still very attractive. The most important and difficult task in prediction is characterization of structure. The structure of fiber assemblies must be understood from several aspects. Basically, information about the structure of a single fiber is needed, including longitudinal and transverse length, and ratio between them, geometry of cross-sections, crimp of fibers, and so on. After that, distribution of fibers and connection between them are required information for the understanding of fiber assembly structures. Depending on applications, fibers may be woven into yarns and woven fabric forms or packed together into nonwoven form. In the modeling process, an appropriate mathematical description has to be introduced to account for different ways of assembling such as the geometrics of yarns for woven fabrics (Dasgupta and Agarwal, 1992; Ning and Chou, 1995a,b) and orientation functions for the random packing of fibers (Pan, 1993, 1994; Fu and Mai, 2003). The structure of interstitial materials among fibers may also contribute to effective thermal conductivity. But no rules can be summarized unless the particular system is given. The simplest term accounting for interaction between the fiber assembly and other components is the volume fraction of each one. Further interaction characterization needs a knowledge of interface properties, such as contact resistance, continuity of thermal flux, and so on.

7.3.2

Prediction of the effective thermal conductivity (ETC)

Due to the importance of effective thermal conductivity, much work has been done in this field. Most of it has concerned research on porous media and composite materials. The first major contribution should be attributed to Maxwell (Bird, Stewart et al., 2002), who predicted the effective thermal conductivity of composite materials with small volume fraction spherical inclusions. During analysis, only one inclusion sphere embedded in an infinite matrix was considered, with the assumption that the temperature field of a


235

sphere is unaffected by presence of other spheres. The result is represented by k eff 3ε =1+ k1  k 2 + 2k1  – ε  k 2 – k1 

[7.20]

where, k1 and k2 are thermal conductivities of the matrix and inclusion spheres, respectively. ε is the volume fraction of spheres. Generally, analysis for dilute particles tries to solve the problem q = –k1∇T, ∇ · q = ∇2T = 0 in each phase

[7.21]

n · k1∇T = n · k2∇T on interface A12 With given particle geometry and boundary conditions, the solution can be found. And for isotropic materials the effective thermal conductivity is given by k eff = –

〈q〉 〈∇T 〉

[7.22]

where 〈 〉 denote the average over the whole domain. For large particle concentrations, Rayleigh (Bird, Stewart et al., 2002) provides the results with spherical inclusions located in a cubic lattice and square arrays of long cylinders. And Batchelor and Obrien (Batchelor and Obrien, 1977) applied ensemble average and field analysis to dealing with particles. Prediction of the lower and upper bound of effective thermal conductivity is the other important category of prediction methods (Miller, 1969; Schulgasser, 1976; Vafai, 1980; Torquato and Lado, 1991). Miller (1969) used an n-point correlation function to characterize the structure of heterogeneous media. He showed that the simple law of mixtures will be achieved when one-point correlation is adopted, i.e. keff = ε k1 + (1 – ε )k2. In the same paper, threepoint correlation is also used to predict boundries for effective transport properties of heterogeneous media with different geometrical inclusions. Torquato and Lado (1991) predicted the effective conductivity tensor boundaries for media, including oriented, possibly overlapping, spheroids, by noticing the scaling relation between the spheroid and the sphere systems. With incorporation of the probability occurrence of four different packing structures, Vafai (1980) predicted the boundaries for microsphere packing beds. The boundries for the transverse effective thermal conductivity of two-dimensional parallel fibers F1, and three-dimensional dispersed fibrous materials F2 are also found by Vafai (1980), given by F1 ( ε , ω , H ) ≥ ( k eff / k1 k 2 ) ≥ F2 ( ε , ω , H )

236


F1 ( ε , ω , H ) =

1 + ε (ω – 1) ω

ε (ω – 1) 2 (1 – ε )   × 1 –  H 3[1 + ( – 1)][1 + ( – 1) + 3( – 1)(1 – 2 ) ] ε ω ε ω ω ε   F2(ε, ω, H) –1

=

[4(ω – 1) 2 (1 – ε )ε ]   [7.23] ω ω – ε (ω – 1) –  H 3[1 + ω + 3(2 ε – 1)( ω – 1) ]  

where k1 is the larger of two component thermal conductivities, ε is the volume fraction of the component with property k1, ω = k1/k2, i.e. ω > 1, and H is the cell geometry factors. H = 1/4 and 1/6 for two-dimensional parallel and three-dimensional dispersed fibers, respectively. An equivalent inclusion method is applied by Hatta and Taya (1985) and by Ehen and Wang (1996) to predict effective thermal conductivity of a misoriented short-fiber composite. The basic idea is replacement of the inhomogeneity domain by a corresponding inclusion domain filled with a uniformly distributed doublet. Then, the relationship between different temperature gradients is given in index form,  ∂T 0  ∂T 0 ∂ T˜ ∂T c ∂T *  ∂T˜ ∂T c  k m δ ij  + + – = k ijf  + +  ∂x j ∂ x j  ∂x j ∂x j ∂x j   ∂x j  ∂x j [7.24] where km and k ijf are thermal conductivities of matrix and fibers, respectively. ∂T 0 is the temperature gradient related to the far field applied heat flux; ∂x j

∂T c is the temperature gradient disturbed by the existence of the ∂x j ∂T * inhomogeneity; corresponds to the uniformly distributed doublet in the ∂x j ∂T˜ inclusion domain and is the temperature gradient related to interaction ∂x j between inhomogeneities. By setting up a relationship between these temperature gradients and applying Fourier’s law for each phase, the effective thermal conductivity of the composite material is given by the relationship, k ijeff

∂T t ∂x j

= k m δ ij

∂T t ∂x j

+ 1 ( k ijf – k ijm δ ij ) VD

∫

Ω

∂T t dV ∂x j

[7.25]


237

∂T t is the total (actual) temperature gradient and is related to the ∂x j temperature gradients mentioned above; Ω denotes the inhomogeneities domain and 〈 〉 means averaging over the whole composite body. Integration in the above equation is performed by consideration of fiber orientation distribution. Hatta and Taya (1985) and Chen and Wang (1996) present the results for three-dimensional and two-dimensional misoriented short-fiber composites with uniform distribution and cosine-type distribution. There are still more methods for predicting effective thermal conductivity of heterogeneous materials (Schulgasser, 1976; Nayak and Tien, 1978; Furmanski, 1992). We will review thermal resistance network models, the volume averaging method and the homogenization method in the following three sections. For more detailed information, please refer to the review for composite systems by Progelhof, Throne et al. (1976) and the review for porous media by Kaviany (1995).

where,

7.4

Prediction of ETC by thermal resistance networks

The thermal resistance network method is based on the similarity between thermal conduction and electrical conduction. By parallel or serial connecting components of the system, a thermal resistance network is built up. This has been successful applied in many multiphase systems. Hsu has predicted the effective thermal conductivity of a packed particle bed by this method. With appropriate treatment of the thermal resistance network, the particle morphology, contacts between particles, and even the bi-porous structure of particles, can all be incorporated into the model and provide fairly good results (Hsu, Cheng et al., 1994; Cheng and Hsu, 1999; Chen, Cheng et al., 2000). Applications of this method to the fibrous system are also found in the literature; such materials as unidirectional fiber-reinforced composites (Springer and Tsai, 1967), fabric-reinforced composites (Dasgupta and Agarwal, 1992; Ning and Chou, 1995a,b; Dasgupta, Agarwal et al., 1996), nonwoven textile fabrics (Woo, Shalev et al., 1994a), and misaligned short-fiber-reinforced composites (Fu and Mai, 2003). In the next part, procedures and results from the application of the thermal resistance network method to the fibrous system will be carefully reviewed. The simplest application of this method to the fibrous system, such as fiber-reinforced composite and textile fabrics, is prediction of the upper and lower bound of effective thermal conductivity by parallel and serial arrangement of each phase: keff,upper = kfVf + kmVm, keff,lower = 1/(Vf /k f + Vm /km)

[7.26]

The bounds resulting from this prediction are generally too wide to apply.

238


The volume fraction alone is not enough to characterize the contributions of the fibers and the matrix and interactions between them. More geometrical description of each phase has to be introduced into the model to get reasonable results. This implies that the structure characterization should be emphasized during the modeling process. As a first step for the thermal resistance network method, a unit cell is chosen from the system. The unit cell is the smallest repeating pattern of the fibrous system and represents all geometrical information at a microscopic level. The thermal resistance network is built up by dividing the unit cell into several components, which can be a single-phase material or a combination of multi-phase materials. Based on certain assumptions of the thermal conduction process and the structure of the unit cell, a thermal resistance network can be built up by serial or parallel connection of the unit cells. For a spatially periodic fibrous system, the effective thermal conductivity of the unit cell is just the bulk effective properties of the system. But, the arrangement of unit cells also contributes to system-level effective thermal conductivity when the system is built up by spatially distributed unit cells. The other important point in application of the thermal resistance network model lies in the assumption of a thermal conduction process inside the unit cell. Due to the geometry of the fibers and the complex packing pattern, many fibrous materials are anisotropic, and effective thermal conductivity has to be predicted for a given direction. Generally, the temperature gradient is applied to the unit cell only along one direction. The surfaces of the unit cell parallel to the one-dimensional heat flux are assumed to be insulated surfaces (Springer and Tsai, 1967; Dasgupta and Agarwal, 1992; Ning and Chou, 1995b; Cheng and Hsu, 1999). By solving this one-dimensional steadystate thermal conduction problem, the effective thermal conductivity of the unit cell in the conduction direction is obtained. Though thermal conduction through the two phases’ interface is a multidimensional process, a onedimensional approximation is valid for most conditions because effective thermal conductivity is an averaged bulk property. Our review of the thermal resistance network method will start from a simple system – a unidirectional fiber-reinforced composite. Springer and Tsai (1967) analyzed composites with filaments arranged in the rectangular periodic array shown in Fig. 7.1. Filaments were uniform in shape and size, also symmetrical about the x- and y-axes. The unit cell was chosen straightforwardly as in Fig. 7.2. Due to the structural symmetry, only two effective thermal conductivities need to be evaluated. One was along the longitudinal direction of the fibers, keff.zz . The other was the transverse effective thermal conductivity keff,t . The longitudinal ETC, keff,zz can be easily predicted by assuming a parallel arrangement of the matrix and the fibers. On the other hand, the transverse ETC keff, t is predicted by applying the thermal resistance network model. With the assumption of one-dimensional thermal conduction, heat flows along the x-direction through


239

Y

2b

X

2a

7.1 Structure of unidirectional fiber-reinforced composites with rectangular filaments arrangement. Adapted from Springer, G.S. and S.W. Tsai, ‘Thermal conductivities of unidirectional materials’. Journal of Composite Materials, 1967. 1: pp. 166–173.

Y

Q1

Q2

h = f (Y )

x

S Fiber

Q3

2a

Matrix

7.2 Unit cell used in effective thermal conductivity prediction. Adapted from Springer, G.S. and S.W. Tsai, ‘Thermal conductivities of unidirectional materials’. Journal of Composite Materials, 1967. 1: pp. 166–173.

240


three parallel components. The thermal resistance of each component in the thermal resistance network is given by Ri =

li Ai k i

[7.27]

where li is the component dimension along the conduction direction; Ai is the cross-sectional area orthogonal to the conduction direction; ki is the thermal conductivity of the component. In the unit cell, three parallel components are easily identified, shown in Fig. 7.2. Components 1 and 3 are composed of purely matrix material and the thermal resistance of them is written by

1 + 1 = (2b – s ) wk m 2a R1 R3

[7.28]

where w is the length in the z-direction, and is constant for a unidirectional system. The component 2 is a combination of matrix material and fiber, i.e. the interphase between the matrix and the fiber, whose thermal resistance R2 may be calculated from the thermal resistance of an infinitely thin slice dy, R2,d y =

1  2a – h ( y ) + h ( y )  wdy  km k f 

[7.29]

Three components are connected in parallel. The thermal effective conductivity of the unit cell is obtained from the relationship

1 = 2a = 1 + 1 + 1 R 2bwk eff R1 R2 R3

k eff = 1 – s  + a b 2b  km 

∫

s

0

dy (2a – h( y )) + h( y )( k m / k f )

[7.30]

[7.31]

The effects of structure are shown in two ways. Firstly, the geometry of the fibers is characterized by two variables: s, the maximum dimension of the fiber in the y direction; and h(y), the width of the fiber at any given y. Both are shown in the equation. Then the rectangular packing pattern of unit cells exhibits its effect by parameters a and b. By choosing appropriate unit cells, other regular packing patterns can be handled in the way similar to the above derivation. Springer and Tsai (1967) predicted the effective thermal conductivity of square fibers and cylindrical fibers in a square packing pattern. k eff,square = (1 – km

Vf ) +

1 V f + B /2

[7.32]


241

k eff,cylinder = (1 – 2 V f /π ) km  + 1 π – B 

1 – ( B 2 V f /π )  4 –1  tan 1 + ( B 2 V f /π )  1 – ( B 2 V f /π )  [7.33]

where k  B = 2  m – 1  kf  These results were compared with numerical calculations from the shear loading analogy and experimental data (Springer and Tsai, 1967). Depending on the thermal conductivity ratio between the fibers and the matrix, the discrepancies between the two models and experiment data are different. But the difference is generally less than 10%. Considering the simple derivation procedure and resulting analytical equations, the thermal resistance network provides a reasonably accurate method for unidirectional composite analysis. As shown in the above example, structure characterization determines effective thermal conductivity prediction. The importance of, and difficulties in, structure modeling are well illustrated in the following reviews of woven fabric composites (Dasgupta and Agarwal, 1992; Ning and Chou, 1995a,b 1998; Dasgupta, Agarwal et al., 1996).

7.5

Structure of plain weave woven fabric composites and the corresponding unit cell

In order to simplify structure characterization, Ning and Chou (1995a,b, 1998) idealized the unit cell by replacement of the yarn crimp with linear segments. Taking account of the symmetry of the unit cell, it is assumed that transverse thermal conductivity can be predicted by analysis on a quarter of the idealized unit cell. This implies that the interaction between the quarters of the unit cell is negligible. In order to predict transverse effective thermal conductivity, thermal conduction in the unit cell is assumed to be onedimensional, and heat flow lines to be all parallel to the z-axis. The unit cell is partitioned into eleven components, with the characteristics that each component is composed of a single material. Taking advantage of this partition and simplified geometry, the thermal resistance of each component can be calculated in simple algebraic form. The effective thermal conductivity of the unit cell is obtained by constructing the thermal resistance network of each component. Based on a structure periodicity assumption, the effective thermal conductivity of the whole woven fabric composite is the same as a single unit cell.

242


k eff

 gf  g af km  gw f + = + a a g h  w f k m hw  hm f   1 + gw   1 + f  + +  h a w   a f    h  k w1 h 

1 hm  k m hw k m + + h  kw 2 h k f 2

gw aw + h f   hm h f  k m h f + + h  kf1 h h   h

     

[7.34]

The parameters in the above equation can be classified into two categories: gw, gf, aw, af, hm, hf, h are geometrical characteristics of the unit cell and are determined by the weave style. km, kd1, k d 2 (d = w, f ) are the thermal conductivities of resin matrix and impregnated warp and fill yarns with mean fiber orientation angle θ di Taking into account the measurement of these parameters, two more steps are needed for prediction closure. Yarn thermal conductivity is predicted by assuming that yarns are unidirectional fiberreinforced composites with certain fiber orientations. Hence, these parameters are predicted by k di = k a sin 2 θ di + k t cos 2 θ di ( d = w , f i = 1, 2)

[7.35]

where ka and kt are the longitudinal and transverse thermal conductivity of the yarns without fiber orientation and are calculated from the fiber and resin thermal conductivity and fiber volume fraction in the yarn (Dasgupta and Agarwal, 1992; Ning and Chou, 1995a,b; Dasgupta, Agarwal et al., 1996). θ di is the mean fiber orientation angle with respect to the x- or y-axis, and is measurable for given woven fabric composites. Considering the geometrical characterization of the unit cell, the matrix volume fraction hm is rather difficult to measure practically. The way to overcome this difficulty is by relating this parameter to the fiber volume fraction, h, in both composites and yarns. Vf hm =1– h Vf y

gf  h f gw gw  hw g f  1 + a  1 + a  + h a + h a  w w f  f

[7.36]

With these two additional equations, the transverse effective thermal conductivity of plain weave woven fabric composites can be predicted from all measurable parameters. The effects of volume fraction and weave style on effective thermal conductivity are discussed for yarn-balanced fabric composites and compared with other numerical and experimental results (Ning and Chou, 1995a). The consistency of these data implies that the


243

thermal resistance network method is robust and that the assumptions made during derivation are valid under pure thermal conduction. Using the same method and assumptions, Ning also successfully predicted the transverse effective thermal conductivities of twill weave, four-harness satin weave, and five- and eight-harness satin weave fabric composites. The results are documented in the literature in their general form (Ning and Chou, 1998). Dasgupta and Agarwal (1992) and Dasgupta, Agarwal et al. (1996) also analyzed the woven fabric composites by a homogenization method and the thermal resistance network model. The unit cell used in this work is shown in Fig. 7.3. Instead of simplification by linear segment, vertical cross-sections and undulation of yarns are approximated by circular arcs in Dasgupta’s work. The effective thermal conductivity of the unit cell has to be calculated from analysis of infinitesimal slices and numerical integration over the whole unit cell domain because of the complex structure of the unit cell. The other important point of this model is incorporation of correction for heat flow lines. Based on observation from the homogenization analysis, Dasgupta allowed the heat to flow preferably from transverse yarns to longitudinal yarns when the resin had high thermal resistance. In-plane and out-of-plane effective thermal conductivity of plain weave fabric composites are all predicted in numerical form based on the thermal resistance network method. Comparison of the homogenization method and experimental data shows good prediction ability for the model. Nonwoven fabric is the other important category of fibrous materials. Fibers are spatially distributed and packed together to form a network structure. The thermal conductivity of a single fiber, fiber volume fraction and orientation of the fibers will determine effective thermal conductivity of nonwoven

h e 2a b 2a e c dd

c

7.3 Unit cell of a balanced plain weave fabric-reinforced composites lamina. The warp yarn and fill yarns are assumed to be identical. Adapted from Dasgupta, A. and R.K. Agarwal, ‘Orthotropic ThermalConductivity of Plain-weave Fabric Composites using a Homogenization Technique’. Journal of Composite Materials, 1992. 26(18): pp. 2736–2758.

244


fabric. Based on analysis of the unit cell, Woo, Shalev et al. (1994a) proposed a model in terms of measurable geometry parameters to predict out-of-plane effective thermal conductivity of nonwoven fabrics. As shown in Fig. 7.4(a), the unit cell is chosen as two touching layers of fiber assembly. The number of fibers oriented along the x- and y-axes are n and m, respectively. Applying the thermal resistance network method, the effective thermal conductivity of this unit cell is given by Z

1

1

y d

d

nd

Xf 1

md

Z

θ

φ

7.4 (a) Idealized unit cell structure of nonwoven fabrics. (b) Orientated unit cells simulating structure of real nonwoven fabrics. Adapted from Woo, S.S., I. Shalev, and R.L. Barker, ‘Heat and Moisture Transfer Through Nonwoven Fabrics.1. Heat-Transfer’. Textile Research Journal, 1994. 64(3): pp. 149–162.


k eff,zz = Po k a +

(1 – Po ) 2 Vf + (1 – Po – V f )/ k a kI

245

[7.37]

k eff ,xx = (0.5 – V f 1 ) k a + V f 1 k 2 +

0.5 (1 – 2V f 2 ) 2V f 2 k1 + ka

[7.38]

k eff,yy = (0.5 – V f 2 )k a + V f 2 k 2 +

0.5 (1 – 2V f 1 ) 2 V f1 k1 + ka

[7.39]

where Vf 1 and Vf 2 are the fiber volume fractions along the x- and y-directions; k1 and k2 are the longitudinal and transverse thermal conductivities of a single fiber; Po is the optical porosity of the unit cell, which corresponds to the area fraction of through pores, given by Po = 1 – nd – md + ndmd = 1 – Vf + (8/π)2Vf 1Vf 2

(7.40)

The nonwoven structure cannot be reconstructed by simply periodic packing of the unit cell. Practically, the behavior of nonwoven fabrics will be better represented by the unit cell with a certain orientation, shown in Fig. 7.4(b). Because orientation distribution function is not introduced in Woo’s model, the polar orientation angle θ and azimuthal orientation angle φ in the following discussion should be considered as average quantities. The out-of-plane effective thermal conductivity of nonwoven fabric is obtained by analysis of this oriented unit cell, keff,oz = keff,xx(cos2 φ cos2θ) + keff,yy(cos2 φ sin2θ) + keff,zz(sin2 θ)

[7.41]

The optical porosity depends on the thickness of the nonwoven fabric. Based on this observation, Woo assumed that unit cells are regularly packed along the fabric thickness direction for predicting whole fabric optical porosity Pi = [1 – (8/π)Vf 1 – (8/π)Vf 2 + (8/π)2Ff 1Vf 2 ]L/(2d)

[7.42]

With this correction, the out-of-plane effective thermal conductivity, i.e. keff,oz is given by keff,oz = ka{sin2 φPi – cos2φ [cos2θ (0.5 – Vf 1) + sin2θ (0.5 – Vf 2)]} + k2 cos2 φ (cos2θ Vf 1 + sin2θVf 2 + 0.5 cos2 φ {cos2θ /[2Vf 2 + /k1 + (1 – 2Vf 2)/k a ]} + sin2θ /[2Vf 1/k1 + (1 – 2Vf 1)/ka] + sin2 φ (1 – P1)2/[Ff /k1 + (1 – Pi – Vf )/k a]

[7.43]

This representation is rather clumsy and some parameters may not be

246


measurable. Woo simplifies the above equation by structuring special nonwoven fabrics in his research. For melt blow or spunbond nonwovens, the average polar orientation angle is approximately zero. Also, an easily measurable anisotropy factor is introduced to take account of the distribution of fibers inside the unit cell,

α = Vf 1/Vf 2

[7.44]

The resulting out-of-plane effective thermal conductivities are given in the form of measurable physical parameters, keff,oz = ka sin2 φPi + k2 cos2 φαVf /(1 + α) + sin2 φ (1 – Pi)2/ + [Vf /k1 + (1 – Pi – Vf )/ka] + cos2 φ (1 + α – α V f )2/ + {(1 + α )[Vf /k1 + (1 – Vf )(1 + α )/ka]}

[7.45]

and Pi = [1 – (8/π)Vf + (8/π)2 V f2 α /(1 + α )2]L /(2d)

[7.46]

In Woo’s work, a series of measurements for different nonwoven fabrics have been made and have validated the prediction model (Woo, Shalev et al., 1994a). It is seen from the above equation that the effective thermal conductivity of nonwoven fabrics is influenced by many physical characteristics, such as fiber volume fraction, anisotropic thermal conductivity of single fibers, orientation of fibers, and so on. The contribution of these effects can be obtained from parameter analysis and validated by experiments. However, the present model is simplified by considering the structure of specific systems. It is better to consider the prediction equation as a semi-experimental approach. In some fibrous materials, such as short-fiber-reinforced composites and textile fiber assemblies, the structure of the system is best described using statistical distribution functions. Compared with mechanical property prediction, analyzing effective thermal conductivity based on a statistical approach is relatively rare (Hatta and Taya, 1985; Chen and Wang, 1996; Fu and Mai, 2003). Among them, Fu and Mai (2003) present a simple model to predict thermal conductivity of spatially distributed, short-fiber-reinforced composites. Depending on the researchers, different statistical distribution functions have been employed to describe fiber distribution. Fu introduced two density functions to account for fiber length and orientation distributions. Fiber length distribution: f (L) = abLb–1 exp(–aLb) for L > 0

[7.47]

Fiber orientation distribution: g(θ, φ) = g(θ) g(φ)/sin θ

[7.48]


247

where g(θ) = (sin θ)2p–1(cos θ)2q–1/

∫

θ max

θ min

(sinθ ) 2 p–1 dθ for 0 ≤ θmin

≤ θ ≤ θmax ≤ π 2

[7.49]

g(φ) is defined in a similar way to g(θ). The parameters a, b, p, q are applied to represent the size and shape of the distribution density function and can be measured for given composites. As Fig. 7.5 shows, Fu’s model tries to predict effective thermal conductivity along direction 1. The laminate analogy approach (Agarwal, 1990) is employed to formulate the model. The original composite with distribution functions f(L) and g(θ, φ) is illustrated in Fig. 7.5(a). Because only the thermal conductivity in direction 1 is concerned, the original composite is first approximated as a hypothetical composite with orientation distribution g(θ, φ) = 0 as in Fig. 7.5(a). The next approximation step is treating the hypothetical composite as a combination of laminates as seen in Figs. 7.5(b) and 7.5(c). Shown in Fig. 7.5(d), the final ‘equivalent’ system is a series of lamina L(Lj, θj), j = 1,2, . . . , m. Each lamina contains fibers with the same length Lj and orientation angle θj. Based on this laminate analogy approach, the thermal conductivity of each laminate is predicted from the results of unidirectional fiber-reinforced composites with a certain orientation angle. The Halpin–Tsai equation (Agarwal, 1990) is applied in Fu’s work.

k1 =

1 + 2α µ1V f km 1 – µ1V f

µ1 =

k f 1 / km – 1 k f 1 / k m + 2α

k2 =

1 + 2µ 2 V f k 1 – µ2Vf m

µ2 =

k f 2 / km – 1 k f 2 / km + 2

[7.50]

where α = L /d f is the aspect ratio of the fibers. Taking account of the orientation of the fibers, the thermal conductivity of each laminate is given by k i j = k1j cos2 θ j + k2 sin2 θ j

[7.51]

Assuming all laminates are connected in parallel with respect to direction 1, the effective thermal conductivity of the composite is predicted by integration with the distribution density functions, M

k eff = Σ k lj h j j =1

=

∫

Lmax

Lmin

∫

θ max

θ min

( k1 cos 2 θ + k 2 sin 2 θ ) f ( L ) g (θ ) dLdθ

[7.52]

248


1

θ 3

φ 2 (b)

(a)

L ( L 1 ) L (L 2 )

(c)

…

L (L n )

L(Ll, θ1 = 0°) L(L2, θ2) … L(L1, θm = 90°)

(d)

7.5 (a) Real misaligned short-fiber-reinforced composites with orientation distribution g ( θ, φ). (b) Hypothetical composite with orientation distribution g (θ, φ = 0). (c) Hypothetical composite treated as combination of laminates L(Lj ), and each laminate contains fibers of same length L j . (d) Each laminate is treated as a stacked sequence of lamina L(L j, θ j ), and each laminae contains fibers with same length L j and orientation angle θ j . From Fu, S.Y. and Y.W. Mai, ‘Thermal conductivity of misaligned short-fiber-reinforced polymer composites’. Journal of Applied Polymer Science, 2003. 88(6): pp. 1497–1505. Reproduced with permission.

Parameter analysis is performed by Fu to evaluate the effects of volume fraction, mean fiber length and mean fiber orientation angle on effective thermal conductivity. For uniform length short fibers, the thermal conductivity of two-dimensional and three-dimensional random fiber distributions is easily predicted by the simplified distribution functions.


249

k eff ,2 D = 1 ( k1 + k 2 ) 2

[7.53]

k eff ,3D = 1 k1 + 2 k 2 3 3

[7.54]

Unfortunately, further discussion concerning distribution function effects is not available in current literature. Improvement of the present statistical model is still needed. In this section, we have reviewed the prediction of the effective thermal conductivity of fibrous materials by the thermal resistance network method. With the assumption of a one-dimensional conduction process and easily built thermal circuits, this method provides a simple and efficient way for thermal conductivity prediction. Comparison with other methods and experimental data also shows that reasonable accuracy can be achieved with appropriate treatment of structures. The numerical, even analytical in some cases, results from this relatively simple method, are believed to be very useful for practical engineering and science applications.

7.6

Prediction of ETC by the volume averaging method

Fibrous materials are not only multiphase but also multiscale systems. With a glance at textile fabrics, several disparate length scales can be identified, such as the diameter of fibers, the length of fibers, the distance between fibers, the size of the whole fibrous system, and so on. Analysis of these multiscale systems may have special challenges due to interactions between different scales. Local volume averaging is a method to upscale the system from the microscale to the macroscale. It has been widely applied in the field of porous media. A well-known example is starting from the microscopic Navier–Stokes equation to arrive at the macroscopic Darcy’s law for creeping flow through porous media (Whitaker, 1969, 1999; Kaviany, 1995). The volume averaging method is well suited for multiphase systems, such as fibrous materials. Textile fibers can form network structures, even with a very low fiber volume fraction. A fiber assembly can be treated as a single continuum, which is called the solid phase in porous media study; the air or water inside the voids between the fibers is referred to as the fluid phase. The length scale, corresponding to the void in fibrous materials, should be the average distance between fibers. Based on basic geometrical fibrous characterization (Pan 1993, 1994), we can get this distance and relate it to the general geometry parameters of the textile fibrous system. Thus, treatment for general porous media may be applied to textile fabrics with appropriate adjustment. In this section, we will review the basic ideas of the local volume averaging method and its application to pure thermal conduction.

250


The first step for the application of the volume averaging method is finding an appropriate representative element volume (REV), also called averaging volume, schematically shown in Fig. 7.6. Generally, averaged properties, such as porosity, will depend on the chosen average volume. The representative element volume in porous media is identified as a volume range, in which averaging properties is independent of volume size, i.e. adding or subtracting pores and solids does not change the average value. Bachmat and Bear (1986; Bear and Bachmat, 1990; Bear, Buchlin et al., 1991) provide detailed discussion about size of REV based on porous media structure and statistical concerns. Representative element volume size is also important for assumptions made during the volume averaging process and will be discussed in following parts. Volume averaged variables are defined by integration of micro-scale variables over the whole REV. For any quantity ψ associated with the fluid, the volume averaged value for the centroid of REV is defined in two ways: superficial averaged ψ is

Iv

R O D

7.6 A typical representative element volume (REV) selected from fibrous materials.


〈ψ 〉 = 1 V

∫

ψ dV

251

[7.55]

Vf

where V = Vf + Vs; and intrinsic averaged ψ is

〈ψ f 〉 f = 1 Vf

∫

Vf

ψ dV .

[7.56]

Generally, intrinsic averaged value is preferred because it is a better representation of properties in the fluid phase. The relationship between them is given by 〈ψ 〉 = ε 〈ψ 〉 f differing by the porosity ε. The same definitions and operations are also applicable for solid phase variables. Throughout the whole fibrous system, we can select REV and perform the volume averaging operation point by point. Thus, new variables over the whole fibrous system are defined. These variables from volume averaging methods have their thermodynamic significance; for instance, discussion about volume averaged temperature is available from Hager’s work (Hager and Whitaker; 2002). Now, one question may be raised – why volume averaged temperature is needed for thermal analysis of fibrous materials. The requirement for these averaged variables lies on two sides, the intrinsic multiscale properties of the fibrous system and the experimental measurement conditions. In previous sections, we discussed only the point temperature field in homogeneous and heterogeneous systems. But, point temperature is a microscale variable in a multiscale system. That means that the characteristic length of a point temperature in a fibrous system will be the size of fibers or the average distance between fibers. From the whole system point of view, i.e. fabrics, the point temperature fluctuates spatially with very high frequency. Detailed information about point temperature will not only depend on boundary conditions imposed on fabrics but also on short-length correlations between fibrous system structures. On the other hand, volume averaged temperature will provide much less frequent fluctuation over the whole fibrous domain by smoothing out fluctuations over the REV, schematically illustrated in Fig. 7.7. Hence, volume averaged temperature is characterized by macroscopic length and is appropriate for analyzing thermal response of whole fibrous materials to certain excitations. The other reason to adopt volume averaged temperature lies in the measurement of temperature fields and setting up boundary conditions. In most scientific and engineering applications, instruments used to measurement temperature must have a measure window. Results from the instruments are volume averaged temperature over the measurement window (Bear and Bachmat, 1990; Bear, Buchlin et al., 1991). Furthermore, boundary conditions in most scientific and engineering applications are not specified as point temperature. They are generally specified in macroscopic variables; for example,

252


O

A

b

l

B

l

b

< T >o

To

l b

l:Fluid b : Fiber

7.7 Schematic illustration of point temperature and volume averaged temperature fluctuation in the REV.

area average temperature and heat flux are specified in heat plate methods (Satsumoto, Ishikawa et al., 1997; Jirsak, Gok et al., 1998; Mohammadi, Banks-Lee et al., 2003). The advantage of applying volume averaging methods is gained by sacrifice of detailed microscopic information. This means that this method is not efficient in predicting behavior at pore and fiber scale. However, the thermal response of the fibrous system to macroscopic boundary and initial conditions are most attractive information for us. Thus, the volume averaging method is appropriate for this purpose. The importance of volume averaging variables has been realized by textile scientists and applied to the analysis of heat and mass transfer through fabrics (Gibson and Charmchi, 1997; a,b; Fohr, Couton et al., 2002; de Souza and Whitaker, 2003). However, the ability of the volume averaging method to upscale the system and predict effective thermal conductivity of the system is rarely found in fibrous materials references. In this section, we will review procedures for the derivation of effective thermal conductivity by the volume averaging method. Following the methods developed by Whitaker (Whitaker, 1991, 1999; Quintard and Whitaker, 1993; Kaviany, 1995), the macroscopic governing equation and a closed solution for effective thermal conductivity will be obtained for the system with special structures. In the following discussion, fibers are assumed to be interconnected to form a continuous phase, referred as the solid phase. Pores are assumed to be fully saturated by air or water, then denoted as the fluid phase. Thermal conduction is assumed to be the only dominant heat transfer process. Based on these assumptions, the point governing equation can be written for each phase as


( ρc p ) s

∂Ts = ∇ ⋅ ( k s ∇Ts ) ∂t

( ρc p ) f

∂T f = ∇ ⋅ ( k f ∇T f ) ∂t

253

Tf = Ts on Afs –nfs · kf∇Tf = –nfs · ks∇Ts on Afs

[7.57]

in which the boundary conditions indicate that the temperature and the normal component of the heat flux are continuous at the fluid–solid surfaces. Thermal conductivity ks for the fibrous phase should be treated as a lumped parameter, which includes bulk heat conductivity of single fibers and thermal contact resistance between fibers. It is clear from observation of these equations that two more boundary conditions at the fabric boundaries and one initial condition are needed to explicitly solve the point temperature field. However, this information is not generally available in the form of point temperature and is not important for derivation of effective thermal conductivity. It will not be shown in the following discussion. Upscaling is achieved by performing volume averaging operations on the above point governing equations. Due to the similarity between solid and fluid phases, we will only discuss procedures for the fluid phase equation. The resulting volume averaged equation for the fluid phase is given by

ε ( ρc p ) f

∂〈T f 〉 f = 〈∇ ⋅ ( k∇T f ) 〉 ∂t

[7.58]

where 〈 〉 denote superficial volume averaging. In order to obtain the macroscopic governing equation, the right-hand side of the above equation must be related to the gradient of the volume averaging temperature. This step is done by application of the spatial averaging theorem, which has already been developed and well discussed by several researchers (Whitaker, 1969, 1999; Gray, 1993; Slattery, 1999). 〈∇ψ 〉 = ∇〈ψ 〉 + 1 V

∫

〈∇ ⋅ ψ 〉 = ∇〈ψ 〉 + 1 V

∫

n sf ψ d A

[7.59]

n sf ⋅ ψ d A

[7.60]

Asf

Asf

After applying the averaging theorem twice to the volume averaged governing equation, the result is given by

254


ε ( ρc p ) f

∂〈 T f 〉 f  = ∇ ⋅  k f  ε ∇〈 T f 〉 f + 〈 T f 〉 f ∇ε + 1 V ∂t    n fs T f d A  + 1 A fs   V

∫

∫

n f s ⋅ k f ∇T f d A

[7.61]

A fs

The last term in above equation corresponds to the interfacial heat flux at the fluid and solid interface and will be handled with the information from the solid phase. Now, the central problem turns out to be the integral of point temperature over the fluid–solid interface. As shown by Slattery (1999) and Whitaker (1999), this problem can solved by introducing spatial decomposition of point temperature as

T f = 〈 T f 〉 f + T˜ f

[7.62]

After substituting decomposition form into the governing equation, the integral n 〈 T 〉 f d A , needs to be term of the volume averaged temperature, 1 V As f sf f noticed. It is clear that this integral is evaluated from the volume averaged temperature other than the centroid of the REV. This is an indication of nonlocal transport phenomena. In order to get the local form-governing equation, Taylor expansion and order of magnitude analysis is applied. The result is given by

∫

1 V

∫

n sf 〈 T f 〉 f d A = – 〈 T f 〉 f ∇ε

[7.63]

As f

with length scale constraints, lf 0.00777 kg/m2⋅s [8.32a] H co ′ = 2.0

W/m2⋅K, m˙ ay ≤ 0.00777 kg/m2⋅s

H mo ′ = 3.408 × 10 m˙ a – 1.2766 × 10 m˙ ay > 0.00777 kg/m2⋅s –3

H mo ′ = 1.3714 × 10 –5

kg/m2⋅kPa⋅s,

–5

[8.32b]

kg/m ⋅kPa⋅s, 2

[8.32c] m˙ ay ≤ 0.00777 kg/m2⋅s [8.32d]

The inner node transport coefficients to be used in the fabric model are as reported by Ghali et al. (2002a) at H ci′ =1.574 W/m2⋅K, and H mi ′ = 7.58 × 10–6 kg/m2⋅kPa⋅s. Internal convection coefficients from the skin to the microclimate air layer Several researchers have empirically estimated the internal convection coefficients between the skin and the trapped air layer under dynamic conditions initiated by motion. Lotens (1993) reported internal mass transport coefficients in two-layer clothing at the skin to the clothing layer, for various garments and apertures. Havenith et al. (1990a) reported data for a clothing ensemble of cotton/polyester workpants, polo shirt, sweater, socks, and running shoes. Their data on dynamic clothing insulation of skin surface air layer were based on measurements of dry heat loss where the subject skin was wrapped tightly with a thin, water-vapor impermeable, synthetic foil. Danielsson (1993) reported internal forced convection coefficients for various parts of the body for a loose-fitting ensemble at walking speeds of 0.9, 1.4 and 1.9 m/s. Ghaddar et al. (2003, 2005b) experimental data on the convective transport coefficients from the skin to the internal air layer were based on the evaporative heat loss and the moisture adsorption in the clothing due only to normal ventilation action of the fabric for both planar and cylindrical geometry of the fabric boundary under periodic ventilation. The dry convective heat transport coefficient from the skin to the lumped air layer hc(skin-air) was found from the Lewis relation for air–water vapor mixtures (ASHRAE, 1997). Ghaddar et al. (2003) experimental findings of convection coefficients are within 8%


293

of the findings of Danielsson, at a walking speed of 0.9 m/s, for the trunk and the arm parts of the body. The mean transport coefficients for a cylindrical geometry are 29% lower than the planar normal periodic flow coefficients reported by Ghaddar et al. (2005b). This is expected due to the reduced normal ventilation rate and increased angular motion parallel to the inner surface within the microclimate air layer annulus of the cylindrical geometry. Table 8.1 presents a summary of transport coefficients reported by various researchers for closed aperture high air permeable cotton clothing at various walking speeds, external winds, or frequencies. When internal ventilation convection coefficients are known at the skin, then the steady periodic time-averaged sensible and latent heat losses per unit area from the skin can be calculated, respectively, as

 QS = hc (skin-air)  1 τ

∫

t +τ

t

 ( Tsk – Ta )dt  + hr 1 τ 

∫

t +τ

( Tsk – To )dt

t

[8.33a]

 Q L = h fg hm (skin–air)  1 τ

∫

t

t +τ

 ( Psk – Pa ) dt  

[8.33b]

In addition, the average overall dry resistance of clothing, IT (clo) and evaporative resistance RE can be determined from the Jones and McCullough (1985) definition of IT =

( Tsk – T∞ ) Cl QS

[8.34a]

RE =

( Psk – P∞ ) QL

[8.34b]

where Cl is the unit conversion constant = 6.45 cloW/m2 °C, and the clo value is a standard unit for comparing clothing insulation. External convection coefficients Many researchers have estimated the heat transfer coefficient at the external exposed surface of clothing subject to elevated air velocities (Nishi and Gagge, 1970; Kerslake, 1972; Fonseca and Breckenridge, 1965; Danielsson, 1993). They suggested formulae for calculating the average convective coefficients from the human body for a range of speeds and body postures in the form of b hc ( o–air) = a ⋅ υ eff

[8.35a]

where υeff is the effective wind velocity in m/s, b is a constant whose value is close to 0.5, and a is a constant evaluated from the characteristic diameter of the whole body, given by Danielsson (1993) as a = 4.8 × d – 0.33

[8.35b]

0 0.694 1.388 0 0.694 1.388

0.2 0.3

7.96 × 10–5 10.69 × 10–5 12.79 × 10–5 9.07 × 10–5 12.68 × 10–5 13.24 ×10–5 0.9

Walking speed (m/s)

hm(skin-fabric) (kg/s·m2·kPa) 0 0.7 4.0 0 0.7 4.0

Wind speed (m/s)

13.7 10.2 11.3

0.9 17.4 13.0 15.0

1.4

8.16 × 10–5 11.9

8.0 × 10–5 11.6

hm(skin-air) (kg/s·m2·kPa) hc(skin-air) (W/m2·K)

2

hm(skin-air) (kg/s·m ·kPa) hc(skin-air) (W/m2·K)

f (rpm)

80 7.54 × 10–5 11.05

60 6.4 × 10 9.4

–5

Internal transport coefficient for cylindrical fabric and skin geometry, Ghaddar et al. (2005b)

37

27

f (rpm)

Transport coefficient for planner oscillating fabric over planner wet skin, Ghaddar et al. (2003)

hc(skin-air) (W/m ·K): [Leg] hc(skin-air) (W/m–2·K): [Trunk] hc(skin-air) (W/m–2·K): [Arm]

–2

Walking speed (m/s)

9.216 × 10–5 13.265

54

19.0 15.1 17.2

1.9

10.093 16.39 31.25 10.31 14.925 38.26

hc(skin-a) (W/m2·K)

Havenith et al. (1990a and 1990b), Ensemble A.

Measured heat transport coefficient from the skin to the air layer, Danielsson (1993)

0.7

Wind speed (m/s)

Walking speed (m/s)

Lotens’ Data (1993)

6.943 × 10–5 11.0 × 10–5 21.9 × 10–5 7.09 × 10–5 10.09 × 10–5 26.3 × 10–5

hm(skin-a) (kg/s·m2·kPa)

Table 8.1 Internal mean heat and mass transfer film coefficients to the air layer as reported by Lotens (1993), Havenith et al. (1990a, 1990b), Danielsson (1993), and Ghaddar et al. (2004, 2005b) for highly permeable cotton fabric


295

where d is 0.16 m. Fonseca and Breckenridge (1965) reported that wind increases the heat transfer coefficient of outer clothing ensembles linearly with the square root of the velocity. Their correlation is given by hc (fabric– ∞ ) = a1 + b1 υ eff

[8.36]

where a1 is due to effective radiation and natural convection and the second term is due to forced convection.

8.3.3

Model extension for fabric–skin contact

The formulation of the periodic microclimate ventilation problem was solved using the 3-D cylinder model of Ghaddar et al. (2005b) for closed and open apertures at amplitudes of periodic motion that are greater than the mean spacing of that between the clothing and the skin (∆Y < Ym), where the amplitude ratio is smaller than unity (ζ < 1). For amplitude ratios greater or equal to unity (∆Y ≥ Ym and ζ ≥ 1), the inner cylinder touches the fabric cylinder. Ghaddar et al. (2005c) suggested additional modifications on the ventilation model to include the region of contact shown in Fig. 8.3. Ghaddar et al.’s (2005c) model assumed that, when the fabric cylinder is in contact with the solid cylinder (skin) at the top (θ = 0°) or the bottom (θ = 180°), both the fabric and the skin remain in touch at zero velocity for an interval of time until the reversal in motion takes place. The contact is not a point contact and is represented by a length of contact of the fabric spanning about 10° around the cylinder surface at (θ = 0°) or (θ = 180°) due to flattening that takes place in the fabric at the contact area as observed in the experiments. Touch region I Fabric II Non-contact air annulus

Non-contact air annulus II

Touch region I

8.3 Fabric–skin contact of Ghaddar et al. (2005c) model.

296


The dimensionless air layer thickness Y′ is defined as Y′ =

Y (t ) = (1 – ζ sin(ω t )) Ym

[8.37]

If Y ′ < 0, then Y ′ is taken as zero. During the touch period, Y ′ is frozen to the value of Y ′ at the time when touch starts in the motion cycle. The modeling of heat and moisture transport covers two regions during contact. The first region is the fabric–skin contact and the second region is a noncontact air layer region that separates the fabric from the skin as shown in Fig. 8.3. During skin fabric contact, the heat and mass transport problem in the fabric of region I is solved as a transient diffusion problem of a thin fabric with one surface suddenly exposed to a step change in temperature. The contact takes place at the skin with both the fabric outer node and the air void temperatures at a lower temperature than the skin surface. The weighted fabric temperature is defined as Tf =

(1 – ε f ) ρs Cs [γ To + (1 – γ ) Ti ] + ε f ρa Ca Tv (1 – ε f ) ρs Cs + ε f ρa Ca

[8.38]

where εf is the fabric porosity, γ is the mass fraction of the fabric in the outer node, Ti is the fabric inner node temperature, To is the fabric outer node temperature, Ri is the fabric inner node regain, and Ro is the fabric outer node regain. The lumping of the fabric inner, outer, and void nodes into one fabric node has permitted the use of the experimentally established properties of the fabric dry and evaporative resistances to estimate the heat and moisture diffusion during the touch period (Jones and McCullough, 1985; McCullough, 1989). The mass and energy balances of the lumped fabric in the contact region yields  ( P∞ – Pf ) ∂R  ( Psk – Pf ) = 1 ∗ + RE RE ρf e f ∂t 1 ∗ h fg +  2 ∗ h fg 2 hm ( f – ∞ )  2 2   + Da  12 ∂ R + ∂ R 2 ∂ x 2    R f ∂θ

[8.39a]

 ∂T f ∂ R had  ( Tsk – T f ) 1 = + ∗ ∗ C e C RD ρ ∂t ∂t pf f f pf  2 

 ∂2Tf ∂2Tf ( Tatm – Tfabric ) + + k a  12 + 2 RD 1 ∂x 2  R f ∂θ + 2 h r + hc ( f – ∞ )

    

[8.39b]


297

where R is the fabric regain (kg of H2O/kg of fabric), RD is the fabric dry resistance which is equal to 0.029 m2⋅K/W for cotton fabric, RE is the fabric evaporative resistance equal to 0.0055 m2⋅kPa/W for cotton fabric, hc(f-∞) and hm(f-∞) are the external heat and mass transfer coefficients with the environment, respectively. When the fabric departs from the skin boundary after contact, the fabric inner node, outer node and void space will be in thermal equilibrium at Tf and R. In the non-contact microclimate air layer region II, the mass and energy balances are given by Mass balance ∂ ( ρ Yw ) = h m (skin–air) [ Psk – Pa ] + hm ( o –air) [ Po – Pa ] ∂t a a

∂ 2 wa  ∂w  + D2 ∂  Y a  + DY ∂x 2 R f ∂θ  ∂θ 

[8.40]

Energy balance ∂ [ ρ Y ( C T + h w )] = h a a c (skin–air) ( Tsk – Ta ) fg a ∂t a

+ hc(o–air)(To – Ta) + Hm(skin–air)hfg(Psk – Pa) + hm(o–air)hfg(Po – Pa) + k a +

Tvoid – Ta P – Pa + Dh f g void e f /2 e f /2

k a ∂  ∂Ta  D ∂  Y ∂w a  Y    + h fg 2 2 ∂θ  ∂ θ   Rf R f ∂θ  ∂θ 

+ ka Y

∂ 2 Ta ∂ 2 wa + h DY fg ∂x 2 ∂x 2

[8.41]

The terms that appear in the energy balance include convective energy transport to the fabric outer node by conduction and moisture adsorption and conduction and mass diffusion terms in angular and axial directions to both the air layer and the fabric void space. The energy balances on the outer nodes and inner nodes of the fabric remain as previously described. The heat and moisture transfer are assumed to occur by diffusion through the void space air at the node in the fabric where the interface between the contact region and noncontact region occurs. The instantaneous sensible heat loss Qs and latent heat loss QL from the skin during the contact interval are given, respectively, in the touch and the non-touch regions by Contact region: Qs =

Tsk – T f RD /2

[8.42a]

298


Ql =

Psk – Pf h RE h fg /2 fg

[8.42b]

Non-contact air layer region: Qs = hc(skin–air)(Tsk – Ta) + hr(Tsk – To) [8.43a] QL = hm(skin–air)hfg(Psk – Pa)

[8.43b]

The contact model assumes that no wicking is present in the fabric.

8.4

Heat and moisture transport results of the periodic ventilation model

Ghaddar et al. (2005) presented results on heat and moisture transport using their 2-D radial and angular flow ventilation model for closed apertures at ambient conditions of 25 °C and 50% RH and at an inner cylinder isothermal skin condition of 35 °C and 100% relative humidity. Simulations were performed for a domain mean spacing Ym = 26 mm at different frequencies and amplitude ratios for Rf = 6.5 cm and Rs = 3.9 cm. Their numerical simulation results of the model predicted for closed and open aperture the transient steady periodic mass flow rates in the radial and angular directions, the fabric regain, the internal air layer temperature and humidity ratio, the fabric temperature, the skin surface temperature, in addition to the sensible and latent heat losses from the skin. For a closed aperture cylinder model, Fig. 8.4a,b shows the Ghaddar et al. (2005) ventilation model predictions as a function of the amplitude ratio ζ of (a) the time–space-averaged total air flow renewal (kg/m2) in the microclimate, (b) the time–space-averaged sensible and latent heat loss in W/m2 at f = 25, 40 and 60 rpm. The air renewal in the microclimate increases with increase of the ventilation frequency and the corresponding sensible and latent heat losses increase with increase in the ventilation frequency. However, at fixed ventilation frequency, the air renewal rate and the total heat loss variation with the amplitude ratio are affected by the fabric–skin contact occurrence during the cycle. The maximum sensible heat loss occurs at ζ = 1 and decreases very slightly with increased contact period within the studied range. Introducing an aperture induces air renewal in the axial direction through the opening. Figure 8.5 presents (a) the total ventilation rate versus the amplitude ratio at different frequencies of motion; and (b) the time and θspace-averaged radial flow rate variation in the axial direction at different amplitude ratios for f = 25, 40, and 60 rpm at ζ = 0.8 and ζ = 1.4. The air renewal through the opening increased with amplitude ratio up to ζ = 1, when fabric–skin periodic contact takes place, and then the change in the opening ventilation rate is negligible for ζ >1 (see Fig. 8.5a). At the opening (x = 0), the radial ventilation rate approaches zero and a high gradient of radial flow rate occurs within the first 10% of the opening even when contact


299

2.00

may · 106 (kg/s · m2)

f = 60 rpm 1.50

f = 40 rpm 1.00

f = 25 rpm 0.50

Time-space-averaged heat loss (W/m2)

0.00 0.00

700 600

0.50

1.00 ζ (a)

1.50

f = 25 rpm f = 40 rpm f = 60 rpm

2.00

Latent

500 400 300 200

Sensible

100 0 0.00

0.50

1.00 ζ (b)

1.50

2.00

8.4 Ventilation model predictions for a closed aperture as a function of the amplitude ratio ζ of (a) the time–space-averaged total ventilation rate in the microclimate, (b) the time–space-averaged sensible and latent heat loss in W/m2 at f = 25, 40 and 60 rpm.

is present for ζ > 1. For most of the domain interior, negligible axial flow exists and the radial flow rate is constant. Figure 8.6 shows the variation of the steady periodic time and angular-space-averaged (a) sensible and (b) latent heat loss as a function of the axial position x for the ventilation frequencies of 25, 40, and 60 rpm at ζ = 0.8 and ζ = 1.4. The maximum latent and sensible heat loss takes place at the opening and the enhancement of the local sensible heat loss at the open aperture compared to the closed end is 27.6%, 17.5%, and 15.1% at f = 25, 40, and 60 rpm, respectively. The local latent heat loss at the opening increases by 17.4%, 12.7%, and 11.6% at f = 25, 40, and 60 rpm, respectively when compared with latent loss at the closed end. The time- and space-averaged sensible and latent heat losses of the open and closed aperture systems reported in Ghaddar et al.’s (2005b,c) work are

300


Total ventilation rate (kg/s)

6.0 × 10–6

f = 60 rpm

5.0 × 10–6 4.0 × 10–6 3.0 × 10

f = 40 rpm

–6

2.0 × 10–6

f = 25 rpm

1.0 × 10–6

Averaged radial flow rate (kg/s.m2)

0.0 0.00

0.00015

0.50

1.00 ζ (a)

1.50

2.00

ζ = 0.8 ζ = 1.4

f = 60 rpm

0.00010

f = 40 rpm 0.00005

0.00000 0.0

f = 25 rpm

0.1

0.2

0.3

0.4 x (m) (b)

0.5

0.6

0.7

8.5 Plot of (a) the total ventilation rate versus the amplitude ratio at different frequencies of motion; and (b) the time and θ-spaceaveraged radial flow rate variation in the axial direction at different amplitude ratios for f =25, 40, and 60 rpm at ζ = 0.8 and ζ = 1.4.

summarized in Table 8.2 at ζ = 1.4 and ζ = 0.8 for a domain of length 0.6 m. The presence of the opening has minimal effect on the overall-time and space-averaged heat loss due to the limited size of the region near the opening where substantial axial flow renewal occurs. For an open aperture system at ζ = 1.4, the overall total heat loss is slightly higher than for closed apertures, giving an increase of 4.4%, 2.8%, and 2.2% at f = 25, 40, and 60 rpm, respectively. Comparing the total heat loss for an open aperture system when no fabric–skin contact is present (ζ = 0.8) to the case when periodic contact occurs (ζ =1.4), it is found that the contact increases the heat loss by 9.6%, 8.6%, and 8.5% at f = 25, 40, and 60 rpm, respectively. At higher frequencies, the effect of the opening on the heat loss is reduced.


301

Sensible heat loss (W/m2)

100 ζ = 0.8 ζ = 1.4

95 90

f = 60 rpm 85 80

f = 40 rpm

75 70

f = 25 rpm

65 60 0.0

0.1

0.2

0.3 0.4 x (m) (a)

0.5

0.6

Latent heat loss (W/m2)

650 ζ = 0.8 ζ = 1.4

600 550

f = 60 rpm 500

f = 40 rpm 450

f = 25 rpm 400 350 0.0

0.1

0.2

0.3 x (m) (b)

0.4

0.5

0.6

8.6 The variation of the steady periodic time and angular-spaceaveraged (a) sensible and latent heat loss as a function of the axial position x for the ventilation frequencies of 25, 40, and 60 rpm at ζ = 0.8 and ζ = 1.4.

8.5

Extension of model to real limb motion

The presented model approach for clothing ventilation systems is fundamental in its consideration of the periodic nature of air motion in the trapped layer between skin and fabric from first principles that capture all the physical parameters of the system. The ventilation model of Ghali et al. (2002c) and Ghaddar et al. (2005b) provides an effective and fast method of providing a solution of ventilation rates at low computational cost. This makes the model attractive for integration with human body thermal models to better predict human response under dynamic conditions. The 3-D motion within the air layer and its interaction with the ambient air through the fabric and the aperture is a complex basic problem. The use of Womersley flow in the axial and angular directions has reduced the complexity of the solution and predicts

302


Table 8.2 The time–space-averaged sensible and latent heat losses for closed and open aperture systems for ζ = 1.4 and ζ = 0.8 Sensible heat loss W/m2 Frequency (rpm)

Closed apertures (2-D flow)

Open aperture at x = 0 (3-D flow)

25 40 60

61.4 72.9 81.5

64.2 75.04 83.31

25 40 60

63.08 73.4 81.81

67.3 77.3 84.36

Latent heat loss W/m2 Closed apertures (2-D flow)

Open aperture at x=0 (3-D flow)

448.16 503.77 541.46

450.14 505.06 543.03

401.1 451.1 492.25

401.97 457.004 492.79

ζ = 1.4

ζ = 0.8

realistic mass flow rates through the apertures. In long domains, the effect of the aperture is localized. The model is not computationally exhaustive since two independent 1-D ventilation models in the polar and axial directions are used in addition to a lumped model of the air layer in the radial direction. The 3-D dynamic ventilation model of the fluctuating airflow in the variable size layer between the fabric and skin can easily be improved to account for rotational (tilting) inner limb motion with respect to the joint within the outer clothing, and the non-uniformity of the inner cylinder. The extension of the model considers variation in the air layer size in the axial direction as well as the angular direction. It should also consider the change in the external pressure around the cylinder due to the combined motion of the fabric and arm. The clothing ventilation model presented in this chapter is flexible, can be used for different conditions and different clothing materials (provided that their physical microscopic properties are known), and can be easily combined with multi-segmented human body models.

8.6

Nomenclature

Af Ai Ao Ca

area of the fabric (m2) inner node area in contact with the outer node (m2) outer-node exposed surface area to air flow (m2) gas concentration in the microclimate measurement location, (m3 Ar /m3 air) fiber specific heat (J/kg K) gas concentration in the distribution system (m3 Ar /m3 air) specific heat of air at constant pressure (J/kg⋅K) specific heat of air at constant volume (J/kg⋅K) water vapor diffusion coefficient in air (m2/s)

Cf Cin Cp Cv D


ef f F H had H ci′

H co ′ hc(f–∞) hc(o-air) hc(skin-air) hfg H mi ′ H mo ′

hm(f-∞) hm(o-air) hm(skin-air) im ka L LR m˙ ay m˙ ax m˙ aθ m˙ ο m˙ vent Pa Pi Po Psk P∞ Q R

303

fabric thickness (m) frequency of oscillation of the inner cylinder in revolution per minute (rpm) stride frequency (steps/min) height of the human subject (m) heat of adsorption (J/kg) normalized conduction heat transfer coefficient between inner node and outer node (W/m2⋅K) normalized convection heat transfer coefficient between outer node and air flowing through fabric (W/m2⋅K) heat transport coefficient from the fabric to the environment (W/ m2⋅K). heat transport coefficient from the fabric to the trapped air layer (W/m2⋅K). heat transport coefficient from the skin to the trapped air layer (W/ m2⋅K). heat of vaporization of water (J/kg) normalized diffusion mass transfer coefficient between inner node and outer node (kg/m2⋅kPa⋅s) normalized mass transport coefficient between outer node and air void in the fabric (kg/m2⋅kPa⋅s) mass transfer coefficient between the fabric and the environment (kg/m2⋅kPa⋅s) mass transfer coefficient between the fabric and the air (kg/ m2⋅kPa⋅s) mass transfer coefficient between the skin and the air layer (kg/ m2⋅kPa⋅s) permeability index thermal conductivity of air (W/m⋅K) fabric length in x-direction (m) Lewis relation, 16.65 K/kPa mass flow rate of air in y-direction (kg/m2⋅s) mass flow rate of air in x-direction (kg/m2⋅s) mass flow rate of air in θ-direction (kg/m2⋅s) net flow rate through the open aperture (kg/s) total ventilation rate (kg/s per m2 of clothed body surface) air vapor pressure (kPa) vapor pressure of water vapor adsorbed in inner node (kPa) vapor pressure of water vapor adsorbed in outer node (kPa) vapor pressure of water vapor at the skin (kPa) atmospheric pressure (kPa) heat loss (W/m2) total regain in fabric (kg of adsorbed H2O/kg fiber)

304

RD RE Rf Rs rpm t T Vvent,a Vvent w Y Ym


fabric dry resistance (m2⋅K/W unless specified in the equation per mm of thickness) fabric evaporative resistance (m2⋅kPa/W) fabric cylinder radius (m) inner cylinder radius (m) revolutions per minute time (s) temperature (°C) ventilation rate through apertures in m3/s ⋅ m2 of clothed body surface ventilation rate through outer fabric in m3/s ⋅ m2 of clothed body surface. humidity ratio (kg of water/kg of air) instantaneous air layer thickness (m) mean air layer thickness (m)

Greek symbols ε fabric emissivity ρ density of fabric (kg/m3) Φ periodic dimensionless flow rate parameter in x-direction ω angular frequency (rad/s) pressure gradient parameter in x-direction (Pa⋅m2/kg) Λx pressure gradient parameter in θ-direction (Pa⋅m2/kg) Λθ α fabric air permeability (m3/m2⋅s) γ fraction of mass that is in the outer node ν kinematic air viscosity (m2/s) υact equivalent air velocity of motion υeff effective wind velocity (m/s) υnatl wind velocity of natural convection, 0.07 m/s for sitting and 0.11 m/s for standing υwalk walking speed (m/s) υwind external wind speed (m/s) τ period of the oscillatory motion (s) θ angular coordinate ζ amplitude ratio Ψtr trace gas mass flux, (m3/s) Subscripts a conditions of air in the spacing between skin and fabric i inner node o outer node L latent s sensible sk conditions at the skin surface


void ∞

local air inside the void environment condition.

8.7

References

305

Acheson D J (1990), Elementary Fluid Dynamics (4th edn), Clarendon Press, New York. Amiri A and Vafai K (1994), ‘Analysis of dispersion effects and non-thermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media’, Int. J. Heat Mass Transfer, 37, 939–954. Amiri A and Vafai K (1998), ‘Transient analysis of incompressible flow through a packed bed’, Int. J. Heat Mass Transfer, 41, 4259–4279. ASHRAE (1997), ASHRAE Handbook of Fundamentals, Atlanta, American Society of Heating, Refrigerating and Air-conditioning Engineers, Chapter 5. ASTM, American Society for Testing and Materials (1983), ASTM D737–75, Standard Test Method for Air Permeability of Textile Fabrics, (IBR) approved 1983. Danielsson U (1993), Convection coefficients in clothing air layers, Doctoral Thesis, The Royal Institute of Technology, Stockholm, Sweden. Fanger P O (1982), Thermal comfort analysis and applications in engineering, New York, McGraw Hill, pp. 156–198. Farnworth B (1986), ‘A numerical model of combined diffusion of heat and water vapor through clothing’, Textile Res J, 56, 653–655. Fonseca G F and Breckenridge J R (1965), ‘Wind penetration through fabric systems: Part I’, Textile Res J, 35, 95–103. Fourt L and Hollies N (1971). Clothing: Comfort and Function, Dekker. Ghaddar N, Ghali K and Harathani J (2005a), ‘Modulated air layer heat and moisture transport by ventilation and diffusion from clothing with open aperture’, ASME Heat Trans J, 127, 287–297. Ghaddar N, Ghali K and Jaroudi E (2005c) ‘Heat and moisture transport through the micro-climate air annulus of the clothing–skin system under periodic motion’, Proceedings of the ASME 2005 Summer Heat Transfer Conference, HT2005-72006, 17–22 July 2005, San Francisco. Ghaddar N, Ghali K and Jones B (2003), ‘Integrated human-clothing system model for estimating the effect of walking on clothing insulation’, Int J Thermal Sci, 42 (6), 605–619. Ghaddar N, Ghali K, Harathani J and Jaroudi E (2005b), ‘Ventilation rates of microclimate air annulus of the clothing–skin system under periodic motion’, Int J Heat Mass Trans, 48 (15), 3151–3166. Ghali K, Ghaddar N and Harathani J (2004), ‘Two-dimensional clothing ventilation model for a walking human’, Proc of the First Int Conf on Thermal Eng: Theory and Applications, ICEA-TF1-03, Beirut-Lebanon, May 31–June 4, 2004. Ghali K, Ghaddar N and Jones B (2002a), ‘Empirical evaluation of convective heat and moisture transport coefficients in porous cotton medium’; ASME Trans, Heat Trans J, 124 (3), 530–537. Ghali K, Ghaddar N and Jones B (2002b), ‘Multi-layer three-node model of convective transport within cotton fibrous medium’, J Porous Media, 5 (1), 17–31. Ghali K, Ghaddar N and Jones B (2002c), ‘Modeling of heat and moisture transport by periodic ventilation of thin cotton fibrous media’, Int J Heat Mass Trans, 45 (18), 3703–3714.

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Harter K L, Spivak S L and Vigo T L (1981), ‘Applications of the trace gas technique in clothing comfort’, Textile Res J, 51, 345–355. Havenith G, Heus R and Lotens W A (1990a), ‘Resultant clothing insulation: a function of body movement, posture, wind clothing fit and ensemble thickness’, Ergonomics, 33 (1), 67–84. Havenith G, Heus R and Lotens W A (1990b), ‘Clothing ventilation, vapour resistance and permeability index: changes due to posture, movement, and wind’, Ergonomics, 33 (8), 989–1005. Hong S (1992), A database for determining the effect of walking on clothing insulation. Ph.D. Thesis, Kansas State University, Manhattan, Kansas. Jones B W and McCullough E A (1985), ‘Computer modeling for estimation of clothing insulation’, Proceedings CLIMA 2000, World Congress on Heating, Ventilating, and Air Conditioning, Copenhagen, Denmark, 4, 1–5. Jones B W and Ogawa Y (1993), ‘Transient interaction between the human and the thermal environment’, ASHRAE Trans, 98 (1), 189–195. Jones B W, Ito M and McCullough E A (1990), ‘Transient thermal response systems’, Proceedings International Conference on Environmental Ergonomics, Austin, TX, 66–67. Kerslake D McK (1972), The stress of hot environments, Cambridge: Cambridge University Press. Kuznetsov A V (1993), ‘An investigation of a wave temperature difference between solid and fluid phases in porous packed bed’, Int. J. Heat Mass Transfer, 37, 3030–3033. Kuznetsov A V (1997), ‘A perturbation solution for heating a rectangular sensible heat storage packed bed with a constant temperature at the walls’, Int. J Heat Mass Transfer, 40, 1001–1006. Kuznetsov A V (1998), ‘Thermal non-equilibrium forced convection in porous media’, Chapter in ‘Transport Phenomena in Porous Media’, D.B. Ingham and I. Pope (Editors), Elsevier, Oxford, 103–129. Lamoreux L W (1971), ‘Kinematic measurements in the study of human walking’, Bulletin Prosthetics Res, 3–86. Lee DY and Vafai K (1999), ‘Analysis characterization and conceptual assessment of solid and fluid temperature differentials in porous media’, Int. J. Heat Mass Transfer, 42, 423–435. Li Y (1997), Computer modeling for clothing systems, M.S. Thesis, Kansas State University, Manhattan, Kansas. Lotens W (1993), Heat transfer from humans wearing clothing, Doctoral Thesis, TNO Institute for Perception, Soesterberg, The Netherlands. Lotens W and Havenith G (1988), ‘Ventilation of rain water determined by a trace gas method’, Environmental Ergonomics eds (Mekjavic I B, Bannister B W, Morrison J B) Taylor and Francis, London, 162–176. McCullough E A, Jones B W and Huck J (1985), ‘A comprehensive data base for estimating clothing insulation’, ASHRAE Trans, 91, 29–47. McCullough E A, Jones B W and Tamura T (1989), ‘A data base for determining the evaporative resistance of clothing, ASHRAE Trans, 95 (2), 316–328. Mincowycz W J, Haji-Shikh A and Vafai K (1999), ‘On departure from local thermal equilibrium in porous media due to a rapidly changing heat source: The Sparrow number’, Int J Heat Mass Trans, 42, 3373–3385. Morris G J (1953), ‘Thermal properties of textile materials’ J Textile Inst, 44, 449–476. Morton W E and Hearle J W (1975), Physical Properties of Textile Fibers. Heinemann, London.


307

Nielsen R, Olesen B W and Fanger P O (1985), ‘Effect of physical activity and air velocity on the thermal insulation of clothing’, Ergonomics, 28, 1617–1632. Nishi Y and Gagge A P (1970), ‘Moisture permeation of clothing – A factor governing thermal equilibrium and comfort’, ASHRAE Trans, 75, 137–145. Rees W H (1941), The transmission of heat through textile fabrics, J Textile Inst, 32, 149– 165. Straatman A G, Khayat R E, Haj-Qasem E and Steinman D E (2002), ‘On the hydrodynamic stability of pulsatile flow in a plane channel’, Phys Fluids, 14 (6), 1938–1944. Vafai K and Sozen M (1990), ‘Analysis of energy and momentum transport for fluid flow through a porous bed’, ASME J Heat Transfer, 112, 690–699. Vokac Z, Kopke V and Kuel P (1973), ‘Assessment and analysis of the bellow ventilation of clothing’, Text Res J, 42, 474–482. Womersley J R (1957), ‘An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries’, Aeronautical Research Laboratory, WADC Technical Report TR, pp. 56–614. Woodcock A (1962), ‘Moisture Transfer in Textile Systems, Part I’, Textile Res J, 32, 628–633.

9 Multiphase flow through porous media P. G I B S O N, U.S. Army Soldier Systems Center, USA

9.1

Introduction

Two decades ago, Whitaker presented a comprehensive theory for mass and energy transport through porous media.1 This model, with some modifications, is also applicable to fibrous materials. Whitaker modeled the solid portion of the solid matrix as a rigid inert material which participates in the transport process only through its thermal properties. In hygroscopic fibrous materials the diffusion of water into the solid is a significant part of the total transport process. The inclusion of the extra transport terms into and out of the solid fibers necessitates extensive modifications of Whitaker’s original derivations.

9.2

Mass and energy transport equations

A typical control volume containing hygroscopic fibers is shown in Fig. 9.1. A typical porous textile material may be described as a mixture of a solid phase, a liquid phase, and a gaseous phase. The solid phase, σ, consists of the Liquid phase (β)

Solid phase (σ) (solid plus adsorbed/absorbed liquid phase)

Averaging volume (V )

Gas phase (γ) (vapor plus inert gas)

9.1 Representative volume containing fibers, liquid, and gas phases.

308


309

solid material (usually a polymer, e.g. wool or cotton) plus any bound water absorbed in the solid matrix. Hence, the solid phase density is dependent on the amount of water contained in the solid phase. The liquid phase β, consists of the free liquid water which may exist within the porous medium. The liquid phase is a pure component, and its density is assumed to be constant. The gaseous phase, γ, consists of water vapor plus the non-condensable gas (e.g. air). The gas phase density is a function of temperature, pressure, and vapor concentration. The general conservation equations are as follows: Continuity equation: ∂ρ v + ∇ ⋅ ( ρv ) = 0 ∂t Linear momentum equation: v Dv v ρ = ρg + ∇ ⋅ T Dt

[9.1]

[9.2]

Energy equation:

ρ

Dh v Dp v = –∇ ⋅ q + + ∇v :τ + Φ Dt Dt

[9.3]

In keeping with Whitaker’s derivation, we will neglect the viscous stress tensor (T).

9.2.1

Point equations

σ-phase – solid The solid σ -phase is a mixture of the dry solid (polymer) and any liquid or vapor that has dissolved into it or been adsorbed onto its surface. This may also result in a volume change for the solid phase (swelling). Swelling causes a small velocity due to displacement, and it can be accounted for by using the continuity equation: ∂ρσ v + ∇ ⋅ ( ρσ vσ ) = 0 ∂t

[9.4]

and for the two components of liquid (1) and solid (2), the species continuity equation is:

∂ρ j v + ∇ ⋅ ( ρ j v j ) = 0, j = 1, 2, ... ∂t

[9.5]

310


The σ-phase density is not constant, since it includes the density of the true solid volume fraction plus the density of the liquid volume fraction contained within the solid phase. The species densities are calculated on the basis of the total phase volume. Hence, for the two species system:

ρ=

m1 + m 2 m m = 1 + 2 = ρ1 + ρ 2 Vσ Vσ Vσ

[9.6]

It is assumed that the dry density of the solid and the density of the liquid are constant. They are denoted as ρS and ρL, respectively. The solid phase can further be divided into the fraction taken up by the liquid, and the fraction taken up by the solid:

εσ L =

Volume of liquid Total σ phase volume

[9.7]

The relations between the species densities and the solid and liquid densities are:

ρσ = εσ L ρL + (1 – ε σ L )ρS = ρ1 + ρ2

[9.8]

ρ1 = εσ L ρL

[9.9]

ρ2 = (1 – εσ L )ρS

[9.10]

The density and velocity of the mixture, in terms of the species densities, are given as:

ρσ = ρ1 + ρ2 v v v ρσ vσ = ρ1 v1 + ρ2 v 2

[9.11] [9.12]

or

ρσ = εσLρL + (1 – εσL)ρS v v v ρσ vσ = ε σL ρL v1 + (1 – ε σ L ) ρS v 2

[9.13] [9.14]

The species velocity is written in terms of the mass average velocity and the diffusion velocity as: v v v v i = vσ + u i [9.15] and therefore, the continuity equation becomes: ∂ρi v v + ∇ ⋅ ( ρi vσ ) = – ∇ ⋅ ( ρi ui ), i = 1, 2, 3, ... ∂t

[9.16]

The diffusion flux may be written in terms of a diffusion coefficient as:

v  ρi  ρ i ui = – ρσ D σ ∇   ρσ 

[9.17]


311

Hence, the continuity equation may be represented as:  ∂ρi v  ρi + ∇ ⋅ ( ρi vσ ) = ∇ ⋅  ρσ D σ ∇ ∂t  ρσ 

  , i = 1, 2, 3, ... 

[9.18]

For the purposes of comparing this model to other models developed for heat and mass transfer through porous materials, it will be convenient to rewrite these equations in terms of concentrations of water (component 1) in the solid (component 2). The concentration of water in the solid (Cs) is defined as:

Cs =

ρ1 m1 Mass of water = = ρσ Mass of the solid phase m1 + m 2

[9.19]

Since liquid water (l) is the only material crossing into or out of the solid phase, it is the most logical basis for the continuity equation: ∂ρ1  v  ρ  + ∇ ⋅ ( ρ1 vσ ) = ∇ ⋅  ρσ D σ ∇ 1   ∂t  ρσ   

[9.20]

Depending on the treatment of the solid velocity, one can rewrite this equation a couple of ways. If solid velocity is included, then the continuity equation can be rewritten as:

 ∂ε σ L v  + ∇ ⋅ ( ε σ L vσ )  = ∇ ⋅ {ρσ D Lσ ∇ ( Cs )} ρL    ∂t

[9.21]

or ∂ε σL v + ∇ ⋅ ( ε σ L vσ ) ∂t

ρS  ρ  = 1 – ∇ ⋅ [ ε σ L D Lσ ∇ ( Cs )] + S {∇ ⋅ [D Lσ ∇( Cs )]} [9.22]  ρL ρL   where

ρ1 = εσLρL and ρσ = εσLρL + (1 + εσL)ρS

[9.23]

If solid velocity is neglected, the continuity equation becomes: ∂ε σ L ρ ρ =  1 – S  ∇ ⋅ [ ε σ L D Lσ ∇ ( Cs )] + S {∇ ⋅ [D Lσ ∇ ( Cs )]} ρL ρL   ∂t

[9.24] Momentum balance is expressed as:

312


ρσ

v v Dvσ v v v   ∂v = ρσ g + ∇ ⋅ Tσ ⇒ ρσ  σ + ( vσ ⋅ ∇ ) vσ  Dt  ∂t  v = ρσ g + ∇ ⋅ Tσ

[9.25]

2

Jomaa and Puiggali neglected the convection term, and hence: v ∂v v ρσ σ = ρσ g + ∇ ⋅ Tσ ∂t

[9.26]

There are two ways to address the mass average solid phase velocity. If the thickness of the material under investigation does not change, then the total volume remains constant, and the change in volume of the solid is directly related either to the change in volume of the liquid phase or the change in volume of the gas phase. Another approach is to let the total volume of the material change with time. As the material dries out, and the total mass changes, the thickness of the material will decrease with time, proportional to the water loss. This total volume change with time can be translated into the solid phase velocity. The two situations are illustrated in Fig. 9.2 for a matrix of solid fibers undergoing shrinkage due to water loss. Initially, the assumption is that the shrinkage behavior is like the first case shown in Fig. 9.2. This means that mass average velocity must be included in the derivations, and that the total material volume (or thickness in one dimension) no longer remains constant. Jomaa and Puiggali also give an equation for the solid velocity,2 in terms of the intrinsic phase average (discussed later) as:

Case 1 Solid fiber shrinkage results in bulk thickness reduction and nonzero mass average solid velocity.

Case 2 Total bulk thickness and volume do not change; shrinkage of solid fiber portion due to water loss does not result in a mass average soild velocity.

9.2 Two methods of accounting for shrinkage/swelling due to water uptake by a porous solid.


〈 vσ 〉 σ =

1 〈 ρ 〉 σ ξ n–1

∫

ξ

0

∂ 〈 ρ 〉 dξ ∂t σ

313

[9.27]

where ξ is the generalized space coordinate, with the origin at the center of symmetry, and n depends on the geometry (n = 1 – plane, n = 2 – cylinder, n = 3 – sphere) according to the paper by Crapiste et al.3 The thermal energy equation is:

ρσ

Dhσ Dp v v = – ∇ ⋅ qσ + + ∇ vσ :τ + Φσ Dt Dt

[9.28]

Some simplifying assumptions can be made at this point by neglecting several effects. For relatively slow flow through porous materials, one can neglect the reversible and irreversible work terms in the thermal energy equation, along with the source term, and expand the material derivative as:

ρσ

Dhσ v v  ∂h  = ρσ  σ + vσ ⋅ ∇hσ  = – ∇ ⋅ qσ Dt   ∂t

[9.29]

It will be assumed that enthalpy is independent of pressure, and is only a function of temperature, and that heat capacity is constant for all the phases. We can replace the enthalpy by: h = cpT + constant, in the σ-, β-, and γ -phases The thermal energy equation can be represented as:

ρσ

∂{( c p ) σ Tσ } v v + ρσ [ vσ ⋅ ∇{( c p ) σ Tσ }] = – ∇ ⋅ qσ ∂t

[9.30]

or

v v  ∂T  ρσ ( c p ) σ  σ + vσ ⋅ ∇Tσ  = – ∇ ⋅ qσ  ∂t  Application of Fourier’s law yields v  ∂T  ρσ ( c p ) σ  σ + vσ ⋅ ∇ Tσ  = kσ ∇ 2 Tσ t ∂   or, for a multi-component mixture:  j= N v  v  ∂T  ρσ ( c p ) σ  σ + vσ ⋅ ∇ Tσ  = kσ ∇ 2 T – ∇ ⋅  Σ ρ j u j h j   j =1   ∂t 

[9.31]

[9.32]

[9.33]

ρj (c p ) j j =1 ρσ and the partial mass heat capacity and enthalpies ( c p ) j , h j are given by the partial molar enthalpy and the partial molar heat capacity divided by the molecular weight of that component. j= N

where

( c p )σ = Σ

314


β-phase – liquid The continuity equation for the liquid phase is:

∂ρβ v + ∇ ⋅ ( ρβ v β ) = 0 ∂t

[9.34]

For the thermal energy equation, as was done earlier, compressional work and viscous dissipation are neglected: Dp v = ∇v β :τ β = Φ β = 0 Dt

[9.35]

This reduces the thermal energy equation to:

 ∂hβ  v v + v β ⋅ ∇hβ  = – ∇ ⋅ q β ρβ  ∂ t  

[9.36]

Assuming enthalpy only depends on temperature, the thermal energy equation for the liquid phase is:

 ∂Τ β  v ρβ ( c p ) β  + v β ⋅ ∇ Tβ  = k β ∇ 2 Tβ ∂ t  

[9.37]

The liquid momentum equation will be discussed later in terms of a permeability coefficient which depends on the level of liquid saturation in the porous solid. γ-phase – gas The gas phase consists of vapor and an inert component (air). Following the assumptions made by Whitaker1 for this phase, the equations are as follows: Continuity equation: ∂ργ v + ∇ ⋅ ( ργ vγ ) = 0 ∂t

[9.38]

and for the two components of vapor (1) + inert component (2), the species continuity equation is: ∂ρi v + ∇ ⋅ ( ρi v i ) = 0, i = 1, 2, ... ∂t

[9.39]

The density and velocity of the mixture are given as:

ργ = ρ1 + ρ2 v v v ργ vγ = ρ1 v1 + ρ 2 v 2

[9.40] [9.41]


315

The species velocity is written in terms of the mass average velocity and the diffusion velocity as: v v v [9.42] v i = vγ + u i Then the continuity equation becomes:

∂ ρi v v + ∇ ⋅ ( ρi vγ ) = – ∇ ⋅ ( ρ i ui ), i = 1, 2, 3, ... ∂t

[9.43]

The diffusion flux may be written in terms of a diffusion coefficient as: v  ρi  ρ i ui = – ργ D∇   ργ 

[9.44]

and the continuity equation may be represented as:

∂ρ i  v  ρi + ∇ ⋅ ( ρ i vγ ) = ∇ ⋅  ργ D∇  ∂t  ργ 

  , i = 1, 2, 3, ... 

[9.45]

Due to incompressibility, the time-dependent term may be omitted. However, the vapor portion may change with time due to condensation, evaporation, or sorption/desorption. Thus, for the vapor component of the gas phase (component 1):

∂ρ1  v  ρ1   + ∇ ⋅ ( ρ1 vγ ) = ∇ ⋅  ργ D∇  ∂t  ργ   

[9.46]

If gas phase convection is neglected (gas is stagnant in the pore spaces), the continuity equation becomes: ∂ρ1   ρ1   = ∇ ⋅  ργ D∇   ∂t  ρr   

[9.47]

The thermal energy equation is given as:  ∂Tγ   i= N v  v ργ ( c p ) γ  + vγ ⋅ ∇Tγ  = kγ ∇ 2 T – ∇ ⋅  Σ ρi ui hi   i=1   ∂t  i= N

where

( c p )γ = Σ

i=1

[9.48]

ρi (c ) , ργ p i

and the partial mass heat capacities and enthalpies ( c p ) i , hi are again given by the partial molar enthalpy and the partial molar heat capacity divided by the molecular weight of that component.

316

9.2.2


Boundary conditions

The phase interface boundary conditions derivation must be extensively modified since the assumption of a rigid solid phase with zero velocity is no longer valid. Therefore, expressions describing the boundary conditions for the solid–liquid and solid–vapor interfaces are no longer simple. The conventions and nomenclature for the phase interface boundary conditions are given in Fig. 9.3. Liquid–gas boundary conditions The appropriate boundary conditions1 for the liquid–gas interface are: r v v v v v ρβ hβ ( v β – w ) ⋅ n βγ + ργ hγ ( vγ – w ) ⋅ nγ β i= N v v v v  v  = –  q β ⋅ n βγ +  qγ + Σ ρi ui hi  ⋅ nγ β  i=1    

and

v v v v v v ρβ ( v β – w ) ⋅ n βγ + ργ ( vγ – w ) ⋅ nγ β = 0

Continuous tangent components to the phase interface: v v v v v β ⋅ λ βγ = vγ ⋅ λ γ β Species jump condition given by: v v v v v v ρi ( v i – w ) ⋅ nγ β + ρβ ( v β – w ) ⋅ n βγ = 0, i = 1 v v v ρi ( v i – w ) ⋅ nγ β = 0, i = 2, 3, ...

[9.49]

[9.50]

[9.51]

[9.52] [9.53]

nγ

w nγ σ

γ -phase (vapor plus inert)

nσ

nσ γ

σ -phase (solid plus liquid)

V

( t) = V σ (t) + V γ (t)

9.3 Typical volume containing a phase interface, with velocities and unit normals indicated. Here, two phases (solid and gas) are shown.


317

Solid–liquid boundary conditions The boundary conditions for the solid–liquid interface are in similar form as above except that the phase interface velocity is given by w2. v v v v v v ρσ hσ ( vσ – w 2 ) ⋅ nσ β + ρβ hβ ( v β – w 2 ) ⋅ n βσ j=N v v v v  v  = –  q β ⋅ n βσ +  qσ + Σ ρ j u j h j  ⋅ nσ β  j=1    

[9.54]

and v v v v v v ρσ ( vσ – w 2 ) ⋅ nσ β + ρβ ( v β – w 2 ) ⋅ n βσ = 0

Continuous tangent components to the phase interface λ: v v v v vσ ⋅ λ σ β = v β ⋅ λ βσ Species jump condition given by: v v v v v v ρ j ( v j – w 2 ) ⋅ n βσ + ρσ ( vσ – w 2 ) ⋅ nσ β = 0, J = 1 v v v ρ j ( v j – w 2 ) ⋅ n βσ = 0, j = 2, 3, ...

[9.55]

[9.56]

[9.57] [9.58]

Solid–gas boundary conditions The boundary conditions for the solid–liquid interface have different expressions compared to the other interfaces because the interface is between two multi-component phases. The phase interface velocity is given by w1: v v v v v v ρσ hσ ( vσ – w1 ) ⋅ nσγ + ργ hγ ( vγ – w1 ) ⋅ nγ σ j= N i= N  v v v  v v  v  = –   qσ + Σ ρ j u j h j  ⋅ nσγ +  qγ + Σ ρi ui hi  ⋅ nγ σ  [9.59] j =1 i=1     

and

v v v v v v ρσ ( vσ – w1 ) ⋅ nσγ + ργ ( vγ – w1 ) ⋅ nγ σ = 0

Continuous tangent components to the phase interface λ: v v v v vσ ⋅ λ σγ = vγ ⋅ λ γ σ Species jump condition given by: v v v v v v ρ j ( v j – w1 ) ⋅ nσγ + ρi ( v i – w1 ) ⋅ nγ σ = 0, i = 1, j = 1 v v v ρ j ( v j – w1 ) ⋅ nσγ = 0, j = 2, 3, ... v v v ρ j ( v j – w1 ) ⋅ nσγ = 0, i = 2, 3, ...

[9.61]

[9.61]

[9.62] [9.63] [9.64]

318

9.2.3


Volume-averaged equations

The volume-averaging approach outlined by Slattery4 is applied. In this approach many of the complicated phenomena occurring due to the geometry of the porous material are simplified. Three volume averages are defined. They are: Spatial average: Average of some function everywhere in the volume:

〈ψ 〉 = 1 V

∫

ψ dV

[9.65]

V

Phase average: Average of some quantity associated solely with each phase: 〈 Tσ 〉 = 1 V

∫

V

Tσ dV = 1 V

∫

Vσ

Tσ dV

[9.66]

Intrinsic phase average: 〈 Tσ 〉 σ = 1 Vσ

∫

V

Tσ dV = 1 Vσ

∫

Vσ

Tσ dV

[9.67]

Volume fractions for the three phases are defined as:

ε σ (t ) =

Vβ ( t ) Vγ ( t ) Vσ ( t ) , ε β (t ) = , ε γ (t ) = V V V

[9.68]

The volume and volume fraction of the solid phase changing with time are now changing with time. It is assumed that the total volume is conserved, or that:

V = Vσ ( t ) + Vβ ( t ) + Vγ ( t )

[9.69]

The volume fractions for the three phases are related by:

εσ (t) + εβ (t) + ε γ (t) = 1

[9.70]

and the phase average and the intrinsic phase averages are related as:

εσ 〈Tσ 〉σ = 〈Tσ 〉

[9.71]

Volume average for liquid β-phase We will first examine the volume average for the β-phase. It is complicated because of the three different phase interface velocities which must now be included in the analysis. The continuity equation for the liquid phase is:


∂ρβ v + ∇ ⋅ ( ρβ v β ) = 0 ∂t

319

[9.72]

Integrate over the time-dependent liquid volume within the averaging volume, and divide by the averaging volume to obtain: 1 V

 ∂ρβ  ∂t 

∫

Vβ ( t )

 1  dV + V 

v ∇ ⋅ ( ρβ v β )dV = 0

∫

Vβ ( t )

[9.73]

The first term of Equation [9.73] may be taken: 1 V

 ∂ρβ  Vβ ( t )  ∂t

∫

  dV 

[9.74]

and the general transport theorem applied5 d dt

∫

ψ dV =

V( s )

∂ψ dV + V( s ) ∂t

∫

v v ψ v ( s ) ⋅ ndS

∫

[9.75]

S( s )

∂ ρβ ∂t and using the modified general transport theorem results in:

Note that Ψ =

1 V

 ∂ ρβ  Vβ ( t )  ∂t

∫

–1 V

∫

Aβγ

 d 1  dV = dt  V  

∫

[9.76]

 ρβ dV  Vβ ( t ) 

v v ρβ w ⋅ n βγ d A – 1 V

∫

Aβσ

v v ρβ w 2 ⋅ n βσ d A

[9.77]

For the second term,

1 V

∫

Vβ ( t )

v ∇ ⋅ ( ρβ v β )dV

[9.78]

We may use the volume averaging theorem as:

〈∇ψ β 〉 = ∇〈ψ β 〉 + 1 V

∫

Aβσ

v ψ β n βσ d A + 1 V

∫

Aβγ

v ψ β n βγ d A [9.79]

to rewrite the term as:

1 V

∫

+ 1 V

Vβ ( t )

∫

v v ∇ ⋅ ( ρβ v β )dV = 〈∇ ⋅ ( ρβ v β ) 〉 = ∇ ⋅ 〈 ρβ v β 〉

Aβγ ( t )

v v ρβ v β ⋅ n βγ d A + 1 V

∫

Aβσ ( t )

v v ρβ v β ⋅ n βσ d A

[9.80]

320


noting that:

d 1  dt  V 

∫

Vβ ( t )

 ρβ dV  = d 〈 ρβ 〉 = ∂ 〈ρβ 〉 ∂t  dt

[9.81]

The continuity equation for the liquid phase is rewritten as: ∂ 〈 ρ 〉 + ∇ ⋅ 〈 ρ vv 〉 + 1 β β V ∂t β

+ 1 V

∫

Aβσ

∫

Aβγ

v v v ρβ ( v β – w ) ⋅ n βγ d A

v v v ρβ ( v β – w 2 ) ⋅ n βσ d A = 0

Liquid density is constant, so that: v v 〈 ρβ v β 〉 = ρβ 〈 v β 〉

[9.82]

[9.83]

〈ρβ 〉 = ε β ρ β

[9.84]

The liquid velocity vector may be used to calculate volumetric flow rates. The flow rate of the liquid phase past a surface area may be expressed by: Qβ =

∫

A

v v 〈v β 〉 ⋅ nd A

[9.85]

The constant-density liquid assumption, Equation [9.84], allows the liquid phase continuity equation to be rewritten as: ∂ε β v + ∇ ⋅ 〈v β 〉 + 1 V ∂t + 1 V

∫

Aβσ

∫

Aβγ

v v v ( v β – w ) ⋅ n βγ d A

v v v ( v β – w 2 ) ⋅ n βσ d A = 0

[9.86]

The thermal energy equation for the liquid phase was given previously as:

 ∂hβ  v v + v β ⋅ ∇hβ  = – ∇ ⋅ q β ρβ  ∂ t  

[9.87]

 ∂ρβ v  + ∇ ⋅ ( ρβ v β )  to the left hand-side of Equation Adding the term hβ   ∂t  [9.87] will result in: ∂ ( ρ h ) + ∇ ⋅ ( ρ h vv ) = – ∇ ⋅ qv β β β β ∂t β β

[9.88]


321

Following the same procedure used previously for the continuity equation yields the following volume averaged equation:

∂ ( ρ h ) + ∇ ⋅ ( ρ h vv ) + 1 β β β V ∂t β β

∫

+ 1 V

Aβσ

∫

Aβγ

v v v ρβ hβ ( v β – w ) ⋅ n βγ d A

v v v v ρβ hβ ( v β – w 2 ) ⋅ n βσ d A = – ∇ ⋅ 〈 q β 〉

+ 〈Φ β 〉 – 1 V

∫

Aβγ

v v q β ⋅ n βγ d A – 1 V

∫

Aβσ

v v q β ⋅ n βσ d A

[9.89]

Note that an additional term is present in comparison to Whitaker’s equations1 due to the solid–liquid interface velocity. The enthalpy of the liquid phase can be expressed as: hβ = hβ° + ( c p ) β ( Tβ – Tβ° )

[9.90]

Accounting for the deviation and dispersion effects from the average properties (marked with a tilde), and writing an expression for the two terms gives: β ∂ 〈 ρ h 〉 + ∇ ⋅ 〈 ρ h vv 〉 = ε ρ ( c ) ∂〈Tβ 〉 p β β β β β β β β ∂t ∂t

 ∂ε β v  + ρβ [ hβ° + ( c p ) β ( 〈 Tβ 〉 β – Tβ° )] + ∇ ⋅ 〈vβ 〉  ∂t  v v + ρβ ( c p ) β 〈 v β 〉 ⋅ ∇〈 Tβ 〉 β + ρβ ( c p ) β ∇ ⋅ 〈 T˜β v˜ β 〉 [9.91]  ∂ε β v  + ∇ ⋅ 〈 v β 〉  is contained in the liquid It is recognized that the term   ∂t  phase continuity equation, hence:

 ∂ε β v  1  ∂t + ∇ ⋅ 〈 v β 〉  + V   + 1 V

∫

Aβσ

∫

Aβγ

v v v ( v β – w ) ⋅ n βγ d A

v v v ( v β – w 2 ) ⋅ n βσ d A = 0

[9.92]

so that:

∂ε β v + ∇ ⋅ 〈 vβ 〉 ∂t  = – 1 V 

∫

Aβγ

v v v ( v β – w ) ⋅ n βγ d A + 1 V

∫

Aβσ

 v v v ( v β – w 2 ) ⋅ n βσ d A   [9.93]

322


v The expression for the two terms ∂ 〈ρβ hβ 〉 + ∇ ⋅ 〈ρβ hβ v β 〉 may be written ∂t as: ∂ 〈 ρ h 〉 + ∇ ⋅ 〈 ρ h vv 〉 + ρ ( c ) 〈 vv 〉 ⋅ ∇〈 T 〉 β p β β β β β β β ∂t β β

v + ρβ ( c p ) β ∇ ⋅ 〈 T˜β v˜ β 〉 – [ hβ° + ( c p ) β ( 〈 Tβ 〉 β – Tβ° )]  × 1  V

+ 1 V

∫

∫

Aβγ

Aβσ


 v v v ρβ ( v β – w 2 ) ⋅ n βσ d A  

[9.94]

Substituting Equation [9.94] back into the thermal energy equation for the liquid phase: ∂〈 Tβ 〉 β v ε β ρβ ( c p ) β + ρβ ( c p ) β 〈 v β 〉 ⋅ ∇ 〈 Tβ 〉 β ∂t

v + ρβ ( c p ) β ∇ ⋅ 〈 T˜β v˜ 〉 + 1 V

∫

+ 1 V

∫

–1 V

∫

– 1 V

∫

Aβγ

Aβσ

Aβγ

Aβσ

v v v ρβ ( c p ) β ( Tβ – Tβ° )( v β – w ) ⋅ n βγ d A v v v ρβ ( c p ) β ( Tβ – Tβ° )( v β – w 2 ) ⋅ n βσ d A v v v ρβ ( c p ) β ( 〈 Tβ 〉 β – Tβ° )( v β – w ) ⋅ n βγ d A v v v ρβ ( c p ) β ( 〈 Tβ 〉 β – Tβ° )( v β – w 2 ) ⋅ n βσ d A

v = – ∇ ⋅ 〈 qβ 〉 + 〈 Φ β 〉 – 1 V – 1 V

∫

Aβσ


∫

Aβγ

v v q β ⋅ n βγ d A

[9.95]

Gray’s definition of the point functions for a phase property6 is defined: Tβ = 〈 Tβ 〉 β + T˜β [9.96]


323

Therefore, the liquid phase thermal energy equation can be written as: ∂〈 Tβ 〉 β v + ρβ ( c p ) β 〈 v β 〉 ⋅ ∇ 〈Tβ 〉 β ∂t v v v v + ρβ ( c p ) β ∇ ⋅ 〈 T˜β v˜ 〉 + 1 ρ ( c ) T˜ ( v – w ) ⋅ n βγ dA V Aβγ β p β β β

ε β ρβ ( c p ) β

∫

+ 1 V

∫

– 1 V

∫

Aβσ

Aβγ

v v v v ρβ ( c p ) β T˜β ( v β – w 2 ) ⋅ n βσ d A = – ∇ ⋅ 〈 q β 〉 + 〈 Φ β 〉 v v q β ⋅ n βγ d A – 1 V

∫


Aβσ

[9.97]

v v Representing the heat flux term – ∇⋅〈 q β 〉 using Fourier’s law ( q β = – kβ ∇Tβ ), and applying the averaging theorem results in: v 〈 q β 〉 = – k β 〈∇Tβ 〉  = – k β  ∇〈Tβ 〉 + 1 V 

∫

Aβσ

v Tβ n βσ d A + 1 V

∫

Aβγ

 v Tβ n βγ d A   [9.98]

It is relevant to use the intrinsic phase average temperature ε β 〈Tβ 〉 β for the temperature field. This leads to: v 〈 q β 〉 = – k β 〈∇Tβ 〉

 = – k β  ∇ ( ε β 〈 Tβ 〉 β + 1 V 

 v Tβ n βγ d A  Aβγ  [9.99] The thermal energy equation for the liquid phase may now be written as:

ε β ρβ ( c p ) β

∫

Aβγ

∫

Aβγ

∫

v v q β ⋅ n βγ d A – 1 V

∫

∫

Aβγ

v v v ρβ ( c p ) β T˜β ( v β – w ) ⋅ n βγ d A

v v v ρβ ( c p ) β T˜β ( v β – w 2 ) ⋅ n βσ d A

  = ∇ ⋅  k β  ∇( ε β 〈 Tβ 〉 β + 1 V   – 1 V

∫

∂〈 Tβ 〉 β v + ρβ ( c p ) β 〈 v β 〉 ⋅ ∇ 〈 Tβ 〉 β ∂t

v + ρβ ( c p ) β ∇ ⋅ 〈 T˜β v˜ β 〉 + 1 V + 1 V

∫

v Tβ n βσ d A + 1 V Aβσ

Aβσ

Aβσ

v Tβ n βσ d A + 1 V v v q β ⋅ n βσ d A

∫

A βγ

  v Tβ n βγ d A     [9.100]

324


Volume average for gas γ -phase The gas phase continuity equation is identical, for the most part, to those developed for the solid and liquid phases:

∂ 〈 ρ 〉 + ∇ ⋅ 〈 ρ vv 〉 + 1 γ γ V ∂t γ + 1 V

∫

Aγ σ

∫

Aγ β

v v v ργ ( vγ – w ) ⋅ nγ β d A

v v v ργ ( vγ – w1 ) ⋅ nγ σ d A = 0

[9.101]

The assumption of constant density for the liquid and solid phases simplified the equations further. However, in the gas phase the density may depend on the temperature and the pressure. Applying Gray’s point functions6 together with the definition of the intrinsic phase average to the gas phase continuity equation gives:

∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) + ∇ ⋅ 〈 ρ˜ vv˜ 〉 γ γ γ γ ∂t γ γ + 1 V

+ 1 V

∫

Aγ β

∫

Aγ σ


v v v ργ ( vγ – w1 ) ⋅ nγ σ d A = 0

[9.102]

The dispersion term in the gas phase can be neglected, hence:

∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) + 1 γ γ V ∂t γ γ + 1 V

∫

Aγ σ

∫

Aγ β


v v v ργ ( vγ – w1 ) ⋅ nγ σ d A = 0

[9.103]

Since the gas is a multi-component mixture, in terms of species the continuity equation is:

∂ 〈 ρ 〉 + ∇ ⋅ 〈 ρ vv 〉 + 1 i i V ∂t i

+ 1 V

∫

Aγ σ

∫

Aγ β

v v v ρi ( v i – w ) ⋅ nγ β d A

v v v ρi ( v i – w1 ) ⋅ nγ σ d A = 0 i = 1, 2, ...

[9.104]

The final form of the gas phase species continuity equation can be written as:


325

∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) i i ∂t γ i

+ 1 V

∫

Aγ β

v v v ρi ( v i – w ) ⋅ nγ β d A + 1 V

∫

Aγ σ

v v v ρi ( v i – w1 ) ⋅ nγ σ d A [9.105]

If only the vapor component (component 1) is considered, the continuity equation can be represented as:

∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) γ 1 ∂t γ 1 + 1 V

∫

Aγ β

  ρ1   v v v ρ1( v1 – w ) ⋅ nγ β d A = ∇ ⋅  〈 ργ 〉 γ D∇  γ   [9.106]  〈 ργ 〉   

The corresponding thermal energy equation for the gas phase may also be written as:  i= N  ∂〈 Tγ 〉 〈 ρp 〉 ( c i ) i   iΣ =1   ∂t + 1 V + 1 V

∫ ∫

γ

 i= N v  +  Σ ( c p ) i 〈 ρi v i 〉  ⋅ ∇〈 Tγ 〉 γ i=1  

i= N Aγ β

i= N Aγ β

v

v

v

v

v

Σ ρi ( c p ) i T˜γ ( v i – w ) ⋅ nγ β d A

i =1

v

Σ ρi ( c p ) i T˜γ ( v i – w1 ) ⋅ nγ σ d A

i =1

i= N i= N v + ∂ Σ ( c p ) i 〈 ρ˜i T˜γ 〉 + ∇ ⋅ Σ ( c p ) i 〈 ρ˜i v˜ i T˜γ 〉 i =1 ∂t i =1

  = ∇ ⋅  kγ  ∇ ( ε γ 〈 Tγ 〉 γ ) + 1 V   + 1 V

∫

– 1 V

∫

Aγ β

Aγ β

∫

  v Tγ nγ β d A  – 1   V

v v q γ ⋅ nγ β d A

Aγ α

∫

Aγ σ

v Tγ nγ σ d A v v q γ ⋅ nγ σ d A

[9.107]

Volume average for solid σ-phase The volume averaging procedure for the liquid phase was made general enough so that the same equations are applicable to the solid phase. The

326


differences are in the interface velocities; w2 is for the solid–liquid interface, and w1 is for the solid–gas interface. Also the species continuity must be accounted for. Since the two components (the liquid and the solid) are assumed to have a constant density, the complications which arose in the gas phase continuity equation will not be encountered here. The appropriate subscripts for the solid phase will be added to the equations. The solid phase density cannot be assumed constant, since this phase is a mixture of the solid and the liquid components and their proportions can change. However, the expressions are less complicated than the gas phase density since it is assumed that each component’s density is constant. The solid phase continuity equation is: ∂ 〈 ρ 〉 + ∇ ⋅ 〈 ρ vv 〉 + 1 σ σ V ∂t σ + 1 V

∫

∫

Aσ γ

v v v ( vσ – w1 ) ⋅ nσγ d A

v v v ( v σ – w 2 ) ⋅ nσ β d A = 0

Aσ β

[9.108]

and the species continuity equation is:

∂ 〈 ρ 〉 + ∇ ⋅ 〈 ρ vv 〉 + 1 j j V ∂t j + 1 V

∫

∫

Aσ γ

v v v ( v j – w1 ) ⋅ nσγ d A

v v v ( v j – w 2 ) ⋅ nσ β d A = 0 j = 1, 2, ...

Aσ β

[9.109]

The same derivation used for the gas phase can be followed, then: ∂ (ε 〈 ρ 〉 σ ) + ∇ ⋅ ( 〈 ρ 〉 σ 〈 vv 〉 ) σ σ ∂t σ σ

+ 1 V + 1 V

∫ ∫

Aσ β

Aσγ

v v v ρσ ( vσ – w 2 ) ⋅ nσ β d A v v v ρσ ( vσ – w1 ) ⋅ nσγ d A = 0

[9.110]

and the final form of the solid phase species continuity equation is: ∂ (ε 〈 ρ 〉 σ ) + ∇ ⋅ ( 〈 ρ 〉 σ 〈 vv 〉 ) + 1 j j V ∂t σ j

+ 1 V

∫

Aσ γ

∫

Aσ β

v v v ρ j ( v j – w 2 ) ⋅ nσ β d A

r r r ρ j ( v j – w1 ) ⋅ nσγ dA

   ρj   v  = ∇ ⋅  〈 ρσ 〉 σ D σ  ∇  – 〈 ρ˜ j v˜ j 〉  j = 1, 2, ... [9.111] σ      〈 ρσ 〉  


327

If one needs to track the liquid component (component 1) only, the continuity equation may be expressed as:

∂ (ε 〈 ρ 〉 σ ) + ∇ ⋅ ( 〈 ρ 〉 σ 〈 vv 〉 ) + 1 1 1 V ∂t σ 1

∫

+ 1 V

Aσ γ

∫

Aσ β

v v v ρ j ( v j – w 2 ) ⋅ nσ β d A

r r r ρ ( v1 – w1 ) ⋅ nσ β d A

  ρ1   = ∇ ⋅  〈 ρσ 〉 σ D σ ∇   [9.112]  〈 ρσ 〉 σ    Furthermore, if the solid velocity is considered to be zero, the solid phase continuity equation may be presented as:

∂ (ε 〈 ρ 〉 σ ) + 1 V ∂t σ 1 + 1 V

∫

∫

Aσ β

v v v ρ1 ( v1 – w 2 ) ⋅ nσ β d A

 r  ρ1   v v ρ1 ( v1 – w1 ) ⋅ nσγ dA = ∇ ⋅  〈 ρσ 〉 σ D σ ∇    〈 ρσ 〉 σ   

Aσγ

[9.113] The corresponding energy equation for the solid phase can be written as:  j= N  ∂〈 Tσ 〉 σ  j= N v  〈 ρj 〉 ( c p ) j  +  Σ ( c p ) j 〈 ρ j v j 〉  ⋅ ∇〈 Tσ 〉 σ  jΣ =1 j =1   ∂t   + 1 V

+ 1 V

j= N

∫

Aσ β

j= N

∫

Aσ γ

v

v

v

v

v

v

Σ ρ j ( c p ) j T˜σ ( v j – w 2 ) ⋅ nσ β d A

j =1

Σ ρ j ( c p ) j T˜σ ( v j – w1 ) ⋅ nσγ d A

j =1

j= N j= N v + ∂ Σ ( c p ) j 〈 ρ˜ j T˜σ 〉 + ∇ ⋅ Σ ( c p ) j 〈 ρ˜ j v˜ j T˜σ 〉 j =1 j =1 ∂t

 = ∇ ⋅  kσ 

+ 1 V

∫

– 1 V

∫

 1 σ  ∇( ε σ 〈 Tσ 〉 ) + V 

Aσ β

Aσ β

∫

  v Tσ nσ β d A   – 1   V v v q σ ⋅ nσ β d A

Aσ γ

∫

v Tσ nσγ d A

Aσ γ

v v qσ ⋅ nσγ d A

[9.114]

328


The continuity and thermal energy equations have been volume averaged for all three phases. The various continuity equations are given in several forms. They cover conditions such as non-zero solid velocity or tracing only the liquid component.

9.3

Total thermal energy equation

The three phases are assumed to be in local thermal equilibrium so that: 〈Tσ〉σ = 〈Tβ〉β = 〈Tγ〉γ = 〈T〉 σ

[9.115]

β

γ

σ

β

〈T〉 ≡ εσ 〈Tσ〉 + εβ 〈Tβ〉 + εγ 〈Tγ〉 = 〈Tσ〉 = 〈Tβ〉 = 〈Tγ〉

γ

[9.116]

Applying the equilibrium condition, the three individual phase equations can be added to present a single energy equation. Except for the addition of extra terms due to the solid–gas and solid–liquid phase interface velocities, this equation is similar to that derived by Whitaker.1 The equation is written in positive flux terms, i.e. liquid is evaporating into the gas phase, rather than condensing.   j= N  i = N   ∂〈 T 〉 〈 ρ j 〉 ( c p ) j  + ε β ρβ ( c p ) β + ε γ  Σ 〈 ρi 〉 ( c p ) i    ε σ  jΣ =1   i=1   ∂t   j= N  j= N v v v  +  Σ ( c p ) j 〈 ρ j v j 〉 + ρβ ( c p ) β 〈 v β 〉 + Σ ( c p ) i 〈 ρi v 〉  ⋅ ∇〈 T 〉 j =1 i=1  

+ 1 V + 1 V

∫ ∫

+ 1 V

∫

+ 1 V

∫

+ 1 V

∫

+ 1 V

∫

Aσ γ

v

Aβσ

Aβγ

v

v

v

v

Σ ρ j ( c p ) j T˜σ ( vσ – w 2 ) ⋅ nσ β d A

j =1

v v v ρβ ( c p ) β T˜β ( v β – w 2 ) ⋅ n βσ d A

v v v ρβ ( c p ) β T˜β ( v β – w ) ⋅ n βγ d A i= N

Aγ β

v

Σ ρ j ( c p ) i T˜γ ( v i – w1 ) ⋅ nγ σ d A

i=1

j= N Aσ β

v

Σ ρ j ( c p ) j T˜σ ( vσ – w1 ) ⋅ nσγ d A

j =1 j= N

Aγσ

v

v

j= N

v

v

v

Σ ρi ( c p ) i T˜γ ( v i – w ) ⋅ nγ β d A

i=1


329

 ∇ [( kσ ε σ + k β ε β + kγ ε γ ) 〈 T 〉    v 1 Tσ nσ β d A   +( kσ – k β ) V Aσ β     v =∇⋅  1 + ( k β – kγ ) Tβ n βγ d A  V Aβγ     v  +( kσ – kγ ) 1 Tγ nσγ d A  V Aσ γ   v v v v – 1 q σ ⋅ nσ β d A – 1 qσ ⋅ n βγ d A V Aσ β V Aβγ

∫ ∫ ∫

∫

+ 1 V

∫

∫

Aγ σ

v v qγ ⋅ nσγ d A

[9.117]

where the averaged density is obtained from: j=N

i= N

j =1

i=1

〈 ρ 〉 = ε σ Σ 〈 ρ j 〉 σ + ε β 〈 ρβ 〉 β + ε γ Σ 〈 ρi 〉 γ

[9.118]

and a mass fraction weighted average heat capacity by:

Cp =

j=N

i= N

j =1

i=1

ε σ Σ 〈 ρ j 〉 σ ( c p ) j + ε β ρβ ( c p ) β + ε γ Σ 〈 ρi 〉 γ ( c p ) i 〈ρ〉 [9.119]

Equations [9.118] and [9.119] allow the first term in the thermal energy equation to be written as:   j=N   i= N  ∂ 〈T 〉 〈 ρ j 〉 ( c p ) j  + ε β ρβ ( c p ) β + ε γ  Σ 〈 ρi 〉 ( c p ) i    ε σ  jΣ =1 i=1     ∂t   ∂〈 T 〉 [9.120] ∂t Then the interphase flux terms in the total thermal energy equation must be considered. Interphase flux terms must include the exchange of mass between the liquid and the gas, between the liquid and the solid, and between the gas and the solid. First the derivation for the liquid–gas interface is presented, and then the other two interfaces are treated. The jump boundary condition for the liquid–gas interface was shown previously to be: v v v v v v ρβ hβ ( v β – w ) ⋅ n βγ + ργ hγ ( vγ – w ) ⋅ nγ β = 〈ρ〉 Cp

i= N v v v v  v  = –  q β ⋅ n βγ +  qγ + Σ ρi ui hi  ⋅ nγ β  i=1    

[9.121]

330


It may be rewritten as: i= N v v v v v v v v v ρβ hβ ( v β – w ) ⋅ n βγ + Σ ρi hi ( v i – w ) ⋅ nγ β = – ( q β – qγ ) ⋅ n βγ i=1

[9.122] The jump boundary condition for the solid–gas interface was expressed previously as: v v v v v v ρσ hσ ( vσ – w1 ) ⋅ nσγ + ργ hγ ( vγ – w1 ) ⋅ nγ σ j= N i= N  v v v  v v  v  = –   qσ + Σ ρ j u j h j  ⋅ nσγ +  qγ + Σ ρi ui hi  ⋅ nγ σ  [9.123] j =1 i =1     

and this may be represented as: v

j= N

v

v

v

v

i =N

v

Σ ρ j h j ( v j – w1 ) ⋅ nσγ + i=Σ1 ρi hi ( v i – w1 ) ⋅ nγ σ j=1 v v v = – ( qσ – qγ ) ⋅ nσγ

[1.124]

The jump boundary condition for the solid–liquid interface was given previously as: v v v v v v ρσ hσ ( vσ – w 2 ) ⋅ nσ β + ρβ hβ ( v β – w 2 ) ⋅ n βσ j= N v v v  v  v = –  q β ⋅ n βσ +  qσ + Σ ρ j u j h j  ⋅ nσ β  j =1    

[9.125]

and may be rewritten as:

v

j= N

v

v

v

v

v

Σ ρ j h j ( v j – w 2 ) ⋅ nσ β + ρβ hβ ( v β – w 2 ) ⋅ n βσ

j =1

v v v = – ( q σ – q β ) ⋅ nσ β

[9.126]

Using Equations [9.122], [9.124] and [9.126], we may write the interphase flux terms in the total thermal energy equation as: – 1 V

∫

v v v ( q σ – q β ) ⋅ nσ β d A – 1 V

– 1 V

∫

v v v ( qσ – qγ ) ⋅ nσγ d A

Aσ β

Aγ σ

∫

Aβγ

v v v ( q β – qγ ) ⋅ n βγ d A


=+ 1 V + 1 V + 1 V

∫

Aσ β

331

 j= N v v v v v v Σ1 ρ j h j ( v j – w 2 )⋅ nσ β + ρβ hβ ( v β – w 2 ) ⋅ n βσ  j= 

 d A 

∫

i= N  v v v v  v v ρ h ( v – w ) ⋅ n + Σ ρi hi ( v i – w ) ⋅ nγ β  d A β β β βγ  i=1  

∫

i= N  j= N v v v v  v v ρi hi ( v j – w1 ) ⋅ nσγ + Σ ρi hi ( v i – w1 ) ⋅ nγ σ  d A  jΣ =1 i=1   [9.127]

Aβγ

Aσ γ

The total thermal energy equation is now written as: 〈ρ〉 Cp

∂〈 T 〉 ∂t

i= N  j= N v v v  +  Σ ( c p ) j 〈 ρ j v j 〉 + ρβ ( c p ) β 〈 v β 〉 + Σ ( c p ) i 〈 ρi v i 〉  ⋅ ∇〈 T 〉 j =1 i=1  

–1 V

–1 V

–1 V

∫ ∫

∫

Aσ β

Aβ γ

Aσγ

 j= N v v  v ρ j [ h j – ( c p ) j T˜σ ]( v j – w 2 ) ⋅ nσ β   jΣ  =1 d A  + ρβ [ hβ – ( c p ) β T˜β ]( vvβ – wv β ) ⋅ nv βσ     ρβ [ hβ – ( c p ) β T˜β ]( vvβ – wv β ) ⋅ nv βγ   i= N   v v d A v ˜ ρi [ hi – ( c p ) i Tγ ]( v i – w ) ⋅ nγ β   + iΣ  =1 

 j =n v v v ρ j [ h j – ( c p ) j T˜σ ]( v j – w1 ) ⋅ nσγ  jΣ =1  i= N  + Σ ρi [ hi – ( c p ) i T˜γ ]( vvi – wv 1 ) ⋅ nvγ σ  i=1

 ∇[( kσ ε σ + k β ε β + kγ ε γ ) 〈 T 〉 ]    v 1 Tσ nσ β d A   +( kσ – k β ) V Aσ β     v =∇⋅  1 +( k β – kγ ) Tβ n βγ d A  V A βγ     v  +( kσ – kγ ) 1 T n dA  V A σγ γ σγ  

∫ ∫ ∫

  d A  

[9.128]

332


Next, the phase interface velocities can be expressed in terms of enthalpies of vaporization, sorption, and desorption. The enthalpies for each phase were previously defined as: h j = h °j + ( c p ) j ( Tσ – Tσ° )

[9.129]

hβ = hβ° + ( c p ) β ( Tβ – Tβ° )

[9.130]

hi = hi° + ( c p ) i ( Tγ – Tγ° )

[9.131]

The intrinsic phase average temperatures, temperature dispersion, and overall average temperatures are related by:

Τ˜σ = 〈Τσ 〉 σ – Τσ

[9.132]

Τ˜β = 〈Τ β 〉 β – Τ β

[9.133]

Τ˜γ = 〈Τ γ 〉 γ – Τ γ

[9.134]

〈Τσ 〉 σ = 〈Τ β 〉 β = 〈Τ γ 〉 γ = 〈Τ 〉

[9.135]

One can use these relations to rewrite the integrands inside the volume integrals on the left-hand side of the total thermal energy equation. The result for the liquid–gas interface is: –1 V

∫ {ρ [ h β

Aβγ

β

v v v – ( c p ) β T˜β ]( v β – w ) ⋅ n βγ

i= N v v  v + Σ ρi [ hi – ( c p ) i T˜γ ]( v i – w ) ⋅ nγ β  d A i=1 

= –1 V

∫

Aβγ

v v v  [ hβ° – ( c p ) β ( 〈 Tβ 〉 β – Tβ° )] ρβ ( v β – w ) ⋅ n βγ  i= N  v v v [ hi° – ( c p ) i ( 〈 Tγ 〉 γ – Tγ° )] ρi ( v i – w ) ⋅ nγ β  + iΣ =1

  dA 

[9.136] From the species jump conditions: v v v v v v ρi ( v i – w ) ⋅ nγ β + ρβ ( v β – w ) ⋅ n βγ = 0, i = 1

v v v ρi ( v i – w ) ⋅ nγ β = 0, i = 2, 3, ...

[9.137] [9.138]

Note that the subscript 1 refers to the component (water) which is actually crossing the phase boundary as it goes from a liquid to a vapor.


From the species jump conditions one may write: v v v v v v ρ1 ( v1 – w 2 ) ⋅ nσ β = – ρβ ( v β – w 2 ) ⋅ n βσ

333

[9.139]

Then, the integral may be restated as:

–1 V

∫

Aβγ

=– 1 V

v v v  [ hβ° – ( c p ) β ( 〈 T 〉 – Tβ° )] ρβ ( v β – w ) ⋅ n βγ   v v d A v ° °  + [ hg 1 – ( c1 )1 ( 〈 T 〉 – Tγ )] ρ1 ( v1 – w ) ⋅ nγ β  v v v  [ hβ° – ( c p ) β ( 〈 T 〉 – Tβ° )] ρβ ( v β – w ) ⋅ n βγ  v v v ° °  – [ hg 1 – ( c1 )1 ( 〈 T 〉 – Tγ )] ρ1 ( v1 – w ) ⋅ n βγ

∫

Aβγ

  hg° 1 – hβ° + ( c p )1 ( 〈 T 〉 – Tγ° )   1 =   ° V – ( c p ) β ( 〈 T 〉 – Tβ )    

∫

Αβγ

 d A 


[9.140] The following definitions can be applied: ∆hvap (at temperature 〈T〉) = {[ hg° 1 – hβ° + ( c p )1 ( 〈 T 〉 – Tγ° ) – ( c p ) β ( 〈 T 〉 – Tβ° )]} [9.141]

〈 m˙ lv 〉 = 1 V

∫

A βγ


[9.142]

to rewrite the integral as: –1 V

∫

Aβγ

 ρβ [ hβ – ( c p ) β T˜β ]( vvβ – wv ) ⋅ nv βγ +   i= N   v v  d A = ∆ hvap 〈 m˙ lv 〉 v ˜ ρi [ hi – ( c p ) i Tγ ]( v i – w ) ⋅ nγ β   iΣ  =1  [9.143]

The corresponding terms for the phase interface between the solid and the liquid are identical, except that the quantity ∆hvap is no longer used. Instead, the differential enthalpy of sorption7 is applied, which is given the notation Ql . The differential heat of sorption is the heat evolved when one gram of water is absorbed by an infinite mass of the solid, when that solid is at a particular equilibrated moisture content. This is very similar to the heat of solution or heat of mixing that occurs when two liquid components are mixed. For textile fibers there is a definite relationship between the equilibrium values of the differential heat of sorption and the water content of the fibers, and those relationships can be used in the thermodynamic equations which will be discussed in a later section.

334


The solid–liquid interface integral term is thus given as:  ρβ [ hβ – ( c p ) β T˜β ]( vvβ – wv 2 ) ⋅ nv βσ   j=N  1 dA – V Aσ β  + Σ ρ [ h – ( c ) T˜ ]( vv – wv ) ⋅ nv  2 p j σ j  j =1 j j σβ    From the species jump conditions one may equate: v v v v v v ρ 1 ( v1 – w 2 ) ⋅ nσ β = – ρβ ( v β – w 2 ) ⋅ n βσ

∫

[9.144]

[9.145]

or rewrite the integral as:

–1 V

∫

Aσ β

∫

= –1 V

v v v  [ hβ° – ( c p ) β ( 〈 T 〉 – Tβ° )] ρβ ( v β – w 2 ) ⋅ n βσ   v v dA v  +[ hs°1 – ( c p )1 ( 〈 T 〉 – Tσ° )] ρ1 ( v1 – w 2 ) ⋅ nσ β 

Aσ β

v v v  [ hs°1 – ( c p )1 ( 〈 T 〉 – Tσ° )] ρ1 ( v1 – w 2 ) ⋅ nσβ   v v dA v  – [ hβ° – ( c p ) β ( 〈 T 〉 – Tβ° )] ρβ ( v β – w 2 ) ⋅ nσ β 

{

}

= [ hs°1 – hβ° + ( c p ) s 1 ( 〈 T 〉 – Tσ° ) – ( c p ) β ( 〈 T 〉 – Tβ° )]

×1 V

∫

Aσ β

v v v ρβ ( v β – w 2 ) ⋅ nσ β d A

[9.146]

One may use the following definitions: Q1 (at temperature 〈T 〉) = [ hs°1 – hβ° + ( c p )1 ( 〈 T 〉 – Tσ° ) – ( c p ) β ( 〈 T 〉 – Tβ° )]

[9.147]

v v v ρσ ( vσ – w 2 ) ⋅ nσ β d A

[9.148]

〈 m˙ sl 〉 = 1 V

∫

Aσ β

to rewrite the original integral as:  j= N v v v ρ j [ h j – ( c p ) j T˜σ ]( v j – w 2 ) ⋅ nσ β  jΣ =1 Aσ β  v v v + ρβ [ hβ – ( c p ) β T˜β ]( v β – w 2 ) ⋅ n βσ d A = Ql 〈 m˙ sl 〉

–1 V

∫

}

[9.149]

For the gas–solid interface, the heat of desorption for the vapor is equal to the energy required to desorb the liquid plus the enthalpy of vaporization, as: Qsv = Ql + ∆hvap

[9.150]

The derivation is exactly the same as for the other two interfaces, where the only component crossing the phase interface is component 1 (water) and hence, the integral is:


–1 V

∫

Aσ γ

335

 j= N v v v ρ j [ h j – ( c p ) j T˜σ ]( v j – w1 ) ⋅ nσγ  jΣ =1 

i= N r r r r  r + Σ ρ i [ hi – ( c p ) i T˜γ ]( v i – w1 ) ⋅ ni – w1 ) ⋅ nγσ  d A i =1 

= ( Ql + ∆ hvap )〈m˙ sv 〉

[9.151]

where 〈 m˙ sl 〉 is the mass flux desorbing from the solid to the liquid phase, 〈 m˙ sv 〉 is the mass flux desorbing from the solid into the gas phase, and 〈 m˙ lv 〉 is the mass flux evaporating from the liquid phase to the gas phase. The total thermal energy equation now becomes: 〈 ρ 〉C p

∂ 〈T 〉 ∂t

i= N  j= N v v v  +  Σ ( c p ) j 〈 ρ j v j 〉 + ρβ ( c p ) β 〈 v β 〉 + Σ ( c p ) i 〈 ρi v i 〉  ⋅ ∇〈 T 〉 i=1  j =1 

+ ∆hvap 〈 m˙ lv 〉 + Ql 〈 m˙ sl 〉 + ( Ql + ∆hvap ) 〈 m˙ sv 〉  ∇[ kσ ε σ + k β ε β + kγ ε γ ) 〈 T 〉 ]    v 1 Tσ nσβ d A   + ( kσ – k β ) V Aσβ     v =∇⋅  + ( k β – kγ ) 1 Tβ n βγ d A  V Aβγ     v  + ( kσ – kγ ) 1 Tγ nσγ d A  V Aσγ  

∫ ∫ ∫

[9.152]

One may simplify the total thermal energy equation based on an effective thermal conductivity, and present the total thermal energy equation in a much shorter form as: 〈 ρ 〉C p

∂ 〈T 〉 ∂t

i= N  j= N v v v  +  Σ ( c p ) j 〈 ρ j v j 〉 + ρβ ( c p ) β 〈 v β 〉 + Σ ( c p ) i 〈 ρi v i 〉  ⋅ ∇〈 T 〉 j =1 i=1  

+ ∆hvap 〈 m˙ lv 〉 + Ql 〈 m˙ sl 〉 + ( Ql + ∆hvap ) 〈 m˙ sv 〉 T = ∇ ⋅ ( K eff ⋅ ∇〈 T 〉 )

[9.153]

The effective thermal conductivity can be expressed in a variety of ways,1

336


depending on the assumptions made with respect to the isotropy of the porous medium, the importance of the dispersion terms, etc. The effective thermal conductivity is also an appropriate place to include radiative heat transfer, by adding an apparent radiative component of thermal conductivity to the effective thermal conductivity to account for radiation heat transfer.

9.4

Thermodynamic relations

The gas phase is assumed to be ideal, which gives the intrinsic phase partial pressures as:

〈 pi 〉 γ = 〈 ρi 〉 γ Ri 〈 T 〉 i = 1, 2, ...

[9.154]

Noting that component 1 is water, and component 2 is air, one can present: 〈 ργ 〉 γ = 〈 ρ1 〉 γ + 〈 ρ2 〉 γ

[9.155]

〈 pγ 〉 γ = 〈 p1 〉 γ + 〈 p 2 〉 γ

[9.156]

The differential heat of sorption, Ql , and the concentration of water in the solid phase must now be connected. An example8 of a general form for Ql (in J/kg), as illustrated in Fig. 9.4, can be expressed as a function of the relative humidity φ:

Ql (J/ kg) = 1.95 × 10 5 (1 – φ )

φ=

  1 1 + ,  0.2 + φ ) 1.05 – φ 

pv 〈 p 〉γ = 1 ps ps

[9.157]

Differential heat of sorption (J/kg)

The differential heat of sorption and the actual equilibrium water content in the solid phase can then be connected further. For the two-component mixture 1.2 × 106

0.9 × 106

0.6 × 106

0.3 × 106

0

0.2

0.4 0.6 0.8 Relative humidity φ

1.0

9.4 Generic differential heat of sorption for textile fibers (sorption hysteresis neglected).


337

of solid (component 2) plus bound water (component 1) in the solid phase, the density is given by: 〈ρσ〉σ = 〈ρ1〉σ + 〈ρ2〉σ

[9.158]

One could make the assumption that mass transport in the textile fiber is so rapid that the fiber is always in equilibrium with the partial pressure of the gas phase, or is saturated if any liquid phase is present. This would eliminate the need to account for the transport through the solid phase. There are a variety of sorption isotherm relationships that could be used, including the experimentally determined relationships for a specific fiber type, but a convenient one8 is given by:   1 1 Regain ( R ) = R f (0.55φ )  +  (0.25 + φ ) (1.25 – φ ) 

[9.159]

Rf is the standard textile measurement of grams of water absorbed per 100 grams of fiber, measured at 65% relative humidity. One may rewrite this in terms of the intrinsic phase averages for both phases as:

R=

〈 ρ1 〉 σ 100 〈 ρ2 〉 2

     〈 p1 〉   1 1  [9.160] = R f  55 + p s      〈 p1 〉 γ  〈 p1 〉 γ     0.25 + p   1.25 – p      s s  γ

If the assumption is that the solid phase is not always in equilibrium, one may use relations available between the rate of change of concentration of the solid phase and the relative humidity of the gas phase, an example of which is given by Norden and David.9 The vapor pressure–temperature relation for the vaporizing β-phase can be given as:    2σ βγ  ∆hvap  1    〈 p1 〉 γ = p1° exp  –   + – 1     R1  〈 T 〉 T°       r ρβ R1 〈 T 〉  

[9.161]

This relation gives the reduction or increase in vapor pressure from a curved liquid surface resulting from a liquid droplet influenced by the surface interaction between the solid and the liquid, usually in a very small capillary. In many cases, the Clausius–Clapeyron equation will be sufficiently accurate for the vaporizing species, and the gas phase vapor pressure may be found from:

338


  ∆hvap  1   〈 p1 〉 γ = p1° exp  –  – 1      R1  〈 T 〉 T°   

[9.162]

This vapor pressure–temperature relation is only good if the liquid phase is present in the averaging volume. However, one may encounter situations where only the solid phase and the gas phase are present. To get the vapor pressure in the gas phase in this situation, one can use the sorption isotherm and assume that the gas phase is in equilibrium with the sorbed water content of the solid phase. One can use any isotherm relation where the solid’s water content is known as a function of relative humidity. The equation given previously is one example:

ε σ l ρl 〈 ρ1 〉 σ = σ (1 – ε σ l ) ρs 〈 ρ2 〉      〈 p1 〉 γ   1 1  = R f  0.55 +  γ γ p  s    〈p 〉  〈p 〉     0.25 + 1  1.25 – 11    ps  ps      

[9.163]

9.5

Mass transport in the gas phase

The volume average form of the gas phase continuity equation was found to be:

∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρv 〉 γ 〈 vv 〉 ) + 1 γ γ V ∂t γ γ

+ 1 V

∫

Aγ σ

∫

Aγ β

v v v ργ ( vγ – w1 ) ⋅ nγσ d A = 0

v v v ργ ( vγ – w ) ⋅ nγβ d A

[9.164]

and the species continuity equation was given as:

∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) + 1 1 1 V ∂t γ 1   〈 ρi 〉 γ = ∇ ⋅  〈 ργ 〉 γ D∇ γ  〈 ργ 〉 

    

∫

Aγ β

v v v ρ1 ( v1 – w ) ⋅ nγβ d A

[9.165]

where the dispersion and source terms were omitted from the equation.


339

If the mass flux from one phase to another is defined as:

〈 m˙ lv 〉 = 1 V

∫

A βγ


[9.166]

or

〈 m˙ lv 〉 = – 1 V

∫

A βγ


[9.167]

the expression for 〈 m˙ sv 〉 is similar. The gas phase continuity equation may now be rewritten as: ∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) = 〈 m˙ 〉 + 〈 m˙ 〉 sv γ γ lv ∂t γ γ

[9.168]

For the two species (1 – water, and 2 – air), the species continuity equations are presented as: ∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) – 〈 m˙ 〉 + 〈 m˙ 〉 sv 1 γ lv ∂t γ 1

  〈 ρ 〉γ = ∇ ⋅  〈 ργ 〉 γ D∇ 1 γ  〈 ργ 〉 

    

γ ∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) = ∇ ⋅  〈 ρ 〉 γ D∇ 〈 ρ2 〉  γ 2 γ  〈 ρ 〉γ ∂t γ 2  γ 

[9.169]     

[9.170] If the effects of the dispersion terms in the diffusion equations are neglected, one may incorporate an effective diffusivity into the species continuity equations, which are now given as: ∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) – 〈 m˙ 〉 – 〈 m˙ 〉 sv 1 γ lv ∂t γ 1

  〈 ρ 〉γ = ∇ ⋅  〈 ργ 〉 γ D eff ∇ 1 γ  〈 ργ 〉 

    

[9.171]

γ ∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) = ∇ ⋅  〈 ρ 〉 γ D ∇ 〈 ρ2 〉  γ 2 2 γ γ eff  γ ∂t  〈 ργ 〉 

    

[9.172] The effective diffusivity will be dependent on the gas phase volume εγ; as the solid volume and the liquid volume fractions increase, there will be less

340


space available in the gas phase for the diffusion to take place. One may define the effective diffusivity as:

D12 ε γ Da ε γ [9.173] = τ τ where the effective diffusivity Deff is related to the diffusion coefficient of water vapor in air (D12 or Da) divided by the effective tortuosity factor τ. A good relation for the binary diffusion coefficient of water vapor in air is given by Stanish et al.10 as: Deff =

 2.23 D12 =  γ γ  〈 p1 〉 + 〈 p 2 〉

  T  1.75 (m K s units)   273.15  

[9.174]

To simplify, one could assume the tortuosity factor is constant, and let the variation in the gas phase volume take care of the changes in the effective diffusion coefficient. Another simplification is to account only for the water vapor movement, and hence the continuity equation becomes: ∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈 vv 〉 ) – 〈 m˙ 〉 – 〈 m˙ 〉 sv 1 γ lv ∂t γ 1   〈 ρ 〉γ D = ∇ ⋅  〈 ργ 〉 γ 12 ∇ 1 γ τ  〈 ργ 〉 

9.6

    

[9.175]

Gas phase convective transport

It is often necessary to include forced convection through porous media – it is an important part of the transport process of mass and energy through porous materials. If gravity is neglected, the gas phase velocity is expressed as:1 v 〈 vγ 〉 = – 1 K γ ⋅ {ε γ [ ∇〈 ργ – ρ0 〉 γ ]} µγ

[9.176]

where the permeability tensor Kγ is a transport coefficient. Equation [9.176] is the general Darcy relation.11 There are other relations which pertain to gas flow through a porous material. For example, the modified form of Darcy’s law: ∇P +

µ v v =0 K γ

[9.177]

The permeability coefficient K can be obtained experimentally. The permeability may be modified to account for the decrease in gas phase volume as the solid swells and/or the liquid phase accumulates. One can


341

account for the variation of K as a function of the gas phase volume – the approach used by Stanish et al.10 γ  εγ  K γ = K dry ε   γ dry 

[9.178]

Relation [9.178] is a very simple model, and may be improved upon. Dullien11 presents a variety of relationships for the dependency of K on porosity; some of his relations may be more realistic in the case of fibrous layers. It is also possible to relate the change in the material permeability to the effective tortuosity function τ. This is useful, because τ is affected by the same factors related to the decrease in gas phase volume, and change in physical geometry, that are needed to account for the Darcy’s law relations defining convective gas flow.

9.7

Liquid phase convective transport

Whitaker’s derivation1 for the convection transport of the liquid phase is one of the most complicated parts of his general theory. He accounts for the capillary liquid transport, which is greatly influenced by the geometry of the solid phase, and the changeover from a continuous to a discontinuous liquid phase. His eventual transport equation, which gives an expression for the liquid phase average velocity is quite complicated, and depends on several hard-to-obtain transport coefficients. His final equation is given as:  ε βξKβ  v 〈 vβ 〉 = –   ⋅ [ k ε ∇ ε β + k 〈 T 〉 ∇ 〈 T 〉 – ( ρβ – ργ )]  µβ 

[9.179]

One advantage of Whitaker’s derivation is that it is almost completely independent of the other transport equation derivations. This means that one may use another expression for the liquid phase velocity if one can substitute a relation that is more amenable to experimental measurement and verification. One such relation is given by Stanish et al.10 The velocity is assumed proportional to the gradient in pressure within the liquid. The pressure in the liquid phase is assumed to be the sum of the gas pressure within the averaging volume minus the capillary pressure (Pc):  kβ  v γ γ 〈 vβ 〉 = –   ∇ ( 〈 p1 〉 + 〈 p 2 〉 – Pc )  µβ 

[9.180]

To use this type of relation, it is necessary to obtain an expression for the capillary pressure as a function of saturation condition. It is also necessary to determine when the liquid phase becomes discontinuous; where, at that point, liquid movement ceases. These types of relations can be identified

342


experimentally for materials of interest, or they may be found in the literature for a wide variety of materials. Capillary pressure Pc is often a function of the fraction of the void space occupied by the liquid. Liquid present in a porous material may be either in a pendular state, or in a continuous state. If the liquid is in a pendular state, it is in discrete drops or regions that are unconnected to other regions of liquid. If liquid is in the pendular state, there is no liquid flow, since the liquid does not form a continuous phase. There may be significant capillary pressure present, but until the volume fraction of liquid rises to a critical level to form a continuous phase, there will be no liquid flow. This implies that there is a critical saturation level, which we can think of as the relative proportion of liquid volume within the gas phase volume that must be reached before liquid movement may begin. Experimentally measured liquid capillary curves often show significant hysteresis, depending on whether liquid is advancing (imbibition) or receding (drainage) through the porous material. A typical capillary pressure curve is shown in Fig. 9.5. We may take a definition for liquid saturation as: S=

Vβ εβ = Vγ + Vβ ε β + εγ

[9.181]

The point at which the liquid phase becomes discontinuous is often called the irreducible saturation (sir).12 When the irreducible saturation is reached, the flow is discontinuous, which implies that liquid flow ceases when:

Capillary pressure

εb < sir[1 – (εds + εbw)]

[9.182]

Pc

Drainage

Imbibition 0 0

Saturation (S) = εβ /(εβ + εβ )

1.0

9.5 Typical appearance of capillary pressure curves as a function of liquid saturation for porous materials.


343

An empirical equation given by Stanish et al.10 suggests a form for the equation for capillary pressure as a function of the fraction of void space occupied by liquid:  kβ  Pc = a    µγ 

–b

, where a and b are empirical constants

[9.183]

For liquid permeability as a function of saturation:10  0;  Kβ =  s   π ( ε β / ε γ ) – s ir  K β 1 – cos  2 (1 – s ir )   

( ε β / ε γ ) < s ir   ; 

( ε β / ε γ ) ≥ s ir

[9.184]

where K βs is the liquid phase Darcy permeability when fully saturated. Another way to construct a liquid phase transport equation is to consider the moisture distribution throughout the porous material as akin to a diffusion process. By combining the conservation of mass and Darcy’s equation, a differential equation for the local saturation S may be written as:13 ∂S = ∂  F ( s ) ∂S  ∂t ∂y  ∂y 

[9.185]

where the ‘moisture diffusivity’ is given by:  K β   dPc   µ   dS   β F( s ) = (ε β + ε γ )

[9.186]

If we rewrite the saturation variable S in terms of its original definition:

S=

Vβ εβ = Vγ + Vβ ε β + εγ

[9.187]

the differential equation for liquid migration under the influence of capillary pressure may be written as:   K β   dPc       εβ εβ   ∂ ∂   µ β   dS  ∂  = ∂t  ( ε β + ε γ )  ∂y  ( ε β + ε γ ) ∂y  ( ε β + ε γ )      

[9.188]

Although we have these relations for the capillary pressures and permeability as a function of saturation and irreducible saturation, it is often difficult to obtain permeabilities for many fibrous materials. Wicking studies on fabrics

344


are usually carried out parallel to the plane of the fabric by cutting a strip, dipping one end in water, and studying liquid motion as it wicks up the strip.14,15 However, wicking through fibrous materials often takes place perpendicular to the plane of the fabric, where the transport properties are quite different due to the highly anisotropic properties of oriented fibrous materials such as fabrics. The usefulness of the relations contained in Equations [9.181]–[9.188] are that they allow one to model the drying behavior of porous materials by accounting for both a constant drying rate period and a falling rate period. In the constant drying rate period, evaporation takes place at the surface of the porous material, and capillary forces bring the liquid to the surface. When irreducible saturation is reached in regions of the porous solid, drying becomes limited by the necessity for diffusion to take place through the porous structure of the material, which is responsible for the ‘falling rate’ period of drying. These effects are most important for materials that are thick, or of low porosity. For materials of the porosity and thickness typical of woven fabrics, almost all drying processes are in the constant rate regime, which suggests that many of the complicating factors which are important for thicker materials can be safely ignored. Studies on the drying rates of fabrics16–19 suggest that simply assuming drying times proportional to the original liquid water content are a good predictor of the drying behavior of both hygroscopic and nonhygroscopic fabrics. Wicking processes perpendicular to the plane of the fabric take place very quickly, and the falling rate period is very short once most of the liquid has evaporated from the interior portions of fabrics.

9.8

Summary of modified transport equations

The set of modified equations which describe the coupled transfer of heat and mass transfer through hygroscopic porous materials are summarized below. Total thermal energy equation:  j=N v  ( c p ) j 〈 ρj v j 〉   jΣ =1 ∂〈 T 〉  v  〈 ρ〉 C p +  + ρβ ( c p ) β 〈v β 〉  ⋅ ∇ 〈 T 〉 + ∆hvap 〈 m˙ lv 〉 ∂t  i= N v  ( c p ) i 〈 ρi v i 〉   + iΣ =1  

+ Q 1 〈 m˙ sl 〉 + ( Q l + ∆h vap ) 〈 m˙ sv 〉 = ∇ ⋅ ( K Teff ⋅ ∇〈 T〉 ) Liquid phase equation of motion:

[9.189]


 kβ v 〈v β 〉 = –   µβ

 γ γ  ∇ ( 〈p 1 〉 + 〈p 1 〉 – Pc ) 

345

[9.190]

Liquid phase continuity equation:

∂ε β v + ∇ ⋅ 〈v β 〉 + 1 V ∂t + 1 V

∫

Aβσ

∫

Aβγ

v v v (v β – w ) ⋅ n βγ dA

v v v (v β – w 2 ) ⋅ n βσ dA = 0

[9.191]

which can be rewritten as: ∂ε β ( 〈 m˙ lv 〉 – 〈 m˙ sv 〉 ) v + ∇ ⋅ 〈v β 〉 + =0 ρβ ∂t

[9.192]

Gas phase equation of motion:  kγ  v γ γ 〈v γ 〉 = –   ∇ ( 〈p 1 〉 + 〈p 2 〉 )  µγ 

[9.193]

Gas phase continuity equation:

∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈vv 〉 ) = 〈 m˙ 〉 + 〈 m˙ 〉 γ γ sv lv ∂t γ γ Gas phase diffusion equations: ∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈vv 〉 ) – 〈 m˙ 〉 – 〈 m˙ 〉 γ sv 1 lv ∂t γ 1   〈 ρ 〉γ = ∇ ⋅  〈 ργ 〉 γ D eff ∇  1 γ  〈 ργ 〉 

    

γ ∂ ( ε 〈 ρ 〉 γ ) + ∇ ⋅ ( 〈 ρ 〉 γ 〈vv 〉 ) = ∇ ⋅  〈 ρ 〉 γ D ∇  〈 ρ1 〉  γ γ 2 eff  〈 ρ 〉γ ∂t γ 2  γ 

[9.194]

[9.195]

     [9.196]

Solid phase density relations: 〈ρσ〉σ = 〈ρ1〉σ + 〈ρ2〉σ

[9.197]

ρ1 = εσLρL

[9.198]

ρ2 = (1 – εσL)ρS

[9.199]

εsS + εσL = 1

[9.200]

Solid phase continuity equation: ∂ ( ε 〈 ρ 〉 σ ) + ∇ ⋅ ( 〈 ρ 〉 σ 〈vv 〉 ) + 〈 m˙ 〉 + 〈 m˙ 〉 = 0 σ σ sv sl ∂t σ σ

[9.201]

346


Solid phase equation of motion (for one-dimensional geometry): ξ

∂ 〈 ρ 〉 dξ 1 σ 〈 ρ 〉 σ ξ n –1 0 ∂t Solid phase diffusion equation (for vaporizing component): ∂ ( ε 〈 ρ 〉 σ ) + ∇ ⋅ ( 〈 ρ 〉 σ 〈vv 〉 ) + 〈 m˙ 〉 + 〈 m˙ 〉 sv 1 1 sl ∂t σ 1 〈v σ 〉 σ =

∫

  〈 ρ1 〉 σ = ∇ ⋅  〈 ρσ 〉 σ D σ ∇  σ  〈 ρσ 〉  Volume constraint:

    

εσ(t) + εβ(t) + εγ(t) = 1

[9.202]

[9.203]

[9.204]

Thermodynamic relations: 〈ρ1〉γ = 〈ρ1〉γ R1〈T〉

[9.205]

〈ρ2〉γ = 〈ρ2〉γ R2〈T〉

[9.206]

〈ργ〉γ = 〈ρ1〉γ + 〈ρ2〉γ

[9.207]

〈ργ〉γ = 〈ρ1〉γ + 〈ρ2〉γ

[9.208]

If liquid phase is present, vapor pressure is given by: ∆hvap  1    2 σ βγ     〈 ρ1 〉 γ = p1° exp  –   + – 1    [9.209]   R T 1  〈T 〉 °     rρβ R1 〈 T 〉    or   ∆hvap  1   〈 ρ1 〉 γ = p1° exp  –  – 1   [9.210]    R1  〈 T 〉 T°    If the liquid phase does not exist, but the liquid component is desorbing from the solid, the reduced vapor pressure in equilibrium with the solid phase must be used. This relation may be determined directly from the sorption isotherm for the solid:

〈 p1〉γ = f ( ps, ρl, ρs, εσ L) at the temperature 〈T 〉, only εσL is unknown [9.211] Sorption relations (volume average solid equilibrium):      〈p 〉  1 1  Ql (J/ kg) = 0.195  1 – 1   + ps     〈 p1 〉 γ   〈 p1 〉 γ     0.2 + p s   1.05 – p s     γ

[9.212]


347

ε σ l ρl 〈 ρ1 〉 σ = σ (1 – ε σ l ) ρs 〈 ρ2 〉      〈p 〉  1 1  = R f  0.55 1   + ps     〈 p1 〉 γ   〈 p1 〉 γ     0.25+ p s   1.25 – p s     γ

[9.213]

The preceding list contains a total of 20 equations and 20 unknown variables, which allow for the solution of the set of equations using numerical methods. The 20 unknown variables are: v v v ε σ , ε β , ε γ , 〈 v σ 〉 , 〈 v β 〉 , 〈 v γ 〉 , 〈 T〉 , 〈 m˙ sl 〉 , 〈 m˙ sv 〉 , 〈 m˙ lv 〉 , Q sl 〈 pγ 〉γ, 〈 p1〉γ, 〈 p2〉γ, 〈 ργ 〉γ, 〈 ρ1〉γ, 〈 ρ2〉γ 〈 pγ 〉σ, 〈 p1〉σ, 〈 p2 〉σ Note that the aforementioned set of equations is accompanied with the appropriate initial and boundary conditions.

9.9

Comparison with previously derived equations

The simplified system of partial differential equations given in the previous section contains many equations with a large number of unknown variables. Even for the simplified case of vapor diffusion, the system of equations is quite confusing, and it is difficult to verify their accuracy other than by checking for dimensional consistency. One way of checking their validity is to see if they simplify down to more well-known diffusion equations for the transport of water vapor in air through a porous hygroscopic solid. Such a system of equations has been well documented by Henry,20 Norden and David,9 and Li and Holcombe,21 who have used them to describe the diffusion of water vapor through a hygroscopic porous material. The same assumptions used by those previous workers will be made here to transform the system of equations for the case of vapor diffusion (no liquid or gas phase convection) to their system of equations. For clarity of comparison, the same variables, notations, and units will be used. The major simplifying assumptions are: (i) there is no liquid or gas phase convection, (ii) there is no liquid phase present, (iii) the heat capacity of the gas phase can be neglected, (iv) the volume of the solid remains constant and does not swell, (v) the solid and gas phase volume fractions are both constant, (vi) the thermal conductivity is expressed as a constant scalar thermal conductivity coefficient, (vii) the gas phase diffusion coefficient is constant, (viii) the transport is one-dimensional (e.g. x-direction).

348


The total thermal energy equation becomes: 〈 ρ〉 C p

∂ 〈T 〉 T + ( Ql + ∆ hvap ) 〈 m˙ sv 〉 = ∇ ⋅ ( K eff ⋅ ∇ 〈T 〉) ∂t

[9.214]

and can be replaced by

〈 ρ〉 C p

∂ 〈T 〉 ∂2 〈T 〉 + ( Ql + ∆ hvap ) 〈 m˙ sv 〉 = k eff ∂t ∂x 2

[9.215]

The gas phase continuity equation becomes:

ε γ ∂ ( 〈 p γ 〉 γ ) = 〈 m˙ sv 〉 ∂t

[9.216]

The gas phase diffusion equation (component 1 – water vapor):

  〈 ρ1 〉 γ ε γ ∂ ( 〈 p1 〉 γ ) – 〈 m˙ sv 〉 = ∇ ⋅  〈 ργ 〉 γ D eff ∇  γ ∂t  〈 ργ 〉  simplified to:

    

[9.217]

∂ 2 〈 ρ1 〉 γ ε γ ∂ ( 〈 p1 〉 γ ) – 〈 m˙ sv 〉 = D eff ∂t ∂x 2 The solid phase continuity equation (component 1 – water):

[9.218]

ε σ ∂ ( 〈 p 1 〉 σ ) + 〈 m˙ sv 〉 = 0 ∂t

[9.219]

For the solid phase diffusion equation (component 1 – water), it is assumed that the diffusional transport through the solid phase is insignificant compared with the diffusion through the gas phase. This is a reasonable assumption since the diffusion coefficient for water in a solid is always much less than the diffusion coefficient of water vapor in air. Therefore, the diffusion equation reduces to the continuity equation:   ρ1   ε σ ∂ ( 〈 ρ1 〉 σ ) + 〈 m˙ sv 〉 = ∇ ⋅  〈 ρσ 〉 σ D σ ∇    = 0 [9.220] ∂t  〈 ρσ 〉 σ    Volume fraction constraint:

εγ + εσ = 1; εσ = 1 – εγ

[9.221]

Thermodynamic relations: 〈p1〉γ = 〈p1〉γ R1〈T 〉

[9.222]

〈p2〉γ = 〈ρ2〉γ R2〈T 〉 γ

γ

〈ργ〉 = 〈ρ1〉 + 〈ρ2〉

[9.223] γ

〈pγ〉γ = 〈p1〉γ + 〈p2〉γ

[9.224] [9.225]


349

One can add Equations [9.218] and [9.219] together to obtain a single continuity equation for water (component 1):  ε ∂ ( 〈 ρ 〉 σ ) + 〈 m˙ 〉  +  ε ∂ ( 〈 p 〉 γ ) – 〈 m˙ 〉  σ sv γ sv 1 1  ∂t   ∂t 

= D eff

∂ 2 〈 ρ1 〉 γ ∂x 2

[9.226]

which can be represented in terms of the gas phase volume fraction as: ∂ 2 〈 ρ1 〉 γ (1 – ε r ) ∂ ( 〈 ρ1 〉 σ ) + ε γ ∂ ( 〈 ρ1 〉 γ ) = D eff ∂t ∂t ∂x 2

[9.227]

Application of the above assumptions reduces the large equation set down to two main equations for the energy balance and the mass balance: 〈 ρ〉 C p

∂ 〈T 〉 ∂2 〈T 〉 + ( Ql + ∆ hvap ) 〈 m˙ sv 〉 = k eff ∂t ∂x 2

∂ 2 〈 ρ1 〉 γ (1 – ε γ ) ∂ ( 〈 ρ1 〉 σ ) + ε γ ∂ ( 〈 ρ1 〉 γ ) = D eff ∂t ∂t ∂x 2

[9.228] [9.229]

To make the comparison easier with the existing equations of Henry,20 Norden and David,9 and Li and Holcombe,21 one can rewrite the intrinsic phase averaged equations in terms of the concentration variables – for water in the solid (CF), and for water vapor in the gas (C):

CF = C=

mass of water in solid phase m1σ = = ρ1σ Vσ solid phase volume

mass of water in gas phase m1γ = = ρ1γ Vγ gas phase volume

[9.230] [9.231]

Since the definition of intrinsic phase average gives the same quantity as the true point value, one may use the relations: 〈ρ1〉σ = 〈CF〉σ = CF

[9.232]

〈ρ1〉γ = 〈C〉γ = C

[9.233]

to rewrite the mass balance equation as: (1 – ε γ )

2 ∂C F + ε γ ∂C = D eff ∂ C ∂t ∂t ∂x 2

[9.234]

The diffusion coefficient for water vapor in air modified by the gas volume fraction and the tortuosity are used to obtain the effective diffusion coefficient as:

350


D aε γ ∂ 2 C ∂C F + ε γ ∂C = τ ∂t ∂t ∂x 2 The thermal energy equation is: (1 – ε γ )

[9.235]

∂ 〈T 〉 ∂2 〈T 〉 + ( Q l + ∆ h vap ) 〈 m˙ sv 〉 = k eff [9.236] ∂t ∂x 2 The energy equation may be modified by recognizing that the mass flux term is contained in the solid phase continuity equation, such as: 〈 ρ〉 C p

∂C F ε σ ∂ ( 〈 ρ1 〉 σ ) + 〈 m˙ sv 〉 = 0 ⇒ 〈 m˙ sv 〉 = – ε σ ∂t ∂t

[9.237]

so that the thermal energy equation may now be rewritten as: 〈 ρ〉 C p

∂ 〈T 〉 ∂2 〈T 〉 ∂C F – ( Q l + ∆ h vap ) ε σ = k eff ∂t ∂t ∂x 2

[9.238]

Referring to the mass fraction weighted average heat capacity, Equation [9.119], j= N

Cp =

i=N

ε σ Σ 〈 ρj 〉 σ ( c p ) j + ε γ Σ 〈 ρj 〉 γ ( c p ) i i =1

j =1

〈 p〉

and spatial average density, Equation [9.118], j= N

i=N

j =1

i =1

〈 ρ〉 = ε σ Σ 〈 ρj 〉 σ + ε γ Σ 〈 ρi 〉 γ the thermal energy equation may be expressed as: {εσ [〈ρ1〉σ(cp)1 + 〈ρ 2〉σ(cp) 2] + εγ [〈ρ1〉γ(cp)1 + 〈 ρ 2〉γ (cp) 2]} – ( Q l + ∆ h vap ) ε σ

∂C F ∂2 〈T 〉 = k eff ∂t ∂x 2

∂〈 T 〉 ∂t

[9.239]

If it is assumed that the heat capacity of the gas phase is negligible, then the thermal energy equation becomes: {ε σ [ 〈 ρ1 〉 σ ( c p )1 + 〈 ρ2 〉 σ ( c p ) 2 ]}

– ( Q l + ∆ h vap ) ε σ

∂〈 T 〉 ∂t

∂C F ∂2 〈T 〉 = k eff ∂t ∂x 2

[9.240]

Dividing the previous equations by the solid volume fraction yields.


[ 〈 ρ1 〉 σ ( c p )1 + 〈 ρ2 〉 σ ( c p ) 2 ]

– ( Q l + ∆ h vap ) ε σ

351

∂〈 T 〉 ∂t

∂C F k ∂2 〈T 〉 = eff ε σ ∂x 2 ∂t

[9.241]

To be consistent with the notation of Li and Holcombe,21 (keff/εσ) is replaced by K. A volumetric heat capacity Cv is defined as: Cv = 〈ρ1〉σ(cp)1 + 〈ρ2〉σ(cp)2

[9.242]

kg  kg Note: Units for 〈ρj〉σ(cp)j are  3   J  ⇒  3  m ⋅ K  m   kg ⋅ K  The final thermal energy equation reduces to:

Cv

∂〈 T 〉 ∂C F ∂2 〈T 〉 – ( Q l + ∆ h vap ) =K ∂t ∂t ∂x 2

[9.243]

The two simplified equations for the mass and energy balance are thus: D aε γ ∂ 2 C ∂C F + ε γ ∂C = τ ∂t ∂t ∂x 2

[9.244]

∂〈 T 〉 ∂C F ∂2 〈T 〉 – ( Q l + ∆ h vap ) =K ∂t ∂t ∂x 2

[9.245]

(1 – ε γ ) Cv

As shown above, the general equations given in Section 9.9, with proper assumptions, can be reduced to the equations derived by Henry,20 Norden and David,9 and Li and Holcombe,21 for describing the diffusion of water vapor through a hygroscopic porous material.

9.10

Conclusions

Whitaker’s theory of coupled heat and mass transfer through porous media was modified to include hygroscopic porous materials which can absorb liquid into the solid matrix. The system of equations described in this chapter make it possible to evaluate the time-dependent transport properties of hygroscopic and non-hygroscopic clothing materials by including many important factors which are usually ignored in the analysis of heat and mass transfer through textile materials. The equations allow for the unsteady capillary wicking of sweat through fabric structure, condensation and evaporation of sweat within various layers of the clothing system, forced gas phase convection through the porous structure of a textile layer, and the swelling and shrinkage of fibers and yarns.

352


The simplified set of equations for heat and mass transfer, where mass transport occurs due to diffusion within the air spaces of the porous solid, was shown to reduce to the well-known coupled heat and mass transfer models for hygroscopic fabrics, as exemplified by the work of Li and Holcombe.21

9.11 A aσβ Am(t) cp Cp CF Cs CV D Deff D Da DLσ r g h h° hi hσβ ∆hvap k kε k〈T〉 K Kβ Kβ Kγ L m 〈 m˙ sl 〉

Nomenclature area [m2] Aσβ /V surface of the σ–β interface per unit volume [m–1] material surface [m2] constant pressure heat capacity [J/kg · K] mass fraction weighted average constant pressure heat capacity [J/kg · K] concentration of water in a fiber [kg/m3] concentration of liquid in the solid phase [kg/m3] volumetric heat capacity [kg/m3 · K] gas phase molecular diffusivity [m2/sec] effective gas phase molecular diffusivity [m2/sec] diffusion coefficient [m2/sec] diffusion coefficient of water vapor [m2/sec] diffusion coefficient of liquid in the solid phase [m2/sec] gravity vector [m/sec2] enthalpy per unit mass [J/kg] reference enthalpy [J/kg] partial mass enthalpy for the ith species [J/kg] heat transfer coefficient for the σ–β interface [J/sec ·m2 · K] enthalpy of vaporization per unit mass [J/kg] thermal conductivity [J/sec · m · K] ∂ 〈Pc〉/∂εβ [N/m2] ∂ 〈Pc〉/∂ 〈T〉 [N/m2 · K] permeability coefficient [m2] Darcy permeability for liquid phase [m2] liquid phase permeability tensor [m2/sec] gas phase permeability tensor [m2/sec] total half-thickness of body model system [0.056 m] mass [kg] mass rate of desorption from solid phase to liquid phase per unit

r r r 1 ρ ( v – w 2 ) ⋅ nσ β d A volume [kg/sec-m3] 〈 m˙ sl 〉 = V Aσ β σ σ 〈 m˙ sv 〉 mass rate of desorption from solid phase to vapor phase per unit volume [kg/sec ·m3]

∫


〈 m˙ lv 〉 r n p pγ pa pv ps Pc p0 p1° Q Q1

Qsv

r q r r r Ri R Rf S sir T T0 T t r ui r v r vi r 〈 vβ 〉 Vσ (t) Vβ (t) Vγ (t) V

(t) rm w r w1

V

353

mass rate of evaporation per unit volume [kg/sec ·m3] outwardly directed unit normal pressure [N/m2] total gas pressure [N/m2] partial pressure of air [N/m2] partial pressure of water vapor [N/m2] saturation vapor pressure (function of T only) [N/m2] pγ–pβ, capillary pressure [N/m2] reference pressure [N/m2] reference vapor pressure for component 1 [N/m2] volumetric flow rate [m3/sec] differential enthalpy of sorption from solid phase to liquid phase per unit mass [J/kg] enthalpy of vaporization from liquid bound in solid phase to gas phase per unit mass [J/kg] heat flux vector [J/sec ·m2] position vector [m] characteristic length of a porous media [m] gas constant for the ith species [N ·m/kg ·K] universal gas constant [8314.5 N·m/(kg·K)] textile measurement (@φ = 0.65), grams of water absorbed per 100 grams of fiber [fraction] saturation, fraction of void space occupied by liquid [fraction] irreducible saturation; saturation level at which liquid phase is discontinuous temperature [K] reference temperature [K] total stress tensor [N/m2] time [sec] diffusion velocity of the ith species [m/s] mass average velocity [m/s] velocity of the ith species [m/s] volume average liquid velocity [m/s] volume of the solid phase contained within the averaging volume [m3] volume of the liquid phase contained within the averaging volume [m3] volume of the gas phase contained within the averaging volume [m3] averaging volume [m3] material volume [m3] velocity of the β−γ interface [m/sec] velocity of the σ−γ interface [m/sec]

354


r w2

velocity of the σ–β interface [m/sec]

Greek symbols

εσ (t) εβ (t) εγ (t) εσL εσS εds εbw(t) Φ φr λ µ µβ µγ ρ ρβ ρi ρds ρw ργ ρv ρa τ τr ξ ξ ξ

Vσ /V , volume fraction of the solid phase Vβ /V , volume fraction of the liquid phase Vγ /v, volume fraction of the gas phase VL/Vσ , volume fraction of the liquid in the solid phase VS/Vσ, volume fraction of the liquid in the solid phase Vds /V , volume fraction of the dry solid (constant) Vbw /V , volume fraction of the water dissolved in the solid phase rate of heat generation [J/sec ·m3] pv/ps, relative humidity unit tangent vector shear coefficient of viscosity [N ·sec/m2] viscosity of the liquid phase [for water, 9.8 × 10–4 kg/m·s at 20 °C] viscosity of the gas phase [kg/m·s] density [kg/m3] density of liquid phase [kg/m3] density of the ith species [kg/m3] density of dry solid [for polymers typically 900 to 1300 kg/m3] density of liquid water [approximately 1000 kg/m3] density of gas phase (mixture of air and water vapor) [kg/m3] density of water vapor in the gas volume (equivalent to mass concentration) [kg/m3] density of the inert air component in the gas volume (equivalent to mass of air/total gas volume) [kg/m3] viscous stress tensor [N/m3] tortuosity factor thermal dispersion vector [J/sec ·m3] dummy integration variable a function of the topology of the liquid phase

Subscripts and superscripts o i l, L s, S σ β γ σβ

denotes a reference state designates the ith species in the gas phase liquid solid designates a property of the solid phase designates a property of the liquid phase designates a property of the gas phase designates a property of the σ –β interface


σγ βγ

355

designates a property of the σ –γ interface designates a property of the β –γ interface

Mathematical symbols d/dt D/Dt ∂/∂t 〈ψ 〉 〈ψβ〉 〈ψβ〉β

ψ˜ β

9.12

total time derivative material time derivative partial time derivative spatial average of a function ψ which is defined everywhere in space phase average of a function ψβ which represents a property of the β phase intrinsic phase average of a function ψβ which represents a property of the β phase denotes dispersion/deviation from the average for that phase or quantity

References

1. Whitaker, S. A., ‘Theory of Drying in Porous Media’, in Advances in Heat Transfer 13, New York, Academic Press, 1977, 119–203. 2. Jomaa, W., Puiggali, J, ‘Drying of Shrinking Materials: Modellings with Shrinkage Velocity’, Drying Technology 1991, 9 (5), 1271–1293. 3. Crapiste, G., Rostein, E. and Whitaker, S., ‘Drying Cellular Material. I: Mass Transfer Theory’, Chem Eng Sci, 1988 43, 2919–2928; ‘II: Experimental and Numerical Results’, Chem Eng Sci, 1988 43, 2929–2936. 4. Slattery, J., Momentum, Energy, and Mass Transfer in Continua, McGraw-Hill, New York, 1972. 5. Slattery, J., Momentum, Energy, and Mass Transfer in Continua, McGraw-Hill, New York, 1972, 19. 6. Gray, W., ‘A Derivation of the Equations for Multi-phase Transport’, Chemical Engineering Science, 1975 30, 229–233. 7. Morton, W. and Hearle, J., Physical Properties of Textile Fibres, John Wiley & Sons, New York, 1975, 178. 8. Lotens, W., Heat Transfer from Humans Wearing Clothing, Doctoral Thesis, published by TNO Institute for Perception, Soesterberg, The Netherlands, 1993, 34–37. 9. Nordon, P. and David, H. G., ‘Coupled Diffusion of Moisture and Heat in Hygroscopic Textile Materials’, Int J Heat Mass Transfer, 1967 10 853–866. 10. Stanish, M., Schajer, G. and Kayihan, F., ‘A Mathematical Model of Drying for Hygroscopic Porous Media’, AIChE Journal, 1986 32 (8) 1301–1311. 11. Dullien, F., Porous Media: Fluid Transport and Pore Structure, Academic Press, London, 1979, Chapters 4 and 6. 12. Kaviany, M., Principles of Heat Transfer in Porous Media, Springer-Verlag, New York, 1991, 428–431. 14. Chatterjee, P., Absorbency, Elsevier Science Publishing Co., Inc., New York, 1985, 46–47. 15. Ghali, K., Jones, B. and Tracy, J., ‘Modeling heat and mass transfer in fabrics’, Int J Heat and Mass Transfer, 1995 38 (1) 13–21.

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16. Gahli, K., Jones, B. and Tracy, J., Modeling Moisture Transfer in Fabrics, Experimental Thermal and Fluid Science, 1994 9 330–336. 17. Crow, R. and Moisture, ‘Liquid and Textiles – A Critical Review’, Defense Research Establishment Ottowa, DREO Report No. 970, June 1987. 18. Crow, R. and Osczevski, R., ‘The Effect of Fibre and Fabric Properties on Fabric Drying Times’, Defense Research Establishment Ottowa, DREO Report No. 1182, August, 1993. 19. Crow, R. and Dewar, M., ‘The Vertical and Horizontal Wicking of Water in Fabrics’, Defense Research Establishment Ottowa, DREO Report No. 1180, July, 1993. 20. Henry, P., ‘Diffusion in absorbing media’, Proceeding of the Royal Society of London, 1939 171A 215–241. 21. Li, Y. and Holcombe, B., ‘A Two-Stage Sorption Model of the Coupled Diffusion of Moisture and Heat in Wool Fabrics’, Textile Research Journal, 1992 62 (4) 211–217.

10 The cellular automata lattice gas approach for fluid flows in porous media D. L U K A S and L. O C H E R E T N A, Technical University of Liberec, Czech Republic

10.1

Introduction

It is appropriate to recollect the meaning of the word ‘automaton’, initially, for a better understanding of the concept of cellular automata. The word ‘automaton’ (plural – ‘automata’) is derived from the Greek word ‘automatos’ meaning ‘acting of one’s own will, self-moving’. In ancient Egypt, the term automaton was utilised for toys to demonstrate basic scientific principles. During the period of the Italian renaissance, automaton was the term used for mechanical devices, which were usually powered by wind or by moving water. The concept of modern automata started with the invention of automated animals (birds in a cage, mechanical ducks, etc.) and humanoids (robots). Therefore, in general, automaton suggests self-operation of activities or functions of an object in the absence of any permanent external governing factor. One of the most popular modern automata, which can be found at any workplace, is a computer, forming an inseparable part of our life. However, this chapter will be mainly focused on a new type of automata, the ‘cellular automata’, which have received a lot of attention recently in the area of modelling and simulation. According to one of the definitions provided by encyclopaedia, a ‘cellular automaton’ is a discrete model studied in computability theory and mathematics. Another definition states that it is a simplified mathematical model of spatial interactions in which each site, i.e. each cell or node of a two-dimensional plane, is assigned with a particular state at every instance of time and it changes stepwise automatically according to specific rules conditioned by its own state and by the states of its neighbouring sites. In Section 10.1.1, the ways by which cellular automata are used for modelling of physical phenomena and for reincarnation of some other models will be discussed. A more detailed definition of cellular automata and the difference between finite and cellular automata will be given in Section 10.1.2. Physical principles of lattice gas cellular automata will be described in Section 10.2. In the next Section, 10.3, 357

358


the reader will be introduced to various lattice gas models based on cellular automata: models of Hardy, de Pazzis and Pomeau (HPP) and Frisch, Hasslacher and Pomeau (FHP), along with their variations. Examples of computer simulations based on the Frisch, Hasslacher and Pomeau models will be presented in the Section 10.4, where physical phenomena such as fluid flow in an empty canal and in a canal with porous fiber-like material will be investigated. Lastly, Section 10.5 contains some suggestions and further information.

10.1.1 Historical overview Cellular automata have been invented independently many times and, as indicated previously (Wolfram, 1983), have been used for different purposes and under different names, viz. ‘tessellation automata’, ‘homogeneous structures’, ‘cellular structures’, ‘tessellation structures’ and ‘iterative arrays’. Some submit that cellular automata were introduced by John von Neumann under the name ‘cellular space’ at the end of 1940s. Others say that cellular automata were introduced by John von Neumann with his co-worker Stanislaw Ulam (Toffoli, 1991; Wolf-Gladrow, 2000). Original and pioneering work in this area was also done by Konrad Zuse around this time. It is mentioned in literature that mainly two journeys took place during the development of cellular automata. The first of them built up cellular automata, originally perceived merely as ‘toy’ tools, into serious systems of biological investigation and monitoring. Based on von Neumann’s works about self-reproducing systems (von Neumann, 1963, 1966), these studies have been developed in Lindenmayer, (1968), Herman, (1969), Ulam, (1974), Kitagawa, (1974) and Rosen (1981), for example. The last one streamed into computer problems (Sarkar, 2000). An excellent instance of the application of cellular automata in biology is the game of ‘Life’, invented by John Conway (Gardner, 1970). Examples of cell patterns obtained by Conway’s game ‘Life’ are shown in Fig. 10.1. A system evolution after 80 time steps or time units (t.u.) from an initial state has been considered there. It has been shown that simple update rules may lead to the formation of complex cellular patterns similar to living cell colonies, and plant and animal tissues. Several theoretical studies and analyses, related to the properties of cellular automata, augured their occurrence in modelling physical problems, especially in the simulation of hydrodynamic phenomena. It has already been noted that, in spite of simple update rules, cellular automata can display complex behaviour, which makes this suitable for use as a simulation tool for the description of many-particle or collective physical phenomena. The fully discrete model of hydrodynamics, based on the cellular automata concept, was first introduced by Hardy, de Pazzis and Pomeau (Hardy et al., 1973), nowadays known as the HPP model. This model led to many interesting results, but it has had

The cellular automata lattice gas approach for fluid flows

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10.1 Sets of patterns obtained in Conway’s game of ‘Life’ for various ˙ time evolution steps T (courtesy of Jakub Hruza).

limited application because of its anisotropic behaviour. It was not refined until 1986, when Frisch, Hasslacher and Pomeau designed their own ‘FHP’ model, based on a triangular lattice. Since then, application of the FHP model in modelling hydrodynamic problems has led to the design of derivative models. In the next sections, examples of such models and their usage in transport phenomena through porous materials will be discussed.

10.1.2 Finite automata, cellular automata, and cellular automata lattice gases The phrase ‘cellular automaton’ usually indicates an infinite set of finite automata, which are interrelated in a specific manner. A lattice gas cellular automaton is a special case of cellular automaton. What do the terms finite automaton, cellular automata, and lattice gas cellular automata mean in general and in the realm of cellular automata? Definitions of these terms are provided below. Finite automata. ‘Finite automata’ refers, in general, to a class of mathematical models of processors, or a special class of programming languages, that are characterised by having a finite number of states (Lawson, 2003), which evolve in time and produce outputs according to rules depending on inputs (Rivet and Boon, 2001). Similar definitions of finite automata can be found in literature sources, which refer to principles of simulation, modelling and programming. Taking this viewpoint, a finite automaton model consists of a finite set of internal states Q = {q0, q1, …, qn}, where q0 is an initial state, of a finite set of possible input signals A = {a1, a2, …, am}, and of a finite set

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of possible output signals B = {b1, b2, …, bp} (Kudryavtsev, originally KyπpRBπeB, 1985). Elements of the aforementioned set Q indicate a state space of the automaton, while sets A and B are the so-called alphabets (Chytil, 1984). It is assumed that the finite automaton works at discrete time moments, i.e. at discrete time steps t, t + 1, t + 2, etc. There exist two functions that drive the work of the finite automaton with respect to time, which are called transition functions. The first of them, denoted as ϕ, determines the state q (t + 1) of a finite automaton at an instant t + 1 if the previous automaton’s state q (t) and actual input signals a(t) are known. Then q(t + 1) = ϕ (q (t), a(t)). The last-mentioned function ψ designates output signals b(t), where b(t) = ψ (q(t), a(t)). An output signal of a finite automaton can be used as an input signal for another automaton. Three possible methods of finite automata representation are shown in Fig. 10.2. The term ‘individual automaton’ is used instead of ‘finite automaton’ in the realm of lattice gas cellular automata models (Rivet, 2001). This notation will be followed hereafter. Cellular automata. According to Wolfram (1986), ‘cellular automaton’ is a set of identical cells located in a regular and uniform lattice. A single cell is considered to be an individual automaton. The main characteristics of a finite automaton, mentioned above, relate to a cell of a cellular automaton. Therefore, a cellular automaton can be represented by a set of synchronized identical finite automata, which exchange their input and output signals with predefined neighbourhoods in accordance to a connection rule, which is the same for all finite automata in a particular model (Rivet, 2001). Purposely, this definition does not contain any reference to the geometrical structure of the lattice, as it is not important to know the distances or angles between neighbours. However, it may be noted that all finite automata in a cellular automaton are identical and frame a homogeneous structure having a uniform internal structure and obeying the same evolution and connection rules. An example of a two-dimensional cellular automaton is presented in Fig. 10.3. Evolution rules are carried out in this case for the concrete transition function. Lattice gas cellular automata. As mentioned earlier (Frisch et al., 1986), the points of view from which a fluid can be described are molecular, kinetic, and macroscopic. The detailed behaviour of a fluid in a continuum at macroscopic level is provided by partial differential equations, e.g. Navier– Stokes equations for the flow of an incompressible fluid. Some other numerical techniques, such as finite-difference and finite-element methods, are used for transforming a continuum system into a discrete one (Chen et al., 1994). The lattice gas models based on cellular automata are newer compared to the numerical methods mentioned above. These models make it possible to describe the behaviour of fluid systems at a molecular level under various microscopic conditions. They are based on detailed information about individual particles,


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10.2 Classical method of finite automata represented with state tree, state diagram and an input–output table. The table of states determines an initial state q0, final states and a transition function ϕ. For instance, from the second table line it is evident that, with the instant state q(t) = q1 and the momentary input signal a(t) = 0, the subsequent output state is q(t + 1) = ϕ (q(t), a(t)) = q3. The original root of the state tree arises from the initial state q0. The number of links that come out from each cusp of the tree is equal to the total number of input and output signals. Successors of each state are created according to the input signals, using the transition functions. Cusps of the state diagram agree with the states of automaton. Links indicate the possible transitions between all possible states.

such as their positions, masses, and velocities and they provide output in terms of molecular dynamics. Thus, lattice gas models entered into the history as an alternative for modelling fluid systems. It is a well-known fact from the molecular theory developed in the last century that, in the equilibrium state, individual molecules in crystals fluctuate around their average locations and that only occasionally do they jump out to other locations; these are considered as fluctuations. These jumps occur due to the molecule’s interaction with other molecules, when the system is shifted from its equilibrium state by some agent. A remarkable idea was to consider that a fluid has a structure similar to a crystal and that every liquid molecule sits at some fixed point, having the same number of neighbouring sites at a definite distance. These sites are either empty or occupied by a molecule (Boublík, 1996). These spatially organized patterns of molecules are in accordance with the term ‘lattice gas model’. Different types of lattice gas models were proposed for a description of simple liquid behaviour.

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Transition function

The state of finite automaton at time t

State of finite Input symbols as states of automaton at time t + 1 neighbourhood finite automata

10.3 Graphical interpretation of a cellular automaton: general appearance of a lattice of cells, detailed configuration providing status of neighbourhood cells of a reference cell, and application of a transition function on input symbols (represented by all the states of the neighbourhood) and an instantaneous state of the cell in question at times t and t + 1.

There are two distinct basic lattice gas models mentioned in the literature: non-interacting and interacting. The non-interacting lattice gas is mentioned in Kittel’s book (Kittel, 1977). This model is represented by a set of N noninteracting atoms distributed over N0 lattice cells. Each cell is either occupied or empty. This system does not have any kinetic energy or any energy due to interaction. In spite of that, it found its application in statistical physics because the non-interacting lattice gas model provides the correct shape of the ideal gas state equation where the pressure is obtained as a partial volume derivation of the system entropy. The interference of non-interacting lattice gas models and models based on cellular automata possibly helped towards a creation of interacting lattice gas models. Models, partly discrete with respect to time and space, were well known from the point of view of biological applications of cellular automata since the end of the 1960s. The first so-


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called classical lattice gases appeared as theoretical models for liquid–gas transition around the late sixties and beginning of seventies (Stanley, 1971). A moment-conserving lattice gas model started to be an object of interest in hydrodynamics and statistical mechanics when Kadanoff and Swift proposed their first discrete-velocity model (Kadanoff and Swift, 1968). They created a version of the Ising model in which positive spins acted as particles with momentum in one of the four directions on a square lattice, while negative spins acted as holes. Particles were then allowed to collide each with other or to exchange their positions with holes satisfying the conservation of energy and momentum (Rothman and Zaleski, 1994). Thus, the first interacting lattice gas models appeared at the beginning of the 1970s. The previously mentioned HPP model (Section 10.1.1) was the first well-known interacting lattice gas model, which reflected inception of current lattice gas models. Lattice gas cellular automata belong to the general class of cellular automata, thus sharing features characteristic to that class: (i)

(ii)

(iii)

Being one of the cellular automata, lattice gas cellular automata consist of identical individual automata which are tied geometrically to the nodes of a Bravais lattice, situated in a Euclidean space of dimension D. Individual automata are also called ‘nodes’ in the purview of cellular automata lattice gases. The instantaneous state of lattice gas cellular automata depends on the states of all individual automata. Each individual automaton can inherit any one of the 2B states. The quantity B represents the number of channels that correspond to the geometry of a lattice. These links play a role of ‘communication channels’ between neighbouring lattice nodes. Each channel may either be occupied by a fictitious particle or remain empty, and so it has two possible states of existence. Consequently, information about the channel’s occupation corresponds to signals fed to individual automata. The elementary evolution process of lattice gas cellular automata takes place in regular discrete time steps and consists of two distinct phases of evolution. The first of them is the collision phase. During this phase, each individual automaton takes the new post-collision state depending on input signals and transition rules. New states of individual automata generate output signals for the next evolution step. During the propagation phase, output signals of one automaton are conveyed to its neighbouring nodes, i.e. neighbouring individual automata, along the channels, thus, becoming a part of the input signals for its neighbours during the next time step. We should emphasise that all the changes in each of the individual automata of the lattice gas cellular automata, transmit output signals simultaneously. The transition rules are the same for all individual automata and do not depend on their position.

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In Fig. 10.4 is sketched the two-dimensional lattice gas cellular automata model based on the triangular Bravais lattice and the state of one individual automaton in a pre-collision phase. Detailed description of the principles and the terms related to the lattice gas cellular automata are furnished in the sections of this chapter to follow.

10.2

Discrete molecular dynamics

At a microscopic level, physical fluids consist of discrete particles. The particles of various fluids have variant shapes, masses, degrees of freedom, chemical structure etc., as shown in Fig. 10.5. That is why the very microscopic guise of collision events between and among them is quite likely to be different. The structure of the individual molecules of physical fluids influences the fluid density and formulates the concrete fashion of molecular interactions, which can affect fluid viscosity. On the other hand, as is well known from previous experiments, the general macroscopic behaviour of a fluid hardly depends on the nature of the individual particles constituting that fluid. From a theoretical point of view, significant variations of the molecular forms do not alter the basic nature of the macroscopic equations governing fluid dynamics. Those universal equations, such as the Navier–Stokes equation describing fluid dynamics or the equation of continuity, are, in fact, quite insensitive to microscopic details (Wolfram, 1986). The next underlying property of fluids is based on the spatial scale relationship between the mean free path of a particle after and before the

3 4 1

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10.4 Two-dimensional lattice gas cellular automata with a selected individual automaton highlighted will all details. The numbers assigned to the highlighted automaton indicate: 1 – the central node; 2 – a link/channel that connects the central node and one of the neighbouring nodes of the individual automaton; 3 – a moving particle; 4 – an arrow representing the particle velocity vector.


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H 2O

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10.5 Water and hexanol molecules have different structures.

succeeding collision and the areas in which collision events occur. As mentioned before (Succi, 2001), in a common collection of gas and liquid molecules, the average inter-particle separation is much greater than the typical size of an individual molecule, as is estimated by the ‘de Broglie length’; λ = h/p, where h is the Planck constant and p is a particle momentum. So the molecules may be treated as point-like particles. Moreover, these point-like particles/ molecules interact via short-range potentials and the effective ranges of interaction potentials are much smaller than the mean inter-particle separation. The universality of fluid dynamics leads one to attempt to extend the universality of the hydrodynamic to model fluids with even simpler microscopic dynamics, molecular structure, and inter-molecular interactions than any real fluid has. The gap between space scales of particles’ free movements and particles’ interactions, i.e. collision events, opens up the possibility of restricting the particle collisions strictly as localized events and of building up this concept as a lattice model, aiming at drastic simplification of classic Newtonian mechanics. From this, one can envisage a splendid fluid model with few assumptions to accomplish it, such as, considering that the particles travel only along the links in regular lattices, and that the inter-particulate collisions occur only at lattice nodes. This super simplification brings about a fully discrete model of hydrodynamics (Rothman and Zaleski, 1994), where the discreteness concerns space, time, particle velocities and any other microscopic observable physical quantities. Lattice gas cellular automata, as these models are generally called, are, in fact, drastically simplified versions of molecular dynamics. The cornerstone for this research has been laid by Frisch et al. (1986) and Hardy et al. (1973). It has been shown that lattice gas cellular automata, having continuity on a large scale, can be described by the partial differential equations of hydrodynamics. The Navier–Stokes system of equations (Landau and Lifschitz, 1987) is introduced below, in Equations [10.1] and [10.2]. The system of continuity equation will be started with the law of mass conservation.

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r r ∂ρ [10.1] + ∇ ( ρv ) = 0 ∂t r r where t is time, v = v ( x , y , z , t ) is the local and momentary liquid velocity vector close to a point having positional coordinate (x, y, z) in a rectangular Cartesianr system, and ρ denotes the fluid’s density derived r The r fromr its mass. k ∂ / ∂z ) ( i ∂ / ∂ x , j ∂ / ∂ y , differential operations symbol ∇ denotes the vector of r r r containing unitary vectors i , j , and k , oriented along x, y and z axes respectively. The intrinsic Navier–Stokes equation relates the fluid’s elementary changes in velocity at particular spatial locations with external forces, such as a force field, a pressure drop, and viscous drag being their origin. Based on Newtonian mechanics, this equation has to reflect conservation laws of momentum and energy. For a non-compressive liquid, the equation takes the form: r r ∂v + (∇r × vr ) × vr + 1 ∇r ( v 2 ) = – ∇p – ∇rU + η ∆vr [10.2] ρ ρ 2 ∂t

where p is the pressure, η is the dynamic viscosity, and U represents the r scalarr potential due r to r an external field. Finally, ∆ is the scalar product of ∇ and ∇ i.e. ∆ = ∇ ⋅ ∇ . Ultimately, to comment briefly on the idea of a creation of a beneficial lattice model of physical fluids with respect to the content of Chapter 14, where formally similar lattice structures of fluids interacting with fibrous materials, so-called ‘auto-models’, are introduced. Auto-models reflect the universal behaviour of liquids with respect to equilibrium thermodynamic laws, where the leading parameter is the surface tension and the underlying microscopic phenomena are attractive and repulsive forces, primarily considered as interaction energies between neighbouring molecules. This universality also leads to the lattice models in Chapter 14.

10.2.1 Lattice as a discrete space The advantages of quite a simple model of hydrodynamics, which has been discussed above, will now be introduced. The spatial structure and the geometry of the fluid model’s discrete space will be introduced at first. Lattices are realised in various dimensions. Here, only one- and especially two-dimensional lattices will be considered. A lattice consists of links, which will be referred to as ‘channels’ henceforth, to evoke traffic paths for particle movements. It also consists of nodes, where particles can collide. The channels connect the neighbouring ‘nodes’. As a rule, several channels meet in one node and the total number of channels that meet in a node is denoted by B, known as the ‘connectivity’. A node in a cellular automaton represents an individual


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automaton, where inputs and outputs are realised through channels with r jumping particles. Nodes will be represented by their radius vectors x i in a desirably chosen coordinate system. The structure of a channel network predescribes the set of allowed particle velocities. Nodes connected directly by a channel are neighbours and the set of all neighbouring nodes of the one in question is called its ‘neighbourhood’. From the above it follows that the set of all possible particle velocity directions destines the system of its neighbourhoods since these directions link the neighbours. The distance between nearest neighbours is denoted ∆l, and this length is called the ‘lattice r unit’, expressed in units of l.u. The channel vector ei is the unitary vector connecting neighbouring nodes through the channel i. In brief, the site at the centre is connected to its B neighbours by channels corresponding to the r r unity vectors ei through e B . It is essential that such lattices be homogeneous and symmetric, as will be explained in detail later on. Additionally, the issue of symmetry of the concerned lattices is the major obstacle standing between the super-simplified discrete lattice gas cellular automata and continuum hydrodynamics, thus drawing one’s attention momentarily towards it. Previous works with lattice models of hydrodynamics, introduced by Hardy, de Pazzis, and Pomeau (Hardy, 1973, 1976), dealt with issues related to problems of statistical mechanics, such as ergodicity and time correlations. Unfortunately, they have only limited application because this class of lattice gas models is limited to anisotropic hydrodynamics. Their anisotropic behaviour will be briefly dealt with in Section 10.2.4, describing the collisions of a lattice gas stream with a straight wall. The anisotropic properties of the HPP model were the direct consequence of the choice of a square lattice. It seems quite surprising that it took one decade to realise the direct consequences of underlying lattice symmetry on the hydrodynamics of lattice models. Fortunately, a very simple extension of the lattice shape to a triangular one with hexagonal symmetry suffices to inspire a discrete model to describe the macroscopic isotropic behaviour of hydrodynamics. The triangular lattice for lattice gas cellular automata was first introduced by Frisch, Hasslacher and Pomeau (Frisch, 1986). The lattice gas cellular automata based on square or on triangular lattices will be explained in detail in Section 10.3. Another necessity originating from the nature of cellular automata pertaining to the discrete fluid models is the structural homogeneity of the underlying lattices with respect to the neighbourhood of each node, which has to be identical. Figure 10.6 depicts two regular and square lattices partly covering a plane. One of these lattices has each of its odd rows shifted by a distance equal to half the length of its elementary side, i.e. half of the lattice unit (l.u.). A lattice without such a shift has identical neighbourhoods. This is the reason behind the fact that only the lattice without any shift fulfils the homogeneity conditions. The homogeneity conditions within a family of

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10.6 Illustration of two rectangular lattices with unlike neighbourhoods: The square lattice on the left-hand side is homogeneous, having identical neighbourhoods surrounding it. The neighbourhoods of the right-hand side rectangular lattice consist of three nodes appearing in two configurations, as highlighted. r a

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regular lattices may be verified from the definition of Bravais lattices (Ashcroft and Mermin, 1976). The complete set of two-dimensional Bravais lattices is introduced in Fig. 10.7. As is mentioned by Rivet (2001), the Bravais lattice is essentially an infinite one. For a lattice gas cellular automaton, it is considered that the lattice is only a subset of the relevant Bravais lattice. The reason behind it is quite simple: the memories of our computers have finite capacities and hence, in practical applications, this lattice subset contains only a finite number of lattice nodes.


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10.2.2 Discrete time It makes sense to speak about time intervals of lattice gases. The aspect of time dependence of lattice gases makes comparison of collective motions in lattice gases with space- and time- dependent local flows in real fluids possible. Time, as well as space of lattice gas cellular automata, is made discrete. The particles jump from their starting nodes to their destination nodes coherently. This synchronisation of jump of all particles constitutes the next step for the fluid model simplification. Each of the pairs of starting and destination nodes is connected by a channel colinear with the velocity vector of the jumping particle. Briefly speaking, in the synchronised time-cycle, particles r hop to the nearest neighbour by the corresponding discrete vector v i . The way to introduce a time unit into the lattice gas cellular automata model is the next problem. According to Rivet (2001), the basic element of a cellular automaton, an individual automaton forming the mathematical model of a processor with a finite number of possible internal states, evolves and produces output data according to a rule depending on input symbols belonging to a finite set of the alphabet. The above definition directs one towards a deterministic evolution rule for the internal state of an individual automaton. Since the internal state of an individual automaton can change, the automaton undergoes some kind of evolution and therefore the underlying notion of the ‘past’ and a ‘future’ is derived. However, these primitive notions do not necessarily imply a temporal structure for the automaton, since the concept of a time interval between events and the evolutionary behaviour of a cellular automaton as a whole is not included in the definition. That is why it is imperative to discuss the consequence of local automata synchronisation in a cellular automaton. Synchronisation of a cellular automata model with respect to time makes time the global parameter for all the nodes simultaneously. Therefore, there must be a single clock for all nodes, which justifies the unified time run for a lattice gas cellular automaton as a whole. The elementary synchronised particle jumps in a lattice gas cellular automaton are repeated at regularly spaced discrete time intervals. The time increment ∆t between successive jumps is called the ‘time step’, which is equivalent to a time unit abbreviated as 1 t.u. For the time step, the relationship ∆ l = v∆t holds true. This relationship r expresses the fact that a particle with velocity v i present in the i th channel r r r at the node x goes to the neighbouring node x + v i ∆t in 1 t.u. The collision phase is considered as an instantaneous one without any consumption of time. It means the time between succeeding collisions is ∆ t. The elementary evolution process of a lattice gas cellular automaton, which occurs at each time step, is a sequence of two distinct phases: the collision phase and the propagation phase. The order of these two phases is immaterial regarding time evolution of the cellular automaton. The aspect

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that is of great importance is the transition between the phases. Section 10.4.2 is devoted to a deeper description of the propagation and collision phases.

10.2.3 Discrete observables Observables, i.e. the basic physical quantities of a lattice gas cellular automaton, may be scalars, vectors or more generally tensors of arbitrary order. Basic observables of lattice gas cellular automata are connected to a channel of an index i and so they will be called ‘channel observables’ henceforth. A typical r channel observable is the number of particles ni ( x ) at a channel i of the r r node x . The ‘value of the observable measured at node x ’ is the total r amount of the observable quantity present at node x . It is called the ‘microscopic density per node’ or simply its ‘microscopic density’ if the observable is a scalar. If the observable is a vector, the value measured at a node is called a ‘microscopic flux’. The essential observable of a lattice gas is the number of particles, namely r the number of particles ni ( x ) at a channel i, i.e. the channel particle density, B r and the total number of particles at a node Σ ni ( x ) , which is the microscopic i =1

particle density at that spot. Commonly, a constraint called the ‘exclusion principle’ is imposed, which resembles Pauli’s exclusion principle in quantum mechanics. The ‘exclusion principle’ of lattice gases says: No two particles sitting at the same node can move along the same direction of the channel at the same time. The existence or non-existence of a particle at a channel i creates a two-bit ‘channel configuration space’ composed of two ‘channel states’. The distribution of a set of particles on various channels of the particular node defines the ‘local configuration space’. Regarding the exclusion principle, the local configuration space consists of 2B various ‘local states’, where B is the number of channels growing from a node. The next scalar observable is the individual mass of a particle. The mass r r assigned to any particle in a channel i at the node x is denoted as m i ( x i ) . r r The total mass m ( x ) at the node x , i.e. microscopic mass density, is given by the following formula: B r r r m ( x ) = Σ mi ( x ) ni ( x ) i =1

[10.3]

r Symbol v i is used to denote ‘velocity vectors’ of particles at a channel i. The velocity vectors must have the same local symmetries as the lattice has; that means the set of velocity vectors includes individual particle velocities that are determined by the structure of the underlying lattice. This set of velocity vectors remains globally invariant for all nodes in the lattice. The number of channels outgoing from a node determines the maximal number of various


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velocity vectors. Moreover, some particles can rest in a node with zero velocity. If the evolution rule involves exchanges of particles only with all the nearest neighbours and if all the velocity vectors are non-zero, then the model is said to be ‘homokinetic’, all velocity vectors having the same r modulus v = | v | . Two homokinetic models HPP and FHP-1 will be introduced shortly in Section 10.3. Assuming a unit time step, the velocity vector of each particle in a homokinetic model is given simply by the vector, r r v i = ei ∆l / ∆t . From the above-mentioned observables, one can easily derive the rest of the scalar and vector observables. To start with the scalars, the total kinetic r r energy E ( x ) at the node x , i.e. the microscopic density of kinetic energy, is obtained from the following formula: B r r r E ( x ) = 1 Σ mi ( x ) v 2 ( x ) 2 i =1

[10.4]

r r The microscopic density of potential energy W ( x ) at a node x holds the following relation: B r r W ( x ) = U ( x ) Σ ni ( x ) i =1

[10.5]

where U(x) is a scalar potential. r Among the vector observables, particle momentum pi at the channel i is given by: r r r r r r [10.6] pi ( x ) = mi ( x ) ni ( x ) v i ( x ) r Component ‘α’ of momentum of a particle at the channel i is piα ( x ) . The r total ‘α’ component of momentum at the node x is then determined by the formula [10.7]: B B r r r r r pα ( x ) = Σ piα ( x ) = Σ m i ( x ) ni ( x ) v iα ( x ) i =1

i =1

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r r The microscopic momentum flux p ( x ) is written as B B r r r r r r r r p ( x ) = Σ pi ( x ) = Σ mi ( x ) ni ( x ) v i ( x ) i =1

i =1

[10.8]

Besides the channel and microscopic observables, there are space-averaged quantities. The space averaging is carried out on a connected subset of the underlying lattice. The set of all nodes in this subset is denoted as φ. After that, the space-averaged mass density m(φ) is defined using the formula m (φ ) =

1 Σ m ( xr ) N (φ ) xr∈φ

[10.9]

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where, N(φ) is the total number of nodes in the lattice subset φ. Finally, it should be noted that macroscopic densities and macroscopic fluxes that are space- or time–averaged are physically relevant. The basic notions, definitions and fundamental visage of a lattice cellular automata serve as equipment sufficient to continue with a description of their kinetics or dynamics.

10.2.4 Propagation, conservation laws, and collision rules The dynamics of lattice gas cellular automata consists of two essential phases: propagation and collision. The propagation phase will be considered first, as it is conceptually much easier to understand. Before the collision phase is dealt with, the basic concept of lattice gas conservation laws will be adopted. As will be shown hereafter, these conservation laws govern the discrete dynamics of lattice gas cellular automata. Propagation phase. During the propagation phase, a particle is shifted from r one node to another, i.e. if a particle is present at any moment t in a node x , it is shifted to the neighbouring node in time t + ∆t. It is notable here that the r neighbourhood is pre-described by all practicable velocity vectors v i , according to a node-independent rule that covers the whole lattice. In practice, the particle at the channel i is transferred during the propagation phase from the r r r node x to the node x + v i ∆t . Consequently, the state of the channel i remains r r r the same, but the node changes from x to x + ei after the propagation. In other words, the propagation phase carries the particle from channel i of the r r r node x to the channel i of the node x + ei . The above description of the propagation phase raises the problem of finite size Bravais lattice subsets that are used for lattice gas cellular automata (as mentioned in Section 10.2.1). Indeed, if the lattice under the consideration r r is finite, the node x + ei may be outside this finite lattice, even if the node r x from which the particle departs is inside. There are various strategies to solve this problem. One of the solutions is to introduce ‘periodic boundary conditions’. More precisely, the part of the lattice on which the cellular automaton for the lattice gas is implemented has to be a finite sub-region of the underlying Bravais lattice, whose opposite sides can be connected to form a loop. This wrapping of opposite sides of a finite lattice leads to a periodic motion of the individual particles. The escaping particles return to the finite lattice on the opposite sides of its boundaries. Periodic boundary conditions influence the propagation phase only. Figure 10.8 gives more details about it. Another solution of the conflict between the theoretically infinite lattices of cellular automata and limited memories of computers that confines one to finite ones is to use ‘reflective boundary conditions’. Reflective boundary


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10.8 Periodic boundary conditions for two-dimensional square lattice gas cellular automaton, as used for the HPP model, result in identical collision and propagation occurring at the boundaries opposite each other. A system with periodical boundary conditions may be represented using fine-drawn joins. These joins transform the originally plan or lattice of nodes into a 3-D body on which surface the originally opposite boundaries of the lattice are joined together.

conditions are based on various types of particle collisions with walls that constitute impenetrable boundaries of the finite subset of Bravais lattice or with obstacles that represent the material of a porous or fibrous media. Since these boundary conditions are collision based, it has been decided to describe them in further detail in the subsection under the heading ‘Collision rules’. It can be summarised that reflective boundary conditions constitute bouncing of a particle from a wall back to the finite Bravais sub-lattice. The wall remains fixed all the time. It absorbs some of the portion of the colliding particle’s momentum, while the particle, after the collision, keeps its original velocity modulus v. The crucial feature of all the introduced boundary conditions is that they keep all the particles in the game. It means that none of the particles in the

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vicinity of the boundary of a finite Bravais sub-lattice may escape; they either reappear on the opposite side in the case of periodic boundary conditions, or bounce back as a result of a collision with a wall satisfying the reflective boundary conditions. Conservation laws. The propagation phase, except for the boundary conditions, is the shapeless part of the lattice gas cellular automaton’s lifetime. Particles move coherently towards their neighbouring nodes through the channels with constant velocities. This phase is purely kinetic. Particle motion is steady and linear during the abrupt and coherent jump. All the physical quantities of the particles, except those depending on the positions of the individual particles, are conserved. The lattice gas time during this phase, as the time step is defined as ∆ t = ∆ l/v. As the particle motion inside the channels has no relevance concerning the channels’ state of cellular lattice gas automata, the time flux is discrete. The next phase is very thrilling, when particles collide in an infinitely small time instant. To obtain the reasonable lattice equivalent of a real fluid dynamic, the conservation of particle numbers and conservation of their momentum are considered. Both these laws are described further for local collisions, i.e. inter-particle collisions at individual nodes. The results of such local collisions are unaffected by any events occurring in other nodes. r For the conservation of the local particle number n and mass m in a node x , the following relations hold true: B

r

r

B

r

B

r

r

B

r

Σ n ni ( x ) = iΣ=1 ni ( x ), iΣ=1 n mi ( x ) n ni ( x ) = iΣ=1 mi ( x ) ni ( x ) [10.10] i =1

r The initial distribution of the colliding particles in the node x at individual r channels i’s is represented by ni ( x ) , while their post-collision state in the r same node and channel is given by ‘new’ n ni ( x ) values. A collision of the particles in a node causes their redistribution possibly at all channels connecting the node in question with its neighbours. The local momentum conservation during the collision phase may be r r expressed using its components n pα ( x ) and pα ( x ) as: B

r

r

r

B

r

r

r

Σ n mi ( x ) n ni ( x ) v iα ( x ) = iΣ=1 mi ( x ) ni ( x ) v iα ( x ) i =1

[10.11]

Therefore, the redistribution of particles in an individual node obeys the rule of keeping the total momentum in this node constant. Rules governing particulate collision depend on the chosen model of the cellular lattice gas. Three such models will be introduced in Section 10.3. Collision phase and collision rules. Particle-conserving and momentumconserving local collision rules safeguard the correspondence of lattice gas


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cellular automata models with Navier–Stokes systems (Landau and Lifschitz, 1987). Their concrete form is elucidated here using the original idea of the first and the simplest lattice gas cellular automaton introduced by Hardy, de Pazzis and Pomeau (Hardy and Pomeau, 1972). The model’s title is also abbreviated to HPP (Frisch et al., 1986; Rivet and Boon, 2001), as has been mentioned earlier. This model is based on the two-dimensional regular square lattice. All the particles have the same unitary modulus of velocity v and they obey the exclusion principle. So the number of particles in a node spans from zero to four. The full set of collision rules for the HPP model can be reconstructed from the reduced set of two collision representatives with the application of lattice symmetry and superposition of the particle distribution obeying the exclusion principle. The representative collision events are depicted in Fig. 10.9. The collision process is said to be ‘microreversible’ (Rivet, 2001) if any collision has the same probability as the reverse one, and this kind of collision symmetry is called ‘detailed balance’. An original collision and the one assigned reverse to it are depicted in Fig. 10.9 as (A) and (C). The next vital notion to be discussed is that of ‘transitional probability’. Transitional probability denotes the probability of an occurrence of a certain post-collision state in the node as the consequence of a particular initial node configuration. As a rule, the symmetric collisions, matching with the lattice symmetry, have equal probabilities. The efficiency of lattice gas models to scatter particles through their mutual collisions is evaluated in terms of ‘effective collision’. A collision is said to be an ‘effective collision’ when a Y

Y

C

C

A D

D B

B

X

X t

t+ ∆ t

10.9 Schematic representation of collision events as applicable for the HPP model: Effective collisions (A) and (C) are microreversible. Collisions involving one (B) and three particles (D) do not change the velocity distribution of particles. The instantaneous positions of the particles, at time t and at a subsequent moment t + ∆t after one time step, are shown.

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post-collision configuration at a node differs from its pre-collision configuration at the same node (Rivet, 2001). To make this notion more lucid, some of the non-effective collisions are also sketched in Fig. 10.9. As has been mentioned in the beginning of this subsection, the symmetry of lattice cellular gas models is a vital issue, as it ensures its resemblance with continuum dynamics. Now, the anisotropic properties of a square, twodimensional, lattice gas automaton may be briefly illustrated using the effect of a particle’s collision with a solid impermeable wall, as shown in Fig. 10.10. To start with, various reflection behaviours of particles colliding with an impermeable obstacle will be introduced. Rivet (2001) introduced three different kinds of reflective boundary conditions. There are called ‘no-slip’, ‘free-slip’, and ‘diffusive’ boundary conditions. No-slip boundary conditions, on a microscopic level, represent a bounceback reflection of a particle colliding with a wall, i.e. with a wall particle. When a particle reaches the wall, its momentum as a vector is changed with central symmetry. In the centre of the symmetry is located a node where the collision occurs. In other words, the gas particle velocity vector goes round the half circle. Such a bounce-back collision conserves particle number and particle kinetic energy, and results in zero average velocity on a slip of a fluid flux in the vicinity of a wall, as each velocity vector at a time t belongs to the same particle velocity vector but with the opposite orientation at a succeeding time step t + ∆ t. Free-slip boundary conditions are realised by ‘specular reflection’. Microscopically, the specular reflection refers to the mirror reflection of a particle on a wall. The vector component of particle momentum, parallel to the wall surface, is conserved during such a collision, while the normal component of it is reversed. As a consequence, the cellular or lattice liquid freely moves along the wall without any change of its velocity component parallel to the wall. A point may be noted here, that it is quite troublesome to find a reflective flat surface on a rugged wall and the reader is referred to the work of Rivet (2001) for more details. The diffusive boundary conditions are stochastic or statistical combinations of bounce-back and specular reflections occurring with chosen probabilities. All previously mentioned boundary conditions with respective types of reflections, i.e. collisions with walls and obstacles, are depicted in Fig. 10.10. Going back to the lattice and lattice hydrodynamics isotropy, non-slip boundary collisions, realised by the bounce-back collision rule, are selected to demonstrate the anisotropic behaviour of a square lattice gas flowing along a flat wall in two-dimensional space. Two cases may be well distinguished. The wall inside the implicit square lattice of the HPP model may be either oriented along the channels in the lattice or inclined to this direction by an angle of 45°. In the first instance, a particle has no chance to slow down the bulk flow because every time, a particle from the gas bulk


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Y t

A

C

B

X Wall

Y

t + dt

p = 1/2 A

C

B

p = 1/2 X

Wall

Y

Y t

Wall

t + dt

X Wall

X

10.10 The upper and middle part of this figure constitutes impermeable walls, angled at 45° from the channel direction of the square lattice of the HPP model. Three various reflective boundary conditions that can appear are: (A) bounce-back reflection, (B) specular reflection, and (C) diffusive reflection. An HPP model with the wall parallel to a system of square lattice channels is depicted at the bottom. All previously mentioned types of reflections are indistinguishable with respect to the orientation of the impermeable wall. Due to the perpendicular direction of the velocity of particles colliding with the wall, there is no change in the particle momentum parallel to the wall before and after collision. Hence, the orientation of such a wall with respect to the lattice channels does not hinder the fluid flux. The initial and subsequent states of the automata are denoted with assigned time moments t and t + dt.

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flying to the wall, attacks the wall perpendicularly. These colliding particles have zero component of velocity parallel to the wall surface. As a consequence, the wall does not get any chance to break or encumber the adjacent tangential flow. That is why a parabolic velocity profile typical to the laminar fluxes of Newtonian fluids near the walls will not be achieved. It is noticeable that, for the mutual orientation between the wall and the square lattice, the bounceback reflection is identical to the specular reflection, and also identical to the diffusive reflections. This situation is depicted in Fig. 10.10. The same HPP lattice gas with underlying square lattice behaves in a different way when a wall is at 45° with one of the directions of the lattice channels. Falling particles on the wall carry both perpendicular and parallel momentum components with respect to the wall plane. During a bounce-back collision, a particle reverses its parallel momentum component. In other words, the wall will hinder a lattice gas flux caused by a prevailing movement of particles along the wall. It is now time to organise the parabolic velocity profile. The above-described behaviour of the HPP model is evidence of unsymmetrical properties of lattice gases living on square lattices. It is intuitively felt that such strict differences among various directions in triangular lattices with hexagonal symmetry do not exist. Therefore, the more advanced lattice gas cellular automata models have been developed on these triangular lattices. Two of them, FHP-1 and FHP-2, are described in the next section and additional details about them are mentioned in the Section 10.4.

10.3

Typical lattice gas automata

This section will introduce three classic lattice gas cellular automata models. The last of them will be used further (in Section 10.4) to demonstrate its utility for computer simulation of fibrous masses. Historically, the first lattice model was introduced in the early 1970s by Hardy, de Pazzis and Pomeau. They focused mainly on aspects of statistical physics. This model was based on a two-dimensional square lattice (Hardy et al., 1973) and had its roots in the earlier work of Hardy and Pomeau (1972). The same research group introduced fifteen years later (Frisch et al., 1986) a lattice gas cellular automata model, FHP-1, based on a triangular lattice with hexagonal symmetry. This was the simplest structure producing proper large-scale dynamics that could mimic the behaviour of a fluid. The last model that will be introduced in this section, abbreviated FHP-2 model, is a variant of the foregoing one. Unlike FHP-1, where all the particles were thought to move with velocities of unitary modulus, FHP-2 model included a possibility of one particle at rest in a node. The common feature of all previously mentioned lattice gas cellular automata models is the choice of basic channel observable values. If mass, velocity, momentum, energy, and time step are non-zero, they are all considered as unitary in their respective units.


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10.3.1 Hardy, de Pazzis and Pomeau model Let a two-dimensional square lattice having four channels at each node be envisaged. Then, the connectivity B is equal to 4, as shown in Fig. 10.9. Thus, each node has four neighbours. The distance ∆l between neighbouring nodes is uniform and equal to 1 l.u. All the particles in the model have the same velocity modulus v of 1 (l.u./t.u). The masses mi of the particles are equal and their value is taken as one unit mass (1 m.u). The model evolves in two phases – propagation and collision. A particle streams from its original r r r node x to its neighbouring one x + v i ∆t in the direction in which its velocity r v i is directed during the propagation phase. During the collision phase, the frontal collisions, i.e. the collisions of particles with opposite velocities, result in a rotation of both the particles by 90°, as illustrated particularly with examples (A) and (C) in Fig. 10.9. Briefly speaking, the horizontal motion of the particles arriving towards each other is changed to a vertical one when they depart from each other after their mutual frontal collisions. These rotations occur with probability one. It is to be noted that all other local states, denoted as (B) and (D) in the same figure, remain unchanged due to the constraint of momentum conservation. There are 24 different local configuration states of this model and only two of them are effective, i.e. two of them lead to the transition of the original state to the next local configuration state. One timestep of the Hardy, de Pazzis and Pomeau model is depicted in Fig. 10.9. The degree of crystallographic isotropy of the model is not sufficient to produce large-scale isotropic dynamics that have been represented above with the Navier–Stokes equations for physical fluids. The shortcomings of this model are highlighted by the atelier of its designers with the following words (Frisch et al., 1986): ‘When density and momentum are varied in space and time, micro-dynamic equations emerge differently, understood for HPP model and from the nonlinear Navier–Stokes equations in three respects. These discrepancies may be classified as (i) lack of Galilean invariance, (ii) lack of isotropy, and (iii) a crossover dimension problem.’ That is why more advanced models had to be sought. Rivet (2001) glosses this historical development as, ‘About ten years after the introduction of the HPP model, the “anisotropy disease” has been cured by models based on the triangular lattice.’ Some of the advanced models, developed initially, are discussed in the next subsection.

10.3.2 Two of the Frisch, Hasslacher and Pomeau models The first member of this group of models with isotropy, producing proper large-scale lattice fluid dynamics, was introduced by Frisch et al. (1986). Several versions of the Frisch, Hasslacher and Pomeau model have been successively developed with the same geometrical lattice structure, but with

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different collision rules. Two of them will be described further, viz. the FHP1 and FHP-2 models. The simplest model of this group, denoted the FHP-1 model, is based on a triangular lattice structure with hexagonal symmetry, having unitary distance ∆l between the neighbouring nodes, and unitary modulus of particle velocities v. Particles obey the exclusion principle. Hence, the maximum number of particles in a node is six, equalling the number of neighbours, i.e. to the connectivity B = 6. This limited number of simultaneous appearances of particles in one node safeguards the implementation of the exclusion principle on the model. The masses mi of all particles at each channel i are equal 1 m.u. The propagation phase in the FHP-1 model proceeds in exactly the same r way as for the HPP model. A particle sitting originally in a node x i with a r r r velocity v i is moved along the channel i to the neighbouring node x i + v i ∆t . A substantial difference with the HPP model appears in the collision phase. In FHP-1, two particles coming from opposite directions undergo a binary collision with an output state rotated by +60° or –60°, with equal probabilities. Another remarkable aspect of the FHP-1 model, compared with HPP, is the inclusion of three-particle collisions. When three particles meet simultaneously in one node, having their mutual velocity vectors at an initial angular disposition of 120°, a collision takes place with a rotatory deflection of the velocity vectors by 60°. The rotation by –60° leads to an identical local state transition. There are 26 (= b) possible various local states of the FHP-1 model and five of them, viz. three two-particle and two three-particle collisions, are effective. Hence, the collision efficiency of the model is 7.81%, as is obvious from Fig. 10.11. FHP-2 is a modification of the model FHP-1. As opposed to HPP and FHP-1, this model includes the possibility of one rest particle at each node. The propagation phase is the same as for the FHP-1 model and it has no Y

Y

C

A

C

A

B

B

D

D

t

X

t + dt

10.11 Typical two- and three-particle collisions in FHP-1 model.

X


381

influence on particles at rest. Particles at rest have zero values of velocity, momentum, and energy. These particles do not belong to any channel and so the exclusion principle is valid for this model too. The collision rules of the FHP-2 model are similar to that of FHP-1 model with the only difference that two additional events are considered in the FHP-2. A moving particle arriving at a node with a rest particle produces a pair of moving particles at angles +60° and –60°, measured from the direction of the incoming particle. The last additional collision event is the reverse to the former. Two colliding particles in a node with their velocity vectors at 120° angle result in one resting particle and in one moving particle moving in the direction of their original pre-collision momentum vector. There exist 27(= b) various local states in the FHP-2 model out of which only 22 are effective, as given in Fig. 10.12. Thus, the collision efficiency of the model is 17.19%. Thanks to the effective collisions with resting particles, FHP-2 does not conserve any kinetic energy. It is assumed that either the energy is exchanged with an adjacent thermodynamic reservoir or the resting particles vibrate with a vibrational energy equalling their original kinetic one.

10.4

Computer simulation of fluid flows through porous materials

In this section, the application of FHP-1 and FHP-2 lattice gas cellular automata models to simulate fluid flows in porous media is introduced. The section is divided into three subsections. To start with, a description of a lattice gas algorithm for general-purpose computers is considered. The text Y

Y A

A

E

E B

B

F

F C

C

G

G D

D

H

t

X

H

t + dt

10.12 Typical two- and three-particle collisions in FHP-2 model.

X

382


then follows with two examples of computer simulations based on the FHP1 lattice gas model. The first of them is devoted to the study of two-dimensional flow in an empty channel, the next one to fluid flow through a porous medium that mimics a fibrous material forced by a certain pressure gradient. Output data are compared with Darcy’s law relating values of flow rate and pressure gradient. The final computer simulation is focussed on the FHP-2 lattice gas model to study a fluid flow in a channel under the influence of outward vibrations transmitted to the fluid environment. Fluid flow through a porous media, and especially through fibrous materials, is a subject of wide interest. The textile industry encounters this phenomenon during many production and finishing processes. In these circumstances, permeability is the physical parameter of prime interest. Moreover, the permeability measurement is one of the most important ways that enables an evaluation of final products, as it provides concrete information about the usability of a material for an application. For example, permeability is a critical parameter for the application of fibrous materials such as filters, barrier materials and sportive clothing. The invention of Gore-Tex materials was based on an idea of combining various layers with different permeabilities to reach optimal comfort with respect to the diffusion of water vapour outwards and exclusion of external liquid droplets. Modelling the generation and propagation of sound wave hangs together with the study of acoustic properties of fibrous materials. New trends are, for instance, looking for ultrasound applications in textile technology to enhance traditional processes (Moholkar, 2002). Newly developing technologies are: (i) application of ultrasound in textile pre-treatment and finishing processes aiming to accelerate diffusion of liquids and gases into fibrous materials; (ii) ultrasound treatment used for reducing the viscosity and surface tension of resin systems involved in the production of fibre reinforced composites; (iii) application of ultrasound for impregnation of fibrous nanomaterials, produced by electrospinning, with highly viscous liquids (Ocheretna and Kostakova, 2005a).

10.4.1 Lattice gas algorithm A large variety of computers ranging from personal computers to powerful parallel processing supercomputers and a wide range of programming languages explain the existence of the quanta of lattice gas algorithms that have been implemented since 1985. The algorithm used in the present work is designed for a general-purpose computer. It includes an unchangeable part that can be used as a basis for each new algorithm, independent of the concrete choice of a lattice gas model. Each node of a lattice in the algorithm is conceived as a box with two main sections. The first of them is intended for registration of an instantaneous


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state. The second one serves as a bin for information about the new state of the cellular automaton in the next time step. When the new state of the system is accepted and becomes the new instantaneous state, data from the second sections are removed to the first ones so that the algorithm is ready for the next evolution step. Each of the sections is divided further into several shelves, where various prices of information about the node, related to the chosen time step, are collected, such as information about the channel occupation by particles, total number of particles in the node, and x- and ycomponents of velocities for all particles in the node. This set of information makes it possible to make all propagation and collision changes ‘simultaneously’ and to have comprehensive information about the system at any moment. The lattice gas algorithm starts with the occupation of chosen lattice nodes with solid stationary particles, which represent walls of a cavity or a channel. They can also in personate the material of a porous medium, particularly a fibrous material. Creation of fluid particles takes place on resting free parts of a lattice, where no solid non-moving particles are present. Each channel in each node takes either the value 1 or 0 at random, with predescribed probability. The value 1 means the occupation of a channel with a fluid particle, while the value 0 marks empty channels. Thereafter, the number of fluid particles and x- and y-components of their total velocity in each node are calculated. This information is stored in different arrays. The main part of the lattice gas algorithm consists of collision and propagation phases that repeat, subsequently. The algorithm starts with the collision phase, which is carried out uniformly and practically simultaneously in each lattice node s, excepting those occupied by a solid non-moving particle. The collision phase consists of the following steps: (i) Selecting the lattice node s0; (ii) Detecting the input information about the number of particles n0 in the node s0. If n 0 = 0 return to the Step 1. For the opposite case, detecting the x-component vx and the y-component vy of the total r particle velocity v in the node s0; (iii) Keeping the new value of the particle number nn0 in the node equal to the input value n0; (iv) Choosing a channel i(i = 1, … , B) of the node s0 at random; (v) If the channel is empty, then, occupying the selected channel i of the node s0 with a particle, i.e. with the value 1, and reducing the parameter n n0 by 1. In case the channel i is settled by a particle, going back to Step 4; (vi) Repeating Steps 4 and 5 till the parameter nn0 equals zero; (vii) Calculating the x-component of the total particle velocity nvx of the newly created configuration in the node s0. If the nvx in the node s0 is not equal to the original input value vx, going back to Step 2;

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(viii) Calculating the y-component of the total particle velocity nvy of the newly-proposed configuration in s0. If the nvy in this node is not equal to the original input value vy, going back to the Step 2; (ix) Registering the new information, i.e. the new configuration parameters’ the occupation of individual channels, nn0,i, nvx,i, and nvy,i, in the node s0, into information fields. The newly obtained configuration conserves the particle number and momentum components and thus can really be considered as a new configuration for the node s0; (x) Repeating the previous steps for all the lattice nodes. The propagation phase comes after the collision phase and consists of the following points in succession: (i) Selecting a lattice node s0 and detecting the input information of this node. Of particular interest now is the channel occupation; (ii) Scanning through the channels of the node s0 subsequently, and looking for the first occupied channel denoted here as i. If all channels are empty, returning to Step 1; (iii) If the channel i is occupied, then detecting the state of the neighbouring node si, which communicates with the node s0 through the channel i. (iv) If the node si is not occupied by a solid, stationary particle, relocating the particle in the channel i from the node s0 to the neighbouring node si so that the new particle number value nni in the node si extends by 1. New values of the x-component nvxi and the y-component nvyi of velocity in the node si are extended by vxi and vyi. If the node si is occupied by a solid, unmoving particle, implementing reflection depending on the chosen type of boundary conditions; (v) Repeating the previous steps for all the other lattice nodes in a chosen sequence. Thus, the basic skeleton of the lattice gas algorithm for a general-purpose computer, which has been used for further introduced simulation experiments, has been detailed. There are also so-called ‘mobile parts’ of the algorithm apart from the previously described skeleton of the algorithm. These mobile parts have not been involved in those aforementioned steps. Each particular simulation experiment includes, for instance, subroutines for the generation of extra conditions. These subroutines provide, for example, pressure gradient, gravity and vibration waves. Subroutines also ensure the formation of special output data files.

10.4.2 Computer simulation of two-dimensional fluid flow in porous materials As mentioned in Section 10.1, the lattice gas cellular automata can describe complex hydrodynamic phenomena in that they can substitute for Navier–


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Stokes equations. It is, then, quite natural to verify if the simulation model fulfils the basic fluid-flow laws, for instance, Darcy’s law. This law was named after the French engineer Henri Darcy (Rothman, 1988), who established it empirically during the middle of the nineteenth century. He found that the flow rate through a porous medium, including a fibrous one, is linearly proportional to the applied pressure gradient. This law is valid for laminar flows, where the Reynolds number is relatively small. In other words, the law is valid for steady Poiseuille flows with parabolic velocity profiles in free channels. An elementary example of a fluid flow satisfying Darcy’s law is the threedimensional flow between two parallel plates. It is a simple model for the flow through a single pore, the channel, which can be reduced to a twodimensional case due to its cross-sectional symmetry. Many researchers have dealt with this problem. For example, Rothman in his work (Rothman, 1988) studied two-dimensional Poiseuille flow as a function of the channel width for various pressure gradients. The same dependence was of Chen’s interest (Chen et al., 1991) for three-dimensional channel flows. Interesting problems were solved by Yang a few years ago (Yang et al., 2000), based on the Lattice–Boltzmann model, where the influence of various interactions between the fluid and the channel walls was considered. In particular, one part of the channel surface was wetted by a liquid while other parts repelled it. The first simulation experiments of the present work are aimed at studying twodimensional fluid flows under the influence of various pressure gradients and under conditions where the laminar character of the flow transits to a turbulent one. Fluid flow in a free two-dimensional channel. The concrete implementation of the lattice gas cellular automata that is used here is based on the FHP-1 model. The following values of channel parameters were chosen: the length L of the channel was chosen to be 550 lattice units (l.u.). In principle, the channel was infinitely long, thanks to the periodic boundary conditions applied on its left and right sides. The width d of the channel was 160 3/2 l.u. Top and bottom channel ends were composed of solid walls to restrict the flow. The bounce-back reflections were pre-set for the fluid particle collisions with solid wall particles. Fluid particles were generated in the free space between the walls. The mass of each particle was one mass unit (m.u.). The r average microscopic mass density m ( x ) was chosen to be 3.5 particles per node. Subsequently, the pressure gradient was varied to study the flow rate versus pressure gradient relationship. A similar method, as used later in this chapter, was exploited previously McNamara and Zanetti (1986) and Rothman (1988), for the creation of a pressure gradient. The pressure gradient in that work was created in terms of reversing particle momentum vectors with the

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chosen probability fx at all nodes of a vertical line of nodes, the length of which was equal to the channel width d, located on the left side of the horizontal channel. In fact, the parameter fx expressed the average change in the x-component of the particle momentum at a particular node during one time step or time unit (t.u.). This flipping mechanism acted merely on particles with negative x-components of velocity pointing leftwards. The ‘total force’ applied on the line of nodes was, then, nfx, where n represented the number of nodes in the line that spanned across the channel width. So the pressure P applied at the left-hand channel side was accordingly (Rothman, 1988; Lukas and Kilianova, 1996) expressed as P = nfx /d. That is why, dimensionally, fx had to have the dimension derived from dimensions of pressure and length, say, m.u. ∗ l.u./t.u.2 The value of the pressure gradient was obtained as the quotient of the ‘total force’ nfx and the product of the channel length and the channel width L ∗ d. During the study, the system was allowed to relax, i.e. to evolve to a steady state flow, after the start of each simulation. The steady flow rate was achieved after about 10 000 t.u. for parameter fx values ranging between 0.005-0.06 m.u. ∗ l.u./t.u.2. The smaller the probability value fx, the longer was the time period needed for achievement of a steady state flow. For example, for fx = 0.005 – 0.012 m.u. ∗ l.u./t.u.2 it took more than 13 000 t.u., as is evident from Fig. 10.13. The x-component of velocity was averaged over the whole channel length L for each horizontal node layer over 5000 time steps in the steady-state region to obtain velocity profiles for various pressure gradients. These computer-simulated outputs are presented in Fig. 10.14, exhibiting parabolic velocity profiles typical for Poiseuille flows. 0.3

fx = 0.063 fx = 0.058 fx = 0.052 fx = 0.047 fx = 0.04

Flow rate, l.u./t.u.

0.25 02

fx = 0.032 0.15

fx = 0.022 0.1

fx = 0.016 fx = 0.012 fx = 0.009

0.05

fx = 0.005 0

2000

4000

6000

8000 Time, t.u.

10000

12000

14000

10.13 Volumetric flow rate of the channel flow as a function of time, with various values of the parameter fx.


fx = 0.063 fx = 0.058 fx = 0.052 fx = 0.047 fx = 0.04 fx = 0.032 fx = 0.022 fx = 0.016 fx = 0.012 fx = 0.009 fx = 0.005

0.35 0.3 0.25

Velocity, l.u./t.u.

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0.2 0.15 0.1 0.05

0

20

40

60

80 100 Axis OY, l.u.

120

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160

10.14 Velocity profiles for various values of the parameter fx in the free two-dimensional channel.

Twelve independent experiments were carried out, for which the parameter fx varied from 0 to 0.06 m.u. ∗ l.u./t.u.2. The pressure gradients corresponding to these fx values were between 0 and 4.6 ∗ 10–4 m.u./(t.u.2 ∗ l.u.). This span of pressure gradients provided flow rates within the interval 0-0.25 l.u./t.u. The flow rate q was considered as a volumetric flow rate and could be easily r r detected as q = v x , where v x is the average x component of velocity per particle space, averaged over the entire lattice. The area where Darcy’s law was valid for the investigated systems is shown in Fig. 10.15. It can be seen that the linear dependence between flow rate and pressure gradient held for low flow rates up to 0.1 l.u./t.u For this region, Darcy’s law was valid. When the flow rate exceeded the value 0.15 l.u./t.u., the laminar flow probably changed into a turbulent one which led to the deviation from the linear relationship. This limit point depends, of course, on the channel width. The wider the channel is, the smaller the pressure gradient value limit for linear behaviour. Fluid flow through two-dimensional fibrous materials. Two-dimensional fluid flow through a porous medium that mimics a fibrous material, represented by a set of parallel pores, was studied in this experiment. The porous material was placed at the middle of a channel of length L = 450 l.u. and of width d = 250 3/2 l.u. The thickness of the model of the fibrous material was 90 l.u. and so it covered approximately one-fifth of the channel length. The width of pores inside the porous material was chosen as 10 l.u., and the distance between these equidistant and parallel pores was 18 l.u. The fluid

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0.3


0.25

0.2

0.15

0.1

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0

0.5*10–4

1*10–4 1.5*10–4 2*10–4 2.5*10–4 3*10–4 3.5*10–4 4*10–4 4.5*10–4 5*10–4 Pressure gradient, mu.*(t.u.)–2*(l.u.)–1

10.15 The extent of linearity of the flow rate’s dependency on the pressure gradient delimits the region of Darcy’s law validity.

flow in the channel was confined by solid walls, i.e. fibre surfaces, with the same boundary conditions as were used in previous computer simulations. The bounce-back reflections were exerted for fluid particle collisions with the fibers of the porous material. The fluid particles were generated again with a density of 3.5 particles per node. A pressure gradient was created in the same way as described previously, with periodic boundary conditions on the left and right sides of the channel. In the first series of computer simulations, the model of the fibrous material was located in a vertical direction, i.e. perpendicular to the direction of the fluid flow and the channel axis. In the final group of experiments, porous material crossed the channel axis at an angle 45°. Pores in the two-dimensional models of a fibrous material pointed, in both the cases, to the natural directions of the underlying triangular Bravais lattice, for more details see Figs. 10.21 and 10.22. They were horizontal in the first case, while they were inclined at 60° in the final one. The two previously mentioned orientations of fibrous materials in channels enabled variation of the inlet area of the fibrous material, keeping its internal geometrical characteristics intact. Several interesting features of the flow through these porous materials were exhibited during the computer simulations. At the beginning of the simulations, the steady fluid flow states were required for the next investigations. From Fig. 10.16, it is evident that the system with vertical orientation of the porous membrane reached its steady state just after 1000 t.u. The time requirement was more than 2000 t.u. when the two-dimensional model of the fibrous material was orientated as shown in Fig. 10.17. The development of temporal peaks of flow rate, which appeared


389

0.3

0.25


fx = 0.634 0.2

fx = 0.524 0.15

fx = 0.414

0.1

fx = 0.308 fx = 0.204

0.05

0

1000

2000

3000

4000

5000 6000 Time, t.u.

7000

8000

9000 10000

10.16 Particle flow rate as a function of time for various values of the parameter fx concerned with the fluid flow through a vertical twodimensional model of a fibrous material with horizontal pores. 0.07


0.06 0.05 0.04

fx = 0,751 fx = 0,602 fx = 0,463

0.03 0.02

fx = 0,334 fx = 0,215

0.01

0

1000

2000

3000

4000

5000 6000 Time, t.u.

7000

8000

9000

10000

10.17 Particle flow rate as a function of time for various values of parameter fx related to the fluid flow through a declined model of a fibrous material.

for low time values, was notable. They came into being as a consequence of the first strike of a group of fluid particles with the fibrous material, when the x-components of momentum had been reversed on the left-hand side of the channel with the probability fx. The flight was not hindered by any porous medium other than the channel walls, which represented a gigantic

390


pore, until they reached the inlet side of the fibrous material. These peaks were higher for simulations with a declined porous membrane. The same method as described for the computer experiments with a free channel was used for both the arrangements of porous membranes to obtain the velocity profiles. Interesting behaviour in the case of vertical as well as declined orientations of the fibrous layer, as demonstrated by computer simulation outputs, can be seen in Figs 10.18 and 10.19. Evidently, the flow was faster for the vertically orientated porous membranes than that for the declined ones, under the same pressure gradient values. As a result, the first case acquired the turbulent character at smaller pressure gradient values. It may also be noted that the local average velocity maxima corresponded to the positions of pores in the porous membrane. This effect is typical for fluids that do not wet pore walls (Yang et al., 2000). In Fig. 10.19, the velocity profiles of the system with the declined membrane may be noted too. The two lower curves predicated a laminar flow since their shapes resembled parabolic profiles. However, with increasing pressure gradient, the fluid flow probably became turbulent. The deformation of the upper curves could be explained quite simply. The declined layer of the fibrous material was in contact with the channel walls on its top and bottom edges. Two blind porous areas arose there. Particles that had been caught inside those areas could not come out easily. Both of those systems behaved in accordance with Darcy’s law, as was confirmed by the computer simulation outputs presented in Fig. 10.20. A 0.16

fx = 0.634

0.14

fx = 0.524

Velocity, l.u./t.u.

0.12

fx = 0.414

0.10

fx = 0.308

0.1

fx = 0.204

0.08 0.06 0.04 0.02 0

25

50

75

100

125 150 Axis OY, l.u.

175

200

225

250

10.18 Velocity profiles for various values of the parameter fx associated with the fluid flow through a vertical porous material. The horizontal axis represents the position across the channel from the axis y.


391

Velocity, l.u./t.u.

0.016 0.014

fx = 0.751

0.012

fx = 0.602 fx = 0.463

0.01

fx = 0.334

0.008 0.006 0.004

fx = 0.215

0.002

0

25

50

75

100

125 150 Axis OY, l.u.

175

200

225

250

10.19 Velocity profiles for various values of parameter fx regarding fluid flow through declined layer of fibrous material. 0.3 Vertical membrane


0.25

0.2

0.15

0.1

0.05

Declined membrane

0

0.001

0.002 0.003 0.004 0.005 Pressure gradient, m.u.*(t.u.)–2*(l.u.)–1

0.006

0.007

10.20 Linearity of flow rate versus pressure gradient relationships validates Darcy’s law for fluid flows through vertical and declined porous materials within the limits of the gradient values used for the present purpose.

nearly perfect linear dependence between the flow rate and the pressure gradient was found in both cases. It seems to be reasonable that the flow rate was higher when the inlet area of a porous medium was smaller, because lower resistance of porous medium was experienced and the flow was not so

392


tortuous. The velocity fields were monitored and expressed graphically for both the systems for better understanding of these phenomena. Particle velocities were space-averaged inside 5 l.u. × 5 l.u. squares and, simultaneously, these space-averaged velocities were time-averaged over 5000 t.u. inside steadystate regions of flows. Velocity vector arrays were obtained for maximal pressure gradients used for both systems. Local fluid flows were nearly parallel to the channel walls at the middle of the channel, as is evident from Fig. 10.21. In the interface between the free channel area and the porous membrane appeared a reorganization of fluid velocity directions, because the flow impacted on the solid parts of the fibrous material and the fluid particles tried to stream to the pores inside the fibrous layer. The reorganization of flow directions was even more evident in the regions of contact between the channel walls and fibrous material than close to the channel axis. An interesting situation appeared in the system with the declined membrane, as is visible from Fig. 10.22. Flow was distorted in this case through a greater part of the channel. The distortions took place on the upper as well as the bottom channel areas, in front of, and behind, the fibrous material layer as well, explained by previously described blind pores. On account of the appearance of tortuous flow, the flow rate decreased compared to the system where the membrane was placed along the vertical direction. It is also evident from Fig. 10.22 that the local fluid flow in blind pores close to channel walls was zero. It has been mentioned in the introduction of this section that the investigation is focused here mainly on the fluid flows through fibrous materials in order to carry out a permeability study. Some interesting problems will be discussed A

A

10.21 The field of velocity vectors for a fluid flow through a vertical fibrous layer. The length of each vector corresponds to the space and the time-averaged speed of the moving particles in a node at the vicinity. The horizontal side of the rectangular figure is parallel to the x-axis, while the vertical one has its direction identical to the y-axis.


393

A

A

10.22 The field of velocity vectors for a fluid flow through a declined layer of fibrous material. The horizontal side of the rectangular figure is parallel to the x-axis while the vertical one is directed towards the y-axis.

in the next part of this section, such as the sound wave motion through the fibrous materials and porous media in general and its attenuation.

10.4.3 Computer simulation of fluid flow through fibrous materials affected by sound vibrations In this subsection, the results of computer simulations for fluid behaviour in a free channel with a porous medium under the influence of vibrations will be presented. Here, an algorithm based on the FHP-2 lattice gas cellular automata model was used. A more detailed description of this model has been given in the Section 10.3.2. The specificity of the algorithm used has been described earlier (Ocheretna, 2005b). This algorithm created sound excitations as harmonic plane waves that travelled through the fluid along the channel and created variations as pressure waves. The pressure is, as a rule, proportional to the particle density in the FHP-2 lattice gas cellular automata model (Rothman, 1988). Firstly, let the focus be on the transmission of a sound wave through a fluid and on detection of attenuation of the sound wave in a free channel with respect to various periods of vibration and densities of the fluid. The free channel was created on a lattice with length L = 350 l.u. and width d = 250 3/2 l.u. The two-dimensional channel was confined within solid walls at its top and bottom sides. Between the walls, liquid particles were generated. Computer simulation trials were performed for particle densities 1.2 and 3.5 particles per node. Specular reflections of fluid particles from solid boundaries were used. A fictitious transmitter of harmonic signals was located on the left-hand side of the channel. These computer experiments

394


(c)

Density deflections from average value (m/u.)

(b)

2 1.5 1 0.5 0 0 –0.5 –1 –1.5 2 1.5 1 0.5 0 0 –0.5 –1 –1.5

0.025 0.023 0.021 50 100 150 200 250 300 350 0.019 Distance (l.u.)

Attenuation coefficient k

(a)

Density deflections from average value (m/u.)

were carried out for five different periods T of sound waves: 10, 15, 30, 45 and 60 t.u. The simulation program included the action of a fictitious sound transducer that was exerted at each time step based on an equation of harmonic vibrations. The action of the transducer was converted into the probability of deflections of fluid particles from their original positions within the transducer area. Fluid particles were considered to bounce in the positive direction of the x-axis if the value of the transducer displacements were positive, and were similarly related for the negative values. The bouncing probability fx inside the transducer area is, in fact, time dependent, and so, the bouncing probability in the x-component of a particle’s momentum at a node during one time step at time t is fx(t) = fx,max sin (2πt/T). As a consequence of the discrete time of lattice gas cellular automata, probabilities fx(t) were coarsegrained. Periodic boundary conditions on the left- and right-hand sides of the channel were used. Information about particle density in each node after the transducer was obtained as an output of the computer simulation. In order to quantify the attenuation coefficient and attenuation in general, the value of particle density obtained for each column of nodes was traced as a function of distance from the transducer (Ocheretna and Lukas, 2005c). Then the attenuation of the pressure wave was clearly visible and the attenuation coefficient was measurable, as shown in Fig. 10.23 (a) and (b). Firstly, maximal deflections of the particle density about their average values were detected, as shown in Fig. 10.23 (a). Then a regression curve was interlarded through the dots obtained from the density profile, and the equation of the regression was found, as shown in Fig. 10.23 (b). The attenuation coefficient k was taken from the regression equation and, in the same way, was found for other

0.017

Density is 1.2 particles per site

0.015 Density is 3.5 particles per site

0.013 0.011

y = e–k *x k

0.009 0.007

50 100 150 200 250 300 350 Distance (l.u.)

0.005

10

20

30 40 Period T (t.u.)

50

60

10.23 Comparison of attenuation coefficients in the free channel with various values of time period T of waves for two different particle densities.


395

waves generated with different periods T. Relationships between the attenuation of sound (pressure) waves and the transducer operational period T, which is a reciprocal of the transducer frequency, are shown for two different fluid densities in Fig. 10.23 (c). It can be seen that the sound waves of different T’s were attenuated quickly in the system with high particle density. It is also evident that waves of smaller period values T have the highest attenuation coefficient, which means a more rapid extinction compared to those with higher values of T. The same channel size and boundary conditions were used for the other computer simulations. Two-dimensional models of fibrous materials, with regular internal structures, were placed adjacent to the transducer area. The residual free part of the cavity was filled up with fluid particles at a density of 3.5 particles per node. Having knowledge of the previous results, it was decided to increase the period T up to 200 t.u. to prolong the life of a wave before it was quenched. Figure 10.24 shows the density profiles of waves which propagated through the regular chessboard-like fibrous material layers of equal thickness but of various porosities: 0.678, 0.736, 0.795, and 0.833. Density profiles of waves that travelled through the porous materials of various thicknesses: 10, 30, 50, and 70 l.u., having the same porosity of 0.678, are presented in Fig. 10.25. It is quite clear that the absorption of a wave depended on the structure and pore size of porous media. The attenuation of a sound wave increased with decreasing porosity or with increasing thickness of a porous material. The concept used here could be used for an investigation into the behaviour of real porous media, including fibrous materials. However, digital images of real fibrous materials have to be carefully analyzed to exactly mimic their internal morphology.

10.5


Interesting facts about cellular automata creation can be found in Hyötyniemi (2004). More generalised information regarding the lattice gas cellular automata may be obtained from some recently published monographs (Rothman and Zaleski, 1997; Chopard and Droz, 1998) and review articles (Chen et al., 1991; Boon, 1992). In this chapter, three basic models of lattice gas cellular automata have been dealt with, but there exist many more. For instance, the FHP-3 model is a further variant of the FHP-2 model (Rivet, 2001), where the collision rules are designed to include as many collisions as possible to achieve a collision efficiency of 59.4 %. The FHP-3 model was later modified (Bernadin, 1990; McNamara, 1990; Hanon and Boon, 1997) in order to study diffusion phenomena. The modifications involved consideration of mixtures of two species of particles that were chemically inert to each other and had identical mechanical properties. The model was called the ‘coloured

Average number of particles per node

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

30 130 180 Axis OX, l.u.

230

280

330

Porosity is 0.678

Porosity is 0.736

Porosity is 0.795

Porosity is 0.833

Without porous medium

A

10.24 Snapshots of particle density in waves that propagate down the x-axis through media of various porosities. The grey rectangle represents the localization of porous media in the channel. The transducer operates in an area just before the channel region is filled up by the model of the fibrous material.

80

A

The cellular automata lattice gas approach for fluid flows Without porous medium

5.5

Average number of particles per node

397

Thickness is 0,678

5.0

Thickness is 0,736

4.5

Thickness is 0,795

4.0

Thickness is 0,833 3.5 3.0 2.5 2.0 1.5 1.0 30

80

130

180 Axis OX, l.u.

230

280

330

10.25 Instantaneous particle densities in waves propagating through porous media of various thicknesses. Different degrees of grey shades represent the gradual growth of the thickness of the model of fibrous material. The transducer operates in an area just before the channel region is filled up by the porous medium.

FHP’ model (i.e. CFHP). Grosfils, Boon and Lallemand (in Boon, 1992) introduced in the beginning of the 1990s a lattice gas cellular automata model with non-trivial thermodynamics that contained thermal effects. The model was abbreviated as GBL following the initials of its developers. All previously mentioned lattice gas cellular automata models were built up on underlying two-dimensional lattices. The next evolution aimed at three dimensions. The frequently used three-dimensional lattice gas cellular automata model with correct isotropy is the ‘face-centred-hyper-cubic’ model, FCHC. More information is provided in papers by Henon (1987, 1989, 1992). One of the main drawbacks of lattice gas cellular automata is their statistical noise, hence, ‘lattice Boltzmann’ models have been developed to quench this noise. The first lattice Boltzmann model was proposed by McNamara and Zanetti (1988) and almost at the same time it was also introduced by Higuera and Jimenz (1989). Some general books on lattice Boltzmann models were written later (Wolf-Gladrow, 1999; Succi, 2001). The most significant application of lattice gas cellular automata is on the flow of heat and mass through porous media. Basic articles in this area have been written by Rothman (1988, 1990) followed by Kohring (1991), Chen et al. (1991a), and Lutsko et al. (1992). The first lattice Boltzmann simulation

398


of porous media was performed on a cubic lattice (Foti et al., 1989). Generally speaking, lattice gas cellular automata and lattice Boltzmann models are considered to be the most suitable for simulating microhydrodynamic flows through porous media (Koponen et al., 1998) and hence through fibrous materials too. Finally, let the two seemingly similar models in this book, viz. the lattice gas cellular automata and the auto-models from Chapter 14, entitled, ‘Computer simulations’, be compared. Lattice gas cellular automata are, in many respects, akin to Markov random field models, especially in those cases where collision rules are governed by transition probabilities (Rivet, 2001). Intuitively, a lattice gas automaton with probabilistic transitions in the collision phase is a spatial stochastic scheme, where the local configuration of a node is influenced by that of its neighbouring nodes. The random variable of lattice gas automata is a numeric integral code representing a local configuration, i.e. the local distribution of particle velocity vector of the node in question. Both the models have nearly identical geometry and formal descriptions of basic notions (Lukas and Chaloupek, 1998) but the construction of their temporal evolution is quite different. In other words, the great difference between the lattice gas cellular automata and the auto-models appears in the rules governing their dynamics. The auto-model dynamics are driven by subsequent alternations of variable values in restricted number of cells/nodes. Generally, the dynamics of auto-models that are used frequently allow only subsequent local changes of a variable in an isolated cell/node or these variable values can be subsequently exchanged in a couple of cells/nodes only. On the other hand, the collision laws of lattice gas cellular automata, reflecting chosen conservation laws, can be run in all lattice nodes simultaneously. The differences between the two aforementioned discrete models reflect discontinuity in recently developed theoretical tools describing equilibrium thermodynamics, such as the above mentioned auto-models, and non-equilibrium thermodynamics, such as the lattice gas cellular automata. Both the models could be used, obviously, for the description of a system in an equilibrium state. Auto-models reflect naturally inter-particle energy exchanges while lattice gas cellular automata mimic conservation laws of chosen scalar as well as vector observables. A more detailed discussion about the mutual relationship between the automodels, represented by the popularly known Ising model, and the cellular automata, in general, can be found in Vichniac’s work (Vichniac, 1984). Lastly, the auto-models and the lattice gas cellular automata may be pointed out to be different from the point of view, purely formal, that the basic element of a cellular automaton is known as a ‘node’, while the term ‘cell’ is used in the realm of the auto-model, as presented in Chapter 14.


10.6

399

References

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Kitagawa T (1974), ‘Cell space approaches in biomathematics’, Math. Biosci., 19, 27. Kittel Ch (1980), Thermal Physics, John Wiley & Sons Inc., New York. Kohring G A (1991), ‘Calculation of the permeability of porous media using hydrodynamic cellular automata’, J. Stat. Phys., 63(1/2), 411–418. Koponen A, Kandhai D, Hellen E, Alava M, Hoekstra A, Kataja K, Niskasen K and Sloot P (1998), ‘Permeability of three-dimensional random fiber webs’, Phys. Rev. Lett., 80(4), 716. Kudryavtsev V B, Aleshin C V, Podkolzin A S (1985), Introduction to the automata theory, Moscow, Nauka. Landau L D and Lifschitz E M (1987), A Course of Theoretical Physics, Fluid Mechanics, 2nd edition, Pergamon Press, Oxford. Lawson M (2003), Finite Automata, Chapman & Hall/CRC Press. Lindenmayer A (1968), ‘Mathematical models for cellular interactions in development’, J. Theoret. Biol., 18, 280. Lukas D and Kilianova M (1996), ‘Modelovani proudeni pomoci bunecnych automatu’, 12th Conference of Czech and Slovak Physicists, Ostrava (Czech Republic), Vol. 2, 729–732. Lukas D and Chaloupek J (1998), ‘Interakcni energie a hybnosti v mrizovych modelech tekutin’, STRUTEX Struktura a strukturni mechanika textilii, Liberec (Czech Republic), 34–38. Lutsko J L, Boon J P and Somers J A (1992), ‘Lattice gas automata simulations of viscous fingering in porous media’, Lecture Notes in Physics, 398, 124–135, Berlin, SpringerVerlag. McNamara G and Zanetti G (1986), ‘Direct measure of viscosity in a lattice gas model’, Cellular Automata ’86 (abstract), MIT Lab. for Comp. McNamara G and Zanetti G (1988), ‘Use of the Boltzmann equation to simulate lattice gas automata’, Phys. Rev. Lett., 61, 2332. McNamara G R (1990), ‘Diffusion in a lattice gas automaton’, Europhys. Lett., 12, 329. Moholkar V S (2002), Intensification of Textile Treatments; Sonoprocesses Engineering, Enschede, Twente University Press. Ocheretna L and Košťáková E (2005a), ‘Ultrasound and Textile Technology – Cellular Automata Simulation and Experiments’, Proceedings of ForumAcusticum, Budapest, Hungary, 29 Aug–2 Sep, 2843–2848. Ocheretna L (2005b), ‘Modeling of generation and propagation of harmonic waves based on a FHP lattice gas model’, Proceedings of 8th International Conference Information Systems Implementation and Modelling, Ostrava (Czech Republic), 313–318. Ocheretna L and Lukas D (2005c), ‘Modeling of ultrasound wave motion by means of FHP lattice gas model’, 5th World Textile Conference AUTEX 2005, Proceedings, Book 2, University of Maribor, 634–639. Rivet J-P and Boon J P (2001), Lattice Gas Hydrodynamics, Cambridge, Cambridge University Press. Rosen R (1981), ‘Pattern generation in networks’, Prog. Theor. Biol., 6, 161. Rothman D G (1988), ‘Cellular automaton fluids: a model for flow in porous media’, Geophysics, 53, 509–518. Rothman D H (1990), ‘Macroscopic laws for immiscible two-phase flow in porous media: results from numerical experiments’, J. Geophys. Res., 95, 8663–8674. Rothman D H and Zaleski S (1994), ‘Lattice-gas models of phase separation: interfaces, phase transitions, and multiphase flows’, Reviews of Modern Physics, 66, 1417–1479.


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11 Phase change in fabrics K. G H A L I, Beirut Arab University, Lebanon N. G H A D D A R, American University of Beirut, Lebanon B. J O N E S, Kansas State University, USA

11.1

Introduction

Phase change in fabrics can result from moisture sorption/de-sorption processes in the fiber, from moisture condensation/evaporation in the fabric air void volume, and from the presence of micro-encapsulated phase-change paraffin inside textile fabrics with melting and crystallization points set at temperatures close to comfort values. The structure of a fabric system consists of a solid fiber and entrapped air. The ability of the fabric to transport dry heat is largely influenced by the amount of entrapped air while the ability to transport water vapor is influenced by the volume of the solid fiber and its arrangement. The solid fiber represents an obstacle to the moving water vapor molecule, and tends to increase the evaporative resistance of the fabric. In addition, the solid fiber serves to absorb or de-absorb moisture, depending on the relative humidity of the entrapped air in the microclimate and on the type of the solid fiber. For example, wool fiber can take up to 38% of moisture relative to its own dry weight. The moisture sorption/de-sorption capability of the fabric influences the heat and moisture transport across the fabric and its dry and the evaporative resistance. When fibers absorb moisture, they generate heat. The released heat raises the temperature of the fiber, which results in an increase of dry heat flow and a decrease in latent heat flow across the fabric. The opposite effect takes place in the case of water vapor de-sorption. When thermal conditions change at the fabric boundaries, the hygroscopic fabric experiences a delayed effect on heat and moisture transport. The water content of the fabric does not only include the absorbed water in the solid fiber and the water vapor in the entrapped microclimate, but also includes the liquid water that can be present in the void space. This liquid water can originate from a moist source in which the liquid water is wicked or it can result from condensation in the case where water vapor continues to diffuse through a fully-saturated solid fiber. Similar to the sorption/de-sorption of moisture, liquid condensation and evaporation influence the flow of heat 402


403

and moisture across the fabric by acting as a heat source or sink in the heat transfer process. In addition, condensation has a significant effect on thermal comfort because of the uncomfortable sensation of wetness by humans. With the advancement of technology, phase change occurrence in fabrics is no longer limited to moisture sorption/de-sorption in the solid fiber and moisture condensation/evaporation in the void space of the fabric, but it also occurs by incorporating micro-encapsulated phase change materials (PCM) inside textile fabrics. The introduction of PCM technology in clothing was developed and patented in 1987 for the purpose of improving the thermal performance of textile materials during changes in environmental temperature conditions (Bryant and Colvin, 1992). PCMs improve the thermal performance of clothing when subjected to heating or cooling by absorbing or releasing heat during a phase change at their melting and crystallization points. Since adsorption/de-sorption is addressed in Chapter 12 of this book, this chapter will mainly take into consideration the effect of condensation and the effect of using PCM in fabrics on the transport of heat and moisture through fibrous medium, and their impacts on clothing properties and comfort.

11.1.1 Mechanism of moisture condensation/evaporation For condensation to take place in a fibrous medium, a temperature gradient should exist across the medium such that one side of the fibrous system is directly exposed to a moist hot air environment or is being sprayed with liquid water, while the other side of the fabric is subject to a low temperature. In addition, the fibrous system should have a low water vapor permeability to achieve condensation. This situation is common in the case of human clothing systems, where clothing can be sandwiched between a hot humid human skin and an outer lining fibrous layer of low water vapor permeability exposed to a cold air stream. When a dry hygroscopic fibrous layer is suddenly exposed to the abovementioned conditions, the water vapor originating at the hot side will diffuse into the fibrous medium. First, there will be a rapid moisture uptake by the dry solid fiber. The heat released as a result of adsorption by the fiber will raise the temperature of the fibers and increase their water vapor pressure. As a result, the vapor pressure gradient between the absorbed water and the microclimate water vapor will be reduced, causing a slow down in the rate of adsorption. The increase in the fiber diameter (swelling) due to moisture uptake will lower the permeability of the fabric system to water vapor (see Chapter 9 for discussion of sorption kinetics). The fabric will remain dry if the water vapor pressure of the microclimate is greater than the water vapor pressure of the bound water, and if the vapor concentration in the microclimate is less than the saturation vapor concentration at the fabric local temperature. When equilibrium between the absorbed water in the solid fiber and the

404


microclimate is established, the diffused water vapor from the humid side will be transferred to the environment without the occurrence of condensation. In this case, the transient effect of absorption ends, but the dry and latent heat transport from the hot humid source continues and the fibrous medium does not become wet. If the vapor concentration is increased to a level such that somewhere within the fibrous system moisture saturation is reached, condensation will occur. The condition of saturation could be attained by increasing the concentration of the water vapor in the microclimate, which can be achieved by either increasing the water vapor concentration at the warm side of the system or by lowering the permeability of the fabric to water vapor. In addition, increasing the temperature gradient across the fabric by lowering the temperature of the colder side will cause the condition of saturation in the microclimate to occur at lower microclimate water vapor concentration. Condensation is a phenomenon that is more likely to take place when the fibrous medium is exposed to large temperature differences and to a high humid source that causes the local relative humidity of the microclimate to reach 100%. Once the microclimate of the fibrous system attains saturation while there is still extra moisture diffusing into it, condensation continues to occur. Therefore, unlike the absorption process, which is transient in nature, the condensation process is continuous. Since condensation takes time, a state of transitory super-saturation may exist in the microclimate causing the relative humidity to exceed 100%. Yet this state of super-saturation does not last, and given enough time, the excess moisture will condense, thus reducing the relative humidity to 100% (Jones, 1992). The condensation process will release the heat of condensation, affecting both temperature and concentration gradients across the fabric. Condensation in a fibrous medium can occur anywhere within the fibrous medium when the local vapor pressure rises above the saturation vapor pressure at that location temperature. The location of the condensation can be predicted by utilizing the saturation vapor line and water vapor pressure line (Keighley, 1985; Ruckman, 1997). Figure 11.1 shows a schematic of water vapor pressure variation against temperature of the fibrous medium (curve A) and the corresponding saturation vapor pressure (curve B). Saturation line curve B shows the water vapor pressure corresponding to 100% relative humidity at a specific temperature. If the microclimate water pressure at that temperature exceeds the saturation temperature, condensation will occur at that location. There is a linear relation between saturated water vapor pressure and temperature. At high temperatures, saturation vapor pressure is already high, and for condensation to occur, the local water vapor pressure should be greater than the saturation pressure. For that reason, condensation is more likely to occur close to the colder boundary of the fibrous system. Contrary to the case for condensation, evaporation of liquid water occurs


405

(A)

Water vapor pressure

Φ = 100 %

(B)

Condensation Saturation line

Temperature

11.1 Schematic of the water vapor pressure distribution in a fibrous medium against its temperature variation (curve A) and the corresponding saturation vapor pressure distribution (curve B).

when the relative humidity of the surrounding microclimate in the void space is less than 100%. When liquid water exists in the fabric void space, a saturated boundary layer is formed at the interface between the liquid and the microclimate air. If the vapor pressure of this boundary is greater than the vapor pressure of the microclimate air, then evaporation occurs. In this case, the rate of moisture leaving the fibrous system is greater than the rate of moisture going into the system. Evaporation of moisture in a fibrous system usually moves from the warm moist boundary of the fibrous medium across the gas-filled void space where it may condense or diffuse out of the fibrous system, depending on the coupled moisture and temperature distributions.

11.1.2 Effect of condensation on clothing heat transfer and comfort Clothing is a crucial factor in determining human thermal comfort. The purpose of clothing is to maintain a uniform body temperature under different body activity levels and different environment temperatures. In addition, clothing keeps the human body skin dry by preventing the accumulation of sweat on the human skin and by allowing the perspired body water to flow to the outside environment. In most comfortable environmental conditions at low activity levels, the perspired sweat from the skin escapes through clothing without the incidence of condensation since the rate of perspiration is low. At higher activity levels, the perspiration increases to a level that may cause

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condensation to occur within the clothing system. The occurrence of sweat in the clothing system is generally affected by the vapor permeability of the different fabric layers constituting the clothing ensemble, the skin vapor concentration, and the environment temperature. Comfortable clothing should not only provide human thermal comfort sensation, but should also give the wearer a minimum awareness of this comfort, as was suggested by Keighley (1985). The condensation of sweat on the clothing layers affects both the human sensation of comfort and the attentiveness of the wearer to the clothing ensemble. When condensation occurs in clothing, the moisture permeability of the fabric decreases, allowing more sweat to accumulate on the skin, thus affecting the human thermal sensation of comfort. In addition, the pressure of the garment on the human skin increases because of its increased weight. As a result, the awareness of the clothing wearer increases and the clothing system will be considered uncomfortable. The condensation process liberates heat of condensation causing the local clothing temperature to increase at the condensation location, thus changing the temperature gradient across the clothing that existed prior to the condensation process. In most cases, the temperature gradient across the clothing system uniformly increases from the human skin to the outside environment. As condensation occurs, the temperature gradient from the skin to the location of condensation decreases and the temperature gradient from the spot of condensation to the outside environment increases (Lotens, 1993). Since the heat of condensation at the human skin does not leave the human clothing system because of the perspired moisture, it may be suggested that the sweating process is thermally ineffective in providing the necessary heat loss from the human body. But as was explained by Lotens (Lotens, 1993), the heat has already left the human skin and passed a good distance in the clothing system away from the human skin, causing an increase in the temperature of the outer clothing layer where condensation is more likely to take place. The increase in temperature of the outer layer causes an increase in the dry heat transport from clothing, which may compensate for the decrease in the latent heat transport from the clothing system. However, in this case, the clothing will be wet and will be considered uncomfortable.

11.1.3 Mechanism of phase change in PCM fabrics Unlike the phase change mechanism in the condensation/evaporation process, which depends on the moisture and temperature gradient across the fabric, the mechanism of the phase change process in PCM fabrics is a temperaturedriven process. It mainly depends on the temperature and the type of the PCM that is encapsulated in a protective wrapping or microcapsules of a few microns in diameter. The microcapsules are incorporated into the fibers of


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the fabric by the wet spinning process or coated onto the surface of the fabric substrate (Pause, 1995). Microcapsules protect the PCM and prevent its leakage during its liquid phase. PCMs are combinations of different types of paraffin (octadecane, nonadecane, hexadecane, etc…), each with a different melting and crystallization point. Changing the proportionate amounts of each paraffin type can yield the desired physical properties (melting and crystallization). When the encapsulated PCM is subject to heating, it absorbs heat energy and undergoes a phase change as it goes from solid to liquid. This phase change produces a temporary cooling effect. Similarly, when a PCM fabric is subject to a cold environment where the temperature is below the crystallization point, the micro-capsulated liquid PCM will change back to the solid phase producing a temporary warming effect.

11.2

Modeling condensation/evaporation in thin clothing layers

The theoretical modeling of the coupled heat and moisture transfer with phase change in a clothing fibrous medium relies on extensive studies performed by many researchers on the heat and mass transfer process in porous media. Coupled heat and mass transfer with condensation/evaporation is of a special importance to the building insulation industry and to the research studies on energy conservation (Vafai and Sarkar, 1986; Vafai and Whitaker, 1986). Condensation can lead to an increase in the thermal conductivity of the insulating material, since the thermal conductivity of water is approximately 24 times that of the conductivity of the air. As a result, the insulating material loses its basic role in the reduction of heat transfer and in conserving energy. In addition, condensation usually results in corrosion and deterioration of the quality of the insulating material. Most research on modeling heat and mass transfer with phase change in porous media is applicable to highly porous thin textile materials. The approach to modeling the condensation/ evaporation process in clothing was based on the fundamental studies of Henry (1948) and the subsequent models that were developed by Farnwoth (1986) and by Lotens et al. (1995) for highly porous media.

11.2.1 Farnworth model Theoretical modeling of the combined heat and water vapor transport through clothing with sorption and condensation started with the model of Farnworth (1986). This model is a simplified expression of Henry’s model with restrictive assumptions limiting the model applicability to a multi-layered clothing system where each layer is characterized by a uniform temperature and moisture content. The assumptions made by Farnworth were as follows. (i) There is no convective airflow and/or convective transport of liquid.

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(ii) The mass of absorbed water is proportional to the relative humidity of the microclimate with a restrictive upper limit of absorbed water vapor. This assumption is important to limit the vapor pressure of the absorbed water to its upper limit, which is the saturation water vapor pressure. (iii) Clothing radiation can be neglected. Based on the above assumptions, Farnworth derived the following conservation equations for mass and heat transport, respectively: P – Pi P – Pi+1 ∂M i = i –1 – i R Re ,i ∂t e ,i –1 Ci =

T – Ti T – Ti+1 ∂Ti = i –1 – i + Qci Rd ,i –1 Rd ,i ∂t

[11.1]

[11.2]

where Mi (kg) is the total moisture in the clothing layer which is the summation of the liquid water present in the void space of the fabric layer and the absorbed water vapor bound to the solid fiber of the fabric layer, Ci (J/kg · K) is the heat capacity per unit area of the clothing layer, Ti (°C) and Pi (kPa) are the temperature and water vapor pressure of the clothing layer respectively, Rd,i (m2 · °C/W) and Re,i (m2 ⋅ kPa/W) are the dry and evaporative resistances characteristic of each clothing layer, Qci (W/m2) is the quantity of heat per unit area which is released in the layer because of moisture adsorption and condensation, and i represents the layer index. The model of Farnworth is easy to use but it is too simplistic to be applied to the whole clothing system. The assumption of linear regain increase with relative humidity presents a serious deficiency in the model. Moisture regain at low and high relative humidity is far from being linear (Chapter 12). If the empirical equilibrium relation between regain and relative humidity is used, the model will still remain limited due to the lumped moisture content and temperature value for each fabric layer. When condensation/ evaporation is taking place, the Farnworth model cannot be used for studying the temperature and moisture distribution inside a fibrous system.

11.2.2 Lotens model The Lotens model is similar to the Farnworth model in its applicability to a clothing ensemble system and in its ability to integrate the clothing model with a nude human model (Lotens, 1993). However, the Lotens model presents a simple physical condensation theory with its associated effects on moisture distribution, temperature, and total heat transfer from the clothing ensemble. The Lotens model can predict the thermal performance of permeable and impermeable garments in cold and hot environmental conditions (Lotens et al., 1995).


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Unlike the absorption phenomenon, which is a transient process lasting for a limited time depending on the fabric hygroscopicity property, condensation is a continuous process. According to Lotens (1993), this continuing nature of condensation can actually simplify the modeling of the condensation process and allow the incorporation of condensation in clothed human body modeling. Lotens’ model divides the clothing system into: (i) an inner underclothing layer; (ii) an outer clothing layer; and (iii) an outer air layer, as shown in Fig. 11.2. The outer layer is characterized by a lower permeability compared to the inner, underclothing layer, to allow condensation to occur. Based on the mass and heat balance between the clothing layers and the outer environmental air layer, the mass and heat transfer resistance network is constructed, neglecting the ventilation mass and heat resistance and the radiative heat transfer resistance. Ps – P1 P – Pa +Ψ= 1 Re 1 Re 2 + Re 3

[11.3]

Ts – T1 T – Ta + Ψ h fg = 1 Rd 1 Rd 2 + Rd 3

[11.4]

where Re is the evaporative heat resistance m2 · kPa/W, Rd is the dry heat transfer resistance m2 · K/W, hfg is the heat of condensation and Ψ is the condensation rate kg/m2 · s. When condensation occurs, P1 = Psat(T1), and the three unknowns in the above equations, Ψ, P1 and T1 can be calculated. Under clothing

Outer clothing

Environmental air

Skin

Tskin

T1 Rd 1

T2 Rd 2

Ta Rd 3

Ψhfg

Pskin

P1 Re 1

P2 Re 2

Ψ

11.2 Lotens clothing system model.

P2 Re 3

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The above simple clothing ensemble model is integrated with the human nude model by Lotens after taking into account the area increase because of the clothing and the ventilation of the inner surface of the outer layer (Lotens, 1993). The simple clothing model developed by Lotens explains the effect of the condensation process on the dry and evaporative resistances of clothing. Dry and evaporative heat transfer leaving the skin, (Qd, Qe), are not the same as the heat dissipated to the outside environment during moisture condensation. During the occurrence of condensation, the rate of moisture leaving the skin is not equal to the moisture leaving the human clothing system, and thus there will be an increase in the temperature of the clothing ensemble at the spot of condensation. Consequently, the dry heat that is dissipated from the human skin is not the same as the dry heat reaching the outside environment. As a result of condensation, the apparent dry and evaporative resistances (Rdt, Ret) can be calculated as follows: Rdt =

Ts – Ta Qd + Ψ h fg

[11.5]

Ret =

Ps – Pa Qe – Ψ h fg

[11.6]

where Rdt is the apparent clothing ensemble dry heat transfer resistance m2 · °C/W, and Ret is the apparent clothing ensemble evaporative heat resistance m2 · kPa/W. Because of condensation, the dry resistance becomes smaller and the evaporative resistance becomes larger. In reality, condensation represents a link between the dry and latent heat that leaves the human skin. Condensation balances the decrease in the latent heat transfer by an increase in dry heat transfer. Experimental verification of Lotens condensation theory. The condensation theory has been validated by the experimental findings of Lotens and other co-authors. Lotens’ aim was to experimentally determine the effect of condensation on the latent and dry heat flows through different clothing ensembles and the resulting effect on the apparent dry and evaporative heat resistances. In the experiment of Van de Linde and Lotens (1983), the condensation effect was tested on human subjects wearing impermeable garments while exercising on a treadmill in the presence and absence of sweat from the skin. The absence of sweat was achieved by wrapping the subjects with plastic foil. Experimental findings showed that, in the absence of sweat, the impermeable garments showed higher dry resistance. The lower resistance of the garment in the presence of sweat is attributed to the presence of sweat condensation. The condensation theory has also been checked by


411

the experimental study of Havenith and Lotens (1984). In this study, impermeable garments were compared to semi-permeable garments in terms of their heat transport ability from human subjects exercising on a bicycle ergo meter in an environment of 14 °C temperature and 90% relative humidity. The experiments showed that the impermeable garments transport more dry heat compared to the semi permeable garments and that their outer surface temperature is higher due to sweat condensation. Van De Linde (1987) tested the condensation theory on the ability of impermeable garments to transport the body-generated heat for different exercise rates and ambient temperature. While exercising in cool environmental conditions at 16 °C, the condensation of sweat generated by the increased human subject work rate was reported to increase the outer garment temperature and to reduce its dry resistance. The same phenomenon was also observed at a higher environmental temperature of 26 °C (Van De Linde, 1987). Lotens (1995) performed numerical simulations to compare the accuracy of his model with the experimental results and to determine the important parameters that evoke condensation. He found that the skin vapor concentration, the vapor resistance of the outer layer, and the air temperature are the important parameters that evoke condensation.

11.3

Modeling condensation/evaporation in a fibrous medium

From the simplified lumped models, it is clear that the effect of condensation on the heat and moisture transfer is captured. These simple models are able to describe the heat and mass transfer with condensation in the clothing ensemble and can be easily integrated with the human thermal model. However, they incorporate only the diffusion of heat and the diffusion of water vapor within the clothing system, and they ignore convection of air and liquid wicking. In addition, the lumped modeling approach relies on the physical dry and evaporative resistance properties of the fabric, which may change when condensation occurs. In the following section, a more accurate mathematical modeling of condensation within fibrous medium is presented.

11.3.1 Mathematical modeling of condensation Figure 11.3 is a schematic of a fibrous porous system model consisting of the following: solid fiber, absorbed water vapor to the solid fiber, gaseous mixture of water vapor and air, and liquid water in the void space. To correctly model condensation/evaporation with sorption in a clothing system, the model should include the following features: • The ability to simulate heat and moisture in space and time without lumping for the heat and concentration parameters.

412

Thermal and moisture transport in fibrous materials Liquid flow

Tb

T∞

Sb

S∞

Pgb

Pg ∞

X=0

X=L

11.3 The fibrous medium system model consisting of the solid fiber, the water vapor absorbed by the solid fiber, the gaseous mixture of water vapor and air, and liquid water in the void space.

• A mechanism of the moisture water vapor movement that could take place due to gradients in the partial water vapor and the convective airflow due to pressure gradients across the clothing system. In situations when there is no total pressure gradient, during sedentary human activity, water vapor diffuses in a clothing ensemble by the driving force of the partial water vapor pressure gradient between the human skin and the outside environment. In movement conditions, pressure gradients can be induced across the fabric leading to bulk moisture movement. • Water liquid transport is driven by capillary forces and surface tension. The inclusion of liquid transport is important for modeling coupled heat and the moisture transfer process with condensation because liquid moisture will affect the pore moisture content and the condition of saturation. In addition, the transport of liquid moisture across textiles increases their thermal conductivity, and thus affects the transport of heat across the clothing system. • The transport of energy that can occur by conduction, as well as convection of the phases that are able to move, i.e. liquid water, water vapor and dry air. The sorption/de-sorption of the hygroscopic fibers with their associated heat of sorption should not be neglected because most textile fibers have a certain degree of moisture absorption ability. The fiber absorption characteristic significantly influences the heat and moisture transfer processes. The above-mentioned inclusions can simultaneously be incorporated with


413

the theoretical development of the coupled heat and moisture processes and condensation, after applying the following simplifying assumptions: (i) The porous system is assumed to be in local thermal equilibrium. Local thermodynamic equilibrium exists if the pore dimension of the fibrous medium is very small; (ii) the volume changes of the fibers due to changes in moisture content, and therefore the porosity, is constant; and (iii) the fibrous media is homogenous and isotropic. With these assumptions, the governing equations of heat and moisture transport with condensation/evaporation can be developed using the considerable research work carried out in the literature by Gibson and Charmachi (1997), Zhongxuan et al. (2004), and Xiaoyin and Jintu (2004). The formulation adapted from Zhongxuan et al. (2004) will be presented in this section. The water vapor conservation distribution is governed by the following equation:

ε

∂C f ∂ [(1 – S ) ρv ] ∂J ∂J + (1 – ε ) – W = – vD – vC ∂t ∂t ∂x ∂x

[11.7]

where S is the liquid water volumetric saturation (liquid volume/pore volume), ε is the porosity of the fabric, ρv is the density of water vapor, W is the evaporation or condensation flux of water in the void space (kg/m3 · s), Cf is the moisture concentration in the fiber (kg/m3), JvD is the mass flux of water vapor by diffusion (kg/m2 · s), JvC is the mass flux of water vapor by bulk flow (kg/m2 · s). The first term on the left-hand side of Equation [11.7] is the storage term of the water vapor in the void space, the second term is the absorbed water vapor stored in the solid fiber, and the third term, W, is the evaporation/condensation term. The right-hand side of Equation [11.7] represents the net diffusive and convective flows of water vapor. The moisture absorbed in the solid fiber can be calculated by using the Fickian law of diffusion as follows: ∂C f =1 ∂ r ∂r ∂t

∂C f    rD f ∂r   

[11.8]

where Df is the fiber diffusion coefficient and r is the radial coordinate. The fiber diffusion coefficient primarily depends on the stage of absorption, the rapid stage of moisture uptake, and the slower stage of absorption. The moisture boundary condition at the fiber surface is determined by assuming instantaneous moisture equilibrium with the microclimate air. Thus, the moisture content at the fiber surface can be determined by the relative humidity of the microclimate air and temperature. It can be obtained directly from the moisture sorption isotherm of the fiber.

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The diffusion of water vapor flux in the voids is described by Stefen’s law (Shuye and Guanyu, 1997) and can be represented by the following expression after substituting for the diffusion coefficient of the water vapor in terms of temperature and gaseous pressure: 0.8

J vD = –1.952 x10 –7 ε (1 – S ) T Pa

∂Pv ∂x

[11.9]

where Pa is the partial pressure of dry air and Pv is the partial pressure of water vapor. The convective water vapor flux in the fibrous medium is JvC = ρvu

[11.10]

Since Darcy’s law holds in the pore of the inter fiber, the convective velocity, u, can be written as

u= –

kk rg ∂Pg µ g ∂x

[11.11]

where k is the intrinsic permeability of the fibrous media, krg is the relative permeability of the gas, µg is the dynamic viscosity of the water vapor, and Pg = Pa + Pv is the gaseous pressure. The condensation/evaporation term W of Equation [11.7] is given by Qing-Yong (2000) as

W = ε (1 – S ) S f hw

Mw P ( T ) – Pv ) RT s

[11.12]

where Sf is the specific area of the fabric, hw is the mass transfer coefficient, Mw is the molecular mass of water vapor, R is the universal gas constant, and Ps(T) is the saturation water vapor. The liquid moisture mass conservation equation is given by

ερw

∂J ∂( S ) +W= – l ∂t ∂x

[11.13]

where ρw is the density of liquid moisture. The first term in Equation [11.13] represents the storage of liquid water in the void, and the second term represents the condensation/evaporation flux. The right-hand side of Equation [11.13] represents the net capillary flow of liquid water and can be written (Nasrallah and Perre, 1988) as

J l = – ρw

kk rw ∂ ( P – Pc ) µ w ∂x g

[11.14]

where Krw is the relative permeability of the liquid water, µw is the dynamic viscosity of the water, and Pc is the capillary pressure of the fabric function of saturation and surface tension.


415

The dry air mass conservation equation is:

ε

∂ [(1 – S ρa ] ∂J ∂J = – aD – aC ∂t ∂x ∂x

[11.15]

The first term in Equation [11.15] represents the dry air storage in the void space, and the right-hand side first and second terms represent the diffusive dry air mass flux and the convective dry air mass flux, respectively. The dry air mass flux JaD is equal in magnitude to the water vapor diffusive mass flux given by JaD = –JvD

[11.16]

and the convective air mass flux JaC can be expressed as J aC = – ρa

kk rg ∂Pg µ g ∂x

(11.17)

where krg is the relative permeability of the gas and µg is the dynamic viscosity of the gaseous phase. The energy equation is represented by the following: ∂C f + Qc = ∂  K c ∂T  Cv ∂T – λ (1 – ε ) ∂t ∂t ∂x ∂x

[11.18]

where Cv is the volumetric heat capacity of the fabric (J/m3 ·K), Kc thermal conductivity of the fabric (W/m ·K), λ heat of sorption (J/kg), and Qc is the heat flux of condensation or evaporation (J/m3 ·s). The first term in the energy equation represents the heat storage term in the fabric, the second term represents energy released by sorption, the third term represents the heat released by condensation, and the right-hand side represents the net conducted heat flow. To solve the conservation Equations [11.7] through [11.18] of liquid moisture, water vapor, dry air, and energy, initial and boundary conditions need to be specified. The initial values of temperature, water vapor concentration, degree of saturation, absorbed moisture in the solid fiber, and the gaseous pressure in the fibrous medium should be known. In most practical cases, the initial conditions are uniform throughout the medium. The boundary conditions can be a constant temperature, saturation, and gaseous pressure or can be a convective air flow condition. Uniform initial conditions for a 1-D system can be expressed as T(x, t = 0) = To, ρv(x, t = 0) = ρvo, S(x, t = 0) = So Pg(x, t = 0) = pgo, Cf (x, t = 0) = f (ρvo, To) while boundary conditions can be written as

[11.19]

416


T ( x = 0, t ) = Tb , S ( x = 0, t ) = Sb , Pg ( x = 0, t ) = p gb , J l | x =l = 0 – k ∂T = hc ( T | x =l – T∞ ), J vD | x =l + J vc | x =l = hm ( ρv | x =l – ρv∞ ) ∂x x =l [11.20] Other boundary conditions can be used depending on the physical system under consideration.

11.4

Effect of fabric physical properties on the condensation/evaporation process

11.4.1 Effect of vapor hydraulic permeability The hydraulic conductivity of the fabric defines the ease with which water vapor passes in the voids of the fibrous media. This factor is determined by the permeability of the fabric to air flow when subject to a pressure difference. The type of yarn count, twist, and weave affect the permeability and thus the hydraulic conductivity of the fibrous media. For very small values of vapor permeability, the moisture movement within the fibrous media is only by diffusion. In such a case, it was found by Xiaoyin and Jintu (2004) that moisture distribution for a fibrous media sandwiched between a hot moist boundary and a cold boundary is close to a convex shape, with a relatively small variation in moisture content. Increasing the vapor permeability will lead to an increase in the amount of condensed water since more water will be transported across the fibrous media. However, with larger values of vapor permeability, the moisture content close to the warm boundary decreases while the moisture content close to the cold boundary increases, resulting in the occurrence of moisture condensation closer to the cold boundary. Fabrics characterized by high porosity are more advantageous for thermal comfort and heat loss than impermeable fabrics, because high porosity makes the wet region of the fibrous media occur away from the skin while minimizing the heat loss from the skin, since no condensation occurs in the fibrous media adjacent to the skin.

11.4.2 Effect of liquid water permeability The transport mechanism of liquid water in a fibrous media is governed by its capillarity and by the liquid permeability of the fibrous medium. The capillarity represents the driving force for the liquid movement, whereas the permeability describes the ease with which water moves through the fibrous medium. For a fibrous medium with zero permeability, the condensate liquid moisture will be immobile. For higher liquid permeability values, the condensate moisture will be mobile and the condensates will move from the region of


417

higher moisture content towards the region of lower water content. The findings of Xiaoyin and Jintu (2004) showed that, with the increase of liquid moisture mobility, the moisture distribution of a fibrous media bounded by the extreme boundary conditions of warm moist and cold dry conditions will shift from concave to almost even. The mobility of the liquid moisture will definitely affect both thermal comfort and heat loss from the skin or warm boundary.

11.4.3 Effect of material hygroscopicity As the hygroscopicity of the fabric increases, its moisture content will increase, mainly due to the water absorbed into the solid fiber. In steady-state conditions, this increase in moisture content leads to a decrease in the insulation value of the fibrous material, and thus more heat loss is observed from the fibrous medium (Xiaoyin and Jintu, 2004). However, during transient conditions, hygroscopic wool battings have shown less condensation when compared to non-hygroscopic battings of polypropylene (Jintu et al., 2004). For the same boundary conditions across the battings assembly, Jintu et al. (2004) showed that condensation starts after a short time for the propylene battings whereas condensation starts to appear in the wool battings after 4 hours. Furthermore, in transient conditions, the hygroscopicity of the fibrous medium decreases the heat loss from the human skin because of the heat liberated by the moisture absorption. Therefore, it is suggested that hygroscopic fabrics can be advantageous for cold protective clothing in transient conditions.

11.4.4 Effect of pressure difference across the fibrous medium During exercise, the human limbs move back and forth forcing the renewal of the microclimate air existing between the skin and the clothing layers. The renewal of the microclimate air is driven by the pressure difference between the microclimate environment and the outside atmospheric motion. The pressure difference alternates between a positive value forcing the microclimate air to be discharged out of the clothing system and a negative value allowing atmospheric air to fill the space between the skin and the human clothing ensemble. The atmospheric pressure gradient developed during the limb motion will definitely affect the fibrous water vapor distribution and to a lesser extent the liquid moisture distribution. The liquid water movement is due to gradients in capillarity and to atmospheric pressures. Fengzhi et al. (2004) found that water vapor concentration in the void space is largely affected by the pressure difference and that the concentration of water vapor was high at the location of the lower pressure. Fengzhi et al. (2004) also found that the liquid water distribution was not significantly affected by

418


atmospheric pressure, as was the case with water vapor when the atmospheric pressure was increased from 1.0135 × 105 Pa to 2.0135 × 105 Pa.

11.5

Modeling heating and moisture transfer in PCM fabrics

The effect of the phase change that takes place in PCM fabrics is transitory. This transitory property is similar to sorption/de-sorption and different from condensation/evaporation phenomena. It lasts for a finite time, determined by the quantity of encapsulated paraffin and the thermal load impending on the PCM fabric. When a PCM fabric is exposed to heating from the sun or a hot environment, it will absorb this transient heat as it changes phase from solid to liquid, and it will prevent the temperature of the fabric from rising by keeping it constant at the melting point temperature of the PCM. Once the PCM has completely melted, its transient effect will cease and the temperature of the fabric will rise. In a similar manner, when a PCM fabric is subject to a cold environment, where the temperature is below the crystallization temperature, it will interrupt the cooling effect of the fabric structure by changing from liquid to solid, and the temperature of the fabric will stay constant at the crystallization temperature. Once all the PCM has crystallized, the fabric temperature will drop, and the PCM will have no effect on the fabric’s thermal performance. Thus, the thermal performance of a PCM depends on the phase temperature, the amount of PCM that is encapsulated, and the amount of energy it absorbs or releases during a phase change. Research studies on quantifying the effect of PCMs in clothing on heat flow from the body during sensible temperature transients were conducted by Shim (1999) and Shim et al. (2001). Shim et al. (2001) measured the effect of one and two layers of PCM clothing materials on reducing the heat loss or gain from a thermal manikin as it moved from a warm chamber to a cold chamber and back again. Their results indicated that the heating and cooling effects lasted approximately 15 min and that the heat release by the PCM in a cold environment decreased the heat loss by 6.5W for the one layer PCM clothing and 13.5W for the two-layer PCM clothing, compared to nonPCM suits. Shim and McCullough (2000) experimentally studied the effects of PCM-ski ensembles on the comfort of human subjects during exercise, and they found no appreciable effect of PCM material on comfort compared to non-PCM-ski clothing. The study of Shim and McCullough (2000) on the effect of PCM-ski ensembles on exercise was done after conditioning the human subjects inside cold environmental chambers. The transport processes of heat and moisture from the human body are enhanced by the ventilating motion of air through the fabric initiated by the relative motion of the human with respect to the environment. Periodic renewal of the air adjacent to the skin by air coming from the environment has a


419

significant effect on the heat loss from the body and on comfort sensations. When sudden changes in the environmental air take place, it is desirable to delay the adjacent air temperature swings to reduce sudden heat loss or gain from the body. During exercise in cold environments, there is a periodic ventilation of the skin adjacent layer. Cold environmental air is pumped inside the clothing ensemble, while warm air heated by the human skin is forced to move out. During the air passage into and out of the clothing system, the moving air is intercepted by the PCM fabrics. It is questionable whether the PCM fabric is actually able to regenerate itself during exercise at steady-state environmental conditions, and whether the PCM fabric can act as a heat exchanger between the incoming cold air and the leaving warm air. The study of Ghali et al. (2004) addressed this question by performing experiments to investigate the effect of PCMs on clothing during periodic ventilation. The study of Ghali et al. (2004) also included a model and a numerical investigation of the transient effect of the phase change material during the sinusoidal motion pattern of the fabric induced by body movement upon exercise. In their work, PCMs were incorporated in a numerical threenode model (Chapter 8), for the purpose of studying their transient effect on body heat loss during exercise when subjected to sudden environmental conditions from warm indoor air to cold outdoor air. In deriving the energy balance for the fabric, the following assumptions were made: (i) the PCM is homogeneous and isotropic; (ii) the thermophysical properties of the PCM are constant in each phase; (iii) the phase change occurs at a single temperature; and (iv) the difference in density between solid and liquid phases is negligible. The study findings of Ghali et al. (2004) indicated that the heating effect lasts approximately 12.5 minutes, depending on the PCM percentage and cold outdoor conditions. The heat released by PCMs decreased the clothedbody heat loss by an average of 40–55 W/m2 depending on the ventilation frequency and the crystallization temperature of the PCM. A typical PCM percentage of the total mass of the fabric is about 20%. It is not recommended by the textile industry to increase the percentage of PCM because it will increase the cost of the fabric as well as its weight. The 20% is actually representative of what is used by industrial manufacturers. The sensitivity of the PCM fabric performance to the amount of the PCM present in the fabric was also considered in the work of Ghali et al. (2004). The PCM percentage, α, was found to affect the length of time of the period during which the phase change process takes place but had negligible effect on the sensible heat loss from the skin when compared to non-PCM fabric. The reported durations of the phase change effect corresponding to α = 0, 20%, 30% and 40% PCM are 0, 8.23 min, 12.26 min and 16.6 min, respectively, due to a change from an indoor environment at 26 °C and relative humidity of 50% to an outdoor environment at 2 °C and relative humidity of 80%. The experimental results of Ghali at al. (2004) revealed that, under steady-state

420


environmental conditions, the oscillating PCM fabric has no effect on the dry fabric resistance, even though the measured sensible heat loss increases with the decreasing air temperature of the environmental chamber. When a sudden change in ambient temperature occurs, the PCM fabric delays the transient response and decreases body heat loss. PCM has no effect on thermal performance of the fabric during exercise in steady-state environmental conditions.

11.6

Conclusions

Phase change is a phenomenon that occurs in a fibrous medium as a result of sorption/de-sorption of fiber moisture, condensation/evaporation of moisture in the void place, and melting/solidification of PCM when incorporated into the fabric structure. Both melting/solidification of PCM and sorption/desorption of fiber moisture processes are transitory in nature. Both are important in the study of transient thermal sensations of human subjects in changing environmental conditions. Their effect on the thermal performance of the fabric primarily depends on the hygroscopicity of the fabric, the amount of encapsulated PCM, and other environmental factors. Modeling the heat and moisture transfer for the sorption/de-sorption phenomena should include the diffusion process of moisture into the fiber, the diffusion of moisture in the void space, and the convective flow of moisture. Other complications are important in modeling sorption/de-sorption and include the change of the fabric permeability due to moisture sorption (Gibson, 1996) and the need to consider different temperatures for the different phases that constitute the fabric structure. The condensation/evaporation phase change process is different from the other phase change phenomena by its steady-state nature. Evaporation and/ or condensation take place depending on the temperature and moisture distribution. The condensation process continues provided that there is a supply of moisture and that the void water vapor pressure exceeds saturation. The condensation phenomenon is relevant to the study of thermal comfort since it leads to the loss of the main role of clothing in keeping the human body dry. It also affects the thermal performance of fabrics by decreasing the dry resistance of the fabric and increasing the fabric’s evaporative resistance. Modeling condensation/evaporation is more complicated than modeling sorption/de-sorption. In addition to including diffusive and convective moisture vapor, modeling condensation should also include the liquid flow of moisture. Current research models describing condensation account for all complicated factors such as hygroscopic sorption, convective and diffusion of moisture, capillary flow of liquid moisture, and coupled diffusion of heat and mass flow. However, efforts to incorporate such a detailed condensation clothing fibrous model with the human thermal model have relied on simple human


421

thermal physiology models (Gibson, 1996) while the detailed human thermal physiology models that are integrated with condensation clothing models have relied on simple clothing condensation models (Lotens, 1993).

11.7

Nomenclature

Cf Ci Cv Df hfg hw JaC JaD Jl JvC JvD k Kc krg Krw Mi Pa Pc Pi Ps Psat Pv Qc Qci Qd Qe Rd,i Rdt Re,i Ret S Sf Ti W

moisture concentration in the fiber (kg/m3) heat capacity per unit area of the clothing layer (J/kg · K) volumetric heat capacity of the fabric (J/m3 ·K) fiber diffusion coefficient (m3/s) heat of vaporization (J/kg) mass transfer coefficient (m/s) convective dry air mass flux (kg/m2 · s) diffusive dry air mass flux (kg/m2 · s) net capillary liquid moisture flow (kg/m2 · s) mass flux of water vapor by bulk flow (kg/m2 · s) mass flux of water vapor by diffusion (kg/m2 · s) intrinsic permeability (m2) thermal conductivity of the fabric (W/m · K) relative permeability of the gas relative permeability of the liquid water total moisture in the clothing layer i (kg) partial pressure of dry air (kPa) capillary pressure (kg/m·s2) water vapor pressure of clothing layer i (kPa) skin vapor pressure (kPa) saturation pressure (kPa) partial pressure of water vapor (kPa) heat flux of condensation or evaporation (J/m3 · s) condensation/absorption heat release (W/m2) dry heat transfer (W/m2) evaporative heat transfer (W/m2) fabric dry resistance of clothing layer i (m2 ·°C/W) apparent fabric dry resistance (m2 · °C/W) fabric evaporative resistance of clothing layer i (m2 kPa/W) apparent fabric evaporative resistance (m2 ·kPa/W) liquid water volumetric saturation (liquid volume/pore volume) specific area (1/m) temperature of the clothing layer (°C) evaporation or condensation flux of water in the void space (kg/m3 · s)

422


Greek symbols

α Ψ ε µg µw λ ρv ρw

PCM percentage of total fabric mass (%) condensation rate (kg/m2 · s) porosity of the fabric. dynamic viscosity (kg/m · s) dynamic viscosity of water (kg/m · s) heat of sorption (J/kg) water vapor density (kg/m3) water liquid density (kg/m3)

11.8

References

Bryant Y G and Colvin D P (1992), ‘Fibers with enhanced, reversible thermal energy storage properties’, Techtextil-Symposium, 1–8. Farnworth B (1986), ‘A numerical model of the combined diffusion of heat and water vapor through clothing’, Tex. Res. J., 56, 653–665. Fengzhi L, Yi L, Yingxi L and Zhongxuan L (2004), ‘Numerical simulation of coupled heat and mass transfer in hygroscopic porous materials considering the influence of atmospheric pressure’, Numerical Heat Transfer, Part B, 45, 249–262. Ghali K, Ghaddar N and Harathani J (2004), ‘Experimental and numerical investigation of the effect of phase change materials on clothing during periodic ventilation’, Textile Res. J., 74(3), 205–214. Gibson P (1996), ‘Multiphase heat and mass transfer through hygroscopic porous media with applications to clothing materials’, Natick/TR–97/005. Gibson P and Charmachi M (1997), ‘Modeling convection/diffusion processes in porous textiles with inclusion of humidity dependent air permeability’, Int. Comm. Heat Mass Transfer, 24 (5), 709–724. Havenith G and Lotens W (1984), What, actually is the advantage of semipermeable over impermeable rain wear?, Report, TNO Institute for Perception, Soesterberg, IZF, 1984–6. Henry P S H (1948), ‘Diffusion of moisture and heat through textiles, Discuss. Faraday Soc., 3, 243–257. Jintu F, Xiaoyin C, Xinhuo W and Weiwei S (2004), ‘An improved model of heat and moisture transfer with phase change and mobile condensates in fibrous insulation and comparison with experimental results’, Int. J. of Heat and Mass Transfer, 47, 2343– 2352. Jones F E (1992), Evaporation of Water with Emphasis on Application and Measurement, Lewis Publishers, MI, USA, 25–43. Keighley J H (1985), ‘Breathable fabrics and comfort in clothing’, J. Coated Fabrics, 15 (10), 89–104. Lotens W (1993), Heat Transfer from Humans Wearing Clothing, Doctoral Thesis, The Royal Institute of Technology, Stockholm, Sweden. Lotens W, Van De Linde F J G and Havenith G (1995), ‘Effects of condensation in clothing on heat transfer’, J. Ergonomics, 38(6), 1114–1131 Nasrallah S B and Perre P (1988), ‘Detailed study of a model of heat and mass transfer during convective drying of porous media’, Int. J. Heat Mass Transfer, 31(5), 957– 967.


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Pause B H (1995), ‘Membranes for building’, Textile Asia, 26 (11), 81–83. Qing-Yong Z (2000), ‘A numerical simulation of drying process in wool fabrics’, Int. Conf. on Applied Fluid Dynamics’, Beijing, China, 621–626. Ruckman J E (1997), ‘An analysis of simultaneous heat and water vapor transfer through waterproof breathable fabrics’, J. Coated Fabrics, 26 (4), 293–307. Shim H (1999), The Use of Phase Change Materials in Clothing, Doctoral research dissertation, Kansas State University, Manhattan, Kansas. Shim H and McCullough E A (2000), ‘The effectiveness of phase change materials in outdoor clothing’ Proceedings of the International Conference on Safety and Protective Fabrics’, Industrial Fabrics Association International, Roseville, MN, April, 26–28, 2000. Shim H, McCullough E A and Jones B W (2001), ‘Using phase change materials in clothing’, Textile Res J, 71(6), 495–502. Shuye L and Guanyu Z (1997), ‘Numerical simulation of heat and mass transfer in wet unsaturated porous media’, (in Chinese), J. Tsinghua Univ., 37, 86–90. Vafai K and Sarkar S (1986), ‘Condensation effects in a fibrous insulation slab’, J. Heat Transfer, 108, 667–675. Vafai K and Whitaker S (1986), ‘Heat and mass transfer accompanied by phase change in porous insulations’, J. Heat Transfer, 108, 132–140. Van De, Linde F J G and Lotens W (1983), ‘Sweat cooling in impermeable clothing’, Proceedings of an International Conference on Medical Biophysics, Aspects of Protective clothing, Lyon, 260–267. Van De Linde F J G (1987), Work in Impermeable Clothing: Criteria for Maximal Strain, Report, TNO Institute for Perception, Soesterberg, IZF, 1987–24. Xiaoyin C and Jintu F (2004), ‘Simulation of heat and moisture transfer with phase change and mobile condensates in fibrous insulation’, Int. J. of Thermal Sciences, 43, 665–676. Zhongxuan L, Fengzhi L, Yingxi L and Yi L (2004), ‘Effect of the environmental atmosphere on heat, water and gas transfer within hygroscopic fabrics’, J. of Computational and Applied Mathematics, 163, 199–210.

12 Heat–moisture interactions and phase change in fibrous material B. J O N E S, Kansas State University, USA K. G H A L I, Beirut Arab University, Lebanon N. G H A D D A R, American University of Beirut, Lebanon

This chapter focuses on phase-change phenomena associated with the adsorption of moisture into fibers, the condensation of moisture onto fibers, and the release or absorption of heat associated with this change of phase. First, a set of mathematical relationships is developed that describes these interactions. These relationships may be somewhat simplified compared to the relationships developed in other chapters so that it is easier to focus on the heat and moisture interactions. However, every effort is made to point out any limitations associated with this simplification. The equations are also developed so that they are based on variables, properties, and other parameters that are readily measured or readily obtained. These equations are then presented in a finite difference form that has been proven effective in modeling heat and moisture interactions in clothing systems.

12.1

Introduction

Each fiber in a fibrous media continually exchanges heat and moisture with the air in the microclimate immediately surrounding it, as shown in Fig. 12.1. In addition, there will be radiation heat exchanges with other fibers and other surfaces. These radiation exchanges are not addressed in the present chapter but may be important in certain situations, especially in fibrous media with a low fiber density or with high temperature gradients. The heat and moisture exchanges between the fiber and the surrounding environment are the focus of this chapter. When there is a temperature difference between a fiber and the air in the surrounding microclimate, a net heat flow results; this exchange is generally well understood, at least in principle. Similarly, if there is difference between the water vapor pressure at the fiber surface and the water vapor pressure in the air in the surrounding microclimate, there will be a net exchange of moisture. For a given fibrous material, the vapor pressure at the surface depends upon the amount of moisture adsorbed onto that surface and the 424

Heat–moisture interactions and phase change

425

Fibrous media

Radiation exchange with other fibers or surfaces outside the media

Moisture exchange with microclimate

Heat exchange with microclimate

12.1 Heat and moisture between a fiber and its microclimate.

temperature of the fiber. The amount of moisture on the fiber is not limited by adsorption, however. When the fiber becomes saturated with respect to the adsorption state, i.e. it has adsorbed as much moisture as it can, additional moisture may condense as a liquid onto the surface of the fiber. Depending on the nature of the fibrous media, large amounts of water condensate may be held on the surface of the fiber. The liquid on the surface may be relatively immobile and trapped in place, or may be transported within the fibrous media by capillary pressure. This capillary pressure transport is not addressed in the present chapter but is addressed in other chapters. Generally, the moisture adsorbed onto a fiber is considered to be immobile and can only move by exchange with the air in the surrounding microclimate. While not well understood or documented, it is possible that the adsorbed moisture becomes mobile when the fiber is nearly saturated with adsorbed moisture. There could then be some transport along the fiber in this situation. There is sometimes confusion with respect to the use of the term ‘saturated’ with regard to moisture in a fibrous media. When a fiber has all of the moisture adsorbed that it can hold in the adsorbed state, it is said to be saturated. Similarly, when a fibrous media is fully wetted with liquid, it is said to be saturated. In the present chapter, both forms may be used with the context making it clear what which form is intended.

426

12.2


Moisture regain and equilibrium relationships

It is customary to refer to the adsorbed moisture content of fibrous material as ‘moisture regain’. The moisture regain is defined as the mass of moisture adsorbed by a fiber divided by the dry mass of the fiber. The dry mass of the fiber is the mass of fiber when it is in equilibrium with completely dry air, even though some fibers may contain a residual amount of moisture in this state. The mass of moisture adsorbed does not include this residual moisture in the dry state (Morton and Hearle, 1993). Mathematically, the regain (R) is defined as

R=

Mass at given condition – Mass at dry condition Mass dry condition

It is customary to express regain as a percentage. The equilibrium moisture regain of most fibrous material depends primarily on the relative humidity of the air in the ambient microclimate surrounding a fiber. That is, the equilibrium regain will be nearly the same at different temperatures if the ambient relative humidity is the same. Ambient temperature and atmospheric pressure can have a small impact independent of relative humidity. However, relative humidity is clearly the dominant variable for most terrestrial applications at common indoor and outdoor environmental temperatures. At more extreme conditions, such as might occur in manufacturing processes, the relationship between relative humidity and regain may not hold. Figure 12.2 presents standardized relationships for moisture regain for a number of common fibers (Morton and Hearle, 1993). In general, natural fibers tend to have higher regains than manufactured fibers, with some of the latter fibers having nearly negligible regain. The regains shown in Fig. 12.2 are for raw fibers. A variety of surface finishes and other treatments are often applied to raw fibers to impart desired properties. While generally not applied for the purpose of changing moisture regain characteristics, some treatments can impact the moisture regain curve and care must be used in applying the equilibrium relationships in Fig. 12.2, especially for fibers that have very low regains in the raw state. The curves in Fig. 12.2 stop at 100% relative humidity, as the regain is defined in terms of adsorbed moisture. Once the ambient microclimate relative humidity reaches 100%, liquid water may condense on the fiber. In terms of actual moisture present on a real fiber, the curves do not terminate at the values shown in Fig. 12.2. Rather, the curves actually become vertical and can extend to very large values, depending on the nature of the fibrous media. For individual fibers, it is difficult to define an upper limit. For fibers in a fibrous media, the upper limit is controlled by a number of factors including the porosity of the media and its structure.


427

0.4

0.35

Regain (fraction)

0.3

0.25

0.2 0.15

l Woo

0.1

Ray

on

Cotton Aceta

0.05

te Polyester

0 0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Relative humidity (fraction)

0.8

0.9

1

12.2 Equilibrium regain for typical fibers (based on data from Morton and Hearle, 1993).

12.3

Sorption and condensation

The heat of adsorption describes the amount of energy that is released when water vapor in the air is adsorbed onto the fiber surface. Similarly, this same amount of energy must be added when moisture is desorbed from the fiber. The heat of adsorption is not a constant, even for a given fiber, but depends on the environmental conditions under which the adsorption or desorption occurs. The primary factor affecting the heat of adsorption is the microclimate relative humidity and, for most applications at normal environmental temperatures and pressures, heat of adsorption can be treated as a function of humidity alone. Figure 12.3 shows the heat of adsorption for several fibers. It is seen that, as the microclimate relative humidity becomes high, the heat of adsorption becomes equal to the heat of vaporization. The heat of sorption is often divided into two components: the heat of vaporization and the ‘heat of wetting’. The heat of wetting is the added heat that is released above and beyond the heat release that would occur if the vapor simply condensed. Or viewed differently, it is the heat that is released if liquid water is added to a fiber. In Fig. 12.3, it is the distance between the heat of adsorption curve and the heat of vaporization line. It is often more convenient to present data in terms of the heat of wetting as it allows the large heat of vaporization, which is the same for all fibers, to be subtracted.

428


4000

l Woo 3500

Nylo

n

Heat of sorption (J/g)

3000 Cotton 2500 2000 Heat of vaporization 1500 1000 500 0 0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Relative humidity (fraction)

0.8

0.9

1

12.3 Heat of adsorption for typical fibers (based on data from Morton and Hearle, 1993).

As can be seen from Fig. 12.3, the heat of sorption for a given relative humidity does not vary greatly from fiber to fiber, especially when one considers the large heat of vaporization component that is common. Given the inaccuracies associated with many fibrous media heat and mass transport calculations, it is often adequate to simply use a common heat of sorption curve for all fibers.

12.4

Mass and heat transport processes

For steady-state conditions where any moisture on the fiber is immobile, there will be no net moisture exchange between the fiber and the air in the surrounding void space in the media. In this steady-state condition, there is no need to address heat–moisture interactions associated with moisture phase change. However, there are many situations where there is a net exchange of moisture between the fiber and the void space and it is necessary to develop mathematical descriptions of these processes. While relationships describing the heat and moisture transport between the fiber and the immediate void space can be developed, these processes are generally not the limiting factors in the transport phenomena. The high surface area associated with the fiber– microclimate interface results in minimal restriction to moisture and heat transport, and local equilibrium between the fiber and the surrounding microclimate is achieved over the time-scale of most applications for fibrous


429

media; or it is at least an acceptable approximation. The factors limiting the heat and moisture interchanges are the restrictions of heat and vapor transport in the bulk fibrous media. A transient, one-dimensional moisture balance gives the following relationship at any location in the media: ∂R ∂m ρ= – ∂t ∂x

[12.1]

where R is the regain (kg H2O per kg of dry fabric), ρ is the bulk density of the dry porous media (kg/m3), m is the vapor moisture flux through the media (kg/s m2), t is time (s), and x is distance along the dimension of interest (m). This formulation ignores the water vapor in the air in the void space in the media. Normally, the amount of moisture stored in this phase is small compared with the regain. Additionally, it does not play an important role in the heat and moisture interactions and thus is ignored in the equations developed in this chapter. The vapor moisture flux is proportional to the vapor partial pressure flux for most fibrous media and the relationship can be written as m = –ϕ

∂P ∂x

[12.2]

where P is the vapor pressure (kPa), and ϕ is the vapor permeability of the media (kg/s m kPa) While it is customary to use concentration gradients rather than vapor pressure gradients as the driving force for vapor diffusion, the vapor pressure gradients are equally valid and are more convenient for this application (Fu, 1995). The vapor permeability, ϕ , is an empirical parameter that describes the overall ability of vapor phase moisture to be transported through the media and is equal to the inverse of the vapor resistance per unit thickness (ASTM, 2005a). Equations [12.1] and [12.2] combine to give a moisture balance in terms of partial pressure: 2 ∂R ρ = ϕ ∂ P ∂t ∂x 2

[12.3]

The right-hand term expands directly to three dimensions, but the onedimensional form is retained here for simplicity. A one-dimension, transient energy balance can be written in similar fashion ∂T c ρ = – ∂q – Q ∂m S ∂t ∂x ∂x

[12.4]

430


where T is the temperature (°C), c is the heat capacitance of the bulk fibrous media (kJ/kg °C), q is the heat flux through the media (kW/m2), and QS is the heat of adsorption (kJ/kg). Several terms in the transient energy balance that are normally negligible have been omitted in Equation [12.4] to yield a relatively simple expression. Equation [12.4] should be acceptably accurate as long as there are no extreme temperature gradients in the porous media. The heat flux through the fibrous media is proportional to the temperature gradient and the relationship can be written as q = –k

∂T ∂x

[12.5]

where k is the thermal conductivity of the fibrous media (W/mK). It should be noted that the thermal conductivity, above, is for the air–fiber combination that makes up the fibrous media and can be determined experimentally (ASTM, 2005b). Equations [12.5] and [12.2] combined with Equation [12.4] allow the energy balance to be expressed in terms of the temperature gradient and the vapor pressure gradient: 2 2 ∂T c ρ = k ∂ T2 + QS ϕ ∂ P ∂t ∂x ∂x 2

[12.6]

Equations [12.3] and [12.6] then describe the transient energy and mass balances at a location within a fibrous media. These equations also describe the transport of heat and vapor through the media. These equations are coupled in that there is a relationship between P, T, and R. Using the approximation that fiber is in moisture and thermal equilibrium with the immediately surrounding void space, this relationship is defined by the curve for the particular fiber in question in Fig. 12.2. Note that relative humidity is a unique function of P and T. Similarly, there is also a relationship between QS and P and T, with that relationship being defined by the appropriate heat of adsorption curve such as is shown in Fig. 12.3. In order to solve Equations [12.3] and [12.6], appropriate boundary conditions, empirical relationships for equilibrium regain, and empirical relationships for heat of adsorption are required. In addition, the values of the bulk density, heat capacitance, thermal conductivity, and vapor permeability must be known. The thermal conductivity and the vapor permeability generally must be determined experimentally for the fibrous media of interest. One way to measure these parameters is to use a sweating hotplate (ASTM, 2005a; ISO, 1995). The bulk density can be measured experimentally (ASTM, 2005b). Thermal capacitance of the media can estimated with reasonable accuracy if the fiber content is known: c = cF + RcL

[12.7]


431

where cF is the thermal capacitance of the fiber (kJ/kg K), and cL is the thermal capacitance of liquid water (kJ/kg K). The air in the void space in the media is again ignored in Equation [12.7] and the equation is valid as long as the bulk density of the media is much greater than the density of air, which is true for nearly all applications. It should also be noted that the liquid term is based on the approximation that the thermal capacitance of a fiber increases with adsorbed moisture as if the adsorbed moisture is in the liquid state. This approximation is sufficiently accurate for all but the most precise calculations.

12.5

Modeling of coupled heat and moisture transport

Modeling the coupled heat flow requires appropriate boundary conditions to be established and Equations [12.3] and [12.6] to be solved. Fortunately, the equations are generally well bounded and well behaved, and the simplest of numerical methods may be used to solve the equations with acceptable accuracy. For modeling purposes, these equations can be written in finite difference form: ∆ Ri ρ = ϕ

P(φi –1 , Ti –1 ) + P(φi+1, Ti+1 ) – 2 P(φi , Ti ) ∆t ∆x 2

∆Ti c i ρ = Qs (φi ) ∆ Ri ρ + k

Ti –1 + Ti+1 – 2Ti ∆t ∆x 2

[12.8] [12.9]

where ∆t is the integration time step (s), ∆x is the distance step in the xdirection (m), φi is the local relative humidity (fraction), i refers to a specific discrete location in the x direction, P(φ, T) is the equilibrium vapor pressure for the fibrous media at the local relative humidity and temperature (kPa), and Qs(φ) is the heat of sorption for the fibrous media at the local relative humidity, (kJ/kg). The local relative humidity, φi is determined from the adsorption equilibrium curve for the media, such as in Fig. 12.2, corresponding to the local regain. This relative humidity value is then used to determine the equilibrium pressure from P(φ, T) = φ (R) Ps(T)

[12.10]

where φ (R) is the relative humidity corresponding to the local regain R from the equilibrium relationship (fraction) and Ps(T) is the saturation pressure of water at local temperature T (kPa). This same value of relative humidity is also used to determine the heat of sorption from the heat of sorption curve for the media, such as in Fig. 12.3. Given initial conditions of temperature and regain, T and R, throughout

432


the media, appropriate boundary conditions, the equilibrium relationships such as in Fig. 12.2, and the heat of sorption information such as in Fig. 12.3, Equations [12.8]–[12.10] can be used to step through time and model the media response fully representing the interactions between heat and moisture. Time steps as small as 0.1 second or less may be required for clothing applications when boundary conditions change rapidly. However, the simplicity of the time-based solution puts little demand on computational capability, and transient solutions for complex systems can be readily solved. For thin fabric layers, it is often sufficient to use only a single increment in the xdirection. For thick fabric layers or fiber fillings, only a small number of increments in the x-direction is generally quite sufficient to obtain solutions of acceptable accuracy; generally, less than ten increments is adequate. Equations [12.8] and [12.9] can be readily expanded to three dimensions. The single dimension form is presented here for simplicity. For many clothing applications, the radial direction from the body is usually the dominant direction for heat and moisture fluxes and local, one-dimensional representations are usually acceptable as long as the local variations in clothing and boundary conditions are addressed. Equation [12.3] and [12.6] and, consequently, Equations [12.8] and [12.9] apply only when the moisture adsorbed or condensed onto the fiber is immobile. This limitation prevents these equations from being considered general representations of mass transport in fibrous media. Once the media contains sufficient moisture for this condensed moisture to become mobile and be transported in significant amounts by capillary pressure gradients, the air in the microclimate surrounding the fiber is saturated, φ = 1, and the heat and moisture interaction phenomenon becomes one of condensation or evaporation. Establishing the necessary boundary conditions is often the most difficult aspect of modeling heat and moisture interactions with fibrous media. Without proper boundary conditions, the equations described previously are of limited value. Each application is unique and it is not feasible to address all boundary condition situations that might be encountered with fibrous media. The following discussion addresses boundary conditions in a layered, cylindrical system which is typical of clothing applications and is depicted in Fig. 12.4. The nomenclature for Fig. 12.4 follows: qc is the conduction or convection heat transfer to/from a surface (W/m2), qr is the radiation between two surfaces or between a surface and the surrounding environment (W/m2), m is the vapor flux to/from a surface (kg/s m2), r is the characteristic radius of the respective layer (m), the i subscript refers to the inner surface of a layer, the o subscript refers to the outer surface of a layer, the s subscript refers to the body surface, and the e subscript refers to the surrounding environment.


Water vapor flux

Heat flux

qr,o2e qr,o2e

433

mo2e

qr,i2 qc,i2 qr,o1

r2 mi2 mo1

qc,o1 qr,i1 qc,i1 qr,s qc,s

r1 mi1 ms r0

Body

12.4 Depiction of boundary conditions for a two-layer radial system.

Each layer of porous media (e.g. fabric) is shown divided into a number of sub-layers that could correspond to ∆x in the finite difference solution. The radius of each layer is characterized by a single value. This simplification is acceptable as long as the layer thickness is less than about one-fourth of the radius. The intervening air layers may present substantial resistance to heat and moisture transport and, consequently, are important in the overall modeling of the system. They do not normally contribute appreciably to the storage of heat or moisture and, thus, simplified modeling is usually acceptable even for transient applications. Figure 12.4 shows all of the boundary conditions for heat and mass transport in a two-layer system. These boundaries can be represented in several ways for finite difference solutions. Figure 12.5 shows one form that is compatible with Equations [12.8] and [12.9]. In the simplest representation, the air can be treated as a single lumped resistance to heat or water vapor transport. For this situation, the boundary conditions shown in Fig. 12.5 take the following form: q o1

r1 r = qi 2 2 = r0 r0

T1,n – T2,1 ∆ x1 r0 ∆ x 2 r0 r0 r0 + + + r1 + r2 r1 + r2 2k1 r1 2k 2 r2 hc ,1 – 2 hr ,1–2 2 2 [12.11]

m 01

r1 r = mi 2 2 = r0 r0

P1,n – P2,1 ∆ x1 r0 ∆ x 2 r0 r0 + + r1 + r2 2ϕ 1 r1 2ϕ 2 r2 hm ,1–2 2

[12.12]

434


∆x2 qi2 qc,i2

qr,i2

mi2

qr,o1

mo1

xa

qc,01

qo1 ∆x1

12.5 Boundary condition detail between layers 1 and 2.

where qo1 is the total heat flux from the outer surface of layer 1 (W/m2), qi2 is the total heat flux to the inner surface of layer 2 (W/m2), hc,1-2 is the overall heat conduction/convection heat transfer coefficient for the air layer (K/W m2), hr,1-2 is the linearized radiation heat transfer coefficient for the air layer (K/W m2) (see ASHRAE, 2005), mo1 is the vapor mass flux from the outer surface of layer 1 (kg/s m2), mi2 is the vapor mass flux from the inner surface of layer 2 (kg/s m2), and hm,1-2 is the mass transfer coefficient for the air layer (kPa m2 s/kg). Note that the r/r0 terms are included to account for the increasing area at increasing distances in the radial direction. Equations [12.8]–[12.10] plus Equations [12.11] and [12.12] for each air layer along with time-dependent values for temperature and vapor pressure for the body surface and the environment allow calculation of the time-dependent heat and vapor flows in the porous media system, fully accounting for the heat and moisture phase change interactions.

12.6

Consequences of interactions between heat and moisture

Equations [12.8] and [12.9] show a clear coupling between moisture and heat in porous media. In particular, Equation [12.9] shows that any increase in regain results in an increase in temperature and vice versa. The heat of sorption is large and, consequently, only small changes in regain can result


435

in large temperature changes. Since heat flows are driven by the temperature gradients, the adsorption and desorbtion of moisture by the media has a large impact on the heat fluxes through the media as well. It has been know for many years that moisture sorption and desorption can impact body heat loss and affect perceptions of the thermal environment (Rodwell et. al. 1965). This effect has been modeled for clothing systems using the above equations and has been measured experimentally as well (deDear et. al., 1989; Jones and Ogawa, 1992). The effect is so large that a person dressed in clothing made of highly adsorptive fibers such as wool or cotton can experience a short-term change in heat loss from the body of the order of 50 W/m2 when going from a dry environment (e.g. 25% rh) to a humid environment (e.g. 75% rh), even when the temperatures of both environments are identical. This effect is relatively short-lived and may only last for 5–10 minutes but is sufficient to elicit a strong change in thermal sensation and plays a large role in the perceived effect of humidity on comfort in many situations. A lesser, but still important, effect can persist for 30 minutes to an hour for some moderately heavy indoor clothing made of highly adsorptive fibers. This interaction is particularly important for the drying of porous media. The transport of adsorbed moisture from a porous media is driven by the vapor pressure gradient. A negative vapor pressure gradient from the media to the surroundings will result in transport of water vapor from the media to the surroundings. The source of this water vapor is moisture adsorbed on the fibers. As the moisture is released and the regain decreases, there is a cooling effect on the media, as quantified by Equations [12.8] and [12.9]. Only a very small decrease in regain results in a large cooling effect. This small decrease in regain has minimal impact on the local equilibrium relative humidity (refer to Fig. 12.2). However, the large change in temperature has a big impact on the saturation pressure. The net result is a big decrease in local vapor pressure (refer to Equation [12.10]). The end result is that the cooling effect nearly eliminates the partial pressure gradient that is driving the moisture removal and, in the absence of a heat source, drying proceeds at a very low rate. The drying of a porous media is almost always limited by heat transfer and this effect is why thick media can take hours of even days to dry. For fibers such as polypropylene or polyethylene that adsorb very little moisture, the interaction of heat and moisture is very minimal unless the conditions are such that condensation occurs. In the case where condensed moisture is present, but still relatively immobile, the equations presented in this chapter still apply and the strong interaction between heat and moisture will be present.

436

12.7


References

ASHRAE (2005), Handbook of Fundamentals, Chapter 8, American Society of Heating, Refrigerating and Air-conditioning Engineers, Atlanta, US. ASTM (2005a), ‘ASTM 1868-02, Standard Test Method for Thermal and Evaporative Resistance of Clothing Materials Using a Sweating Hot Plate,’ 2005 Annual Book of ASTM Standards, Vol. 11.03, American Society for Testing and Materials, West Conshohocken, PA, US. ASTM (2005b), ‘ASTM D 1518–85(2003), Standard Test Method for Thermal Transmittance of Textile Materials,’ 2005 Annual Book of ASTM Standards, Vol. 7.01, American Society for Testing and Materials, West Conshohocken, PA, US. deDear R.J., Knudsen H.N., and Fanger P.O. (1989) ‘Impact of Air Humidity on Thermal Comfort during Step Changes,’ ASHRAE Transactions, Vol. 95, Part 2. Fu G. (1995), ‘A Transient, 3-D Mathematical Thermal Model for the Clothed Human,’ Ph.D. Dissertation, Department of Mechanical Engineering, Kansas State University, Manhattan, US. ISO (1995), ‘ISO 11092, Textiles – Physiological Effects – Measurement of Thermal and Water Vapour Resistance Under steady-State Conditions (Sweating Guarded Hotplate Test), International Organization for Standardization, Geneva, Switzerland. Jones B.W., and Ogawa Y. (1992), ‘Transient Interaction Between the Human Body and the Thermal Environment’, ASHRAE Transactions, Vol. 98, Part. 1. Morton W.E., and Hearle J.W.S. (1993), Physical Properties of Textile Fibres, 3rd Edition, The Textile Institute, Manchester, UK. Rodwell E.C., Rebourn E.T., Greenland J., and Kenchington K.W.L. (1965) ‘An Investigation of the physiological Value of Sorption Heat in Clothing Assemblies,’ Journal of the Textile Institute, Vol. 56, No. 11.

13 Heat and moisture transfer in fibrous clothing insulation Y. B. L I and J . F A N, The Hong Kong Polytechnic University, Hong Kong

13.1

Introduction

Heat and moisture transfer with phase change in porous media is a very important topic in a wide range of scientific and engineering fields, such as civil engineering, energy storage and conservation, as well as functional clothing design, etc. Such processes have therefore been extensively studied by experimental investigation and numerical modeling.1–10 For clothing systems used in subzero climates, heat and moisture transfer is complicated by various factors. Heat transfer takes place through conduction in all of the phases, radiation through the highly porous fibrous insulation, and convection of moist air. Mass transport occurs not only through diffusion and convection, but also through moist absorption or desorption between the fibres and the surrounding air as well as the movement of condensed liquid water as a result of external forces, such as capillary pressure and gravity. The moisture absorption or desorption and phase change within the fibrous insulation absorbs or releases heat, which further complicates the heat transfer process. The difficulty in studying these processes is further aggravated by the fact that the transport properties of the material involved vary considerably with the moisture or liquid water content. In this chapter, past literature and our recent work will be reviewed and discussed, which include experimental investigations and development of theoretical models, as well as numerical simulation of the effects of material properties and environmental parameters.

13.2

Experimental investigations

13.2.1 Experimental methods Thomas et al.11 studied the diffusion of heat and mass through wetted fibrous insulation of medium density. The experiment consisted of uniformly wetting six layers of insulation and stacking them together to form a continuous slab. 439

440


The slab was then heat-sealed in a plastic film. The test sample was inserted into a protected hot plate apparatus and subjected to one-dimensional temperature gradients. The temperature profile inside the slab was monitored with thermocouples and the liquid content was measured at regular intervals through disassembling the slab and measuring the weight of each of the six layers. Farnworth12 reported the use of a sweating hot plate, by which water is fed into the hot side of the fibrous insulation using a syringe pump. The temperature and heat loss was measured during one sweating on and off cycle. Shapiro and Motakef13 conducted an experiment in which a fiberglass test sample with a known liquid content distribution was placed inside a hot– cold box, and the cold side of the specimen was covered by a vapor barrier. The hot–cold box consists of two temperature- and humidity-controlled chambers, connected through the specimen. The temperature profile in the sample at different times and the final liquid content distribution were measured. Wijeysundera et al.14 conducted two series of experiments in which a heat flow meter apparatus based on the ASTM guidelines was built. In the first series, water was sprayed on the hot face of the slab and, in the second series, the hot face was directly exposed to a moist airflow. Transient temperature changes were monitored and the total amount of moisture absorption and/or condensation after a period of time was measured. A similar experiment to Wijeysundera’s second series was conducted by Tao et al.15 except that the cold side was subjected to the temperature below the triple point of water. Murata16 built an apparatus in which mixture of dry air and distilled water vapor was preheated to desired temperature (89 °C), then the mixture was blown through the fibrous insulation and stopped by an impermeable glass plate at a low temperature (24–62 °C). The temperature and heat flux were monitored during the testing. In order to resemble many practical situations where the fibrous insulation is sandwiched in between two layers of moisture retarders, such as is the case in clothing and building insulation, Fan17,18 and his coworkers investigated coupled heat and moisture transfer through fibrous insulations sandwiched between two covering lining fabrics, using a sweating guarded hot plate under a low temperature condition. The details of this experiment will be elaborated in the following sections.

13.2.2 Instrumentation The sweating guarded hot plate specified in the ISO 11092:1993(E) was improved for use under frozen conditions. The device is shown schematically in Fig. 13.1. The device had a shallow water container 1 with a porous plate 3 at the top. The container was covered by a man-made skin 2 made of a waterproof, but moisture permeable (breathable) fabric. The edge of the breathable fabric was sealed with the container to avoid water leakage. Water

Data input

9 8 7 6 5

4

3

2

1

24

441

23

Water level 22

Power output

Computer

Heat and moisture transfer

21 20 19 10 11 12 13

18 14

1. 2. 3. 4. 5. 6. 7. 8.

Shallow water container Menmade skin Porous plate Water Measuring sensor Layers of specimen Temperature sensor Heating element

9. 10. 11. 12. 13. 14. 15. 16.

15

Insulation foam Insulation pad Temperature sensor Temperature sensor Heating element Water supply pipe Insulation layer Electronic balance

16

17 17. 18. 19. 20. 21. 22. 23. 24.

Water pump Water tank Insulation foam Heating element Warm water Water level adjustor Cover Temperature sensor

13.1 Schematic drawing of the sweating hot-plate.

was supplied to the container from a water tank 16 through an insulated pipe 14. The water in the water tank was pre-heated to 35 °C. The water level in the water tank was maintained by a pump 17 which circulated the water between the two halves of tank. Between the two halves was a separator 22 whose height could be adjusted to ensure that the water was in full contact with the breathable skin at the top of the container. The water temperature in the container 1 was controlled at 35 °C, simulating the human skin temperature. The amount of water supplied to the water container was measured by the electronic balance 16. To prevent heat loss from directions other than the upper right direction, the water container was surrounded with a guard having a heating element 13. The temperature of the guard was controlled so that its temperature difference from that of the bottom of the container was less than 0.2 °C. The whole device was further covered by a thick layer of insulation foam. All temperatures were measured using RTD sensors (conforming to BS 1904 and DIN43760, 100 Ω at 100 °C) and the heating elements were made of thermal resistant wires. Temperature control was achieved by regulating the heat supply according to a Proportional–Integral–Derivative (PID) control algorithm19. To ensure the accuracy in the measurement of heat supply and stability of the system, the power supply was in DC and was stabilized using a voltage stabilizer.

442


13.2.3 Experimental procedure The samples in the experiment consisted of several thin layers of fibrous battings sandwiched with inner and outer layers of covering fabric, simulating the construction of a ‘down’ jacket. Two types of covering fabrics and fibrous battings were used in the testing, and their properties are listed in Tables 13.1 and 13.2: The resistance to air penetration was tested using a KES -F8-AP1 Air Permeability Tester.20 The moisture absorption and condensation under cold condition was measured according to the following instructions: (i) Condition the covering fabric and the fibrous battings in an air-conditioned room, with temperature at 25.0 ± 0.5 °C and humidity at 65 ± 5%, for at least 24 hours. (ii) Start the temperature control and measurement system of the sweating hot plate in the same conditioned room until the temperature and power supply is stabilized. (iii) Weigh and record the weights of each layer of fibrous battings. (iv) Sandwich multiple layers of fibrous batting with top and bottom layers of covering fabric, and place the ensemble on top of the man-made skin of the instrument. Immediately place the sweating guarded hot plate in a cold chamber with the temperature controlled at –20 ± 1°C. (v) After a pre-set time (e.g. 8, 16 or 24 hours), take out each layer of the fibrous battings and weigh them immediately using an electronic balance. (vi) Record the temperatures, consumed water and power supply with time, continuously and automatically. Table 13.1 Properties of covering fabric Composition

Nylon

Three-layer laminated fabric

Construction Weight (kg/m2) Thickness (m) Thermal resistance (Km2/W) Water vapour resistance (s/m) Resistance to air penetration (kPa.s/m)

Woven 0.108 2.73E-04 3.15E-02 64.99 0.524

Woven + membrane + warp knit 0.22 5.15E-04 3.16E-02 143.79 Impermeable

Table 13.2 Properties of fibrous batting Composition

Viscose

Polyester

Weight (kg/m2) Thickness (m) Fibre density (kg/m3) Porosity Resistance to air penetration (kPa.s/m)

0.145 1.94E–03 1.53E+03 9.51E–01 0.062

0.051 4.92E–03 1.39E+03 9.93E–01 0.0061


443

(vii) Calculate the percentage of moisture or water accumulation due to absorption or condensation on each layer of fibrous battings by Wc i =

Wai – Woi × 100% Woi

13.2.4 Experimental findings and discussion Temperature distribution. The temperature distributions within the fibrous battings are plotted against the thickness from the inner layer of the covering fabric in Figs. 13.2–13.5 for two types of fibrous battings and covering fabrics. In general, the temperature of the inner battings next to the warm ‘skin’ increases quickly in the first few minutes and may even exceed the ‘skin’ temperature of 35 °C before it drops to a stable value. However, the temperature at the outer battings close to the cold environment reduces gradually. Most of the changes of temperature distribution occurred within about 0.5 hour, unrelated to the type of battings and covering fabrics. Comparing Figs. 13.2 and 13.3, which are for the same non-hygroscopic polyester battings but with differing covering fabrics, there is no significant difference in the stabilized temperature distribution, but the one with the more permeable nylon covering fabric reached stabilization faster. As for the hygroscopic viscose batting (see Figs. 13.4 and 13.5), a significant difference 35 0.1 hr

Initial

30 8 hrs

Temperature (°C)

25 20 4 hrs

15

0.5 hr 10 5 0

0

0.5

1

1.5

2

2.5

–5 Thickness (cm) –10

13.2 Temperature distribution for 6 plies polyester batting sandwiched by two layers of nylon fabric.

3

444

Thermal and moisture transport in fibrous materials 40

0.1 hr

35 Initial

30

Temperature (°C)

25 20

0.5 hr

15 4 hrs

10 5

8 hrs

0 0

0.5

1

1.5

2

2.5

3

–5 –10 Thickness (cm)

13.3 Temperature distribution for 6 plies polyester batting sandwiched by two layers of laminated fabric. 0.1 hr 35

Initial

30 25

Temperature (°C)

4 hrs 0.5 hr

20 15 8 hrs

10 5 0 0 –5

0.5

1

1.5

2

2.5

3

Thickness (cm)

13.4 Temperature distribution for 15 plies viscose batting sandwiched by two layers of nylon fabric.

in temperature was found in the middle of the battings. This was caused by the differences in the moisture absorption within the fibrous battings. When covered with the highly permeable nylon fabric, more moisture was transmitted into the viscose battings within the same period and a greater rate of moisture absorption took place in the initial period, which released a greater amount


445

0.1 hr 40

0.5 hr Initial

Temperature (°C)

30

20

4 hrs

10 8 hrs 0 0 –10

0.5

1

1.5

2

2.5

3

Thickness (cm)

13.5 Temperature distribution for 15 plies viscose batting sandwiched by two layers of laminated fabric.

of heat during moisture absorption and consequently caused a higher temperature. After about 4 hours, the viscose battings covered with either the nylon fabric or the laminated fabric was almost saturated, resulting in a smaller temperature difference. Heat loss. The changes of power supply or heat loss with time for the two types of battings and covering fabrics are shown in Fig. 13.6. The initial fluctuation of the curves is understandably due to the PID adjustment used for controlling the temperature of the water within the shallow container. It is clear from Fig. 13.6 that the heat loss after stabilization through the polyester battings covered with the more permeable nylon fabric was about 5% greater than that through the battings covered with the less permeable laminated fabric, which may be attributed to the greater loss in latent heat of moisture transmission. It can also be seen from Fig. 13.6 that the heat losses through the viscose battings are similar, irrespective of whether they are covered with the nylon or laminated fabric. Moreover, it can be seen that clothing assemblies with hygroscopic viscose batting will lose more heat than those with nonhygroscopic polyester batting after stabilization. When condensation takes place, hygroscopic batting may not be as warm as non-hygroscopic batting at the same thickness. Water content distribution. Figures 13.7 and 13.8 show the distribution of water content within the fibrous battings after 8 and 24 hours for the two types of battings and covering fabrics, respectively. Here water content within the fibrous battings is a combination of moisture absorption and condensation. As can be seen, the water content in the batting next to the ‘skin’ was nearly zero for the non-hygroscopic polyester and about 18% for the hygroscopic viscose. It remained almost unchanged from after 8 hours to after 24 hours

446


8

Power supply (W)

7

6

5

Polyester batting + nylon fabric Polyester batting + laminated fabric Viscose batting + nylon fabric Viscose batting + laminated fabric

4

3 0

5

10

15

20

25

Time (hr)

13.6 Power supply for different configurations of battings and cover fabrics.

for the polyester batting and only increased slightly for the viscose batting. It is therefore reasonable to believe that there was no condensation in the batting next to the ‘skin’, and the accumulation of water was only because of moisture absorption. Polyester batting absorbs little moisture and hence its water content remained zero. However, viscose batting absorbs much moisture at the beginning because the saturated water content is about 30% in a 95% RH environment. The water content increases from the inner region to the outer region of the batting due to the increased amount of condensation. The water content also accumulates with time. At the outer region, the water content after 24 hours is about 4 times that after 8 hours for the polyester batting, and about 2 times that after 8 hours for viscose batting. The possible reason is that the polyester batting is more porous and permeable, thus allowing more moist air to be transmitted or diffuse from its inner region to the outer region, where condensation takes place. Another reason is that viscose is hydrophilic and polyester is hydrophobic. The condensed water on the hydrophilic fibre surface tends to wick to regions where the water content is lower. It can also been seen from the graphs that, although the water content at the outer region was greater, the greatest water content may not occur at the outermost layer of the battings. This may be due to the complex interaction of heat and moisture transfer in the battings. Condensation within the fibrous

Heat and moisture transfer 100

447

6 plies polyester batting sandwiched by two nylon fabrics

90 6 plies polyester batting sandwiched by two laminated fabrics

80

Water content (%)

70 15 plies viscose batting sandwiched by two nylon fabrics

60

50 15 plies viscose batting sandwiched by two laminated fabrics

40

30

20

10

0 0

0.5

1

1.5 Thickness (cm)

2

2.5

3

13.7 Water content distribution within the fibrous batting after 8 hours at –20 °C environment temperature.

battings is a result of the moisture content within the air space exceeding the saturated moisture content, which is a function of temperature. As the temperature reduces from the inner region to the outer regions, the saturated moisture content initially reduces sharply and then more gradually from the inner region to the outer region (because the effect of temperature on the saturated moisture content is less pronounced at lower temperatures). On the other hand, from the inner region to the outer region, the moisture content reduces at an increasing rate due to increased condensation in the outer region. The greatest difference between the moisture content of the fibers and the saturated moisture content of the air in the void space may not occur at the outermost boundary of the battings.

448


400

6 plies polyester batting sandwiched by two nylon fabrics

350 6 plies polyester batting sandwiched by two laminated fabrics

Water content (%)

300 250

15 plies viscose batting sandwiched by two nylon fabrics

200 150

15 plies viscose batting sandwiched by two laminated fabrics

100 50 0 0

0.5

1

1.5 Thickness (cm)

2

2.5

3

13.8 Water content distribution within the fibrous batting after 24 hours at –20 °C environment temperature.

13.3

Theoretical models

Apart from experimental study, a number of theoretical models have been proposed to simulate the coupled heat and moisture transfer in fibrous insulations. Henry21 first established a model of coupled heat and moisture transfer with sorption/desorption in cotton bales. Phase change was, however, not considered in this early model. Since then, little progress was made in terms of rigorous theoretical modeling until Ogniewicz and Tien22 proposed a model, assuming that heat is transported by conduction and convection and that the condensate is in a pendular state. The analysis was limited to a quasisteady state, viz. the temperature and vapor concentration remained almost unchanged with time before the condensates became mobile. Motakef and El-Masri23 first considered the quasi-steady state corresponding to mobile condensate, under which the condensate diffuses towards the wet zone’s boundaries, as liquid, and re-evaporates at these boundaries leaving the timeinvariant temperature, vapor concentration and liquid content profiles. This theoretical model was later extended by Shapiro and Motakef13 to analyze the unsteady heat and moisture transport processes and compare the analytical results with experimental ones under some very limited circumstances. This analysis was only valid when the time scale for the motion of the dry–wet boundary in porous media is much larger than the thermal diffusion time scale, which may, however, not be the case with frosting and small moisture accumulation.15


449

The first dynamic model of coupled heat and moisture transfer with sorption and condensation was presented by Farnworth.24 This model was rather simplified and only appropriate for multi-layered clothing, as it was assumed that the temperature and moisture content in each clothing layer were uniform. Vafai and Sarkar25 first modeled the transient heat and moisture transfer with condensation rigorously. For the first time, the interface between the dry and wet zones was found directly from the solution of the transient governing equations. In this work, the effects of boundary conditions, and of the Peclet number and the Lewis Number, on the condensation process were numerically analyzed. Later, Vafai and Tien26 extended the analysis to two-dimensional heat and mass transport, accounting for phase change in a porous matrix. Tao et al.15 first analyzed the frosting effect in an insulation slab by applying Vafai and Sarker’s model to the case with a temperature below the triple point of water. Tao et al.27 also for the first time, considered the hygroscopic effects of insulation materials in the modeling. Murata16 first considered the falling of condensate under gravity and built the phenomena into his steadystate model. Fan and his co-workers28,29 first combined the dynamic moisture absorption process and radiative heat transfer as well as the movement of liquid condensates in their transient models. These models were further improved by introducing moisture bulk flow, which is caused by the vapor pressure gradients and super-saturation state.30–32 The following description is based on Fan’s theoretical model and is combined with Finite Volume Method (FVM) for the numerical solution.

13.3.1 Model formulation In modeling the heat and moisture transfer within a clothing system, we assume that (i) The porous fibrous batting is isotropic in fiber arrangement and material properties. (ii) Volume changes of the fibers due to changing moisture and water content can be neglected. (iii) Local thermal equilibrium exists among all phases and, as a consequence, only sublimation or ablimation is considered in the frozen region. (iv) The moisture content at the fiber surface is in sorptive equilibrium with that of the surrounding air. The mass conservation equation for the water vapor in the inter-fiber void can be given by

ε

∂Ca ∂Ca D ε ∂ 2 Ca = – εu + a – Γ ( x, t ) τ ∂x 2 ∂t ∂x

[13.1]

450


where u is the bulk movement speed of moist air, which is calculated by Darcy’s law: u= –

K x ∂p µ ∂x

[13.2]

where p is the pressure of water vapor in the inter-fiber void, calculated by p = psat · Rhf. Even when there is no condensation on the surface of a fiber in the porous batting (i.e. the relative humidity is less than 100%), fibers absorb or desorb moisture; the absorption or desorption rate is of the form:

Γs ( x , t ) = ρ (1 – ε )

∂C f ( x , t ) ∂t

[13.3]

where Cf (x, t) is the moisture content within the fiber, which can be integrated by33 C f ( x , t ) = {2/( ρR 2f )}

∫

Rf

C ′f rdr

[13.4]

0

where Rf is the radius of the fiber. C ′f is the volumetric moisture concentration in the fiber, which can be determined by the Fickian diffusion law11

∂C ′f  ∂C ′f  = 1 ∂  dr r ∂r  ∂t ∂r 

[13.5]

Water condensation and evaporation are modeled using the Hertz–Knudsen equation.34 The condensation or evaporation rate per unit surface area of fiber covered with condensate (liquid water or ice) is Γsce ( x , t ) = – E M /2 πR ( Psat / Ts – Pv / Tv )

[13.6]

From Equation [13.6], we can get Γsce ( x , t ) = – E M /2 π R (1 – Rhf ) Psat / T

[13.7]

since the surface area of the fiber covered by condensate in the control 2 (1 – ε˙ )(1 – ε ) . Therefore, the condensation or evaporation volume is Rf rate per unit volume is Γce ( x , t ) = –

2 E (1 – ε˙ )(1 – ε ) Rf

M /2 π R (1 – Rhf ) Psat / T [13.8]

Therefore, the total water accumulation rate Γ(x, t) is


Γ = Γs + Γce

451

[13.9]

The free water, i.e. the water on the fiber surface, may diffuse when it is in liquid form and its content exceeds a critical value. According to the mass conservation, we have 2 ˜ ˜ ρ (1 – ε ) ∂W = ρ (1 – ε ) d l ∂ W + Γce ( x , t ) ∂t ∂x 2

[13.10]

where W˜ = W(x, t) – Wf (x, t) is the free water content. Wf (x, t) = Cf (x, t)/ρ

∫

t

is the water absorbed within the fiber; W ( x , t ) = 1 Γ ( x , t ) dt is the ρ 0 ce total water content including that absorbed by the fibers and on the fiber surface. dl is defined phenomenologically, and depends on water content, temperature and properties of the fiber batting. dl = 0 when the condensate is immobile, which is the case when the water content is less than a critical value Wc, or when the free water is frozen. Based on the conservation of heat energy and applying the two-flux model of radiative heat transfer, at position x and time t, we obtain the heat transfer equation

Cv ( x , t ) ∂T = – εu Cva ( x , t ) ∂T + ∂  k ( x , t ) ∂T  ∂t ∂x ∂x ∂x +

where

∂FL ∂FR – + λ ( x, t ) Γ ( x, t ) ∂x ∂x

[13.11]

∂FL = β (x)FL – β (x)σ T 4(x, t) ∂x

[13.12]

∂FR = –β (x)FR + β (x)σ T 4(x, t) ∂x

[13.13]

where the effective thermal conductivity k(x, t) is a volumetric average calculated by k(x, t) = εka + (1 – ε)(kf + ρWkw), the effective volumetric heat capacity of the fibrous batting is calculated by Cv = εCva (1 – ε)(Cvf + ρWCvw), and the porosity of fiber plus condensates (liquid water, or ice) is calculated by ε = ε′ – (ρ/ρw)W(1 – ε′). Since the conductive heat transfer and moisture transport at the interfaces between the inner covering fabric and the batting, as well as between the batting and the outer covering fabric, should be continuous, we have k (0, t ) ∂T ∂x

k ( L , t ) ∂T ∂x

x =0

= 1 ( T | x =0 – T0 ) Rt 0

[13.14]

T1 – T | x = L Rt 1 + (1/ ht )

[13.15]

= x=L

452


Da ε ∂Ca τ ∂x Da ε ∂Ca τ ∂x

=

Ca| x = 0 – Ca 0 Rd 0

[13.16]

=

Ca1 – Ca| x = L Rd 1 + (1/ hc )

[13.17]

x =0

x=L

Considering the radiative heat transfer at the interface between the inner thin fabric and the fibrous batting and that between the outer thin fabric and the fibrous batting, we have initial conditions for Equations [13.12] and [13.13] as follows: (1 – ζ1)FL(0, t) + ζ1σ T 4(0, t) = FR(0, t)

[13.18]

(1 – ζ2)FR(L, t) + ζ2σ T (L, t) = FL(L, t)

[13.19]

4

13.3.2 Numerical solution with finite volume method (FVM) Due to the natural formulation of the FVM, the integral conservation of quantities such as mass, momentum and energy is exactly satisfied over any group of control volumes and, definitely, over the whole calculation domain.35 Therefore, in the FVM, the mass conservation exists for any number of grid points; even the coarse-grid solution exhibits exact integral balances. Moreover, it is easier to handle the mass transfer over the boundary of control volumes with many different properties such as variable porosities and media materials. For each control element, as shown in Fig. 13.9, the vector quantities, such as moisture flow speed, heat resistance, vapor resistance and Darcy flow resistance, are placed on the boundary; the scalar quantities, such as energy, temperature, moisture pressure and vapor concentration, are on the center of the element. This treatment leads to a clear physical interpretation of the moisture and heat transport. From Fig. 13.9, it is easy to derive the discretized formulation for the moisture mass conservation equation. In order to make the calculation more efficiently, an upstream method for the convection item is also introduced

ui ki

Ti, Pi Cai

u i +1 k i +1

13.9 Location of element properties.


453

into the numerical formation. This means that the physical properties transferred along the convection flow are dominated by the donating element’s physical properties. Therefore, the discretized convection item in Equation [13.1] can be written as follows:

εu

∂Ca i = 1 {Min {ε i , ε i+1 ) × ( Ca in+1 × Max ( ui+1 , 0) ∆xi ∂x i

– Ca in+1+1 × Max (– ui+1 , 0)) – Min {ε i , ε i –1 ) × ( Ca in–1+1 × Max ( ui , 0) – Ca in+1 × Max (– ui , 0))}

[13.20]

where symbol Min means the minimum value of the two parameters and Max means the maximum value. Min() ensures that the lower porosity of the neighboring elements is chosen to calculate the moisture flow between the two neighboring elements. Max() identifies whether the moisture flow is moving out or into the element. Such treatment ensures the mass transport over the entire calculation domain is conserved, viz. no mass is lost during calculation. For calculation of u by Darcy’s law, if we use K to replace Kx /µ, then Darcy’s law can be expressed as u = K ∆P = ∆P ∆x ∆x K

[13.21]

Then ∆x can be regarded as the Darcy flow resistance RD. This resistance K on the element boundary between element i and i + 1 can be calculated: ∆x i+1  ∆x RD = 1  i + K i+1  2  Ki

[13.22]

where ∆xi is the dimension for ith element. We can take advantage of the FVM and introduce a diffusion resistance Rd, which is located on the element boundary, and we have 1 ∆x τ 1 ∆x τ i i –1 2 Rd i = + 2 Da i ε i Da i –1ε i –1

[13.23]

Then the diffusion item in Equation [13.1] can be described as: n+1 n+1 n+1 n+1 Da ε ∂ 2 Ca 1  Ca i+1 – Ca i – Ca i – Ca i –1  =   τ ∂x 2 Rd i+1 Rd i ∆x i  

Using the implicit scheme over the time step, we have:

[13.24]

454


Ca in+1 – Ca in ∂Ca i = ∆t ∂t

[13.25]

Submit Equations [13.20], [13.24] and [13.25] into Equation [13.1], and we get the following discretization formation for the mass conservation equation:

A˜ in Ca in–1+1 + B˜ in Ca in+1 + C˜ in Ca in+1+1 = D˜ in

(i = 1, 2, …, N – 1) [13.26]

where A˜ in = –

Min( ε i –1 , ε i ) 1 ⋅ Max ( ui , 0) – ∆x i ⋅ Rdi ∆x i

Min( ε i –1 , ε i ) ε 1 1 B˜ in = i + ⋅ Max (– ui , 0) + + ∆t ∆x i ⋅ Rd i ∆x i ⋅ Rd i+1 ∆x i +

C˜ in = –

Min( ε i , ε i+1 ) ⋅ Max (– ui+1 , 0) ∆x i

Min( ε i , ε i+1 ) 1 ⋅ Max (– ui+1 , 0) – ∆x i ⋅ Rd i+1 ∆x i

ε D˜ in = i Ca in – Γin ∆t Next, the discretized formation of Equation [13.11] will be discussed. It is believed that heat radiation inside the batting plays an important role in the total heat transfer because of the high porosity. From Equations [13.12] and [13.13], we get  FL ( x , t ) = – βσ e βx  

∫

x

0

 FR ( x , t ) = – βσ e – βx  

∫

 e – βx T 4 ( x , t ) dx + c 2  

[13.27]

 e βx T 4 ( x , t ) dx + c1  

[13.28]

x

0

Submit Equations [13.27] and [13.28] into Equations [13.12] and [13.13], and we have

 ∂FL = – β 2 σ e βx  ∂x 

∫

 ∂FR = – β 2 σ e – βx  ∂x 

 e – βx T 4 ( x , t ) dx + c 2  – βσ T 4 ( x , t ) [13.29] 

x

0

∫

x

0

 e βx T 4 ( x , t ) dx + c1  + βσ T 4 ( x , t ) [13.30] 


455

By using Equations [13.27] and [13.28] and the initial conditions [13.18] and [13.19], we obtain c1 =

ζ1 4 T (0, t ) – (1 – ζ 1 ) c 2 β

c2 =

 1 (1– ζ 2 ) βe – βL – βL βL  (1– ξ 2 ) β (1– ζ 1 ) e – βe 

+ βe βL

∫

L

0

[13.31]

∫

L

e βk T 4 ( k , t ) dk

0

 e – βk T 4 ( k , t ) dk + (1– ξ 2 ) ζ 1 e – βL T 4 (0, t ) + ζ 2 T 4 ( L , t ) 

[13.32] Submit Equations [13.29] and [13.30] into Equation [13.11], and we obtain Cv ( x , t ) ∂T = – εuCva ( x , t ) ∂T + ∂  k e ( x , t ) ∂T  + Θ ∂t ∂x ∂x ∂x

where

 Θ = β 2 σe βx  

∫

x

0

 – β 2 σe – βx  

[13.33]

 e – βx T 4 ( x , t ) dx + c 2  

∫

x

0

 e βx T 4 ( x , t ) dx + c1  

+ 2βσT4(x, t) + λ(x, t)Γ(x, t) A positive integer N is selected, and inscribed into the strip such that {(x, t): x ∈ [0, L], t ≥ 0}. The values of the temperature in xi at time n∆t are denoted by Ti n . Since Θ appears in nonlinear form, we replace it by the corresponding value at the previous time step Θ in . Consequently, at each time step we only need to solve a linear tri-diagonal system. The same manipulation can be used in discretizing the energy equation. The moisture concentration Ca is replaced with temperature T and diffusion resistance is replaced with thermal resistance Rt: 1 ∆x 1 ∆x i i –1 2 Rti = + 2 [13.34] k ( x, t )i k ( x , t ) i –1 Similarly, the discretized formation for Equation [13.11] is derived: n+1 n+1 Ain Ti –1 + Bin Ti n+1 + Cin Ti+1 = Din

(i = 1, 2, …, N – 1)

where Ain = –

Min( ε i , ε i –1 ) 1 Cvai –1 Max ( ui , 0) – ∆x i ⋅ Rti ∆x i

[13.35]

456


Bin =

Cvin C + vai [ Min ( ε i , ε i –1 ) ⋅ Max ( ui+1 , 0) ∆t ∆x i + Min ( ε i , ε i –1 ) ⋅ Max (– ui , 0)] +

Cin = – Cin =

13.4

1 1 + ∆x i ⋅ Rti+1 ∆x i ⋅ Rti

Min( ε i , ε i+1 ) 1 Cvai+1 Max (– ui+1 , 0) – ∆x i ⋅ Rti+1 ∆x i

Cvin n T + Θ in ∆t i

Numerical simulation

Numerical simulation has been applied to investigate the effects of material properties and environmental conditions on heat and moisture transfer through fibrous clothing insulations.30,31 In this section, the simulation results of Fan’s model solved with a finite volume scheme are presented. The initial condition is 25°C and 65% RH for the clothing assembly. The numerical values used as boundary conditions are given as follows: T0 = 35 °C, RH0 = 96% T1 = –25 °C, RH1 = 90% The numerical values of the other physical parameters used in the calculation are shown in Tables 13.2 and 13.3. The numerical results of temperature and water content distribution in fibrous battings are plotted in Figs. 13.10– 13.13, with comparison to experimental data. As can be seen, there is generally good agreement between the numerical results and the experimental ones. The theoretical model can provide a good mapping of the mechanisms involved in the coupled heat transfer through clothing assemblies with moisture absorption and phase change. Table 13.3 Common properties used in calculation for two kinds of battings ρice

λ Dry region

Wet region

Freezing region

920.0 β

2522.0 ht

2260.0 Da

2593.0

2.5

12.7

Wc 0.5

ζ1 0.9

hc

ka

kf

kw

0.1149 R

0.025 Cva

0.1 Cvf

0.57 Cvw

8.31

1.169

1300

4200

RH0 99%

RH1 90%

T0 308

T1 253

1.8

 Ti  2.16 × 10–5 ×    273.15  ζ2 σ 0.9 5.672 × 0–8


457

35 30 25

Temperature (°C)

20 15 10 5 0 0 –5 –10

0.5

1

1.5

2

2.5

3

Thickness (cm)

Experiment (0.1 hr) Experiment (0.5 hr) Experiment (4 hrs)

Calculation (0.1 hr) Calculation (0.5 hr) Calculation (4 hrs)

13.10 Comparison of temperature distribution between simulation and experiment (6 plies polyester batting sandwiched by two layers of nylon fabric).

In this section, the effects of various material properties on condensation within the fibrous battings are investigated through numerical simulation. This is important for the optimization of clothing design for thermal comfort. The following simulations are conducted based on the configuration of the viscose batting and two layers of nylon covering fabrics. In Tables 13.1– 13.3, one parameter is considered as a changing variable and the rest are kept unchanged.

13.4.1 Effect of radiative sorption constant of the fibers β The effect of radiative sorption constant of the fibers β on the water content distribution in the batting is shown in Fig. 13.14. The water content reduces with increase of β, but the reduction at the outermost region is less than that in the inner and middle regions. From the view of thermal comfort of clothing wearers, sweating occurs when the human body cannot release its heat through dry heat loss alone. During sweating, less accumulation of water in clothing is desirable, and hence a greater value of β is preferred. β is related to the fractional fiber volume, fiber emissivity and fiber radius:36

Thermal and moisture transport in fibrous materials 400 350

Water content (%)

300 250 200 150 100 50 0 0

0.5

1

1.5 2 Thickness (cm)

Experiment (24 hrs) Experiment (8 hrs)

2.5

3

Calculation (24 hrs) Calculation (8 hrs)

13.11 Comparison of water content distribution between simulation and experiment (6 plies polyester batting sandwiched by two layers of laminated fabric). 40 35 30 25

Temperature (°C)

458

20 15 10 5 0 0 –5

–10

0.5

1

1.5 2 Thickness (cm)

Experiment (0.1 hr) Experiment (0.5 hr) Experiment (4 hrs)

2.5

3

Calculation (0.1 hr) Calculation (0.5 hr) Calculation (4 hrs)

13.12 Comparison of temperature distribution between simulation and experiment (15 plies viscose batting sandwiched by two layers of nylon fabric).


459

250

Water content (%)

200

150

100

50

0 0

0.5

1

1.5 Thickness (cm)

Experiment (24 hrs) Experiment (8 hrs)

2

2.5

3

Calculation (24 hrs) Calculation (8 hrs)

13.13 Comparison of water content distribution between simulation and experiment (15 plies viscose batting sandwiched by two layers of laminated fabric).

250.0

β = 250 β = 200

200.0

Water content (%)

β = 100 150.0

100.0

β = 300

50.0

0.0 0.0

β = 400

0.5

1.0

1.5 2.0 Thickness (cm)

2.5

3.0

13.14 Water content distribution within the batting thickness after 24 hours while having different radiative sorption constants of the fibers.

460


β=

(1 – ε ) ζ f Rf

where Rf is the fiber radius, ε porosity of the batting and ζf the emissivity of fiber. Therefore, higher fiber content, finer fiber, and greater emissivity of fiber are preferred for less condensation within the fibrous batting.

13.4.2 Effect of diffusion coefficient of moisture in the fiber df As can be seen from Fig. 13.15, water content increases with increasing df. However, the increase is limited by the saturation of moisture regain within the fibres. Therefore, when the diffusion coefficient is greater than 1.512 × 10–16 m2/s, there is little further increase in water content, even if the coefficient increases 100 times (see Fig. 13.15). These results indicate that hygroscopic fibers with hydrophilic surfaces having a high value of df (e.g. viscose) may be detrimental to thermal comfort in cold conditions because, when exercising and sweating, there is more and quicker accumulation of water within clothing. The accumulated water content may be a source of ‘after-chill’ discomfort when the wearer stops exercising.

250.0

df = 6.0e-13

Water content (%)

200.0

150.0

df = 1.512e-14

df = 4.536e-16

100.0

df = 1.512e-16

df = 0.0

50.0

0.0 0.0

0.5

1.0


2.5

3.0

13.15 Water content distribution within the batting thickness after 24 hours while having different diffusion coefficients of moisture (unit: m2/s) in the fiber.


461

13.4.3 Effect of disperse coefficient of free water in the fibrous batting, dl The effect of the disperse coefficient of free water in the fibrous batting dl on the water content distribution is shown in Fig. 13.16. When d l is small or approaches zero (corresponding to a hydrophobic fiber or ice condensate), there is no movement of liquid water on the fiber surface, and the water content will accumulate at the outer region of batting. With an increase of dl, when the amount of liquid condensate exceeds a certain value, the liquid water overcomes the surface tension and moves to the region with a lower water content. When dl = 5.4 × 10–8 m2s–1, the distribution of water content is almost even, except the outmost part, because its temperature is below zero and the condensed water is in a frozen state.

13.4.4 Effect of air permeability of fibrous batting, kx The effect of the air permeability of the fibrous batting (i.e. coefficient of K Darcy’s Law k x = x ) on the water content distribution, assuming the porosity µ of the batting has not changed, is shown in Fig. 13.17. When kx = 0.0 (i.e. there is no flow of moist air within the fibrous batting, which may be achieved 250.0

dl = 5.4e-11 200.0

dl = 0

Water content (%)

dl = 2.7e-10 150.0

dl = 5.4e-9 100.0

50.0

dl = 5.4e-8 0.0 0.0

0.5

1.0

1.5 Thickness (cm)

2.0

2.5

13.16 Water content distribution within the batting thickness after 24 hours while having different disperse coefficients of free water (unit: m2/s) in the fibrous batting.

3.0

462

Thermal and moisture transport in fibrous materials 300.0

kx = 4.393e-5

250.0

Water content (%)

kx = 3.16e-5 200.0

kx = 1.757e-5 150.0

kx = 8.787e-6 100.0

50.0

kx = 0.0

kx = 4.393e-7

0.0 0.0

0.5

1.0


2.5

3.0

13.17 Water content distribution within the batting thickness after 24 hours while having different coefficients of Darcy’s Law kx (unit: m2/(Pa s).

by interlacing the layers of battings with air-impermeable fabrics), the distribution of water content is convex. With increase of kx, the shape of the distribution is gradually changed to concave. It is clear that a higher kx can cause a greater difference between the water content at the inner region and that at the outer region of the batting. The reduction of the air permeability kx, achieved as indicated by interlacing the layers of the battings with less permeable fabrics or by using finer fibres with greater surface area to volume ratio, can create more even distribution of water content.

13.4.5 Effect of moisture vapor resistance of inner covering fabric When the moisture vapor resistance of the inner covering fabric is varied from 64.7 to 2087.0 s m–1, its effect on the water content distribution within the fibrous batting can be seen from Fig. 13.18. Obviously, the moisture vapor resistance of the inner covering fabric has a significant influence on the water content within the batting. This is due to the fact that fabric having a higher vapor resistance will retard the moisture entering the batting, hence resulting in less condensation. This, however, will cause discomfort to the wearer, as perspiration cannot be dissipated from the body effectively. The


463

250

Water content (%)

200

w0 = 1064.7

w0 = 564.7

150

w0 = 264.7 100

w0 = 64.7 w0 = 32.5 50

0 0

0.5

1

1.5 Thickness (cm)

2

2.5

3

13.18 Effect of moisture vapor resistance (unit: s/m) of inner covering fabric on water content distribution after 24 hours.

increase of moisture vapor resistance of the inner covering fabric as a method for reducing condensation is hence not recommended.

13.4.6 Effect of moisture vapor resistance of outer covering fabric The water content distribution within the batting is also significantly affected by the moisture vapor resistance of the outer covering fabric, as shown in Fig. 13.19. With an outer covering fabric of lower moisture vapor resistance, less condensation will take place within the batting. This is because more moisture will be transmitted by convection or diffusion into the environment through a more permeable outer covering fabric. The vapor permeability of the outer fabric is therefore critical and should be maximized while meeting the requirements of protection.

13.5

Conclusions

From experiments at –20 °C using a novel sweating guarded hot plate, it was found that most of the changes in temperature distribution within fibrous battings sandwiched by an inner and outer layer of thin covering fabric took place within 1/2 hour of exposure to the cold condition. The distributions of

464


w 1 = 130.0 200

Water content (%)

w 1 = 97.5 150

w 1 = 32.5

100

w 1 = 1.3

w 1 = 6.5 w 1 = 13.0

50

0 0

0.5

1

1.5 Thickness (cm)

2

2.5

3

13.19 Effect of moisture vapor resistance (unit: s/m) of outer covering fabric on water content distribution after 24 hours.

temperature and water content within the battings are affected by the moisture absorption properties of the fibers and the density or porosity of the battings. Moisture absorption of the fibers increases the temperature in the battings. And greater permeability of the battings results in greater accumulation of condensed water in the outer region of the battings. A theoretical model of coupled heat and moisture transfer within clothing assemblies considering moisture absorption, phase change, moisture bulk flow, radiative heat transfer and mobile condensates has been presented. Numerical solution was accomplished by using the Finite Volume Method. The theoretical results were compared with the experimental ones and found good agreement. Numerical simulation was conducted to investigate the effects of various clothing parameters on the condensation and moisture absorption within clothing. Based on the numerical analysis, it is believed that, for clothing consisting of fibrous battings sandwiched by two layers of thin fabrics, inner fibrous battings having higher fiber content, finer fiber, greater fiber emissivity, higher air permeability, lower disperse coefficient of surface free water, and lower moisture absorption rate would cause less condensation and moisture absorption, which is beneficial to thermal comfort during and after exercising in cold weather conditions. It is also shown that covering fabrics have a significant influence on the water content within the fibrous batting.


13.6 Ca Cf C ′f Cv Cva Cvf Cvw Da df dl E FL FR hc ht k Kx Krx kx L Li M p psat pv R Rf Rti Rwi Rdi r RHi Rhf T Ti

465

Nomenclature water vapor concentration in the inter-fiber void space (kg m–3) mean water vapor concentration in the fiber (kg m–3) volumetric moisture concentration in the fiber (it varies over the radius of the fibers) (kg m–3) effective volumetric heat capacity of the fibrous batting (kJ m–3 K–1) volumetric heat capacity of dry air (kJ m–3 K–1) effective volumetric heat capacity of the fiber (kJ m–3 K–1) volumetric heat capacity of water (kJ m–3 K–1) diffusion coefficient of water vapor in the air (m2 s–1) diffusion coefficient of moisture in the fiber (m2 s–1) disperse coefficient of free water in the fibrous batting (m2 s–1) condensation or evaporation coefficient, dimensionless total thermal radiation incident traveling to the left (W) total thermal radiation incident traveling to the right (W) convective mass transfer coefficient (m s–1) convective thermal transfer coefficient (W m–2 K–1) effective thermal conductivity of the fibrous batting (W m–1 K–1) permeability of porous batting (m2) relative permeability of porous batting (m2) coefficient of Darcy’s Law (m2 (Pa.s)–1) thickness of the fabric batting (m) thickness of the inner and outer covering fabrics (m) (i = 0: inner fabric; i = 1: outer fabric) molecular weight of the evaporating substance, M = 18.0152 (g.mol–1) for water pressure of water vapor in the inter-fiber void (Pa) saturated water vapor pressure at temperature Ts (Pa) vapor pressure in vapor region at Tv (Pa) the universal gas constant, R = 8.31 (J K–1 mol–1) radius of fibers (m) resistance to heat transfer of inner or outer covering fabric (Km2 W–1) (i.e. i = 0: inner fabric, i = 1: outer fabric) resistance to water vapor (i.e. i = 0: inner fabric, i = 1: outer fabric) (s m–1) diffusion resistance to water vapor radial distance (m) relative humidity of the surroundings (%) (i.e. i = 0: surface next to human body, i = 1: surrounding air) relative humidity of the air space within the porous batting (%) temperature (K) temperature of the boundaries (K) (i.e. i = 0: surface next to human body, i = 1: surrounding air)

466

Ts Tv t u W

Wai W0i Wc Wf WCi x β ε

ε′ λ µ ρ ρw σ τ

Γ Γce Γs

13.7


temperature at the interface of condensates and vapor (K) temperature in the vapor region (K) time (s) velocity of water vapor (m s–1) water content of the fibrous batting (%), which is defined as the weight of water divided by the weight of the dry fibrous batting, and may be greater than 100% weight of the ith layer of the batting after placing on the instrument in the cold chamber (g) weight of the ith layer of the batting before placing on the instrument in the cold chamber (g) critical level of water content above which the liquid water becomes mobile (%) water content of the fibers in the porous batting (%) water content of the ith layer of the batting (%) distance from the inner covering fabric (the warm side) (m) radiative sorption constant of the fibers (m–1) porosity of fibrous batting considering condensates (liquid water, or ice) in the batting porosity of the dry fibrous batting (ε = cubic volume of interfiber space/total cubic volume of batting space) latent heat of (de)sorption of fibers or condensation of water vapor (kJ kg–1) dynamic viscosity of dry water vapor (kg m–1s–1) density of the fibers (kg m–3) density of liquid water or ice (kg m–3) Boltzmann constant σ = 5.6705 × 10–8 (W K–4 m–2) effective tortuosity of the fibrous batting. The degree of bending or twist of the passage of moisture diffusion due to the bending or twist of fibers in the fibrous insulation. It normally changes between 1.0 and 1.2, depending on the fiber arrangements. total rate of (de)sorption, condensation, freezing and/or evaporation (kg s–1 m–3) rate of condensation, freezing and/or evaporation (kg s–1 m–3) rate of (de)sorption (kg s–1 m–3)

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23. Motakef S and El-Masri M A, ‘Simultaneous heat and mass transfer with phase change in a porous slab’, J. Heat Mass Transfer 1986 29(10) 1503–1512. 24. Farnworth B, ‘A numerical model of the combined diffusion of heat and water vapor through clothing’, Tex. Res. J. 1986 56 (11) 653–665. 25. Vafai K and Sarkar S, ‘Condensation effects in a fibrous insulation slab’, J. Heat Transfer 1986 108 (8) 667–675. 26. Vafai K and Tien H C, ‘A numerical investigation of phase change effects in porous materials’, Int. J. Heat Mass Transfer 1989 32 (7) 1261–1277. 27. Tao Y X, Besant R W and Rezkallah K S, ‘The transient thermal response of a glassfiber insulation slab with hygroscopic effects’, Int. J. Heat Mass Transfer 1992 35 (5) 1155–1167. 28. Fan J, Luo Z and Li Y, ‘Heat and moisture transfer with sorption and condensation in porous clothing assemblies and numerical simulation’, Int. J. Heat Mass Transfer 2000 43 (12) 2989–3000. 29. Fan J and Wen X, ‘Modelling heat and moisture transfer through fibrous insulation with phase change and mobile condensates’, Int. J. Heat Mass Transfer 2002 45 4045–4055. 30. Cheng X Y and Fan J, ‘Simulation of heat and moisture transfer with phase change and mobile condensates in fibrous insulation’, Int. J. of Thermal Sciences 2004 43 665–676. 31. Fan J, Cheng X Y, Wen X and Sun W, ‘An improved model of heat and moisture transfer with phase change and mobile condensates in fibrous insulation and comparison with experimental results’, Int. J. of Heat and Mass Transfer 2004 47 2343–2352. 32. Fan J and Cheng X Y, ‘Heat and moisture transfer with sorption and phase change through clothing assemblies: Part II: Theoretical modeling, simulation, and comparison with experimental results’, Textile Research Journal 2005 75 (3) 187–196. 33. Gibson P W and Charmchi M, ‘Modeling convection/diffusion processes in porous textiles with inclusion of humidity-dependent air permeability’, International Comm. Heat Mass Transfer 1997 24(5) 709–724. 34. Jones E F, Evaporation of Water – With Emphasis on Application and Measurements, Michigan, Lewis Publishers, 1992, 25–43. 35. Patankar S V, Numerical Heat Transfer and Fluid Flow, Levittown, Taylor & Francis, 1980, 31–40. 36. Ghali K, Ghaddar N and Jones B, ‘Modeling of heat and moisture transport by periodic ventilation of thin cotton fibrous media’, International Journal of Heat and Mass Transfer 2002 45 3703–3714.

14 Computer simulation of moisture transport in fibrous materials D. L U K A S, E. K O S T A K O VA and A. S A R K A R Technical University of Liberec, Czech Republic

The present chapter is oriented towards the field of computer simulation of some selected transport phenomena associated with fibrous materials. Particularly, the phenomena covered here are transportation of liquid and moisture along individual fibres, fibre bundles, and in fibrous masses. A unified approach to describe these physical processes, based on stochastic processes in the lattices with spatial interactions amongst the lattice cells, has been sought. Unlike the classical fashion of analysis, generally incorporating dynamics of heat and fluids with an application of differential calculus in continuum, this method is characterised by discrete cells that represent nibbles of solid fibres, liquids and gases, mutually bonded together with various types of interactions. Unfortunately, this distinctive endeavour has not covered the phenomena associated with liquid viscosity in detail, but has created an excellent theoretical base, reflecting manifold structural aspects of fibrous masses. Briefly, the first section, 14.1, will be primarily concerned with a description of random fields and Markov random fields, following a short introduction to one of the most famous spatial schemes used for computer simulations, known as the Ising model. These mathematical objects will be illustrated with a few elementary examples. As will be explained there, random and Markov random fields delineate the richness of manifold configurations, which mutate with time and exhibit the properties of their constituent cells, thus demonstrating the net geometry of their interactions. Subsequently, the remarkable proposition of Besag will be formulated in Section 14.2 to bridge the gap between joint and local conditional probabilities of random fields. The mathematical conceptualisation of the Besag proposition will then serve as the general equation encompassing all kinds of spatial interactions in the model. The interactions dealt with there will be described using the notions of ‘neighbourhood’ and ‘clan’. In the process of finding a general solution for the Besag proposition, the theorem of Hammersley and Clifford that will form, thenceforth, the foundation to categorise stochastic models with spatial 469

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interactions, will be encountered. Amongst them, the auto-model will be of prime concern here and will be used for computer simulation of wetting phenomena. The fact that the Hammersley and Clifford theorem rationalises an additive nature of energies in physical systems, which are well explained with the concept of random fields, is also estimable. The dynamics of the stochastic processes and their time evolution in the domain of random fields will be managed by Markov chains, based on various exchange dynamics. Eventually, constraints associated with the Besag proposition, from the viewpoint of conservation laws of canonical systems, will be mentioned. Unlike the preceding sections, which mainly concentrate on the theoretical concepts of stochastic spatial processes, Section 14.3 contains three case studies of computer simulation used for reproducing wetting phenomena in a fibre mass. To begin with, individual fibres will be chosen to use a wellestablished fact pertaining to interfacial fluid dynamics, called ‘Rayleigh instability’, to explain the behaviour of liquid films coating individual fibres. This instability has a significant influence on the processes of fibre spinning from polymer solutions or melts. The theory justifies the spontaneous creation of equidistantly spaced droplets from liquid films covering individual cylindrical fibres, and the detachment of liquid jets into drops. The description will be prolonged thereafter, describing morphological transitions of shapes of the liquid body within bundles of parallel fibres with various inter-fibre distances. In addition, transition from long cylindrical to unduloid structures of liquids for pairs of parallel fibres, will be dealt with. The third case study of this section is related to the behaviour of moisture imbibitions in disordered fibrous material.

14.1

Introduction

The random evolution of spatial systems with spatially interacting variables has attracted the attention of numerous disciplines, viz. plant ecology (Besag, 1974), chemistry (van Kampen, 1992), physics (Swishchuk, 1997), cybernetics (Korolyuk and Swishchuk), 1995; Ermakov and Nekrakin, 1989), and modelling of images and textures (Chen and Dubes, 1989). Probably, the first physical model built on a spatial concept was the Ising model. It was introduced by Lenz (1920) and Ising (1925) at the beginning of the twentieth century. A more recent survey of this classic model has been given by Prum and Fort (1991). The Ising model consists of a periodic lattice of cells wherein are situated elementary magnets with either ‘up’ or ‘down’ orientation with respect to a preferred direction. Physically, two neighbouring magnets require less energy to be orientated in the same orientation as compared to a mutually opposite one. It was primarily assumed that each magnet was sensitive only to the orientation of its nearest neighbouring magnets. So the first spatial scheme


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was created, having a lattice with spatially interacting elements represented by magnets at each of the lattice cells, defined by random variables also known as the Ising variables, ascribed with values +1 and –1. The basic concept of the Ising model is displayed in Fig. 14.1. This model was developed to explain the thermal dependence of magnetism. Such dependence originates due to thermal fluctuations causing, for instance, magnets to be flipped randomly and subsequently to new orientations in discrete steps. The new local configuration of an elementary magnet in a cell is then determined by comparison of the values of ‘local conditional probabilities’. Although this notion will be detailed subsequently, for the time being, it can be mentioned that the first ‘local conditional probability’ refers to the probability of the original configuration of the chosen elementary magnet with respect to a certain configuration of its neighbours, i.e. the configuration of its neighbouring elementary magnets creates the condition. The last local conditional probability belongs to a newer orientation of the magnet under consideration, subjected to the same condition. Both the local conditional probabilities depend on the sum of the interaction energies between the elementary magnet having a certain orientation, and its neighbours, and, of course, on the temperature. 1

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The lower the interaction energy of a configuration, the higher is the probability of its occurrence. Temperature also influences these transitional probabilities of flipping. An elevated temperature causes the system to resist a shift towards a higher ordered configuration of its array of elementary magnets. The Ising model was designed for studying the phase transition between the ferro- and para-magnetic state of metals. Spatial schemes, built on approaches more generalised than that of the Ising model (having random variables attributable by more than two values, representing spatial dislocation of various materials, such as, gas, liquid, and solid fibre in the case of wetting and liquid transport phenomena) will be employed. The Ising model was designed chiefly for a thorough investigation of phase transitions such as the previously mentioned transition between ferro- and para-magnetic states. As opposed to that approach, this chapter emphasises the pertinence of the spatial schemes, namely auto-models, to analyse phenomena far away from a threshold of phase transition given by a critical temperature. The present attempt is mainly aimed at an investigation of the influence of the boundary conditions on the system configuration. As an example, an interaction of a fibre surface with a binary mixture of a liquid and a gas to evoke wetting and a subsequent transfer of the liquid into the fibrous material may be thought of. The liquid starts to take conformations of films and droplets to coat the solid fibres. The manifold morphology of liquid bodies, interacting with fibres and fibrous masses, hence conforms to the complexity of the superficial geometry of fibrous matters. In this chapter, an elaborate microscopic account of such complex structure-based phenomena, using computer simulation, is presented.

14.1.1 Random fields Two fundamental mathematical concepts of random field and Markov random field, significantly involved in the context of the present chapter, will be described in this subsection. The concept of ‘random field’ forms the foundations of the Besag proposition. The construct of Markov random field facilitates ‘knowing the ropes’ in a quantum of existing and unrealized interactions between elements/cells belonging to a random field, thus simplifying procedural aspects of model design with all the required essence kept intact for explaining the physics of the phenomena. The notations for random fields used in this chapter will be kept as it was used by Geman (1991) and Paget (1999). Spatial schemes, which will be dealt with, consist of a set S of cells, denoted by si where i is an index identifying each of the cells individually. Set S is represented as S = {s1, s2,…, sn}, where n is the total number of cells in a particular system. For any of the cells si belongs a random variable Xi assigned with values xi from a local configuration state space Λi. Any particular configuration of a lattice


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composed of those elementary cells is given by a set of values of related r random variables, conveniently represented as x with elements {x1, x2,…, xn} and mostly referred to as the ‘configuration vector’. The global configuration space of the system Ω is then created by the product of all the local configuration state spaces Λi’s. Ω = Π Λi 1≤ i ≤ n

[14.1]

Thus, the geometry of a random field has been presented. To make it more vivid, let it be illustrated with some more examples, such as a bed of carrot plants in a yard. Using a plant peg, the bed is divided into areas, each of which represents a cell. The cells of this system need not be regular. A usable variable Xi to evaluate a horticultural achievement may be considered to be assigned with the number of plants xi, in a cell si. The number of plants xi may vary stochastically with time, i.e. over various seasons. Here, the number xi is a particular value of the random variable Xi. The local configuration space Λi is directly connected to the number ni of seeds embedded in a cell si as the random variable Xi may have values {0, 1, 2, … , ni}, ranging from an absolute failure or, ‘zero success’, i.e. when none of the seeds germinate, to an entire success, i.e. when all ni seeds germinate. Here, the global configuration space Ω represents all the eventualities of carrot plant r distributions, for instance x = (3, 02, 53, … , 2n) as given by the local fertility of the bed and the interactions among plants. Over a series of seasons, the variable Xi behaves randomly because of the fact that the yield depends on contingencies such as weather, activity of pests, intensity of sunlight and competition among all the carrots for water, mainly dependent on watering and weeding of the bed. The next example of a random field is more close to the subject of our interest. Envisage that a three-dimensional space in the vicinity of a fibre, coated with a liquid film, is divided into regular cubic cells. The set of cubes in a lattice, in this case, may be related to the set S of cells, as defined before. Local variable Xi in a cell si is then considered to acquire three possible values {0, 1, 2} i.e. xi ∈ {0, 1, 2}, depending on the prevailing material in it, viz. a gas, a liquid, or a solid fibre. The related local configuration state space Λi then either consists of values {0, 1} for areas with liquid–gas binary mixture, or is composed of a value of 2, signifying a solid fibre, considered fixed in space and stationary over time. The global configuration space Ω is then made up of all the possible configurations given by redistribution of gas and liquid cells within a fixed fibrous mass. The concept of this type of random field is pictorially depicted in Fig. 14.4 of Section 14.1.2, which will follow the forthcoming discussion concerning intercellular interactions of a system. With the random fields are consorted two generic notions of ‘joint’ and ‘local conditional’ probabilities. The joint probability is associated with the

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r appearance of a particular configuration P ( x ) of a system as a whole, and the ‘local conditional probability’ concerns that of a local value xi in a cell, si, conditional to a certain fixed configuration to which the remaining cells are tuned up. A local conditional probability will be denoted as p(xi| x1, x2,…, xi–1, xi+1,…, xn), wherein the part of the argument of p before the partition represents the concern about the probability of an appearance of xi in the cell si, while the set of concrete random variable values {x1, x2,…, xi–1, xi+1,…, xn}, beyond the partition, appropriately expresses the fixed configuration of all the remaining cells of the system that represents the condition. Now, to explain vividly the concept of global, i.e. joint and local conditional probabilities, a modified Ising model will be utilised. Let a part of a regular two-dimensional square lattice, comprising nine cells arranged in a square array, be as shown in Fig. 14.2. In each of the cells is embedded an elementary magnet. It may be supposed that this random field of magnets is not influenced by any external magnetic field and so configurations involve, exclusively, mutual interaction among the elementary magnets. Deviating from the classic Ising model, it is further assumed that the mutual interactions exist among all the nine elementary magnets. Now, if the temperature of the system is

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very low, i.e. if it tends towards absolute zero, the joint probability for the total order in the system becomes very high due to identical orientations of the elementary magnets. There exist two such completely unified configurations, characterised by all the elementary magnets orientated either upwards or downwards. It is due to the choice of a temperature close to absolute zero that the two entirely ordered arrangements of the random field of nine magnets share a unitary value of the total probability of appearance of all possible configurations having identical cellular orientations. Denoting the upward orientation of an elementary magnet in a cell si with a value of the random variable (xi) as +1 and the downward orientation as –1, the joint probabilities pertaining to two different instances when all the magnets simultaneously point upwards or downwards are represented as P(+ 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18, + 19) and P(–11, –12, –13, –14, –15, –16, –17, –18, –19), respectively, each having a value of 0.5. It is quite obvious that any other configuration, with at least one magnet oppositely orientated with respect to the others, will certainly have a joint probability nearly equal to zero. For instance, an alternately orientated array of elementary magnets, P(+11, –12, +13, –14, +15, –16, +17, –18, +19), has a value equalling zero. Let a chosen local probability of an elementary magnet in the field, be considered keeping the rest of the eight unaltered. First of all, appropriate conditions are selected to evaluate the local probabilities of its two possible orientations. In this process the elementary magnet at the centre of the array in the cell s5 will be attended. Considering an upward orientation of all the other elementary magnets, the set of eight variables, as stated before, may be arranged as {+11, +12, +13, +14, +16, +17, +18, +19}. The upward orientation of the central magnet, under this condition, will be highly probable, i.e. it can be written that p(+15| +11, +12, +13, +14, +16, +17, +18, +19) has a value of 1. It should be noted here that the state of the central magnet is not the condition of itself. Accordingly, the index 5 with its assigned variable x5 is excluded from the conditional part of the argument of the aforementioned local conditional probability p. For the same system there exists another instance of the local conditional probability related to the cell at the fifth position under the same condition. It refers to the downward orientation of the central magnet. For this particular configuration, the local conditional probability, p(–15| +11, +12, +13, +14, +16, +17, +18, +19) diminishes as the temperature approaches absolute zero. An upward or downward orientation of the central elementary magnet may be chosen for quite an improbable configuration, such as the one having an alternate arrangement of upwards and downwards orientations of the rest of its eight elementary magnets, represented as {+11, –12, +13, –14, –16, +17, –18, +19}. The relevant local conditional probability, p(+15| +11, –12, +13, –14, –16, +17, –18, +19) for the upward orientation of the central magnet then attains a value of 0.5, as the central elementary magnet, acted upon by

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equally numerous groups of neighbours, has an equal chance to be orientated either upwards or downwards. A similar result is obtained for the downward orientation of the central magnet, i.e. p(–15| +11, –12, +13, –14, –16, +17, –18, +19) too has a value of 0.5. Hence, the sum of the local conditional probabilities associated with two possible local configurations of the cell s5 under the same condition is equal to one. Obviously, the sum of local conditional probabilities for all nodes si of the random field for a particular system configuration is not normalised and so may be, occasionally, greater than unity. This is well exemplified, when a coordinated upward orientation of all the magnets with a joint probability value of 0.5 is considered. Each of the local conditional probabilities with xi having a value of +1, i.e. p(xi| +11, . . . +1i–1, . . . , +19) then approaches a value of 1, mainly due to the interaction of the elementary magnet in the observed cell si having an orientation identical to the predominant orientation of the system. Hence, the sum of all local conditional probabilities for an entire upward orientation, i.e. Σ p ( x i | + 11 , . . . , +1i –1 , . . . , +19 ) is 9. 1≤ i ≤ 9 Interactions between all the members of the set S are permitted in random fields, whereas restrictions in this regard are exerted in Markov random fields, as will be described in the next subsection.

14.1.2 Markov random fields A Markov random field is a random field with a slightly modified characteristic concerning the condition that influences the value xi attributed to the variable Xi at a cell si. The value of xi may be any one from a local state space Λi, but the probability that Xi assumes a value xi is strictly conditional to the values xj of the neighbouring cells of si. This assumption, as a rule, causes that random field to have independent regions, at least by groups of two. In other words, a random field is composed of groups of cells wherein local probabilities are not conditioned by the configurations in the independent regions. This condition is referred to as the Markovian condition and the pertinent field is known as the Markov random field. Moussouris (1973) mentioned that this Markovian assumption resembles the definition of the famous Markov random chains in which the likelihood of a particular configuration of a system at any instant depends on the state of its immediate past only and not on any other of its historical details (Van Kampen, 1991). Markov chains will be involved more deeply later in Section 14.2.5 to inspire dynamics in some models. A Markov random field is, therefore, a space–time analogy of a Markov chain where some events, characterised by a certain space, depend on chosen spatial neighbours only at the same instant of time and not on the whole spatial array of other events. This comparison has a slight limp with time, having a constantly grooving value, while on the other hand, interactions in space schemes are distributed in three dimensions and are, as a rule,


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bilateral. Nevertheless, the analogy with their preceding spatial distribution of interactions has caused these random fields to be named Markov random fields. The notion of a Markov random field is documented here taking support from the following example. Resurrecting the previous example of a carrot bed and selecting one plant in it, let it be indexed with ‘i’ for denoting its cell. The system of cells consists, at this moment, of individual plants. As is seen from Fig. 14.3, the plant in this concrete Markov random field has just five close neighbours with whom it competes for water and sunshine. Here, a mutual independence between growth of the chosen plant in the cell si and the plants situated further on may be considered as a supposition, i.e. the development of the plant in question and its ‘non-neighbours’ are independent of each other. To be more precise, for the Markov random field model of the bed, the corresponding local conditional probabilities of ‘non-neighbours’ are independent of each other. The last illustration underlines that a neighbourhood does not necessarily mean a group of cells from the vicinity of a chosen one. Considering a dissociated liquid wetting a fibre and subsequently treating this system as a Markov random field, two kinds of interactions of a cell have to be accounted for. Short-range intermolecular interaction, caused by van der Waals forces is one. The other involves long-range interaction caused by Coulomb

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electrostatic forces between charged particles, i.e. between dissociated ions. Such a neighbourhood is sketched in Fig. 14.4. Undoubtedly, mathematical objects of random and Markov random fields are so universal that they can be used widely to model systems, especially those concerning many-body issues. On the other hand, random and Markov random field are such general objects that it seems no massive and special rules govern them, but the contrary is surprisingly true, as will be shown in the next section.

14.2

Auto-models

To demonstrate the rules and laws governing random fields and Markov random fields, let the Besag proposition, introduced in the first half of the 1970s by Besag (1974), be considered. Besag’s speech, read before the Royal Statistical Society in the year 1974, had been inspired by the area of plant ecology, which was then supposed to benefit from the potential applications of this newly developed theory. As a typical illustration of a stochastic spatial situation in plant ecology, it had been mentioned beforehand in Cochran’s work (1936) on patterns of infection in an array of tomato

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plants. It is due to the general nature of random fields that the Besag proposition, has recently become useful for versatile applications. As will be shown later in this section, there lies a strong link between the Besag proposition, its consequences, and some substantial ideas of statistical physics that are further employed to simulate cooperative wetting phenomena in fibrous masses.

14.2.1 The Besag proposition It was only in the 1960s that the relationship between joint and local conditional probabilities of random fields was purely realized. White (1963) remarked that equilibrium processes in statistical mechanics have a natural construction in terms of joint probabilities rather than in local probability formulations. Again, it was found in the process of formulating the Ising model of ferromagnetism that the conditional probabilities completely generated its joint probabilities. This could well be explained by the fact that Markov chains were designed merely on the basis of the local conditional probabilities related to the local changes in Hamiltonian, as will be shown in Section 14.2.5. But, as has already been shown, the original Ising model considered exclusively the interactions with the nearest neighbours. At the start of the 1970s, two works (Spitzer, 1971; Besag, 1972) threw light on the joint-local conditional probability relationship for Markov random fields for systems based on concept of nearest neighbour in k-dimensional finite cubic lattices. The scope, consequently, was restricted neither to the nearest neighbour interactions nor to the lattice schemes. In fact, the Besag proposition is even valid for fairly arbitrary systems with the cells distributed randomly in space. The Basag proposition was formulated for random fields, without any limitation on the opulence of the intercellular interactions. As will be seen immediately, the Besag proposition is a direct consequence of Bayes’ theorem. p(xi| x1, x2,…, xi–1, xi+1,…, xn =

P ( x1 , x 2 , . . . , x n ) Π ( x1 , x 2 , . . . , x i –1 , x i+1 , . . . , x n )

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More information about Bayes’ theorem is provided by Van Kampen (1992) in the section ‘Multivariate distributions’. Two terms of Bayes’ rule, i.e. in Equation [14.2], are frequently encountered, viz. the conditional probability p on the left-hand side and the joint probability P in the numerator of the right-hand side of the equation. The probability Π(x1, x2,…, xi–1, xi+1,…, xn) in the denominator is, according to Van Kampen (1991), referred to as the ‘marginal probability’, which is the sum of the joint probabilities for all the possible variable values xi in a chosen cell si, keeping the variable values in all the remaining cells unchanged.

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Π(x1, x2,…, xi–1, xi+1,…, xn) =

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[14.3] where (x1, x2,…, xi–1, xi+1,…, x1) are fixed and only xi changes. The above-cited Bayes’ rule may be verified using the example connected to the modified Ising model consisting of nine cells, as described earlier in Section 14.1.1. Previously, the joint and the local conditional probabilities for a temperature approaching absolute zero were evaluated. Recalling that the calculated joint probabilities for the coordinated orientation of all the magnets constituting the lattice, directed either upwards or downwards, were 0.5 separately, the joint probability of the configuration with the downwards orientation solely being associated to the central magnet, i.e. P(+11, +12, +13, +14, –15, +16, +17, +18, +19) has to have a value close to zero, as is r evident from the normalisation condition rΣ P ( x ) = 1. Consequently, the x ∈Ω marginal probability, according to Eq. [14.3], comes out to be Π(+11, +12, +13, +14, +16, +17, +18, +19) = 0.5 + 0 = 0.5

[14.4]

It is notable at this juncture that the local configuration space Λ5, related to the cell s5, comprises only two possible configurations of upward and downward orientations. Accordingly, Bayes’ rule furnishes identical values of the local conditional probabilities for the cell s5 when the remaining magnets point upwards. It can also be seen that the local conditional probability p for x5 = +1 is, in agreement with Eq. [14.2], equal to p(+15| +11, +12, +13, +14, +16, +17, +18, +19) = 0.5 = 1 0.5

[14.5]

while, under the same condition, the local conditional probability p with the central magnet pointing downwards, i.e. x5 = –1, appears to be p(–15| +11, +12, +13, +14, +16, +17, +18, +19) = 0 = 0 0.5

[14.6]

The above results, Equations [14.5] and [14.6], are in agreement with those obtained previously in Section 14.1.1. The derivation of the Besag proposition begins with a trivial mathematical manipulation of Bayes’ rule to enable one to rewrite the joint probability r P ( x ) of the system, taking into account events relating to cell s1, as given below: r P ( x ) = p(x1 | x2, x3,…, xn) Π (x2, x3,…, xn) [14.7] Let another value, y1, of the variable X1 in the same cell s1 be conceived, such that x1, y1 ∈ Λ1. This means that the local state space Λ1 is composed of at least two local configurations with two values of its local variable X1 = x1


481

or X1 = y1. Using this notion, Bayes’ rule for random variable value y1 in the cell s1 may be written as P(y1, x2, x3,…, xn) = P(y1 | x2, x3,…, xn) Π(x2, x3,…, xn)

[14.8]

The marginal probability Π in Equation [14.8] is identical with the one in Eq. [14.7]. Hence these two equations may be used to obtain the following relation:

r p ( x1 | x 2 , x 3 , . . . , x n ) P( x ) = P ( y1 , x 2 , x 3 , . . . , x n ) p ( y1 | x 2 , x 3 , . . . , x n )

[14.9]

Therefore, the Bayes’ rules for the joint probabilities P(y1, x2, x3,…, xn) and P(y1, y2, x3,…, xn), having marginal probabilities identical to the one obtained, when cell s2 is attended Π(y1, x3, x4,…, xn), take the following forms: P ( y1 , x 2 , x 3 , . . . , x n ) =

p ( x 2 | y1 , x 3 , x 4 , . . . , x n ) , Π ( y1 , x 3 , x 4 . . . , x n )

P ( y1 , y 2 , x 3 , . . . , x n ) =

p ( y 2 | y1, x 3 , x 4 , . . . , x n ) Π ( y1 , x 3 , x 4 . . . , x n )

[14.10]

Implementing substitutions successively for P(y1, x2, x3,…, xn) in Eq. [14.9] from the first relation in Eq. [14.10] and then for the marginal probability Π(y1, x3, x4,…, xn) from the second relation in Eq. [14.10], the following relation is derived:

r p ( x1 | x 2 , x 3 , . . . , x n ) p ( x 2 | y1 , x 3 , x 4 , . . . , x n ) P( x ) = p ( y1 | x 2 , x 3 , . . . , x n ) p ( y 2 | y1, x 3 , x 4 , . . . , x n ) × P(y1, x2, x3, x4,…, xn)

[14.11]

Undoubtedly, after ‘n’ such substitutions, the right-hand side of Equation [14.11] will have all the values of xi replaced by new yi values in the argument of the joint probability P as well as in arguments of the last fraction of conditional probabilities p, to take an ultimate form as exhibited in Eq. [14.12]. The most vital role played by the chain of substitutions is that the marginal probabilities, crucial in Bayes’ rule, disappear from the Besag proposition. Therefore, the shape of the final relation can be predicted as r p ( x1 | x 2 , x 3 , . . . , x n ) p ( x 2 | y1 , x 3 , x 4 , . . . , x n ) P( x ) = p ( y1 | x 2 , x 3 , . . . , x n ) p ( y 2 | y1, x 3 , x 4 , . . . , x n )

×

p ( x 3 | y1 , x 2 , x 4 , . . . , x n ) p ( y 3 | y1 , y 2 , x 4 , . . . , x n )

× ...

r p ( x n | y1 , y 2 , y 3 , . . . , y n–1 ) P( y ) p ( y n | y1 , y 2 , y 3 , . . . , y n–1 )

[14.12]

482


Arranging and concisely writing the terms of the above obtained relation, the Besag proposition is framed in the following way: r p ( x i | y1 , y 2 , . . . , y i –1 , x i+1 , x i+2 , . . . , x n ) P( x ) r = Π [14.13] P ( y ) 1≤ i ≤ n p ( y | y , y , . . . , y , x , x , . . . , x ) i

1

2

i –1

i+1

i+2

n

where the symbol Π denotes the product of all the fractional terms having 1≤ i≤ n indices ranging from 1 to n. The Besag proposition states that the joint r probability distribution P ( x ) of random variables Xi is uniquely determined by its local conditional probabilities p. It is quite obvious that each conditional probability appearing in the denominator of Equation [14.13] has to be nonzero. It implies that each configuration of the system has to have the chance r r to appear, i.e. P ( y ) > 0 for each ( y ) ∈Ω . This assumption was referred to as the ‘positivity condition’ by Hammersley and Clifford (1971). It will be seen later that the positivity condition gives a number of problems when dealing with some constraints imposed by using canonical and micro-canonical systems that conserve the numbers of particles and/or energy. To comment on the mathematical formalism involved above, it may be observed that all the equations used there are basic algebraic equations constructed with ordinary operations of multiplication and division. The richness of these relations is, however, hidden in manifold forms of the joint and the local configurations that echo in complex forms of the arguments of joint and local conditional probabilities. To ‘know the ropes’ in an intricate juxtaposition of conditions caused by numerous intercellular interactions in the models aimed at in the present chapter, application of appropriate tools becomes indispensable. The next sub-section will include an elaborate account of such tools. For the time being, it can merely be said that the tools will conceptually encompass the forms of ‘neighbourhoods’ and ‘clans’. These tools will be instrumental in the formulation of the Hammersley and Clifford theorem, which, in turn, will give birth to generalised ‘energy functions’. The Hammersley and Clifford theorem will be formulated through the general solution of the Besag proposition.

14.2.2 Neighbourhoods and clans The fundamental aspect of Markov random fields, demarcating the domain of interactions among cells in random fields, has already been described. Local probabilities for Markov random fields are conditional only on the values of xj’s referring to the predefined cells neighbouring to a selected cell si. As a supposition, Ni may be considered as the set of all the sj cells that construct the neighbourhood of the cell si. Representing the relational facet of the neighbourhood Ni, formed by sj cells having a set of random variables xj, the notation {xj; sj ∈ Ni} is applied. Using this notation, a more concise representation of conditional probabilities of either Markov random fields or random fields is accomplishable:


p(xi| x1, x2,…, xn) = p(xi|xj; sj ∈ Ni)

483

[14.14]

Accordingly, the argument of the conditional probability on the right-hand side of Equation [14.14] is read in the following manner. Conditional probability that belongs to the appearance of a random variable Xi, assigned with a value xi and associated to a referred cell si, depends on the condition that the cells {sj; sj ∈ Ni} create a neighbourhood Ni of the cell at reference with a definite configuration {xj; sj ∈ Ni}. A set of all the existing neighbourhoods in a set of S cells make the neighbourhood system N. Drawing an analogy with the classic physical principle of action and reaction, the models will involve, hereafter, only symmetric neighbourhoods which adhere to the following criteria: (i) si ∉ Ni (ii) sj ∈ Ni ⇔ si ∈ Nj

[14.15]

The first criterion claims that a cell si is not its own neighbour and the last one alleges that, if a cell sj is an element of the neighbourhood Ni of the cell si then the cell si belongs to the neighbourhood Nj of the cell sj, and vice versa. An example each of a symmetrical and an unsymmetrical neighbourhood is depicted in Fig. 14.5. On regular lattices, some symmetrical neighbourhoods have translational symmetry. The geometrical shape of such r a neighbourhood Ni coalescences with that of a Nj after a translation, i.e., t : s i → s j , as also shown in Fig. 14.5. In this chapter, symmetric neighbourhoods will be predominantly dealt with. An important property pertaining to the idea of neighbourhoods is the order ‘o’, introduced by Geman and Geman (1984) and Geman (1991) while dealing with the recognition of pictures. In the process, these authors sought appropriate symmetric as well as translational symmetric neighbourhoods. They chose an integer ‘o’ as the order of neighbourhood and constructed a neighbourhood N io of order ‘o’, associated to a cell si as the set of cells sk’s fulfilling the condition

N io = {sk ∈ S; 0 < /si – sk/2 ≤ o}

[14.16]

The mathematical representation of the definition of a neighbourhood N io of a cell si, having an order o, is interpreted as follows. The neighbourhood N io of a cell si, having an order of o, is the set of cells sk that belong to the set S of all cells, where the square of the distance between the cells si and sk, abbreviated as /si–sk/2, is equal to or less than the neighbourhood order o. The first-order neighbourhood is also called the ‘nearest neighbour neighbourhood’. The neighbourhoods of different orders are shown in Fig. 14.6. Generally, distances in regular lattices are measured in lattice units, l.u. One lattice unit is the distance between the nearest neighbouring cells. In physical systems, the scope of inter-particle forces predefines the neighbourhood order. The order o is then the second power of the distance over which interaction energies are non-negligible. Moreover, various kinds

484


(b)

(a)

Ni si

r t

Nj sj

(c)

14.5 Symmetrical (a) and unsymmetrical (b) neighbourhoods of cells i and j in a Markov random field. Interactions of the cell si are depicted with a bold line while neighbouring cells of the cell sj interact with it via thin bonds. On regular lattices, some symmetrical neighbourhoods have translational symmetry. The geometrical shape of r neighbourhood Ni coalesces with Nj after the translation t : si → s j .

of physical interaction, such as short-distance intermolecular forces together with long-distance selective Coulomb forces, can render sparseness toneighbourhood. As an eventuality, the neighbourhood system loses the translational symmetry and so ceases to obey the relation [14.16], mainly accounted for by some uncharged cells at a distance D with magnitude less than that of the square root of the order o, i.e. o from the ‘central’ charged one si, characterised by either not having any long-range electrical interaction with the cell si or being beyond the reach of any short-distance forces from the referred cell to act upon. For a graphical interpretation of this kind of neighbourhood, Fig. 14.6(b) is presented. A general solution of the Besag proposition will be sought in the next section, where the notion of a clan will be instrumental in accomplishing the


0=1

485

0=2

0=4

(a)

0=5

(b)

14.6 Neighbourhoods of the order o = 1, 2, 4, and 5 (a) are sketched, together with a Markov random field that contains a combination of short- and long-range interactions (b). Distances in regular lattices are measured, as a rule, in lattice units, l.u. One lattice unit is the distance between the nearest neighbouring cells.

present target. Clans are uniquely determined by a system of neighbourhoods N on a set S of all cells. According to Paget (1999), a ‘clan’ in a given a neighbourhood system N is a set ‘c’ of cells, where c ⊂ S. Every couple of distinct cells in the set c are mutual neighbours. That is, for each couple of cells si and sj belonging to the set c, i.e. si, sj ∈ c such that, si ≠ sj, the relation si ∈ Nj implies sj ∈ Ni. As opposed to the definition of a neighbourhood, where the cell at reference si is excluded from its own neighbourhood, the definition of a clan says that a cell si creates its own clan, known as one-clan. Henceforth, the set of all possible clans in S with respect to the system of neighbourhoods N will be defined by C, while Ci will refer to the local set of the clans of a cell si, i.e. it will include all the clans where a cell si participates. The number of clans grows almost exponentially with an increase in the order o of neighbourhoods in regular lattices of cells, as has been shown by

486


Paget (1999). In other papers and works, clans are called ‘cliques’ (Paget, 1999; Besag, 1974; Geman and Geman, 1984), but here will be used the notion of Moussoris (1973), who argued that the term clique is generally used in the graph theory for maximal complete sub-graphs, whereas clans are not only complete sub-graphs but, at times, are single sites and even empty sets, too. The count of cells in a clan will be introduced here as ν, to denote a zero-clan – an empty clan; one-clan – a clan containing one cell only; two-clan – a clan containing two cells; and a ν-clan for a clan consisting of ν cells. Being equipped with all the essential prerequisites, it is now time to seek the general solution of the Besag proposition. It will be realised in the next subsection with the endeavour perceived through the attainment of the Hammersley and Clifford theorem.

14.2.3 Hammersley and Clifford theorem The Besag proposition [14.13], derived earlier in Section 14.2.1, has the striking feature of being independent of labelling of individual cells. This has rendered the flexibility to jumble up wildly the indices of the cells with the variable values assigned to them unaltered and retain the original form of the Besag proposition. Arbitrary labelling of cells evidently justifies the existence of severe restriction on the functional form of joint probabilities, as was asserted by Besag (1974). It is here that the functional form is looked for. The functional form is based on an expansion of functions having arguments comprising random variable values, assigned to the cells of ν-clans, where ν = 1, 2, 3,…, n. To realise this idea, let a Markov random field, characterised by the sole existence of one-clans, be conceived. In the present enterprise, an ideal gas-like system with various particles (or cells) will be dealt with. These particles are, in the parlance of physics, considered not to interact mutually. The probability of the appearance of a particle in a certain part of a space, then, depends merely on external fields, for instance, gravitation. The existence of exclusively one-clans in the system under consideration, i.e. in the Markov random field, considerably simplifies the conditional part of local conditional probabilities appearing in the Besag proposition. For one-clans in Markov random fields, the term p(xi|x1, x2,…, xi–1, xi+1,…, xn) reduces to p(xi|Ø) = p(xi), where Ø denotes an empty set of variables. The Besag proposition for one-clan systems is hence simplified to r p( x i ) P( x ) r = Π [14.17] 1 ≤ i ≤ n p( yi ) P( y ) with a trivial solution r P( x ) = Π p( x i ) 1≤ i ≤ n

[14.18]


487

This result is in accord with the familiar statement about the probability of an event composed of mutually exclusive sub-events. Stimulated by the above trivial solution, an attempt to solve the Besag proposition for a Markov random field comprising only of two-clans will be made. The Besag proposition is a simple algebraic equation with complex r r arguments ascertained by combinations of configurations ( x , y ) from a joint r r configuration space, defined by x , y ∈Ω . Moreover, this proposition is valid for each labelling of cells. While exploring a general solution of the Besag proposition, attainment of a more physically realisable form of the arguments of the joint and the local conditional probabilities in Eq. [14.13] by a systematic approach, would be emphasised. Structurally, the Besag proposition is quite similar to that of the functional equations in which the arguments play a vital role also. Let a simple and well-known instance of an equation of the following functional form be considered: f(x) = f(x + a), 0 < a = const.

[14.19]

The above equation, for a particular function f, is satisfied only for certain discrete values of a in the argument, as the function f repeats its value with steps of a in the argument. Hence, the solution of Eq. [14.19] is a set of all the functions with a periodicity of a. One can anticipate a similar inference drawn from the prospective solution of the Besag proposition. In fact, as will be revealed, not only a sole concrete function but a family of functions accomplishing it will surface as a consequence. Similar qualitative information was incurred while solving the one-clan Markov field, Eq. [14.18]. At the onset of the forthcoming exercise to bring out a solution to the Besag proposition for a two-clan system with the backdrop of the Markov random field (used to describe a stochastic process involving spatial interactions), an intention towards designing graphical tools rather than the commonly encountered mathematical formalism, to simplify the procedural aspect of seeking the solution, is thought to be beneficial. The relevant model here is supposed to be composed of six cells with six bonds or interactions between the couples of neighbouring cells, i.e. for the neighbourhood order, o of 1. Cells are, solely for convenience, arranged on a fraction of a square lattice as shown in Fig. 14.7. The figure demonstrates the system of neighbourhoods N of this Markov random field, and the set C of all the two-clans with the same elements. The variable Xi, here, is connected to the number of entities, such as carrots, in the cell si. Two different joint r r configurations of systems having configuration vectors x and y will be dealt with. The elements of these vectors are not considered to be under any compulsion to be uniquely assigned to individual cells. Two such configurations are portrayed in Fig. 14.7. It will be appropriate for further r graphical interpretation of the Besag proposition to express the x configuration r with black cells and that of the y with white ones. It is now time to draw

488 1


3 (a)

4

5

6

1

2

1

2

3

2

3

2

3 (b)

5

6

4

5

4

4

5

5

6

5

6

5

6

The set of all neighbourhoods N .

1

2

2

3

2

3

(c) 5

6

The set C of all two-clans.

(d)

r y

r x

(e)

14.7 A fraction of a square lattice with six cells where bonds among cells represent mutual, i.e. symmetrical, cell interactions (a). The set N of all neighbourhoods in the Markov random field is depicted in (b). Neighbours of particular cells are depicted here as empty circles. The set C of all two-clans is sketched at in (c). The meaning of the plants random variable in a node si can be the number r of carrot r growing in a cell. Two various configurations x and y of a carrot bed that represents the Markov random field are shown in (d). The random field with the complete set of interactions/bonds is depicted in (e).


489

a graphical scheme for the representation and solution of the Besag proposition. The pertinent Markov random field, comprising solely two-clans, has a significant reduction of the number of cells remaining in the conditional part of the local conditional probabilities. Instead of the notation p(x1|x2, x3, x4, x5, x6), typical for random fields with all the possible intercellular bonds, as displayed in Fig. 14.7, the local conditional probabilities for particular configurations of the chosen Markov random field will be presented through notations that look like p(x1|x2), p(y2| y1, y3, x5) or p(x6|x3, x5). Thus, in this context, the part of the argument revealing the condition of a local conditional probability merely constitutes all the neighbours of the referred cell to which the local conditional probability applies. On the right-hand side of the Besag proposition [14.13] appear local conditional probabilities with ‘mixed’ conditions, contributed by local configurations from the configuration vector r y to an index i–1, then followed by local states from the configuration r vector x starting with an index i + 1. To exemplify, let p(x3|y1, y2, x4, x5, x6) for a particular cell with an index, i of 3, belonging to the random field, as shown in Fig. 14.7, be considered. The analogous argument for the local conditional probability is reduced to p(x3 | y2, x6) in the case of a Markov random field with two-clans and a system of neighbourhoods with order o of 1. In this chapter, the argument of these local conditional probabilities pertaining r r to configuration vectors x and y is symbolised by black and white cells, respectively. An instance of a particular distribution of interactions is illustrated in Fig. 14.8. In the same figure are sketched schemes for some local conditional probabilities, such as p(x3|y2, x6) p(x1|x2) from the numerator, and p(y5|y2, x4, x6) from the denominator of the Besag proposition. Subsequently, an effort is made to extend the aforementioned procedure of graphical interpretation of local conditional probabilities to depict the entire right-hand side of the Besag proposition. In Fig. 14.9, the multiplication operator is denoted by a little ‘ex’ and the dividing line is sketched as a long horizontal line separating elementary pictures derived from the numerator of the Besag proposition from those depicting representing terms from the denominator. The condition in the local conditional probability belonging to a cell si, defines all two1

4

2

3

1

2

3

1

2

3

5 6 p (x 3 / y 2 , x 6 )

4

5 6 p (y 5 / y 2 , y 4 , x 6 )

4

5 p (x 1 / x 2 )

6

14.8 Schemes for local conditional probabilities: p(x3|y2, x6) and p(x1|x2) are from the numerator of the Besag proposition, while p(y5|y2, y4, x6) belongs to its denominator.

490

Thermal and moisture transport in fibrous materials A

B

x

x

x

x

C

x

A

E F

D B x

C

x

E x

D

Graphical scheme for the Besag proposition

x

x F

(a)

Heterogeneous two-clans

Homogeneous two-clans (b)

14.9 A graphical scheme of the Besag proposition depicted using the particular Markov random field and its two configurations black and white (a). Heterogeneous and homogeneous two-clans are sketched in (b).

clans containing the cell si. The clans in Figs. [14.7], [14.8] and [14.9] are couples of neighbouring cells interconnected to each other. Two-clans may be classified into two groups. The first will be called here ‘homogeneous’ two-clans consisting either completely of variable values from the configuration r r vector x , i.e. the black cells or from that of the configuration vector y , i.e. the white cells. The last class of the two-clans consists of the ‘heterogeneous’ one, which is characterised by a combination of a black and a white cell. These heterogeneous two-clans are of prime interest. They are distinguished in Fig. 14.9 with capital letters and, as is it evident from the figure that they appear with the same frequency in the numerator and in the denominator, the heterogeneous terms share a high risk of getting wiped out. Consequently, with the prospect of having only the homogeneous terms left behind, the analysis of two-clan Markov Random field may be deemed to have been simplified. In fact, then, the homogeneous clans consisting of black cells and r r white cells will correspond to the joint probabilities P ( x ) and P ( y ) , respectively. Before moving on to the final stage of the present analysis, it is worth recalling, that Eq. [14.17] had expressions for joint probabilities of r r one-clan system as P ( x ) = Π p ( x i ) and P ( y ) = Π p ( y i ) , located in the 1≤ i ≤ n 1≤ i ≤ n numerator and denominator, respectively. To understand the two-clan affair, it should be said beforehand that homogeneous two-clan functions may be mathematically laid out as g(xi, xj), g(yi, yj). Incidentally, it is notable that such a convention, when applied to heterogeneous clans, reveals a symmetry between the functions g(yi, xj) and


491

g(xj, yi). Using these g-functions, each of the local conditional probabilities may be factorised, e.g. p(y5|y2, y4, y6) = g(y5, y2)g(y5, y4) g(y5, x6) and p(x6|y3, y5) = g(x6, y3)g(x6, y5). It is unnecessary to substitute the entire graphical scheme of Fig. 14.9 by such factors; such step would cause nothing but confusion. It is sufficient to have in one’s mind that each of the two-clans is a factor of the Besag proposition. Heterogeneous clans, being symmetric and appearing with the same frequency in the numerator and the denominator, cancel out altogether. After removal of all the heterogeneous clans, denoted by capital letters in Fig. 14.9, only homogeneous two-clan g-functions remain. Moreover, all those appearing in the numerator are black homogeneous clans and those appearing in the denominator are white ones. Accordingly, the ultimate form of the Besag proposition is delivered as: r g( xi , x j ) P( x ) r = Π [14.20] 1 ≤ i ≤ n g( y , y ) P( y ) i

j

where the product notation runs over all the pairs of cells interconnected with a bond, i.e. over all two-clans. Structurally, the terms in the numerator r associated with the configuration vector x are identical to that of the r denominator belonging to the configuration vector y in Equation [14.20]. Hence, it follows that the general solution of the Besag proposition for a r two-clan Markov random field can be constructed as P ( x ) = Π g ( x i , x j ) . 1≤ i ≤ n This result is, however, not a general one to represent instances where various v-clans ν ∈ {1, 2,…, z} are juxtaposed. The freedom in the choice of the gfunction allows one to write the solution, accompanied by an arbitrary multiplication factor F: r P( x ) = F Π g ( x i , x j ) [14.21] 1≤ i ≤ n

r Apparently, the normalisation condition rΣ P ( x ) = 1 implies that the gx ∈Ω functions in Eq. [14.21] have values different from those in Eq. [14.20], but they will not be denoted by any other notation. The physical illustration of a one-clan Markov random field is, for instance, given by the system of an ideal gas, as has already been mentioned. A pure two-clan system may be conceived as a system composed of elements, particles or groups of particles interacting in pairs, where the individual cells are not influenced by any external field. Let, now, the above-obtained solutions for a one-clan and a two-clan Markov random field on a regular lattice with a neighbourhood order, o, of 1 be merged. Having a Markov random field with one and two-clans, it is quite meaningful to see if the solutions [14.20] and [14.21], corresponding to a one-clan and a two-clan system, contribute in constructing a solution for a mixed system containing both types of clans. The factor F from Eq. [14.21] will be instrumental for this purpose. It is meaningful, too, to assume the solutions

492


of the systems concerning each of the clan types, derived on independent platforms, will be useful to solve the Besag proposition for a mixed system, when Π p ( x i ) from Eq. [14.18] is substituted for F in Eq. [14.21]. The | ≤i ≤ n coexistence of one- and two-clans in a mixed system does not interfere with arriving at the solution [14.21], i.e. one-clans do not affect factorisation of the local probabilities using two-clan g-functions. The only way to express the factor F is to use the one-clan probabilities p(xi) from Eq. [14.18] and to divide them by a constant value to fulfil the normalisation condition r rΣ P ( x ) = 1 . These functions will, subsequently, be denoted as one-clan g x ∈Ω functions. To differentiate between one- and two-clan g-functions, corresponding symbols will be indexed as g1 and g2. After substituting the factor F with multiple g1 functions, the long-awaited solution for the Besag proposition, applicable for systems with one and two-clans, is realised. r p ( x ) = F Π g 2 ( x i , x j ) = Π g1 ( x i ) Π g 2 ( x i , x j ) [14.22] 1≤ i ≤ n

1≤ i ≤ n

1≤ i ≤ n

Returning to the primitive idea of drawing the simplest picture of a complex situation, an instance of a Markov random field solely occupied by threeclans with an order of neighbourhoods, o, of 2, is now considered in the present context. It would be apposite now to highlight the fact that a threeclan system, even with fewer cells, is far richer than a pure two-clan system. Presently, a Markov random field comprising only four cells with two mutually independent cells s3 and s4, as is shown in Fig. 14.10, will serve the purpose. In the said field, a local conditional probability, p, is presented in terms of three-clan g functions. This has been demonstrated in the same figure with the local conditional probability of the cell s2 to establish p(x2|y1, x3, x4) = g3(y1, x2, x3) g3(y1, x2, x4). Functions of g3, such as g3(yi, xj, xk) and g3(xk, xj, yi), maintain a symmetry among themselves for all permutations of their arguments. Unlike Fig. 14.9, where two-clans are easily visible from one picture belonging to the local conditional probability, the g3 functions in Fig. 14.10 are individualised to show their joint correspondence with the local conditional probabilities. Concerning the heterogeneous clans, identical configurations with the same frequency appear in the numerator and the denominator, and are marked with the same upper-case letters in Fig. 14.10. Accordingly, the Besag proposition [14.13] for a purely three-clan system reduces to: r g3 ( x i , x j , x k ) P( x ) r = [14.23] Π P ( y ) 1≤ i < j < k ≤ n g 3 ( y i , y j , y k ) with the solution r P( x ) =

Π

1≤ i < j < k ≤ n

g3 ( x i , x j , x k )

[14.24]

Computer simulation of moisture transport in fibrous materials 1

493

2 (a)

3

4

1

2 3-clans (b)

3

4

1

2 A x

3

C

B

x

x

x

D x

4 (c) B

x

A

x

C

x

D

x

x

14.10 A graphical scheme of the Besag proposition for a purely three-clan system: The Markov random field (a). There exist only two three-clans in it (b). All local conditional probabilities of the type p(xi|yj, xk) are depicted in (c) The symbol × represents multiplication and the horizontal line the division of the right-hand side of the Besag proposition [14.23].

where, the nodes {si, sj, sk} create clans. The solution of the Besag proposition for a system composed of a mixture of one, two, and three-clans will be brought about in a way dissimilar to the one previously used for the construction of Equation [14.22]. A Markov random field with a mixture of one-, twoand three-clans has local conditional probabilities typically represented as p(x2|x1), p(x2|x1, x3), and p(x2|x1, x3, x4), i.e. conversely, the associated local conditional probabilities can be factorised with the help of those clans, as is shown in Fig. 14.10. Such a factorisation, when carried out in the node s2, attributed with a random variable x2 and a local set of clans C2, acquires a form of p(x2|y1, x3) = g1(x2)g2(x2, y1)g2(x2, x3)g3(y1, x2, x3)g3(y1, x2, x4). Complete nullification of all the heterogeneous two-and three-clans causes the Besag proposition to be expressed as: r P ( x ) = Π g1 ( x i ) Π g 2 ( x i , x j k ) Π g3 ( x i , x j , x k ) 1≤ i ≤ n

1≤ i < j ≤ n

1≤ i < j < k ≤ n

[14.25] The above multiplicative expansion may be further extended to any arbitrary count ν of nodes of a clan in a Markov random field, as will be encountered in Eq. [14.26].

494


Now let the formulation of the Hammersley and Clifford theorem be initiated. This theorem is indispensable to the construction of valid spatial schemes through conditional probability. As the original methodology of its proof was rather circuitous and necessitated development of an operational calculus, the ‘blackening algebra’, the theorem was never published by its authors, Hammersley and Clifford (1971). An alternative approach was brought about by Besag (1974) and the formulation of Hammersley and Clifford theorem could be found in the work of Paget (1999). A great deal of work, preparatory to the present study, regarding the formulation of this theorem, was carried out to achieve the general solution of the Besag proposition. As a result, the first part of the Hammersley and Clifford theorem, as stated below, will be fully apprehensible. (i) Consider a random field or a Markov random field with a given neighbourhood of each cell that predetermines the set C of all its clans. r Then there exists an observation that any joint probability P ( x ) of the system can be written down as the multiplicative expansion [14.26], subject to the ‘positivity condition’. r P ( x ) = Π g1 ( x i ) Π g 2 ( x i , x j ) Π g3 ( x i , x j , x k ) 1≤ i ≤ n

×…

1≤ i < j ≤ n

Π

1≤ i< j U ( y )

= U(x1,…, yi,…, xj,…, xn) [14.45] r r r r The terms U ( x ), U ( y ) are Hamiltonians of the configurations x and y . Glauber dynamics were originally designed for a two-state Ising model. Kawasaki dynamics. Kawasaki dynamics was initially proposed for a nearestneighbour exchange procedure (Kawasaki, 1972), as shown in Fig. 14.14(b). Two neighbouring cells, si and sj, which are allowed to commute their variable values, are chosen at random. With an original local configurations having variable values xi and yj and a subsequent trial carried out for destined values of yi and xj, the transition probability is typified as W(xi ↔ yi), where r r  U( y ) – U ( x )  W ( x i ↔ y j ) = exp  –  τ  


505

(a)

r x k , step k

r x k +1, step k+1

r x k , step k

r x k+1, step k +1

r x k , step k

r x k +1, step k+1

(b)

(c)

14.14 Glauber’s dynamics (a) and Kawasaki’s dynamics for short (b) and long (c) distance. All neighbourhoods are considered here of the order o = 1.

where

r r U ( x ) = U(x1,…, xi,…, yj,…, xn) ≤ U ( y )

= U(x1,…, yi,…, xj,…, xn) W(xi ↔ yj) = 1 r r where U ( x ) = U(x1,…, xi,…, yj,…, xn) > U ( y ) = U(x1,…, yi,…, xj,…, xn) [14.46] r r The terms U ( x ), U ( y ) are, respectively, the original and the final total energies/ r r Hamiltonians for the global configurations x and y . Kawasaki dynamics have a version for long-distance exchange between a chosen pairs of cells. In that case si and sj are not necessarily mutual neighbours as shown in Fig. 14.14.

506


14.2.6 Markov random fields with constraints Positivity conditions postulated for deriving the Besag proposition and, consequently, for the Hammersley and Clifford theorem, seem awkward for some assemblies of statistical physics normally obeying conservation laws, such as canonical or micro-canonical assemblies, when some of the r configurations, x , of a system violate conservation laws to produce relevant r joint probabilities, P ( x ) , and local conditional probabilities, p(xi| x1,…, xi–1, xi+1,…, xn), with values equalling zero. It is also well established both from theory and experiments that the choice of a statistical assembly does not affect the additive structure of the corresponding Hamiltonians resembling the Hammersley and Clifford expansion [14.27]. Hence, there exists a possibility to preserve the newly introduced theory for physical systems, where violation of conservation laws works against the positivity condition. As far as it can be traced back, pioneering work dealing with Markov random fields having non-zero joint probabilities assigned only to some subset Ω+ of a configuration space Ω, were carried out by Moussouris (1973). Configurations in a configuration space Ω outside the region Ω+, i.e. in the region Ω – Ω+, were forbidden and so they were assigned zero probabilities. In those systems, Markov conditions, i.e. the independence of local probabilities r in some areas of cells, still made sense but the positivity condition P ( x ) ≠ 0 r for each configuration x ∈ Ω was no longer conformed to. Moussouris suggested the treatment of systems with such constraints using a finite value of a continuous limit of an energy function for forbidden configurations r r U ( x ) → ∞ for x ∈ Ω – Ω+ [14.47] Clan configurations that were energetically forbidden were called ‘barriers’ and all the configurations with zero probabilities created a ‘barrier system’. More information about these considerations can be obtained from Moussouris’s paper (1973). The approach followed here will be based on the structures of the Besag proposition and the Hammersley and Clifford theorem. Initially it is worth pointing out that the Hammersley and Clifford expansion belonging to a r joint probability P ( x ) may also be expressed as an expansion of a certain chosen set of non-zero individual homogeneous clan functions, g. For this reason, the Hammersley and Clifford theorem is a statement about particular configurations and so works, even when configurations are chosen from Ω+ without accounting for the whole configuration space Ω. These findings elevate the suitability of the described theory in analysing the Markov random fields also with constraints. Nevertheless, the Hammersley and Clifford theorem was deduced from r the Besag proposition, based on a metamorphosis of configurations, from x r to y , effected by change of some variable values at individual cells in


507

accordance with a chosen process to label the cells. It may be noted that the Besag proposition [14.3] was derived in a similar fashion from the Bayes’ rule [14.7]. Some of the local conditional probabilities p(yi|y1,…, yi–1, xi+1,…, xn), appearing in the Besag proposition may belong to forbidden configurations r x and, therefore, violate conservation laws. It may be overambitious to safeguard the Besag proposition for each of the random fields, but may not be so for Markov random fields with a large number of total cells much greater than the count of the cells in its particular neighbourhoods. The energy content along with the spectrum of clans regarding random variable values outside the chosen neighbourhood and within Ω+ will be supposed to be large as well as rich enough to buffer during manipulations of the local configurations, i.e. the local conditional probabilities p(xi|y1,…, yi–1, xi+1,…, xn) in the Besag proposition, wherein {s1,…, si–1, si+1,…, sn} ∈ Ni. Briefly speaking, a conservation of the total energy or the total particle number in a system of canonic or micro-canonic assemblies is not susceptible to manipulations with random variable values in a cell and its neighbourhood because beyond the separator the system configuration is rich enough to maintain the conservation laws. To be more specific, for each such local change, depending only on the configuration of the neighbourhood Ni of a cell si, there exists a configuration of cells outside Ni fulfilling the conservation laws of the whole system. The situation is sketched in Fig. 14.15. In other words, outside the neighbourhood Ni and among configurations of Ω+, the joint configuration can be tuned up to meet the conservation laws.

j

k l

i

Ni m

14.15 A local conditional probability p (xi |yj, xk, xl, ym) of a random node xi with the neighbourhood Ni = {sj, sk, sl, sm} does not depend on random variable values behind the separator. The separator is represented by the neighborhood Ni. Changes of random variable values in the cell si and its neighbourhood can be carried out if they belong to Ω+.

508


For canonical assemblies having the total number of particles conserved, e.g. the total number of cells with a particular value of a random variable conserved, a more rigorous process may be undertaken. The dynamics of these systems are generally governed by Kawasaki dynamics. The modified Bayes’ rule written for random variable values exchanged between two chosen cells si and sj, takes the form: P(x1,…, xi,…, xj,…, xn) = p(xi, xj|x1,…, xi–1, xi+1,…, xj–1, xj+1,…, x1) × Π (x1,…, xi–1, xi+1,…, xj–1, xj+1,…, x1)

[14.48]

This is a modified form of the relation [14.7]. The local conditional probability of the appearance of random variable values xi and xj in corresponding cells si and sj is denoted above as p(xi, xj|x1,…, xi–1, xi+1,…, xj–1, xj+1,…, x1). Permitted by the local configuration spaces Λi and Λj, an exchange of variable values xi ↔ xj takes place to arrive at a final configuration (x1,…, xi–1, yi, xi+1,…, xj–1, yj xj+1,…, x1) from Ω+. Following the exchange between the cells si and sj , the new corresponding random variables are denoted here as r yi and yj, respectively. The original configuration x starts to get transformed r to the one comprised solely of y when the variables yi and yj gradually substitute the xj and xi, in the given order. The analogue of Equation [14.7], representing the modified Bayes’ rule, just after the first exchange may be organised as: P(x1,…, xi–1, yi, xi+1,…, xj–1, yj, xj+1,…, xn) = p(yi, yj|x1,…, xi–1, xi+1,…, xj–1, xj+1,…, x1) × Π (x1,…, xi–1, xi+1,…, xj–1, xj+1,…, x1)

[14.49]

where yi = xj and yj = x. Marginal probabilities, Π, in both Equations [14.48] and [14.49], are identical. So, Eq. [14.48] and [14.49] may be used in a way very similar to that employed in Eq. [14.9] to get: P ( x1 , x 2 ,…, x n ) =

p ( x i , x j | x1 ,…, x i –1 , x i+1 ,…, x j –1 , x j +1 ,…, x n ) p ( y i , y j | x1 ,…, x i –1 , x i+1 ,…, x j –1 , x j +1 ,…, x n )

× P(x1, x2,…, xi–1, yi, xi+1,…, xj–1, yj, xj+1,…, xn)

[14.50]

One may repeat the previous step for another pair of cells sk and sl to obtain an equation analogous to Eq. [14.11].

Computer simulation of moisture transport in fibrous materials P ( x 1 , x 2 ,…, x n ) = ×

509

p ( x i , x j | x 1 ,…, x i –1 , x i+1 ,…, x j –1 , x j +1 ,…, x n ) × p ( y i , y j | x 1 ,…, x i –1 , x i+1 ,…, x j –1 , x j +1 ,…, x n )

p ( x k , x l | x 1 , x 2 ,…, x i –1 , y i , x i+1 ,…, x k –1 , x k +1 ,…, x j –1 , x j , y j +1 ,…, x l –1 , x l +1 ,…, x n ) × p ( y k , y l | x 1 , x 2 ,…, x i –1 , y i , x i+1 ,…, x k –1 , x k +1 ,…, x j –1 , y j , x j +1 ,…, x l –1 , x l +1 ,…, x n )

× P(x1, x2,…, xi–1, yj, xi+1,…, xk–1, yi, xk+1,…, xj–1, yj, xj+1,…, xl–1, yl, xl+1,…, xn)

[14.51] By continuing the process still further, a modified Besag proposition is framed r ∗ p ( x i , x j | condition ) P( x ) r = Π [14.52] P( y ) 1≤ i ≤ n p ( y i , y j | condition ) ∗ r r where x and y naturally belong to Ω+. The symbol Π , standing for the product of all the fractional terms, runs only once over each pair of cells to r ensure a transition from an original configuration x to a final configuration r y . A graphical representation of the modified Besag proposition for the chosen Markov random field is sketched in Fig. 14.16. It is shown that the heterogeneous two-clans cancel out from the right-hand side of the modified Besag proposition. Therefore, similar processes of factorisation with g functions and successive implementation of q functions to write in terms of the Hammersley and Clifford expansion, based on a change of variable values of a chosen pairs of cells, are made possible. Each of the exchange procedures fulfil the law of particle conservation and so a changeover from an original configuration to the next one obeys the constraints of a canonical assembly, r i.e. using a sequence of configurations x from Ω+. The procedure to factorise conditional probabilities p(xi, xj | condition) with one-, two-, three- and other clans is the same as that for the local conditional probabilities p(xi, | condition). The next section is devoted to applications of the Markov random field theory and the auto-models for computer simulation of the wetting of a fibrous mass.

14.3

Computer simulation

This section is divided into three parts. A computer simulation of the wetting phenomenon of individual fibres, describing Rayleigh instability of liquid jets and liquid films on a highly curved fibrous surface, is followed by a subsection devoted to the simulation of liquid columns in parallel fibrous bundles. The final subsection covers the simulation of liquid imbibition in a fibrous mass. The first two simulations study equilibrium or near equilibrium states of liquid bodies, while the final one concerns multiphase flows in fibrous materials.

510


1

2

3

1

2

3 (a)

4

5 r x

6

4

5 r y

6

(b)

B ×

×

A

C (c)

B A

D

×

D

C

×

14.16 Graphical expression of the Besag proposition (c) for the Markov random field (a) composed purely of two-clans. Local conditional probabilities belong to pairs of cells that exchange their positions obeying Kawasaki dynamics (b).

Capillary phenomena associated with the presence of a liquid in a fibrous media, where the fibres are nearly cylindrical, are of great importance in a number of fields and products, such as wetting and wicking in textiles and paper. Out of these products, absorbents form a major category including baby diapers, training pants, sanitary napkins, tampons and incontinence products (White, 2001). The interaction of liquids, polymer melts and solutions with fibrous systems of highly curved surfaces is critical in numerous industrial manufacturing processes, such as fibre formation by various spinning technologies, dyeing and finishing of fabrics, protective coating of wires, and non-woven fabrications. These interactions govern performance of the final products including baby diapers, surgical wipes, feminine hygiene products, hair cosmetics, and paper printings. Given such wide and common applications, it is puzzling to learn that the theoretical interpretation of the wetting of fibrous system is still in its early stage of development (Neimark, 1999).


511

14.3.1 Computer simulation of the Rayleigh instability This section will attend to two physical processes, viz. the breaking down of a continuous liquid film covering a fibre and the detachment of a fragmented liquid film from a liquid jet. Individual interactions of liquid with flat and cylindrical surfaces are dissimilar. When a fibre is covered with a thin, macroscopic liquid film, the film spontaneously breaks down into small droplets. Excepting extremely thin liquid films stabilised by intermolecular interactions like the van der Waals forces, the droplets have roughly uniform spacing along the fibre. The formation of such a droplet chain from an initially continuous liquid film occurs even in the case of a zero contact angle that favours a complete wetting of a surface, especially when it is plane (Roe, 1957). In other words, the wetting behaviour of fibres practically displays an instability or breakdown of liquid columns, crucial for all spinning technologies. This was first observed by Plateau (1873) and theoretically explained later by Rayleigh (1878), giving use to the term ‘Rayleigh instability’. In about 1870, Plateau experimented with oil-drops in water, mixed with alcohol, to form cylinders and found the instability to occur when the cylinder length, expressed as a wavelength λ, was between 1.99πro and 2.02πro, where ro is the radius of a cylindrical fibre with a liquid film. Several other theoretical as well as experimental studies on the same topic are traceable. Papers by deGennes et al. (2003) and Roe (1957) present a similar condition, λ >2πro, for cylindrical liquid bodies. Here, two of such studies will be involved, viz. the ones by Tomotika (1935) and by Meister (1967). The Rayleigh instability of long liquid jets is a consequence of a temporal development and intensification of original tiny perturbations, also known as capillary waves (Adamson and Gast, 1997; deGennes, 2003). The exact value of the Rayleigh wavelength is obtained by using Navier–Stokes equations. The Rayleigh wavelength λ has a value higher than the limiting value given in the aforementioned work of Plateau and according to the result obtained by Rayleigh (1878) has a value of 2.88πro. Studies of de Gennes (2003) may be consulted for more details. The well-known auto-model employed here to investigate the Rayleigh instability is composed of a three-dimensional simulation box of cuboid structure having n cells. The number of cells n is determined by the product of a chosen numbers of cells along the three axes x, y, and z, written as, nx ∗ ny ∗ nz. The present simulation is connected to a particular value of n equalling 600 × 20 × 20 = 24 × 104 cells. In a cell si the random variable Xi is assigned the variable value xi, equivalent to 0 for a gas, 1 for a liquid and r 2 for a solid filling it. In the present context, function U ( x ) is interpreted as the system Hamiltonian, i.e. the total system energy. This Hamiltonian is of the previously stated relation [14.40].

512


r U ( x ) = Σ C( xi , x j )

[14.53]

1≤ i ≤ n

where C(xi, xj) represents the interaction energies between the neighbouring cells si and sj, having corresponding variable values xi and xj. τ is the statistical temperature, assigned a value of 50 e.u., where e.u. is the energy unit of the model. The values of C are provided in Table 14.1. One-clans do not contribute to energetic interactions because influences of the external gravitational field, the electrical field and other fields are presently neglected. Thus, the Hamiltonian [14.53] misses the first term of gravity of the auto-model, meant for describing the wetting phenomena. The freshly introduced Hamiltonian has a form similar to that of Eq. [14.40], mentioned earlier. For the current purpose, free boundary conditions (Binder, 1997) are engaged. The neighbourhood of a cell, depicted in Fig. 14.17, is composed of 26 neighbouring cells that form a supercube of 27 cells, including the one at reference. So, the order o of this neighbourhood in a three-dimensional cuboid lattice is 2, according to the notation introduced in the Section 14.2.2 Table 14.1 Interaction energy values used for the simulation of the Rayleigh instability. Interaction energies are symmetric functions if their two variables C(xi, xj) = C(xj, xi) and represent the energies belonging to the mutual interaction of neighbouring nodes i and j where random variables Xi and Xj have values xi and xj Interaction energies in e.u. Rayleigh instability

Gas, Xi = 0

Liquid, Xi = 1

Fibre, Xi = 2

Gas, Xj = 0 Liquid, Xj = 1 Fibre, Xj = 2

C(0, 0) = –40 C(1, 0) = –10 C(2, 0) = 20

C(0, 1) = –10 C(1, 1) = –26 C(2, 1) = –10

C(0, 2) = 20 C(1, 2) = –10 C(2, 2) = 0

i Gas

Gas

Gas

Gas

Liquid

Liquid

Fibre

Fibre

14.17 A cell i in the centre to form a supercube with its neighbouring cells. On the front surface can be seen various kinds of media that occupy the cells: the white colour denotes air, the grey color denotes liquid, and fibre cells are black.


513

and in Paget’s paper (1999). Regarding the initial configurations of cylindrical liquid films and jets, all liquid and fibre cells are arranged in chains parallel to the x-axis so that their cross-sections perpendicular to the x-axis are nearly circular. Figure 14.18 may be consulted for an illustration of this. For simulations considering a liquid coating on a fibre, the central chain of cells represents a solid fibre with all xi’s of 2, and surrounded by chains of liquid cells with xj’s as 1. The cross-section of a studied system is also shown in Fig. 14.18. The simulation in this chapter uses Monte Carlo dynamics, based on Gibbs–Geman sampling. The Markov chain is based on Kawasaki dynamics for short distances, described earlier in Section 14.2.5, where the proposed cells involved in an exchange are among the neighbours. In this method, a pair of cells is chosen at random using a random number generator. The r0=4.259 l.u.

r0=3.949 l.u. r0=3.613 l.u. r0=3.430 l.u. r0=3.040 l.u. r0=2.821 l.u. r0=2.585 l.u. r0=2.034 l.u. r0=1.693 l.u. r0=0.5641 l.u.

S=57 l.u.2

S=49 l.u.2

S=45 l.u.2

S=37 l.u.2

S=29 l.u.2

S=25 l.u.2

S=21 l.u.2

S=13 l.u.2

S=9 l.u.2

S=5 l.u.2

(a)

5 l.u.

20 l.u.

5 l.u.

20 l.u .

600 l.u. (b)

14.18 Initial configuration of a liquid film on a fibre.

514


random variables in a chosen pair of cells then exchange their positions to form a new configuration. The Hamiltonian values are calculated for the configurations both before and after the exchange. A configurational change r r from an original x to a new one y is accounted if it satisfies one of two conditions. One of the conditions, (A), is fulfilled, when the total energy for r r the new configuration U ( y ) is lower than that of the original one U ( x ) . In r r that case the exchange probability W ( x → y) is equal to one. If it is otherwise, r r i.e. U ( y ) > U ( x ) , the exchange is carried out only under condition (B), which r r postulates that claims the transition probability W ( x → y) allows such an r r r r exchange when W ( x → y) = exp[– (U ( y ) – U ( x ))/ τ ] . These values of transition probabilities are in accordance with [14.46] in Section 14.2.5. This probabilistic transition is realised by the generation of a random number z inside the uniform interval 〈0, 1〉 and the acceptance of a new system r r configuration meets the condition z < W ( x → y) . More details about the model can be found in some recent work (Binder, 1997; Lukas et al., 2004). After creation of an initial joint system configuration, the simulation algorithm iterates the following steps to create the Markov chain of successive configurations of the system: (i) Selecting at random two lattice cells si and sj at which the random variables xi and xj are ascribed for the exchange xi → xj and xj → xi. The cell si contains liquid, i.e. xi has a value of 1, and sj is filled by a gas. Moreover, as both the cells si and sj are to be located in the interface between liquid and gas, the cell at the liquid interface is defined as the one whose neighbourhood of the order o is 2, consisting of at least one gas cell. Correspondingly, for the gaseous interface, the cell has at least one liquid cell in its neighbourhood. This type of random variable exchange between distant cells si and sj is based on spin exchange of long-range Kawasaki kinetics (Kawasaki, 1972). (ii) Computing the total energy change, ∆U, according to a chosen Hamiltonian associated with the exchange of the random variables. The values of the mass of the liquid cell, m, of unity, and the acceleration due to gravity, g, equalling 10 are considered here. r r (iii) Calculating the transition probability W ( x → y) according to Equation [14.45]. In the present simulations, the value of the statistical temperature, τ, i.e. the product of the Boltzmann constant with the absolute temperature, is preset to 50 e.u. (iv) Realising the probabilistic nature of the spin exchange by drawing a random number z, that is uniformly distributed in the interval 〈0, 1〉. r r (v) Exchanging the spin cells, when z < W ( x → y) . (iv) The spin configuration obtained at the end of step (v) is counted as a new configuration after which the next iteration starts from step (i).


515

After an adequate number of repetitions, the registered set of simulated configurations is analysed both numerically and graphically to yield the shape of the liquid body as well as the sum of the total energy changes in the system. The chain of succeeding configurations is broken as soon as the liquid film or jet disintegrates to independent droplets. The unit of time used in Monte Carlo dynamics is the Monte Carlo step per site (MCSPS), which corresponds, approximately, to one exchange trial for each cell in the simulation box. In the present case, one MCSPS is equivalent to 24 × 104 trials of cell exchange, since the simulation box of the dimensions 600 × 20 × 20 [l.u.3] has 24 × 104 cells. One of the chosen original configurations of a liquid film coating a fibre is exhibited in Fig. 14.18. The system dynamics are nearly perpetual because the neighbouring droplets, detached from the original liquid film, communicate ceaselessly with others, resulting in exchange of liquid particles either via evaporation and condensation or via liquid films stabilised between droplets on a fibre. Accordingly, in spite of being a relatively sluggish process, substantial physical significance of the temporal system development draws one’s attention in an attempt to observe the complete breakdown of the original cylindrical liquid bodies. Such an accomplishment is done through a series of graphs obtained at equal intervals on the MCSPS time scale. These graphs are obtained by plotting the count of liquid cells in the cross-sections perpendicular to the x-axis as a function of the corresponding x-axis coordinates. The liquid cells are counted at individual cross-sections, perpendicular to the xaxis, for every point on the x-axis. One such instance is given in Fig. 14.19. Pertaining to the present simulation, time-dependent developments of droplet count both for a film covering a fibre surface and for a jet having a crosssection the same as that given in Fig. 14.20, are displayed in Fig. 14.21. It is quite discernible from the figure that the number of detached droplets experiences a temporal change propelled by minimisation of the total system energy. The system is dissipative and the released energy flows from it to a reservoir. Once the liquid jet is fragmented into droplets, the droplet number decreases as fusions of the neighbouring droplets take place. The timedependent development of the droplet number, averaged from five computer simulations, is recorded in Fig. 14.20. It may be seen that the temporal

5 l.u. MCSPS 20 000

X 16 Droplet no.

14.19 Liquid cell count inside individual cross-sections perpendicular to the x-axis.

516

Thermal and moisture transport in fibrous materials n

35

30

25

20

15

10 0

5000

10000 MCSPS

15000

20000

15000

20000

In(n) 3.5 In(n) = –0.000039 MCSPS + 3.364 R = 0.985

3.4 3.3 3.2 3.1 3 2.9 2.8 2.7 2.6 2.5 0

5000

10000 MCSPS

14.20 Temporal development of the droplet number is the consequence of a subsequent droplet fusing.

development of the droplet number, no, is well described by a logarithmic dependence ln (no) = A × MCSPS + B, with a correlation coefficient, R, as high as 0.97. For comparing the simulated results with the λ - ro scaling for the Rayleigh instability, i.e. λ = 2.88πro, pertinent measurements from the computergenerated outputs are taken at the primary stages of the process of droplet detachment. The radii of the liquid film and of the liquid jet, both expressed as ro, are varied. The original cross-sectional configurations of the films and the jets are shown in Fig. 14.18(a). The equivalent radii, ro, of the circular


517

Droplet no. MCSPS 20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 0

100

200

300 x (l.u.)

400

500

16 16 16 16 16 17 19 22 25 30 600

MCSPS 110 100 90 80 70 60 50 40 30 20 10 0

100

200

300 x (l.u.)

400

500

600

14.21 The perpetual dynamics of the system are caused by the coalescence of neighbouring droplets.

cross-sections are calculated from the equivalent cross-sectional areas, S, of the discrete models using the relation ro = S / π . The simulated systems maintain areas, S, in terms of the number of cells measured in (l.u.)2, where l.u. refers to a lattice unit, given by the distance between the nearest cells in the cubic lattice of the investigated model, as has been already mentioned in Section 14.2.2. One l.u. is also the natural length scale of the discrete lattice model under consideration. Cross-sectional areas S of the models and corresponding radii of the cylinders are also shown in Fig. 14.18(a). Right from the primal phases of detachment of the liquid film or the jet, the number of droplets, formed no, is recorded. To estimate the Rayleigh wavelength, the initial length of the film or jet (590 l.u.) is divided by no. The wavelength obtained from the computer simulation is denoted as λsim. After averaging the set of values of the wavelength received from several successive executions of the computer program, it is compared with the analytical expression of λ as 2.88πro. Each of the computer simulations is repeated five times for a chosen set of input data. Simulation results associated with liquid films and jets for various values of ro are portrayed in Fig. 14.22. The slopes of both the curves are nearly the same.

518


λ [l.u.] 40 35 30 25 20 15 10 5 0 0

0.5

1

1.5

2 r0 [l.u.]

2.5

3

3.5

4

λ [l.u.] 40 35 30 25 20 15 10 5 0 0

1

2

3

4

5

r0 [l.u.]

14.22 The dependence of the Rayleigh wavelength λ on the original radius of liquid films (a) and liquid jets (b).

Thus, it is established that the Monte Carlo method applied to an automodel to examine the instability of liquid films and jets provides simulation outputs very similar to those of classic theoretical analysis. It may be anticipated that this kind of simulation will also imitate the behaviour of the soft mass in experimental arrangements under more complex conditions. It is highly probable that the auto-models are effective in describing liquid film and liquid jet instability even when the system loses its axial symmetry, i.e. for bent fibres. It may also be applicable to incorporate aspects of external fields, roughness and chemical inhomogeneity of fibre surfaces, mixtures of liquids, electrically charged liquids and polymer solutions, and so on. This


519

approach is not appropriate, however, for phenomena affected predominantly by momentum or viscosity. A simple and straightforward approach to using an auto-model for accounting for the wetting of more complex fibrous structures will be shown in the next subsection. Interactions of liquids with fibre bundles will be taken up first.

14.3.2 Computer simulation of liquid columns in fibre bundles Through a series of papers presented at the end of the 1960s, Princen (1969) considered a capillary rise between two vertical cylinders, and, subsequently, between three-, four-, and multi-cylinder systems (Princen, 1969a). The last paper in the series (Princen, 1970) was devoted to a study of equilibrium configurations of limited amounts of liquid in horizontal cylindrical assemblies, where gravitational effects were neglected. All of Princen’s works were focused on contact angles ranging between 0° and 180°. Princen claimed in one of his works (Princen, 1970) that a liquid column between two closely separated horizontal cylinders ceased to maintain a uniform cross-section as soon as the distance between the cylinders increased to reach a certain critical value, and the liquid acquired a shape similar to that of an unduloid droplet on a single cylinder caused by Rayleigh instability, as described in the previous subsection. The abrupt change in the liquid body shape was mentioned as a morphological transition. The same literature (Princen, 1970) also maintained that liquid columns within a higher number of equidistantly placed cylinders exhibited other unusual features. The systems were then found to undergo morphological transitions, marked by two different changes in shape of the liquid phase, depending on the inter-cylinder separation. For a sufficiently small inter-cylinder distance, the liquid bridged only each pair of adjacent cylinders, with the central channel left open. Three separate columns were observed in the case concerned. Experiments and computer simulation for such morphological changes were carried out by Lukas and Chaloupek (2003a). When the distance was greater than a certain critical value, the liquid filled the inter-cylinder channel completely. The shape of the liquid body was termed the channel-filling column. A further increment of the distance between the cylinders carried the system to a second critical region, characterised by a transition of the channel-filling column to a state akin to the unduloid on a single cylinder. All these phenomena could be investigated using automodels. The cylinder system considered by Princen was simply a general geometrical model of fibrous bundles and fibrous materials. The venture to apply an auto-model to generate column morphological transitions, predicted previously by Princen for systems composed of pairs of parallel cylinders, will now be considered. The main focus will be on long liquid column–unduloid transitions

520


in bundles composed of two cylindrical fibres. Before the computer simulations are dealt with, an analytical description of the problem will be given. Let a limited amount of a liquid, denoted as L, coexistent with another fluid phase represented for instance by a gas/vapour, say V, be considered. Air may be envisaged to constitute phase V. The liquid phase L is supposed to be located within the gap between two narrowly spaced cylinders S. Real wetting structures with dimensions in the order of microns will be involved, so as to neglect the line tension and the gravitational effects. It may be assumed that the liquid phase L forms a short bridge, the surface of which possesses a constant mean curvature 1/R, as is sketched in Fig. 14.23. When the bridge contains a sufficient amount of the phase L, a long column with a constant cross-section, excluding the terminal menisci, will be constructed. For convenience, the notations of various quantities used here are kept similar to those used by Princen (1970). The distance between the solid cylinders, having individual radii of r, is 2d. Phases L (a liquid), V (a vapour) and S (a solid fibre) are characterised by a liquid contact angle θ. Referring to the cross-sectional geometry in Fig. 14.24, the angle between the line connecting the cylinder centres and the radial distance to the L – V – S line is α, shown as the covering angle. The radius R represents the curvature of

14.23 A liquid phase forms a short bridge between two cylindrical fibres. T

π / 2– Θ– α

R A

B x

r S

V-phase, air

α +Θ

y

α

Θ

Solid 2d

Liquid

C

D

14.24 A cross-section of a long liquid column between two cylindrical fibres.


521

the liquid surface. Its value with respect to the Laplace pressure is considered positive if it has an outward concave surface; negative otherwise. The present objective is to find the threshold value of the dimensionless ratio of the aforementioned quantities, d/r, at which the transition occurs. The complexity of the situation may be reduced by considering the contact angle θ tending to zero. To be more rigorous, the value of spreading, i.e. the Harkinson parameter S = γSV – γLS – γ must approach zero, and so the transition to complete wetting is dealt with. Different shapes of cross-sections of the long liquid columns between the two cylinders/fibres are depicted in Fig. 14.25 for the case of θ having a value of 0°. The forces, at equilibrium, on the front edge of liquid column are FSV = FLS + FLV + Fp

[14.54]

where FSV is the force of solid–vapour interface, generated by a corresponding surface tension γSV. This force acts outward in the liquid body and tends to spread the liquid along the cylinders. The force FSV is also called the ‘line force’ as it is proportional to the length of the line along which γSV acts. A transitional shape of the liquid body between the conformations of long column and unduloid is the cylindrical column as sketched in Fig. 14.25. For a circular cross-section of a long liquid column having an effective liquid area, responsible for Laplace pressure, of A = π (R2 – 2 r 2), Equation [14.54] takes the following form: γ [14.55] 4γ SV r = 4γ SV r + 2γ R – ( R 2 – 2 r 2 ) R Using Young’s equation, γSV – γSL = γ cos θ and making the assumption that cos θ equals 1, a quadratic relationship for the dimensionless parameter,

α = 180° r

α = 135° 2r r 0

α = 90°

2r

α = 45°

α = 10°

r r 0 0

d/r=0.23243

d/r=0.03376

d/r=0.57083 d/r=1.02930

Θ=0°

d/r=1.41421

14.25 Various shapes of cross-sections of long liquid columns between two cylinders for spreading coefficient equal to zero, i.e. for contact angle just being equal to zero. The transitional shape between long columns and unduloids is the cylindrical long liquid column.

522


R/r, is obtained by multiplying Equation [14.55] by R . This is given below: r 2γ 2  R – 4 R + 2 = 0 [14.56] r  r The solution for R/r in the above Equation [14.56] is 2 + 2 , arrived at with an omission of the other root 2 – 2 to satisfy the relation between R and 2r as evident from Fig. 14.25. Particularly, the relation R = 2r + d holds true. So the threshold of the dimensionless parameter d/r for the appearance of the long cylindrical liquid columns has the value d/r = (R – 2r)/r = 2 = 1.41421. The morphologies of a liquid body that are generated through computer simulation, built on the auto-model and Kawasaki dynamics identical to those described in the previous subsection, are depicted in Fig. 14.26. The d/r = 0.5 1

2

3

4

5

6

7

8

d/r = 1.0 d/r = 1.5 d/r = 2.0 9

10 (a) 1 l.u.2

1 2

1

1 2 2 Original configuration

Subsequent configuration (b)

14.26 Long liquid column–unduloid transition. Computer simulation outputs carried out for zero contact angle θ and various values of dimensionless distance d /r between fibres (a) (courtesy of Jana Holubova, Technical University of Liberec). The original configuration and the subsequent one after the slab displacement are used for an estimation of surface tension values when the system temperature is low (b).


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Table 14.2 Interaction energy values C(xi, xj) used for the simulation of morphological transitions of liquid columns in a pair of parallel fibres. The choice of these values ensures the zero contact angle between the liquid and the solid fiber material since E0,2 ≥ E1,2 + E0,1, i.e. γ VS ≥ γ LS + γ , thanks to C(0, 2) ≥ C(1, 2) + C(0, 1) – C(1, 1) Interaction energies in e.u. morphological transitions of liquid columns

Gas, Xi = 0

Liquid, Xi = 1

Fibre, Xi = 2


C(0, 0) = –40 C(1, 0) = –20 C(2, 0) = –10

C(0, 1) = –20 C(1, 1) = –28 C(2, 1) = –20

C(0, 2) = –10 C(1, 2) = –20 C(2, 2) = 0

values of C from the Hamiltonian [14.53] are given in Table 14.2. These values are so chosen to ensure almost negligible contact angles. The calculus for choosing C’s follows. Imagine a prismatic simulation box partially filled up by two unlike Materials 1 and 2, with a flat interface in-between; let a new configuration be created simply by moving a slab of Material 2 to a space originally occupied by Material 1. The substitution is supposed to obey a conservation of the total number of cells for both the materials so that the original configuration with two material slabs has one interface while the latter, consisting of four slabs, has three interfaces. Both the configurations are sketched in Fig. 14.26(b). The two-slab configuration has an interfacial energy per unit area [l.u.2] of 1, corresponding to C(X1, X2), while the four-slab configuration possesses three interfacial interaction energies of the same value. The first of them substitutes the interaction energy C(X1, X1) in the layer made of Material 1 and the next substitutes the original interaction energy C(X2, X2) in that made of Material 2 of the original configuration. Finally, the third interface boundary between 1 and 2 with the interaction energy C(X1, X2) located somewhere in the middle of the simulation box remains unchanged. All the other corresponding horizontal layers of cell of both the configurations of the simulation box have equal contents of interaction energies. Subtracting the extra interfacial energy 3C(X1, X2) of the second configuration from the first one, i.e. C(X1, X2) + C(X1, X1) + C(X2, X2), the increase of interaction energy per unit surface area, 2C(X1, X2) – [C(X1, X1) + C(X2, X2)], is obtained. This increase is due to the creation of the two new interfaces between Materials 1 and 2 and hence the surface energy E1,2, numerically equal to the surface tension γ1,2, is half the increment of the interaction energy per unit area, i.e. E1,2 = C(X1, X2) – 12 [C(X1, X1) + C(X2, X2)] A more advanced and general but more complicated method to determine the interfacial tension can be found, for instance, in the work of Aloisi et al. (1987). In Table 14.2 the interaction energies are expressed in e.u./l.u.2, where

524


e.u. is the energy unit. There, the indices 0, 1, 2 denote the gas/vapour, the liquid, and the fibre, i.e. the triad (0,1,2) corresponding to the former (V, L, S). The interaction energies, i.e. C(Xi, Xj)’s are chosen to keep the contact angle between the liquid and the solid fibre negligible. Hence, the relations E0,2 ≥ E1,2 + E0,1, i.e. γVS ≥ γLS + γ are obtained, in agreement with the condition for a complete wetting given by Young’s equation. One obtains the inequality C(0, 2) – 12 [C(0, 0) + C(2, 2)] ≥ C(1, 2) – 12 [C(1, 1) + C, (2, 2)] + C(0, 1) – 12 [C(0, 0) + C(1, 1)] after substituting the previously derived expression of E1,2 into the aforementioned Young’s condition for a zero contact angle. It is proper to point out that the inequality does not depend on C(0, 0) and C(2, 2). So, the final form of the inequality is C(0, 2) ≥ C(1, 2) + C(0, 1) – C(1, 1). For the particular C(Xi, Xj) values of Table 14.2, the inequality gives –10 e.u./l.u.2 > –12 e.u./l.u.2, safeguarding the zero contact angle of the investigated system and maintaining the spreading coefficient S close to zero. The appearance of long liquid columns and unduloids for various distances and fibre radii are depicted in Fig. 14.27. The threshold values of d/r estimated from the computer simulation are near to 1.414, the same as predicted by Princen in his approach.

(d/r )tr1,2 1.6 1.41421 1.36598 1.4 1.23623 1.34052 1.2 1.04741 1.11716 1 [1]

“Unduloid”

0.82249

0.8

0.73205 0.58522

0.6 Long liquid column

0.4

[2] 0.35988 0.17161

0.2 0.21141

0.04491

0 0

10

20

30

40

50 θ

60

70

80

0 90

100

14.27 The dependence of the dimensionless distance d/r, (r is the fibre radius) between two parallel cylindrical fibres and the contact angle θ. The curve in the graph represents the transition boundary between long liquid columns and unduloids.


525

14.3.3 Computer simulation of three-dimensional liquid transport in fibrous materials Liquid movement in three-dimensional fibrous materials is studied in this subsection by means of Monte Carlo simulation, based on the auto-model with Kawasaki kinetics. Related algorithms are developed in accordance with the standard liquid wicking rate tests from both the European Disposables and Nonwovens Association (EDANA), and the International Nonwovens and Disposables Association (INDA). The simulation provides information about liquid wicking into computer-generated fibre assemblies as a function of time. The analysis mainly focuses on the connection between the fibre orientation and the liquid wicked into computer-generated fibre assemblies, as a function of time. Factors such as the fibre orientation and the liquid wicking rate are varied, keeping other geometrical parameters of the fibrous material unaltered. This simulation also presents dynamic data of both the mass of drawn liquid and energy changes, i.e. the energy dissipation of the system. The results are in agreement with some known experimental evidence. More information about this kind of simulation is available in recent papers of the authors Lukas et al. (2004). A flow involving more than a single phase is classified as a multiphase or inhomogeneous one – such as liquid flow in fibrous media. The dynamics of the evolved macroscopic interface of the distinct phases during such inhomogeneous flow through fibrous masses is of interest. The dynamics are qualified by surface tension, fibrous media anisotropy and inhomogeneity, fibre volume fraction, and fibre wetting behaviour. An uncertainty of structural conformation in fibrous media, accompanied by susceptibility to small loads, tortuous connectivity of their open pores and poorly defined boundaries, results in complex local inhomogeneous flows and interfacial evolution (Hentschel, 1994). This complexity becomes prohibitive for development of analytical theories describing these phenomena in many cases. The wetting and wicking of fibrous materials constitute a class of flows that have critical scientific and practical significance, on which technologies such as fibre lubricating and processing, fibre-reinforced composite manufacturing, and fibre web bonding and dyeing are based. Wetting and wicking behaviours of many consumer products, such as baby diapers, female hygiene products, and sport and other protective garments, are vital in determining their commercial success. The chosen method makes it possible to investigate the wetting and wicking phenomena, as well as to predict the system parameters that are comparable with experimental ones at a macroscopic level. One such parameter is the wicking rate, which can be measured, simulated, and theoretically derived from the Lucas–Washburn theory. The wicking rate in fibrous systems is heavily influenced by several complex geometrical parameters, as mentioned

526


previously. Hence, at present, exclusively one aspect of the effect of fibre orientation is attended to here. Other morphological features, such as heterogeneity, fibre volume fraction, fibre aspect ratio, fibre curl and fibre surface geometry, are kept constant in the bulk of the modelled specimens. All fibres are represented strictly by their linear geometry. In other words, the main subject in this subsection is a detailed study of the relationship between the wicking rate and fibre orientation. Wicking and absorption rate in fibrous media are conveniently described using the Lucas (1918) and Washburn (1921) approach. The simple theory of Lucas and Washburn deals with the rate at which a liquid is drawn into a circular tube by capillary action. Such a capillary is, in reality, a grossly simplified model of a pore in a fibrous medium with a highly complex structure. The derivation of the following equation may be found in a recent work of Lukas et al. (2004). 2 dh = rγ cos θ – r ρg cos β 8µ dt 4 µh

[14.57]

Parameters and quantities visible in the Lucas and Washburn equation are introduced further. A newtonian liquid with viscosity µ wets a tube of radius r. The contact angle of the liquid against the tube wall is denoted as θ, and the parameter β is the angle between the tube axis and the vertical direction, as shown in Fig. 14.28. The symbol γ stands for the liquid surface tension, ρ for the liquid density, g for the gravitational acceleration, and h for the distance travelled by the liquid, as measured from the reservoir along the tube axis. The distance, obviously, is a function of time, i.e. h = h(t). For a given system, as is shown in Fig. 14.28, parameters r, γ, θ, µ, ρ, g, and β remain constant. The following two constants may be introduced

K′ =

r ρg cos β rγ cos θ and L ′ = 4µ 8µ

[14.58]

to simplify the Lucas–Washburn Equation [14.58] to get

dh = K ′ = L ′ [14.59] dt h The above relation is an ordinary non-linear differential equation that is solvable in terms of function h(t) after ignoring the parameter L′ physically by considering either of the cases when liquid penetration is horizontal (β = 90°), or r is small to render L′ → 0. L′ is also comparatively negligible when the rising liquid height h is low, so that K ′ >> L ′ . It also becomes insignificant h with a diminishing effect of the acceleration due to gravity g. The Lucas– Washburn Equation [14.59] thus reduces to: h=

2 K ′t

[14.60]


527

1

r

3

Θ

h

β

2

14.28 A tube (1) of a radius r is suspended in a liquid source (2). The distance traveled by the liquid along the capillary axis is h. The angle θ is the contact angle between the liquid surface and the capillary wall, while β denotes the angle between the tube and the vertical axis (3).

The result satisfies an initial condition h = 0 for t = 0. Gupta’s approach (Gupta, 1997) concerns a similar work pertaining to a study of the wicking rate of a fluid spreading radially outward from a point source at the centre of a substrate instead of a vertical movement of the liquid in which the fibrous substrate is partially immersed. This is illustrated in Fig. 14.29. It is now time to speak about modifying the Lucas–Washburn equation by expressing the distance h in terms of the liquid mass uptake m. Such an alteration is detailed in some recent literature (Ford, 1933; Hsieh, 1995; Lukas et al., 2004). The applied manipulation does not affect the fundamental shape of the Equation [14.59] as h and m are linearly related for a circular tube with a fixed cross-section. Furthermore, for radial spreading, the liquid mass mR is expressed as π r2TρVL, and for an ascending movement the mass mA becomes whTρVL, where T is the thickness of the substrate and VL is the liquid volume fraction inside the substrate of width w. For a radial liquid spreading in a flat textile specimen, the following equation may be written using Equation [14.60]: Q=

mR = 2 πK ′TρVL t

[14.61]

528

Thermal and moisture transport in fibrous materials T

W

1

h

2

T

h

3

(a)

(b)

14.29 Two different experimental arrangements for wicking (absorbency) rate measurement: (a) Liquid source initiates in the centre of a flat specimen and is spread radially outward, and (b) Liquid ascends in a specimen perpendicular to the reservoir liquid surface.

where Q is the liquid wicking (absorbency) rate used by Gupta (1997). It is independent of time during the spreading process. Now, let liquid mass uptake mA be substituted into the original Lucas–Washburn Equation [14.59], resulting in: dm A = K –L mA dt

[14.62]

The new constants K and L are K = (wTρVL)2K′, L = wTρVLL′

[14.63]

It is obvious that the constant K in the modified Lucas–Washburn Equation [14.62] is proportional to the wicking/absorption rate Q, defined in [14.61]. 2π K. Hence the parameter From [14.61] and [14.63], it follows that Q = 2 w TρVL K can be used as a measure of the wicking spread rate Q in those experiments where a fabric is hung vertically in a liquid. The values of K and L may be derived from the slope and intercept of the dmA/dt versus 1/mA graph, as mentioned by Miller and Jansen (1982). On the other hand, Equations [14.59] and [14.62] are solvable in terms of functions t(h) or t(mA) without dropping the gravity term, g, as shown in some earlier works (Lucas, 1918; Washburn, 1921; Lukas and Soukupova, 1999). For the liquid mass uptake, i.e. for the ascending liquid movement, the Lucas–Washburn Equation [14.62] yields: t (mA) = –

L mA – K2 ln  1 – m A  K K   L

[14.64]

However, a theory like that of Lucas–Washburn has innumerable limitations.


529

Firstly, the assumed cylindrical capillary tubes are far from the reality from the standpoint of open porous structures of fibrous masses. Also, because of the increasing mass of ascending liquid in a vertical sample, the wicking rate is, in fact, a variable and not a constant, as assumed theoretically. Furthermore, this kind of theory is unable to deal with the effects of several fundamental system parameters, including fibre orientation and fibre volume fraction. Therefore, development of a more robust technique is desirable in this field, and the following is such an attempt. A three-dimensional stochastic method, based on the auto-model with Kawasaki dynamics on a cubic lattice, is presented to obtain the shape of a liquid body wicked into a fibrous material with known fibre orientations. The auto-model used here consists of a rectangular simulation box of size nW ∗ nH ∗ nL = W × H × L (width × height × length), subdivided into numerous lattice cells. Each cell is occupied by only one type of media: gas, liquid, or fibre. The cells interact with their neighbours through an energy exchange. A neighbourhood of order, o, of 2 is formed of 26 cells, arranged into a super-cube, surrounding a cell at the centre, as seen in Fig. 14.17. The energy exchange values of C in the Hamiltonian [14.65] depend on the types of interactions. To make the description concise and the notations appropriate for computer processing, the lattice cells may be located using indices i and j and designated with an entity, such as the random variable value xi. Indices i and j may vary in the range i, j ∈ {1, 2,…, n}, where n is the total number of cells. The variable value x assumes a value of 0 when occupied by a gas, a value of 1 when occupied by a liquid, and a value of 2 when occupied by a fibre. The exchange energies, C, are then symmetrical functions of two variables, i.e. C(xi, xj) = C(xi, xj). The possible energy values assigned to two interacting cells are shown in Table 14.3. There are nine possible combinations for pairs of the three variables. However, the allowable combinations reduce to six due to symmetry. Furthermore, as the fibre structure or the fibre cells are immovable during the simulation, the interaction energy value C(2,2) does not play a role in the model and hence is set to 0. Table 14.3 Interaction energy values C(xi, xj) used for the simulation of threedimensional liquid transport in fibrous materials Interaction energies in e.u., Liquid transport in fibrous materials

Gas, Xi = 0

Liquid, Xi = 1

Fibre, Xi = 2


C(0, 0) = –40 C(1, 0) = 5 C(2, 0) = 20

C(0, 1) = 5 C(1, 1) = –26 C(2, 1) = –30

C(0, 2) = 20 C(1, 2) = –30 C(2, 2) = 0

530


Generally, systems tend to minimise their total internal energy. This leads to an expectation, that an attractive intercellular interaction corresponds to a low-exchange energy while a repulsive one corresponds to a high-exchange energy. Chosen values of C(xi, xj) ensure such interactions by encouraging aggregation of gas and liquid cells. Commonly, the liquid cells will have higher preference than gas cells to adhere to the fibres, due to the assigned interaction energy values, thus reflecting the physics behind spontaneous imbibition of a liquid into a fibrous material. As an assumption, the interaction with the gravitational field is considered significant only for liquid cells; the influence of gravity on the mass of the gas is small enough to be neglected. The presence of gravity is represented by a uniform field Cg. The gravitational potential of a liquid particle of mass m in a cell si, located at the hi-th level of the lattice along the vertical axis H, is mghi. With this information, the r system Hamiltonian U ( x ) may be written as r [14.65] U ( x ) = Σ Cg h ( x i ) + Σ C( x i , x j ) 1≤ i ≤ n

1≤ i < j ≤ n

where the first sum applies to liquid cells only, with m having a value of 1 and Cg equivalent to acceleration due to gravity, g. Each couple of cells is counted once in the second sum term of the right-hand side of Eq. [14.65]. It may be recollected, that this Hamiltonian was introduced earlier in Eq. [14.40] while examining auto-models in Section 14.2.4. A simple cuboid lattice of size 37 × 300 × 150 (nW × nH × nL), having a total number of cells n of 1 665 000 under free boundary conditions, is conceived for the present simulation. To convey the initial spin cell configuration, it is proper to specify that the first ten bottom layers at hi ∈ {0, 1, 2,…, 9} are immersed in a liquid pool, with variables, xi , assigned values of 1, and the rest of the lattice space is filled by gas spins having variable values, xi, of 0. The direction of a fibre inside the cuboid lattice is determined by a unit r vector a , making angles α and β with the perpendicular axes H and W of the three-dimensional cuboid lattice system, as shown in Fig. 14.30. The orientation r of a can be chosen up or down the line, determined by the couple of angles (α, β). A vertical cross-section of each fibre consists of one fibre cell only. The entire fibrous mass is, then, simulated through generation of individual random lattice cells, acting as origins of the fibres. Consequently, the vector r a , as described above, is used to generate the fibre directions through a random-number generator related to the angle α and fibre origin, along with a chosen constant value β. Each fibre is defined by an abscissa, having an r equal probability in upwards or downwards directions along the vector a , starting from the fibre origin. The fibre length is equal to 50 lattice units, and the diameter is equal to 1 lattice unit. The total number of fibres is 10 000 (M = 10 000). Fibres thus generated are arranged throughout the cubic lattice and can be considered as chains of cells that are occupied by them (i.e. the


531

H r a

H r a

β 1

β = 45°

L α

W

14.30 A fibre system is generated via the location of its origin o of a fibre containing a unit vector a inside the simulation box W × H × L. On the right-hand side is a thin section of the simulation box (one cell thick) with a part of a fibre with the deflection angle β =45°.

fibre cells). The average fibre volume density Vf (Wiebel,1979), hence equals the proportion of fibre cells out of the total number of cells, n. Simulations are carried out for Vf = 0.22. After creating the initial configuration, the simulation is carried out following all the steps other than the fifth one of the algorithm, previously described in Section 14.3. This conditional step is substituted with the one stating that as soon as a liquid particle is moved from the initial liquid layer, i.e. hi ∈ {0, 1, 2, …, 9}, its position is always refilled with other liquid particles. This simulates the liquid absorbing from an inexhaustible liquid reservoir. The described computer simulation model delineates the dynamic behaviour. r r Apparently, the evolving configurations x m → x m+1 in the algorithm differ only by an exchange of a couple of cells. That is the reason behind a strong correlation between the physical properties of the sequential configurations of the system. This correlation forms the theoretical basis for the application of Kawasaki dynamics in simulating dynamic processes (Manna et al., 1992; Lukkarinen, 1995; Binder, 1997). The dynamic interpretation of the method r r lies in the association of a time, t, with a subsequent configuration x m → x ( t ) in a Markov chain. The time scale is often normalised to a relative unit, during which exchange, trials of n variable values are performed. The time unit is 1 MCSPS, as was introduced in Section 14.3.1. r For the joint probability distribution function, P ( x , t ) at a time t, a r configuration x occurs in the Monte Carlo process, satisfying the Markovian master equation (Binder, 1997): r r r r r r r dP ( x ) [14.66] = – Σr W ( x → y) P ( x , t ) + Σr W ( y → x) P ( y , t ) x y dt

532


In the above equation, the first term on the right-hand side represents all the r processes that push the system away from the configuration x . The second term contains all the reverse events that lead to an enhancement of the joint r r probability P ( x ) , corresponding to the configuration x . In the case of thermal equilibrium, the two sums cancel each other because the system is supposed r to fulfil the principle of detailed balance. Therefore, Peq( x , t)/dt = 0, where r Peq( x , t) is the steady-state distribution of the master Equation [14.66]. In general, the time associated with the variable exchange procedure cannot be related to the physical time through which a real system evolves. The fact that the time evolution in a real system is governed by a deterministic equation, not by a stochastic master Equation [14.66], accounts for the disparity. But for an Ising-like Hamiltonian [14.65], the stochastic kinetics provided by Eq. [14.66] may be explained physically in terms of a very weak coupling of the spins in a heat bath, which induces random spin exchanges in the system (Binder, 1997). In the simulation, an initial investigation is done to study the influence of the fibre orientation on the dynamics of liquid wetting and wicking in a fibrous mass. Thereafter, the fibre declination β from the vertical axis H is varied by steps of 10°, so that the simulation is carried out for 11 different fibrous systems with β ranging between the inclusive limits of angles 0° and 90°, in addition to β = 45°. After each Monte Carlo step per site/cell (MCSPS), the outputs recorded are: the number or the mass mA of the wicked liquid particles, the liquid particles above the original liquid surface, and the changes of the total system energy U. The simulation is terminated after 600 MCS, and information from each subsequent configuration is preserved for further analysis. The results of this process are provided in Fig. 14.31, where side views and top views of wetting patterns of cross-sections of the fibrous mass at different values of β are paired. The horizontal cross-sections are all cut at a distance of 100 cells from the liquid surface. The pictures present two extreme situations. The first one is vertical to the (WL) plane or β = 0°, where the ascending liquid moves at the highest rate but is mostly scattered. The other one is parallel to the (W L) plane or β = 90°, with the lowest wetting rate but the liquid pattern is highly aggregated, as evident from Fig. 14.32(a). From Fig. 14.31, one can intuitively evaluate the volume of the liquid wicked into the fibrous mass. Yet from Fig. 14.32(a), it is clear that, at initial periods (i.e. MCSPS < 200), the absorbed liquid body is generally more voluminous as angle β reduces, because the fibre assemblies with smaller β values apparently start with greater wicking rates as the liquid climbs faster. Nevertheless, such trend in the calculation remains true only for β ≥ 20°, whereas for β < 20° the climbing of the liquid stagnates and become independent of the time; the smaller the β value, the earlier the climbing stops. The above discussions reveal that there will be either one or


0°

70°

10°

80°

20°

533

30°

90°

14.31 Wetting patterns from both vertical and horizontal crosssections of a fibre mass at different orientation angles β after 600 MCSPS (Monte Carlo step per site particle). The fibres in the individual samples have strictly identical declination β. From the leftto the right-hand side, the β values are 0, 10, 20, 30, 70, 80, and 90, as denoted in each sample. Only those fibres that lie in the crosssection plane are depicted in full length with the corresponding declination angle β. The fibre system is uniformly random.

a range of optimal combinations of β and MCSPS at which the fibre mass will absorb a maximum amount of liquid, resulting from an optimal wicking rate and wicking duration, as illustrated in Fig. 14.32(b). It is apparent from the figure that, at a value of β equalling 20°, the greater the MCSPS value, the more the liquid is absorbed into the specimen. To evaluate the wicking rate K, the time derivation of the wicked liquid mass, dmA/dt, is plotted against the reciprocal value of the mass 1/mA at different values of β, as given in Fig. 14.33. As mentioned earlier, the value of K may be determined from the slope of the plot. In general, a wicking rate may be defined as K = tgα, where α is, presently, the slope of the dmA/dt – 1/mA plot. It is clear from Fig. 14.33 that a fibrous mass with a smaller value of β does yield a higher slope or a greater wicking rate K. However, according to Equation [14.62], when the other parameters are given, the dmA/dt versus 1/mA characteristic is expected to be linear. The results of the simulation, consistent with experimental practice, demonstrate otherwise. This is clear

534

Thermal and moisture transport in fibrous materials (a) m.(10–5) 8 7 6 5 4 3 2 1 0 0 100

(b)

m.(10–5) 8 7 6 5 4 3 2 1 0 0 10

30° 45° 60° 90°

0°

200

300 MCS

400

500

600

MCS 600 500 400 300 200 100

20 30 40 50 60 70 80 β (degree)

90 100

14.32 (a) The dependency of a liquid mass wicked into a specimen on MCS (Monte Carlo step per particle) at different fibre orientation angles β. (b) The dependency of a liquid mass wicked into a specimen on orientation angle β after different MCS values.

from Fig. 14.33. In reality, the rate dmA/dt cannot maintain a linear relationship as the liquid weight will slow down and eventually stop the wicking process (i.e. the wicking rate dmA/dt decreases gradually with time until zero). This, on the other hand, provides strong evidence for validating the simulation methodology. A detailed analysis of the relationship between the wicking rate K and the fibre orientation angle β based on the simulated data is summarised in Fig. 14.34. It may be observed that the dependence between the two is also non-linear, showing a rapid decrease of the wicking rate K in the interval β ∈ 〈0, 30°〉, while for the rest of the graph, i.e. β > 30°, there is, rather, a constant or even a slight increase of K. Computer simulations based on this method, realistically and quantitatively depict the dynamic ascending liquid wicking process. Moreover, a parametric study has been conducted in this article to examine the influences of the important factors involved in the simulation. The simulated results indicate that the relationship between the fibre orientation β and the generalised wicking rate K is non-linear, and a greater wicking rate K is achieved for the structures with fibre orientation with respect to the axis perpendicular to the original liquid surface, i.e. β smaller than 30°.

Computer simulation of moisture transport in fibrous materials dm/dt [m.u./MCS] 10000 8000

β = 20°

6000 4000 2000 α 0 0

0.00005

dm/dt [m.u./MCS]

0.0001 l/m [m.u.–1] (a)

0.00015

0.0001 l/m [m.u.–1] (b)

0.00015

10000 8000

β = 45° 6000 4000 2000

α 0 0

0.00005

dm/dt [m.u./MCS] 10000 8000 6000

β = 90°

4000 2000 α 0 0

0.00005

0.0001

0.00015

l/m [m.u.–1] (c)

14.33 Time derivation of wicked liquid mass dmA/dt against the reciprocal mass 1/mA: (a) fibre orientation β = 20°, (b) fibre orientation β = 45°, and (c) fibre orientation β = 90°.

535

536

Thermal and moisture transport in fibrous materials K.10–8 2.5 2

1.5

1 0.5 0 0°

10°

20°

30°

40°

50°

60°

70°

80°

90°

β

14.34 The generalised wicking rate K versus fibre orientation angle β.

The effects of β on the total liquid absorption are not monotonic; a smaller β leads to a greater wicking rate K or a quicker liquid wicking process, as stated above. The wicking process, however, stagnates as time increases. Although a greater β results in a slower wicking rate, the wicking process can prolong further. An optimal condition can be explored. Anisotropy of liquid transport in fibrous layers is of great interest for producers of disposable items such as diapers and feminine hygiene products. The liquid distribution in such commodities has to be selective so as to protect consumers against lateral outflow. This is achieved by the so-called acquisition layers, whose function may be controlled by using the wicking rate–fibre orientation relationship.

14.4


The field of inhomogeneous flows, i.e. flows described in Section 14.3, has been an active area for systematic research since the mid-nineteenth century. The basic theory was the equation of capillarity, given by Young (1972) and De Laplace (1806). One of the earliest works is that of Lamb (1879), dealing with the motion of solids through liquids. In the 1970s, some pioneering works on the dynamics of liquid spreading on solids were published; Huh and Scriven (1971) suggested a singularity in the dissipation for such flows. A useful distinction was revealed between simple fluids, e.g. liquids that were found to spread through a ‘rolling motion’ (Dussan and Davis, 1974), and polymeric melts, which often tended to slip on a solid surface (Brochard and de Gennes, 1984). More recently, de Gennes (1985) published a review of wetting research. The first attempt to understand capillary-driven inhomogeneous flows for


537

practical applications was made by Lucas (1918) and Washburn (1921). Lucas–Washburn theory had been used and developed further for the textile area by few authors. Chatterjee (1985) dealt with these kinds of flows in dyeing. Pillai and Advani (1996) conducted an experimental study of the capillarity-driven flow of viscous liquids across a bank (a container filled by a fibrous mass) of aligned fibres. Hsieh (1995) discussed theories and applications of wetting and capillary principles to analyse liquid wetting and transport in fibrous materials. Several techniques employing fluid flow to characterise the structure of fibrous materials were also presented by Hirt et al. (1987). Lukas and Soukupova (1999) carried out data analysis to test the validity of the Lucas–Washburn approach for some fibrous materials. Inhomogeneous flows have also been studied using stochastic simulation. Manna et al. (1992) presented a stochastic simulation that generated the shape of a two-dimensional liquid drop, subjected to gravity, on a wall. The system was based on the modified Ising model, i.e. the auto-model in the context of the present chapter, with Kawasaki dynamics. They located a phase transition between a hanging and a sliding droplet. Lukkarinen (1995) studied mechanisms of a fluid droplet spreading on flat solids and found, that the spreading initially behaved almost linearly with time, and the spreading of the liquid precursor film was dominated by a surface flow on the bulk of a solid. Afterwards, however, such dynamics of spreading depended on the square root of time. A similar study of a fluid droplet spreading on a porous surface was recently reported by Starov et al. (2003). The first attempt to simulate the liquid wetting dynamics in fibre structures using the auto-model was done by Lukas et al. (1997) and Lukas and Pan (2003) followed by Zhong, Ding and Tang (2001, 2002), although the simulation was restricted to two-dimensional systems only. EDANA, the European Disposables and Nonwovens Association, and INDA, the International Nonwovens and Disposables Association have formulated tests (ERT Absorption 10.2-96 and IST Absorption 10.1-92) to measure the vertical speed at which liquid moves upwards in a fabric, i.e. the capillarity of the test material. Gupta (1997) defined absorption rate as the quantity that was characterised by modifying the Lucas–Washburn equation, and then applied it to a flat, thin circular fabric on which fluid diffused radially outwards. Berg (1989) dealt also with the Lucas and Washburn theory. The theory was actually a special form of the Hagen–Poiseuille law (Landau and Lifschitz, 1988) for laminar viscous flows. It was shown that the Lucas and Washburn equation was valid for a variety of fibrous media, including paper and textile materials (Berg, 1989; Evert et al., 1978). Finally, a brief discussion to compare the Monte Carlo-based simulation of auto-models, and final element analysis may be added. One may also refer to Section 10.5 of Chapter 10 for more information about the relationship

538


between lattice gas cellular automata and auto-models. The finite element method (Nicholson, 2003), also referred to as finite element analysis, is a computational technique for approximate solutions of boundary value problems, which is preferably used in engineering. A boundary value problem is, generally, a mathematical problem in which one or more dependent variables must satisfy a differential equation everywhere within a known domain of independent variables and should satisfy specific boundary conditions at the boundary of this domain. Depending on the type of physical problem being analysed, the dependent variables may include physical displacement, temperature, heat flux, fluid velocity, and so on. Thus, the basic difference between the auto-model approach and finite element analysis is that automodels are based on discrete structures, statistical physics and simulated through Monte Carlo algorithms, while finite element analysis is governed by continuum mechanics framed with differential equations. Auto-models are governed totally by Hamiltonians with interaction energies and are not able to reflect easily mass inertial effects. On the other hand, the extremely simple ones involve manifold and complex boundary conditions and can tackle systems composed of fluid mixtures. Finite element methods are much more widespread in application and are employed more frequently as compared to the alternative approach.

14.5

References

Adamson A W and Gast A P (1997), Physical Chemistry of Surfaces, John Wiley & Sons, New York. Aloisi G, Guidelli R, Jackson R A and Barnes P (1987), ‘A new method to determine the interfacial tension in Monte Carlo simulations’, Chemical Physics Letters, 133, 343– 345. Berg J C (1989), ‘The use of single-fibre wetting measurements in the assessment of absorbency’, Nonwovens Advanced Tutorial, edited by F. T. Albin and L.V. Tyrone. Atlanta, GA: TAPPI Press. Besag J E (1972) ‘Nearest neighbour systems and the auto-logistic model for binary data’, J. R. Statist. Soc. B, 34, 75–83. Besag J (1974), ‘Spatial interaction and statistical analysis of lattice systems’, J. R. Statist. Soc. B, 36, 192–236. Binder K and Heermenn D W (1997), Monte Carlo Simulation in Statistical Physics; An Introduction, Berlin, Heidelberg, New York, Springer Verlag. Binder K (1986), Monte Carlo Methods in Statistical Physics, Topics Curr. Phys., Berlin Heidelberg, Springer Verlag. Brochard F and de Gennes P G (1984), ‘Spreading laws for liquid polymer droplets – interpretation of the foot’, Journal de Physique (Paris) Lettres, 45, L597–L602. Brush S (1967), ‘History of the Lenz–Ising model’, Review of Modern Physics, 39, 883– 893. Chatterjee, P K (1985) Absorbency. Amsterdam: Elsevier. Chen C C and Dubes R C (1989), ‘Experiments in fitting discrete Markov random fields to textures’, CVPR’89 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, San Diego, IEEE Comput. Soc. Press.


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Cochran W G (1936), ‘The statistical analysis of the distribution of field counts of diseased plants’, J. R. Statist. Soc. Suppl., 3, 49–67. Cox D R (1970), Analysis of Binary Data, London, Meuthen. Cross G C and Jain A K (1983), ‘Markov random field texture models’, IEEE Trasactions on Pattern Analysis and Machine Intelligence, 5, 25–39. Derin H and Elliott H (1987), ‘ Modelling and segmentation of noisy textured images using Gibbs random fields’, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI–9(1), 39–55. Dobrushin R L, Kryukov V I and Toom A L (1978), ‘Locally interacting systems and their applications in biology’, Lecture Notes in Mathematics, 653. Dubes R C and Jain A K (1989), ‘Random field models in image analysis’, Journal of Applied Statistics, 16(2), 131–164. Dussan V E and Davis S (1974) ‘On the motion of a fluid–fluid interface along a solid surface’, Journal of Fluid Mechanics, 65, 71–95. de Gennes P G (1985), ‘Wetting: statics and dynamics, Reviews of Modern Physics, 57(3, pt. 1), 826–863. de Gennes P-G, Wyart-Brochard F and Quere D (2003), Capillarity and Wetting Phenomena; Drops, Bubbles, Pearls, Waves, New York, Berlin, Springer-Verlag. De Laplace P S (1806), Mechanique Celeste. Supplement to Book 10. Paris, Durat. Ermakov S M and Nekrutkin V V (1989), Random Processes for Classical Equations of Mathematical Physics, Dordrecht, Boston, London, Kluwer Academic Publishers. Everet D H, Haynes J M and Miller R J (1978), ‘Kinetics of capillarity imbibition by fibrous materials’ Fibre–Water Interactions in Papermaking, edited by the Fundamental Research Committee. London, Clowes. Ford L R (1933), Differential Equations, NewYork, McGraw-Hill. Fricot J (1985), ‘Champs aleatoires de renouvellement’, thesis, University of Grenoble Geman S and Geman D (1984), ‘Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images’, IEEE Transaction of Pattern Analysis and Machine Intelligence, 6(6), 721–741. Geman D (1991), ‘Random fields and inverse problems in imaging’, Lecture Notes in Mathematics, 1427, 113–193. Glauber R J (1963), ‘Time development statistics of the Ising model’, J. Math. Phys., 4, 294–307. Gupta B S (1997), ‘Some recent studies of absorbency in fibrous nonwovens’, XXV International Nonwovens Colloquium, Brno, Czech Republic. Hammersley J M and Clifford P (1971), Markov fields on finite graphs and lattices (unpublished). Hentschel, H G E (1994), Encyclopaedia of Applied Physics: Inhomogeneous Flows. Vol. 11. New York, VCH Publishers. Hirt D G, Adams K L, Prud’Hommer R K and Rebenfeld L (1987), ‘In-plane radial fluid flow characterization of fibrous materials’, Journal of Thermal Insulation, 10, 153– 172. Hsieh Y L (1995), ‘Liquid transport in fibrous assemblies’, Textile Research Journal, 65, 299–307. Huh C and Scriven L E (1971), ‘Hydrodynamic model of steady movement of a solid/ liquid/fluid contact line’, Journal of Colloid and Interface Science, 35, 85–101. Ising E (1925), ‘Beitrag zur Theorie des Ferromagnetismus’, Z. fur Phys., 31, 253–58. Kawasaki K (1972), Phase Transitions and Critical Phenomena, ed. By C. Domb and M.S. Green, 5b(2), New York, Academic.

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Kittel C (1980), Thermal Physics, San Fransisco, W.H. Freeman. Korolyuk V and Swishchuk A (1995), Semi-Markov Random Evolutions, Dordrecht, Boston, London, Kluwer Academic Publishers. Lamb H (1945) [1879], Hydrodynamics. Reprint, NewYork: Dover. Landau L D and Lifshitz E M (1988), Theoretical Physics: Vol. 6. Hydrodynamics, Moscow, Nauka. Lenz W (1920), ‘Baitrag zur Verstandnis der magnetischen Erscheinung in festen Korpern’, Z. fur Phys., 21, 613–15. Liggett T M (1985), Interacting Particles Systems, Berlin, New York, Springer Verlag. Lucas R (1918), ‘Uber das Zeitgesetz des kapillaren Aufsteigs von Fussigkeiten’, Kolloidn Zhurnal, 23, 15–22. Lukas D, Glazirina E and Pan N (1997), ‘Computer simulation of liquid wetting dznamics in fibre structures using the Ising model’, Journal of the Textile Institute, 88, 149–161. Lukas D and Soukupova V (1999), ‘Recent studies of fibrous material wetting dynamics’, INDEX 99 Congress, Geneva, Switzerland. Lukas D and Pan N (2003), ‘Wetting of a fibre bundle in fibrous structures’, Polymer Composites, 24, 314–322. Lukas D and Chaloupek J (2003a), ‘Wetting between parallel fibres; column-unduloid and column disintegration transitions’, Journal of Engineering in Medicine, 217, 273– 277. Lukas D, Pan N, Soukupova V and Parikh D V (2004), ‘Computer simulation of 3/D liquid transport in fibrous materials’, SIMULATION, 80, 547–557. Lukkarinen A (1995), ‘Mechanisms of fluid spreading: Ising model simulations’, Physical Review E: Statistical Physics, Plasma, Fluids, and Related Interdisciplinary Topics, 51, 2199–2202. Manna S S, Herrmann H J and Landau D P (1992), ‘A stochastic method to determine a shape of a drop on a wall, Statistical Physics, 66, 1155–1163. Meister B J and Scheele G F (1967), ‘Generalized Solution of the Tomotika Stability Analysis for a Cylindrical Jet’, AICHE J., 13, 682. Metropolis N, Rosenbluth A W, Rosenbluth M N, Teller A and Teller E (1953), ‘Equation of state calculation by fast computing machines’, Journal of Chemical Physics, 21, 1087–1092. Miller B and Jansen S H (1982), ‘Wicking of liquid in nonwoven fibre assemblies: Advances in nonwoven technology’, 10th Technical Symposium, New York, 216–226. Moussouris J (1973), ‘Gibbs and Markov random systems with constraints’, Journal of Statistical Physics, 10(1), 11–33. Neimark A V (1999), ‘Thermodynamic equilibrium and stability of liquid films and droplets on fibres’, J. Adhesion Sci. Technol., 13, 1137–1154. Nicholson D W (2003), Finite Element Analysis: Thermodynamics of Solids, London, New York, CRC Press. Paget R D (1999), ‘Nonparametric Markov random field models for natural texture images’, Doctoral thesis, Queensland, Australia, The University of Queensland. Pillai K M and Advani S G (1996), ‘Wicking across a fibre-bank, Journal of Colloid and Interface Science, 183, 100–110. Plateau J (1873), Statique Experimentale et Theoretique des Liquids Soumis aux Seriles Forces Molecularies, Paris, Gauthier-Villars. Preston C (1976), ‘Random fields’, Lecture Notes in Mathematics, 534. Princen H M (1969), ‘Capillary phenomena in assemblies of parallel cylinders: Capillary rise between two cylinders’, J. Colloid Interface Sci., 30, 69–75.


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Princen H M (1969a), ‘Capillary phenomena in assemblies of parallel cylinders: II. Capillary rise in systems with more than two cylinders’, J. Colloid Interface Sci., 30, 359–371. Princen H M (1970), ‘Capillary phenomena in assemblies of parallel cylinders: III. Liquid columns between horizontal parallel cylinders, J. Colloid Interface Sci., 34, 171–184. Prum B and Fort J C (1991), Stochastic Processes on a Lattice and Gibbs Measures, Dordrecht, Boston, London, Kluwer Academic Publishers. Rayleigh L (1878), On the Instability of Jets, Proc. London Math. Soc., 10(4). Roe R-J (1957), ‘Wetting of fine wires and films by a liquid film’, Journal of Colloid and Interface Science, 50, 70–79. Rowlinson J S (2002), Cohesion, A Scientific History of Intermolecular Forces, Cambridge University Press. Ruelle D (1969), Statistical Mechanics; Rigorous Results, New York, Benjamin. Seymour L (1993), ‘Parameter Estimation and Model Selection in Image Analysis Using Gibbs-Markov Random Fields’, PhD thesis, The University of Carolina, Chapel Hill. Spitzer F (1971) ‘Markov random fields and Gibbs ensembles’, Amer. Math. Monthly., 78, 142–154. Starov V M, Zhdannov S A, Kosvinstev S R, Sobolev V D and Velarde M G (2003), ‘Effect of interfacial phenomena on dewetting in dropwise condensation’, Advances in Colloid and Interface Science, 104, 123–190. Swishchuk A (1997), Random Evolutions and Their Applications, Dordrecht, Boston, London, Kluwer Academic Publishers. Tomotika S (1935), ‘On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid’, Proc. Roy. Soc., A 150, 322–337. van Kampen N G (1992), Stochastic Processes in Physics and Chemistry, Amsterdam, New York, North-Holland. Washburn E W (1921), ‘The dynamics of capillary flow’, Physical Review, 17, 273–283. White C F (2001), ‘Driving innovation’, Nonwovens Report International, November, 368, 8. White P (1963), ‘Stochastic processes in several dimensions’, Bull. Int. Statist. Inst., 40, 974–994. Young T (1972), Miscellaneous Works. Vol. 1. Edited by G. Peacock. NewYork: Johnson Reprint Corporation. Zhong W, Ding X and Tang Z L (2001), ‘Modeling and analyzing liquid wetting in fibrous assemblies’, Textile Research Journal, 71, 762–766. Zhong W, Ding X and Tang Y L (2002), ‘Analysis of fluid flow through fibrous structures’, Textile Research Journal, 72, 751–755.

15 Computational modeling of clothing performance P. G I B S O N, U.S. Army Soldier Systems Center, USA J. B A R R Y and R. H I L L, Creare Inc., USA P. B R A S S E R, TNO Prins Maurits Laboratory, The Netherlands, M. S O B E R A and C. K L E I J N, Delft University of Technology, The Netherlands

15.1

Introduction

Protective clothing provides laboratory and hazardous materials workers, fire fighters, military personnel, and others with the means to control their exposure to chemicals, biological materials, and heat sources. Depending on the specific application, the textile materials used in protective clothing must provide high performance in a number of areas, including impermeability to hazardous chemicals, breathability, light weight, low cost, and ruggedness. Models based on computational fluid dynamics (CFD) have been developed to predict the performance of protective clothing materials. Such models complement testing by enabling property data from laboratory material testing to be used in predictions of integrated multilayer garments under varying environmental conditions. Advances in computational fluid dynamics (CFD) have made it possible to create accurate engineering models which can include the irregular shapes of a clothed human, as well as the extremely different length and time scales present in a typical computation (e.g. thin clothing layers over a relatively large human body and irregular air spaces). Body-fitted coordinates or unstructured grids are used to model shapes such as a clothed human arm covered by a permeable clothing layer, or simpler geometries such as a fabric-covered cylinder. This capability can help to design more comfortable and effective protective clothing. Modern chemical protective garments provide high levels of protection against battlefield chemical threats, but also cause debilitating heat stress under certain environmental conditions. Accurate modeling and analysis techniques for heat and mass transfer aid in designing more comfortable protective clothing systems. Air flow, heat transfer, and mass transfer through clothing can be studied on different scales: (i) macroscale of the whole human body in protective clothing, (ii) mesoscale of a single limb covered by clothing material and (iii) microscale, focusing on transport phenomena in the clothing material at the scale of individual fibers. Clothing material properties are important at 542


543

all of these scales. Transport through the clothing system involves diffusion of heat and moisture, convective airflow, and liquid water capillary wicking. Hygroscopic fibers absorb water in vapor or liquid form and release the heat of sorption within the clothing. Water can condense or evaporate in outer layers of clothing. Many modern protective clothing systems include polymeric membranes, which may be a microporous hydrophobic polymer or a very thin solid layer of a hydrophilic polymer. Modeling the sorption of liquid water or vapor into the membrane, diffusion through the structure, and desorption from the other side, are complicated by the polymer’s concentrationdependent permeation properties. Nearly all transport phenomena in clothing systems are time dependent. Equilibrium does not take place within a matter of seconds, but may require time scales of minutes to hours. Since humans rarely work at a sustained constant level for hours on end, the use of steadystate approximations to determine quantities such as total moisture accumulation within the clothing, or total heat and mass transferred through the clothing, are often inaccurate. Steady-state heat and mass transfer properties do not describe the true situation. With the appropriate assumptions, CFD is useful at both the material and system level in clothing design. CFD provides a framework to model the diffusive and convective transport of heat and gases/ vapors; capillary transport of liquids; vapor and liquid sorption phenomena and phase change; and the variable properties of the various clothing layers. It can also model the effects of sweating and humidity transport on the thermal stress imposed upon the wearer of the clothing.

15.2

Material modeling

A porous material may be described as a mixture of a solid phase, a liquid phase, and a gaseous phase (Fig. 15.1(a)). In protective clothing the solid phase consists of several materials (usually polymers and carbonaceous adsorbents), plus any bound liquid absorbed in the solid matrix, or on the surface of a solid absorbent (such as activated carbon). Hence, the solid phase density is dependent on the amount of liquid contained in the solid phase. Solid polymer layers, such those present in laminates and membranes, are treated by assuming the gas phase volume fraction for that layer is zero. The liquid phase consists of the free liquid that may exist within the porous medium. The liquid phase is a pure component, and its density is assumed to be constant. In protective clothing, the liquid phases are liquid water (sweat or rain) and liquid chemical agents. The gaseous phase consists of vapor plus the non-condensable gas (e.g. air). The gas phase density is a function of temperature, pressure, and vapor concentration. The general governing equations for energy, mass, and momentum transport

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Liquid phase

Solid phase (solid plus adsorbed/absorbed liquid phase)

Averaging volume

Gas phase (vapor plus inert gas) (a)

Convection Diffusion

Evaporation Condensation

Wicking Liquid

Vapor

Sorption Evaporation Condensation

Sorption

Bound liquid

(b)

15.1 (a) Material model and (b) Mass transport interactions.

in porous media are obtained by volume-averaging techniques,1,2 using definitions for intrinsic phase average, phase average, and spatial average for porous media given by Whitaker.3,4 Material models must account for vapor phase transport (convection and diffusion), liquid phase transport (wicking), heat transfer (convection, conduction, and radiation), liquid evaporation and condensation, and sorption/diffusion of vapor and liquid through the solid phase (Fig. 15.1(b)). Complications due to variable porosity caused by swelling or shrinkage of the porous matrix are accounted for by various source terms in the transport equations.


15.3

545

Material modeling example

Temperature rise above initial temperature (°C)

The system of governing equations may be solved to simulate the case of transient diffusion/sorption. Heat is released as water vapor diffuses through and is absorbed by a bed of hygroscopic fibers. The heat released can be physiologically significant for clothing layers incorporating hygroscopic fibers such as cotton or wool, or polymer membranes such as polyurethane, which are often components in protective clothing systems. This sorption process can also buffer the effect of rapid changes in environmental humidity. CFD methods have been used to model this transient sorption process for a variety of hygroscopic clothing materials,5 and an example is given as follows. Experimental results for the transient diffusion/sorption case were obtained using two layers of fabric instrumented with three thermocouples sandwiched between the two layers, to record temperature changes as the fibers absorb or desorb water vapor from a gas stream flowing on the two sides of the fabrics. The sample was initially equilibrated with a dry gas flow on both sides under constant nitrogen flow. The relative humidity was then changed to 100% on both sides, and the temperature rise and fall due to vapor sorption was recorded as a benchmark for the computer simulations. Figure 15.2 shows the computed temperatures and the experimentally measured temperatures of three fabrics as a function of time. The numerical results match the experimental results, illustrating that the coupled diffusion of heat and moisture in hygroscopic materials can be successfully modeled using these computational techniques.

15 Wool Cotton Silk 10

5

0

0

5

10 Time (minutes)

15

15.2 Numerical (solid lines) and experimentally measured (symbols) temperature transients for hygroscopic fabrics subjected to a step change in relative humidity.

546

15.4


Modeling of fabric-covered cylinders

Fabric-covered cylinders provide a convenient geometry to study some of the system-level effects important for clothing systems. The effects of variable air spacing between fabric layers, or between the fabric and the human skin surface, can be important in determining how much heat and mass is transferred into or out of the clothing. In many cases, approximating the human body as an assemblage of fabric-covered cylinders provides sufficiently accurate results for engineering purposes. In general, an external air flow due to wind or body motion impinges on the clothed human, and some air flows around the body, while some air penetrates through the clothing system and into the gap between the clothing and body. This is illustrated in Fig. 15.3, which is a typical CFD simulation of flow over a cylinder covered by a permeable fabric. For a given external air velocity, the amount of air which flows around the body, and the amount which penetrates through the clothing layer is determined by the air-flow resistance (air permeability) of the clothing layer. Materials with a low air-flow resistance allow a relatively high flowrate through the fabric, with a correspondingly low pressure drop. Materials with a high air-flow resistance allow less flow through the fabric, and have a higher pressure drop across the fabric layer (up to the limit of the stagnation pressure for the particular environmental flow conditions). The simple flow geometry shown in Fig. 15.3 is useful for answering some very basic questions about the interaction of the different transport properties characterized individually by laboratory test methods, but which all operate simultaneously in a clothing system. For example, protective clothing systems can be designed to protect from aerosol particles present in the environment. But clothing aerosol barrier performance cannot be measured in the same way as industrial filters are evaluated. Aerosol particle filters are

External air flow over body

Clothing layer Air gap Body

Y Z

X

Forward stagnation point (maximum pressure drop)

15.3 Flow conditions for a fabric-covered body.


547

commonly placed in systems that have a well-defined flow rate or pressure drop across the filter material. In clothing systems, however, the aerosol barrier is incorporated into a clothing system covering the human body, and the actual flow rates and pressure drops for a particular set of conditions are dependent on the air-flow resistance properties of the fabric layer itself. For a truly valid comparison between aerosol barrier materials which differ in their air permeability properties, it would be desirable to test at a unique volumetric flow rate/filter velocity and pressure drop which corresponds to that produced by a given external air velocity on a typical clothing system. A simple cylinder model is useful in defining a reasonable set of laboratory test conditions for comparing different material candidates for new protective clothing systems.6–11 Similar questions arise for interactions between the convective properties of air permeability, or convective air flow resistance, and the diffusive property of thermal conductivity/thermal resistance. Laboratory evaluations of individual transport properties yield a variety of material responses due to a wide range of thermal resistance and air permeability properties. A CFD simulation of the performance of the materials in the fabric-covered cylinder geometry can help to determine how the total heat and mass transfer coefficients are influenced by a particular combination of fabric properties. An illustration of typical CFD modeling results for fabric-covered cylinders is shown in Fig. 15.4. A body-fitted mesh is created for a cylindrical geometry. A thermal insulation layer, and an air space, is placed around the cylinder. The air-flow resistance property of the insulation layer is varied (while keeping thermal insulation constant). In this particular case the cylinder diameter was 0.187 m (corresponding to the diameter of a human thigh), and the air

Thermal resistance (m2-K/Watt)

1 Fabric air flow resistance (RD) = 1 × 109 m–1

RD = 1 × 108 RD = 1 × 107

0.1

0.01 1

RD = 1 × 106 Bare cylinder (no fabric layer) 2

5

10

Wind speed (m/s)

15.4 Overall heat transfer resistance of fabric-covered cylinders in cross-flow conditions at various wind speeds.

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space between the fabric and the cylinder surface was 0.01 m. The cylinder surface temperature was 35 °C (nominal human skin temperature) and the air temperature was 5 °C. Simulations were carried out over a range of air-flow velocities from 1 to 40 m/s, although only the results up to 10 m/s are shown in Fig 15.4. The curve for the bare cylinder case in this figure agrees with heat transfer correlation experiments for gas flow over heated cylinders. Figure 15.4 shows that the effect of fabric air flow resistance on heat transfer is non-linear in nature; above a certain value of resistance, the fabric-covered cylinder behaves like a solid cylinder, with little air penetration of the system. These types of simulations are useful in efforts to design the proper mix of fabric transport properties to maximize comfort while still providing adequate protection from the environment, and from chemical/biological warfare agents. These modeling techniques were applied to commercially available knit fleece layers that were under consideration for cold-weather applications. Considerable effort is expended to develop clothing that is ‘breathable’ to sweat vapor, yet that will also retain heat even in windy conditions. Modeling simulations were carried out for a variety of new insulating fleece fabrics produced for the outdoor clothing market that address the need for materials which breathe well, but which also keep out enough wind in cold conditions to keep people warm. These fabrics are becoming available in a wide variety of material properties, and it is difficult to rank or evaluate them based on material properties alone. CFD modeling of the behavior of these materials in a simple clothing analog of a fabric-covered cylinder can be helpful in distinguishing significant differences between some of these fabrics. Thermal resistance, water vapor diffusion resistance, and air permeability were determined on flat samples using laboratory test methods. The material properties were then input into a CFD model of a fabric-covered cylinder, with an air space, under conditions of several different wind speeds. The CFD model provides the ability to go directly from laboratory tests of material properties to a system simulation that approximates real-world conditions. An example comparison for three of the fabrics is shown in Fig. 15.5. The baseline bare cylinder results are essentially the cooling effect and evaporative heat loss due to ‘wind chill.’ The presence of a clothing layer modifies the wind chill effect significantly, depending on the air permeability, thermal resistance, and water vapor diffusion resistance of the clothing layer, and the air space between the cylinder surface and the clothing layer. The air permeability of the fabric layer was found to be the most significant parameter affecting the performance of the various materials evaluated. Differences in thermal resistance that seemed important under the stagnant test conditions present in the laboratory tests were much less significant when the materials were compared under moderate wind speeds which produced significant air flow through the model fabric-covered cylinder system.


549

Wind barrier fleece Bare cylinder Wind-resistant fleece Extremely air-permeable fleece 7

5

4 3

Water vapor flux (g/m2-day)

Thermal insulation (Clo)

6

1.2 × 105

Air temperature = 0 °C Relative humidity = 20%

1.0 × 105 0.8 × 105 0.6 × 105 0.4 × 105 0.2 × 105 0

2

0

5

10 15 20 Wind speed (miles/hour)

25

1 0 0

5

10 15 Wind speed (miles/hour)

20

25

15.5 Comparison of heat transfer and water vapor flux for three varieties of insulating fleece.

Of equal importance for military protective clothing systems is the transport of chemical and biological warfare agents into a clothing system. The cylinder model is useful as a starting point for looking at the importance of a property such as fabric air flow resistance on the performance of a particular protective clothing system design. More systematic studies of flow around fabric-covered cylinders use sophisticated turbulence models incorporating vortex shedding in the wake of the cylinder.12,13 Experimental measurements conducted in parallel with CFD computations developed single correlations which predict heat and mass transfer properties based on nondimensional scaling relations between fluid flow rate, fabric air-flow resistance, and the inner gap between the fabric and the cylinder surface. Scaling rules have been deduced as a function of the Reynolds number, the Darcy number and the Damkohler number, representing the wind speed, clothing permeability, and adsorptivity of poisonous trace gases, respectively. The range of values of the studied parameters was based on applications in heat and mass transfer to a clothed human limb in outdoor wind. For a wide range of conditions, air was found to penetrate the outer porous cylinder in the upstream region, down to an angle of approximately 50 degrees from the front stagnation point. In this region, heat and mass transfer are high. Further downstream, heat and mass

550


transfer are dictated by conduction/diffusion through the air layer in-between the two cylinders. For large Reynolds numbers and high fabric air permeability, the flow easily penetrates the porous sheath and heat transfer approaches that of a bare cylinder. For low Reynolds numbers, on the other hand, the boundary layer around the pair of concentric cylinders becomes very thick and limiting for heat transfer, and again heat transfer approaches that of a bare cylinder. The dimensionless parameters for heat transfer and mass transfer can be summarized in a single correlation under a wide range of conditions. Within its range of validity, this correlation is accurate within 5%, compared to the numerical simulations. A battlefield hazard for the soldier is the presence of droplets of liquid chemical agents that have contaminated the outer surfaces of protective clothing. These droplets evaporate and the vapor can diffuse into the clothing system. The droplets can also be carried into the clothing system by convective air flow due to body motion or an external wind. Figure 15.6 is an example of a CFD simulation of a liquid droplet placed onto the outer surface of the fabric-covered cylinder model. All fabric properties are constant with the exception of the fabric air flow resistance. What is interesting about this particular example is that the fabric with the highest air flow resistance does not produce the lowest total exposure to the chemical vapor. The fabric with low air-flow resistance provides a standoff of the

Fabric with high air permeability -Agent swept in and out quickly

Liquid drop

Fabric with low air permeability Agent diffuses in, not swept out

15.6 CFD simulation of liquid droplet evaporating on outside surface of an air-permeable fabric layer covering a cylinder (shade contours refer to vapor concentration, scale not shown).


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droplet from the cylinder surface, while the ventilating air flow carries vapor quickly through the system and away from the cylinder. For the case of the fabric with a high air-flow resistance, the total vapor exposure is higher. This is due to the fact that, although there is little convective flow into the air space under the fabric layer, the vapor is able to diffuse through the clothing and build up to higher concentrations. In this case there is no ventilating air flow to sweep the vapor out of the system, and the total exposure to the vapor is much higher than for the well-ventilated system. Modeling fabric properties in this simple cylindrical geometry provides much of the information required for system-level decisions about various transport property trade-offs between different material candidates for protective clothing materials. However, there are some situations, particularly those having to do with body movement and motion, as well as interfaces and closures between different clothing system components, where more detailed models are desirable. A variety of geometric parameters affect the transport of agent vapor from a liquid surface drop.14,15 These parameters include orientation of the drop relative to the flow direction, gap size between the cylinder and the clothing layer, non-uniform gap size, and multiple clothing layers. The variation in parameters was assessed at two imposed wind speeds: 5 and 20 mph. Figure 15.7 shows the base geometry assumed for these two-dimensional transient simulations. The model is that of a cross-section of a 10 cm diameter Possible locations for agent droplets

90°

45°

0°

135°

Simplified 2-D OD = 10

180°

Gap Single clothing layer 0.5

15.7 Schematic of base geometry for two-dimensional simulations of evaporating surface agent (with droplet locations shown).

552


arm covered with a single 0.5 mm thick clothing layer having a 1.1 cm gap between the surface of the arm and the clothing layer. Fabric permeability was assumed to be a cotton shell (permeability approximately 2.4 × 10–12 m2). Vapor properties correspond to GB (Sarin) (molecular weight of 140.1 kg/kg mol, vapor pressure of 2.9 mm Hg, and vapor diffusivity of 7.5 × 10–6 m2/s). The transient simulations were performed with a time step chosen to resolve the shedding frequency of the vortex street for flow over a circular cylinder. The following sets of geometric parameters were considered: (i) Base geometry with an incident wind direction of 0°, 45°, 90°, 135°, and 180° relative to the droplet. A condition of 0° corresponds to the droplet located at stagnation point. (ii) Wind direction at 0° relative to droplet with single clothing layer having uniform gap spacing of 0.2, 0.6, 1.1, and 2.1 cm. (iii) Wind direction at 0° relative to the droplet with single clothing layer eccentric to the arm with non-uniform gap spacing of 4.1:1, and 6.8:1 (maximum/minimum of 1.767 cm/0.433 cm and 1.918 cm/0.282 cm, respectively with minimum thickness located at the stagnation point). (iv) Wind direction at 0° relative to the droplet with two clothing layers having uniform gap spacing. First case: inner surface of inner and outer layers located 0.55 cm and 1.1 cm from arm surface, respectively. Second case: inner surface of inner and outer layers located 1.1 cm and 2.1 cm from arm surface, respectively. All simulations were run until they reached a stationary oscillatory state in which the solution variables varied with time around a constant asymptotic mean (physical times of approximately 80 seconds and 20 seconds for 5 mph and 20 mph, respectively). Once this condition was reached, the agent concentration was assessed at the arm to determine the maximum concentration (mg/cm3) as a function of time. The area-averaged concentration over the arm surface as a function of time was also computed. For the conditions under consideration here, the amplitudes of the time varying concentrations at the arm were typically small relative to the time average values once the stationary state was reached. Table 15.1 provides a summary of the maximum and area averaged agent concentrations observed at the surface of the arm for these simulations. Note that the saturation concentration imposed at the droplet is 2.19 × 104 mg/m3. The trends are generally consistent with expectations. Droplets located at the stagnation point produce higher exposure than when located at other angles relative to the wind direction. For the 5 mph wind speed, maximum and average concentrations are the same order of magnitude for all orientations. The lowest maximum is observed for the 45° orientation with an increase in maximum concentration as the droplet is


553

Table 15.1 Summary of maximum and area average agent concentration (mg/m3) at the arm surface for various 2-D arm cases Simulation

Case

5 Mph Maximum

20 Mph

Average

Maximum

Average

9386 3238 3778 4914 5641

1769 545 449 542 519

12 850 135 101 69 51

5379 29 15 6 3

Droplet orientation

0° 45° 90° 135° 180°

Non-uniform gap width

4:1:1 offset 6:8:1 offset

14 428 16 900

1582 1512

16 058 17 925

4899 4679

Uniform gap width

0.2 0.6 1.1 2.1

gap gap gap gap

16 867 11 867 9379 7417

1200 1464 1765 2173

14 13 12 11

814 871 842 626

3339 4601 5374 6243

Multiple clothing layers

1.1 cm and 0.55 cm 2.1 cm and 1.1 cm

8787

1357

11 816

4299

6735

1512

10 858

5014

cm cm cm cm

moved around to the back of the arm (180°). The average concentration is similar for all orientations other than the stagnation point droplet location, indicating that diffusion of agent from the droplet location is of primary importance for these wind and fabric conditions. For the 20 mph wind speed, the maximum and average concentrations are several orders of magnitude higher for the stagnation point location, with asymptotically decreasing concentrations as the droplet is moved away from this location. For the fabric conditions considered here, the higher velocity increases exposure for stagnation point conditions, but assists in sweeping agent away when the droplet is located away from the stagnation point. There is an increase in maximum concentration as the droplet is moved around to the back of the arm (180°). The average concentration is similar for all orientations other than the stagnation point droplet location, indicating that diffusion of agent from the droplet location is of primary importance for these wind and fabric conditions. For droplets located at the stagnation point, increased gap widths and multiple clothing layers reduce the observed maximum agent concentration at the arm surface. For this flow configuration, the maximum concentration is always directly under the droplet location. Increasing the distance from the droplet or adding additional resistance to agent transport in the form of an additional clothing layer allows the surrounding air to dilute the agent concentration before reaching the arm surface. The results in Table 15.1 also indicate that increased gap widths tend to increase

554


area-averaged agent concentrations at the arm surface. Increasing the distance between the arm and the outer clothing layer has the effect of increasing the resistance to agent diffusion transport out of the gap region and allowing higher concentrations to be reached within the gap away from the droplet location. As an example of the agent concentration results, Fig. 15.8 shows the agent concentration at the surface of the arm for different uniform gap thicknesses with a droplet located at the stagnation point. The larger gaps show a lower peak concentration. However, the larger gaps show higher concentrations over most of the arm surface.

15.5

Full-body modeling

Detailed computational simulations of clothing systems are not necessary at all steps of an analysis. Use of a body that is a solid geometric shape (no porous clothing layers or air spaces) can be useful to define the air flow pattern and pressure field developed over a human form. This information may then be used with the simpler body segment models (such as an arm or torso) to define exposure levels. Figure 15.9 shows the geometry of a soldier modeled as a solid body (no porous clothing layers). Two calculations were performed with the model, both for a 4.5 m/s headwind. Figure 15.9(a) shows the calculated steadystate flow field around the soldier. The recirculation region immediately behind the soldier is readily visible. In the second calculation (transient), a 1.6E + 04 0.2 0.6 1.1 2.1

Agent concentration (mg/m3)

1.4E + 04 1.2E + 04

cm cm cm cm

gap gap gap gap

1.0E + 04 8.0E + 03 6.0E + 03 4.0E + 03 2.0E + 03 0.0E + 00 –180

–120

–60 0 60 Position on arm (degrees)

120

180

15.8 Agent concentration at the surface of the arm using different uniform gap thicknesses (wind speed 20 mph).


555

(a)

4.00e–01 3.67e– 01 3.33e– 01 3.00e– 01 2.67e– 01 2.33e– 01 2.00e–01 1.67e–01 1.33e–01 1.00e– 01 6.67e–02 3.33e– 02 0.00e+ 00 (b)

15.9 (a) Streaklines for 4.5 m/s headwind (b) Surface concentrations of tracer gas: 0.5 s after release of 1 m spherical cloud at location 2 m in front of soldier.

556


spherical cloud of tracer gas 1 m in diameter is released at a location 2 m in front of the soldier. Figure 15.9(b) displays surface concentrations of gas at 0.5 seconds after release. The results show high concentrations of gas in the soldier’s midsection where the cloud is centered. Figures 15.10(a) and 15.10(b) depict more complex 3-D models of an arm and torso, respectively. Two layers of fabric clothe the arm. The undulations

(a)

(b)

15.10 (a) 3-D Model of arm with two clothing layers. (b) Model of torso (clad in T-shirt).


557

visible on the inner arm near the elbow are in the outer layer of fabric only. The torso model here is clad in a single fabric layer, a crew-neck T-shirt. Both arm and torso models are based on laser scans of humans. Scanned points are brought into computer-aided design software for creation of the body surface and generation of clothing layers. The geometry is then exported to the CFD software’s preprocessor for grid generation. 3-D simulations are generally similar to the 2-D fabric-covered cylinder studies, particularly for the fabric-covered arm segment. Problems arise in the computational requirements for the large number of grid points for a human body, the irregular geometry, and the differences in scale between the size of the human body and the thickness of the mesh required for thin clothing layers. Three-dimensional models of a human torso having one or more layers of clothing have been developed.14,15 Results from a simulation using the computational mesh of Fig. 15.10(b) were performed under steady-state conditions of a 5 mph wind imposed on the front of the torso. Skin conditions are assumed to be a sweat flux of 3 × 10–5 kg/m2 s and heat flux of 100 W/m2, with the shirt modeled as a 1 mm thick cotton fabric. Figure 15.11 shows the temperature at the skin surface and velocity vectors in a plane around the torso for conditions of full closure between the layer of clothing and skin at the bottom of the shirt, ends of the sleeves, and at the 3.09e+02 3.08e +02 3.07e +02 3.06e +02 3.05e+02 3.04e+02 3.04e+02 3.03e+02 3.02e+02 Z

3.01e+02 X

Y

3.00e+02

15.11 Simulation results of clothed 3-D human torso: Temperatures at skin surface and flow around torso with complete closure (snug fit) at neck, sleeves, and waist.

558


neck (i.e. snug fit at neck, sleeves, and waist). As might be expected, the highest temperatures are in the vicinity of under the arm. Furthermore, the presence of the closures and their effect of limiting the ability of flow from the environment to enter the area under the shirt results in higher temperatures. These results represent a first step in performing simulations of thermal and agent transport in protective clothing on a 3-D human torso model.

15.6

Conclusions

Modeling offers a powerful companion to experiments and testing in the development of protective clothing materials and other textiles. Detailed material models for vapor and liquid phase transport within textile fabrics have been developed and integrated with CFD software. Material-level modeling can account for transient processes such as aerosol transport and deposition, liquid wicking, and phase changes due to evaporation, condensation, and sorption/desorption. It is desirable to include these more complicated phenomena in the system-level three-dimensional body modeling efforts. General non-dimensional scaling correlations of heat and mass transfer from cylinders covered with a porous layer are very valuable for application to the problems of clothing comfort and protection from chemical agents. Modeling and experiments for fabric-covered cylinders will continue to be an efficient pathway to understanding the interactions between materials properties and their performance in a protective clothing system. Modeling applications involve assessment of thermal comfort/stress on wearers of protective clothing, effects of layering on protective performance, and sensitivity to textile permeability and wicking properties.

15.7

References

1. P.W. Gibson, Governing Equations for Multiphase Heat and Mass Transfer in Hygroscopic Porous Media with Applications to Clothing Materials, Technical Report Natick/TR-95/004 (U.S. Army Natick Research, Development and Engineering Center, Natick, MA, 1994). 2. P.W. Gibson, Multiphase Heat and Mass Transfer Through Hygroscopic Porous Media With Applications to Clothing Materials, Technical Report Natick/TR-97/005 (U.S. Army Natick Research, Development, and Engineering Center, Natick, MA, 1996). 3. S. Whitaker, in Advances in Heat Transfer 13 (Academic Press, New York, 1977) p. 119. 4. S. Whitaker, in Advances in Heat Transfer 31 (Academic Press, New York, 1998) p. 1. 5. P.W. Gibson, M. Charmchi (1997) Journal of Applied Polymer Science 64, p. 493. 6. P. Brasser and J. Kaajik, Modelling of the Protective Performance of NBC-Clothing 1: Profile Between Clothing and Skin, Technical Report PML 1998-A107 (TNO Prins Maurits Laboratory (The Netherlands) June, 1999).


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7. P. Fedele, W. Bergman, R. McCallen and S. Sutton (1986) Proceedings of the 1986 Army Science Conference, Vol. I p. 279. 8. J. Hanley and P. Fedele (1988) Proceedings of the 1987 U.S. Army Chemical Research, Development, and Engineering Center Scientific Conference on Chemical Defense Research, p. 444. 9. J. Hanley, D. Van Osdell and P. Fedele (1989) Proceedings of the 1988 U.S. Army Chemical Research, Development, and Engineering Center Scientific Conference on Chemical Defense Research, p. 347. 10. R. Kind and C. Broughton, Textile Research Journal 70 (2000), p. 171. 11. R. Kind, J. Jenkins and F. Seddigh (1991) Cold Regions Science and Technology 20 p. 39. 12. M.P. Sobera, C.R. Kleijn and P. Brasser, H.E.A. Van den Akker, presented at the 2002 ASME Pressure Vessel and Piping Conference, Vancouver, British Columbia, Canada, August 4-8, 2002, preprint. 13. M.P. Sobera, C.R. Kleijn and P. Brasser, H.E.A. (2002) Van den Akker, ‘Convective Heat and Mass Transfer to a Cylinder Sheathed by a Porous Layer,’ unpublished manuscript. 14. J.J. Barry and R.W. Hill 2002, presented at the International Nonwoven Technical Conference INTC 2002, Atlanta, GA, September 24-26, preprint. 15. J.J. Barry and R.W. Hill (2002) in Proc. Fall 2002 Fiber Society Annual Technical Conference, Natick, MA, October 16-18, 2002 [CD-ROM] (The Fiber Society, Raleigh, NC, p. 124.

16 The skin’s role in human thermoregulation and comfort E. A R E N S and H. Z H A N G, University of California, Berkeley, USA

16.1

Introduction

This chapter is intended to explain those aspects of human thermal physiology, heat and moisture transfer from the skin surface, and human thermal comfort, that could be useful for designing clothing and other types of skin covering. Humans maintain their core temperatures within a small range, between 36 and 38°C. The skin is the major organ that controls heat and moisture flow to and from the surrounding environment. The human environment occurs naturally across very wide range of temperatures (100 K) and water vapor pressures (4.7 kPa), and in addition to this, solar radiation may impose heat loads of as much as 0.8 kW per square meter of exposed skin surface. The skin exercises its control of heat and moisture across a 14-fold range of metabolisms, from a person’s basal metabolism (seated at rest) to a trained bicycle racer at maximum exertion. The skin also contains thermal sensors that participate in the thermoregulatory control, and that affect the person’s thermal sensation and comfort. The body’s heat exchange mechanisms include sensible heat transfer at the skin surface (via conduction, convection, and radiation (long-wave and short-wave)), latent heat transfer (via moisture evaporating and diffusing through the skin, and through sweat evaporation on the surface), and sensible plus latent exchange via respiration from the lungs. Dripping of liquid sweat from the body or discharge of bodily fluids cause relatively small amounts of heat exchange, but exposure to rain and other liquids in the environment can cause high rates of heat loss and gain. Clothing is used outside the skin to extend the body’s range of thermoregulatory control and reduce the metabolic cost of thermoregulation. It reduces sensible heat transfer, while in most cases permitting evaporated moisture (latent heat) to escape. Some clothing resists rain penetration, both to prevent the rain from directly cooling the skin, and to prevent the loss of insulation effectiveness within the clothing. Wet clothing will have a higher heat transfer than dry: depending on design, it can range from almost no 560


561

difference to a 20-fold increase. Clothing is nearly always designed to allow the wearer’s breath to enter and exit freely in order to keep the temperature and humidity of inhaled air low, and to avoid moisture condensation within the clothing. Bedclothes are a form of clothing used for sleeping. Because the metabolic rate during sleep (0.7 met) is lower than the basal rate, and the body’s skin temperature tends to be higher during sleep, bedclothes typically have a higher insulation value than clothing. Bandages and other medical coverings may also be a special case of clothing, controlling the heat, moisture, and biotic transfer above a damaged skin. This chapter gives a brief description of the body’s skin structure and thermoregulatory system, followed by a more detailed description of how heat and moisture are transferred at the skin’s outer boundary, and finally, the comfort implications of skin temperature and humidity. Since skin characteristics are not evenly distributed across the surface of the entire body, it is useful for clothing design to have this information presented by individual body part, wherever possible.

16.2

Body–environment exchange

Over time, heat gains and losses must balance to maintain homeothermy – maintaining the body’s core temperature within its narrow range. Figure 16.1 illustrates the full range of core temperatures and environmental temperatures encountered by humans.

16.2.1 Heat gains Most of the body’s heat production is in the liver, brain, and heart, and in the skeletal muscles during exercise. This heat is transferred, through the network Environmental temperature (°C) Very hot sauna 100 90 Hot desert sand 80 70 Hot beverage 60 July air temp., Death Valley 50 Hot bath 40 Mean July temp., Yuma, Arizona 30 Cool bath 20 10 Mean Jan. temp., New York City 0 –10 –20 –30 Mean Feb. temp., South Pole Station –40

Core temperature (°C) 44 42 40 38 36 34 32 30

Upper limit for survival Impaired thermal regulation Heat stroke, brain damage Extreme physical exercise and fever Normal range Intense shivering and impaired coordination Violent shivering; speech and thought impaired Decreased shivering; erratic movements; incoherent

16.1 Ranges of environmental and human body temperatures (adapted from Brooks et al., 1996).

562


of blood vessels and tissue, to the skin, from whence it is lost to the environment. The amount of metabolic heat generation depends on the level of muscular exercise, and to a lesser degree on factors such as illness and time in the menstrual cycle. A base level of metabolism has been defined as the metabolism of a seated person resting quietly. For a man of typical height and surface area, this amount is about 100 W. To normalize among people of different sizes, metabolism is typically expressed in per unit skin surface area. A specialized unit, the ‘met’, has been defined in terms of multiples of basal metabolism: 1 met is 58.15 W/m2. A sleeping person has a rate of 0.7 met, and reclining awake is 0.8 met. Office work is 1.2 met: a mostly seated activity but one that involves occasional moving about. Walking slowly (0.9 m/s, or 2 mph) is 2 met, moderate walking (1.2 m/s or 2.7 mph) is 2.6 met, and fast walking (1.8 m/s or 4 mph) is 3.8 met (ASHRAE Handbook of Fundamentals, 2005). Swimming ranges from 4 to 8 met, and jogging 8 to 12 met (Brooks et al., 1996). The work efficiency of muscles is about 15%, with 85% of total energy released as heat. Brain metabolism consists mostly of the energy required to pump ions through neuron cell membranes (Guyton and Hall, 2000). This takes place at a rate per unit mass that is 7.5 times that of non-nervous system tissues. Although the brain only comprises 2% of the body mass, it produces about 15% of the body’s total metabolism. During high mental activity, this neuron metabolism can more than double. The head has specialized thermoregulatory physiology to assure the high rates of heat loss needed to keep the brain temperature constant. Heat may also be gained from the environment through the skin. Solar radiation, and long-wave radiation from surfaces warmer than skin temperature, warm the skin as a function of its color and surface emissivity. Although in most conditions convection and evaporation carry metabolic heat away from the body, hot winds may cause the skin to warm, when the body’s sweat supply rate is insufficient to keep up with evaporation, and sensible gains exceed evaporative losses.

16.2.2 Heat losses The body’s heat losses are through radiation, convection, conduction, evaporation, and through respiration. Figure 16.2 shows heat transfers above and below the skin surface. In a neutral environment, where the body does not need to take thermoregulatory action to preserve its balance, evaporation provides about 25% of total heat loss, and sensible heat loss provides 75%. During exercise, these percentages could be reversed. In general, the heat transfer by conduction through the soles of the feet or to a chair is small, around 3%. In normal indoor environments with still air, the convective and radiation heat transfer are about equal (McIntyre and Griffiths, 1972). In the

The skin’s role in human thermoregulation and comfort 75%

Convection

Skin

Radiation

563

25%

Conduction

Vasoconstriction and vasodilatation

Evaporation

Sweat glands

Respiration Metabolic heat production

16.2 Heat transfer through and above the skin.

outdoors, wind strongly affects convective heat loss or gain, and radiation (solar and long-wave) can also cause large losses and gains. These forces act asymmetrically on the body, affecting some parts more than others.

16.2.3 Thermal regulation Thermoregulation generally refers to four mechanisms: sweating, shivering, vasodilatation, and vasoconstriction. Sweating increases body heat loss by increasing sweat evaporation. Shivering produces heat by involuntary movement of muscle. Vasodilatation and vasoconstriction refer to changes in blood vessel diameter, which affect skin temperature by changing the rate of blood exchange with the interior. In the heat, increased conductance below the skin surface (due to increased blood flow) facilitates heat transfer from body interior to the skin. Then convection and evaporation of sweat carries the heat away from the surface of the body to the environment. In the cold, muscle tensing and shivering increase heat production and body temperature. Decreased conductance (due to decreased blood flow) keeps the heat from escaping to the cold environment. This combination of heat loss and heat gain control mechanisms is able to maintain human body core temperature within a very small range in spite of variation in metabolic output that can exceed an order of magnitude above the base value, and similar variation in the heat loss rate from body to the environment. A comprehensive overview of the thermoregulatory control system is found in Guyton and Hall (2000) and Gagge and Gonzalez (1996). The control system senses the body’s thermal state with sensory organs in the hypothalamus (within the brain), within the skin, and in the spine and some

564


abdominal organs. The thermal sensors within the anterior hypothalamus sense the core temperature of the body, especially that of the brain, by measuring the temperature of blood passing through it. The anterior hypothalamus’s warm sensors outnumber its cold sensors by three to one, and are most active when the body core is too hot. The anterior hypothalamus primarily acts as a controller of the body’s heat loss; any rise in hypothalamus temperature above its set point causes it to send out nerve impulses to activate vasodilatation and sweating, the body’s heat loss mechanisms. The mechanism is precise: the setpoint for vasodilatation and sweating is only a couple of tenths of a degrees higher than the set point for vasoconstriction, and the setpoint for shivering is just below 36 °C (Sessler, 2006). These setpoints are raised during exercise or fever. The skin temperature also plays a secondary role in controlling cooling in the heat: at the same core temperature, a warmer skin temperature enhances the sweat rate, and a colder skin inhibits it (Stolwijk et al., 1971; Nadel et al., 1971). Cold- and warm-sensitive nerve endings located in the skin send signals, through the sympathetic nerve system to the anterior hypothalamus, that are passed on to the posterior hypothalamus, which acts a controller of body temperature during cold. The skin has many (ten times) more cold sensors than warm, and the cold sensors are closer to the surface than the warm, so these peripheral sensors are more dedicated to the rapid detection of cold than of warmth. There are some cold-sensitive temperature sensors in the anterior hypothalamus, and in the spine and abdomen, that also alert the posterior hypothalamus to body cooling. The posterior hypothalamus emits nerve signals to the periphery, stimulating vasoconstriction and shivering, and it also initiates the release from the medulla of hormonal messengers such as norepinephrine that rapidly initiate vascular contraction throughout the body. If a local part of the body is warmed or cooled, sweating or vasoconstriction can be locally initiated and controlled for that particular area, even if the rest of the body is being centrally controlled for a different temperature. The relative contributions to sweating from core and skin temperatures are about 10 to 1 (Nadel and Stolwijk, 1973; Nadel at al., 1971; Benzinger et al., 1961). The core threshold for sweating decreases by 0.6 °C as the skin temperature is warmed from 29 °C to 33 °C. Similarly, with the hypothalamus temperature constant, heating a local body part can induce local sweating (Nadel et al., 1971; Randall, 1946).

16.3

Skin

16.3.1 Skin surface area The area of skin on the body can be estimated from the body’s height and weight, using a relationship developed by DuBois and DuBois (1915):


ADubois = 0.202 M0.425 L0.725 m2

565

[16.1]

where ADubois is the skin area in m2, M is the mass in kg, and L the person’s height in m. A 1.65 m person weighing 73 kg will have a skin surface area of 1.8 m2, a commonly used figure for ‘standard’ men. The range of surface areas from school-age children through large adults is 0.8 through 2.4 m2. The surface areas of local body segments vary among individuals, but it can be useful to know the relative percentages of total surface area that they cover. Table 16.1 presents such percentages for a detailed female thermal manikin with a total surface area of 1.588 m2.

16.3.2 Skin structure The skin provides a first barrier between the organism and its environment. It keeps the uncontrolled loss or gain of water through the skin at a low constant level. In addition to that, it contains complex vascular systems and sweat glands that allow it to change its conductance in response to thermoregulatory demands of the body. It also contains four types of thermallysensitive nerve endings (to cold, warmth, and hot and cold pain) that sense the skin’s temperature and transmit the information to the brain. Although there are some regional variations in skin thickness, in most places the skin is about 2 mm thick. It includes two main layers, the epidermis and dermis (Fig. 16.3). The epidermis is thin, mostly about 0.075–0.15 mm (except for the soles and palms, which are thicker). The outermost layer of the epidermis is the Table 16.1 Body surface areas for a detailed female thermal manikin Body part

Area (m2)

Percentage (%)

Head Chest Back Pelvis L-Upper arm R-Upper arm L-Lower arm R-Lower arm L-Hand R-Hand L-Thigh R-Thigh L-Calf R-Calf L-Foot R-Foot Whole-body

0.117 0.143 0.135 0.143 0.093 0.093 0.063 0.063 0.039 0.039 0.143 0.143 0.125 0.125 0.048 0.048 1.588

7.5 9.2 8.6 9.2 5.9 5.9 4.1 4.1 2.5 2.5 9.2 9.2 8.0 8.0 3.1 3.1 100

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Thermal and moisture transport in fibrous materials Sweat pore Hair Epidermis

Stratum corneum Stratum granulosum

Stratum basale

Capillaries

Dermis

Eccrine sweat gland Hair follicle Nerve Subcutaneous fat

16.3 Cross-sectional view of the skin (Image courtesy of LifeART.com). Stratum corneum > 25 cell layers Stratum granulosum 2–3 cell layers

Stratum spinosam 2–7 cell layers

Stratum basale 1–2 cell layers Basal membrane

Dermis

16.4 Layers of the epidermis (Copyright (2004) from Skin, Hair and Nails: Structure and Function by Forslind and Lindberg, Reproduced by permission of Routledge, Taylor and Francis Group LLC).

stratum corneum (Fig. 16.4), an assemblage of overlapping plate-like cells (corneocytes), interleaved with hydrophobic layers of lipids. The stratum corneum is 0.01 to 0.1 mm thick, and serves as the skin’s primary barrier to water diffusion. Because the corneocytes are impervious to water transmission, whatever moisture passes the stratum corneum barrier has to travel around them through the lipids, following a long tortuous path back and forth among the plates. The stratum corneum is well described in Forslind and Lindberg (2004), who make the memorable point in their introduction that this waterproof


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barrier protecting our bodies is thinner than the plastic cling wrap used to protect our sandwiches. The corneocytes are non-viable, having lost their nucleus and organelles. They are continuously shed from the surface as they are replaced from below. The shedding is accomplished by the dissolution of small rivetlike structures called desmosomes that hold the plates together. The dissolution is mediated by enzymes controlled by the moisture gradient in the stratum corneum layer. Corneocytes consist of a protein cell wall and a matrix of keratinous fibrils within, which stiffen the structure. When immersed in water or exposed to high levels of atmospheric humidity, they absorb moisture and thicken by as much as 25%; this is thought to smooth the outer skin surface and protect it from tearing when wet (Forslind and Lindberg, 2004). Below the stratum corneum, at the bottom of the epidermis, is a basal layer of stem cells (‘stratum basale’), which generates epidermal cells continuously. Above it are two layers in which the upward-migrating cells transform themselves into the interleaved plates and lipids of the stratum corneum. The basal level has an undulating lower contour to provide mechanical shear resistance, connecting the epidermis to the dermis layer below it. The dermis is much thicker than the epidermis, varying by body part (Fig. 16.5, Rushmer et al., 1966). It contains vascular systems, sweat glands, and thermoregulatory nerves at different depths in the layer. These will be described in the following sections. The dermis also houses nail and hair follicles, which produce keratinized structures physiologically related to the stratum corneum. Sebaceous glands within the dermis serve the functions of smoothing and moisture-proofing the outer surface of the skin, and coating hair to reduce tangling. Beneath the dermis lies the subcutaneous or fat layer, whose thickness is highly variable among individuals (for a normal person, it is, on average,

mm. 4

Epidermis

3

Dermis Thick stratum corneum

2

1 Scalp

Forehead

Bock

Abdomen Thigh

Wrist

Palm

16.5 Regional variations in thickness of skin (From Rushmer et al., 1966, with permission from the American Association for Advancement of Science, Washington, D.C.).

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about 17 times the thickness of the dermis – Stolwijk and Hardy, 1965). It serves the functions of insulating the underlying musculature against conductive heat transfer to the outer skin, as well as of storing food energy for the body.

16.3.3 Thermoreceptors Human beings can perceive different levels of cold and warmth (including pain) through four discrete types of sensory organs – cold, warmth, and cold and hot pain receptors (Guyton and Hall, 2000; Craig, 2003). The relative degrees of stimulation of the nerve endings determine the person’s perception of the intensity of thermal sensation. The discovery of discrete thermoreceptors was made independently in 1884/1885 by Blix in Sweden, Goldscheider in Germany, and Donaldson in America. All three investigators, and many since, have reported that, when touched with small (punctate) warm and cold stimulators, some spots on the skin feel warm and/or cold, others do not. Each receptor is activated in a specific range (Fig. 16.6). At high temperatures perceived as painfully hot, warmth receptors are inactive, and pain receptors are simulated. The same is true for painfully cold temperatures. If a warm stimulus is applied to a cold thermoreceptor, no signal is produced. Thermoreceptors are located mainly in the skin and in the hypothalamus, but are also found in places such as the spinal cord, abdominal viscera, and in or around the great veins in the upper abdomen and thorax.

Freezing cold

Cold

Cool

Indiffer- Warm ent

10

Impulses per second

Burning hot

Hot

Warmth receptors

8 Cold receptors

6

Heat pain

Cold pain 4 2

5

10

15

20

25 30 35 40 Temperature (°C)

45

50

55

60

16.6 Discharge frequencies of a cold receptor, a warmth receptor, and cold and hot pain nerve fibers at different temperatures (From Guyton and Hall, 2000: Textbook of Medical Physiology, with permission from W.B. Saunders Company, Philadelphia).


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The thermoreceptors are located in the dermis at an average depth of 0.15 to 0.17 mm for cold receptors and 0.3 to 0.6 mm for warmth receptors (Bazett and McGlone, 1930; Bazett et al., 1930; Hensel, 1982). These depths indicate that the layer of cold receptors is immediately beneath the epidermis, and the site of warmth receptors is within the upper layer of the dermis. The number of cold thermoreceptors far exceeds the number of warmth receptors. In general, there are about ten times more cold receptors than warmth receptors in skin (Guyton and Hall 2000). The distribution of the cold and warm receptors is shown in Table 16.2. Figure 16.7 displays examples from classic studies: the warm and cold receptors on the dorsal forearm (Strughold and Porz, 1931), and warm receptors on the fingers (Rein, 1925). The preponderance of cold spots over warm spots, and the shallower depth of cold spots relative to the skin surface, suggest that humans are more sensitive to danger from cold than from heat. The dynamic characteristics of thermoreceptors determine thermal sensation and comfort responses. A thermoreceptor is capable of a great deal of adaptation. When it is subjected to an abrupt change in temperature, it is strongly stimulated at first, sending impulses at a high frequency, but this stimulation fades rapidly during the first minute following the temperature change, and then progressively more slowly until it reaches a steady level (Fig. 16.8 – Hensel, 1982). Thermoreceptors respond to steady temperature states at this lower rate. A person feels much colder or warmer when the temperature of the skin is actively falling or rising than when the temperature remains at the same Table 16.2 Number of cold and warm spots per cm2 in human skin Body parts

Cold spots (Strughold and Porz 1931)

Forehead Nose Lips Other parts of face Chest Abdomen Back Upper arm Forearm Back of hand Palm of hand Finger dorsal Finger volar Thigh Calf Back of foot Sole of foot

5.5–8 8 16–19 8.5–9 9–10.2 8–12.5 7.8 5–6.5 6–7.5 7.4 1–5 7–9 2–4 4.5–5.2 4.3–5.7 5.6 3.4

Adapted from Hensel, 1982

Warm spots (Rein 1925) 1 1.7 0.3

0.3–0.4 0.5 0.4 1.7 1.6 0.4

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Dorsal nerves

Radial nerves

(a)

1 cm2

(b)

16.7 (a) Warm and cold receptors on the dorsal forearm, and (b) warm receptors on the fingers.

Impulse frequency


571

STATIC

DYNAMIC

Warm receptor

Warm receptor

T1

T2

Cold receptor

Cold receptor

T1 T2 Temperature

Temp

T2 T1

T1 Time

16.8 General properties of thermoreceptors. Static and dynamic responses of warm and cold receptors to constant temperature and temperature changes (From Hensel, 1982: Thermal Sensation and Thermal Receptors in Man, courtesy of Charles C Thomas Publishers, Springfield, Illinois.).

level. This explains the stronger sensation of coolth or warmth felt upon entering a cold pool or a hot tub. The overreaction during transient exposures has been termed ‘overshoot’ (deDear et al., 1993; Gagge et al., 1967; Zhang, 2003). The dynamic response of thermoreceptors to changes in temperature in essence predicts the body’s steady-state response to a new thermal environment well before the body’s heat content has had time to alter significantly. Such a capability has clear adaptive value for survival.

16.3.4 Vascular system Figure 16.9 illustrates the vascular system in the skin. The primary function of blood circulation is to deliver nutrients and oxygen to tissues and organs. In addition to that, blood circulation assists the principal mechanisms of thermal homeostasis. It keeps the heat within the body when it is cold by reducing blood circulation (vasoconstriction) to the skin, or enhances the outward flow of heat to the skin by vasodilatation. In the outer region of the skin (epidermis and outer dermis), the thermal resistance of the tissues determine heat flow, and the variation of blood flow within the small dermal capillaries is not thermally important. Below these, however, the subcutaneous region contains the venous plexus, a dense vascular network that strongly affects skin temperature and heat transfer from the

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Epidermis

Epidermis Dermis

Artery Veins Venous plexus

Subcutaneous tissue

Arteriovenous anastomosis Artery

16.9 Vascular system in the skin (From Guyton and Hall, 2000: Textbook of Medical Physiology, with permission from W.B. Saunders Company, Philadelphia, p 823, with permission from Elsevier).

skin to the environment. Blood flow into the venous plexus is fed by arterioles, which vasodilatate and vasoconstrict significantly. Since blood flow varies to the fourth power of vessel diameter, a doubling in diameter corresponds to a 16-fold increase in the blood supply volume. In cold, the blood supply to the venous plexus can be effectively as low as zero, resulting in a local gradient across the skin of 10 K. In heat, dilatation can cause an eight-fold increase in the skin conductance, producing a gradient from the body’s central core temperature to skin surface temperature that is less than 1 K. Vascular control takes place through the sympathetic nervous system. When stimulated in the posterior hypothalamus or skin, the rate of nerve impulses transferred to the periphery activates vasoconstriction or vasodilatation. The relative contributions to the control signal from the hypothalamus (core) temperature and from the skin temperature are in the order of 10:1, so the system is heavily weighted toward representing the overall body thermal state. Highly exposed areas of the body, such as the fingers, hands, feet, and ears, have an additional vascular control mechanism that can vary their temperature and heat loss across a wide range. Arterio-venous anastamoses (AVAs) (Sherman, 1963) are present in these areas in large numbers. These are valves that, when open, shortcut the normal route of the blood from the arterioles to the venous plexus. The diameters of AVA are 20–150 µm, about 20 times larger than that of capillaries (1–8 µm) and five times larger than that of arterioles (15–30 µm) (Hales, 1984, 1985). When the body is hot, AVAs are stimulated by sympathetic nerves to open and rapidly supply arterial blood to the venous plexus, which acts as a warm reservoir close to the skin


573

surface. This promotes heat loss by conduction through the overlying tissues to the surroundings. Warm-blooded animals keep their core temperature fairly constant using another important vascular feature: counter-current heat exchange between arteries and veins, where the warm outbound arterial blood transfers heat to the cold inbound venous blood. The counter-current vascular structures can be categorized into three types. The first has one artery and one vein in parallel, as exist down the lengths of our arms and legs, and in most birds’ legs. The second has one artery surrounded by many veins, as in human fingers and relatively uninsulated body parts such as whales’ fins and some animals’ tails. The third consists of a net where 20–40, or sometimes several hundred, small arteries and veins run parallel and are intermingled (SchmidtNielsen, 1972). An example of this is the rete mirabile, found between the nasal passages and the brain of non-sweating animals like dogs, which rely on panting for keeping their brains cool. Unlike tissues in other body parts, the brain stores very small amounts of energy and oxygen. Due to the high level of metabolic rate of neurons, the brain needs second-by-second delivery of glucose and oxygen from the blood (Guyton and Hall, 2000), a relatively constant blood flow, 750 ml/min, about 13–15% of the total cardiac output. Since this amount cannot be reduced, the head does not have a vasoconstriction mechanism. If the body is cooling in a cold environment, the head should be wrapped in order to protect it from excessive heat loss. In heat, selective brain cooling (SBC, discussed below under evaporative cooling systems) is able to keep the brain temperature lower than the nearby core temperatures measured in the esophagus (Cabanac, 1993).

16.3.5 Evaporative control systems Because the temperature gradient between the skin surface and the environment diminishes in hot weather, sensible heat transfer becomes insufficient to remove the body’s metabolic heat. Evaporation of body moisture is a highly efficient heat removal process, and therefore complex physiological mechanisms have evolved to encourage evaporation under conditions of heat stress, and to minimize it when not, both to avoid overcooling and to minimize the amount of water lost by the body. Insensible evaporative heat losses There is always a constant amount of trans-epidermal loss of water vapor directly diffused through the skin, resulting in heat loss by ‘insensible evaporation’. In addition, the breathing cycle involves humidifying exhaled air, producing another evaporative heat loss. The transdermal moisture diffusion

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is about 100 to 150 mL per day per m2 of skin surface, representing a heat loss 6% as great as the evaporation from a fully wetted surface. The respiratory portion of the body’s total heat loss is estimated as 8%, depending on the metabolic rate. Although both these modes of evaporation contribute to heat loss from the body, neither is controlled for the purpose of temperature regulation. Thermoregulatory sweating mechanism The body’s eccrine sweat glands primarily serve the purpose of thermoregulation, although emotions can also stimulate them. Figure 16.3 shows an eccrine sweat gland and its opening onto the skin surface. When the body becomes overheated, sweat is secreted onto the surface of the skin and is evaporated by the heat supplied by the skin surface. If the atmosphere is dry, evaporation is effective, and high sweat rates can occur without wetting much of the skin around the sweat gland opening, so sweating may not be perceived. If the atmosphere is moist, the sweat-covered area around the sweat gland must increase in order to evaporate the quantity of sweat coming out of the gland. The term ‘skin wettedness’ alludes to this area. It is the fraction of the skin covered with water that would account for a total amount of observed evaporation (ASHRAE Fundamentals, 2005). Perspiration as secreted has a lower salt concentration than interstitial body fluid or blood plasma. Over short intervals of sweating, sweat evaporates indistinguishably from water (Kerslake, 1972; Berglund and McNall, 1973). With extended sweat evaporation, salt may accumulate on the skin, reducing the sweat’s vapor pressure and evaporative efficiency. However, this is somewhat offset by the sweat glands reducing the salt concentration of sweat after prolonged heat exposure, presumably to conserve salt. Eccrine sweat glands are regulated by the autonomic nervous system. For thermoregulation, they are activated through nerve fibers that stimulate the release of the neurotransmitter acetylcholine. Warming in the anterior area of the hypothalamus excites sweating through the whole body. However, as mentioned in Section 16.2.3, the threshold of the hypothalamus temperature for sweating is somewhat modified by skin temperature (Nadel et al., 1971; Benzinger, 1961). An increase in sweat production is brought about by both increasing the number of participating sweat glands and by increasing the output of each active gland. The primary response to heating a local skin area is to increase the output of individual glands, rather than stimulating a larger number of glands to sweat (Randall, 1946, 1947). Sweat does not begin simultaneously all over the body. On the onset of sweating, the first area is generally the forehead, followed in order by the upper arms, hands, thighs, feet, and back and abdomen (Houdas and Ring, 1982); a little different order is presented


575

by McIntyre (1980) after Kuno (1956) and Randall (1946). The high skin temperature and small number of sweat glands on the zygomatic and buccal regions of the face have been hypothesized to help the evaporation of sweating droplets running down over these regions from the forehead (Randall, 1946). With repeated intermittent heat exposure, the set point for the onset of sweating decreases and the proportional gain or temperature sensitivity of the sweating system increases (Hensel, 1982). This is acclimatization, and it takes place over a time period of about a week. However, under long-term exposure to hot conditions, the sweating set point increases, perhaps to reduce the physiological effort of sweating. Sweat gland distribution The distribution of eccrine sweat production across the body is described in Kuno (1956) and is shown in Table 16.3. The average density of active glands is around 125–200 glands/cm2, depending on the individual (Kuno, 1938; Randall, 1946), although considerable variation exists in different areas of the body (Randall, 1946). Table 16.4 gives the sweat gland distribution in various areas. Table 16.3 Distribution of eccrine sweat production Body parts Greatest

Forehead, neck, back of hand and forearm, back and front of trunk

Middle

Cheeks, arms and legs, lateral surface of trunk

Least

Inside of thighs, soles, palms, armpits

Kuno, 1956

Table 16.4 Sweat gland distribution Segment

Sweat glands (functional pores/cm2)

Forearm, extensor surface Upper arm, over biceps Dorsum of hand Trunk – anterior chest Trunk – scapular region of back Leg (over gastrocnemius) Thenar eminence Face – forehead Face – zygomatic arch (temple) Face – buccal (cheek, jaw)

213 177 377 151 30 99 284 167 21 16

Adapted from Randall, 1946

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Acclimatization and sweat dripping People are born with a considerable excess of sweat glands, but if they do not spend their early years in a hot climate, many of the glands become permanently inactive. Whether or not people live in hot climates, accumulated exposure to heat or exercise (acclimatization) will cause more of their available sweating glands to be active, so that their sweat more evenly covers their bodies, making their evaporative heat loss more effective (Kuno, 1956; Guyton and Hall, 2000). In the heat, if sweat beads up and drops off the skin, it normally means that the person is not well acclimatized. An acclimatized person, in fact, looks drier, even though the sweat rate may be greater. Dripping sweat is adaptively undesirable because it provides no evaporative cooling to the skin while dehydrating the body. The maximum sweat rate for an unacclimatized person seldom reaches 1 liter/hour. However, a well-acclimatized person can sweat as much as 2–3 liter/hour, which, when evaporated, removes about ten times the basal body heat production. It is equivalent to a 4–6 pounds of body weight loss per hour (Guyton and Hall, 2000). The concentration of sodium chloride in the sweat is also smaller for an acclimatized person. Hidromeiosis When a local area of skin is thoroughly wet for periods exceeding two hours, the sweat rate abruptly drops off, an effect called ‘hidromeiosis’ (Sargent, 1961; Hertig et al., 1961; Candas et al., 1980). It can be brought on by a large amount of unevaporated sweat on the skin surface, or by exposure to very high ambient humidities (such as produced by waterproof coverings over the skin). If the skin is dried (by a towel, wicking fabric, or by convection), the sweat-rate dropoff is rapidly reversed. The causal mechanism has been the subject of debate for many years and has not yet been resolved. Proposed mechanisms include squeezing the sweat pores by swelling in the epidermal cells, changes in the water/salt balance around the sweat glands affecting the secretion rate, and changes in the transmission of nerve impulses at the junction between nerve and gland. Because a fully wetted skin surface cannot further increase its evaporative heat removal rate, and is likely to be experiencing sweat drip, the hidromeiosis mechanism has adaptive benefit to the body by reducing unproductive (in heat removal terms) water loss (Ogawa et al., 1984). Salt will accumulate on the surface of the skin as the water in sweat evaporates. Salt lowers the vapor pressure of water and thereby reduces its evaporation rate. Accumulated salt therefore results in increased skin wettedness for a given rate of evaporative cooling. The ASHRAE Handbook (2005) suggests that some of the pleasure of washing after a warm day comes from the restoration of a hypotonic sweat film and decreased skin wettedness.


577

Emotional sweating The palms of hands and soles of feet have a large number of eccrine sweat glands, but these do not respond during thermal stimulation or play a substantial role in thermoregulation. They do, however, sweat profusely as a result of emotional excitement and strong mental activity (Kuno, 1956). The sweat glands on palms and soles are stimulated by epinephrine or norepinephrine circulation in the blood. This emotional sweat starts and stops very rapidly, and the resulting changes to electrical resistance of the skin (galvanic skin response) are exploited by lie detectors to detect psychological stress. Apocrine sweat glands There is another type of sweat gland, the apocrine. The apocrine glands in humans are mainly located in the armpits and pubic region, always exiting to the skin surface via hair follicles. They are stimulated by adrenergic nerves, part of the sympathetic nervous system. Apocrine sweat contains a mixture of substances unlike that of the more watery eccrine sweat (Goldsmith, 1991); the decomposition this mixture by bacteria in the skin creates its odour (McIntyre, 1980). Due to their locations on the body, apocrine sweat glands serve little thermoregulation purpose. Guyton and Hall (2000) suggests that apocrine sweat, together with sebum exiting from the hair follicles, serves a lubrication function in these areas of the body where skin surfaces touch each other. Selective brain cooling The brain, with its high rate of internal heat production, needs to be kept cool within a narrow range to avoid damage. The lack of vasoconstriction in the scalp blood vessels has already been noted. Brain temperature, as measured on the tympanum (inner ear), is typically higher than or equal to other core temperatures in the cold, but in the heat, a mechanism called selective brain cooling (SBC) keeps it lower than the other core temperatures (Caputa and Cabanac, 1988). SBC cooling occurs through upper airway respiration and high heat losses from the surfaces of the head, via convection, radiation, and sweating. Sweating from the surface of the head is maintained at the expense of warming the rest of the body: in mild hyperthermia with the body mildly dehydrated, the sweat rate on the trunk will be depressed while remaining constant on the head (Cabanac, 1993).

16.3.6 Shivering In cold environments, the body first conserves its internal temperature by vasoconstricting its blood vessels. If this passive insulative measure proves

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insufficient, the body begins to actively generate additional metabolic heat through tensioning its muscles, starting with ‘muscle tone’ in the skin, and then leading to involuntary shivering. Shivering can raise metabolic heat production by as much as three times the normal sedentary value. It begins in the trunk region and spreads to the limbs (Spurr et al., 1957). In the cold, skin temperature has a more important role in thermoregulation than it does in the heat. The relative contribution from core and skin temperatures in eliciting shivering is between 4 and 5 to 1 when the skin is cold (Tikuisis and Giesbrecht, 1999; Stolwijk, 1971), compared to the 10 to 1 ratio seen for sweating. When the core becomes very cold, below 30 °C, the metabolic heat produced by shivering diminishes (Bristow and Giesbrecht, 1988); it is thought that the body does this in order to conserve energy for survival. Shivering onset requires simultaneous cold signals from both the skin and core. Shivering is not activated when a person is exercising in a cold environment with a cold skin temperature but high core temperature. People with higher body fat shiver less at a given temperature due to the insulation provided by the fat. Shivering heat production is inversely proportional to the square root of body fat (Tikuisis and Giesbrecht, 1999). In addition, fatter people tend shiver less because they produce more metabolic heat by carrying the additional mass.

16.4

Heat exchange at the skin surface

16.4.1 Conduction Conduction takes place primarily to solid surfaces in the environment, since conduction from the skin surface to fluid or gaseous surroundings is rapidly replaced by convective processes. A standing person has roughly 3% of his/ her body surface area in contact with the floor. For a seated or reclining person, the surface contact area to the seat or bed is in the order of 8 to 12% of total body surface area, depending on how yielding the contact surface is. The overall heat transfer from the body via conduction is usually less than these percentages suggest, because the conductivity of bedding and seating materials tends to be small. However, if the contact surface conductivity is high (such as the earth), conduction can be a dominant path of heat exchange with the environment: K = hk (Tskin – Tsurface) (W/m2)

[16.2]

where K is conductive heat transfer from the skin surface to a contacting surface. In a transient state, the heat flux to and from a contacting environmental material will depend on the thermal inertia of that material, quantified as its volumetric heat capacity multiplied by its conductivity. The thermal inertia determines the amount of heat that the skin will gain or lose from contact with


579

Table 16.5 Values of thermal inertia of, and sensation of heat or cold given by, various substances Material

Thermal Inertia (J/m4 K2 sec)

Steel Concrete Rubber Oak Pine Cork

0.5 1 1.2 7.3 2.4 6.0

× × × × × ×

10–2 10–2 10–3 10–4 10–4 10–5

Threshold of pain (cold)

Temperature (°C) range of comfort

Threshold of pain (hot)

14 4 –12 –20 –53 –140

29–32 27–34 24–35 22–35 17–39 5–42

45 54 67 74 84 150

Houdas and Ring, 1982

that material over short-term periods. Houdas and Ring (1982) present a table of common materials, and the temperature ranges at which contact with them is comfortable, or produces hot or cold discomfort or pain (Table 16.5). For comparison, the thermal inertias of the body tissues themselves are: skin, 30–80 × 10–4, fat 10–15 × 10–4, muscle 30–60 × 10–4, and bone 25–40 × 10–4. Each of these is temperature dependent, reflecting blood flow and content.

16.4.2 Convection Convective heat loss from the body surface is often expressed as a heat transfer coefficent and the difference between the mean temperature of the outer surface of the body and that of the surrounding air: C = hc (Tskin – Ta) (W/m2)

[16.3] 2

where hc = convective heat transfer coefficient (W/m · K) Still-air convective heat transfer coefficients If there is no air motion over the skin surface, a relatively thick layer of heated air will lie adjacent to the surface, and the temperature gradient across this layer is determined by the conductivity of air (k = 0.024 W/m K). However, in nature, the thermal gradient produces a buoyancy gradient in the air, which, depending on the air’s viscosity and the orientation of the surface relative to gravity, will cause bulk motion that decreases the conductance of this boundary layer. The heat transfer through such a naturally convecting boundary layer is governed by the Nusselt, Prandtl, and Grashof numbers, which take into account the diffusion and bulk convection of heat. Still-air convective heat exchange from the human body is dominated inevitably by a slow-moving buoyant plume of air that develops adjacent to the heated body surfaces, and rises along the body carrying heat, water

580


vapor, and bioeffluents with it. The plume usually envelops the head, and affects the subject’s breathing and olfactory senses, giving it an important effect on thermal sensation. The convective heat transfer to the plume depends on posture, clothing, and the local temperature of the air at various points in the plume. This has recently become the subject of computerized fluid dynamics (CFD) simulations (Murakami, 2004), and detailed laboratory visualizations using particle-tracing anemometry. Since the free-convection process is complex, still-air convective dry heat loss coefficients have been obtained empirically. deDear et al. (1997) used a detailed thermal manikin whose individual body segments were electrically heated to maintain realistic skin temperatures, measuring the required wattage to obtain the heat transfer coefficient for each segment. Radiant losses were decoupled by varying the surface emissivity of the manikin. (This manikin’s surface areas are shown in Table 16.1, Section 16.3.1.) Table 16.6 gives some natural convective heat transfer coefficients (hc) for a nude thermal manikin standing and sitting in still air (velocity < 0.1 m/s), from deDear et al. (1997). These were obtained at a fixed skin-to-air temperature gradient of 12 K. Using the same manikin, Oguru et al. (2002a) found the still-air coefficients to vary with the temperature difference between air and skin. He presents power functions for calculating them for each body part. For the overall body, seated, the coefficient is hc = 0.78 (Tskin – Ta)0.56

[16.4]

and for the standing body, hc = 1.21 (Tskin – Ta)0.43

[16.5]

Table 16.6 Natural convective heat transfer coefficients (hc) for a nude thermal manikin standing and sitting in still air (velocity < 0.1 m/s), Obtained at a fixed skin-to-air temperature gradient of 12K Manikin segment

Seated natural convective coefficients (W/m2 K)

Standing natural convective coefficients (W/m2 K)

Foot (L and R) Lower leg (L and R) Thigh (L and R) Pelvis region Head Hand (L and R) Forearm (L and R) Upper arm (L and R) Chest Back Whole body

4.2 4.0 3.7 2.8 3.7 4.5 3.8 3.4 3.0 2.6 3.3

5.1 4.1 4.1 3.4 3.6 4.1 3.7 2.9 3.0 2.9 3.4

From deDear et al. (1997).


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Moving-air convective heat transfer coefficients Moving air increases heat transfer from the skin surface by reducing the thickness of the heated layer adjacent to the skin, and thereby increasing its conductance (W/m2 K). The body’s boundary layers are fundamentally characterized by the properties of air and the dimensions of the body’s surfaces (Nusselt, Prandtl, and Reynolds numbers), with body segments represented with empirical values for cylinders. However, with the complexities of the human shape, such values are usually empirically determined on humanshaped manikins. With the manikin in a wind tunnel, deDear (1997) and Oguro et al. (2002b) separately developed expressions for heat transfer coefficients as a function of air velocity. For the overall body, their values are close, as shown in Table 16.7. They also obtained expressions for each body part, repeated for upwind, downwind, and sideways wind orientations. In general, the expressions are quite similar, regardless of direction, unless a given body part passes into the wind shadow of another body part (e.g. the arm and its adjacent torso, or the lower limbs while seated). The insensitivity to direction can be explained to some extent by the fact that most body segments are cylindrical and vertically oriented. Air motion across the skin surface can also be caused by the motion of the body itself. For a walking or running person, the limbs experience higher and lower air velocities than the mean speed of the body as they swing back and forth. Chang et al. (1988) quantified values for various body parts using an articulated moving manikin mounted on a treadmill in a wind tunnel. They found that the convective coefficients for the outer limbs were lower than those for body parts closer to the trunk. This could be due to the nonlinear relationship of convection to velocity; if for half of the swing cycle the arm or calf is almost stationary relative to the ground and the surrounding air, the reduction in convective loss will be great, and the doubled velocity during the other half of the cycle will not make up for it. In the outdoors, it is common to express forced-convective heat transfer in terms of an ambient temperature at which equivalent heat transfer would occur; this is termed a ‘wind-chill index’ or ‘wind-chill equivalent temperature’ (WCET):

Table 16.7 Moving air convective heat transfer coefficients Standing

Seated

deDear

Oguro

deDear

Oguro

hc = 10.4 V0.56

hc = 9.41 V0.61

hc = 10.1 V0.61

hc = 9.43 V0.63

582


WCET = Tskin – hactual/hcalm × (Tskin – Ta)

[16.6]

Shitzer (2005) has reviewed the historical formulations of the wind chill index. The index is usually developed for a simple geometrical shape such as a cylinder. In cold weather, the skin develops bumps (gooseflesh) around hair follicles, and the hair itself ‘stands on end’ – pilo-erection. For fur-bearing animals, this serves to insulate the skin. For humans, the hair density on most of the skin has become thin, and both the hair and the muscular apparatus that erects it have evolved to be insignificant for thermoregulatory purposes. The hair on the head, however, is often thick enough to form a substantial insulative layer.

16.4.3 Long-wave radiant exchange The radiation emitted from a surface is proportional to the fourth power of absolute temperature, but it is possible to approximate radiant exchange with a linear coefficient when the surfaces are within a limited range of temperatures. R = hr × ε (Tskin – Tr) (W/m2)

[16.7]

where hr = radiative heat transfer coefficient (W/m2 K), ε = emissivity, and Tr = the temperature of the surrounding surfaces. Tr is also represented by the ‘mean radiant temperature’ (MRT). The linearized radiative heat transfer coefficient can be calculated by: hr = 4 εσ

Ar AD

3  273.2 + Tskin + Tr    2

[16.8]

where σ = Stefan–Boltzmann constant, 5.67 × 10–8 W/m2 K4 Ar = effective radiation area of the body AD = total area of the body (Dubois area) The ratio A/AD is 0.70 for a sitting person and 0.73 for a standing person (Fanger, 1967). Emissivity is close to unity for the skin surface (typically 0.95). The coefficient hr is nearly constant for typical indoor temperatures, and a value of 4.7 W/m2 · K suffices for most calculations (Fanger, 1972). Using the thermal manikin with its radiative and convective loss fractions separated, deDear et al. (1997) obtained hr values for each segment of the unclothed body (Table 16.8). These apply to uniform radiant surroundings. Although the individual segment values differ substantially, the whole-body values are close to the Fanger value of 4.7 W/m2 K. When exposed to an asymmetric radiant environment, where the surfaces surrounding the body have different temperatures from each other, e.g. a cold or hot window in an otherwise neutral room, it is necessary to know the angle factors between the body and the surrounding surfaces to determine the radiative heat exchange with them. Angle factors require knowledge of


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Table 16.8 Body segment radiative heat transfer coefficients (hr) for a nude thermal manikin standing and sitting in still air Manikin segment

Seated radiative coefficients (W/m2 K)

Standing radiative coefficients (W/m2 K)

Foot (L and R) Lower leg (L and R) Thigh (L and R) Pelvis region Head Hand (L and R) Forearm (L and R) Upper arm (L and R) Chest Back Whole body

4.2 5.4 4.6 4.8 3.9 3.9 5.2 4.8 3.4 4.6 4.5

3.9 5.3 4.3 4.2 4.1 4.1 4.9 5.2 4.5 4.4 4.5

From deDear et al. (1997).

the projected area of the body from the direction of the surface with which the body is exchanging radiation. The whole body’s projected areas were measured by Fanger (1967, 1972) for the full spherical surroundings of standing and seated persons. From these he calculated angle factors for a wide range of window sizes at varying distances from standing and seated occupants; these are universally used in standards and design (ASHRAE, 2005). Oguro et al. (2001b,c) extended this to measuring projected areas for each individual body part, again using the manikin in both standing and seated postures.

16.4.4 Short-wave (solar) gain to skin The absorptivity of skin to solar radiation varies with skin color in the visible and the near-infrared spectra. For visible wavelengths (0.4–0.7 µm), white skin is about 0.5 absorptive, while black skin has been measured at 0.74 (Houdas and Ring, 1982). In the near infrared from 0.8 to 1.4 µm, white skin is 0.6 and black 0.7, while from 1.4 to 2 µm they are almost the same at 0.82, and above 2 µm, they both approach unity. For ultraviolet ( 0.2 is perceived as uncomfortable. Skin wettedness can theoretically approach 1.0 while the body still maintains thermoregulatory control, but in practice it is difficult to exceed 0.8 (Berglund and Gonzalez, 1977).


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16.5.1 Control of evaporation from damaged skin Wounds (burns and scrapes) that remove the stratum corneum expose a fully wetted saturated surface. Evaporation from a wounded surface may cause heat loss 3 times that of dry heat loss and equivalent to vigorous sweating (Maglinger et al., 2005). This creates a serious problem in cool operating theaters because anesthetic-induced inhibition of normal thermoregulation is likely to cause patient hypothermia. Surgical drapes and bandages are used to reduce moisture and heat loss. Ordinary surgical drapes reduce cutaneous dry heat loss by 30% during operations and prior skin preparation (Sessler et al., 1991). Making the surgical drapes impervious to moisture may reduce evaporative heat loss, as well as preventing contamination of the surgical sites from fluid passing through the drape. Bandages were traditionally cotton and polyester but have expanded into numerous types of porous and nonporous fabrics, foams, and films (van Rijswijk and Beitz, 1998) including bioactive dressings that accelerate healing and grow with the skin (Bhowmick et al., 2003). They provide a moisturized microenvironment, reduce fluid loss, wick away bacteria and exudates, and prevent bacteria from penetrating. Bandages help to fulfil the functions of the natural skin.

16.6

Typical skin temperatures

The core temperature is maintained within a small range, about 36–38 °C (Fig. 16.1). The skin temperature may change significantly in order to keep the core temperature in that range, and its temperature will be sensed by the thermoreceptors in the skin (Fig. 16.6). However, the skin temperature is not uniform across the different segments of the body, due to a variety of physiological factors. The inter-segment temperature variation has entirely different patterns in the cold versus in the heat, when the body is either attempting to conserve or reject metabolic heat. Skin temperature distributions are described below for three conditions: neutral, cold, and warm. Skin temperature distribution under conditions perceived by the subjects as neutral are provided in Table 16.9. The UC Berkeley data were taken with the subject wearing a thin leotard over the temperature sensor (Zhang, 2003). The data provided by Olesen and Fanger (1973) were taken with subjects wearing office clothing. It is unclear why the Berkeley data are consistently about 1K higher than the Olesen and Fanger data, but the distribution patterns are consistent. In cold environments, skin temperature varies widely across the body as a whole, and even within many individual body parts, due to the effects of vasoconstriction. In the cold, the fingers and nose are the coldest, and finger

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Thermal and moisture transport in fibrous materials Table 16.9 Local skin temperatures (°C) in neutral stable condition Segment

Skin temperature (°C) – UC Berkeley

Skin temperature (°C) – Olesen and Fanger

Forehead Cheek Front neck Back neck Chest Back Abdomen Upper arm Lower arm Hand Left finger Thigh Shin Calf Foot Average

35.8 35.2 35.8 35.4 35.1 35.3 35.3 34.2 34.6 34.4 35.3 34.3 32.9 32.7 33.3 34.45

34.2

34.5 34.4 34.9 33.5 32.7 33.5 33.7 32.6 32.2 32.2 33.38

Table 16.10 Local skin temperatures in a cold stable condition (°C) Segment

Skin temperature (°C)


30.7 27.7 33.5 34.5 30.9 32.4 28.7 24.7 27.3 23.1 21.1 27.0 26.5 24.3 21.4 26.8

temperature is 9 K colder than the forehead temperature. The neck temperature is the warmest and can represent a significant source of heat loss (Table 16.10). In warm environments, skin temperatures are more uniform than in cold. The skin temperatures are evenly distributed, with only a small variation of 2.7 K (Table 16.11). Unlike the cold, the fingers and feet are dilated, and


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Table 16.11 Local skin temperatures in a warm stable condition Segment

Skin temperature (°C)


36.5 36.3 36.8 36.1 36.1 36.3 36.2 36.4 36.1 36 36.7 35.6 34.4 34.1 36.4 35.8

have some of the warmest surface temperatures on the body. The upper torso and extremities are very uniform. The lower extremities (thigh, lower leg) are colder than the areas above the pelvis, because in sitting they have not moved as frequently as the hands and arms. A following section describes how movement increases finger and hand skin temperatures.

16.6.1 Finger skin temperature in extreme cold environments Finger dexterity is decreased when the finger skin temperature is below 15 °C. If finger tissue temperature reaches freezing, frostbite occurs, which can cause permanent damage to the tissue and permanently impair circulation. The body has a defense mechanism in cold-induced vasodilatation, which periodically delivers warm blood to the freezing tissue. The vasodilatation occurs because the smooth muscle in the blood vessel walls becomes fatigued or paralyzed and cannot continue the vasoconstriction. A similar effect can be obtained by heating the chest or torso, which increases hand and finger skin temperatures by triggering increased circulation of blood to the extremities (Bader and Macht, 1948; Brajkovic et al., 1998, 2003).

16.6.2 Neck The neck has the highest skin temperature of any body part when a person is cold. In the IR images taken in a cold environment, the warm neck is very

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noticeable, like a bright collar. This justifies turning up jacket collars or putting on a scarf in cold weather. (Zipping up the collar of a jacket also restricts the pumping effect of the air through the opening of the collar, which removes heat from the larger torso area).

16.6.3 Hand The hand is probably the most active body part in responding to the body’s thermoregulation requirements. In warm conditions, the hand is fully vasodilated and the fingertips are the warmest areas of the hand. This pattern is reversed when cooling. Constriction of the hand blood vessels causes the skin temperature of the hand to vary on the order of 8 °C. When the hand is cold, it ceases to transfer much body heat to the environment. The hand is very sensitive to the body’s overall thermal state. An IR image taken a few minutes after the subject moved from a warm environment (30 °C) to a slightly cool (22.6 °C) environment, showed that, although the rest of the upper body temperature has not changed much yet, the blood vessels of the hand were already well constricted. A 3 K difference in finger skin temperature has been observed in slightly cool environments, caused by the difference in muscular exertion between typing vs holding the computer mouse (Table 16.12 – Huizenga et al., 2004).

16.6.4 Skin temperature during sleep Skin temperature increases, and core temperature decreases, prior to and during sleep onset (Van Someren, 2004; van den Heuvel, et al., 1998; Gilbert et al., 2000). To preserve heat, people normally would cover themselves with blankets to create a microclimate of 34 °C (Muzet et al., 1984), which is much warmer than the normal environment for comfort when awake (28–30 °C for a nude person). Measures that increase skin temperature may also help people fall asleep (Van Someren, 2004), as long as the high skin temperature does not evoke warm discomfort. Warming the extremities (feet) is especially effective (Krauchi et al., 1999).

Table 16.12 Effect of using a computer mouse on fourth finger temperature (°C) in cold, neutral, and warm environments. Skin temperatures were measured at the end of a two-hour test

Left Right (using mouse)

Tair: 15.6

Tair: 19

Neutral

Tair: 30

21.1 17.8

21.1 19.3

35.4 34.7

36.4 36.2


16.7

589

Sensation and comfort

16.7.1 The distribution of thermal sensitivity Much research exists on how thermal sensitivity is distributed across the body, and across specific areas of the body. We will focus here on sensitivity differences found across the body as a whole. Regional sensitivity to warmth (Stevens et al. 1974) and cold (Stevens 1979) has been examined. For warmth, thermal stimulation was provided by heat lamps, which eliminated any touch stimulation. The skin surfaces measured are shown in Fig. 16.10. The subjects judged the degree of warmth sensation using the method of magnitude estimation, i.e. the assignment of numbers to match sensation. The forehead and cheek are the most sensitive, followed by: chest, abdomen, back, thigh, and calf. At the higher levels of radiation, the differences were less pronounced. In the cold sensitivity test, an aluminum contact stimulator was used on the same body surface locations (Fig. 15.11). For cold, the trunk region (including the thighs is the most sensitive, followed by the limbs; the forehead and cheek are the least sensitive.

1000

500

Thigh

Upper arm

Back

Chest

Abdomen

20

Forehead

30

Cheek

50

Forearm

100

Calf

200

Shoulder

Milliwatts/cm2

300

16.10 Equal-warmth profile. Each contour shows the irradiance level needed to produce a given constant level of warmth (From Stevens et al., 1974 with permission from Elsevier).

590


16° 20°

3 2

1

Forehead

30°

Lowerback

Chest

Abdomen

Calf

27°

Upperarm

Thigh

5

Forearm

23°

Upperback

10

Cheek

Magnitude estimation of cold

20 0° 9°

20.3 cm2 stimulated area

16.11 Magnitude estimation of cold sensation, arranged in order from the least sensitive (forehead) to the most sensitive (lower back), for different levels of cold stimulation (From Stevens, 1979, with permission from Elsevier).

16.7.2 Thermal sensation and comfort distributions under uniform environments Sensation and comfort for local body parts vary greatly in subjects exposed to uniform environments (Arens et al., 2006a,b; Zhang, 2003). In cool environments, the sensation difference from the least cold sensation (breathing zone, ‘slightly cool’) to the coldest (hand, ‘very cold’) is more than 2.5 units on the sensation scale. The hands and feet feel colder than other body parts (Fig. 16.12), and their skin temperatures are also the lowest. A local skin temperature distribution in a cold room (15.6 °C, Table 16.10) shows hand and feet skin temperatures to be about 9 °C lower than the forehead skin temperature. The head, insensitive to cold but sensitive to warm, feels warmer than the rest of the body in cold environments. The subjects’ overall sensation follows that of the coldest (hands and feet) in cool environments. The comfort for all body parts varies considerably – almost four comfort scale units (from ‘very uncomfortable’ to ‘comfortable’. Although the wholebody sensation is cold and the local sensations for the head region (head, face, breathing zone, neck) are also cool, the local comfort for all the head parts is on the comfortable side. The comfort for the remaining body parts is all uncomfortable. The overall comfort follows the worst local comfortable

The skin’s role in human thermoregulation and comfort Very uncomfortable

591 Very hot

Comfort Sensation

Comfortable

Hot Warm Sl. warm

Just comfortable Just uncomfortable

Neutral Sl. cool

Uncomfortable

Cool Cold

Very uncomfortable Ov

er

al

l

Very cold

B

a re

th

a He

d N

ec

k

c Fa

e

o Fo

t

n Ha

d P

v el

is

c Ba

k

e Ch

st

Le

g

m Ar

16.12 Local and overall thermal sensation and comfort in a uniform/ cold environment. Very uncomfortable

Very hot Hot Warm

Comfortable

Sl. warm

Just comfortable Just uncomfortable

Neutral Cool

Uncomfortable

Sl. cool Sensation Comfort

Ov

er

al

l

Very uncomfortable

Br

e

h at

H

d ea

N

k ec

F

e ac

F

t oo

Ha

nd

Pe

i lv

s B

k ac

C

s he

t

Cold Very cold

g Le

m Ar

16.13 Local and overall thermal sensation and comfort in a uniform/ warm environment.

votes closely (marked by circles), the head region comfort exerting little apparent influence. In the warm environment, there is still a considerable variation in sensation between body parts. The sensation distribution (Fig. 16.13) shows the variation to be about 1.5 scale units from the head or face (sensation above ‘warm’ at

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2.3) to the feet (sensation ‘slightly warm’ at 0.8). The variation in skin temperature is much smaller (2.7 °C, Table 16.11) than in the cold environment, and the sensation distribution does not follow the skin temperature as closely. Unlike in the cold environment, where the overall sensation is driven by the vasoconstricted extremities, in the warm environment the overall sensation follows the head region sensation (the warmest local sensation) closely. Unlike the distribution of local comfort seen in the cold environment, all the local comfort levels are uncomfortable, with the head region perceived as the most uncomfortable. Again, the overall discomfort value is close to the level of the head region discomfort, and could be predicted by the worst one or two local comfort values. The sensation of the head – insensitive to cold but sensitive to warm – matches the thermal sensitivity presented by Stevens (1974, 1979) described earlier. From the above two figures, we see that head is not only relatively insensitive to cold, but comfortable with the cool sensation.

16.7.3 Sensation and comfort related to skin temperature Although the contribution of core temperature to the body’s thermoregulation is much stronger than the contribution of skin temperature (e.g. about 10:1 for sweating, 4:1 for shivering), for determining subjective thermal sensation, skin and core temperatures have equal importance (Frank et al., 1999). Within a small range of skin temperatures around neutral, thermal sensation does not change. In this range the setpoint for sensation adapts to the current skin temperature, and people are comfortable (McIntyre, 1980). The range is considerably greater for the extremities than the torso (Zhang, 2003). This is shown in Table 16.13. The set point adaptation occurs over periods of time, Table 16.13 Ranges of comfortable skin temperatures by body part Body part

Tlow

Thigh

Head Face Breath Front neck Chest Back Pelvis Upper arm Lower arm Hand Thigh Lower leg Foot

33.8 32 31.7 36.0 33.8 33.8 32.6 31 31 30 31.6 31.8 30.8

35.8 36.2 35.2 36.6 35.4 35.8 34.8 34.6 36.5 36 34.8 35.1 35


593

varying from seconds for small surface areas, to minutes when the areas are larger.

16.7.4 Thermal sensation related to finger skin temperature Figure 16.14 presents subjects’ overall thermal sensation votes and corresponding finger temperatures obtained during the final 10 minutes of hour-long stable condition tests, in chamber air temperatures from 15.6– 31.5 °C (Wang et al., 2006). Each subject registered 6 votes. When subjects felt warm or hot (their overall sensation between 0.5 and 2.8), their finger temperatures were closely bunched around 37 °C, close to the core temperature. On the cool to cold side (overall sensation less than –0.5), finger temperature ranged widely between 20–30 °C. Whenever finger temperature was above 30 °C, the overall sensation was above –0.5 (the lower boundary of the neutral sensation zone), and there was therefore no cool discomfort (p < 0.001). When finger temperature was below 30 °C, whole-body sensation was (with the exception of a few data points) always below –0.5 (p < 0.001), and therefore, cool discomfort was a possibility. 30°C is a clear threshold, separating warm from cool. All the actual discomfort votes occurred below 28 °C (overall thermal sensation < –1.5).

Sensation vs. finger temperature 40

Tfinger (°C)

35

30 Chamber temperature (°C) 25

29 ~ 31.5 28 ~ 29 25.8 ~ 27.1 17.5 ~ 20.7 15.6 ~ 17

20

15 Very cold

Cold

Cool

Sl. cool Neutral Sl. warm Warm

Hot

16.14 Whole-body sensation vs finger skin temperature.

Very hot

594


16.7.5 Upper and lower back Figure 16.15 compares upper back and lower back thermal sensations. The thermal sensation for the upper back is warmer than for the lower back (Arens et al., 2006a). When the lower back sensation is neutral (scale value 0), the upper back feels warm (scale value 0.3). When the upper back is neutral, the lower back is already cool. This is also consistent with the thermal sensitivities which show the lower back to be more sensitive to cold than the upper back (Section 16.7.1). These sensation results apply also to comfort (not shown here). In warm environments, the upper back feels warmer (and more uncomfortable) than the lower back. In cool environments, the lower back feels colder (and more uncomfortable) than the upper back. A heated chair (Knudsen and Melikov, 2005) can be an effective solution to this thermal sensitivity.

16.7.6 Neck, face, ankles In cold, the neck always has a high skin temperature. Consequently, people feel particularly susceptible to drafts impinging on the back of the neck. Yet air movement from the side of the head is generally regarded as pleasant and effective at cooling in warm environments. This directional asymmetry suggests that lightweight clothing designed to shield the back of the neck (with a relatively high collar) could be very adaptable for thermoregulation, especially in environments such as offices, where people tend to be oriented in a fixed direction for long periods of time. Very hot Hot

Upper back

Warm

Upper back sensation = 0.29 + 0.94 Lower back sensation, R2 = 0.93

Sl. warm Neutral Sl. cool Cool Cold

Very cold

Cold

Cool

Sl. cool Neutral Sl. warm

Warm

Lower back

16.15 Comparison between upper and lower back thermal sensations.

Hot

Very hot


595

16.7.7 Moisture effects on comfort Wet skin discomfort Humans have no known sensors that directly detect humidity, but they are sensitive to skin moisture caused by perspiration, and skin moisture is known to correlate with warm discomfort and unpleasantness (Berglund and Cunningham, 1986). It is rare for a sedentary or slightly active person to be comfortable with a skin wettedness greater than 25%. The proposed mechanisms for discomfort are related to the swelling of the epidermis as it absorbs moisture (Kerslake, 1972). Berglund (1995) suggested that the skin’s swelling may stimulate the skin’s tactile mechanoreceptors in some way and be perceived as uncomfortable. In addition, mechanoreceptors are clearly stimulated by the friction of clothing moving across the skin surface. With moisture absorption, the stratum corneum outer layer softens, allowing clothing fibers to dig in and increase friction. The additional friction is perceived as fabric coarseness and cling (Gwosdow et al., 1986). Moisture-induced cling effect also occurs with architectural and furniture surfaces, particularly smooth, non-hygroscopic materials (ASHRAE, 2005). Fabrics with different moisture absorbance properties are potentially perceived differently, but there is little experimental evidence for this at present. Toftum et al. (1998a) studied knitted and woven cotton and polyester clothes under controlled levels of skin relative humidity ranging from 10 to 70%, and found that fabric type had no effect on comfort, or on perceived humidity of skin or fabric. He found the acceptance of skin humidity decreases as the skin’s relative humidity increases, and produced a predictive model of this. It should be noted that, for normal environments where air temperature is less than skin temperature, the effect of the air’s relative humidity is much less than that of the skin’s relative humidity. For cool environments, very high air relative humidity produces almost no perceived skin comfort effect; even at the warm limit of the comfort zone, 70% air relative humidity causes less than 15% of subjects to perceive discomfort due to skin humidity. Dry eye syndrome Dry eye discomfort is caused by thinning or sometimes rupturing of the precorneal tear film (PTF) which defends the ocular surface from exposure to the environment. Dry eye can be caused by high temperature, low relative humidity, and indoor pollutants affecting the PTF (Wolkoff et al., 2003, 2005). The body’s thermal plume, clinging to the face, acts to protect the PTF from excessive heat and moisture losses, but air movement across the face can disrupt the plume and hasten PTF evaporation (Murakami, 2004). Dry eye stimulates eye blinking in order to restore the moisture on the PTF;

596


indeed eye-blinking frequency is used as a measure to investigate dry eye discomfort (Nojgaard et al. 2005). Respiratory tract comfort Insufficient evaporative and convective cooling of the mucous membranes in the upper respiratory tract may cause local warm discomfort and a sense of stuffiness and staleness when temperature and humidity are high. This effect is many times more noticeable immediately after a step-change in humidity than after a period of exposure to the humidity. In determining the sensation of staleness, 1K drybulb temperature is equivalent to 1K dewpoint temperature after 15 sec exposure, whereas 1K drybulb is equivalent to 6 K dewpoint after 60 minutes of exposure (Berglund and Cain, 1989; Toftum, 1998b).

16.8

Modeling human thermal regulation and comfort

There are numerous models of human thermoregulation and of the resulting perceived thermal sensation and comfort. They can be divided into whether they are static or dynamic, whole-body or multisegment, physical or empirical. The most commonly used comfort model is the Fanger Comfort Equation, or ‘predicted mean vote’ (PMV) model, which now forms the basis of the ASHRAE and ISO indoor environmental comfort standards (Fanger, 1972). The model predicts the mean thermal sensation (PMV) and the ‘predicted percentage dissatisfied’ (PPD) of a large population. It bases the PMV and PPD on a physical prediction of the heat flow from the body to the surrounding environment, relative to the heat flow required for comfort (this is a function of metabolic rate). The prediction is applicable only in steady-state conditions, and the whole body and its clothing are treated as a uniform object. The thermal sensation and discomfort outcomes are empirically based on data from chamber studies of a large number of subjects. A model capable of dynamically simulating transient conditions was developed by Gagge et al. (1971, 1986). This ‘two-node model’ treats the body as a core and a skin shell, with the whole body surface treated uniformly. Skin and core temperatures are simulated by a physiological model of the heat transfers between core, skin and the environment, using dynamic thermoregulatory control functions for sweating, vasodilatation and constriction, and shivering. The skin and core temperatures are then the basis for a prediction of thermal sensation, which in turn is combined with skin wettedness to predict comfort. The ASHRAE Handbook of Fundamentals provides the underlying equations for the PMV and the two-node models, and the models are compared in


597

detail in Doherty and Arens (1988). Both models are incorporated in userfriendly software available from ASHRAE. Multi-segment models have been developed to account for the differences in clothing, heat transfer, and thermal sensitivity that occur on a real body, especially when the body is exposed to a non-uniform environment. Like the two-node model, they are dynamic and are based on simulations of the physiology in the body. Most multi-segment models originated with Stolwijk and Wissler in the early 1970s (Wissler, 1964; Stolwijk, 1971). The body is divided into segments (e.g. head, trunk, leg, foot, arm, hand), and each segments into four layers (core, muscle, fat, skin). The models vary in their detail and treatment of the thermophysiology, but most of them (Smith, 1991; Wang, 1992; Tanabe, 2002; Fiala, 1998) calculate the thermal sensation outcome on a whole-body basis. The University of California at Berkeley developed a multiple-segment physiology and comfort model (Huizenga et al., 2001; Zhang et al., 2003) that calculates thermal sensation and comfort for each body segment, based on segment-specific human subject tests. It predicts whole-body sensation and comfort by integrating the sensations from all the body parts. In addition to the physical models above, deDear and Brager (2002) developed an entirely empirical model of human adaptation to the environment. It is based on extensive studies of thermal sensation and comfort carried out in office buildings around the world, in which the range of acceptable environments varied widely, especially for buildings without air conditioning. The concept of the adaptive model is that when people are subjected to a given thermal environment over time, they adapt physiologically, psychologically, and behaviorally. The comfort ranges for these people are different from people who have not adapted to the environment. This model is now incorporated in the ASHRAE (2004) comfort standard, applicable for buildings with openable windows.

16.9

References

Arens E, Zhang H and Huizenga C (2006a), ‘Partial and whole body thermal sensation and comfort. Part I: Uniform environmental conditions’, Journal of Thermal Biology, 31, 53–59, Also presented in The 2nd Symposium on Physiology and Pharmacology of Thermal Biology and Temperature Regulation, Phoenix US, March, 2006. Arens E, Zhang H and Huizenga C (2006b), ‘Partial and whole body thermal sensation and comfort. Part II: Non-uniform environmental conditions’, Journal of Thermal Biology, 31, 60-66, Also presented in The 2nd Symposium on Physiology and Pharmacology of Thermal Biology and Temperature Regulation, Phoenix US, March, 2006. ASHRAE (American Society of Heating, Refrigerating, and Air Conditioning Engineers) (2004), ANSI/ASHRAE Standard 55–2004, Thermal Environmental Conditions for Human Occupancy. Atlanta, ASHRAE, Inc.

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Index

adhesive forces 138–42 acid-base interactions 141–2 Lifshitz-van der Waals forces 138–41 adrenergic nerves 577 adsorption 108–9, 427–8 aerosol barrier performance 546–7 agent concentration 553, 554 air 595 convection and 579–80, 581–2 dry 103 mass balance 279–81, 285–7 motion 581 permeability 442, 461–2, 548 ‘pockets’ 168 trapped 273, 284 American Society of Heating, Refrigerating and AirConditioning Engineers (ASHRAE) standards 289, 596–7 American Society for Testing and Materials (ASTM) guidelines 440 anisotropy 11–14, 536 ‘disease’ 379 grooves and 174–5 planar 78–82 ankles 594 aperture circles 11, 15, 21, 27–8 area and area density 64–7 Arrhenius-type equations 104 arterio-venous anastamoses (AVAs) 572 arterioles 572 ASHRAE (American Society of Heating, Refrigerating and Air-

Conditioning Engineers) standards 289, 596–7 aspect ratio 3 ASTM (American Society for Testing and Materials) guidelines 440 atmosphere and partial pressures 103–5 atomic force microscopy (AFM) 152 auto-models 398, 478–509, 511, 518, 538 auto-binary 498 auto-binomial 499 Besag proposition 479–82 Derin-Elliott 499 Hammersley and Clifford theorem 486–96 ideal gas 486–8 with Kawasaki dynamics 525, 529, 537 Markov chains 470, 500–6, 513–14, 531 Markov random fields 469–79 passim, 482, 484–97, 500–10 passim neighbourhoods and clans 482–6 for wetting phenomena 499–500 see also Ising model automata see lattice gas automata autonomic nervous system 574 averaging volume 250 back, upper and lower 594 bandages 541, 585 Bayes’ rule 507–8 theorem 5, 479–81 bedclothes 561 Bernoulli’s equation 281

603

604

Index

Besag proposition 469–70, 478, 479–82, 484 Besag coding 503–4 Hammersley/Clifford theorem and 486–7, 489, 491–3 random fields and 472, 506–7, 509, 510 Bessel function 203 BET-Kelvin method 33–6, 123, 125 ‘blackening algebra’ 494 body segment heat transfer 583 body-environment exchange 561–4 heat gains 561–2 heat losses 562–3 thermal regulation 563–4 Boltzmann constant 140, 514 law 495 models 397–8 bounce-back collision 378 boundary conditions 316–17 2 layer radial system 433, 434 condensation and 449 diffusive 376 free-slip 376 layers 581 liquid-gas 316 no-slip 376 solid-gas 317 solid-liquid 317 value problems 538 brain 562, 573, 577 Bravais lattice 363–4, 368, 372–4, 388 British Standards (BS) 441 Brunnauer, Emmett and Teller (BET) equation 113 model 121–5 see also BET-Kelvin method Buffon’s needle 45–6, 67–71 capillarity 160–7 flow 161–2 wicking fabrics 164–5 fibrous materials 162–3 yarns 163–4 capillary adhesion 191 condensation 111–13 length 191

number, spreading speed and 190–1 ‘penetration’ 162 phenomena 510 pressure curves 342 rise in fibrous material 213–14 in tubes 192 ‘sorption’ 162 see also psychrometry and capillary hydrostatics Cartesian coordinate system 9, 201, 231–2, 366 Cassie equation 174 state 169, 171, 177 Cauchy’s formula 44 Cavalieri’s principle 43–4, 45 cellular automata 360, 362, 363 definition 357, 360 features of 363 lattice gas 360–4 space and 358 structures and 358 synchronization of 369 see also lattice gas automata chains of successive configurations 514–15 steps to create 514 characterization theorem 54–5 see also fibrous material characterization chemical and biological warfare agents 549 chemically-textured surfaces 170–1 circular granulometry 74–8 clans, neighbourhoods and 469, 478, 482–6, 488 Clausius-Clapeyron equation 132, 337 closed aperture cylinder model 298–9 clothing 560–1 ‘breathable’ 548 performance see computational fluid dynamics (CFD) cold sensation 590 collision bounce-back 378 detailed balance 375 effective 375–6 micro reversible 375

Index particle 380, 381 phase rules and 374–8 steps in 383–4 colour combinations 49 ‘coloured FHP’ (CFHP) model 395, 397 comfort see sensation and comfort compact sets 50–1 ‘composite contact’ 168 computational fluid dynamics (CFD) 542–59, 580 fabric-covered cylinder modeling 546–54 full body modeling 554–8 material modeling 543–4, 545 computer simulation 509–36 fluid flows 381–95 moisture transport 509–36 in fibrous materials 525–36 liquid columns in fiber bundles 519–24 Rayleigh instability 511–19 condensation capillary 111–13 heat transfer and 405–6 sorption and 427–8 condensation/evaporation models fibrous media 411–16 thin clothing layers 407–11 physical properties and 416–18 liquid water permeability 416–17 material hygroscopicity 417 pressure differences 417–18 vapour hydraulic permeability 416 conduction see thermal conduction conservation laws 374 contact angles 128, 129, 149 convection 579–82 moving air 581–2 still-air 579–80 convection and ventilation 271–307 dry/evaporative resistances 271–2 heat/moisture transport 272–3 models 283–98 extension to real limb motion 301–2 periodic ventilation 298–301 insulation 273–4 skin-adjustant air layer 274–5

605

ventilation rates 275–83 mathematical models 277–8 models Lotens’s empirical 275–7 mathematical 277–8 periodic ventilation 279–83 convex bodies 51–3 convex ring sets 53, 54–5, 74–8 Conway’s game of ‘Life’ 358, 359 corneocytes 567 Coulomb forces 477–8, 484 curvature 158–60 of cylinders 180–1 droplets and wet-fiber filters 160 length estimation 67–71 surface wetting 193–5 torsion and 94–6 wetting 145–8 curved surfaces 158–9 fiber surfaces 159–60 planar surfaces 158 cylinder system (Princen) 519–22 Dalton’s law of Partial Pressures 103 Damkohler number 549 Darcy’s equation 343 flow resistance 452 law 147, 166, 249, 340–1, 453, 461 computer simulation and 382, 385, 387, 388, 390, 391 number 549 defects 150, 151 Delesse’s principle 85–6 density area 64–7 feature count 71–4, 96–8 length 90–3 surface area 86–90 volume 83–6 Derin-Elliott model 499 Derjaguin disjoining pressure 199 dermis 567–8, 569 desorption isotherm 109 see also heat/moisture transfer with sorption/desorption detailed balance collision 375 dewetting process 182 dewpoint temperature 105–6 diffusion 544, 545

606

Index

boundary conditions and 376 direct and indirect water 118–19 discrete molecular dynamics 364–78 collision phase and rules 374–8 conservation laws 374 lattice as discrete space 366–8 observables 370–2 propagation phase 372–4 space 366–8 time 369–70 disjoining pressure 215 distribution planar and harmonic 25–32 pore 14–17 BET-Kelvin method 33–6, 123 random 19–25 droplet adherence 191 evaporation 550, 551 fusing 516 neighbours 517 shapes 146, 159, 160, 189 spreading 196–8 see also liquid dry eye syndrome 595–6 dry resistances 271–2 drying rates 344 du Nouy ring 142 EDANA (European Disposables and Nonwovens Association) 209, 525, 537 effective collision 375–6 ‘effective material properties’ 233 ellipsometry 217 embedded objects 84, 85 empirical ventilation relationships 283 energy balance 286, 297 ‘energy function’ 495–6 environment 561–4, 590–2 epidermis 565–7, 569 equal-warmth profile 589 equilibrium 192, 426–7 moisture content (EMC) 107–8 probabilities 500 Euclidean space 363 Euler-Poincaré characteristic 58, 64, 71–4 European Disposables and Nonwovens Association (EDANA) 209, 525, 537

evaporative control systems 573–7, 584 damaged skin and 585 hidromeiosis 576 insensible heat losses 573–4 resistances 271–2 selective brain cooling (SBC) 573, 577 sweating 564 acclimatization and dripping 576 apocrine glands 577 emotional 577 gland distribution 575 thermoregulatory 574–5 see also condensation/evaporation exclusion principle 370 external convection coefficients 293–5 fabric layers 546 fabric-air layer-skin system 279 fabric-covered cylinder modeling 546–54 face 594 face centred cubic packing 37 ‘face-centred-hyper-cubic’ model (FCHC) 397 Fanger Comfort Equation 596 value 582–3 Fan’s theoretical model 449 Farnworth model 407–8, 449 feature count 71–4, 96–8 fiber -reinforced composite materials 8 arrangement 7–8 assemblies 8 bundles 180, 183–4, 194, 203–8, 519–24 coating 182–3 coordinates 6 surfaces 159–60 total amount 4–5 volume fractions 20 fiber orientation determination 33–7 BET-Kelvin method 33–6, 123 Fourier transformation method 36–7 planar and harmonic distribution 25–32 random distribution 19–25 special 19–32

Index fibrous material characterization 7–10 fiber arrangement 7–8 internal structures 6, 7, 9–10 finger temperature 587, 588, 593 finite automata 359–60, 361 finite element method 538 Finite Volume Method (FVM) 449 fluid flows simulation 381–95 lattice gas algorithm 382–4 sound vibrations and 393–5 two-dimensional 384–93 four-layer ventilation model (Lotens) 284 Fourier’s law 33, 36–7, 228, 231, 236 free-slip boundary conditions 376 Frisch, Hasslacher and Pomeau (FHP) models 358–9, 371, 378–82, 385, 393, 395 full body modeling 554–8 functional, properties of 54–5 Galileo 43, 379 gas phase 336, 339 continuity equation 345, 348 convective transport 340–1 diffusion equations 345, 348 equation of motion 345 mass transport 338–40 porous material 543 see also lattice gas automata gas-solid interface 334 Gauss distribution 500 geometry global 170 parameters and 552 see also structure and geometry Ghaddar model 285–6, 292–3, 295, 298, 301 Ghali model 285, 288–92, 295, 301 Gibbs distribution 495–6 Gibbs-Geman sampling 504, 513 Glauber dynamics 502, 504 global geometry of surfaces 170 Gore-Tex materials 382 Grammetric Absorbency Test System (GATS) 167, 174 granulometry, circular 74–8 Gray’s point functions 322, 324 Green functions methods 230 grooves 174–5

607

Grosfils, Boon and Lallemand (GBL) model 397 ground sections 55–60 Guggenheim-Anderson-de Boer (GAB) model 125 theory 122–4 Hagen-Poiseuille law 210 hair 582 Halpin-Tsai equation 247 Hamaker constant 139–40, 144, 215 Hammersley and Clifford theorem 469–70, 482, 486–96, 497, 498 expansion 499, 501, 506, 509 hand temperature 588 hard solids 115 Hardy, de Pazzis and Pomeau (HPP) model 358, 363, 367, 371, 375–8, 380 Harkinson spreading 194, 208, 521 harmonic distribution 25–32 heat gains 561–2 losses 445, 562–3, 573–4 sorption and 130–2 thermodynamic relations 336–8 see also thermal conduction heat exchange at skin surface 578–84 conduction 578–9 convection 579–82 long-wave radiant 582–3 short-wave (solar) gain 583–4 heat, moisture and phase change 424–36 coupled heat and moisture 431–4 heat and moisture interaction 434–5 mass and heat transport 428–31 moisture regain and equilibrium 426–7 sorption and condensation 427–8 heat transfer 544, 547, 549 body segment 583 convective 580, 581 through skin 563 heat/moisture transfer with sorption/ desorption 439–68 air permeability 461–2 diffusion coefficient 460 disperse coefficient 461 investigations 439–48 findings 443–8

608

Index

instrumentation 440–2 methods 439–40 procedure 442–3 moisture vapour resistance 462–3, 464, 548 numerical simulation 456–63 radiative sorption constant 457–60 theoretical models 448–56 finite volume method (FVM) 452–6 formulation 449–52 see also phase change heat/moisture transport 272–3 air layer mass and heat balances 285–7 heat losses 302 models 283–98 extension for skin contact 295–8 three-node ventilation 287–95 see also computer simulation hemi-wicking 172–4 Henry’s model 117, 407, 448 heterogeneity of surfaces 168–70, 217–18 hexanol molecules 365 hidromeiosis 576 high-energy surfaces 115–16 homogeneous structures 358 homokinetic models 371 Hot-Cold Box 440 Hough transform 79 humidity 105, 106, 107, 121, 126, 545, 595 ‘hydrophilic’ surfaces 118, 137 ‘hydrophobic’ surfaces 137 hygroscopicity 263, 403, 417, 543, 545 hypothalamus 564, 568, 574 hysteresis 109, 176–7 categories 127–30 characterization of 176–7 on fibrous materials 177 on micro-textured surfaces 177 ideal gas model 486–8 INDA (International Nonwovens and Disposables Association) 209, 525, 537 ‘individual automaton’ 360 instantaneous sensible heat loss contact region 297 non-contact region 298

insulation 273–4 see also heat/moisture transfer with sorption/desorption internal convection coefficients 292–3 internal mean heat coefficients 294 internal structures 6, 7, 9–10 International Nonwovens and Disposables Association (INDA) 209, 525, 537 International Standards Organization (ISO) standards 440, 596 interphase flux terms 329 intersections 90 probability of 87 rose of 81, 83 intrinsic phase 312, 318, 336, 544 Ising model 148, 209, 398 computer simulation and 469, 470–2, 474, 479, 498–9, 537 ISO (International Standards Organization) standards 440, 596 isotropic sections 62–3 ‘iterative arrays’ 358 joint probabilities 479–82, 495 jump conditions 332–3 boundary 329–31 Jurin’s height 192 Kawasaki dynamics 148, 209, 502, 504–5, 508 computer simulation and 510, 513, 522, 529, 531, 537 Kawasaki kinetics 514, 525 Kelvin equation 112, 113 Kepler conjecture 37 Landau-Levich-Derjaguin (LLD) law 178, 195–6 Langmuir -type trapping 117 model 109, 119 Laplace equation 146, 147, 158, 178, 188, 191, 521 latent heat loss contact region 298 non-contact region 298 lattice gas automata 357–401 definitions 359–64

Index discrete molecular dynamics 364–78 historical overview 358–9 lattice-Boltzmann model 385, 397–8 simulation of fluid flows 381–95 typical 378–81 Frisch, Hasslacher and Pomeau models (FHP) 379–81 lattices and test systems 60–4 length and length density 90–3 Lennard-Jones interactions 147–8 potential 138–9 Lewis acid/base (AB) interaction 141–2 number 449 ratio 584 relation (air-water vapour mixtures) 292 Lifshitz-van der Waals (LW) forces 138–41, 141–2 liquid columns in fiber bundles 519–24 evaporation and condensation 544 film on fiber 513 flow instability 180–3 curvature of cylinders 180–1 dewetting process 182 fiber coating 182–3 fluid jets 181 ‘Marangoni effect’ 181–2 morphology of 183–4 phase convective transport 341–4 water permeability 416–17 liquid and fibrous material interaction 188–222 curved surface wetting 193–5 fundamentals 188–92 Lucas-Washburn theory 203–14 Rayleigh instability 199–203 spreading dynamics on solids 195–8 wetting and 214–19 liquid phase continuity equation 345 equation of motion 344–5 porous material 543–4 liquid-gas 316, 332 Liquid/Air Displacement Analyser (LADA) 209

609

local conditional probabilities 471, 474, 476, 481–3, 489, 492, 507 ‘local configuration space’ 370 ‘local states’ 370 long range force effects 215 long-wave radiant exchange 582–3 Lotens model 275–7, 284, 408–11 ‘lotus-effect’ 175 low-energy surfaces 116–18, 143–4 Lucas-Washburn theory 161, 163, 173, 203–14 capillary rise in fibrous material 213–14 computer simulation and 525–8 liquid climbing fiber bundle 203–8 radial spreading of liquid 211–13 ‘Marangoni effect’ 181–2 marginal probability 479, 481 Markov chains 470, 500–6, 513–14, 531 master equation 531–2 random fields 469–79 passim, 482, 484–97, 500–10 passim mass balance 297 transfer film coefficients 294 transport coefficients 291–2 mass and energy transport equations 308–28 boundary conditions 316–17 point equations 309–15 material modeling 543–4, 545 MCSPS see Monte Carlo Step per Site (MCSPS) mean fiber cut ends 29 ‘mean radiant temperature’ (MRT) 582 meniscus 178–80 on multifilament 179–80 on single fiber 178–9, 189–90 ‘meso-scale’ approach 148 micro reversible collision process 375 micro-structural unit 10 micro-textured surfaces 177 microclimate air layer 292–3 ‘microscopic density’ 370 microscopic fabric heat 291–2 military protective clothing systems 549–50

610

Index

‘mobile parts’ of algorithms 384 models auto see auto-models BET-Kelvin 33–6, 123 Boltzmann 140, 397–8, 495 Brunnauer, Emmett and Teller (BET) model 113, 121–5 closed aperture cylinder 298–9 ‘coloured FHP’ (CFHP) 395, 397 condensation/evaporation 407–11 Farnworth 407–8, 449 Lotens 275–7, 284, 408–11 mathematical 411–16 fabric-covered cylinder 546–54 features 50–3 Frisch, Hasslacher and Pomeau (FHP) 358–9, 371, 378–82, 385, 393, 395 full body 554–8 GAB 125 Ghaddar 285–6, 292–3, 295, 298, 301 Ghali 285, 288–92, 295, 301 Grosfils, Boon and Lallemand (GBL) 397 Hardy, de Pazzis and Pomeau (HPP) 358, 363, 367, 371, 375–8, 380 heat/moisture transfer 448–56 transport 283–98 Henry’s 117, 407, 448 homokinetic 371 Langmuir 109, 117, 119 lattice gas 358, 361–2, 381–2 lattice-Boltzmann 385, 397–8 material 543–4, 545 mathematical 277–8 multiple-segment physiology and comfort 597 open aperture cylinder 299 parallel exponential kinetics (PEK) 125 PCM fabric 418–20 periodic ventilation 279–83, 298–301 extension 301–2 Poisson 15, 500 porous material 173 structural features of 47–53 thermal regulation 596–7 three-node ventilation 287–95 extension 295–8

3-D dynamic ventilation 301–2 two-equation 255 Vafai and Sarker’s 449 see also computational fluid dynamics (CFD); computer simulation moisture diffusivity 343 vapour resistance 462–3, 464, 548 moisture effects on comfort 595–6 dry eye syndrome 595–6 respiratory tract 596 wet skin 595 moisture exchange at skin surface 584–5 evaporation control 585 moisture in a medium 106–15 diffusion 226, 262–5 kinds of 114 moisture sorption isotherms 108–11, 119–32 behaviors 110 heat and temperature 130–2 hysteresis categories 127–30 main regions 111 theories on 120–7 pore size effects 113–15 regain and content 106–8 water activity capillary condensation and 111–13 sorption types and 113 see also heat, moisture and phase change molecular dynamics (MD) 147–8 see also discrete molecular dynamics Monte Carlo Step per Site (MCSPS) 147, 515–16, 518, 531–3, 534 dynamics 513, 515 simulations 525, 537–8 morphology of liquid 183–4, 523 multi-component systems 262 multi-layered clothing 449 multifilament, meniscus on 179–80 multiphase flow through porous media 308–56 comparisons 347–51 gas phase convective transport 340–1 mass transport 338–40 liquid phase convective transport 341–4 mass and energy transport 308–28

Index modified transport summary 344–7 thermo-dynamic relations 336–8 total thermal energy 328–36 multiple-segment physiology and comfort model 597 nature of the material system 112 Navier-Stokes equations 102, 249, 511 lattice gas automata and 360, 364–6, 375, 379, 384–5 ‘nearest neighbour neighbourhood’ 483 neck temperature 587–8, 594 neighbourhoods 367, 469 clans and 478, 482–6, 488 symmetric 503 Newtonian factors 147, 210, 230, 366, 378 no-slip boundary conditions 376 non-equilibrium joint probabilities 502 non-hygroscopic fibers 262 nuclear magnetic resonance (NMR) 119 ‘onto mapping’ 54 open aperture cylinder model 299 orientation distribution function (ODF) 36–7 probability density function 7–8 ‘overshoot’ of thermoreceptors 571 packing problem 37–8 parallel exponential kinetics (PEK) model 125 partial pressures 103–5 particle flow rate 389 Pauli’s exclusion principle 370 Peclet number 287, 449 periodic ventilation model 298–301 permeability 416–17 air 461–2 phase average 318, 544 phase change 402–23, 448 condensation and heat transfer 405–6 condensation/evaporation 403–5 models fibrous media 411–16 thin clothing layers 407–11 physical properties and 416–18 material (PCM) fabrics 403, 406–7 models 418–20 see also heat, moisture and phase

611

change; heat/moisture transfer with sorption/desorption phase property 322 Pierce’s theory 121 planar anisotropy 78–82 distribution 25–32 surfaces 158 Planck’s constant 140 Plateau, J. 199, 201–2 point -bonded non-woven fabric 47, 48 counting method 64–7 equations 309–15 functions, Gray’s 322, 324 Poiseuille flows 385 Poisson’s function 15 model 500 pore distribution 14–17 BET-Kelvin method 33–6, 123 size effects 113–15 porous material model 173 porous media 4–6 gas flows see lattice gas automata pore shape 6 porosity 5–6 tortuosity 6 total fiber amount 4–5 see also multiphase flow through porous media ‘positivity condition’ 482 precorneal tear film (PTF) 595 precursor film 197 ‘predicted mean vote’ (PMV) model 596 ‘predicted percentage dissatisfied’ (PPD) 596 pressures differences 417–18 partial 103–5 probability equilibrium 500, 502 of intersections 87 joint 479–82, 495, 502 local conditional 471–6 passim, 482, 489, 492, 507 marginal 479, 481 orientation 7–8 theory 7–8

612

Index

transitional 375 propagation phase 372–4 points in 384 Proportional-Integral-Derivative (PID) control algorithm 441 psychrometry and capillary hydrostatics 102–35 essentials of 103–4 atmosphere and partial pressures 103–5 dewpoint temperature 105–6 saturation and humidity 105 moisture in a medium 106–15 moisture sorption isotherms 119–32 wettability 115–19 radial spreading of liquid 211–13 random fields 472–6 Markov 469–79 passim, 482, 484–97, 500–10 passim random and isotropic sections 62–3 Raoult’s law 113 Rayleigh instability 159, 181–2, 199–203, 215, 470, 519 computer simulation of 511–19 dynamic approaches 201–3 large particle concentrations 235 static analysis 199–201 wavelength 517 reflective boundary conditions 372–4 regulation thermal 563–4, 596–7 see also sweating relative fiber cut ends 30 representative element volume (REV) 250–1, 252, 254, 257 resistance dry/evaporative 271–2 moisture vapour 462–3, 464, 548 thermal 237–41, 548 respiratory tract comfort 596 retention of water 118–19 Reynolds number 181, 385, 549–50 transition flow region 184 ‘rolling motion’ 536 rose of intersections 81, 83 roughness see surface roughness of solids Royal Statistical Society 478 RTD sensors 441

Rule of Mixtures 103 salt 576 saturation and humidity 105 scanning force microscopy (SFM) 152 sections and ground sections 55–60 selective brain cooling (SBC) 573, 577 ‘self-cleaning’ surfaces 168 sensation and comfort 589–96 finger temperature and 587, 588, 593 moisture effects 595–6 neck, face and ankles 594 skin temperature and 592–3 thermal sensitivity 589–90 uniform environments and 590–2 upper and lower back 594 separator 503 shivering 563, 577–8 short-wave (solar) gain 583–4 shrinkage 312 sigmoid-shaped adsorption isotherm 109 simulation computer see computer simulation full body modeling 554–8 numerical 456–63 single fibers 3–4, 210 aspect ratio 3 meniscus on 178–9, 189–90 specific surface 3–4 skin 292–3, 560–602 -adjustant air layer 274–5 body-environment exchange 561–4 contact 295–8 evaporative control systems 573–7 sensation and comfort 589–96 shivering 577–8 structure 565–8 surface area 564–5 heat exchange 578–84 moisture exchange 584–5 temperatures 585–8, 592–3 thermal regulation modeling 596–7 thermoreceptors 568–71 thickness 567 vascular system 571–3 wettedness 574, 584 sleep temperature 588 solid phase 336 continuity equation 345, 348 density relations 345

Index diffusion equation 346 equation of motion 346 porous material 543–4 solid–gas boundary conditions 317 phase 328 solid–liquid boundary conditions 317 interface 334 jump boundary condition 330–1 phase 328 solids hard 115 yarn 288 sorption 544, 545 relations 346–7 condensation and 427–8 isotherms 107 see also under moisture in a medium types of 113 see also heat/moisture transfer with sorption/desorption sound vibrations 393–5 Sparrow number 287 spatial average 318, 544 spatial schemes 472–3 spatial systems, evolution of 470–2 species jump conditions 332–3 specific surface 3–4 sphere packing problem 37 spin exchange 514 spreading 195–8 droplet 196–8 Harkinson parameter 194, 208, 521 Landau-Levich-Derjaguin (LLD) law 178, 195–6 radial 211–13 speed 190–1 wetting and 214–19 experimental investigation 215–16 heterogeneity 217–18 liquids other than water 218–19 long range force effects 214–15 scale effects 216–17 Stefen’s law 414 Steiner compact steps 80–2 stereology basic principles 54–64 convex ring sets 54–5

613

lattices and test systems 60–4 sections and ground sections 55–60 development of 42–53 features and models 47–53 structural features and models 47–53 of three-dimensional fibrous mass 64–82 curvature and torsion 94–6 density of feature count and 96–8 length and 90–3 surface area and 86–90 volume and 83–6 of two-dimensional fibrous mass 64–82 Buffon’s needle 67–71 convex ring sets 74–8 Euler-Poincaré characteristic 71–4 planar anisotropy 78–82 point counting method 64–7 ‘stratum basale’ 567 stratum corneum 566–7, 595 structure and geometry 3–41 anisotropy 11–14 characterization 7–10 features and models 47–53 fiber orientation determination 33–7 special 19–32 influences 113 packing problem 37–8 parameters for porous media 4–6 pore distributions 14–17 single fibers 3–4 tortuosity distributions 17–18 see also three-dimensional structure surface area 86–90, 564–5 body 565 Cauchy’s formula 44 heat exchange 578–84 moisture exchange 584–5 surface interactions 113 surface roughness of solids 167–76 anisotropy and grooves 174–5 chemically-textured 170–1 fibrous materials 175 global geometry 170 hemi-wicking 172–4 heterogeneity 168–70 wetting and 148–52 textured fabrics 175–6

614

Index

wicking 171–2 surface tension 116–18, 156–8 liquid 139 wettability and 143–5 surfaces single defect 129–30 textured 169, 175–6, 177 sweating hot-plate 440–1 see also under evaporative control systems swelling 312 Tanner’s law 216 temperature 112 dewpoint 105–6 distribution 443–5, 457, 458 skin 585–8, 592–3 body 561 during sleep 588 finger 587, 588, 593 hand 588 neck 587–8 sorption and 130–2 transients 545 tessellation automata 358 test systems area density 67 lattice 60–4 particle numbers 75 textured surfaces 169, 175–6, 177 thermal conduction 225–70 analysis 226–33 at skin surface 578–9 effective thermal conductivity (ETC) 233–7 ETC prediction 234–7 by thermal resistance networks 237–41, 548 by volume averaging 249–59 future research 265 homogenization 259–61 moisture diffusion 226, 262–5 plain weave composites 241–9 sensory contact 266 thermal energy equation 348, 350–1 thermal regulation 563–4, 596–7 see also sweating thermal sensitivity 589–90 thermodynamic relations 336–8, 346

thermoreceptors 568–71 three-dimensional structure 42–101 classes of analysis 46–7 stereology basic principles 54–64 development of 42–53 features and models 47–53 of three-dimensional fibrous mass 64–82 of two-dimensional fibrous mass 64–82 three-node ventilation model 287–95 extension of 295–8 3-D dynamic ventilation model 301–2 torsion 94–6 tortuosity 6 distributions 17–18 length 23–5, 31–2 total energy change 514 total thermal energy equation 328–36, 344 transitional probabilities 375, 500–2, 514 trapped air layer 273, 284 two-dimensional fibrous mass 64–82 two-equation models 255 uniform random and isotropic sections 62–3, 91, 93 unit cells 239, 241–9 Vafai and Sarker’s model 449 van der Waals actions 199 forces 138–41, 477, 511 moisture sorption and 109, 113, 115, 117 interactions 140, 143, 215 liquids 218 vapour hydraulic permeability 416 phase transport 544 resistance, moisture 462–3, 464 water 405, 548, 549 variable air spacing 546 vascular system 571–3 vasoconstriction 563–4, 571 vasodilatation 563, 571 velocity profiles 387, 390, 391 vectors 370–1, 392, 393

Index venous plexus 571–2 ventilation see convection and ventilation vibrations, sound 393–5 volume 44, 83–6 Cavalieri’s principle 45 constraint 346 volume average for gas [GAMMA] phase 324–5 for liquid b phase 318–23 for solid s phase 325–7 solid equilibrium 346 techniques 544 Von Koch curve and flake 51 von Mises function 33 warmth 589 water activity, sorption types and 113 content distribution 445–8, 459, 460–2 direct and indirect 118–19 molecules 365 retention 118–19 transport categories 119 vapour 405, 548, 549 weak molecular crystals 115 Weber number 181 Wenzel equation 174 state 169–70, 171–2, 177 wet skin discomfort 595 wettability 115–19 high-energy surfaces 115–16 low-energy surfaces 116–18 retention of water 118–19 surface tension and 143–5 wetting 136–7 curvature and 145–8, 158–9, 193–5 dewetting process 182 fiber surfaces 159–60 fibrous media 203–14, 525–6 liquid spreading and 214–19

615

patterns 533 planar surfaces 158 rough surfaces 148–52 textured fabrics 175–6 types of 145 wetting phenomena 156–87 capillarity 160–7 curvature effect 158–60 hysteresis effects 176–7 liquid flow instability 180–3 morphology of 183–4 surface roughness 167–76 tension 156–8 Whitaker model 308, 328, 341 wicking 137–8, 343–4, 534–5 fabrics 164–5 fibrous materials 162–3, 525–6 hemi-wicking 172–4 rate 528, 536 surface 171–2 yarns 163–4 Wilhelmy plate 142 technique 177 ‘wind-chill’ 548 ‘equivalent temperature’ (WCET) 581–2 ‘index’ 581 Womersley flow model 281–3 X-ray reflection ellipsometry 218 yarns, wicking in 163–4 Young’s contact angle 171 equation 142, 148, 151, 157, 193, 524 force 197 law 128–9 Zisman plot 144

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