Transport Phenomena in Food Processing (Food Preservation Technology)

Transport Phenomena in Food Processing © 2003 by CRC Press LLC Library of Congress Cataloging-in-Publication Data Tr...

Author: Jorge Welti-Chanes | Jorge F. Velez-Ruiz

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Transport Phenomena in Food Processing

© 2003 by CRC Press LLC

Library of Congress Cataloging-in-Publication Data Transport phenomena in food processing / edited by Jorge Welti-Chanes, Jorge F. Vélez-Ruiz, Gustavo V. Barbosa-Cánovas p. cm.— (Food preservation technology series) Includes bibliographical references and index. ISBN 1-56676-993-0 (alk. paper) 1. Food—Effect of heat on. 2. Food industry and trade. I. Welti-Chanes, Jorge. II. VélezRuiz, Jorge F. III. Barbosa-Cánovas, Gustavo V. IV. Series. TP371.2.T73 2002 664—dc21

2002073736

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. 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 prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press LLC, provided that $.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 0-8493-0458-0/02/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC 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 CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2003 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-56676-993-0 Library of Congress Card Number 2002073736 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper


Series Preface Transport phenomena is one of food process engineering’s most important pillars, and this new addition to the CRC Food Preservation Technology Series offers a sound combination of the fundamental and applied aspects of this classic engineering topic. Transport Phenomena in Food Processing plays an important role in the series because it offers the tools for quantifying many important operations, classic or novel, using state-of-the-art calculation approaches. I am particularly impressed with the analytical depth of the chapters and the wide spectrum of covered topics. For me, this is a very inspiring book that hopefully will become a key reference. This book is also important because it is the first in the CRC series to cover fundamental aspects of food processing. In short, I anticipate that this book will be in good company with upcoming books in the series on other fundamental aspects of food preservation. Gustavo V. Barbosa-Cánovas


Preface The latest applications of new and improved traditional food preservation processes have generated the need for increased knowledge of the phenomenological and engineering principles that are the basis of the correct application of factors that produce stability and maintain the quality of transformed and processed products. This need for knowledge has given the field of food engineering a new identity at both the research and industrial levels. Understanding the transport phenomena that govern the engineering analysis and design of food preservation processes is a key element in improving processing conditions and the employment of energy resources, and to increasing the quality of the product. This book presents the state of the art in the transport phenomena area as applied to food preservation and transformation. It is divided into four sections containing a total of 33 chapters, each written by prestigious scientists from institutions and universities around the world. The first section reviews the fundamental concepts of mass, heat, and momentum transfer, while the remaining three sections discuss specific applications for a large variety of processes and products where the predominant transfer phenomenon is mass or heat, or processes employing more than one transport mechanism. The mass transfer section focuses on phenomena controlling osmotic dehydration and hot-air drying processes. However, the themes related to water transfer in superficial films placed on foods, and pre-evaporation, ultrasound, and spinning cone columns are also included. The seven chapters that constitute the heat transfer section study the effects of product shape and process equipment on the phenomenon’s efficiency. The chapters in the last section deal with the study of the combination of two or three transfer phenomena in frying, sterilization, and drying processes. We trust this book on transport phenomena will make a meaningful contribution in facilitating the understanding, design, and implementation of food processing unit operations that will result in the production of safer, higher quality, and more convenient foods. Jorge Welti-Chanes Jorge Vélez-Ruiz Gustavo V. Barbosa-Cánovas


About the Editors Jorge Welti-Chanes is a Professor in the Departments of Chemical and Food Engineering and of Chemistry and Biology at the Universidad de las Americas– Puebla in Mexico. He earned his B.S. degree in Biochemical Engineering and his M.S. in Food Engineering from the ITESM, Mexico and his Ph.D. from the Universidad de Valencia, Spain. Dr. Welti-Chanes was president of the International Association of Engineering and Food and is a member of the editorial boards of five international journals. He is the author and co-author of more than 80 scientific papers and the editor of five books with the food technology, water activity, drying, emerging technologies, and minimal processing areas. Dr. Jorge F. Vélez-Ruiz was born in Puebla City, Mexico. He received a B.S. in Biochemical Engineering in 1977 and an M.S. in Food Science in 1981, both from the Instituto Tecnológico y de Estudios Superiores de Monterrey (ITESM), Mexico, and a Ph.D. in Food Engineering from Washington State University, Pullman in 1993. He began his professional career working in the food industry and in 1979 joined the Food Science Department, ITESM. Since 1980, he has been with the Department of Chemical and Food Engineering, University of the Américas, Puebla (UDLA,P), serving as chairman from 1987 to 1990. In 1990, he was recognized as Food Engineering Researcher by the National System of Researchers in Mexico and in 1999 was named titular professor. Dr. Vélez-Ruiz’s research activities are focused on rheology of foods; dairy products and milk processing; evaporation and dehydration of fluid foods; osmotic concentration of fruits; heat and mass transfer through the frying process; and physical properties of foods. He is the author of approximately 45 scientific publications in international journals, more than 90 presentations at national and international professional meetings, and three book chapters, and he is an editor of three food science and engineering journals. Gustavo V. Barbosa-Cánovas earned his B.S. in Mechanical Engineering at the University of Uruguay and his M.S. and Ph.D. in Food Engineering at the University of Massachusetts–Amherst. He then worked as an Assistant Professor at the University of Puerto Rico from 1985–1990. Next, he went to Washington State University (WSU), where he is now a Professor of Food Engineering and Director of the Center for Nonthermal Processing of Food (CNPF). His current research areas are nonthermal processing of foods, physical properties of foods, and food powder technology.


Acknowledgments The editors would like to acknowledge each one of the researchers who kindly agreed to participate in this project with their contributions. The support of the Universidad de las Américas, Puebla, of Washington State University, and of Texas Christian University is also acknowledged. For the manuscript’s revision and correction process, the editors counted on the valuable work of M.S. Reyna León and M.S. Daniela Bermúdez, who were supported by Ing. Luz del Carmen López, to help the complete book. We express our gratitude to them.


Contributors A.G. Abdul-Ghani Food Science and Process Engineering Group Department of Chemical and Materials Engineering The University of Auckland Auckland, New Zealand E.A.A. Adell Departamento de Engenharia Química e Alimentos Escola de Engenharia Mauá Instituto Mauá de Tecnologia — Praça Mauá São Paulo, Brazil S.M. Alzamora Departamento de Industrias Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Buenos Aires, Argentina A. Andrés Department of Food Technology Universidad Politécnica de Valencia Valencia, Spain A. Angersbach Department of Food Biotechnology and Food Process Engineering Berlin University of Technology Berlin, Germany J. Arul Department of Food Science and Nutrition and Horticulture Research Center Laval University Sainte-Foy, Quebec, Canada


P.M. Azoubel Faculdade de Engenharia de Alimentos UNICAMP Campinas, São Paulo, Brazil E. Azuara-Nieto Instituto de Ciencias Básicas Universidad Veracruzana Xalapa, Veracruz, México M.O. Balaban Food Science and Human Nutrition University of Florida Gainesville, Florida J. Barat Department of Food Technology Universidad Politécnica de Valencia Valencia, Spain G.V. Barbosa-Cánovas Department of Biological Systems Engineering Washington State University Pullman, Washington A.F. Baroni Departamento de Engenharia Química e Alimentos Escola de Engenharia Mauá Instituto Mauá de Tecnologia — Praça Mauá São Paulo, Brazil F.H. Barron Department of Packaging Science Clemson University Clemson, South Carolina

J. Benedito Food Technology Department Universidad Politécnica de Valencia Valencia, Spain C.I. Beristain-Guevara Instituto de Ciencias Básicas Universidad Veracruzana Xalapa, Veracruz, México J.M. Bunn Department of Packaging Science Clemson University Clemson, South Carolina

M.S. Chinnan Center for Food Safety and Quality Enhancement Department of Food Science and Technology University of Georgia, Griffin Campus Griffin, Georgia A. Chiralt Department of Food Technology Universidad Politécnica de Valencia Valencia, Spain

J. Cárcel Department of Food Technology Universidad Politécnica de Valencia Valencia, Spain

P. Coronel Department of Food Science North Carolina State University Raleigh, North Carolina

F. Castaigne Department of Food Science and Nutrition and Horticulture Research Center Laval University Sainte-Foy, Quebec, Canada

J.G. Crespo Department of Chemistry Faculdade de Ciências e Tecnología Universidade Nova de Lisboa Caparica, Portugal

M.A. Castro Departamento de Ciencias Biológicas Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Buenos Aires, Argentina K.V. Chau Biological and Agricultural Engineering University of Florida Gainesville, Florida X.D. Chen Food Science and Process Engineering Group Department of Chemical and Materials Engineering The University of Auckland Auckland, New Zealand


S.L. Cuppett Department of Food Science and Technology University of Nebraska-Lincoln Lincoln, Nebraska F. Erdoˇgdu Department of Food Engineering University of Mersin Ciftlikkoy, Mersin, Turkey M.M. Farid Food Science and Process Engineering Group Department of Chemical and Materials Engineering The University of Auckland Auckland, New Zealand

H. Feng Department of Biological Systems Engineering Washington State University Pullman, Washington P. Fito Department of Food Technology Universidad Politécnica de Valencia Valencia, Spain A.L. Gabas Departamento de Engenharia de Alimentos Universidade Estadual de Campinas Campinas, São Paulo, Brazil M.A. Garcia-Alvarado Departamento de Ingeniería Química y Bioquímica Instituto Tecnológico de Veracruz Veracruz, México C. González-Martínez Department of Food Technology Universidad Politécnica de Valencia Valencia, Spain S. Grabowski Food Research and Development Centre Agriculture and Agri-Food Canada St. Hyacinthe, Quebec, Canada G.F. Gutiérrez-López Departamento de Graduados e Investigación en Alimentos Escuela Nacional de Ciencias Biológicas — I.P.N. México, México F. Hamouz Department of Nutritional Science and Dietetics University of Nebraska-Lincoln Lincoln, Nebraska


J.A. Hernández-Pérez Joint Research Unit Food Process Engineering Cemagref, ENSIA, INAPG, INRA Massy, France B. Heyd Joint Research Unit Food Process Engineering Cemagref, ENSIA, INAPG, INRA Massy, France M.V. Karwe Department of Food Science Rutgers University New Brunswick, New Jersey D. Knorr Department of Food Biotechnology and Food Process Engineering Berlin University of Technology Berlin, Germany H. Krishnamurthy Department of Food Science and Agricultural Chemistry McGill University, Macdonald Campus Ste. Anne de Bellevue, Quebec, Canada R. León-Cruz Departamento de Ingeniería Química y Alimentos Universidad de las Américas-Puebla Santa Catarina Mártir Cholula, Puebla, México M. Marcotte Food Research and Development Centre Agriculture and Agri-Food Canada St. Hyacinthe, Quebec, Canada

J. Martínez-Monzó Department of Food Technology Universidad Politécnica de Valencia Valencia, Spain

K. Niranjan School of Food Biosciences The University of Reading Reading, Berkshire, U.K.

F.C. Menegalli Departamento de Engenharia de Alimentos Universidade Estadual de Campinas Campinas, São Paulo, Brazil

N. Nitin Department of Food Science Rutgers University New Brunswick, New Jersey

M.R. Menezes Departamento de Engenharia de Alimentos Universidade Estadual de Campinas Campinas, São Paulo, Brazil F.J. Molina-Corral Graduate Program in Food Science and Technology University of Chihuahua Chihuahua, México H. Mújica-Paz Facultad de Ciencias Químicas Universidad Autónoma de Chihuahua Chihuahua, México A. Mulet Food Technology Department Universidad Politécnica de Valencia Valencia, Spain F.E.X. Murr Faculdade de Engenharia de AlimentosUNICAMP Campinas, São Paulo, Brazil A. Nieto Departamento de Industrias Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Buenos Aires, Argentina © 2003 by CRC Press LLC

R. Olivas-Vargas Advanced Materials Research Center Complejo Industrial Chihuahua Chihuahua, México E. Ortega-Rivas Graduate Program in Food Science and Technology University of Chihuahua Chihuahua, México K.S. Pannu Food Process Engineering Delta British Columbia, Canada A. Pérez-Hernández Advanced Materials Research Center Complejo Industrial Chihuahua Chihuahua, México D.L. Pyle School of Food Biosciences The University of Reading Reading, Berkshire, U.K. H.S. Ramaswamy Department of Food Science and Agricultural Chemistry McGill University, Macdonald Campus Ste. Anne de Bellevue, Quebec, Canada A.L. Raoult-Wack CIRAD, Agri-Food Program Montpellier, France

N.K. Rastogi Department of Food Engineering Central Food Technological Research Institute Mysore, India E.P. Ribeiro Departamento de Engenharia Química e Alimentos Escola de Engenharia Mauá Instituto Mauá de Tecnologia—Praça Mauá São Paulo, Brazil C. Rosselló Chemistry Department Universitat de les Illes Balears Palma de Mallorca, Spain S.S. Sablani Department of Bioresource and Agricultural Engineering Sultan Qaboos University Muscat Sultanate of Oman K.P. Sandeep Department of Food Science North Carolina State University Raleigh, North Carolina T. Schäfer Department of Chemistry-CQFB Faculdade de Ciências e Tecnología Universidade Nova de Lisboa Caparica, Portugal M. Schnepf Department of Nutritional Science and Dietetics University of Nebraska-Lincoln Lincoln, Nebraska D.R. Sepúlveda Department of Biological Systems Engineering Washington State University Pullman, Washington © 2003 by CRC Press LLC

S. Simal Chemistry Department Universitat de les Illes Balears Palma de Mallorca, Spain R.P. Singh Department of Biological and Agricultural Engineering University of California, Davis Davis, California M.E. Sosa-Morales Chemical Engineering and Food Engineering Department Universidad de las Américas-Puebla Santa Catarina Mártir, Cholula Puebla, México J. Tang Department of Biological Systems Engineering Washington State University Pullman, Washington J. Telis-Romero Departamento de Engenharia e Tecnologia de Alimentos Universidade Estadual Paulista São José do Rio Preto, São Paulo, Brazil R.F. Testin Department of Packaging Science Clemson University Clemson, South Carolina G. Trystram Joint Research Unit Food Process Engineering Cemagref, ENSIA, INAPG, INRA Massy, France A. Valdez-Fragoso Facultad de Ciencias Químicas Universidad Autónoma de Chihuahua Chihuahua, México

M.E. Vargas-Ugalde Departamento de Ingeniería y Tecnología Facultad de Estudios Superiores Cuautitlán Izcalli, Edo. de México, México School of Food Biosciences The University of Reading Reading, Berkshire, U.K. Departamento de Graduados e Investigación en Alimentos Escuela Nacional de Ciencias Biológicas - I.P.N. México, México J.F. Vélez-Ruiz Chemical Engineering and Food Engineering Department Universidad de las Américas-Puebla Santa Catarina Mártir, Cholula Puebla, México P.J. Vergano Department of Packaging Science Clemson University Clemson, South Carolina

J. Welti-Chanes Departamento de Ingeniería Química y Alimentos Universidad de las Américas-Puebla Santa Catarina Mártir, Cholula Puebla, México S. Wichchukit Department of Biological and Agricultural Engineering University of California, Davis Davis, California J.L. Wiles Department of Packaging Science Clemson University Clemson, South Carolina Y. Wu Department of Research and Development The Wright Group Crowley, Louisiana

O. Vitrac INRA - Food Packaging Unit Centre de Recherches Agronomiques Reims, France

M.R. Zareifard Department of Food Science and Agricultural Chemistry McGill University, Macdonald Campus Ste. Anne de Bellevue, Quebec, Canada

C.L. Weller Industrial Agricultural Products Center and Department of Biological/Systems Engineering University of Nebraska-Lincoln Lincoln, Nebraska

S.E. Zorrilla Instituto de Desarrollo Tecnológico para la Industria Química (INTEC) Consejo Nacional de Investigaciones Científicas y Técnicas Universidad Nacional del Litoral Santa Fe, Argentina


Table of Contents Part I 1 2 3

Fundamentals of Mass Transport J. Welti-Chanes, H. Mújica-Paz, A. Valdez-Fragoso, and R. León-Cruz Heat Transfer in Food Products D.R. Sepúlveda and G.V. Barbosa-Cánovas Introductory Aspects of Momentum Transfer Phenomena J.F. Vélez-Ruiz

Part II 4

5 6 7

8

9

10

11

12

Fundamental Concepts

Mass Transfer

Structural Effects of Blanching and Osmotic Dehydration Pretreatments on Air Drying Kinetics of Fruit Tissues S.M. Alzamora, A. Nieto, and M.A. Castro Pretreatment Efficiency in Osmotic Dehydration of Cranberries S. Grabowski and M. Marcotte Mass Transfer Description of the Osmodehydration of Apple Slabs E. Azuara-Nieto, G.F. Gutiérrez-López, and C.I. Beristain-Guevara Combined Effect of High Hydrostatic Pressure Pretreatment and Osmotic Stress on Mass Transfer during Osmotic Dehydration N.K. Rastogi, A. Angersbach, and D. Knorr Hydrodynamic Mechanisms in Plant Tissues during Mass Transport Operations P. Fito, A. Chiralt, J. Martínez-Monzó, and J. Barat Effect of Pretreatment on the Drying Kinetics of Cherry Tomato (Lycopersicon esculentum var. cerasiforme) P.M. Azoubel and F.E.X. Murr Determination of Concentration-Dependent Effective Moisture Diffusivity of Plums Based on Shrinkage Kinetics A.L. Gabas, F.C. Menegalli, and J. Telis-Romero Modeling Dehydration Kinetics and Reconstitution Properties of Dried Jalapeño Pepper R. Olivas-Vargas, F.J. Molina-Corral, A. Pérez-Hernández, and E. Ortega-Rivas Application of an Artificial Neural Network for Moisture Transfer Prediction Considering Shrinkage during Drying of Foodstuffs J.A. Hernández-Pérez, M.A. García-Alvarado, G. Trystram, and B. Heyd


13

14

15

16

17

18 19

Modeling of Prato Cheese Salting: Fickian and Neural Network Approaches A.F. Baroni, M.R. Menezes, E.A.A. Adell, and E.P. Ribeiro Influence of Vacuum Pressure on Salt Transport during Brining of Pressed Curd A. Chiralt, P. Fito, C. González-Martínez, and A. Andrés Effects of Water Concentration and Water Vapor Pressure on the Water Vapor Permeability and Diffusion of Chitosan Films J.L. Wiles, P.J. Vergano, F.H. Barron, J.M. Bunn, and R.F. Testin Water Vapor Permeability, Water Solubility, and Microstructure of Emulsified Starch–Alginate–Fatty Acid Composite Films Y. Wu, C.L. Weller, F. Hamouz, S.L. Cuppett, and M. Schnepf Mass Transport Phenomena during the Recovery of Volatile Compounds by Pervaporation T. Schäfer and J.G. Crespo Ultrasonic Mass Transfer Enhancement in Food Processing A. Mulet, J. Cárcel, J. Benedito, C. Rosselló, and S. Simal Mass Transfer and Residence Time Studies in Spinning Cone Columns M.E. Vargas-Ugalde, K. Niranjan, D.L. Pyle, and G.F. Gutiérrez-López

Part III 20 21

22

23

24

25 26

Heat Transfer

Transport Phenomena during Double-Sided Cooking of Meat Patties S.E. Zorrilla, S. Wichchukit, and R.P. Singh Thermal Processing of Particulate Foods by Steam Injection. Part 1. Heating Rate Index for Diced Vegetables K.S. Pannu, F. Castaigne, and J. Arul Thermal Processing of Particulate Foods by Steam Injection. Part 2. Convective Surface Heat Transfer Coefficient for Steam K.S. Pannu, F. Castaigne, and J. Arul Modeling of Heat Conduction in Elliptical Cross Sections (Oval Shapes) Using Numerical Finite Difference Models F. Erdog˘du, M.O. Balaban, and K.V. Chau Heat Transfer Coefficient for Model Cookies in a Turbulent Multiple Jet Impingement System N. Nitin and M.V. Karwe Flow Dynamics and Heat Transfer in Helical Heat Exchangers P. Coronel and K.P. Sandeep Relating Food Frying to Daily Oil Abuse. Part 1. Determination of Surface Heat Transfer Coefficients with Metal Balls K.S. Pannu and M.S. Chinnan


Part IV 27

28 29

30

31

32

33

Combined Transfer Phenomena

Relating Food Frying to Daily Oil Abuse. Part 2. A Practical Approach for Evaluating Product Moisture Loss, Oil Uptake, and Heat Transfer K.S. Pannu and M.S. Chinnan Heat and Mass Transfer during the Frying Process of Donuts J.F. Vélez-Ruiz and M.E. Sosa-Morales Influence of Liquid Water Transport on Heat and Mass Transfer during Deep-Fat Frying O. Vitrac, A.L. Raoult-Wack, and G. Trystram Numerical Simulation of Transient Two-Dimensional Profiles of Temperature, Concentration, and Flow of Liquid Food in a Can during Sterilization A.G. Abdul-Ghani, M.M. Farid, and X.D. Chen Heating Behavior of Canned Liquid/Particle Mixtures during End-over-End Agitation Processing S.S. Sablani, H.S. Ramaswamy, and H. Krishnamurthy Dimensionless Correlations for Forced Convection Heat Transfer to Spherical Particles under Tube-Flow Heating Conditions H.S. Ramaswamy and M.R. Zareifard Heat and Mass Transfer Modeling in Microwave and Spouted Bed Combined Drying of Particulate Food Products H. Feng and J. Tang


Part I Fundamental Concepts


1

Fundamentals of Mass Transport J. Welti-Chanes, H. Mújica-Paz, A. Valdez-Fragoso, and R. León-Cruz

CONTENTS 1.1 1.2

Introduction Mass Transfer Variables 1.2.1 Concentration 1.2.2 Velocity 1.2.3 Flux 1.2.4 Flux Relations for Binary Systems 1.3 Mass Transfer by Diffusion 1.3.1 Steady State Diffusion 1.3.2 Molecular Diffusion in Gases, Liquids, and Solids 1.3.2.1 Molecular Diffusion in Gases 1.3.2.2 Molecular Diffusion in Liquids 1.3.2.3 Molecular Diffusion in Solids 1.3.3 Unsteady State Diffusion 1.3.3.1 Solutions of Fundamental Equations 1.4 Mass Transfer by Convection 1.4.1 Film Theory and Mass Transfer Coefficient 1.4.2 Two-Film Theory and Mass Transfer Coefficient 1.4.3 Dimensionless Numbers for Mass Transfer 1.4.4 Transport Analogies 1.4.5 Mass Transfer Coefficients and Correlations 1.4.6 Mass Transfer Units Nomenclature References

1.1 INTRODUCTION Mass transfer can be defined as the migration of a substance through a mixture under the influence of a concentration gradient in order to reach chemical equilibrium. Biochemical and chemical engineering operations, such as absorption, humidification, distillation, crystallization, and aeration, involve mass transfer principles.


In food processing, mass transfer phenomena are present in freeze-drying, osmotic dehydration, salting or desalting, curing and pickling, extraction, smoking, baking, frying, drying of foods, membrane separations, and the transmission of water vapor, gases, or contaminants across a packaging film. Food stability and the preservation of food quality are also affected by mass transfer of environmental components that can affect the rate of reactions. Among the components involved in these mass transfer processes are water, sugars, salt, oils, proteins, acids, flavor and aroma substances, oxygen, carbon dioxide, residual monomers or polymer additives, and toxins or carcinogens produced by microorganisms. Furthermore, mass transfer phenomena are important in the scale-up of processes to pilot- or commercial-scale plants and in the control and optimization of the processes. This chapter presents the basic principles of mass transfer. First, the variables that occur in mass transfer are reviewed. Then the mechanisms of mass transfer, diffusion and convection are discussed. Analogies among momentum, heat, and mass transfer are described; convective mass transfer coefficients and correlations are derived by analogy with convective heat transfer. Finally, the concept of transfer units is presented.

1.2 MASS TRANSFER VARIABLES Mass transfer processes involve concentration, velocity, and flux variables, which are defined and related by a set of basic equations (White, 1988).

1.2.1 CONCENTRATION The concentration of a mixture and its components may be expressed in terms of 3 mass and mol. In terms of mass, the mass concentration of the mixture (ρ, kg/m ), 3 the mass concentration of a component i (ρi, kg/m ), and the mass fraction of component i (wi) are given by: ρ= m/V

(1.1)

ρi = m i / V

(1.2)

w i = m i / m = ρi / ρ

(1.3)

where m and mi are the mass flux of the mixture and component i, respectively. 3 The bulk molar concentration (C, kg mol/m ), the molar concentration of com3 ponent i (Ci, kg mol/m ), and the mole fraction of component i (xi) are defined by: C = n/V

(1.4)

Ci = n i / V

(1.5)

x i = n i / n = Ci / C

(1.6)

where n and ni are the mol of the mixture and component i, respectively.


According to the previous definitions, it can be easily shown that: n

m=

∑

n

m i and ρ =

i =1

∑

∑C

i

(1.8)

i =1

n

∑

(1.7)

n

n i and C =

i =1

w=

i

i =1

n

n=

∑ρ

n

w i = 1 and x =

i =1

∑x =1 i

(1.9)

i =1

where xi is the mole fraction of component i, and ρi and Ci are related through the molecular weight of constituent i (Mi, kg/kg mol): ρi = M i C i

(1.10)

1.2.2 VELOCITY In mass transfer phenomena, the velocity of a bulk mixture and of its components can be measured with respect to fixed coordinates. In addition, the velocity of the components can also be measured relative to the bulk velocity. Figure 1.1 illustrates these velocities in a binary system of components A and B in the z direction.

fixed coordinate

vB – v = diffusion velocity of B = UB

vB

v= wA vA + wB vB vA

z vA – v = diffusion velocity of A = UA FIGURE 1.1 Scheme of individual and bulk velocities of a binary mixture.


The mass bulk velocity of the mixture (v, m/sec) relative to fixed coordinates is defined as: n

v=

∑

n

wi vi =

i =1

∑ i =1

mi v = m i

n

ρi

∑ ρv

(1.11)

i

i =1

where vi is the velocity of component i with respect to stationary coordinates. In a similar manner, a molar bulk velocity ( V, m/sec) measured relative to stationary coordinates can be defined as: n

n

V=

∑x V = ∑ i

i =1

i

i =1

ni V = n i

n

∑ CV Ci

(1.12)

i

i =1

The velocity of the constituent i relative to the bulk velocity of the mixture is: u i = vi − v

(1.13)

Ui = Vi − V

(1.14)

where ui (m/sec) and Ui (m/sec) are the mass and molar diffusion velocities, respectively.

1.2.3 FLUX 2

2

The mass bulk flux (n, kg/m sec) and the molar bulk flux ( N , kg mol/m sec) of a mixture relative to fixed coordinates are: n

n = ρv =

∑ρ v

(1.15)

i i

i =1 n

N = CV =

∑C V i

i

(1.16)

i =1

The flux of the components of a mixture can also be expressed relative either to fixed coordinates or to the bulk average velocity. The flux of the component i relative to stationary coordinates is: n i = ρi v i

(1.17)

N i = Ci Vi

(1.18)

The diffusion fluxes of the constituents i of the mixture with respect to the 2 2 average bulk velocity are ji (kg/m sec) for the mass flux and Ji (kg mol/m sec) for the molar flux.


ji = ρi u i = ρi ( vi − v)

(1.19)

J i = C i U i = C i (V i − V)

(1.20)

The use of concentration, velocities, and fluxes in mass or molar terms is subject to preferences and convenience. Nevertheless, concentration and flux expressed in molar units and average molar velocity are preferred. In discussing the fundamentals of mass transfer, systems are frequently assumed as a two-component mixture to facilitate the understanding of multi-component systems found in engineering applications.

1.2.4 FLUX RELATIONS

FOR

BINARY SYSTEMS

For a binary mixture of components A and B, Equations (1.18) and (1.20) become: N A = CA VA

(1.21)

J A = C A U A = C A (V A − V)

(1.22)

Substituting V from Equation (1.12) into Equation (1.22), J A = CA V A − CA V = CA VA −

CA (C A V A + C B V B ) C

(1.23)

Since N A = C A V A and N B = C B V B , for component A, Equation (1.23) becomes: NA = JA +

CA (N A + N B ) C

(1.24)

A similar mathematical derivation gives for component B: NB = JB +

CB (N A + N B ) C

(1.25)

Equations (1.24) and (1.25) show that the absolute molar flux (NΑ or NB) results from a concentration gradient contribution or a molar diffusion flux (JA or JB) and a convective contribution ( C A V or C B V). The molar diffusion flux is described by Fick’s law, which for component A is written as: J A = − D AB

dC A dz

(1.26)

where DAB is the diffusion coefficient of A through B and dCA/dz is the change of the concentration A with respect to the position z. In terms of mass, the mass fluxes for components A and B are, respectively:


n A = jA +

ρA (nA + nB ) ρ˜

(1.27)

n B = jB +

nB (n + n B ) ρ˜ A

(1.28)

Two important simple cases can be considered in Equation (1.24): 1. Diffusion of A through stagnant B, NB = 0 NA =

JA D dC A = − AB 1 − xA 1 − x A dz

(1.29)

2. Equimolar counter-diffusion NA + NB = 0, N A = J A = − D AB

dC A dz

(1.30)

where xA is the mole fraction of component A in the case of dilute systems xA 2.5 mm

Sh = 0.42 + (Gr ) (Sc)

Used for rising bubbles, d < 2.5 mm

Membranes

k =1 dD

Used in actual or hypothetical membranes

Packed bed of pellets

jD = 1.17 Re −0.415

Re f (particle diameter and superficial velocity in the bed)

Bubbles

1/ 3


1/ 2

1.4.6 MASS TRANSFER UNITS The number of transfer units is another approach in the application of pilot plant results to the design of mass transfer equipment, such as cooling towers, packed columns, extractors, or driers. This method can be illustrated in the determination of the height of a drier (Z), which is obtained as the product: Z = HTU C NTU D

(1.60)

where HTUc is the height of a transfer unit and NTUD is the number of mass transfer units given by: C2

NTU D =

∫ C −C

C1

dC

i

(1.61) b

Subscripts 1 and 2 refer to inlet and outlet values. The integral is evaluated from considerations of bulk (b) and interphase (i) conditions at any cross-section of the drier, absorption tower, etc., the former being determined by mass balances and the latter by equilibrium data. The height of a transfer unit is given by: HTU =

G k m aA tρg

(1.62)

where km is the mass transfer coefficient, a is a specific area, At is a cross-sectional area of the drier, ρg is the density of the gas phase, and G is the gas flow rate. In practice, it is found that HTU is a property of the type of drier and the material being dried but is independent of size and operating conditions. Therefore, HTU can be calculated from pilot plant experiments in which the height of the drier and the number of transfer units are known. Thus, the design of a similar drier can be obtained by scaling up the pilot plant results. Here again, the Colburn analogy is of great utility. From the arguments used in the section about transfer analogies, a similar expression for the Colburn analogy has been proposed (Le Goff, 1980): NTU DSc2 / 3 NTU H Pr 2 / 3 = = NEU LfD LfH

(1.63)

where the quantity NEU is the number of energy units, and the Le Goff number (Lf) represents the deviation of the Colburn analogy from unity, the Lf number being in the range 1–0.02.


Equations for NTUH and NEU are similar to Equation (1.58) for NTUD: T2

NTU H =

∫ T −T

(1.64)

P1 − P2 ρv2

(1.65)

T1

NEU =

dT

i

b

The number of transfer units is a basic parameter that appears frequently in the design of heat and mass exchangers (Van den Bulck et al., 1985; Khodaparast, 1992; Treybal, 1981).

NOMENCLATURE a At bn C, c * C Cp d D f g G h HTUD j J Jo J1 km k Km K m m m M n n

2

2

Specific area; cm or m 2 Cross-sectional area of drier; m Roots of Bessel function 3 Molar concentration; kg mol/m 3 Molar concentration at equilibrium; kg mol/m Specific heat capacity; kJ/kg K Characteristic dimension of a solid body; m or cm 2 2 Diffusion coefficient; m /sec or cm /sec Friction factor; dimensionless 2 Acceleration constant for gravity, m/sec 2 Flow rate; kg/m sec 2 Heat transfer coefficient; W/m K Height of a mass transfer unit, m The mass flux of the mixture with respect to the average bulk velocity; 2 kg/m sec Molar diffusion flux of the mixture with respect to the average bulk velocity; 2 kg mol/m sec Bessel function of first kind and order zero Bessel function of first kind and order one 2 Mass transfer coefficient; m /sec Thermal conductivity; W/m K Overall mass transfer coefficient based on the gas system concentration driving force Partition coefficient; dimensionless Mass of the mixture; kg in Equation (1.1) Solubility constant between the two phases in Equation (1.48) 2 Mass flux of the mixture relative to fixed coordinates, kg/m sec in Equation (1.11) Molecular weight; kg/kg mol Mole of the mixture; kg mol in Equation (1.4) Relative position in Equations (1.7) and (1.12)


n N NEU NTUD p P q r R R S′ T t u U v V V V w x x X z Z

Mass bulk flux of the mixture relative to stationary coordinates, kg 2 mol/m sec 2 Molar bulk flux of the mixture relative to fixed coordinates; kg mol/m sec Number of energy units Number of mass transfer units Partial pressure; atm Pressure; atm Heat flow rate; W Radial coordinate Radius; cm or m in Equations (1.42) and (1.43) Gas constant; joules/g mol K in Equation (1.38) Shape factor Temperature; K Time; sec Mass diffusion velocity of the system relative to the mass bulk velocity; m/sec Molar diffusion velocity of the system relative to the molar bulk velocity; m/sec Mass bulk velocity of the mixture relative to stationary coordinates; m/sec 3 Volume of the mixture, m in Equation (1.1) 3 Molal volume of the system; m /kg mol in Equation (1.34) Molar bulk velocity measured with respect to fixed coordinates; m/sec Mass fraction of the mixture Rectangular coordinate Mole fraction of the mixture in Equation (1.29) Dimensionless time Rectangular coordinate Height of a drier; m

GREEK SYMBOLS α β δ ε µ ν ρ σAB τ ΩD ϕ

2

Thermal diffusivity; cm /sec Coefficient of thermal expansion; 1/K Thickness of a hypothetical stagnant film; cm or m Porosity Viscosity; centipoises 2 Kinematic viscosity; m /sec 3 Mass concentration of the mixture or fluid density; kg/m Collision diameter; °A Tortuosity Collision integral for molecular diffusion; dimensionless Association parameter

REFERENCES Crank, J., The Mathematics of Diffusion, 2nd ed., Oxford University Press, London, 1975. Cussler, E.L., How we make mass transfer seem difficult, Chem. Eng. Educ., 18(3), 124–127, 149–152, 1984.


Geankoplis, C.J., Transport Processes and Unit Operations, 3rd ed., Prentice-Hall., London, 1993. Gekas, V., Transport Phenomena of Foods and Biological Material, CRC Press, Boca Raton, FL, 1992. Khodaparast, K.A., Predict the number of transfer units for cooling towers, Chem. Eng. Progress, 4, 67–68, 1992. Le Goff, P., Performance energétique des echangeurs de matière et de chaleur: interprétation energétique des analogies de Reynolds et de Colburn, Chem. Eng. J., 20, 197–209, 1980. Sherwood, T.K., A review of the development of mass transfer theory, Chem. Eng. Educ., 3, 204–213, 1974. Sherwood, T.K., Pigford, R.L., and Wilke, C.R., Mass Transfer, McGraw-Hill Kogakusha, Ltd., Tokyo, 1975. Treybal, R.E., Mass Transfer Operations, 3rd ed., McGraw-Hill International Editions, 1981. Van den Bulck, E., Mitchell, J.W., and Klein, S.A., Design theory for rotary heat and mass exchangers II. Effectiveness number of transfer units method for rotary heat and mass exchangers, Int. J. Heat Mass Transfer, 28(8), 1587–1595, 1985. Welty, J.R, Wicks, C.E., and Wilson, R.E., Momentum, Heat and Mass Transfer, 3rd ed, John Wiley & Sons, New York, 1984. White, F.M., Heat and Mass Transfer, Addison-Wesley, Reading, MA, 1988.


2

Heat Transfer in Food Products D.R. Sepúlveda and G.V. Barbosa-Cánovas

CONTENTS 2.1 2.2

Introduction General Background 2.2.1 Thermal Properties of Foods 2.2.1.1 Specific Heat 2.2.1.2 Thermal Conductivity . 2.2.1.3 Thermal Diffusivity 2.2.1.4 Surface Heat Transfer Coefficient 2.2.2 Heat Transfer by Conduction 2.2.2.1 Steady State 2.2.2.2 Nonsteady State 2.2.3 Heat Transfer by Convection 2.2.3.1 Natural Convection 2.2.3.2 Forced Convection 2.2.4 Heat Transfer by Radiation 2.3 Conclusions References

2.1 INTRODUCTION Heating and cooling are common activities in food processing. Several operations involving heating of raw foods are performed for different purposes, such as reduction of the microbial population, inactivation of enzymes, reduction of the amount of water, modification of the functionality of certain compounds, and of course, cooking. On the other hand, heat is removed from foods to reduce the rate of its deteriorative chemical and enzymatic reactions and to inhibit microbial growth, extending shelf life by cooling and freezing. Heat transfer plays a central role in all of these operations; therefore, food engineers need to understand it in order to achieve better control and avoid under- or over-processing, which often results in detrimental effects on food characteristics. In practice, heat transfer to or from foods can be attained either by indirect or direct methods. Indirect methods involve the use of heat exchangers that isolate the product from the medium used as a source or sink


TABLE 2.1 Examples of Direct and Indirect Heating and Chilling

Direct Indirect

Heating

Chilling

Frying, boiling, baking, and grilling solid foods Fluid food pasteurization; canned products sterilization

Fluidized bed individual quick freezing (IQF) Fluid food cooling; ice production

of heat. Direct methods allow contact between the food and the heating/cooling medium. Examples of these methods can be found in Table 2.1. Indirect heating methods use gases and vapors, such as steam or air, and liquids, such as water or organic compounds, as a source of thermal energy. Chilling by  indirect methods involves the use of coolant gases, such as ammonia or Suva , or the use of coolant liquids, such as water or ethylene glycol. Direct heating can be attained by means of hot air, oil, infrared energy, and dielectric or microwave methods, among others. Direct chilling can be achieved by the use of cold air or by the application of the Peltier effect.

2.2 GENERAL BACKGROUND The first step in understanding heat transfer is to define what heat is and how it diffuses through a single body or is transferred from one body to another. Heat is a nonmechanical form of energy transferred between regions of different temperature. Heat transfer, therefore, is a natural energy transfer process in which energy tends to travel from a hotter point to a colder point in order to reach an equilibrium temperature. Now that we know that the temperature gradient is the driving force in heat transfer processes, and therefore that if a temperature difference exists, energy transference will occur, a couple of questions arise. How fast will this energy transfer process occur? How will the energy diffuse through foods? The answers to these questions are not easy. Great efforts have been made to try to answer these questions, and several models have been developed to describe heat transfer behavior in different systems under different conditions. The heat transfer mode governing the process is defined by the physical state of the bodies and their relative position. If a heat gradient exists between two solid bodies in contact, the heat transfer will proceed by conduction. If the same gradient exists between two fluids, or between a fluid and a solid, the energy will be transferred by convection. Finally, any body with a temperature above absolute zero will radiate energy in the form of electromagnetic waves transferring heat by radiation. Besides the physical state or relative position, other physical properties of the bodies involved in these processes influence the heat transfer rate. Characteristics such as form, size, structure, thermal conductivity, specific heat, density, and viscosity, among others, are of paramount importance in the definition of the behavior of a system. Early developments in the study of heat transfer in the chemical engineering field assumed controlled situations dealing with well-defined substances with fixed


physical characteristics and clearly defined heat transfer modes. However, actual situations in food engineering involve more than one mode of heat transfer simultaneously, and frequently some of the physical characteristics of food, such as density, form, or viscosity, change as heat modifies the chemical structure, affecting the food’s thermal behavior. Furthermore, foods usually have neither regular form nor homogeneous or isotropic behavior. Finally, some particular features of food being heated, such as nonuniform evaporation of water, crust formation, or closing or opening of pores, are of such complexity that they make the modeling of this process difficult or impracticable. Nevertheless, some of these drawbacks have been overcome, and the modeling of several specific practical situations is possible, mainly due to the development of knowledge of empirical relations that properly suit these specific processes. Present-day analytical techniques, such as the finite element method, allow for the modeling of situations characterized by nonuniform thermal properties that change with time, temperature, and location, so that great developments can be expected in the modeling of heat transfer processes in foods. The objective of this chapter is not to describe all of these specific models, but to introduce the reader to a practical approach to the study of heat transfer in foods. Classic models constructed over assumptions of homogeneity and isotropy will be introduced in order to provide a basic knowledge that will enable the modeling of simple systems and the understanding of more elaborate and specific models. Now we will focus on some of the most important engineering properties of foods and how they can be measured or calculated in order to use them in further modeling of heat transfer processes.

2.2.1 THERMAL PROPERTIES

OF

FOODS

As stated before, the modeling of heat transfer processes is dependent on some of the physical properties of the foods involved. As mathematical techniques become more elaborate, a higher accuracy is needed in the measurement or calculation of properties such as specific heat, thermal conductivity, thermal diffusivity, and surface heat transfer coefficient. These properties will be discussed below. 2.2.1.1 Specific Heat Specific heat (Cp) is an exclusive property of every substance, and it is defined as the amount of energy needed to increase the temperature of one kilogram of matter by one degree Celsius; therefore, its units are J/kg°C. This property is not dependent upon temperature or mechanical structure (e.g., density). It has been found that a strong correlation exists between a food’s composition and its specific heat. Derived from the definition of specific heat, the amount of heat Q required to increase the temperature of a body with mass m from an initial temperature T1 to a final temperature T2 is: Q = m Cp (T2 – T1)

(2.1)

In order to determine the Cp value for a specific substance or food, differential scanning calorimetry (DSC) is used. Comprehensive data have been gathered, and tables containing Cp values for many products are available in the literature (American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., 1985).


However, as the water content and composition of foods have been found to affect the Cp value, the data presented in the literature are bound to specific water contents and formulations, thus reducing their usefulness. For a more practical way to find the Cp value, several empirical mathematical expressions have been developed. These expressions can vary from product to product and sometimes even from author to author. Some of these expressions are based only on the water content, having the form Cp = C1 + C2W, where W stands for water content and the constants are defined depending on the situation and product. Examples of these expressions are: Cp = 0.837 + 3.349W

Siebel, 1982

(2.2)

Cp = 1.382 + 2.805W

Dominguez et al., 1974

(2.3)

Cp = 1.470 + 2.720W

Lamb, 1976

(2.4)

More complex expressions including other food components besides water have been developed, and some examples are shown below. Cp = 2.309Xf + 1.256Xs + 4.187W

Charm, 1971

(2.5)

Cp = 1.424Xc + 1.549Xp + 1.675Xf + 0.837Xa + 4.187W Heldman and Singh, 1981 (2.6) where Xc stands for mass fraction of carbohydrates, Xp for proteins, Xf for fat, Xa for ash, and W for water. Generic expressions for mixtures are also available: Cp =

∑ (C Cp ) i

i

Choi and Okos, 1986a

(2.7)

i

where Ci is the mass concentration of each constituent i and Cpi its corresponding Cp. 2.2.1.2 Thermal Conductivity Thermal conductivity (k) is a characteristic that tells us how effective a material is as a heat conductor. As stated in Fourier’s law for heat conduction, this constant is a proportionality factor needed for calculations of heat conduction transfer processes. This physical property can be directly measured from a food material using a thermocouple and a heater, as described in detail by Mohsenin (1980). The suited units for this property are W/m°C. Some data have been gathered and can be found in the literature (American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., 1985). Thermal conductivity of foods depends mainly on the composition of the food, but factors such as fiber orientation and void spaces have an influence on the heat flow paths through food; therefore, it is important to describe the bulk condition in addition to the composition when reporting a k value. As in the case of specific heat, some empirical expressions to calculate thermal conductivity values for different foods have been developed. Some examples are: For fruits and vegetables: k = 0.148 + 0.493W where W is the mass fraction of water.


Sweat, 1974

(2.8)

For sucrose solutions, fruit juices and milk: −3

2

k = 1.73 × 10 (326.8 + 1.0412T – 0.00337T )(0.44 + 0.54W)

Riedel, 1949 (2.9)

where T is temperature expressed in degrees Celsius. Other applicable more general expressions for mixtures are of the type: k=

∑ (k X ) i

v i

(2.10)

i

v

th

where ki is the thermal conductivity of the i component and Xi its volume fraction. When heat is flowing through wrappings or compound layer materials, a total thermal conductivity coefficient is needed. This expression can be calculated if the individual thermal conductivity coefficients are known. For heat flowing parallel to two layers, the total thermal conductivity coefficient is given by kT = k1 (1 – c) + k2c

Hallstrom et al., 1988

(2.11)

where c is the volume fraction of material two. On the other hand, if the heat flow is perpendicular to the material’s layer orientation: kT =

k1k 2 ck1 + (1 − c)k 2

Hallstrom et al., 1988

(2.12)

In the case of material mixtures with random size and orientation particles, the value for the total thermal conductivity value will be found between the value for parallel flow and the value for perpendicular flow. For mixtures of more than two components, the same method is followed, taking two materials at a time. 2.2.1.3 Thermal Diffusivity Thermal diffusivity (α) is a compounded thermal property of materials and is calculated from the values found for thermal conductivity, specific heat, and density of a particular product. α=

k Cpρ

(2.13)

where k is thermal conductivity, Cp specific heat, and ρ mass density. This thermophysical property links conductivity of the materials with their ability to store heat, thereby showing how heat will diffuse throughout the materials when heated. The units 2 for thermal diffusivity are m /sec. Although the recommended method to determine thermal diffusivity is a calculation based on experimentally measured values of thermal conductivity, specific heat, and density, other heat diffusivity measurement methods have been developed (Choi and Okos, 1983). Expressions based on water content and temperature of foods have also been developed. Some examples are: –8

–10

α = 5.7363 × 10 W + 2.8 × 10 T


Martens, 1980

(2.14)

where W is the mass fraction of water and T is the temperature expressed in degrees Kelvin, and –8

α = 8.8 × 10 (1 – W) + α wW

Dickerson, 1969

(2.15)

where α w is the thermal diffusivity of water at the studied temperature. As in the cases of other properties, the thermal diffusivity of mixtures can be calculated with a general expression: α=

∑ (α X ) i

i

Choi and Okos, 1986b

(2.16)

i

th

where αi is the thermal diffusivity of the i component and Xi its mass fraction. As thermal diffusivity can be calculated from other thermal properties, data are not frequently found in the literature. 2.2.1.4 Surface Heat Transfer Coefficient The heat transfer coefficient (h) is not a property of materials themselves, but rather a property of convective heat transfer systems involving a solid surface and a fluid. This coefficient is used as a proportionality factor in Newton’s law of cooling, adjusting for the characteristics of the system under study. To define the value of this factor, it is necessary to characterize the convective medium and the surface involved in the convective heat transfer process. Some of the characteristics involved in the calculation of the heat transfer coefficient are the fluid’s velocity, viscosity, density, thermal conductivity, and specific heat. The form and surface texture of the solid involved are also important. As can be determined from a dimensional analysis of Newton’s cooling law, the units for the heat transfer 2 coefficient are W/m °C. Since the heat transfer coefficient is a property of the system rather than of the material, its measurement is difficult, and several empirical expressions have been developed to overcome this problem. Some of these expressions will be reviewed below in the section dealing with heat transfer by convection. A compilation of surface heat transfer coefficient data and empirical expressions to calculate this coefficient can be found in the 1985 edition of the ASHRAE Handbook of Fundamentals. Further information regarding measurement of the heat transfer coefficient can be found in the literature (Arce and Sweat, 1980). Now that we have shown methods for determining the most relevant thermal properties of foods, we will find out how to use them to model heat transfer processes. The particulars of the different heat transfer modes will be dealt with in the following sections.

2.2.2 HEAT TRANSFER

BY

CONDUCTION

Heat transfer in solids or highly viscous materials takes place by conduction. In this mode, energy is transferred among particles touching each other with no movement of material. As a solid material is heated on one of its faces, a gradient is established between the hot face and the opposite cooler face. This gradient is the driving force promoting the heat flow from one face to the other. As heat penetrates the body,


the interior temperature changes from point to point and across time. This period of time is known as the unsteady state period. Later, when heat has traveled all the way across the body and equilibrium in temperatures has been reached, the interior temperature of each point will remain the same with respect to time and will only depend on its relative position inside the body. At this moment, the steady state transfer regime has been reached, and the body is working as a heat conductor with a determined heat flux going through it. Thus, the study of heat conduction can be divided into two main areas: the study of heat conduction in the steady state and the study of heat conduction in the nonsteady state. 2.2.2.1 Steady State The study of heat conduction in the steady state is useful in modeling the performance of insulators, heat exchangers, and other equipment used to transfer heat from one point to another, such as containers, pans, and walls. In these, the initial variations in internal temperature dependent on time have settled, and the temperature profile inside the material is stationary. The important issue here is to determine the amount of heat a particular material of a given thickness will allow to flow through it. This mechanism can be modeled using Fourier’s law for heat conduction (Equation 2.17), which establishes that the heat flux Qx transmitted through a solid in the direction x is inversely proportional to the thickness x and directly proportional to both the perpendicular transmission area A and to the temperature difference between its two opposite faces ∆T. The proportionality constant needed by this model is the thermal conductivity (k), one of the physical characteristics previously described. In some materials, thermal conductivity may vary with temperature. In these situations, the value corresponding to the average temperature should be used. Q x = − kA

∂T(x) ∂x

(2.17)

The negative sign represents the heat flow from the hottest to the coolest surface, thereby rendering a positive value for the heat flux. As can be seen, this equation describes heat flow only in one direction. For a complete mathematical description, we would need to write similar equations for the other two directions in a three-dimensional system and to integrate over the entire volume. However, most steady state processing applications involve heat conduction in only one direction, as when heat flows across walls or heat exchangers. For practical purposes, these materials can be considered as infinite slabs, so we do not need to be concerned with a general mathematical solution. Qx = − k

A∆T x

(2.18)

Equation (2.18) is the integrated form of Fourier’s law for unidirectional steady state heat conduction over a path of constant cross-sectional area in a parallelepiped. It can be used directly to calculate the heat flux through a body. Besides heat flowing through flat surfaces, another common situation in food engineering is the use of


cylindrical containers or pipes; therefore, an expression for Fourier’s law on cylindrical surfaces is necessary. That expression is shown in Equation (2.19), Qx = − k

A lm ∆T x

(2.19)

where all the terms stand for the same characteristics as in the previous expression except for Alm, which in this case is the logarithmic mean of the inner and outer areas: A lm =

A1 − A 2 A ln 1 A   2

(2.20)

Another common situation is the presence of compounded walls. The introduction of the concept of thermal resistance can be of help when studying heat conduction in this kind of system. If thermal resistance R is defined as the ratio of the thickness x over the area A and thermal conductivity k, R=

x kA

(2.21)

Fourier’s law takes a form identical to Ohm’s law for electric current flow, and all mathematical manipulations can be performed as in electrical calculations. I=

∆V R

(2.22)

Q=

∆T R

(2.23)

Here, the flow of electric current I is directly proportional to the voltage difference ∆V and inversely proportional to the electric resistance R. In the same way, the heat flow Q is directly proportional to the temperature difference ∆T and inversely proportional to the thermal resistance R. This analogy shows that in the case of several different layers placed in series (normal heat flow), the overall temperature difference is the sum of the individual temperature differences, and the same Q flows through all the resistances. Therefore, the total thermal resistance is simply the sum of the individual resistances. On the other hand, when there is parallel heat flow through several layers, the analogy indicates that we have a parallel system. Therefore, the total Q is the sum of the Qx through the individual resistances, and the temperature differences for the individual resistances are all the same and equal to the overall temperature difference. In this case, the reciprocal of the total thermal resistance is equal to the sum of the reciprocals of the individual resistances. The use of these analogies allows us to predict heat transfer in compound systems. 2.2.2.2 Nonsteady State The study of heat conduction in the nonsteady state is pertinent when calculating processes in which the focus is to heat or cool a body instead of using it as a heat conduction medium. Such processes include freezing, cooking, and thermal sterilization


of foods. In these cases, the main concern is to find out how long it will take the coldest/hottest point in the body to reach the desired temperature. Moreover, the study of this phenomenon allows us to determine the time needed to process foods at a desired temperature or the temperature at a determined point at a given time. In nonsteady state heat transfer processes in which more than one heat transfer mode applies, conduction usually governs the process, as it is the slowest heat transfer mode. It is important to consider that heat conduction in foods is frequently a threedimensional phenomenon, as the food has finite dimensions and sometimes is immersed in the heat transfer medium, with heat being transferred through all its surfaces (as in oven baking or deep fat frying). Fourier’s second law of heat transfer for three-dimensional nonsteady state heat conduction states that:  ∂2 T ∂2 T ∂2 T  ∂T = α 2 + 2 + 2  ∂t ∂y ∂z   ∂x

(2.24)

where T stands for temperature, t for time, x, y, and z for the distance on the x, y and z axes, respectively, and α for thermal diffusivity, which is a physical characteristic of the materials, as previously discussed. To arrive at the analytical solution to this complex expression can be difficult or impracticable. However, the analysis of some practical situations may be simplified through a couple of useful assumptions. The first assumption supposes that we are dealing with a semi-infinite body, also known as a thick solid. This semi-infinite body is defined as one with infinite width, length, and depth. If a body with these characteristics is immersed in the heating medium, we can assume that heat will be transferred just from the surface toward the interior, following a straight trajectory to the center; therefore, Fourier’s law can be transformed into: ∂T ∂2 T =α 2 ∂t ∂x

(2.25)

where the x-axis corresponds to any one of the dimensions on a parallelepiped or to the radius on a sphere-like body. The solution to Equation (2.25) becomes simpler now and may be obtained by applying the boundary conditions, which are: T0 Initial temperature of the body at time 0 Tm Heating medium temperature acquired at the surface when immersed in the fluid. The solution takes the form of Equation (2.26), where the temperature T at any point x, measured from the surface, can be determined using a dimensionless temperature ratio: hx

Tm − T  x  k = erf   +e  4αt  Tm − T0

2

 h +   αt  k

  x  h αt  +  erf   4αt  k  

(2.26)

where erf is the Gauss’ error function, which can be obtained from Table 2.2, and k is the thermal conductivity.


TABLE 2.2 Gauss’ Error Function η

η) erf (η

η

η) erf (η

η

η) erf (η

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52 0.56 0.60 0.64

0.00000 0.45110 0.09008 0.13476 0.17901 0.22270 0.25670 0.30788 0.34913 0.38933 0.42839 0.46622 0.50275 0.53790 0.57162 0.60386 0.63459

0.68 0.72 0.76 0.80 0.84 0.88 0.92 0.96 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70

0.66278 0.69143 0.71754 0.74210 0.76514 0.78669 0.80677 0.82542 0.84270 0.88020 0.91031 0.93401 0.95228 0.96610 0.97635 0.98379

1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.20 3.40 3.60

0.98909 0.99279 0.99532 0.99702 0.99814 0.99886 0.99931 0.99959 0.99976 0.99987 0.99993 0.99996 0.99998 0.999994 0.999998 1.000000

Since, in this case, the heat is transferred from a fluid to a solid surface, the surface heat transfer coefficient h is incorporated into the solution. For situations where the heat transfer between the fluid and the solid proceeds very efficiently, that is, if h is infinite, the solid surface takes the medium temperature instantaneously, and Equation (2.26) can be simplified to: Tm − T  x  = erf    4αt  Tm − T0

(2.27)

Now that we know how to handle semi-infinite bodies, a problem arises. The definition of a semi-infinite body describes a body with infinite dimensions, which in reality is not a possible situation. When can a body be considered as a semi-infinite body for practical purposes? The concept of a semi-infinite body in practice is better related to the system’s behavior than to the body dimensions. The surface heat transfer coefficient, along with the thermal conductivity, plays a fundamental role in determining whether a body will exhibit a thick body response. If a finite body of a given thickness is studied in the early steps of heat penetration, it will exhibit a semi-infinite body response as heat has only flowed from the outside to the center; therefore, the ratio of thickness to time plays an important role as well. Schneider (1973) found a critical Fourier number to define when a body ceases to exhibit a thick body response: –0.3

Focritical = 0.00756 Bi


+ 0.02

for 0.001 ≤ Bi ≥ 1000

(2.28)

The Fourier and Biot numbers are: Fo =

αt ( L )2

(2.29)

Bi =

hL k

(2.30)

where L is half the thickness of the body and t is the time the body has been exposed to the conditions studied. If the actual Fourier number exceeds the critical value, the body can no longer be considered to be a semi-infinite body, and a different approach must be taken. Another possible approximation used to solve practical problems regarding heat conduction in a nonsteady state involves the assumption of infinite bodies with simple geometry. Again, the concept of an infinite body is just theoretical, as in practice no such bodies occur and, therefore, a practical definition is needed. An infinite body is a body in which two of its dimensions are much larger than the third. In this situation, it can be expected that heat will reach the center of the body first traveling the shortest way; that is, following the axis corresponding to the thinner dimension. Heat transfer through the remaining axes becomes negligible. This assumption applies to infinite slabs with thickness 2xo and infinite longitude and width, infinite cylinders with radius r and infinite length, and spheres. Analytical solutions for this kind of problem can be developed using dimensionless ratios. Expressions for temperature Y and position n are: Y=

Tm − T Tm − T0

(2.31)

x x0

(2.32)

n=

where Tm is the temperature of the cooling or heating fluid surrounding the solid, T0 is the initial temperature of the body, T is the temperature at a point x, x0 is the distance from the center or midplane to the surface, depending on the body’s shape, and x is the distance from the center or midplane to the selected point. Time and thermal diffusivity are introduced using Fourier module τ, while surface heat transfer coefficient and thermal conductivity are both included using the Biot module in m: τ = Fo =

αt x 20

(2.33)

1 k = Bi hx 0

(2.34)

∂T ∂2 Y =α 2 ∂t ∂n

(2.35)

m= Modifying Equation (2.25), we obtain:


which is the dimensionless form for Fourier’s second law. Applying boundary conditions, Equation (2.35) can be analytically solved by partial derivatives for different situations. However, the analytical solutions for an infinite slab, an infinite cylinder, and a sphere have already been developed and can be found in graphic form as shown in Figures 2.1, 2.2, and 2.3. Using these figures, it is possible to find the Y value for a given set of conditions. First, the correct group of lines is chosen based on the calculated m value. Next, one of the lines is picked depending on the n value. Finally, by finding the intersection between the selected line and the calculated Fourier modulus τ, the proper Y value will be determined. Once the Y value is obtained, only a simple substitution from Equation (2.31) is needed. A deviation from the assumption of an infinite body is apparent in several cases where the ratio of length or width to thickness is not too large. In these cases, the application of Newman’s rule can be of help. Newman demonstrated that for a finite solid body being heated or cooled, the internal variation of temperature as a function of time and position could be stated as: Y = YxYyYz

(2.36)

1 m=∞ m=6

m m

x

x

=2

=2 n = 1.0 0.8 0.6 0.4 0.2 0.0

m = 1

m

0.1

0

= 1 m= 0.5

m= 0.5

Y m= 0

m= 0

n = 1.0 0.8 0.6 0.4 0.2 0.0

0.01 n = 0.8 0.6 0.4 0.2 0.0

n = 1.0 0.8 0.6 0.4 0.2 0.0

m=0 n=1

0.001 0

0.5

1

1.5

2

2.5

3

τ = (Fo) FIGURE 2.1 Dimensionless temperature chart for an infinite cylinder.


3.5

1

m=∞

x x?

m=6

m= m=

x?

n = 1.0 0.8 0.6 0.4 0.2 0.0

2

2

0.1 m = 1

m =

n = 1.0 0.8 0.6 0.4 0.2 0.0

1

Y m .5 =0

m .5

=0

m= 0

m=

0.01

0

n = 1.0 0.8 0.6 0.4 0.2 0.0

n = 0.8 0.6 0.4 0.2 0.0

m=0 n=1

0.001 0

1

2

3

4

5

6

τ = (Fo) FIGURE 2.2 Dimensionless temperature chart for an infinite slab.

where Y is the dimensionless temperature defined in Equation (2.31), and values Yx, Yy and Yz are the dimensionless temperature values with respect to the x, y, and z axes, respectively, which can be obtained from Figures 2.1, 2.2, and 2.3. Newman’s rule can be applied in the calculation of heat transfer processes on finite parallelepipeds and cylinders, as it can be understood that a finite body results from the conjunction of three infinite bodies.

2.2.3 HEAT TRANSFER

BY

CONVECTION

Heat transfer within a fluid will occur by convection. In this heat transfer mode, energy flows as a result of bulk movement of the fluid due to a temperature gradient. The molecules in the fluid will move due to density changes and will interact with each other at different points of the fluid, exchanging energy. Since the efficiency of this process depends on the extent of mixing within the fluid, flow characteristics are as important as thermal properties in determining heat flow rate. Most processing applications of heat transfer by convection involve steady state heat transfer from a solid surface to a fluid in contact with it or vice versa. Heat transfer is defined as the rate of heat flowing through the interface between the fluid and the solid. The amount of energy flowing through this interface is proportional to the temperature gradient and to the interchange surface. This relationship is defined by Equation (2.37),


1

m=∞

x m=

m m

0.1

=

x

6

=

0

2

2

m= 1

m=

m=

1

0.5

m=

Y

n = 1.0 0.8 0.6 0.4 0.2 0.0

0.5

n = 1.0 0.8 0.6 0.4 0.2 0.0

m=0

m=0

n = 1.0 0.8 0.6 0.4 0.2 0.0

0.01 n = 0.8 0.6 0.4 0.2 0.0

m=0 n=1

0.001 0

0.5

1

1.5

2

2.5

3

3.5

τ = (Fo) FIGURE 2.3 Dimensionless temperature chart for a sphere.

known as Newton’s law of cooling: Q = hA ∆T

(2.37)

where Q corresponds to heat flux, A to the interchange area, ∆T to the difference between the fluid and solid temperatures, and h to the proportionality coefficient, known as the heat transfer coefficient. Bulk molecular motion can be induced by differences in the fluid’s temperature at different points within it, developing a buoyant force. This process is known as natural convection. The coefficient of thermal expansion of a fluid (β) is the main property governing natural convection. Without the existence of gravity and thermal expansion, natural convection would not be possible. Mixing efficiency in natural convection depends on the temperature gradient. A small temperature difference between the fluid and solid in contact will promote weak currents with a low heat transfer coefficient. On the other hand, when a fluid is forced to flow past a surface by mechanical means such as a pump or a fan, higher velocities are obtained and strong currents are induced regardless of natural convection. High heat transfer coefficients can be reached due to more efficient mixing. This regime is known as forced convection. In practice, forced convection is the most used convective heat transfer mode, as greater heat transfer can be accomplished and better control of the system can be easily established.


As stated by Newton’s cooling law, heat flux is proportional to the contact area and temperature gradient, both easily measurable. However, several problems arise when determining the proportionality constant h. This constant depends on several factors, including the fluid’s properties such as viscosity, density, specific heat, and thermal conductivity; the nature of the flow (i.e., its velocity, natural or forced convection, streamline or turbulent flow); and other physical characteristics of the system such as size and shape. In practice, exact derivation of heat transfer coefficients from the basic fluid flow and heat conduction equations is not possible, and the use of empirical correlations is needed. These empirical correlations, which have been developed from a theoretical basis and experimental observations, are expressed in dimensionless numbers, allowing us to apply these generic expressions to different fluids and systems. A basic general dimensionless expression that connects the heat transfer coefficient to particular characteristics of fluids and systems has been proposed. This expression states that the Nusselt number Nu is proportional to the product of the Reynolds (Re), Prandtl (Pr), and Grashof (Gr) numbers, as shown in Equation (2.38): Nu = φ [(Re)(Pr)(Gr)]

(2.38)

The Nusselt number is a dimensionless form for the heat transfer coefficient. The other factors involved in this ratio are the thermal conductivity of the fluid k and the characteristic dimension of the system d, which is the diameter for a round pipe or the length for a flat surface. Nu =

hd k

(2.39)

The Reynolds number introduces information about the fluid characteristics and flow regime. It can be considered as the ratio of macroscopic flow to internal friction. When this ratio exceeds a certain value, the inertial force predominates, converting flow from a laminar form into a turbulent one. This number involves the characteristic dimension d, and the fluid’s properties density ρ, viscosity η, and velocity v. For physical properties such as viscosity or density that vary with temperature, the value at the arithmetic mean temperature between the inlet and outlet points will be used. Re =

ρvd η

(2.40)

The Prandtl ratio deals only with physical properties of the fluid and behaves as a physical constant. This expression involves specific heat Cp, viscosity η, and thermal conductivity k. Pr =

Cpη k

(2.41)

Finally, the Grashof number represents the ratio of buoyancy to internal friction. This quantity involves the characteristic dimension of the system d, the acceleration


due to gravity g, the thermal expansion coefficient β, the density of the fluid ρ, the viscosity η, and the temperature difference between a surface and the fluid ∆T: Gr =

β g ∆T d 3ρ2 η2

(2.42)

As can be seen, all the process variables included in these dimensionless ratios, with the exception of h, can be measured; therefore, a heat transfer coefficient value for a given set of conditions can be calculated using Equation (2.38). However, a difficulty remains, as the exact proportionality factor has not been stated. This exact proportionality factor varies from situation to situation, and empirical values have been found for several specific situations, as a general expression is not possible. Equation (2.38) can be further modified into two generic expressions. From simple examination of the meaning of each of the dimensionless ratios, it is apparent that when dealing with natural convection, the buoyancy effect governs the process, and the importance of the Reynolds number subsides under the importance of the Grashof number, leading to: Nu = φ [(Pr)(Gr)]

(2.43)

On the other hand, Equation (2.38) can be practically modified for forced convection as the Reynolds number turns out to be more important than the Grashof number under this flow regime; hence: Nu = φ [(Pr)(Re)]

(2.44)

Now we will review some empirical expressions developed for particular, though frequent, situations. More information about expressions regarding heat transfer by convection in different situations is found extensively throughout the specialized literature. 2.2.3.1 Natural Convection The most common expression used for natural convection in Newtonian fluids is: a

Nu = K [(Pr)(Gr)]

(2.45)

where values for constants K and a in some specific situations are given in Table 2.3. 2.2.3.2 Forced Convection The relationship most commonly used for turbulent flow inside tubes is known as the Dittus–Boelter equation: 0.8

n

Nu = 0.023 (Re) (Pr)

(2.46)

where n = 0.4 for a fluid being heated or n = 0.3 for a fluid being cooled. This 4 expression is valid for Re > 10 and 0.7 < Pr < 160.


TABLE 2.3 Values of K and a for Equation (2.45) Configuration

(Pr)(Gr)

K

a

10 –5 10 5 7 10 < (Pr)(Gr) < 2 × 10 (FU) 7 10 2 × 10 < (Pr)(Gr) < 3 × 10 (FU) 5 10 3 × 10 < (Pr)(Gr) < 3 × 10 (FD)

1.36 0.59 0.13 0.49 0.71 1.09 1.09 0.53 0.13 0.54 0.14 0.27

1/5 1/4 1/3 0 1/25 1/10 1/5 1/4 1/3 1/4 1/3 1/4

4

Vertical surfaces L = vertical dimension 0.9 m Horizontal cylinder L = diameter 0.2 m

Horizontal flat surfaces

Note: FU = Facing upward; FD = Facing downward. Data from Heldman, D.R., and Singh, R.P., Food Process Engineering, 2nd ed., AVI, Westport, CT, 1981.

For liquid flowing inside tubes in a laminar flow (Re < 2100), a suitable equation is 1

d  3 η   Nu = 1.86 (Pr)(Re)     L    ηW  

0.14

(2.47)

where d/L is the diameter/length ratio and η/ηW is the ratio of the bulk viscosity to the viscosity at the wall. The equation used for a liquid flowing in turbulent flow outside a solid body is 0.52

Nu = [0.35 + 0.47(Re)

0.3

] (Pr)

(2.48)

and the equation for a liquid flowing in laminar flow (Re < 200) outside a solid body is 0.43

Nu = 0.86(Re)

2.2.4 HEAT TRANSFER

BY

0.3

(Pr)

(2.49)

RADIATION

Heat radiation is another way to dissipate energy. Any body with a temperature above absolute zero emits electromagnetic radiation. The wavelength of this radiation varies as a function of the body temperature. The higher the temperature, the more


energetic the radiation; therefore, the smaller the wavelength. Heat radiation does not need a material in which to propagate and can travel through a vacuum. Additionally, heat transfer by radiation does not require a temperature gradient to proceed and therefore occurs constantly throughout nature. However, for practical purposes, use of heat transfer by radiation in food engineering is limited to the use of infrared radiation, dielectric heating, and microwave heating. Use of infrared radiation implies the use of electromagnetic radiation with wavelengths between visible light and radio waves (0.76–1000 µm). Often, the infrared source has a high temperature (500–3000°C), and heat transfer by convection is also taking place and cannot be ignored unless the process occurs under vacuum, as in the case of freeze drying. Examples where heat transfer by infrared radiation is present are grilling and the use of electrical resistances. As penetration of this kind of radiation is poor, the heating effect of infrared radiation has an impact only on the surface of the body, and heat transfer through the body proceeds by conduction or convection. Although infrared radiation is almost always present in heat transfer processes, generally its effect is not as important as the effect of convection, and its contribution has already been taken into account by the empirical equations used to model heat transfer by convection. Dielectric heating such as microwave or radio frequency heating is generally used when large objects need to be quickly and uniformly heated. The principle underlying the operation of these methods lies in the ability of some wavelengths to induce vibration in some dipolar molecules such as water, producing intermolecular friction and thereby increasing the temperature. Using these methods, food products are heated volumetrically, that is, at all points at the same time. The frequencies used for radio frequency heating are in the range of 10 to 100 MHz, while the frequencies used for microwave heating are a little higher, normally 915 to 2450 MHz. The dielectric heating methods described above involve the “instantaneous” generation of heat within the product. Conduction and convection take command of the heat transfer process after heat has been generated. The study of heating processes such as microwave heating, radio frequency heating, and ohmic heating is beyond the scope of this chapter, and no further details will be discussed here.

2.3 CONCLUSIONS The modeling of heat transfer in foods is a difficult task. Often, two or even three heat transfer modes proceed in a system at the same time. Some thermal properties of foods can vary with changes in temperature, and further problems arise because food is seldom regular in shape or homogeneous. However, some assumptions and empirical relations allow the development of practical models that describe certain processes with sufficient precision. Nowadays, some efforts are being made to apply finite element method to take into account the changing properties of foods and to better model complex food heating or cooling processes. Most empirical heat transfer models applicable to foods are highly specific, and general analytical models are seldom encountered.


REFERENCES American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., ASHRAE Handbook of Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Inc., Atlanta, 1985. Arce, J.A. and Sweat, V.E., Survey of lashed heat transfer coefficients encountered in food refrigeration processes, ASHRAE Trans., 86(2), 235–260, 1980. Charm, S.E., Fundamentals of Food Engineering, AVI, Westport, CT, 1971. Choi, Y. and Okos, M.R., The thermal properties of tomato juice concentrates, Trans. ASAE, 26, 305–311, 1983. Choi, Y. and Okos, M.R., Effects of temperature and composition on the thermal properties of foods, in Food Engineering and Process Applications, vol. 1, Maguer L. and Jelen, P., Eds., Elsevier, New York, 1986a. Choi, Y. and Okos, M.R., Thermal properties of liquid foods: review, in Physical and Chemical Properties of Foods, Okos, M.R., Ed., American Society of Agricultural Engineers, St. Joseph, MI, 1986b. Dickerson, R.W., Thermal properties of foods, in The Freezing Preservation of Foods, vol. 2, 4th ed., Tressler, D.K., Van Arsdel, W.B., and Copley, M.J., Eds., AVI, Westport, CT, 1969. Dominguez, M., De Elvira, C., and Fuster, C., Influence of air velocity and temperature on the two-stage cooling of perishable large-sized products, Annexe Bressanone, Bull. IT Regrig., Annexe 83–90, 1974. Hallstrom, B., Skjoldebrand, C., and Tragardh, C., Heat Transfer and Food Products, Elsevier, New York, 1988. Heldman, D.R. and Singh, R.P., Food Process Engineering, 2nd ed. AVI, Westport, CT, 1981. Lamb, J., Influence of water on the thermal properties of foods, Chem. Ind., 24, 1046–1048, 1976. Martens, T., Mathematical model of heat processing in flat containers, in Operaciones Unitarias en la Ingenieria de Alimentos, Ibarz, A. and Barbosa-Canovas, G., Eds., Technomic Publishing Company, Lancaster, PA, p. 317. Mohsenin, N.N., Thermal Properties of Foods and Agricultural Materials, Gordon and Breach Science Publishers, Inc., New York, 1980. Riedel, V.L., Wärme leitfähigkeitsmessungen an Zuckerlösungen, Frucht säften und Milch, Chem. Ing. Technik., 21, 340, 1949. Schneider, P.J., Conduction, in Handbook of Heat Transfer, Rohsenow, W.M. and Hartnett, J.P., Eds., McGraw-Hill, New York, 1973. Siebel, J.E., Specific heat of various products, Ice Refrigeration, 2, 256–257, 1982. Sweat, V.E., Experimental values of thermal conductivity of selected fruits and vegetables, J. Food Science, 39, 1080, 1974.


3

Introductory Aspects of Momentum Transfer Phenomena J.F. Vélez-Ruiz

CONTENTS 3.1 3.2

Introduction Momentum Transfer Variables 3.2.1 Flux of Momentum 3.2.2 Mass Conservation 3.2.3 Energy Balance 3.3 Laminar and Turbulent Flow 3.3.1 Laminar Flow 3.3.2 Turbulent Flow 3.3.3 Transportation of Fluids 3.4 Rheological Properties of Foods 3.4.1 Rheology 3.4.2 Rheological Models 3.4.3 Temperature and Concentration Variables 3.5 Simultaneous Transfer Phenomena 3.6 Final Remarks Nomenclature References

3.1 INTRODUCTION Momentum transfer is a physical phenomenon present in almost any process of the food industry; often the heat and/or mass transfer phenomena occur in association with flow, and it is difficult to imagine a unit operation without moving streams. This phenomenon necessarily involves a convection mechanism between molecules or groups of molecules, or the momentum gradient would be null. In other words, momentum transfer is characterized by the tendency toward equilibrium. When momentum transfer is analyzed in the food fluid field, some particular considerations should be taken into account. Foodstuffs are biological materials consumed by humans and animals for nutritional purposes; in order to process the


final food products, the raw food material must be transformed to achieve the proper and desirable properties, providing a diversity of satisfactory factors. A food may be considered a multi-component mixture in which carbohydrates, lipids, proteins, and water are the macro constituents, and minerals and vitamins, among others, are the micro components. The type of component, its concentration, and its specific characteristics play important roles in the structure and engineering properties of each food product, particularly in fluid motion. Food materials are influenced by many aspects such as variety, time of harvesting, ripeness stage, handling during transport, and of course, by process treatments. The nonNewtonian behavior of many food fluids makes transfer momentum a complicated phenomenon. To explain transfer phenomena, a rigorous mathematical approach leads to differential equations that often are too difficult to be solved analytically, especially when a transient instead of a steady state condition is involved; furthermore, the presence of many molecules or groups of molecules in natural or forced convection complicates analytical solution. For this reason, empirical and numerical methods have been developed to reach faster, more practical, and useful solutions. In food processing, the utilized raw materials are exposed to process conditions that transform, to a minor or major degree, the food’s native state. Differences in concentration, mechanical force, pressure, temperature, and/or velocity, which act as driving forces of the three transport phenomena, importantly affect the food product’s characteristics. Thus, mathematical and/or engineering modeling favors the predictability of those transport phenomena, particularly of momentum transfer, in order to control food quality and to produce a well specified end product. The fundamentals and applications of momentum transfer are themes widely covered by many authors. Therefore, the main purpose of this chapter is to present the basic principles and main process variables involved in momentum transfer phenomena. Two major fluid motions are introduced: laminar and turbulent flows. Rheological properties and simultaneous transfer phenomena are commented on briefly, and finally, some general comments are given.

3.2 MOMENTUM TRANSFER VARIABLES The momentum of a fluid may be defined as the product of its mass and velocity at a given point; hence, changes in velocity per unit mass can result in momentum transport. Therefore, momentum transfer or fluid motion depends upon the interrelation of three fundamental variables: mass, velocity, and time (Bennet and Myers, 1983; Fahien, 1983).

3.2.1 FLUX

OF

MOMENTUM

Considering that the velocity is a vector and the velocity gradient is a second-order tensor, the corresponding moment flux or shear stress is also a second-order tensor,


which for an incompressible fluid is given by the following equation (Brodkey and Hershey, 1988): τ = − µ[∇v + (∇v)T ]

(3.1)

Equation (3.1) shows that the stress tensor (τ) is a function of the shear rate T tensor (∇v) and its transpose (∇v) . Velocity, a vector quantity, has three components, and any of these can vary in three directions. Since Equation (3.1) must be homogeneous, the stress tensor (ττ) must be a second-order tensor:  τ xx τ xy τ xz    τ =  τ yx τ yy τ yz       τ zx τ zy τ zz 

(3.2)

This stress tensor has a complicated physical interpretation due to the fact that it is utilized both as a momentum flux and as a shear stress. The application/interpretation of this stress tensor, or force imposed on a unit of surface area, for a given material is conducted as a function of the specific conditions controlling a physical situation; for instance, in a steady simple shear flow the number of components of the stress tensor is reduced to five (Rao, 1999; Steffe, 1992). Because the complete analysis of momentum transfer is too complicated, the solution for each particular physical situation has been studied and solved by many researchers, using both simple and complex analytical methods (Gekas, 1992; Toledo, 1991). And even though the solution for many practical situations has been developed, many physical phenomenon problems involving food fluids remain to be solved. Furthermore, whatever the physical problem, the elementary variables are dimensions of the system, involved forces, particular properties of the process material, shear and normal stresses, time, and velocities, among the most important. Most of these variables are combined in those driving force balances known as dimensionless numbers, in which the variable’s participation depends on the physical phenomenon. Table 3.1 includes the common dimensionless numbers related to momentum transfer of Newtonian fluids; the list could be extended if non-Newtonian behavior were considered (Bennet and Myers, 1983; Fahien, 1983; Welty et al., 1976; Vélez-Ruiz, 2001).

3.2.2 MASS CONSERVATION Sometimes it is not necessary to formulate a complete force balance in the solution of a new flow problem; instead, it is more practical to start with the equations of mass conservation, momentum balance, and mechanical energy conservation, also known as equations of change (Bird et al., 1960).



TABLE 3.1 Momentum Transfer Dimensionless Numbers Variables Density, drag force, surface area, velocity Density, pressure, velocity Density, stress, velocity Characteristic dimension, gravity, velocity Agitator speed, diameter, density, power Agitator speed, diameter, volumetric flow Characteristic dimension, density, velocity, viscosity Characteristic dimension, density, velocity, stress Same variables, including rheological properties

Relationship 2

FD/ρv A 2 P/ρv 2 τ/ρv 2 v /dg 3 5 P/ρN D 3 Q/ND vdρ/µ 2

v dρ/σ

Involved Forces

Name

Symbol

Drag force/inertial force Pressure force/inertial force Wall shear force /inertial force Inertial force /gravity force Power/inertial force Flow force /inertial force Inertial force /viscous force

Drag coefficient Euler Friction factor Froude Power Pumping Reynolds

CD Eu f Fr Po P Re

Kinetic energy force/surface force

Weber

We

Generalized Hedstrom (GHe) and Reynolds (GRe), Hedstrom (He), and Plasticity (Pl) numbers

The developed mass conservation over a stationary volume element generates the equation of continuity:  ∂  ∂ρ ∂ ∂ = − ρv x + ρv y + ρvz  = −(∇ ⋅ ρv) ∂t ∂ x ∂ y ∂ z  

(3.3)

This equation describes the density rate of change resulting from the changes in the mass velocity vector (ρv) at a given point, the right hand term being the point product of ρv. The equation of motion for a volume element at a fixed point, even though it is complicated for the x, y, and z directions, takes a single vector expression form (Bird et al., 1960; Liu and Masliyah, 1998): ∂ ρv = −[∇ ⋅ ρvv] − ∇P − [∇τ] + ρg ∂t

(3.4)

in which the different mathematical terms represent the contributions to the momentum balance: ∂ ρv = rate of momentum increase ∂t [∇ ⋅ ρvv] = rate of momentum gained by convection ∇P = pressure force on the element ∇τ = rate of momentum gained by viscous transfer ρg = gravitation force For an incompressible Newtonian fluid, the x-component of the Navier–Stokes equations for the three components of the velocity vector for x, y, z coordinates appears as (Molerus, 1997):  ∂2 v  ∂v ∂v ∂v ∂v  ∂2 vx ∂2 vx  ∂P + µ 2x + + 2  ρ x + v x x + v y x + vz x  = − ∂x ∂y ∂z  ∂x ∂y 2 ∂z   ∂t  ∂x

(3.5)

3.2.3 ENERGY BALANCE Similarly, the mechanical energy balance can be applied to a stationary volume element through which the fluid flows, generating the corresponding equation with several terms: ∂  1 2 1 ρv = − ∇ ⋅ ρv 2 v − (∇ ⋅ Pv) − P( −∇ ⋅ v) − (∇ ⋅ [τ ⋅ v]) − ( − τ: ∇v) + ρ(v ⋅ g)     ∂t 2 2 (3.6) In which the meaning of each term is: ∂  1 2 ρv , rate of increase in the kinetic energy  ∂t  2 1  ∇ ⋅ ρv 2 v , net rate of input of kinetic energy due to bulk flow  2 


(∇ ⋅ Pv), rate of work done by surrounding pressure P( −∇ ⋅ v), rate of reversible conversion to internal energy (∇ ⋅ [τ ⋅ v]), rate of work done by viscous forces ( − τ: ∇v), rate of irreversible conversion to internal energy ρ(v ⋅ g), rate of work done by gravity force As may be noted, the three preceding relationships were obtained in rectangular coordinates; thus, if the flow problem involves curvilinear coordinates, the cylindrical or spherical corresponding equations should be applied. Additionally, the specific characteristics of the physical system and its properties will have to be considered.

3.3 LAMINAR AND TURBULENT FLOW Among the many types of flow developed by fluids, laminar and turbulent flows are very important and commonly found, no matter what the specific physical situation. Several geometries, such as cylinder, coil, cube, helix, oven, parallel plates, shell and tubes, tube, and vessels containing food fluids are frequently utilized in food process operations. It can be observed from Table 3.1 that the ratio of the inertial variables to the viscous force (Re) is determinant in the presence of streamline or turbulent flow; therefore, the rheological nature of the food plays an outstanding role in flow definition. From a practical viewpoint, the type of flow may be established with the evaluation of the Reynolds number for Newtonians, whereas for non-Newtonian behavior the Generalized Reynolds and the critical Generalized Reynolds numbers should be computed.

3.3.1 LAMINAR FLOW Of the several geometries involved in food processing, tube flow is the most important due to easy pipe manufacturing and a very high rate of handled volume to amount of construction material. In it, the shear stress and the velocity profile are functions of the rheological behavior. The corresponding equations are introduced next and are extremely useful for most food fluid systems: For shear stress: τ rz =

r( ∆P ) 2L

(3.7)

For the velocity distribution, the type of fluid implies different relationships: 1. In the case of a Newtonian fluid: 1  ∆P  2 vx =    (R − r 2 )  2   2µL 


(3.8)

2. In cases of non-Newtonian behavior: a. Power law model: n   ∆P  vx =   n + 1  2 KL 

( 1n)

 R( n +1 n ) − r ( n +1 n )   

(3.9)

b. Herschel–Bulkley model (when R0 ≤ r ≤ R):

vx =

2L  1  1n ∆P + 1 K H  nH  H

 1    n +1  1   H    n +1  H   r∆P − τ   ( τ − τ ) − 0 0  2L   w    

(3.10)

c. Bingham plastic model (when R0 ≤ r ≤ R): vx =

1 µP

 ∆P R 2 − r 2 − τ R − r  ) 0(  4 L 

(

)

(3.11)

Similar equations can be developed or have been obtained for other geometries. For the aforementioned equations, four assumptions were considered: 1. 2. 3. 4.

Steady flow Constant rheological properties and density No entrance or exit effects Prevalence of laminar flow

Even though most industrial process flows are turbulent, food materials may be the exception during transport due to their non-Newtonian nature.

3.3.2 TURBULENT FLOW There are many considerations and equations for flows under turbulent conditions; in this type of flow, the average and the instantaneous/point velocity are normally different as a consequence of the velocity fluctuation for a given volume element of fluid. Several approaches, such as the Boussinesq theory, Prandtl mixing length, analogies between the transport phenomena, and film and penetration theories, among others, have been proposed for the modeling of turbulence (Brodkey and Hershey, 1988). The mathematical relationships for velocity distribution, when turbulent flow is present, are known as “universal velocity distribution” equations. They are valid inside channels and smooth tubes in which the flow has been fully developed, with most of the boundary layer arising from Prandtl’s theory:


+

For the wall region or viscous sublayer, y ≤ 5: v+ = y+

(3.12)

+

For the buffer or generation zone, 5 < y < 30: v + = 5.0 ln y + − 3.05

(3.13)

+

For the turbulent zone, y ≥ 30: v + = 2.5 ln y + + 5.5

(3.14)

These velocity profiles, also known as the law of the wall, have been verified experimentally for Newtonian fluids with very good results, but such is not the case for non-Newtonian fluids, in which some differences have been observed. Both groups of velocity equations, for viscous and turbulent flows, may be used in the involved unit operation not only for velocity profiles mass and volumetric flows, but also for computation of residence time, design parameters, and dimensions inside process units. Furthermore, temperature distribution, heat transfer area, and concentration profiles in heat exchangers or in mass transfer related equipment may be obtained with incorporation of momentum transfer considerations.

3.3.3 TRANSPORTATION

OF

FLUIDS

As previously mentioned, there are process conditions involving different geometries, but the cylindrical shape is the most commonly encountered flow situation. Also, it is the most studied in relation to momentum phenomena and related topics, such as pump needs, pressure losses, and friction factors through pipe systems. Accurate prediction of pressure drop/flow relationships for non-Newtonian fluids in ducts is consequently of great importance. Additionally, when fluid flows through fittings (bends, elbows, entrances, expansions, reductions, and valves), the flow patterns become more complex than those in a straight tube and are characterized by altered velocity distributions and the presence of centrifugal forces and secondary flows (Das et al., 1991). In the unit operation of fluid foods’ transportation, there is a lack of information for non-Newtonian materials. Some studies related to the prediction of friction factors, pressure losses, and power consumption evaluations as part of the mechanical energy have been completed. Bernoulli’s equation (Equation 3.15), in which the input of mechanical energy is conserved through the output for uncompressible, isothermal, and steady flow in one direction, is the fundamental mathematical model for the aforementioned evaluations. v12 P Zg v22 P Zg + 1 + 1 + ηp Wp = + 2 + 2 + Ef 2α1g c ρ gc 2α 2 g c ρ gc

(3.15)

Some advances for non-Newtonian behavior, particularly those related to the correction factor for the kinetic energy, friction factor, and related coefficients have


been made recently; they are summarized in Table 3.2 for both Newtonian and nonNewtonian fluids. There are many proposed explicit and implicit equations for Newtonian fluids as a function of Reynolds number and relative roughness that were not included in Table 3.2, and although some contributions have been proposed to characterize non-Newtonian food fluid transportation, it remains a complex flow and an incompletely understood phenomenon.

3.4 RHEOLOGICAL PROPERTIES OF FOODS 3.4.1 RHEOLOGY Rheology, defined as the science of flow and deformation of material, is a fundamental interdisciplinary science that has been gaining increasing importance in the field of foods. According to Rao (1986), Steffe (1992), Kokini (1992) and VélezRuiz (1996), among other authors, there are numerous topics of interest to the food industry related to rheological behavior, such as: 1. Process engineering applications involved in equipment and process design 2. Development of new products or reformulation of existing products 3. Quality control of intermediate and final products 4. Correlation with sensory evaluation 5. Understanding of food structure Rheology can be used to characterize not only flow behavior of foods and other materials, but also structural characteristics. The knowledge of flow food properties, such as consistency, pourability, thickness, viscoelasticity, viscosity, and yield stress, contributes substantially to facilitating transport and commercial processing, as well as to promoting consumer acceptance. Insight into structural arrangement helps to predict the behavior or stability of a given material with storage, change in humidity and/or temperature, and handling (Motyka, 1996). Consequently, basic rheological information on materials is important not only to engineers, but also to food scientists, processors, and others who may utilize this information and find new applications. Many foods are neither solid nor liquid, but exist in an intermediate state of aggregation known as semiliquid or semisolid. As a consequence of the complex nature and lack of precise boundaries between solids/semisolids and liquids/semiliquids, many foods may exhibit more than one rheological behavior depending on their specific characteristics and the measuring conditions used during the physical characterization. Due to the complex composition and particular structure of foods, the approaches for characterizing a food’s rheological behavior may be classified into five methods: empirical, phenomenological (including fundamental and imitative), linear viscoelastic, nonlinear viscoelastic, and micro-rheological. In the case of foods, empirical and phenomenological approaches have been predominant in studies associated with flow properties (Rao, 1992; Vélez-Ruiz and BarbosaCánovas, 1997b). With respect to viscoelastic nature, the approaches mainly include



TABLE 3.2 α) Friction Factors (F) and Related Coefficient (K) for Newtonian and Non-newtonian Fluids Kinetic Energy Factor (α Factor

Regimen

Considerations

Kinetic Energy Correction Factor Newtonian fluids, Re ≤ 2100

References

α = 0.5

Laminar

(2n + 1)(5n + 3) α= 3(3n + 1) 2

Laminar

Power law fluids, and evaluation of GRec

Heldman (1975) Ibarz et al. (1996)

Laminar

Bingham plastic fluids, and evaluation of GRec

Skelland (1967) Ibarz et al. (1996)

α=

α=

1 2−

τ0 τw

Charm (1971)

[(1 + 3n + 2 n 2 + 2 n 2 C + 2 nC + 2 n 2 C 2 )3 (2 + 3n )(3 + 5n )(3 + 4 n )] [[(1 + 2 n) (1 + 3n ) ][18 + n(105 + 66 C) + n (243 + 306C + 85C 2 ) + n 3 (279 + 522C + 350C 2 ) + n 4 (159 + 390C + 477C 2 ) + n 5 (36 + 108C + 216C 2 )]] 2

2

2

Laminar

Herschel-Bulkley fluids, and evaluation of GRec

Osorio and Steffe (1984) Ibarz et al. (1996)

ln(2α ) = 0.168C − 1.062 nC − 0.954 n 0.5 − 0.115 C 0.5 + 0.831

For 0.06 ≤ n ≤ 0.38

Briggs and Steffe (1995)

ln(2α ) = 0.849C + 0.296 nC − 0.600 n

For 0.38 ≤ n ≤ 1.60

Briggs and Steffe (1995)

Turbulent

Newtonian fluids, c:6 − 10 depending of Re

Ibarz et al. (1996)

Laminar and turbulent

Values obtained as a function of Re

Foust et al. (1980)

Laminar

Newtonian fluids, fD = 4 fF

Foust el al. (1980)

Laminar flow

Power law Fluids, GRe instead of Re

Ibarz et al. (1996)

α=

4c (3 + c)(3 + 2c) ≈ 1.0 (1 + c) 3 (1 + 2c) 3

0.5

− 0.602C

0.5

+ 0.733

4

0.5 < α < 1.0

Friction factor 16 64 fF = ;f = Re D Re fF =

16 G Re


f 1 (He) 4 He = F − + Re B 16 6(Re B )2 3fF3 (Re B )8

Laminar flow

Implicit equation for Bingham plastic fluids

Ibarz et al. (1996)

Laminar flow

Herschel-Bulkley fluids, GRe and GHe need to be considered

Garcia and Steffe (1986)

 1.255 1 1 ε = −4 log  +  1/ 2 (fF )1/ 2 Re( f ) 3 .7 D  F 

Turbulent flow

Newtonians, Re: 4000 − 10

ε 1  5.62 = −2 log  0.9 + 0.27  ( 4 fF )1/ 2 D  Re

Turbulent flow

Newtonians, rough pipes

1 4 0.4 log G Re ff(1− n / 2 ) − 1.2  = (fF )1/ 2 n 0.75 n  

Turbulent flow

fF =

16 ψGR ε

 (1 − C)2 2C(1 − C) C 2  ψ = (3n + 1) n (1 − C)1+ n  + + 2n + 1 n + 1   3n + 1

n

1 = 2.27 log(1 − C) + 2.27 log[Re B f F0.5 ] − 1.15 ( 4 f F ) 1/ 2 fF from figures

k from graphical representations −0.896

k = 191 G Re −0.504 k = 29.4 G Re −0.492 k = 30.3 G Re

Laminar and Turbulent flow

8

Power law fluids, GRe: 2900-36000

Levenspiel (1993)

Levenspiel (1993) Dodge and Metzner (1959)

Implicit equation for Bingham fluids

Ibarz et al. (1996)

Values obtained as a function of GRe and GHe for n = 0.2 and 0.5

Garcia and Steffe (1986)

Friction Loss Coefficients Different fittings Values obtained as a function D and equivalent length. Newtonians 90° short radius elbow Laminar flow of power law fluids Tee, line to branch Laminar flow of power law fluids Fully open three-way plug valve Laminar flow of power law fluids

Foust et al. (1980) Steffe et al. (1984) Steffe et al. (1984) Steffe et al. (1984)

fundamental and linear. Most food systems are considered to be linear viscoelastic materials at small strains. Any of the available commercial instruments that allow objective characterizations are appropriate for measuring the rheological properties and identifying the rheological behavior of food materials. The accuracy of these rheometers has been improved due to the incorporation of modern mechanical devices and microcomputer technology.

3.4.2 RHEOLOGICAL MODELS The relationship between applied stress and resulting strain shown by any food material can be expressed either empirically or in terms of a rheological equation of state, also known as a rheological model. There exists a particular approach in which the rheological behavior of a material is analyzed on a simplified deformation called single shear or uniaxial deformation. This approach is the basis for many rheological measurement techniques and permits the analysis of many food materials. Fluid, solid, and viscoelastic behaviors have been observed in food products. Juices, milk, vegetable oils, and water behave as Newtonian fluids; but dressings, fruit concentrates, ketchup, and yogurt behave as non-Newtonian. Gels, meat emulsions, and some types of cheese have been characterized as solids with elastic behavior. Other food products such as ice cream, mayonnaise, and whipped cream have been characterized by their viscoelastic nature. Among the many mathematical relationships proposed to characterize the rheological nature of liquid materials, four models are particularly useful for describing the behavior of fluid foods. One is Newton’s model for Newtonians, expressing the concept of viscosity, and the other three (Power Law, Bingham Plastic and Herschel and Bulkley models) are for non-Newtonian behavior; the corresponding equations are: Newtonian model Power law model Bingham model Herschel–Bulkley model

τ = µγ˙

(3.16)

τ = K( γ˙ )n

(3.17)

τ = τ 0 + ηP γ˙

(3.18)

τ = τ 0 + K( γ˙ )n

(3.19)

Many other more complex or more specific models have been proposed for different fluid materials. Consequently, these models are utilized for particular products. Some of them are included in Table 3.3. In addition, if time is considered as another variable, the fluids may show a thixotropic or a rheopectic behavior. Thixotropy refers to a reversible decrease in apparent viscosity with time at constant shear rate. This is generally due to a reversible change in structure of the material with time under shear, with a limiting viscosity ultimately being approached. Rheopexy is, in essence, the reverse of thixotropy; that is, it represents an increase of apparent viscosity with time at a constant


TABLE 3.3 Additional Rheological Models for Time-Independent Fluids Model

Application

Casson

τ 0.5 = τ 00.5 + K( γ˙ ) 0.5

Vocadlo

1   τ =  ( τ 0 ) n + Kγ˙   

Modified Casson

τ 0.5 = τ 00.5 + K( γ˙ ) n

Generalized H-B Power Series

⋅

Casson, 1959

Fruit and puree concentrates

Food fluids

Parzonka and Vocadlo, 1968 Mizrahi and Berk, 1972 Ofoli et al., 1987

Food fluids

Whorlow, 1992

Concentrated orange juice

γ = K1τ + K 2 τ + K 3 τ ... ⋅

Chocolate, fruit dispersions and products

n

τ n1 = τ 0n1 + K( γ˙ ) n 2 3

Reference

5

τ = K1 γ + K 2 ( γ˙ ) + K 3 ( γ˙ ) ... 3

5

shear rate. To model thixotropic and rheopectic behavior, several models have been developed. The model expressed by Equation (20) is a modification proposed by [Mason et al. (1987)] to another previously established model (Barbosa-Cánovas et al., 1993).   t  Σb i exp −     λi   τ = K( γ˙ )n 1 + ( b 0 t( γ˙ ) − 1)   Σb i    

(3.20)

Sometimes the rheological behavior of foods depends on the experimental conditions; for example, evaporated milk has been characterized as Newtonian by Fernández-Martín (1972), Bloore and Boag (1981), and Wayne and Shoemaker (1988), and as a non-Newtonian fluid by Randhahn (1973) and Schmidt et al. (1980), and during its flow has shown thixotropic behavior (Ibarz et al., 1987). Also, sweetened condensed milk has been characterized as non-Newtonian by Alvarez De Felipe et al. (1991), as a thixotropic fluid by Higgs and Norrington (1971), and as a viscoelastic material by Patil and Patel (1992). The same material may even develop different flow behaviors depending on the applied range of shear rate or shear stress. A viscoelastic material may show either linear or nonlinear behavior. A linear viscoelastic material has properties that are dependent upon time alone and not upon the magnitude of the stress that is applied to the material. Nonlinear viscoelastic materials exhibit mechanical properties that are a function of time and the magnitude of stress used. To express the viscoelasticity of a linear food material, mechanical models incorporating two basic elements have been utilized. These elements are the elastic spring, which represents the instantaneous elastic nature, and the dashpot, which expresses the pure viscous flow. These two elements when in a series constitute a Maxwell body, and when parallel, a Kelvin–Voigt body (Vélez-Ruiz, 1996).


3.4.3 TEMPERATURE

AND

CONCENTRATION VARIABLES

The influence of temperature on the viscosity of Newtonian and the consistency coefficient of non-Newtonian fluid foods is commonly expressed in terms of an Arrhenius type equation (Equation 3.21). Furthermore, the effects of both variables, concentration and temperature at a given shear rate range, can be combined in terms of single relationships (Vélez-Ruiz and Barbosa-Cánovas, 1997a). Fernández-Martín (1972) and Bloore and Boag (1981) have proposed a polynomial relationship (Equation 3.22) to fit the viscosity of concentrated milk (considered a Newtonian fluid) as a function of total solids and temperature, which also has been fitted to other food fluids.

η, µ a , K = A 0 exp 2

 Ea   RT 

(3.21)

2

2

2

log η = A0 + A1t + A2t + (B0 + B1t + B2t )S + (C0 + C1t + C2t )S

(3.22)

Vitali and Rao (1984) used an exponential relationship to model the apparent viscosity of concentrated orange juice influenced by temperature and concentration: B

η = φ (T, C) = KT,C exp (Ea/RT)⋅C

(3.23)

Reddy and Datta (1994) employed a combined exponential potential functionality to relate the consistency coefficient, absolute temperature, and concentration: X

K = B1 exp (C2/T) 1.25

(3.24)

Generally, the effect of concentration and temperatures on rheological behavior has been expressed by power and/or exponential equations for different types of food fluids (Vélez-Ruiz and Barbosa-Cánovas, 1997a).

3.5 SIMULTANEOUS TRANSFER PHENOMENA Industrial equipment often involves simultaneous transfer phenomena, hence flow effects must be considered when designing and evaluating heating/cooling processes and mass transfer operations. The velocity profile of any phase will depend on the rheological properties of the food fluid, used geometries, and other specific factors, such as the solids loading, particle diameter, and density differences in the case of solids–food liquids, and density differences and interface in the case of oil–water and gas non-Newtonian liquid mixtures. Other important factors should be considered, including temperature and thermal properties for heat transfer operations and concentration and density for mass transfer processes.


Since non-Newtonian fluids are often very viscous, most thermal treatment methods assume the flow of foods in tubular ducts as a laminar regime. Thus, the process calculations need to be very accurate in order to avoid under- or overprocessing; in most cases the Arrhenius relationship has been used to describe the effect of temperature on the rheological behavior of non-Newtonian food fluids. To take account of the rheological behavior, Pereira et al. (1989), for instance, considered the rheological parameters as part of the Generalized Reynolds (Equation 3.25), Generalized Prandtl (Equation 3.26), and Generalized Grashoff (Equation 3.27) numbers and the “viscosity correction term” (Kb/Kw), in a double tube heat exchanger modeling for laminar flow:

G Re =

8v 2 − n D n ρ n 3n + 1 2 n K  n 

  3n + 1 n n −1  2  K Cp   n    G Pr = 1− n n −1 k t  4v D    G Gr =

gβρ2 D3∆T   3n + 1 n    K n    1− n n −1   4v R  

2

(3.25)

(3.26)

(3.27)

Flow properties are also closely related to those unit operations involving heat transfer phenomena. Cooling, heating, pasteurization, sterilization, and evaporation are the most common heat transfer operations utilized in food processing, and whatever the main purpose, heat transfer implies flow of food fluids inside the exchanger, mostly with non-Newtonian behavior. Since the flow profile controls mixing and thermal–times effects in the heat equipment, it is essential to know how flow regime influences the heat transfer phenomenon. On the other hand, there are mass transfer operations in which food fluids are handled and flow behavior will affect the operation performance importantly. Therefore, engineers who work with air and spray drying, fermentation processes, membrane separations, and vacuum drying should consider the flow properties of the fluid materials. Other unit operations, such as extraction, frying, leaching, and osmotic dehydration, have less dependency on rheological properties. Table 3.4 presents a summary of studies in which the heat and mass transfer operations have been related to momentum transfer, such as rheological properties or velocity profiles.


TABLE 3.4 Examples of Heat and Mass Transfer Operations Related to Rheological Properties of Food Fluids Application/Fluids

Purpose

Important Remarks

Continuous sterilization/Bingham plastic fluids

Modeling continuous a sterilization for a NN fluid

Gauriguata et al., 1979

Thin film evaporation/water and sugar solutions Double tube heat exchanger/guar gum b and CMC solutions c UHT process/starch solutions and water

Simulation of fruit juices evaporation

Understanding the effect of temperature on the rheology is necessary for good sterilization process Evaporation regimes were related to flow Correlations were developed for a nonNewtonian fluid d h for particles heated in e N fluids were 20% higher than in NN fluids

Pereira et al., 1989

Aseptic treatment/particles in CMC solutions Scraped surface heat/NN fluids Spray drying/power law liquid Drum drying/power law fluids

Spray drying/soy protein milk a b c d e

Predicting heat exchange during laminar flow Evaluations of the particle/fluid interface convective heat transfer coefficient Determination of fluid convective heat transfer coefficient Effect of heat transfer on flow profiles and numerical simulation Prediction of drop diameter; theoretical deprivation Film thickness of drying material for a drum dryer; theoretical approach Effect of viscosity on drying rates of soy protein milk

d

h was significantly affected by fluid viscosity Figures of velocity profiles for three NN fluids An equation for drop diameter of NN fluids was developed Thickness of the film was equated as a function of velocity ratio of cylinders and flow behavior index The viscosity must be controlled for good atomization and milk quality

Reference

Stankiewics and Rao, 1988

Awuah et al., 1993

Bhamidipati and Singh, 1995 Wang et al., 1999 Weberschinke and Filková, 1982 Daud, 1989

Hayashi, 1996

NN = non-Newtonian CMC = carboxy methylcellulose UHT = ultra high temperature h = convective heat transfer coefficient N = Newtonian

3.6 FINAL REMARKS This chapter is an overall review of an extensive topic in which many aspects of the complex phenomena remain unknown. Therefore, the main momentum transfer parameters and variables have been only briefly commented upon. In process design, momentum transfer is most likely present; thus, some simple or complex approach


must be taken into consideration, not only with respect to the flow nature, but also with respect to heat and mass transfer, as well as biochemical reactions. Rheological behavior has some impact on all unit operations involving food liquids. Nowadays, several computer programs have been developed to solve various transport equations, and they are commercially available. Furthermore, continuous advances in computer capacity in terms of computational speed, disposability, and memory will enable application of rigorous analytical models to specific momentum transfer problems. Also, the utilization of modern instrumental techniques, such as nuclear magnetic resonance and magnetic resonance imaging, will contribute to the experimental approach for measuring velocity profiles and rheological behavior of non-Newtonian fluids, including fluid/particle mixtures, and therefore to a better understanding of the momentum phenomenon.

NOMENCLATURE A A0, Ai b0, bi B, B0, Bi c C C0 , Ci CD Cp d D Ea Ef Eu f fD fF FD Fr g g GGr GHe GPr GRe GRec He k kt K

2

Surface area or transfer area (L ) Constants model Constants model Constants model Constant value (with a magnitude between 6 and 10) Constant ratio (τ0/τw) (dimensionless); solids content (ºBx) Constants model Drag coefficient (dimensionless) Fluid specific heat (E/MT) Characteristic dimension (L) Tube diameter (L) Activation energy (E/Mol) Friction losses energy (FL/M) Euler number (dimensionless) Friction factor (dimensionless) Darcy’s friction factor (dimensionless) Fanning’s friction factor (dimensionless) Drag force (F) Froude number (dimensionless) 2 Gravitational constant (L/t ) 2 Vector representing the acceleration due to a gravitational field (L/t ) Generalized Grashoff number (dimensionless) Generalized Hedstrom number (dimensionless) Generalized Prandl number (dimensionless) Generalized Reynolds number (dimensionless) Critical generalized Reynolds number (dimensionless) Hedstrom number (dimensionless) Friction loss coefficient Thermal conductivity (E/LtT) Consistency coefficient, or model constant for rheological relationships n 2 (Ft /L )


K1, K2, K3 L n np N P Pl Po Q r R R0 Re S t ∆T T Tbav Tw v +

v ∇v T (∇v) We X + y Z

Constants model Pipe length (L) Flow behavior index or in general, power law exponent (dimensionless) Pump efficiency Agitator speed or rotational velocity (rev/t) 2 Pressure (F/L ) or power (FL/t) Plasticity number (dimensionless) Power number (dimensionless) 3 Volumetric flow (L /t) Radial coordinate (L) Pipe inner radius (L); universal gas constant (Equation 21) (E/MolT) Plug radius (L) Reynolds number (dimensionless) Solids content (%) Time; temperature (°C) Temperature difference (Tw – Tbav) Absolute temperature (K) Bulk average fluid temperature (T) Wall temperature (T) Flow velocity vector (L/t); v is the magnitude of v, which is composed of vx, vy, and vz; velocity components in directions x, y, z Time averaged velocity (dimensionless) Shear rate tensor (1/t) Transpose of the shear rate tensor (1/t) Weber number (dimensionless) Milk concentration (% w/w) Time averaged distance, distance from the wall (dimensionless) Vertical position (L)

GREEK SYMBOLS α β ρ ε γ λi Ψ σ τ τ0 τrz τw µ, η

Kinetic energy correction factor (dimensionless) Thermal expansion coefficient (1/T) 3 Fluid density (Μ/L ) Relative roughness (dimensionless) Shear rate (1/t) (dimensionless) Time constant (t) Constant for NN friction factor Surface force (F/L) 2 Shear stress tensor or momentum flux (F/L ) which is composed of τxy , τyz, etc., the components of the shear stress tensor 2 Yield stress (F/L ) 2 Shear stress (F/L ) 2 Shear stress at wall (F/L ) 2 Viscosity coefficient or absolute viscosity (Ft/L or M/Lt)


2

µa Apparent viscosity (Ft/L or M/Lt) 2 µp, ηp Plastic viscosity (Ft/L or M/Lt)

REFERENCES Alvarez De Felipe, A.I., Melcón, B., and Zapico, J., Structural changes in sweetened condensed milk during storage: an electron microscopy study, J. Dairy Res., 58, 337–344, 1991. Awuah, G.B., Ramaswamy, H.S., and Simpsom, B.K., Surface heat transfer coefficients associated with heating of food particles in CMC solutions, J. Food Process Eng., 16, 39–57, 1993. Barbosa-Cánovas, G.V., Ibarz, A., and Peleg, M., Rheological properties of fluid foods: review, Alimentaria, April 1993, pp. 39–89. Bennet, C.O. and Myers, J.E., Momentum, Heat and Mass Transfer, McGraw-Hill, Tokyo, 1983. Bhamidipati, S. and Singh, R.K., Determination of fluid-particle convective heat transfer coefficient, Trans. ASAE, 38, 857–862, 1995. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley and Sons, New York, 1960. Bloore, C.G. and Boag, I.F., Some factors affecting the viscosity of concentrated skim milk, New Zealand J. Dairy Sci. Technol., 16, 143–154, 1981. Briggs, J.L. and Steffe, J.F., Kinetic energy correction factor of a Herschel-Bulkley fluid, J. Food Process Eng., 18, 115–118, 1995. Brodkey, R.S. and Hershey, H.C., Transport Phenomena: A Unified Approach, McGraw-Hill, New York, 1988. Casson N., A flow equation for pigmented-oil suspensions: the printing ink type, in Rheology of Dispersed Systems, Mill, C.C., Ed., Pergamon Press, London, 1959, pp. 84–104. Charm, S., The Fundamentals of Food Engineering, AVI, Westport, CT, 1971. Das, S.K., Biswas, M.N., and Mitra, A.K., Non-Newtonian liquid flow in bends, Chem. Eng. J., 45, 165–171, 1991. Daud, W.R.B.W., Theorical determination of the thickness of film of drying material in a top loading drum dryer, in Drying ’89, Mujumdar, A.S. and Roques, M., Eds., Hemisphere Publishing Corporation, New York, 1989, pp. 496–500. Dodge, D.W. and Metzner, A.B., Turbulent flow of non-Newtonian systems. AIChE J., 5, 189–204, 1959. Fahien, R.W., Fundamentals of Transfer Phenomena, McGraw-Hill, New York, 1983. Fernández-Martín, F., Influence of temperature and composition on some physical properties of milk and milk concentrates. II. Viscosity, J. Dairy Res., 39, 75–82, 1972. Foust, A.S., Wenzel, L.A., Clump, C.W., Maus, L., and Andersen, L.B., Principles of Unit Operations, John Wiley and Sons, New York, 1980. García, E.J., and Steffe J.F., Comparison of friction factor equations for non-Newtonian fluids in pipe flow, J. Food Process Eng., 9, 93–120, 1986. Gekas, V., Transport Phenomena of Food and Biological Materials, CRC Press, Boca Raton, FL, 1992. Guariguata, C., Barreiro, J.A., and Guariguata, G., Analysis of continuous sterilization processes for Bingham plastic fluids in laminar flow, J. Food Sci., 44, 905–910, 1979. Hayashi, H., Effect of viscosity on spray drying of soy protein milk, in Proceedings of the 5th World Congress of Chemical Engineering at San Diego, Vol. II, 1996, 98–102.


Heldman, D., Food Process Engineering, AVI, Westport, CT, 1975. Higgs, S.J. and Norrington, R.J., Rheological properties of selected foodstuffs, Proc. Biochem., May 1971, pp. 52–54. Ibarz, A., García, J.L., and Puy Llorens, J., Thixotropic analysis of condensed milks, Alimentaria, Oct. 1987, pp. 35, 37–38. Ibarz, A., Vélez, J., and Barbosa, G., Transport of Food Fluids through Pipes, unpublished document, 1996. Kokini, J.L., Rheological properties of foods, in Handbook of Food Engineering, Heldman, D.R. and Lund, D.B., Eds., Marcel Dekker, New York, 1992, pp. 1–38. Levenspiel, O., Engineering Flow and Heat Exchange, Plenum Press, New York, 1993. Liu, S. and Masliyah, J.H., On non-Newtonian fluid flow in ducts and porous media, Chem. Eng. Sci., 53, 1175–1201, 1998. Mason, P.L., Bistany, K.L., Puoti, M.G., and Kokini, J.L., A new empirical model to simulate transient shear stress growth in semi-solid foods, J. Food Process Eng., 6, 219–233, 1983. Mizrahi, S. and Berk, Z., Flow behavior of concentrated orange juice: mathematical treatment, J. Texture Stud., 3, 69–79, 1972. Molerus, O., Appropriately defined dimensionless groups for the description of flow phenomena in dispersed systems, Chem. Eng. Sci., 53, 753–759, 1997. Motyka, A.L., An introduction to rheology with emphasis on application to dispersions, J. Chem. Educ., 73, 374–380, 1996. Ofoli, R.Y., Morgan, R.G., and Steffe, J.F., A generalized rheological model for inelastic fluid foods, J. Texture Stud., 18, 213–230, 1987. Osorio, F.A. and Steffe, J.F., Kinetic energy calculations for non-Newtonian fluids in circular tubes, J. Food Sci., 49, 1295–1296, 1315, 1984. Parzonka, W. and Vocadlo, J., Méthode de la caractéristique du comportement rhéologique des substances viscoplastiques dápres les mesures au viscosimétre de coutte (modéle nouveau a trois paraméters), Rheol. Acta., 7, 260–265, 1980. Patil, G.R. and Patel, A.A., Viscoelastic properties of sweetened condensed milk as influenced by storage, Milchwissenschaft, 47, 12–14, 1992. Pereira, E.C., Bhattacharya, M., and Morey, R.V., Modeling heat transfer to non-Newtonian fluids in a double tube heat exchanger, Trans. ASAE, 32, 256–262, 1989. Randhahn, H., Flow properties of milk and milk concentrates, Milchwissenschaft, 28, 620–628, 1973. Rao, M.A., Rheological properties of fluid foods, in Engineering Properties of Foods, Rao, M.A. and Rizvi, S.S.H, Eds., Marcel Dekker, New York, 1986, pp. 1–47. Rao, M.A., Rheology of Fluid and Semisolid Foods. Principles and Applications, Aspen Publishers, Inc., Gaithersburg, MD, 1999. Rao, V.N.M., Classification, description and measurement of viscoelastic properties of solid foods, in Viscoelastic Properties of Foods, Rao, M.A. and Steffe, J.F., Eds., Elsevier Applied Science, London, 1992, pp. 3–48. Reddy, C.S. and Datta, A.K., Thermophysical properties of concentrated reconstituted milk during processing, J. Food Eng., 21, 31–40, 1994. Schmidt, R.H., Sistrunk, C.P., Richter, R.L., and Cornell, J.A., Heat treatment and storage effects on texture characteristics of milk and yogurt systems fortified with oilseed proteins, J. Food Sci., 45, 471–475, 1980. Skelland, A.H.P., Non-Newtonian Flow and Heat Transfer, John Wiley and Sons, New York, 1967. Stankiewics, K. and Rao, M.A., Heat transfer in thin-film wiped-surface evaporation of model liquid foods, J. Food Process Eng., 10, 113–131, 1988.


Steffe, J.F., Rheological Methods in Food Process Engineering, Freeman Press, East Lansing, MI, 1992. Steffe, J.F., Mohamed, I.O., and Ford, E.W., Pressure loss in valves and fittings for pseudoplastic fluids in laminar flow, Trans. ASAE, 27, 616–619, 1984. Toledo, R.T., Fundamentals of Food Process Engineering, Chapman and Hall, New York, 1991. Vélez-Ruiz, J.F., Rheological Properties of Concentrated Milk, Ph.D. thesis, Washington State University, Pullman, 1996. Vélez-Ruiz, J.F., Notes of Food Engineering I, unpublished document, 2001. Vélez-Ruiz, J.F. and Barbosa-Cánovas, G.V., Rheological properties of concentrated milk as a function of concentration, temperature and storage time, J. Food Eng., 35, 177–190, 1997a. Vélez-Ruiz, J.F. and Barbosa-Cánovas, G.V., Rheological properties of selected dairy products, CRC Crit. Rev. Food Sci. Nutr., 37, 311–359, 1997b. Vitali, A.A. and Rao, M.A., Flow properties of low-pulp concentrated orange juice: effect of temperature and concentration, J. Food Sci., 49, 882–888, 1984. Wang, W., Walton, J.H., and McCarthy, K.L., Flow profiles of power law fluids in scraped surface heat exchanger geometry using MRI, J. Food Process Eng., 22, 11–27, 1999. Wayne, J.E. and Shoemaker, C.F., Rheological characterization of commercially processed fluid milks, J. Texture Stud., 19, 143–152, 1988. Weberschinke, J. and Filková, I., Apparent viscosity of non-Newtonian droplet on the outlet of wheel atomizer, in Drying ’82, Mujumdar, A.S., Ed., Hemisphere Publishing Corporation, New York, 1982, pp. 165–170. Welty, J.R., Wilson, R.E., and Wicks, C.E., Fundamentals of Momentum, Heat and Mass Transfer, John Wiley and Sons, New York, 1976. Whorlow, R.W., Rheological Techniques, Ellis Horwood Ltd., Chiclester, U.K., 1992.


Part II Mass Transfer


4

Structural Effects of Blanching and Osmotic Dehydration Pretreatments on Air Drying Kinetics of Fruit Tissues S.M. Alzamora, A. Nieto, and M.A. Castro

CONTENTS 4.1 4.2

Introduction Materials and Methods 4.2.1 Experiment 4.2.2 Moisture Diffusivity Calculation 4.3 Results 4.3.1 Effect of Blanching 4.3.2 Effect of Osmotic Treatment 4.4 Conclusions Nomenclature References

4.1 INTRODUCTION Typical commercial fruit processing currently involves predrying treatments aimed at either quality improvement of the final product or improved drying kinetics (Lewicki, 1998). The most popular pretreatments are blanching and sugar or salt osmotic dehydration. Blanching prior to drying is sometimes performed to inactivate enzymes that adversely affect product quality, denature cell membranes to reduce drying times, remove intercellular air, and/or reduce microbial load. Osmotic pretreatment results in substantial dewatering; it has also been reported to prevent excessive shrinkage of the tissue undergoing drying and to improve rehydration of dry material. Literature results pertaining to the drying behavior of fruit and vegetable tissues as affected by blanching and the osmotic process are apparently contradictory, as


reviewed by Alvarez et al. (1995). For example, moisture diffusivity of water in sugar beet root plates, carrot cubes, and potato slices was reported to decrease because of the uptake of sugar and/or salt. In contrast, sucrose osmotic treatment did not affect the dehydration rate of pineapple. Blanching increased the rate of drying of sugar beet and carrot, decreased the moisture transport rate in potato, and had only a small effect on avocado dehydration behavior. This dissimilarity can be ascribed to the intricate simultaneous heat and mass transfer drying process and to the wide variation in physical and structural properties among vegetables and fruits. A schematic view of the various transfer mechanisms that can be present during drying of fruit tissues is presented in Figure 4.1a. A significant (a)

Molecular diffusion CONDUCTION

Capillary motion Knudsen flow Vapor diffusion in pores

CONVECTION

Liquid diffusion Evaporation - condensation

(b)

tt

cwp

tt st

cl = chloroplast; cw = cell wall; cwp: the cell wall pathway; g = Golgi; is = intercellular space; nu = nucleolus; n = nucleus; ml = middle lamella; pl = plasmodesmata; pm = plasmalemma; rer = rough endoplasmic reticulum; st: the symplastic; tt: the transmembrane transport; v = vacuole

FIGURE 4.1 (a) Major mechanisms during drying of parenchymatous fruit tissue. (b) Possible pathways for water transport in a parenchymatous fruit cell (courtesy of Dr. Ancíbor).


number of mechanisms for moisture transport within porous foods has been proposed, including molecular diffusion, capillary motion, liquid diffusion through solid pores, Knudsen flow, vapor diffusion in air-filled pores, and hydrodynamic flow, none of them prevailing throughout the whole drying process (Zogzas et al., 1996). Usually, a series, parallel, and/or series–parallel combination of the aforementioned mechanisms contributes to the total flow, and the relative contributions show product and drying system specificity as well as drying time dependence. Structural heterogeneity of biological materials, such as fruit, brings an additional complexity to the water migration in a porous solid system.As Figure 4.1b shows, cellular tissues are multiphase systems (i.e., vacuole, cytoplasm, tonoplast and plasmalemma membranes, cell wall, intercellular spaces), and water can migrate out of cells in three possible ways: transmembrane transport (through tonoplast and plasmalemma membrane boundaries), symplastic transport (via cytoplasmic strands or plasmodesmatas), or cell wall pathway (the apoplastic way), this last being the preferred pathway for small nonionic species such as water (Molz and Ikenberry, 1974; Tyree, 1970). Because of this complexity, the macroscopic (also called the “black box”) approach to quantifying the transport of water during the first falling rate period of air drying commonly involves the determination of an empirical single experimental effective water diffusivity (Deff). This parameter, obtained by applying Fick’s second law for species diffusion in a single phase, takes into account the internal porosity and the tortuosity of the sample, but it also lumps together the deformation of the fruit matrix, the volume changes, the composition and structure of the fruit, the material–water interactions, and the multiphase water transport (Zogzas et al., 1996; Achanta and Okos, 1996). Hence, Deff is an empirical parameter useful as a quick engineering design tool but without physical significance. A phenomenological understanding of the dehydration kinetics and the effect of pretreatments, however, requires, among others, a microscopic analysis to address the structural and ultrastructural changes produced at the cellular level and their potential influence on drying rates and quality of the final products. Food structure is a process variable usually neglected in engineering design and transport kinetics analysis. But cell wall resistance to water flux, as well as membrane integrity, are modified to different degrees due to the predrying treatments, depending on the type of fruit considered. In this chapter, ultra-, micro- and macrostructural effects of blanching and osmotic dehydration (atmospheric or in vacuum) pretreatments on the moisture transport rate during the first falling rate period of air drying of fruits are illustrated. Drying behavior of strawberry, mango, and apple is discussed under the following headings: shrinkage of fruit piece, porosity, sugar content, cell membrane integrity, and cell wall architecture.

4.2 MATERIALS AND METHODS 4.2.1 EXPERIMENT The experimental process has been well described by Alvarez et al. (1995) and Nieto et al. (1998, 2001). Briefly, apples (Malus pumila, Granny Smith cv.; ≅85–88% w/w moisture content, wet basis) and mangoes (Mangifera indica Linn, Keitt var.; ≅85– 88% w/w moisture content, wet basis) were hand peeled and cut into an infinite plate


shape (dimensions at the end of the pretreatment ≅4 × 4 × 0.4 cm). Strawberries (Pájaro or Tioga Leico var.; ≅86% w/w moisture content, wet basis) were washed, decapped by hand, and selected to obtain samples of uniform size and maturity. For steam blanching, apple and mango plates were exposed to steam for 1 min at atmospheric pressure and then cooled in water at 5°C. Whole strawberries were exposed to steam for 3 min and cooled in similar conditions. For osmotic dehydration at atmospheric pressure, samples were immersed into different glucose aqueous solutions with (mango and apple) or without (strawberry) forced convection at 25°C until the desired final water activity (aw) value was reached (≅3 h for mango and apple and 48 h for strawberry). For vacuum glucose impregnation, fruit samples were immersed in a 59.0% w/w glucose aqueous solution (aw = 0.84) at 25°C, and a pressure equal to 60 mmHg was applied to the system for 10 min. After the vacuum treatment, the system was placed at atmospheric pressure for 10 min; the final aw value reached by fruits was 0.97. After osmosis, fruit samples were drained and dried at 60°C (mango and apple) or 55°C (strawberry) with air at high constant velocity (about 15 m/sec) to eliminate or minimize external resistance to moisture loss.

4.2.2 MOISTURE DIFFUSIVITY CALCULATION The Deff value was obtained by applying Fick’s second law for species diffusion in a single phase, with boundary conditions of internal resistance controlling and uniform initial moisture content, integrated over the volume of the slab (apple, mango) or sphere (strawberry) (Luikov, 1968). The Deff values for spheres were affected by a shape factor to take into account that strawberries are ellipsoids (Becker, 1959; Aguerre et al., 1987). Uniform internal fruit temperature was assumed due to the low Biot number for heat transfer usually found for conventional air drying of fruits (Alzamora et al., 1979). Negligible external heat transfer effects were also considered. Assuming diffusion in an isotropic medium, constant diffusion coefficient, and isothermal process, and using the moisture concentration converted to moisture content on a dry basis, Fick’s second law for one-dimensional unsteady diffusion is given as:  ∂2 m C ∂m  ∂m = Deff  2 +  ∂t x ∂x   ∂x

(4.1)

where x is the spatial coordinate, C is a constant (0 for infinite plate; 2 for sphere), and Deff is the effective moisture diffusivity. For uniform moisture distribution (assuming no constant drying period takes place) and internal control to mass transfer, initial and boundary conditions are: m(x, 0) = m0

at t = 0

(4.2)

m(xS, t) = me

at x = xS (at the surface)

(4.3)

m(0, t) = finite


at x = 0 (at the center)

(4.4)

Solutions for the different geometrical configurations can be expressed as: m − me 8 = m0 − me π2

∞

∑ n =1

2   1 2 π D eff t exp ( 2 1 ) − n +  2 2  (2 n + 1) 4 l0  

(4.5)

for a slab, where l0 is the half-thickness of the slab; m − me = m0 − me

∞

∑ n 6π 2

2

n =1

 D t exp − n 2 π 2 eff2  Re  

(4.6)

for a sphere, where Re is the radius. For long drying times, only the first term of Equations (4.5) and (4.6) is significant, and the values of Deff may be estimated from the linear relationship m − me/m0 − me vs. time on semilogarithmic coordinates. Strawberries were assumed to be ellipsoids, having three characteristic diameters (2 rM1 ≅ 2 rM 2 ≤ 2 rM ). For bodies having an ellipsoid shape, the Fourier number may be extended to (Aguerre et al., 1987; Becker, 1959): F0 =

Deff t  Vp  3 S   p

2

(4.7)

where Vp is the volume and Sp is the surface of the body. Generally, the equivalent spherical radius (Re) is used as the characteristic length. F0′ =

Deff t R 2e

(4.8)

where Re = 3(VS/SS), and SS and VS are the surface area and the volume of the sphere, respectively. If VS = VP = V, the relationship between F0 and F0′ is given by 2

Deff t  Sp  Deff t F0′ F0 = 2 = 2   = Ψ2 R2 = Ψ2 S  V  V  S e 3 S  3 S   S  p Deff t

(4.9)

where Ψ is defined as SS/Sp, and SS is the surface area of a sphere of volume equal to that of the fruit with surface area Sp, which is assumed to be an ellipsoid. The 2 intrinsic diffusivity Deff is given by Ψ D′eff. Ψ is the shape factor of a solid, commonly known as the sphericity (Becker, 1959). The diffusion coefficient calculated from 2 Equation (4.6) is D′eff, and it must be corrected by the factor Ψ when the product shape can be assumed as an ellipsoid: Expressing the surface area of an ellipsoid as Sp = 2 π rm2 + 2 π


 rm R m  sin −1 e  e 

(4.10)

and calculating the eccentricity, e, as (Aguerre et al., 1987): e = 1 − ( rm / R m )2

(4.11)

the shape factor results in: Ψ=

SS = Sp

4 π R 2e r R  2 π rm2 + 2 π m m sin −1 e  e 

(4.12)

4.3 RESULTS 4.3.1 EFFECT

OF

BLANCHING

Table 4.1 summarizes the main structural features that may be produced by fruit scalding (Ilker and Szczesniak, 1990; Alzamora et al., 1997). Depending on the preponderant phenomenon/a, drying rates of blanched tissues could be decreased or increased compared to the moisture transport rate for the fresh fruit. The photomicrographs in Figure 4.2 compare the structures of the fruits with and without blanching. All raw fruits (Figure 4.2A, C, F, G) exhibited parenchymatous cells containing intact membranes, cytoplasm, and organelles and had intact cell walls without degradation, disruption, or dissolution. The plasma and tonoplast membranes were closely associated with the cell wall, and the middle lamella was clearly seen cementing adjacent cells. On the contrary, in the three blanched fruits (Figure 4.2B, D, E, H, I), plasmalemma and tonoplast were disrupted, and numerous vesicles of the cytoplasm had been formed. In blanched strawberry, the electronic density of the cell wall was much lower than that of fresh fruit walls (Figure 4.2B). The middle lamella practically disappeared, and microfibrils appeared distorted. Reduction in fruit volume was small ( 1 are small, and the following approximation can be made (Hawlader et al., 1991): 2  w − we   8  π Deff t ln   = ln 2  − π L2  wo − we 

(9.10)

where Deff can be measured from the slope of the plot of ln ww −−wwe against t or t/L . o e 2


The mean relative deviation modulus (E, %) was used as a criterion to evaluate the fit of the tested model, as applied to the experimental data. Values of E were calculated using the following equation:

E=

100 N

∑

ν0 − ν p ν0

(9.11)

where ν0 and νp are the observed and the predicted values, respectively. Values of E less than or equal to 10% are considered to fit the experimental data satisfactorily.

9.4 MATERIALS AND METHODS Fresh cherry tomatoes harvested between the end of January and the middle of May were purchased from a local market. The fruits were sorted visually for color (completely red), size (approximately 2.8 cm diameter), and lack of physical damage, and the physicochemical characteristics were determined according to Ranganna (1977). As the cherry tomato waxy skin presents a high resistance to mass transfer, the fruits were washed and perforated with needles (1 mm diameter) (Shi et al., 1997) 2 to pin hole density of 16 holes/cm . The perforated cherry tomatoes were immersed in NaCl and NaCl + sucrose (3:2) solutions of different concentrations (10 and 25% w/w) at 25°C (room temperature) and agitation of 70 r/min, maintained in a temperature– agitation controlled shaker (Tecnal, TE-421). In order to avoid any significant dilution effect on the osmotic solution, a 1:10 fruit-to-solution ratio was used. At different processing times, samples were withdrawn from the solution, rinsed with cold water to remove adhering osmotic solution, and gently blotted to remove surface moisture. The water content of cherry tomato samples was determined gravimetrically by vacuum oven drying at 70°C for 24 h. The salt content was determined by Mohr’s titration method (Ranganna, 1977). Three replications were made. Desorption isotherms of the samples were determined using the static gravimetric method. Saturated salt solutions were prepared to give defined constant water activity (Greenspan, 1977). Three replications of the same experiment were carried out. After equilibrium was reached, the equilibrium moisture content was determined (vacuum oven at 70°C for 24 h). The adequacy of the mathematical models of BET, GAB, Halsey, and Oswin was verified. For air drying experiments, a cabinet dryer was used, and the tests were conducted at three different temperatures (50, 60, and 70°C) and two air velocities (0.75 and 2.60 m/sec). The tomatoes were cut into quarters along the longitudinal axis and the seeds were removed. The samples (length of 1.8 mm) were spread uniformly on a perforated stainless steel tray in a fixed bed dryer. The processing temperature was controlled using precalibrated cooper-[constantan] thermocouples and the air flow rate was monitored by an anemometer (TSI, 8330-M). The drying curves were determined by periodic weighing of the tray on a semianalytical scale.


9.5 RESULTS AND DISCUSSION 9.5.1 CHERRY TOMATO CHARACTERIZATION The physicochemical characteristics of the cherry tomatoes used in the experiments are shown in Table 9.1. The total solids content, acidity, and soluble solids content were similar to the values obtained by Gould (1974) for tomato, but the NaCl content was higher and the reducing sugars content was lower. Sugars and organic acids were the majority of the total dry matter content of the tomato fruit. Similar results for cherry tomato were obtained by Picha (1987). The pH value was similar to the results of PetroTurza (1987) for tomato and the density to tomato pulp (Ranganna, 1977).

9.5.2 OSMOTIC DEHYDRATION Characterizing the effect of osmotic dehydration on the air drying kinetics of cherry tomatoes was the main objective of this study. The effect of the osmotic agent (NaCl and the NaCl–sucrose mixture) and solution concentration (10 and 25% w/w), at room temperature (25°C), was evaluated by determining moisture and salt contents, as shown in Figure 9.1. The more concentrated the solution, the higher the percentages of water loss and solid uptake, due to an increase in osmotic pressure gradient resulting in increased mass transfer. This finding is in agreement with the results of Rastogi and Raghavarao (1994) for the osmotic dehydration of carrots in sucrose solutions, and of Vijayanand et al. (1995) for cauliflower in salt solutions. The rate of water loss was faster in the first 2 h of the process, decreasing gradually while approaching the end of the experiment, which was not sufficient in length for equilibrium to be reached, except for the 10% solution. Figure 9.1(b) shows the salt gain during the osmotic dehydration of cherry tomato. It was observed that an increase in the concentration of the osmotic solution gave higher salt gain within 6 h of processing. By fixing the concentration of the

TABLE 9.1 Physicochemical Characteristics of Cherry Tomatoes Analysis Moisture content (%) Acidity (% citric acid) NaCl (%) Reducing sugars (%) pH Brix Ascorbic acid (%) Density (g/ml)


Mean Value 94.39 0.64 0.65 1.03 4.04 5.53 31.39 0.99

TABLE 9.2 Average Water Apparent Diffusion Coefficient (Davg) Obtained from the Osmotic Dehydration Process 2

Sample Sodium Sodium Sodium Sodium

S1(10 ) chloride chloride chloride chloride

(10%) (25%) + sucrose (10%) + sucrose (25%)

1.76 2.28 0.40 1.75

100

10

2

m /sec)

E (%)

11.69 14.42 2.17 11.66

8.85 0.12 3.68 10.01

10 NaCl 10% NaCl 25% NaCl+sucrose 10% NaCl+sucrose 25%

96

8

Salt gain (%)

Water content (%)

Davg (10

92

88

NaCl 10% NaCl 25% NaCl+sucrose 10% NaCl+sucrose 25%

6

4

2 84 0

60

120

180

240

300

360

0 0

60

120

180

240

Time (min)

Time (min)

(a)

(b)

300

360

FIGURE 9.1 Moisture (a) and salt gain (b) profiles of osmotically dehydrated cherry tomato.

solution, the effect of the osmotic agent can be evaluated. Our results showed that the salt gain was lower when the mixed NaCl–sucrose solution was used. The experimental water loss results were used to estimate the apparent water diffusion coefficients. The time used for prediction by the proposed model was 180 min. Table 9.2 presents the obtained average water apparent diffusivities (Davg) and S1 values. Increasing the osmotic solution concentration (10 to 25% w/w) caused an increase in Davg. Changing the osmotic medium from salt to a mixed salt–sucrose solution, at the same temperature, resulted in a decrease of the diffusion coefficient. This is due to sodium chloride ionization in solution, and as the molecular weight of sodium chloride is lower than that of sucrose, its rate of penetration into vegetable tissues is higher. Higher values for S1 indicate a higher diffusion of water per unit of time. The model was able to predict the entire osmotic dehydration process up to equilibrium, using data obtained over a short period of time, with satisfactory mean relative modulus. Consumer demand for fresh, convenient, and safe vegetables has promoted interest in processed products with fresh-like qualities (Shi et al., 1997). The osmotic dehydration process was carried out within a short period of time in order to achieve


we (kg/kg)

0.5 0.4 0.3

T= 50°C (fresh) T= 70°C (fresh) T=50°C (NaCl 10%)

0.2 0.1 0.0 0.0

0.2

0.4

aw

0.6

0.8

1.0

FIGURE 9.2 Sorption isotherms for fresh and osmoticly dehydrated cherry tomato.

a high degree of dewatering with relatively small solids gain. Pretreatment with sodium chloride, 10% w/w for 2 h, was chosen for the air drying experiments, and desorption isotherms of these samples were determined.

9.5.3 SORPTION ISOTHERMS The results of the experimental measurements of the equilibrium moisture content (we) of cherry tomato with and without pretreatment, at relative humidity ranging from 10.75 to 81.20%, are shown in Figure 9.2. The pretreated sample obtained lower values of water activity than the fresh fruit. Sloan and Labuza (1976) noted that components such as glycerol or salt are particularly effective in reducing the water activity. No temperature dependence of the experimental data of the fresh fruit can be seen. Similar behavior was observed by Bolin (1980) for prunes. The results for 50°C were considered for all samples, and temperature independence was assumed for the osmotically pretreated samples. It can be seen that the desorption isotherms do not intersect. For products with high sugar content, the intersection of sorption isotherms can be observed (Saravacos et al., 1986). For products with low sugar, high protein, or high starch, there is no intersection point with increased temperature (Benado and Rizvi, 1985). The results of direct nonlinear regression analysis of fitting the GAB, BET, Halsey, and Oswin models to the experimental points (Tables 9.3, 9.4, and 9.5) showed that the GAB equation was satisfactory in predicting the equilibrium moisture content of fresh cherry tomato, and that the Halsey model presented the best fit for samples pretreated in 10% NaCl osmotic solution for 2 h (Figure 9.3).

9.5.4 AIR DRYING The effect of air temperature (50, 60, and 70°C) at air velocities of 0.75 and 2.60 m/sec on the drying kinetics of cherry tomato is illustrated in Figure 9.4. Increasing the temperature of the drying medium increased the drying potential and the moisture removal rates. When the flow rate was increased at constant temperature, similar behavior was observed. The effect of osmotic pretreatment was clearly verified at a temperature of 50°C and air velocity of 2.60 m/sec. In this condition, the drying time was reduced, with higher drying rates and effective diffusion coefficients (Table 9.6). For the other conditions studied (60 and 70°C at both air velocities), the drying time of pretreated samples with 10% NaCl solution was not faster when compared


TABLE 9.3 Estimated Parameters for Fresh Cherry Tomato ο at 50 C Parameters

Model BET GAB Halsey Oswin

Xm 0.133 Xm 0.138 A 0.026 A 0.227

C 20.087 C 95.615 B 2.205 B 0.335

R

2

E (%)

n S K 0.781

0.979

3.221

0.990

2.278

—

0.995

2.341

—

0.990

2.171

TABLE 9.4 Estimated Parameters for Fresh Cherry Tomato ο at 70 C Parameters


Xm 0.145 Xm 0.135 A 0.036 A 0.231

C 11.154 C 51.868 B 1.988 B 0.370

R

2

E (%)

n S K 0.822

0.97

8.022

0.99

3.747

—

0.99

6.617

—

0.99

3.353

TABLE 9.5 Estimated Parameters for Osmosed Cherry ο Tomato at 50 C Parameters



Xm 0.575 Xm 0.074 A 0.069 A 0.120

C 0.11 C 5.973 B 1.099 B 0.751

R

2

E (%)

n S K 0.991

0.98

6.060

0.99

2.573

—

0.99

2.211

—

0.99

3.353

we (kg/kg)

0.5 0.4

fresh (GAB)

0.3 0.2 0.1 0.0 0.0

NaCl 10% (Halsey)

0.2

0.4

0.6

0.8

1.0

aw

FIGURE 9.3 Equilibrium moisture content at 50°C and the predictions of the GAB and Halsey models for fresh and preteated cherry tomato, respectively.

(w – we)/(wo – we)

1.0

fresh (T = 50°C) 10% NaCl (T = 50°C)

0.8

fresh (T = 60°C) 10% NaCl (T = 60°C)

0.6

fresh (T = 70°C)

0.4

10% NaCl (T = 70°C)

0.2 0.0

0

20

40

60

80

100

120

140

Time (min) (a)

(w – we)/(wo – we)

1.0

fresh (T = 50°C) 10% NaCl (T = 50°C)

0.8

fresh (T = 60°C) 10% NaCl (T = 60°C)

0.6

fresh (T = 70°C)

0.4

10% NaCl (T = 70°C)

0.2 0.0

0

20

40

60

80

100

120

140

Time (min) (b)

FIGURE 9.4 Effect of temperature on water removal for fresh and osmotically dehydrated cherry tomato at constant air velocities of 0.75 m/sec (a) and 2.60 m/sec (b).

to the untreated samples at the same air temperature and velocity, The effective diffusion coefficients were lower, indicating a less favored diffusional process. The differences in the effective diffusion coefficients can be attributed to the compositional changes that occur following osmosis. The uptake of salt and the loss of water that occur in osmosis give increased internal resistance to moisture movement. However, the dried cherry tomato in 10% NaCl solution presented a flexible structure, smaller shrinkage, and a more natural coloration when compared with the dried


TABLE 9.6 Effective Diffusion Coefficients of Air Dried Cherry Tomatoes 11

2

Deff (10 m /sec) o

T ( C) 50 60 70

v (m/sec)

Fresh

10% NaCl

0.75 2.60 0.75 2.60 0.75 2.60

12.02 13.66 17.49 26.78 20.77 37.70

12.02 17.49 14.21 19.67 18.03 21.86

TABLE 9.7 Activation Energy for Diffusion in Cherry Tomatoes Sample Fresh 10% NaCl

v (m/sec)

Ea (kJ/mol)

R

0.75 2.60 0.75 2.60

25.39 46.61 18.31 10.56

0.99 0.99 0.99 0.99

fruit with no preliminary treatment. The same behavior was found by Lenart (1996) for osmodehydrated dried apples. Figure 9.5 shows the variation in the drying rate for all samples as a function of moisture content with air velocity and temperature. For the given experimental conditions, the samples did not show a constant rate of drying. Higher temperature, higher air velocity and the pretreatment for the lower temperature increased the potential for the transport of moisture, thus increasing the drying rate. For temperatures of 60 and 70°C, the untreated samples showed higher drying rates in the beginning, due to a higher free water content. The activation energy was estimated by an Arrhenius type equation, using the value of effective diffusivity for each temperature from Table 9.6: Deff = A exp

 − Ea   RT 

(9.12)

where Ea is activation energy (kJ/mol), R is the universal gas constant (8.314 J/mol K), A is the integration constant, and T is temperature (K). The values of activation energy and the correlation coefficient are presented in Table 9.7. The pretreated samples’ activation energy was much lower than that of the fresh samples, indicating that temperature has less influence on drying rate for


0.04 0.03

2

–dw/dt (10 kg/kg sec)

0.05

0.02

fresh (T = 50°C) 10% NaCl (T = 50°C) fresh (T = 60°C) 10% NaCl (T = 60°C) fresh (T = 70°C) 10% NaCl (T = 70°C)

0.01 0.00 0.0

0.2

0.4

0.6

0.8

1.0

0.8

1.0

w (kg/kg)

2

–dw/dt (10 kg/kg sec)

(a) 0.08 0.06 0.04

fresh (T = 50°C) 10% NaCl (T = 50°C) fresh (T = 60°C) 10% NaCl (T = 60°C) fresh (T = 70°C) 10% NaCl (T = 70°C)

0.02 0.00 0.0

0.2

0.4

0.6

w (kg/kg)

(b)

FIGURE 9.5 Drying rate vs. moisture content at constant air velocities of 0.75 m/sec (a) and 2.60 m/sec (b).

salt osmosed fruits. These results are similar to those of Islam and Flink (1982) for osmotically dehydrated potatoes in NaCl solution.

9.6 CONCLUSIONS The rate of moisture removal and solids gain in the osmotic dehydration of cherry tomato was directly related to the concentration of the solution, the osmotic agent and the immersion time. In order to obtain a processed product with fresh-like qualities, pretreatment with 10% NaCl solution for 2 h was the condition used prior to drying. The water apparent diffusivity for osmotic dehydration ranged from 2.17 × –10 –10 2 10 to 14.42 × 10 m /sec and the calculated effective diffusivity ranged from –11 –11 2 –11 12.02 × 10 to 37.70 × 10 m /sec for fresh fruit and from 12.02 × 10 to 21.86 × –11 2 10 m /sec for cherry tomatoes pretreated in 10% NaCl solution for 2 h. For the given experimental conditions, the samples did not show a constant rate of drying. Osmotic dehydration of cherry tomato in 10% NaCl solution for 2 h before air drying was efficient in increasing the water removal rate and the effective diffusion o coefficient, and in decreasing the drying time at 50 C with air velocity of 2.60 m/sec.


For all conditions studied, the osmodehydrated dried samples presented a flexible structure, smaller shrinkage, and a more natural coloration when compared to dried cherry tomato with no preliminary treatment. The activation energy for effective diffusivity was lower for osmosed samples, indicating less influence of temperature on drying rate.

NOMENCLATURE Davg Deff Ea L R R t w we w0 WL WL∞ T

2

Moisture apparent diffusion coefficient, m /sec 2 Moisture effective diffusion coefficient, m /sec Activation energy, kJ/mole Thickness, m Universal gas constant, J/mol K Radius, m Time, sec Moisture content, kg/kg Equilibrium moisture content, kg/kg Initial moisture content, kg/kg Water loss, g water/100 g sample Amount of water leaving the solid at equilibrium, g water/100 g sample Temperature, K

ACKNOWLEDGMENT The authors gratefully acknowledge the financial support of CAPES.

REFERENCES Azuara, E., Beristain, C.I., and Garcia, H.S., Development of a mathematical model to predict kinetics of osmotic dehydration, J. Food Technol., 29, 239, 1992. Benado, A.L. and Rizvi, S.S.H., Thermodynamic properties of water on rice as calculated from reversible and irreversible isotherms, J. Food Sci., 50, 101, 1985. Bolin, H.R., Relation of moisture to water activity in prunes and raisins, J. Food Sci., 56, 1190, 1980. Brunauer, S., Emmet, T.H., and Teller, F., Adsorption of gases in multimolecular layers, G. Am. Chem. Soc., 60, 309, 1938. Crank, J., Mathematics of Diffusion, Clarendon Press, Oxford, 1975. Dincer, I. and Dost, S., An analytical model for moisture diffusion in solid objects during drying, Drying Technol., 13, 425, 1995. Folquer, F., El Tomate: Estudio de la Planta y su Produccion Comercial, Editorial Hemisferio Sur, Buenos Aires, 1976. Gal, S., Recent developments in techniques for the determination of sorption isotherms, in Water Relations of Foods, Duckworth, R.B., Ed., Academic Press, London, 1972, p. 89. Gould, W.A., Tomato Production, Processing and Quality Evaluation, AVI, Westport, CT, 1974.


Greenspan, L., Humidity fixed points of binary saturated aqueous solutions, J. Res. Stand. A. Phys. Chem., 81, 89, 1977. Halsey, G., Physical adsorption on uniform surfaces, J. Chem. Phys., 16, 931, 1948. Hawlader, M.N.A., Uddin M.S., Ho, J.C., and Teng A.B.W., Drying characteristics of tomatoes, J. Food Eng., 14, 259–268, 1991. Iglesias, H.A. and Chirife, J., Handbook of Food Isotherms: Water Sorption Parameters for Food Components, Academic Press, New York, 1982. Islam, M.N. and Flink, J.N., Dehydration of potato II. Osmotic concentration and its effect on air drying behaviour, J.Food Technol., 17, 387, 1982. Kaur-Sawhney, R., Applewite, P.B., and Galston, A.W., Formation in vitro of ripe tomato fruits from thin layer explants of flower pedicels, J. Fruit Nuts, 18, 191, 1996. Lenart, A., Osmo-convective drying of fruits and vegetables: technology and application, Drying Technol., 14, 391, 1996. Lerici, C.R., Pinnavara, G., Dalla Rosa, M., and, Bartolucci, L., Osmotidehydration of fruits: influence of osmotic agents on drying behavior and product quality, J. Food Sci., 50, 1217–1219, 1985. Oswin, C.R., The thickness of package life. III. Isotherm, J. Chem. Ind., 65, 419, 1946. Petro-Turza, M., Flavour of tomato and tomato products, Food Ver. Int., 2, 309, 1987. Picha, D.H., Sugar and organic acid content of cherry tomato fruit at different ripening stages, Hort. Sci., 22, 94, 1987. Ranganna, S., Manual of Analysis of Fruit and Vegetables Products, McGraw-Hill, New Delhi, 1977. Raoult-Wack, A.L., Lafont, F., Rios, G., and Guilbert, S., Osmotic dehydration: study of mass transfer in terms of engineering properties, in Drying of Solids, Mujumdar, A.S. Roques, M.A., Eds., Hemisphere Publishing Company, New York, 1989, p. 487. Raoult-Wack, A.L., Rios, G., Saurel, R., Giroux, F., and Guilbert, S., Modeling of dewatering and impregnation soaking process (osmotic dehydration), Food Res. Int., 27, 207, 1994. Rastogi, N.K. and Raghavarao, K.S.M.S., Effect of temperature and concentration on osmotic dehydration of coconut, Lebensm.-Wiss. Technol., 27, 564, 1994. Saravacos, G.D., Tsiourvas, D.A., and Tsami, E., Effect of temperature on the water adsorption isotherms of sultana raisins, J. Food Sci., 51, 381, 1986. Shi, J.X., Le Maguer, M., Wang, S.L., and Liptay, A., Application of osmotic treatment in tomato processing: effect of skin treatments on mass transfer in osmotic dehydration of tomatoes, Food Res. Int., 30, 669, 1997. Sloan, A.E. and Labuza, T.P., Investigating alternative humectants for use in foods, Food Prod. Devel., 9, 75, 1976. Uddin, M.S., Hawlader, M.N.A., and Rahman, M.S., Evaluation of drying characteristics of pineapple in the production of pineapple powder, J. Food Process. Preserv., 14, 375, 1990. Vagenas, G.K., Marinos-Kouris, D., and Saravacos, G.D., An analysis of mass transfer in airdrying of foods, Drying Technol., 8, 323, 1990. Van den Berg, C. and Bruin, S., Water activity of and its estimation in food systems: theoretical aspects, in Water Activity: Influences on Food Quality, Rockland, L.B. and Stewart, G.F., Eds., Academic Press, New York, 1981. Vijayanand, P., Nagin, C., and Eipeson, W.E., Optimization of osmotic dehydration of cauliflower, J. Food Process. Preserv., 14, 391, 1995. Zhang, X.W., Liu, X., Gu, D.X., Zhou, W., Wang, R.L., and Liu, P., Desorption isotherms of some vegetables, J. Sci. Food Agric., 70, 303, 1996.


10

Determination of ConcentrationDependent Effective Moisture Diffusivity of Plums Based on Shrinkage Kinetics A.L. Gabas, F.C. Menegalli, and J. Telis-Romero

CONTENTS 10.1 Introduction 10.2 Mathematical Model 10.3 Methodology 10.3.1 Material 10.3.2 Drying Apparatus and Conditions 10.3.3 Shrinkage Kinetics 10.4 Shrinkage and Effective Diffusivity 10.5 Conclusions Nomenclature Acknowledgment References

10.1 INTRODUCTION Air drying is one of the most used unit operations in food processing, and mathematical modeling of the dehydration process is very useful in the design and optimization of dryers (Madamba, 1997; Arrouz et al., 1998). Nevertheless, theoretical simulations of drying processes require a substantial amount of computing time as a consequence of the complexity of diffusion equations governing the process. A drying process can be described completely using an appropriate drying model made


up of differential equations of heat and mass transfer in the interior of the product and its interface with the drying agent (air). Knowledge of transport and material properties is necessary to apply any transport equation. Such properties as diffusivity and mass and heat transfer coefficients, as well as thermal conductivity, must be accurately estimated. Knowledge of additional properties such as equilibrium moisture content, shrinkage, bulk density, specific volume, and porosity, is also required. Many articles have recently been published concerning the estimation of the abovementioned properties (Zogzas et al., 1994; Madamba et al., 1994; Barbanti et al., 1994; Newman et al., 1996; Sabarez et al., 1997; Karathanos, 1999; Telis et al., 2000). Shrinkage is common during drying, and volume changes may affect the moisture removal rate. Study of the shrinkage phenomenon is essential to better understand drying kinetics and to improve the reliability of diffusivity based models (Raghavan et al., 1995; Simal et al., 1996; Sanjuán et al., 1996). Drying of plums (Prunus domestica) in drying tunnels produces prunes generally at about 14–19% moisture content (wet basis), which have sufficiently low water activity to avoid the problems of microbial spoilage, allowing long-term storage of the fruit. These fruits are then rehydrated (35–40%) prior to packing and sale. Many countries have a flourishing prune industry, including the U.S., France, Italy, Turkey, Chile, Argentina, and Australia (Price et al., 1997). The drying kinetics of plums have been investigated as a function of temperature in the range of 50 to 80°C using a laboratory pilot plant. The volume change or shrinkage of grapes during thin layer convective air drying has been described by a correlation with respect to water content (Gabas et al., 1999). The method for evaluating effective diffusivity as a function of moisture content and shrinkage during drying of plums is based on numerical solution to Fick’s diffusion equation. A number of studies have focused on modeling the drying process of fruits and vegetables and solving the mass transfer equation analytically. Córdova-Quiroz et al. (1996) obtained a simplification of boundary conditions of the mass transfer equation in cylindrical slabs of carrots. They developed an alternative simplification for the mathematical description of interfacial mass transfer, introducing a modified Biot number. Hernández et al. (2000) proposed an analytical solution of a mass transfer equation with concentration dependence of shrinkage and constant average water diffusivity for mango slices and cassava parallelepipeds. Although the drying behavior of a range of fruits and vegetables has been extensively studied, there is relatively little work on the kinetics of drying of plums, especially simultaneously considering shrinkage and diffusivity as a function of moisture content.

10.2 MATHEMATICAL MODEL A mass balance for the drying process of plums can be applied by considering the fruit as a sphere (Figure 10.1), according to the system of coordinates presented in Figure 10.2. The partial differential equation resulting from the mass balance at a differential 2 volume element r (senφ)∆φ∆ψ∆r is shown as (Bird et al., 1960): → → → ∂ ρA  → + ∇⋅ ρA v  =  ∇⋅ D AB ∇ ρA  + q A     ∂t


(10.1)

r(senφ)∆ψ r ∆φ

∆r

flesh stone rs rp

2

FIGURE 10.1 Differential volume element r (senφ)∆φ∆ψ∆r by considering the plum as a sphere.

∆r r∆φ r

∆φ φ

r(sen φ)∆ψ

∆ψ r senφ 2

FIGURE 10.2 System of spherical coordinates with the volume element r (senφ)∆φ∆ψ∆r for the mass balance.

The following hypotheses are generally used to describe the mass transfer in solid drying: unsteady state system, constant temperature and pressure, constant mass density and diffusion coefficient, negligible convective mass transfer, and absence of chemical reactions. Thus, assuming that internal resistance controls the drying rate, the differential equation for spherical coordinates in terms of Fick’s


second law is (Bird et al., 1960): r ∂ρA = D AB∇2ρA ∂t

(10.2)

Most cereals (rice, corn, wheat, etc.) experience little change of volume during the drying process. Therefore, the analytical solution of Equation (10.2) applies satisfactorily to the study of these materials. On the other hand, for high moisture content foods (potato, apple, persimmon, grape, plum, etc.) the volume variation is important, and the analytical solution of Equation (10.2) obtained for diffusivity and volume constant is not applicable, since the shrinkage and values of diffusivity as a function of moisture content should be taken into account. The drying process of plums can be studied by solving the second Fick’s law by numerical methods, considering moving boundary conditions and diffusion coefficient as a function of moisture content (Murray and Landis, 1959; Gabitto and Aguerre, 1985; Sobral, 1987). The plum is viewed as a composite spherical body comprised of two concentric materials (stone and flesh). The flesh component of the spherical plum is divided into N concentric spherical shells of equal thickness. The radial length of the fruit decreases due to shrinkage; consequently, the thickness of each shell reduces as drying progresses. The following dimensionless groups are necessary in order to have Equation (10.2) in a dimensionless form: X − Xe Xo − Xe

(10.3)

r , in (r > rs) rpo

(10.4)

Deff t r2

(10.5)

M= ξ=

Fo =

Substituting and rearranging the differential of Equations (10.3) and (10.4), and using the definition of Fourier mass number (Equation (10.5)), the second Fick’s law in spherical coordinates is developed and presented in the following form: ∂M 2 ∂M ∂2 M = + 2 ∂Fo ξ ∂ξ ∂ξ

(10.6)

with the initial and boundary conditions: Fo = 0, ξ = 1, ξ=


rs , rpo

M=1

(10.6a)

∂M = − Bi m Ms ∂ξ

(10.6b)

∂M =0 ∂ξ

(10.6c)

Equation (10.6) cannot be integrated, since it does not represent a moving boundary system, and shrinkage was not considered. In order to allow its integration, the following substantial derivative is defined: dM ∂M ∂M ∂ξ = + dFo ∂Fo n ∂ξ ∂Fo n

(10.7)

as well as the expression that relates the position of an arbitrary nodal point with the interface position: ξ=

n ξ N p

(10.8)

Equation (10.8) can be derived to give: ∂ξ n ∂ξ p = ∂Fo N ∂Fo

(10.9)

Substituting equations (10.7), (10.8) and (10.9) in Equation (10.6), the representative differential equation of the moving boundary system is: dM  2 N n  ∂ξ p   ∂M ∂2 M = + + 2 dFo  nξ p N  ∂Fo   ∂ξ ∂ξ

(10.10)

with the initial and boundary conditions: Fo = 0, ξ = ξp, ξ=

M=1

(10.10a)

∂M = − Bi m Ms ∂ξ

(10.10b)

∂M =0 ∂ξ

(10.10c)

rs , rpo

Equation (10.10) represents the second Fick’s law expressing the drying process considering shrinkage and diffusivity varying with average moisture content. Considering that this equation can be expressed in finite differences, it is solved using Equations (10.11), (10.12), and (10.13):


dM M n ,i +1 − M n ,i = dFo ∆Fo

(10.11)

∂M M n +1,i +1 − M n −1,i +1 = ∂ξ 2 ∆ξ

(10.12)

∂2 M M n −1,i +1 − 2 M n ,i +1 + M n +1,i +1 = ∂ξ 2 ∆ξ 2

(10.13)

Combination of equations (10.10), (10.11), (10.12) and (10.13) leads to: + M n −1,i +1  M M n ,i +1 − M n ,i = λ 2 k  n +1,i +1  + λ1 (M n −1,i +1 − 2 M n ,i +1 + M n +1,i +1 ) (10.14)   2 where: λ1 =

∆Fo ∆ξ 2

(10.14a)

λ2 =

∆Fo ∆ξ

(10.14b)

k=

2 N n  ∂ξ p  + nξ p N  ∂Fo 

(10.14c)

i = temporary coordinate index; n = space coordinate index; with the following initial and boundary conditions: Fo = 0, ξ = ξp, ξ=

rs , rpo

M=1

(10.14d)

(M N − M N −1 ) = − Bi m M N ∆ξ

(10.14e)

M0 = M1

(10.14f)

A rearrangement of Equation (10.14) will result in: λ k λ k   − λ1 + 2  M n −1,i +1 + (1 + 2λ1 )M n ,i +1 −  λ1 + 2  M n +1,i +1 = M n ,i (10.15)   2  2  In this way, the integration of Equation (10.2) consists of the solution of a system constituted by the development of Equation (10.15) in a matrix form (n = 1 to n = N – 1). For evaluation of Equation (10.14c), the particle radius and its derivation are determined by applying the finite difference: ∂ξ p ∂Fo

=

rpi − rpi −1 ∆Fo ⋅ rpo

(10.16)

The particle radius at a time ti and ti−1 is calculated from the relation between volume and moisture content experimentally obtained for the total drying time. The ∆Fo term is evaluated as follows: ∆Fo =

Deff ( x ) ∆t 2 rpo

(10.17)

where ∆t is a time interval between the sample collection in the drying experiments, and Deff ( x ) (effective moisture diffusivity) is calculated through an iterative


procedure aiming to adjust the experimental and calculated values of average residual moisture. The group of equations (10.14a, b, c) was completed considering ∆ξ equal to 1/N, since the nodal points number is constant. In order to apply the boundary condition corresponding to Equation (10.14e), it is necessary to evaluate the Biot mass number (Bim) by assuming a linear relationship between the food material and the air moisture at equilibrium (Córdova-Quiroz et al., 1996): k c K eq rpo

Bi m =

(10.18)

Deff

Y∗ = K eq X

(10.18a)

The parameter Keq represents an average partition constant. It may be evaluated with the equation suggested by Hernández et al. (2000):

K eq

∫ =

Xe

(Y∗ / X)dX

Xo

(10.18b)

Xo − Xe

where Y∗ =

a w ( Pvap / P )

18 1 − a w ( Pvap / P ) 28

(10.18c)

Data on water activity can be obtained from Gabas et al. (2000), showing the desorption isotherms of plum pulp described by the Guggenheim–Anderson–de Boer (GAB) model. The mass transfer coefficient in Equation (10.18) can be calculated by an empirical equation (Equation (10.19)) applied to mass transfer between spheres and air for turbulent flow in fixed bed (Saravacos, 1986): 1

Sh = 2 + 0.6 Re p 2 Sc

1

3

(10.19)

The water vapor diffusivity in the air is evaluated by the Schmidt correlation (Dãscãlescu, 1969): D vap = 0.083

(Ta + 273)1.81 273

(10.20)

A computational program written in Fortran is applied to the numerical integration of Equation (10.2). Equation (10.15) is written in a matrix form and solved using the Gauss-elimination procedure, obtaining the moisture profile. The spaceaveraged moisture content in a plum material is given by:

X =


∫

rp

rs

X drp rp − rs

(10.21)

and, in the dimensionless form:

∫ M =

ξp

ξs

M d ξp

ξ p − ξs

(10.22)

Equation (10.22) is integrated by using the composite Simpson’s rule to calculate the average moisture content.

10.3 METHODOLOGY 10.3.1 MATERIAL All the experimental measurements were made with plums of the Angeleno variety acquired in a local market and kept at 7°C prior to use. The plums were pretreated in a solution of 1.5% ethyloleate for 1 min at 50°C and then rinsed in a current of water for 2 min.

10.3.2 DRYING APPARATUS

AND

CONDITIONS

The dryer used to conduct the experiments is a pilot-scale piece of equipment consisting of three basic parts: an airflow rate control system, a drying air heating section, and a drying chamber; it is equipped with a process control system based on Fieldbus technology supplied by SMAR Industrial Equipment Ltd. As indicated in Figure 10.3, a centrifugal fan (1) was used to force the air through the drying chamber. The fan was driven by an electric motor with the airflow rate being controlled by a frequency modulator (2) (Siemens, model MMV). An orifice plate (3) connected to a pressure transmitter (SMAR, model LD302) (4) was installed after the fan to measure the airflow during the process. Dry bulb (5) and wet bulb (6) temperatures of the air stream were measured online using temperature transmitters (SMAR, model TT302). The drying air was heated by passing through electric resistances (7) controlled by a power converter (8) (Therma, model TH 6021A/80). Before the drying chamber, a “beehive” (9) was installed for better distribution of air in the product. The drying compartment (10) consisted of two square metal trays, placed perpendicular to the airflow. These compartments were designed for experiments of through drying. Thermocouples connected with temperature transmitters (SMAR, model TT302) were placed before and after the trays (11,14), and at the center (12) and surface (13) of the plums to analyze the evolution of temperature in the product during the experiments. In order to integrate the instruments with the Fieldbus system, a 4–20 mA current converter (SMAR, model IF302) (15) was used. The equipment operation was monitored and controlled by means of a computer  (16) running the software AIMAX -WIN. The drying experiments were carried out in air at temperatures in the range of 50 to 80°C and with an air velocity of approximately 1.5 m/sec.


14

16

10 13 12 11

9

15

6

5

4 2

7

8 3 1

FIGURE 10.3 Experimental drying apparatus: (1) centrifugal fan, (2) frequency modulator, (3) orifice plate, (4) pressure transmitter, (5) dry bulb temperature, (6) wet bulb temperature, (7) electric resistance heaters, (8) power converter, (9) beehive, (10) square metal trays, (11) thermocouple before the trays, (12) thermocouple at the center of the plum, (13) thermocouple at the surface of the plum, (14) thermocouple after the trays, (15) current converter, (16) computer.

10.3.3 SHRINKAGE KINETICS The shrinkage kinetics of the particles was used to correlate instantaneous size (spherical equivalent diameter) as a function of material moisture content. The dimensional change of the plum was assumed to occur only in the flesh section (rs < r < rp), and the volume of the stone component was considered constant throughout the drying process. Samples were placed on the tray in a single layer. The volume change was determined with about 48 plums, from which samples of three were taken out periodically from the dryer, to measure the total volume by the water displacement method. Moisture content of the samples was determined by drying at 60°C under vacuum for 48 h (AOAC, 1990).

10.4 SHRINKAGE AND EFFECTIVE DIFFUSIVITY The volume change or shrinkage during thin layer convective air drying of plums is described as a linear function of moisture content by the following equation: V/Vo = A + BX

(10.23)

Table 10.1 shows the constants A and B at different temperatures and the coefficient 2 of determination (r ), which presented satisfactory values. This equation is used to estimate the particle radius of the plum in the numerical solution described previously.


TABLE 10.1 Constants of Equation (23), Which Describes the Volume Change during Drying of Plums at Different Temperatures o

T ( C)

A

50 60 70 80 a

a

B

0.174 0.181 0.191 0.193

a

r

0.172 0.174 0.175 0.178

2

0.998 0.988 0.997 0.998

All fits had a significance level less than 5%.

1.0

0.8

V/Vo

0.6

0.4

T = 80°C T = 70°C T = 60°C

0.2

T = 50°C

0.0 0

1

2

3

4

5

Moisture content (dry basis)

FIGURE 10.4 Shrinkage during drying of plums.

Figure 10.4 shows the volume change of the plums at 50–80°C, where considerable shrinkage is observed, although with little influence of temperature. This can be attributed to the viscoelastic nature of the material, which suffers a structural collapse during drying. Similar results were obtained for thin layer drying of Thompson seedless grapes by Rhagavan et al. (1995), who found a linear relationship of volume change as a function of the moisture content of the berries. Simal et al. (1996) employed a diffusivity model with moving boundary conditions to simulate the drying kinetics of seedless grapes of the Flame variety, with the volume change represented by plotting V/V0 vs. X/X0 and by fitting a linear equation with similar constants. The initial moisture content was similar for all samples of plums. The influence of air drying temperature on the drying curves can be observed in Figure 10.5. As expected, increases in the air temperature substantially shortened the drying time in o comparison with that observed at 50 C. The drying time required to evaporate 90% o of water was shorter by 39.8, 57.7, and 70.5% at 60, 70, and 80 C, respectively, o compared to that at 50 C. Therefore, it is evident that at the same water content, the


4.0


3.5 3.0 T = 80°C

2.5

T = 70°C

2.0

T = 60°C T = 50°C

1.5 1.0 0.5 0.0 0

500

1000

1500

2000

2500

Drying time (min)

FIGURE 10.5 Drying curves of plums at different temperatures; symbols refer to experimental ) numerical solution from data; (— —) analytical solution from Equation (10.2); ( Equation (10.15).

drying rate is dependent on air temperature. Induction periods or constant drying rate periods were not observed, and the drying was characterized only by the falling rate period. Barbanti et al. (1994), writing about air drying kinetics of cultivar plums, also observed that there was no constant drying rate period in this process. The existence of only the falling rate period could be indicative of a diffusion-controlled mechanism of drying. It can be also seen in Figure 10.5 that the analytical model of Fick’s Law, considering radius and effective diffusivity constants, is not applicable to fit the experimental data in the final drying period. However, during the drying period fitted by this model, the plums had not reached the commercially desirable moisture content (approximately 0.15 kg water/kg dry matter). In addition, according to Figure 10.4, the radius of the plum is a function of moisture content, showing that shrinkage has to be taken into account during the drying process. The apparent agreement between the analytical solution of Equation (10.2) applied to plums and the experimental data in a great part of the drying curves (Figure 10.5) could be explained by supposing a simultaneous decrease in the effective diffusivity and in the radius, in such a way that these effects compensated for each other. The numerical solution fitted to the experimental data can be also seen in this figure. As expected, a good adjustment was obtained for all drying curves, including for the final drying period. From the numerical solution of Fick’s second law, the effective diffusivity as a function of simultaneous shrinkage and moisture content can be calculated. Figure 10.6 presents the effective diffusivity behavior with decreasing moisture content of the plums, and it can be observed that the effective diffusivity parameter is far from being constant during the process. The effective diffusivity is more influenced by o moisture content at 80 C than at other temperatures. On the other hand, neglecting o the initial inductive period, the effective diffusivity at 60 and 50 C changes almost


1.00E-009

Effective diffusivity (m2/s)

T = 80°C

8.00E-010

T = 70°C

6.00E-010 T = 60°C

4.00E-010

T = 50°C

2.00E-010

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5


FIGURE 10.6 Variation of effective diffusivity with moisture content of plums at different temperatures. 90 T = 50°C T = 60°C T = 70°C T = 80°C

Temperature (°C)

80 70 60 50 40 30 20 0

1

2

3

4


FIGURE 10.7 Evolution of temperature inside plums during drying.

linearly with decreasing moisture content. The increase of effective diffusivity in this initial period could be attributed to the time necessary for the temperature of the plum to increase until it reaches the drying process temperature. According to Figure 10.7, in general, the temperature of the fruit begins to reach equilibrium with the drying air at a moisture content of approximately 3.0 kg/kg dry mass. Sabarez et al. (1997) reported that the resistance to moisture transfer through the skin layer is important during drying of plums, particularly in the early stages of drying. These authors concluded that the skin layer limits the maximum effective evaporation rate from the plum surface and provides significant resistance to initial moisture loss.


Activation energy (kJ/mol)

39 38 37 36 35 34 33 32 0.0

0.5

1.0

1.5

2.0

2.5

3.0


FIGURE 10.8 Relationship between activation energy for water diffusion and moisture content in plums undergoing drying

Raghavan et al. (1995) obtained effective diffusivity values varying with moisture content under convective and microwave drying of grapes, and these values were found to decrease with decreasing water content. Effective diffusivity coefficients considering shrinkage but not varying with moisture content were reported by Hawlader et al. (1991) for tomatoes and by Simal et al. (1996), who obtained diffusivities of around –10 –9 2 –1 4.7 × 10 to 1 × 10 m sec for grapes. Sabarez and Price (1999) studied the drying of plums of the d’Agen variety and obtained Deff values in the range of 4.3 to 7.6 × –10 2 –1 10 m sec at 70, 75, and 80°C. The temperature dependence of the diffusivity can be represented by an Arrhenius type equation (Simal et al., 1996; Lewick et al., 1998; Simal et al., 2000). The activation energy for diffusion is calculated by taking a log plot of effective diffusivity at constant moisture content, against the reverse of the absolute air drying temperature. The relationship between activation energy and moisture content is presented in Figure 10.8. The tendency of increasing activation energy with decreasing moisture content is expected, and the majority of researchers have recognized this effect, especially for the low moisture content range. Satisfactory values of the determination coefficient (0.99) were obtained during the fitting procedure.

10.5 CONCLUSIONS The volumetric shrinkage of plums results in a linear correlation with respect to the moisture content. The finite difference method can be used for more accurate predictions and simulation of the drying process, and effective diffusivity values can be obtained as a function of moisture content, taking into account the shrinkage of the material. In general, these values are found to decrease with decreasing moisture content. The reported moisture diffusivity data at different temperatures falls between


–10

–10

2

1.2 × 10 and 8.9 × 10 m /sec. These results lead to the conclusion that shrinkage and moisture content cannot be neglected in establishing reliable values of effective diffusivity. The activation energy for diffusion is also dependent on the moisture content. It increases with decreasing water content in plums undergoing drying.

NOMENCLATURE A aw B Bim d DAB Deff Dvap Ea Fo Keq kc m M Ms N n P Pvap qA R Rep rs rp rpo Sc Sh T Ta t v V Vo X Xo Xe ∗ Y

Constant in Equation (10.23) Water activity Constant in Equation (10.23) Biot mass number ([kc ⋅Keq ⋅rpo]/Deff) Sphere diameter (m) 2 Diffusion coefficient (m /sec) 2 Effective diffusivity (m /sec) 2 Water vapor diffusivity in the air at a temperature Ta (m /h) Activation energy (kJ/mol) 2 Fourier mass number ([Deff ⋅t]/[rpo] ) Average partition constant Mass transfer coefficient (m/sec) Mass (kg) Residual moisture content Dimensionless moisture content at the surface of the plums Number of nodal points Arbitrary nodal point Total pressure (atm) Vapor pressure of water (atm) Chemical reaction in Equation (10.1) Universal gas constant (8.314 J/mol K) Modified Reynolds number ([ρ⋅v⋅d]/µ) Radius of the stone (kernel + shell) component (m) Radius of the whole plum (m) Radius of the plum at t = 0 (m) Schmidt number (µ/ρ⋅Dvap) Sherwood number ([kc ⋅d]/Dvap) Absolute temperature (K) Air temperature (°C) Time (sec) elocity (m/sec) 3 Volume (m ) 3 Initial volume (m ) Moisture content dry basis (kg/kg dry matter) Initial moisture content (dry basis) Equilibrium moisture content (dry basis) Molar fraction at equilibrium


GREEK SYMBOLS ξ Radial coordinate (m) ξp Dimensionless particle radius (rp(t)/rpo) 3 ρ Density (kg/m ) µ Viscosity (kg/m sec)

ACKNOWLEDGMENT The authors acknowledge financial support from São Paulo State Research Fund Agency, FAPESP (Processes: 98/12283–5; 98/08738–7 and 98/05130–8).

REFERENCES AOAC, Official Methods of Analysis, 15th ed., Association of Official Analytical Chemists, Washington, DC, 1990. Arrouz, S., Jomaa, W., and Belghith, A., Drying Kinetic Equation of Single Layer of Grapes, Drying ’98 — Proceedings of the 11th International Drying Symposium, August 19–22, 1998, B, pp. 988–997. Barbanti, D., Mastrocola, D., and Severine, C., Air drying of plums: a comparison among twelve cultivars, Sci. Aliment., 14, 61–73, 1994. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley & Sons, Inc., New York, 1960, pp. 554–560. Córdova-Quiroz, A.V., Ruiz-Cabrera, M.A., and García-Alvarado, M.A., Analytical solution of mass transfer equation with interfacial resistance in food drying, Drying Technol., 14, 1815–1826, 1996. Dãscãlescu, A., Le Sechage et ses Applications Industrielles, Dunod, Paris, 1969. Gabas, A.L., Menegalli, F.C., and Telis-Romero, J., Effect of chemical pretreatment on the physical properties of dehydrated grapes, Drying Technol., 17, 1215–1226, 1999. Gabas, A.L., Menegalli, F.C., and Telis-Romero, J., Water sorption enthalpy-entropy compensation based on isotherms of plum skin and pulp, J. Food Sci., 65, 680–684, 2000. Gabitto, J.F. and Aguerre, R.J., Solucion numerica del processo de secado com cambio de volumen, Revista Latinoam. Transf. Cal. Mat., 9, 231–240, 1985. Hawlader, M.N.A., Uddin, M.S., Ho, J.C., and Teng, A.B.W., Drying characteristics of tomatoes, J. Food Eng., 14, 259–268, 1991. Hernández, J.A., Pavón, G., and García, M.A., Analytical solution of mass transfer equation considering shrinkage for modeling food-drying kinetics, J. Food Eng., 45, 1–10, 2000. Karathanos, V.T., Determination of water content of dried fruits by drying kinetics, J. Food Eng., 39, 337–344, 1999. Lewick, P.P., Witrowa-Rajchert, D., and Nowak, D., Effect of drying mode on drying kinetics of onion, Drying Technol., 16, 59–81, 1998. Madamba, P.S., Optimisation of the drying process: an application to the drying of garlic, Drying Technol., 15, 117–136, 1997. Madamba, P.S., Driscoll R.H., and Buckle, K.A., Shrinkage, density and porosity of garlic during drying, J. Food Eng., 23, 309–319, 1994. Murray, W.D. and Landis, F.L., Numerical and machine solutions of transient heat conduction problems involving melting or freezing, J. Heat Transfer, Trans. ASME, May 1959, 106–112.


Newman, G.M., Price, W.E., and Woolf, L.A., Factors influencing the drying of prunes 1. Effects of temperature upon the kinetics of moisture loss during drying, Food Chem., 57, 241–244, 1996. Price W.E., Sabarez, H.T., Laajoki, L.G., and Woolf, L.A., Dehydration of prunes: kinetic aspects, Agro-Food-Industry Hi-Tech, 8, 29–33, 1997. Raghavan, G.S.V., Tulasidas, T.N., Sablani, S.S., and Ramaswamy, H.S., A method of determination of concentration dependent effective moisture diffusivity, Drying Technol., 13, 1477–1488, 1995. Sabarez, H.T. and Price W.E., A diffusion model for prune dehydration, J. Food Eng., 42, 167–172, 1999. Sabarez, H.T., Price W.E., Back, P.J., and Woolf, L.A., Modelling the kinetics of drying of d’Agen plums (Prunus domestica), Food Chem., 60, 371–382, 1997. Sanjuán, N., Bermejo, M.V., Vivanco, D., Cañellas, J., and Mulet, A., Drying Kinetics of Moscatel Cultivar Grapes, Drying ’96 — Proceedings of the 10th International Drying Symposium, July 30 – August 2, 1996, B, pp. 1069–1076. Saravacos, G.D., Mass transfer properties of foods, in Engineering Properties of Foods, Rao, M.A. and Rizvi, S.S.H., Eds., Marcel Dekker, New York, 1986, pp. 169–221. Simal, S., Femenía, A., Llull, P., and Rosselló, C., Dehydration of aloe vera: simulation of drying curves and evaluation of functional properties, J. Food Eng., 43, 109–114, 2000. Simal, S., Mulet, A., Catalá, P.J., Cañellas, J., and Rosselló, C., Moving boundary model for simulating moisture movement in grapes, J. Food Sci., 61, 157–160, 1996. Sobral, P.J.A., Secagem de sangue bovino incorporado a proteína texturizada de soja, em leito fluidizado e em leito fixo, thesis, FEA, UNICAMP, 1987. Telis, V.R.N., Gabas, A.L., Menegalli, F.C., and Telis-Romero, J., Water sorption thermodynamic properties applied to persimmon skin and pulp, Thermochim. Acta, 343, 49–56, 2000. Zogzas, N.P., Maroulis, Z.B., and Marinos-Kouris, D., Densities, shrinkage and porosity of some vegetables during air drying, Drying Technol., 12, 1653–1666, 1994.


11

Modeling Dehydration Kinetics and Reconstitution Properties of Dried Jalapeño Pepper R. Olivas-Vargas, F.J. Molina-Corral, A. Pérez-Hernández, and E. Ortega-Rivas

CONTENTS 11.1 Introduction 11.2 Materials and Methods 11.3 Results and Discussion 11.4 Conclusions Nomenclature Acknowledgments References

11.1 INTRODUCTION Drying is a physical separation process that has the objective of removing a liquid from a solid phase by means of thermal energy. The liquid is generally water and is liberated by vaporization rather than by breaking chemical bonds between the liquid and the solid; i.e., the liquid is not chemically bound to the solid. In most industrial drying applications, it is neither necessary nor economically feasible to remove every vestige of water from the solid; thus, commercially dry solids will usually contain a certain amount of residual moisture, the amount of which is determined by a compromise between product quality and economic factors. In food and biological materials, low moisture levels are necessary to stop or slow the growth of spoilage microorganisms, as well as the occurrence of undesirable biochemical and enzymatic reactions. The fundamental principles underlying drying processes are, in general, those typical of the science of heat and mass transfer. Drying, as a unit operation in food and chemical engineering, is characterized by the separation, usually partial, of a


liquid contained within a solid by the process of vaporization of the liquid into a gas phase. The mechanism of the drying process, as controlled by the principle of heat and mass transfer, consists of the transport of mass from the interior of the solid to the surface, the vaporization of the liquid at or near the surface, and the transport of the vapor into the bulk gas phase. Simultaneously, heat is transferred from the bulk gas phase to the solid phase, where all or a portion of it provides for vaporization, and the remainder accumulates in the solid as a sensible heat. The overall rate by which the above sequence of steps takes place defines the drying rate and is inversely related to the drying time. Regardless of how heat is provided, the drying cycle generally consists of three steps: 1. Before any evaporation takes place, sensible heat must be added to the drying mass until the boiling point of the liquid under the given operating conditions is reached. 2. Once the boiling point is reached, evaporation takes place at a rate related to the moisture level in the solid, and this rate is normally constant over a certain moisture range. 3. At some condition, a critical moisture point is reached and the drying rate begins to fall. Constant rate drying is the first stage, in which drying occurs at or near the boiling point of the liquid. There is no resistance to vaporization, since the moisture appears on the surface of the solid, completely wetting the outer surface. The constant drying rate is proportional to the difference between the vapor pressure of the liquid covering the surface of the solid and that of the vapor surrounding the wetted solid. It has been found that the vapor pressure of the wetting liquid is very close to that of the pure liquid at the same temperature. At the critical moisture content of the solid, the drying rate begins to fall because there is not enough moisture to completely cover the solid surface. The reduced amount of moisture being evaporated comes from the interstices of the solid through the porous structure. Falling rate drying is controlled by the physical properties of the liquid and solid. The rates of movement of the liquid and its vapor depend on capillary size, glazing of the solid, pressure gradients between trapped liquid and vapor, and the environs of the solid, as well as on cracking, checking, etc. At the same time, the heat transfer rate to the interior of the solid is being slowed because of the receding boundary of the liquid-wetted portion of the particle. This boundary movement increases the resistance to heat flow because it reduces the thermal conductivity within the solid. Cracking and checking disrupt the paths of heat transfer, further reducing the rate. Moisture migration during drying has been explained by different models. Four mechanisms are recognized: capillarity, concentration gradient diffusion, vapor diffusion by pressure difference, and layer diffusion across solid–fluid interfaces (Brennan, 1994). Capillary forces are responsible for water retention in porous solids or those of rigid construction, while osmotic pressure is responsible in aggregates of fine powders and on the surface of the solid (Toei, 1983). It has been reported that


moisture migration by diffusion is the predominant mechanism in drying of many food products, such as vegetables (Geankoplis, 1983). Drying data are usually expressed as total weight of the material as a function of time during the drying process. These data can also be expressed in terms of drying rate by recalculation of some values. The moisture content is defined as the ratio of the amount of water in the food to the amount of dry solids, i.e.,: Xt =

Wt − Fs Fs

(11.1)

where Xt is the moisture expressed as weight of water/weight of dry solids, Wt is the total weight of the material at time t, and Fs is the weight of the dry solids. An additional important quantity used in designing drying processes is the free moisture content X, which can be evaluated considering the equilibrium moisture content Xeq, by the simple relation: X = X t − X eq

(11.2)

As previously mentioned, moisture contents can be recalculated to obtain drying rates. The rate of drying R can be expressed proportional to the change in moisture content as a function of time t, by the following relationship: R∝

dX dt

(11.3)

Individual values of dX/dt as a function of time t can be obtained from tangent lines drawn to the curve of X vs. t. Replacing the proportionality condition in Equation (11.3) by Fs /A, the drying rate can be represented by: R=−

 Fs   dX   A   dt 

(11.4)

where A is the drying surface area. Drying of fruits and vegetables follows conventional dehydration theory, which was adapted from drying of inorganic materials and other raw materials in the chemical processing industry (Sarvacos and Charm, 1962). The advantages of reducing moisture content below 5% in biological materials are well known: microbial and enzyme activity are practically stopped, and transport costs are greatly reduced. However, fruits and vegetables can also suffer undesired changes due to the drying process. Shrinkage, case hardening, texture damage, and reconstitution difficulty can be caused by drying (Brennan, 1994). Dehydrated vegetables, when reconstituted, cannot match the original texture even if precautions are taken during the entire drying process. The large scale of industrial drying makes the analysis of the process difficult. Operators and process engineers often have only a few variables to use in formulating


any changes needed to control the operation. Typically, the feed rate is held constant and the controlling variable is the drying medium temperature. The operator may use exaggerated safety factors to prevent product build-up or over-drying. Mathematical modeling may be considered an efficient tool to overcome inefficiencies of drying plants such as damaged product, wasted energy, wear on the dryer, or decreased throughput. Models to describe dehydration of foods are necessary for process design, improvement, and minimization of energy while keeping quality as unimpaired as possible. Development of mathematical models to describe drying of porous solids has been the topic of many research studies for many years. Heat and mass transfer take place in a porous solid when it contains moisture and is subjected to any or all of the gradients of concentration, partial vapor pressure, temperature, total pressure, and external force fields. There has been no general agreement among researchers on which driving forces predominate, considering different materials or media and external drying conditions. More and more sophisticated drying models are becoming available based on either a mechanistic classical approach or on nonequilibrium thermodynamics. However, a major question that still remains is the determination of the best driving forces and assumptions to be made for drying model formulation. Another concern should be the determination of the coefficients and parameters used in the model. The measurement or determination of the necessary coefficients should be feasible and practical for general applicability of a drying model. The research described here was directed towards developing a predictive mathematical model of jalapeño pepper drying and thermal damage, due to the economic importance of this commodity, as well as its potential commercialization value as a processed product.

11.2 MATERIALS AND METHODS Fresh green jalapeño peppers (Capsicum annuum L.) obtained from local markets were used for the experiments. They were selected manually in order to obtain peppers in good condition of uniform size (6–7 cm), and then sliced to an approximate thickness of 5 mm using an electrical cutting machine (S.A. Bertuzzi; Brugherio, Milano, Italy). The jalapeño slices were washed to remove seeds and allowed to drain for 10 min. Samples of 180 g were placed over nine trays of a locally manufactured dryer, diagrammatically illustrated in Figure 11.1. Air was heated by a gas burner model HP225B LP (Adams Manufacturing Company; Cleveland, OH) and directed in parallel direction towards the peppers at 5 m/sec and temperatures of 60, 70, and 80°C. The drying time was 150 min. Dry bulb temperatures were recorded using thermometers located at different positions, as shown in Figure 11.1. Air velocity was measured using a Pitot tube (Dwyer Instruments, Inc.; Michigan City, IN) connected to a differential inclined manometer according to ASTM (1995) norm D3154. Moisture changes were determined by weight difference according to AOAC (1990) method 6.004. To determine a rehydration ratio, samples of 5 g dried slices were immersed in 400 ml water baths. The water baths were set at 75°C, and the immersion time was 6 min.


198 cm Trays

Thermometers

Air valve

206 cm

Exhaust

Burner

64 cm

83.5 cm

49.5 cm

FIGURE 11.1 Schematic diagram of the tray dryer used for the experiments.

The results obtained were used to derive drying and rehydration kinetics predictive models by means of regression analysis, using the Statistica 5.5 software (StatSoft, Inc., Tulsa, OK).

11.3 RESULTS AND DISCUSSION In fundamental drying research of food materials, ideally a model would contain a minimum of restrictive assumptions, and all material property values needed for the model would be obtained. The prediction of the drying characteristics based on knowledge of composition and structure is complex due to the nature of solid structures and the limited understanding of the moisture transport mechanism. An alternative is the identification of the relevant driving forces and the use of independent experimental tests that isolate the different driving forces to determine the necessary model coefficients. The model should have the most possible theoretical meaning and require the minimum of experimental tests to determine the coefficients. The model developed in this work was derived according to the characteristics described above. Drying kinetics were determined and the data were handled in a mathematical way to evaluate the best fitting equations describing the drying curves. To achieve this, drying conditions had to be controlled very carefully to ensure constant velocity, constant moisture content, and constant temperature of the flowing air. When these conditions were set, it was assumed that the drying rate represented by Equation (11.4) could also be established entirely in terms of the moisture gradient


between the sample and the surrounding drying air, i.e., dX R = K y (X R − Xe ) dt

(11.5)

where XR is the moisture content on a dry basis of the vegetable, Ky is a coefficient governing the mass transfer, and Xe is the moisture content of the air. To give the model a generalized characteristic, the moisture content was normalized using the following dimensionless relation: Θ=

XR − Xe X0 − Xe

(11.6)

where X0 is the moisture content of the material at the initial time, i.e., t = 0. It can be demonstrated (Molina-Corral, 1988) that Equation (11.5) can be rearranged by incorporating the normalized moisture content described by Equation (11.6), to give the following relationship:  X − Xe  d  XR − Xe  = Ky  R    dt  X 0 − X e   X0 − Xe 

(11.7)

The normalized moisture content Θ can be made explicit and transposed to give: dΘ = K y dt Θ

(11.8)

Integrating Equation (11.8), the following relationship is therefore obtained: ln Θ = K y t + C

(11.9)

where C is the integration constant. Taking natural logarithms on both sides of Equation (11.9) transforms it into the following exponential expression: Θ=e

K y +C

(11.10)

Using logarithm’s laws it can be demonstrated (Molina-Corral, 1988) that Equation (11.10) simply becomes the one presented below, which relates the variation of moisture content of the material Θ as a function of time t and temperature T, i.e., Θ=e


Kyt

(11.11)

TABLE 11.1 Regression Parameter Values for the Dehydration and Rehydration Models from Experimental Data Parameter

Dehydration Model

β1 β2 β3 β4 β5 β6

5.209 × 10 −2 3.730 × 10 −2 1.550 × 10 0.0 −4 1.000 × 10 −4 6.000 × 10

−1

Rehydration Model −2

−7.3340 × 10 −5 1.6000 × 10 −3 2.5004 × 10 0.0 −5 −2.4360 × 10 0.0

Equation (11.11) represents a one-exponential model known as the thin film drying model. Such a model basically states that the drying constant, Ky in this case, is a combination of the transport properties involved in drying and describes with reasonable accuracy the drying kinetics of hygroscopic materials. When the previously mentioned constant conditions of velocity, moisture, and temperature of the drying air are ensured, the transport properties most influential in drying of vegetables are time and temperature. The correlation exercises practiced in accordance with the experimental data demonstrated that the best fitting model for the coefficient governing the mass transfer Ky was a quadratic one expressed as: K y = β1 + β2 t + β3T + β 4 t 2 + β5T 2 + β6 tT

(11.12)

where the parameters β1, β2,…, β6 are those presented in Table 11.1. The model for predicting moisture content as a function of time and temperature was therefore given by: Θ = e(β1 +β2 t +β3T +β4t

2

+β5T 2 +β6 tT ) t

(11.13)

Figure 11.2 shows the experimental drying curves contrasted with the correspondinc correspondent ones obtained using Equation (11.13) above at the three temperatures tested. As can be seen, the mathematical model fit the experimental 2 data very well, judging by the high given values of the coefficient of determination R . As stated earlier, moisture migration by diffusion is the predominant mass transfer mechanism in drying of fruits and vegetables. It has also been reported (Toledo, 1991) that diffusivity may be constant if cells do not collapse and pack together. This behavior would be identified in firm solids such as grains, or in high moisture products such as fruits and vegetables when physical changes such as water removal are minimal. As shown in Figure 11.3, the presence of only a falling rate period for all the conditions tested was confirmed. This trend was in agreement with the work reported by Turhan


1 0.8 0.6 S = 0.023

0.4

R2 = 0.99

0.2 0 0

30

60

90

120

150

(a)

Moisture content

1 0.8 0.6 S = 0.031

0.4

R2 = 0.99

0.2 0 0

30

60

90

120

150

(b)

1 0.8 0.6 0.4

S = 0.020 R2 = 0.99

0.2 0 0

30

60

90

120

150

(c)

Time (minutes) FIGURE 11.2 Experimental (䉱) and calculated (—) drying curves for 60 (A), 70 (B), and 80°C (C) dehydration temperatures.


1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

Drying rate (kg/h·m2)

(a)

2 1.5 1 0.5 0 0

0.2

0.4

(b)

2.4 2 1.6 1.2 0.8 0.4 0 0

0.2

0.4

(c)

Moisture content

FIGURE 11.3 Experimental drying rate curves for 60 (A), 70 (B), and 80°C (C) dehydration temperatures.

and Turhan (1997), who found that peppers dehydrated without previous blanching presented only a falling rate period in their drying kinetics. The drying curves presented in Figure 11.2 also show the difference in drying velocities according to the applied temperature. It can be observed that drying


kinetics were higher, with a similar trend, for the processes carried out at 70 and 80°C, but slower for the minimum temperature of 60°C. Thus, it can be argued that drying kinetics were optimal at any of the higher temperatures, and that these levels of heat were moderate enough to prevent undesirable phenomena such as case hardening. It has been observed that in drying of some fruits, vegetables, meat, and fish, as drying temperatures approach the water boiling range, a hard and impermeable film develops on the surface (Brennan et al., 1979). Case hardening can be more severe during final stages of drying, where more than one falling rate period is presented. Since there was only one falling rate period in this study, case hardening was not a problem and drying kinetics developed quite well. However, as more undesirable phenomena can be caused by high drying temperatures, it would be advisable to use 70°C as the dehydration processing temperature in further applications, as drying behavior was similar at either 70 or 80°C. In addition to case hardening, shrinkage and texture damage can be caused by drying of vegetables, resulting in poor appearance and loss of some sensory attributes. The amount of thermal damage in drying of vegetables is difficult to evaluate, but it may be related in some way to structural alteration of the tissues due to excessive heat. The degree to which a dehydrated sample will rehydrate is influenced by such structural and chemical changes. Rehydration is maximized when cellular and structural disruption is minimized. Several measures can be taken to minimize structural alterations in order to improve the reconstitutability of dried food products. The drying method and adjustment of drying conditions can result in a product with good rehydration properties. For example, it has been reported that freeze-drying causes fewer structural changes than any other method, with minimal changes to the product’s hydrophilic properties (Heldman and Singh, 1981). This procedure results in obtaining food pieces with an open pore structure that will absorb water easily when reconstituted. However, freeze-drying is an expensive form of dehydration for foods due to the slowness of the drying rate and the use of vacuum. Since the vapor pressure of ice is very small, freeze-drying requires very low pressures or very high vacuum. For this reason, it is more feasible to try to obtain food products with good reconstitution properties by manipulating the operating variables of dryers. One possibility is to partially dehydrate fruits by immersion in sugar solutions to promote water migration by osmotic pressure difference, then give them a gentle drying cycle in order to obtain crisp products that rehydrate easily (Beltrán-Reyes et al., 1996). Also, the blanching method and drying temperatures can have an effect on reconstitution properties of vegetables (Ortega-Rivas et al., 1997). In this work, the amount of damage produced by the different drying conditions was determined by estimating the rehydration ratio RR using the following relation: RR =

w0 wf

(11.14)

where w0 is the weight of the dry sample and wf is the weight of the rehydrated sample.


Rehydration relation

1 0.9

60°C

0.8 0.7

70°C

0.6 80°C

0.5 0.4 0.3 0.2 0.1 0 0

20

40

60

80

100

120

140

Drying time (minutes)

FIGURE 11.4 Experimental rehydration curves.

Rehydration rates, estimated by Equation (11.14), are shown in Figure 11.4. The best fitting model for the rehydration ratio, obtained using a procedure similar to that in deriving Equation (11.13), was: RR = e(β1 +β2 t +β3T +β4t

2

+β5T 2 +β6 tT ) t

(11.15)

The parameters β1, β2,…, β6 are given in Table 11.1. As can be observed in Figure 11.4, rehydration kinetics showed a trend opposite to that of the drying process, i.e., the dried peppers rehydrated better at the lower drying temperature and at the shorter treatment times. In general terms, the worst rehydration characteristics were demonstrated by the peppers dried at 80°C. This behavior could be due to the extent of tissue damage, which is expected to be more severe at higher drying temperatures. It has been reported (Charm, 1971) that excessive thermal damage, due to high drying temperatures, may be responsible for poor appearance and loss of texture of dehydrated biological materials.

11.4 CONCLUSIONS A mathematical model to describe dehydration of jalapeño pepper at normal operating conditions was developed. The model describes the drying process in terms of the moisture migration as a function of time at different temperatures. The effect of thermal damage as a function of rehydration capacity is also included. The regression equations describing the model were highly correlated. The model can be used to opt for the best compromise between drying capability and product quality. Application of the model to a material similar to the one tested in the experiments would indicate that the use of relatively high temperatures would make a product practically bone dry, but with irreversible thermal damage, as its capacity to absorb water in a reconstitution process would be impaired. A moderate drying temperature, on the other hand, would result in a product with low moisture content and appropriate rehydration properties. Since the model development was based on careful experimental


conditions and the use of fresh materials of random composition, it can be applied with confidence to conditions different from those used at the University of Chihuahua. The model is easy to adapt, and simple calculations could provide guidance in choosing processing times and temperatures in advance as a function of the expected properties of the product, avoiding waste of time, energy, and raw materials.

NOMENCLATURE A β C Fs Θ Ky R RR t T w0 wf Wt X Xe Xeq XR Xt X0

2

Drying surface area, m Parameter of the regression model Integration constant Weight of dry solids, kg Normalized moisture content 2 Coefficient governing the mass transfer, kg mol/sec m 2 Rate of drying, kg water/h m Rehydration relation Time, min Temperature, °C Weight of dry sample, g Weight of rehydrated sample, g Total weight of wet material, kg Free moisture content, kg water/kg dry solid Moisture content of the air, kg water/kg dry air Equilibrium moisture content, kg water/kg dry solid Moisture content of vegetable, kg water/kg dry solid Total moisture content, kg water/kg dry solid Initial moisture content, kg water/kg dry solid

ACKNOWLEDGMENTS The authors thank the National Council for Science and Technology (CONACyT) as well as the Program for Academics Improvement (PROMEP), México, for providing funding to the project.

REFERENCES AOAC, Official Methods of Analysis, 15th ed., Association of Official Analytical Chemists, Washington, DC, 1990. ASTM, Standard test method for average velocity in a duct (Pitot tube method), Annual Book of ASTM Standards, Vol 11.03, 1995, pp. 89–92. Beltrán-Reyes, B., Ortega-Rivas, E., and Anzaldúa-Morales, A., Characterization of reconstituted apple paste in terms of rehydration and firmness, Food Sci. Technol. Int., 2, 307–313, 1996. Brennan, J.G., Food Dehydration: A Dictionary and Guide, Butterworth-Heinemann, Stoneham, MA, 1994.


Brennan J.G., Butters, J.R., Cowell, N.D., and Lilly, A.E.V., Food Engineering Operations, 2nd ed., Applied Science, London, 1979. Charm, S.E., The Fundamentals of Food Engineering, 2nd ed., AVI, Westport, CT, 1971. Geankoplis, C.J., Transport Processes and Unit Operations, Allyn and Bacon, Boston, MA, 1983. Heldman, D.R. and Singh, R.P., Food Process Engineering, Van Nostrand Reinhold, New York, 1981. Molina-Corral, F.J., Development of a Mathematical Model of Jalapeño Pepper Drying in a Tray Dehydrator with Gas Heating Source, thesis, University of Chihuahua, México, 1988. Ortega-Rivas, E., Moreno-Pérez, L.F., and Gasson-Lara, J.H., Effect of blanching and drying temperatures on rehydration rate and texture of sweet potatoes, in Advances in Food Engineering, Narsimhan, G., Okos, M.R., and Lombardo, S., Eds., Purdue Research Foundation, West Lafayette, IN, 1997, pp. 81–85. Sarvacos, G.D. and Charm, S.E., A study of the mechanism of fruits and vegetables dehydration, Food Technol., 16, 78–81, 1962. Toei, R., Drying mechanism of capillary porous bodies, in Advances in Drying, vol. 2, Mujumdar, A.S., Ed., Hemisphere, New York, 1983. Toledo, R.T., Fundamentals of Food Process Engineering, 2nd ed., Van Nostrand Reinhold, New York, 1991. Turhan, M. and Turhan, K.N., Drying kinetics of red pepper, J. Food Process. Preserv., 21, 209–223, 1997.


12

Application of an Artificial Neural Network for Moisture Transfer Prediction Considering Shrinkage during Drying of Foodstuffs J.A. Hernández-Pérez, M.A. Garcia-Alvarado, G. Trystram, and B. Heyd

CONTENTS 12.1 Introduction 12.2 Materials and Methods 12.2.1 Neural Network Systems 12.2.2 Learning Algorithm 12.2.3 Database Preparation 12.3 Results and Discussion 12.3.1 Learning Bases of the Model 12.3.2 Validation of the Proposed Model 12.4 Conclusions Nomenclature References

12.1 INTRODUCTION Air drying is an essential activity in food processing industries. On-line state estimation and control of the air drying operation require the mathematical description of food moisture evolution during the process. The dynamics of the food drying process involve simultaneous heat and mass transfer, where water is transferred by diffusion from the inside of food material towards the air–food interface and from the interface to the air stream by convection;


heat is transferred by convection from the air to the air–food interface and by conduction to the interior of the food (Balaban and Pigott, 1988; Karathanos et al., 1990; Kiranoudis et al., 1993; Zogzas and Maroulis, 1996). This phenomenon has been modeled with different levels of complexity; existing models do not permit adequate control of the air drying process in industrial applications, mainly because physical dynamic models (or detailed models), considering the complexity of the process, usually result in coupled nonlinear differential equations with partial derivatives, where numerical simulations require specialized software and are very time consuming. It is important to mention that these detailed models are able to predict within a wide range. The dynamic models can be simplified (Courtois et al., 1991; Hernández et al., 2000), but these models cannot take into account the complexity of the process, and some still contain ordinary nonlinear differential equations that take too long to simulate for applications of control (Trelea et al., 1997b). Empirical model representations approximate the drying kinetic by several line segments (Daudin, 1982), high order polynomials (Techasena et al., 1992), neural networks (Dornier et al., 1993), etc. These models have a narrower validity range but require only a limited number of simple arithmetic operations for simulation and can be easily incorporated in control software. Shrinkage is an important factor in mathematical models of heat and mass transfer for the prediction of food drying kinetics. Mulet (1994) tested models with different levels of complexity, concluding that the main factor that must be considered in models is product shrinkage. Models that take into account the shrinkage of the product are models in which the volume is considered as a function of the content of moisture. The shrinkage of biological products during drying is not perfectly homogeneous (Ratti, 1994; Hernández, et al., 2000). It is interesting to mention that there is a deviation of some predictions during food drying kinetics considering shrinkage at lower moisture levels (Ψ < 0.05); this is due to the possible conclusion of the shrinkage and the heterogeneous characteristics of the product that could be the product, and this phenomenon’s behavior is typical for foodstuffs (Hernández et al., 2000). The progress of neurobiology has allowed researchers to build mathematical models of neurons to simulate neural behavior. Neural networks are recognized as good tools for dynamic modeling and have been extensively studied since the publication of the perceptron identification method (Rumelhart and Zipner, 1985). The interest in such models lies in their modeling without any assumptions about the nature of underlying mechanisms and in their ability to take into account nonlinearities and interactions between variables (Bishop, 1994). Recent results establish that it is always possible to identify a neural model based on the perceptron structure, with only one hidden layer, for either steady state or dynamic operations (with recurrent models) (Hornik et al., 1989; Hornik, 1993). An outstanding feature of neural networks is the ability to learn the solution of the problem from a set of examples and to provide a smooth and reasonable interpolation for new data. Also, in the field of food process engineering, neural networks are a good alternative to conventional empirical modeling based on polynomial and linear regressions. For food processes, the neural computing application keeps on growing (Linko and Zhu, 1992; Huang and Mujamdar, 1993; Zhu et al., 1996; Linko et al., 1997; Trelea et al., 1997a; Hernández-Pérez et al., 2000).


The aim of the present work was to test the ability of neural networks to model and predict moisture transfer during air drying in foodstuffs. The model was validated with experimental drying data of cassava parallelepipeds and mango slices.

12.2 MATERIALS AND METHODS 12.2.1 NEURAL NETWORK SYSTEMS Neural networks are composed of simple elements operating in parallel. As in nature, the network function is determined largely by the connections between elements (neurons); each connection between two neurons has a weight coefficient attached to it. A neuron is grouped into distinct layers and interconnected according to a given architecture. The standard network structure for function approximation is the multiple layer perceptron (or feedforward network). The feedforward network often has one or more hidden layers of sigmoid neurons followed by an output layer of linear neurons. Multiple layers of neurons with nonlinear transfer functions allow the network to learn nonlinear and linear relationships between input and output vectors. The linear output layer lets the network produce values outside the range –1 to +1 (Demuth and Beale, 1998; Limin, 1994). For multiple layer networks, we use the number of layers to determine the superscript on the weight matrices. The appropriate notation is used in two-layer networks. A simple view of network structure and behavior is given in Figure 12.1. This network can be used as a general function approximator. It can approximate any function with a finite number of discontinuities, arbitrarily well, given sufficient neurons in the hidden layer. The numbers of neurons in the input and output layers are given by the number of input and output variables in the process under investigation. In this work, the input layer consisted of five variables in the process (air temperature, air velocity,

IW1{1,1}

Σ

n1

In2

Σ

n2

In k

Σ

ns

In1

LW2 {2,1}

b1 Input layer

Σ

Out

b2 Hidden layer

Output layer

FIGURE 12.1 The neural network computational model. k = number of inputs, In = inputs, Out = output, thick lines = weights and biases.


shrinkage function of the moisture, time, and air humidity), one for each input. The optimal number of neurons in the hidden layer(s) ns is difficult to specify and depends on the type and complexity of the task; this is usually determined by trial and error. Each neuron in the hidden layer has a bias b, which is added with the weighted inputs to form the neuron input n. This sum, n, is the argument of the transfer function f. n = IW{1,1}In1 + IW{1,2}In2 + ··· + IW{1, k}Ink + b The coefficients associated with the hidden layer are grouped into matrices IW1 (weights) and b1 (biases). The output layer contains one variable (moisture of the product) for mango and cassava, respectively, which computes the weighted sum of the signals provided by the hidden layer; the associated coefficients are grouped into matrices LW2 and b2. Using the matrix notation, the network output can be given by Equation (12.1). 2

1

Out = f (LW2f (IW1*In + b1) + b2)

(12.1)

Hidden layer neurons may use any differentiable transfer function to generate their 1 2 output. In this work we used as f Tan-Sigmoid Function and for f Linear Transfer Function. The number of network coefficients (weights and biases) is given by Equation (12.2). m = n(In + 1) + Out(n + 1)

(12.2)

12.2.2 LEARNING ALGORITHM A learning algorithm is defined as a procedure that adjusts the coefficients (weights and biases) of a network in order to minimize an error function (usually a quadratic one) between the network outputs for a given set of inputs and the correct outputs already known (this procedure may also be referred to as a training algorithm). If smooth nonlinearities are used, the gradient of the error function can be easily computed by the classical backpropagation procedure (Rumelhart et al., 1986). Previous learning algorithms used this gradient directly in a steepest descent optimization, but recent results have shown that second order methods are far more effective. In this work, the Levenberg–Marquardt (trainlm), optimization procedure in the Neural Network Toolbox of Matlab (Demuth and Beale, 1998) was used. The algorithm of Levenberg is an approximation of Newton’s methods; this algorithm was designed to approach second order training speed without having to compute the Hessian matrix (Hagan and Menhaj, 1994). Despite the fact that computations involved in each iteration are more complex than in the steepest descent case, the convergence is faster, typically by a factor of 100. The root mean square error (RMSE) between the experimental values and network predictions was used as a criterion of model adequacy, as shown in Figure 12.2.


Xa t Lv

Process

Va

Ψ

T°C

+ Shrinkage

+

RMSE

Optimization Method (Levenberg-Marquardt)

Neural Network (weights and bias)

ˆ Ψ

FIGURE 12.2 Recurrent network architecture for the drying kinetic considering the shrinkage of the product and procedure used for learning of neural network.

12.2.3 DATABASE PREPARATION Experimental data provided by Hernández et al. (2000), consisting of mass transfer kinetics of mango slices, 4 × 4 cm with three slab thicknesses (0.50, 0.75, and 1.0 cm), and cassava parallelepipeds, 10 cm long with three thicknesses (1.0, 2.0, and 3.0 cm), were used. The experimental drying kinetics were carried out at three different air temperatures and air velocities with a time of 10 h for each kinetic. The arrangement was a 3*3*3 (slab thickness * air temperature * air velocity) complete factorial design for mango and 3*3 (thickness * air temperature) complete factorial design for cassava at constant air velocity, with two replicates. Tables 12.1 and 12.2 give the experimental conditions studied for each kinetic of mango and cassava. Eperimental files were split into learning and test databases. The inputs of the network were air temperature (T), air velocity (Va), shrinkage (Lv), time (t), and air humidity (Xa); the output of the network was the food moisture content in dimenˆ ) for mango. The same network was used for cassava but without sionless form (Ψ the input (Va). The food moisture evolution during drying was calculated by sample weight loss of the product at times t = 0, 0.25, 0.50, 0.75, 1.0, 1.25, 1.50, 1.75, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, and 10.0 h. Wet and dry bulb temperatures of the ambient air were measured with glass thermometers to estimate the air moisture content (Xa). Equation (12.3) was used to determine the variation of shrinkage during kinetic drying of mango and cassava, as proposed by Hernández et al. (2000). This equation


TABLE 12.1 Experimental Conditions Studied for Mango File Number

Initial Thickness Lo (cm)

Air Temperature T (°C)

Air Velocity Va (m/sec)

Time t (h)

0.5 0.75 1.0 0.5 0.75 1.0 0.5 0.75 1.0 0.5 0.75 1.0 0.5 0.75 1.0 0.5 0.75 1.0 0.5 0.75 1.0 0.5 0.75 1.0 0.5 0.75 1.0

50 50 50 60 60 60 70 70 70 50 50 50 60 60 60 70 70 70 50 50 50 60 60 60 70 70 70

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 3 3 3 3 3 3 3 3 3

0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19 M20 M21 M22 M23 M24 M25 M26 M27

is a function of moisture content of the product.   X  Lv = L 0 ∆L f + (1 − ∆L f )    X 0   

(12.3)

where Lv represents the thickness (cm), L0 is the initial thickness (cm), ∆Lf represents the fraction of initial slab thickness at the end of drying (∆Lf = 0.55 for mango and ∆Lf = 0.68 for cassava), and X and X0 are the moisture and initial moisture of the product (g water/g dry matter). The selected experimental files were split into learning and test databases to obtain a good representation of the situation diversity. In each case:


TABLE 12.2 Experimental Conditions Studied for Cassava File Number

Initial Thickness Lo (cm)

Air Temperature T (°C)

Air Velocity Va (m/sec)

Time t (h)

1 1 1 2 2 2 3 3 3

50 60 70 50 60 70 50 60 70

3 3 3 3 3 3 3 3 3

0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10 0–10

C1 C2 C3 C4 C5 C6 C7 C8 C9

Mango Eighteen experimental kinetics in learning database (M1, M3, M5, M6, M7, M8, M10, M11, M14, M15, M16, M18, M20, M21, M22, M23, M25, and M27) Nine experimental kinetics in test database (M2, M4, M9, M12, M13, M17, M19, M24, and M26) Cassava Six experimental kinetics in learning database (C1, C2, C4, C6, C8, and C9) Three experimental kinetics in test database (C3, C5 and C7) The learning database was obtained to optimize the neural network, and the test database was reserved for the validation of the predictive capability of the model.

12.3 RESULTS AND DISCUSSION The proposed model for mango (Equation (12.4)) involved three neurons ns = 3 in the hidden layer (18 weights and 4 bias) to determine the moisture transfer in a dimensionless form. The same model, but without the input of air velocity (Va), was used for cassava (15 weights and 4 bias). This model includes the effect of shrinkage (Lv) as a function of the last moisture content. It is important to mention that the mango slices were dried on one side, and cassava parallelepipeds were suspended so drying took place on the four sides (two dimensions). The parallelepipeds were of the same size and drying was homogeneous on the four sides, so we used as input one only Lv for mango and cassava. ˆ represents the predicted moisture of the product in dimensionless form; X Ψ e is the moisture in equilibrium (g water/g dry matter). This moisture (Xe) was determined using the relationship of air moisture and the sorption isotherms that were experimentally obtained for mango and cassava in the laboratory (Equations (12.5) and (12.6)). The moisture levels (X/X0) of Equation (12.3) were determined for the last output network, which makes one closed boucle for (Lv), as represented in Equation (12.4).


Lv = L 0 ∆L f + (1 − ∆L f

)

T°

X X0

V L

(X − Xe ) ˆ = Ψ (Xo − Xe )

t

(12.4)

X bia Mango:

aw = 1 – exp[–exp(0.9154 + 0.5639lnX)]

Cassava:

aw = 1 – exp[–exp(–44.8 + 8.66lnT – 6.41lnX −5 2

+ 0.049TlnX – 6.7 × 10 T lnX)]

12.3.1 LEARNING BASES

OF THE

(12.5)

(12.6)

MODEL

Figure 12.3 gives the trial root mean square error (RSME) against the iteration number in the case of mango and cassava, for one to six neurons in the hidden layer. These results showed that the typical learning error decreased when the number of neurons in the hidden layer increased, but for three or more neurons in the hidden layer for mango, and two or more neurons for cassava, respectively, an additional increase in structure complexity did not strongly decrease the RSME. One problem that occurs during neural network training feedforward is called over-fitting. The RSME on the training set is driven to a very small value, but when new data are presented (e.g., test database) to the network, the RSME is large. Therefore, the RSME on the test database is a good criterion to optimize the number of iterations and avoid over-fitting. Figure 12.3 shows that for three (RMSE = 0.001408) or four (RMSE = 0.001323) neurons in the hidden layer, the error is very small for mango, whereas for cassava the same error is obtained with two (RMSE = 0.00149) or three (RMSE = 0.00148) neurons. In order to determine the number of neurons in the hidden layer, we plotted the RMSE (learning and test bases) against the number of neurons, as shown in Figure 12.4. This figure presents over-fitting because the error on the test database can increase with the number of neurons. The optimal number of neurons in the hidden layer is three for both mango and cassava. Figure 12.5 presents the simulated results against experimental data for test and learning bases of mango and cassava; in all cases the prediction was correct, whatever the level of moisture. However, at lower moisture contents an evident deviation between experimental and simulated data was observed for mango. In order to confirm these deviations, a plot of some drying experimental and predicted kinetics


1

10

Root-mean square error training

0

10

10

10

10

10

- one neuron

-- two neurons

-. three neurons

.. four neurons

- - five neurons

-o- six neurons

-1

MANGO -2

-3

-4

0

20

40

60

80

100

Levenberg-Marquardt 1

Root-mean square error training

10

- one neuron

-- two neurons

-. three neurons

.. four neurons

- - five neurons

-o- six neurons

0

10

10

-1

CASSAVA 10

10

-2

-3

0

20

40

60

80

100

Levenberg-Marquardt

FIGURE 12.3 Test root mean square error (RMSE) vs. number of iterations and various numbers of hidden neurons for mango and cassava.

(experimental M25, M27, C2, C8; see drying conditions in Tables 12.1 and 12.2) are presented in semilog scale in Figure 12.6. This figure shows that at Ψ < 0.05 the model cannot predict the drying curves. These deviations at lower moisture content are also observed with detailed models (Hernández et al., 2000).


10-1

MANGO

Root-mean square error

-o- Learn 10

-2

- - Test

10-3

10-4 1

2

3

4

5

6

5

6

Number of neurons 10-1

Root-mean square error

CASSAVA -o- Learn - - Test 10

-2

10-3

1

2

3

4

Number of neurons

FIGURE 12.4 Comparison of RSME learn and test vs. the number of neurons in the two cases (mango and cassava).

12.3.2 VALIDATION

OF THE

PROPOSED MODEL

Figure 12.7 depicts the ability of the models to predict drying kinetics at different thicknesses, temperature, and air velocities in the experimental domain. This figure shows some simulated results generated and experimental data obtained by the test database (M4, M12, M26, C3, C5, and C7; see Tables 12.1 and 12.2). It is evident that


1

Mango

learn

0.9

o test

0.8

Simulated

0.7 0.6 0.5 0.4

y = 0.0101 + 0.9838x

0.3

R=1

0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Experimental

1 learn

0.9 0.8

o test

Simulated

0.7 0.6 0.5

Cassava

0.4

y = 0.0144 + 0.9763x

0.3

R=1

0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Experimental

FIGURE 12.5 Simulated vs. experimental data for test and learning bases of mango and cassava.

the model was successful in predicting the experimental drying kinetics, showing the importance of the artificial neural network to simulate the drying curves of foodstuffs. These models are not complex, and they can be used for on-line estimation of air drying processes in industrial applications.

10 0

PSI = (X–Xe)/(Xo–Xe)

Mango

10 -1

T = 70°C

10 -2 o Lvo = 0.5 cm (M25)

* Lvo = 1 cm (M27) - Predict 0

2

4

6

8

10

8

10

Time (hr)

0


10

Cassava

-1

10

T = 60°C -2

10

o Lvo = 1.0 cm (C2) * Lvo = 3.0 cm (C8) -3

10

- Predict

0

2

4

6

Time (hr)

FIGURE 12.6 Experimental data and predicted results of the learning bases for mango and cassava.

12.4 CONCLUSIONS This study shows that neural network modeling can be used to obtain accurate simulation of the moisture transfer during food drying over a wide experimental range. This neural network modeling was validated with experimental drying data. Technological interest in this kind of modeling must be related to the fact that it is elaborated without any preliminary assumptions about the underlying mechanisms, and also to its implementation facility and speed of simulation. The applications of


1 * Lvo = 0.5 cm T = 60°C Va = 0.5 m/sec + Lvo = 1 cm T = 50°C Va = 1.75 m/sec o Lvo = 0.75 cm T = 70°C Va = 3m/sec - Prediction

0.9


0.8 0.7 0.6 0.5 0.4 0.3 0.2

Mango

0.1 0 0

2

1

6 Time (hr)

8

10

* Lvo = 1 cm T = 70°C Va = 3 m/s + Lvo = 2 cm T = 60°C Va = 3 m/s o Lvo = 3 cm T = 50°C Va = 3m/s - Prediction

0.9 0.8


4

0.7 0.6 0.5 0.4

Cassava

0.3 0.2 0.1 0

0

2

4

6 Time (hr)

8

10

FIGURE 12.7 Experimental data and simulated curves generated with the proposed model in the drying kinetics of mango and cassava. The symbols represent experimental moisture evolution at different conditions.

artificial neural networks can be used for on-line state estimation and control of drying processes.

NOMENCLATURE k In

Number of input Input variables


Out ns, n IW1, LW2 b1, b2 T Va Lv L0 t Xa X, X0 and Xe aw

Output variables Neuron in the hidden layer Matrix weight Matrix bias Air temperature (°C) Air velocity (m/sec) Thickness (cm) Initial thickness (cm) Time (h) Air moisture content (g water/g dry air) Food moisture content: in each time, initial, and in equilibrium, respectively (g water/g dry matter) Food water activity

GREEK SYMBOLS ˆ Dimensionless variables (experimental and predicted moisture) Ψ, Ψ ∆Lf Fraction of the initial characteristic length at the end of drying period (cm)

REFERENCES Balaban, M. and Piggot, G.M., Mathematical model of simultaneous heat and mass transfer in food with dimensional changes and variable transport parameters, J. Food Sci., 53, 935–939, 1988. Bishop, C.M., Neural networks and their applications, Rev. Sci. Instrum., 65, 1803–1832, 1994. Courtois, F., Lebert, A., Duquenoy, A., Lasseran, J.C., and Bimbenet, J.J., Modeling of drying in order to improve processing quality of maize, Drying Technol., 9, 927–945, 1991. Daudin, J.D., Modélisation d’un séchoir à partir des cinétiques expérimentales de séchage, Ph.D. thesis, ENSIA, Massy, France, 1982. Demuth H. and Beale M., Neural Network Toolbox for Matlab—User’s Guide, version 3, The MathWorks Inc., Natick, MA, 1998. Dornier, M., Rocha M.T., Trystram, G., Bardot, I., Decloux, M., and Lebert, A., Application of neural computation for dynamic modeling of food processes: drying and microfiltration, Artificial Intelligence for Agriculture and Food (AIFA Conf.), Nimes, France, 27–29 October, 1993, pp. 223–240. Hagan, M.T. and Menhaj M.B., Training feedforward networks with the Marquardt algorithm, IEEE Trans. Neural Networks, 6, 989–993, 1994. Hernández, J.A., Pavon, G., and Garcia, M.A., Analytical solution of mass transfer equation considering shrinkage for modeling food drying kinetics, J. Food Eng., 45, 1–10, 2000. Hernández-Pérez, J.A., Ramírez-Figueroa, E., Rodriguez-Jimenes, G., and Heyd, B., Spray Dried Yogurt Water Sorption Isotherms Prediction Using Artificial Neural Network, 12th International Drying Symposium, 28–31 August 28–31, 2000, p. 56. Hornik. K., Some new results on neural network approximation, Neural Network, 6, 1069–1072, 1993.


Hornik, K., Stinchcombe, M., and White, H., Multilayer feedforward networks are universal approximations, Neural Network, 2, 359–366, 1989. Huang B. and Mujumdar, A., Use of neural networks to predict industrial dryers’ performances, Drying Technol., 11, 525–541, 1993. Karathanos, V.T., Villalobos, G., and Saravacos, G.D., Comparison of two methods of estimation of the effective moisture diffusivity from drying data, J. Food Sci., 55, 218–223, 1990. Kiranoudis C.T., Maroulis Z.B., and Marinos-Kouris D., Heat and mass transfer modeling in air drying of foods, J. Food Eng., 26, 329–348, 1993. Limin, F., Neural Networks in Computer Intelligence, McGraw-Hill, New York, 1994. Linko, P. and Zhu, Y.H., Neural networks for real time variable estimation and prediction in the control of glucoamylase fermentation, Prog. Biochem., 27, 275–283, 1992. Linko S., Luopa, J., and Zhu, Y.H., Neural networks as “software sensors” in enzyme production, J. Biotechnol., 52, 257–266, 1997. Mulet, A., Drying modeling and water diffusivity in carrots and potatoes, J. Food Eng., 22, 329–348, 1994. Ratti, C., Shrinkage during drying of foodstuffs, J. Food Eng., 23, 91–105, 1994. Rumelhart D. and Zipner D., Feature discovering by competitive learning, Cognitive Sci., 9, 75–112, 1985. Rumelhart, D.E., Hinton, G.E., and Williams, R.J., Learning internal representations by error propagation, Parallel Data Process., 1, 318–362, 1986. Techasena, O., Lebert, A., and Bimbenet, J.J., Simulation of deep bed drying of carrots, J. Food Eng., 16, 267–281, 1992. Trelea, I.C., Courtois, F., and Trystam, G., Dynamic models for drying and wet-milling quality degradation of corn using neural networks, Drying Technol., 15, 1095–1102, 1997a. Trelea, I.C., Raoult-Wack, A.L., and Trystam, G., Note: application of neural network modeling for the control of dewatering and impregnation soaking process (osmotic dehydration), Food Sci. Technol. Int., 3, 459–465, 1997b. Zogzas, N.P. and Maroulis, Z.B., Effective moisture diffusivity estimation from drying data. a comparison between various methods of analysis, Drying Technol., 14, 1543–1573, 1996. Zhu, Y.H., Rajalahti, T., and Linko, S., Application of neural networks to lysine production, Biochem. Eng, J., 62, 207–214, 1996.


13

Modeling of Prato Cheese Salting: Fickian and Neural Network Approaches A.F. Baroni, M.R. Menezes, E.A.A. Adell, and E.P. Ribeiro

CONTENTS 13.1 Introduction 13.1.1 Mass Transfer Phenomena 13.1.2 Artificial Neural Networks 13.2 Materials and Methods 13.2.1 Cheese Processing 13.2.2 Cheese Sampling 13.2.3 Analytical Methods 13.3 Mathematical Modeling 13.3.1 Fickian Diffusion Model 13.3.2 Average Concentration 13.3.3 Neural Network Model 13.4 Results and Discussion 13.4.1 Cheese Composition 13.4.2 Salting Process 13.4.3 Performance of the Model Approaches 13.4.4 Statistical Analysis 13.4.5 Effective Diffusion Coefficients 13.5 Conclusions Acknowledgments References

13.1 INTRODUCTION The production of cheese in Brazil is quite recent, having been consolidated from an industrial standpoint in the beginning of the twentieth century, especially after the arrival of Danish and Dutch immigrants in the state of Minas Gerais in 1920


(Furtado, 1991). Prato cheese holds a leading position in Brazilian production with 57,758 tons produced yearly. Within the universe of Brazilian cheese, Prato has the highest technological standardization; it is the best characterized commercial cheese (Oliveira, 1986; Ribeiro, 1996). The expected average composition of ripened Prato cheese is 42 to 44% moisture, 26 to 29% fat, pH ranging from 5.2 to 5.4, and salt concentration 1.6 to 1.9% (Furtado and Neto, 1994). Schiftan and Komatsu (1980) carried out experiments on 50 commercial Prato cheese samples and obtained average values of 40% moisture, 47.5% fat in the dry matter, 1.7% salt, and 3.5% ash. Furtado and WolfschoonPombo (1979) reported the average composition of Prato cheese as 53.0 to 59.2% dry matter, 40.8 to 47.0% moisture, 25.3 to 30.5% fat and 47.7 to 51.5% fat in the dry matter.

13.1.1 MASS TRANSFER PHENOMENA Salting is one of the main steps of cheese production; thus, salt concentration distribution is an important parameter affecting cheese quality and acceptability. Salting plays a number of important roles in cheese, such as controlling the biochemistry of cheese ripening and the water activity of young cheeses. Salting affects the growth and survival of bacteria and the activity of cheese enzymes. Nowadays, brine salting is the most widely used salting technique in the Brazilian cheese industry. It is performed in an empirical way, and many times the distribution of the salt in the final product is irregular. Salt is absorbed and penetrates into the cheese, while an outer flux of water takes place. The amount of salt retained and water removal by the cheese depend, mostly, on brine concentration and brining time (Furtado and Neto, 1994). Fick’s law of diffusion is usually used to represent mass changes during brine salting of cheese. However, simplifications imposed on the model may affect its accuracy. Geurts et al. (1974) modeled the brine salting of Gouda cheese applying Fick’s law of diffusion. They obtained one-dimensional data on salt concentration and moisture profiles in cheese pieces, measured as a function of time. The authors noticed that the effective diffusion coefficient of salt within the cheese was considerably lower than in free solution, due to factors such as tortuosity, viscosity, and matrix restriction effects. However, Fick’s first law was not able to describe moisture flux during the brining step. An empirical linear relationship between water content and salt concentration was applied, resulting in a coefficient of proportionality of the dewatering process. Guinee and Fox (1983, 1987) found similar results for water and salt diffusion in Romano-type cheese. The same authors published a comprehensive review of salt and moisture diffusion of various cheeses in 1987. Luna and Chavez (1992) and Turhan and Kaletunç (1992) also investigated water movement during brine salting of cheese. Payne and Morison (1999) used a Maxwell–Stefan multi-component approach to model salt and water diffusion in cheese based on published data. This refined approach, taking into account diffusive interactions among salt, water, and cheese matrix, still requires a large number of assumptions during the modeling.


13.1.2 ARTIFICIAL NEURAL NETWORKS Artificial neural networks (ANNs) have been of interest because they enable direct modeling of nonlinear processes without requiring a prespecified detailed relationship. ANNs have been increasingly used in food and chemical engineering for several purposes. They have been applied recently in the prediction of quality attributes in beef (Park et al., 1994), the snack frying process (Huang et al., 1998b), and cheddar cheese (Muir et al., 1997); improvement of control loops in food processing (Huang et al., 1998a; Benne et al., 2000); classification problems, such as classification of beef (Park et al., 1994); and prediction of protein functionality (Arteaga and Nagai, 1993), moisture attained in a rehydration process of dried rice (Ramesh et al., 1996), pressure drop of non-Newtonian fluids in tubes (Adhikari and Jindal, 2000), and drying in a fluidized bed (Balasubramanian et al., 1996), among other applications. Artificial neural networks consist of a large number of simple interconnected nonlinear processing units (neurons) arranged in a layered structure, having adjustable connection weights. These weights are the internal parameters of the network. The input neurons are connected to the output neurons through layers of hidden nodes. Each neuron receives information in the form of inputs from other neurons and processes it through some function, typically nonlinear, called the activation function (Psichogios and Ungar, 1992). The general structure of a multilayered neural network with one output is shown in Figure 13.1. Due to the parallel structure and the ability to learn by adjusting the connection weights, ANNs can handle nonlinear relationships where mathematical models are not available without need of any previous knowledge on the relationships of the process variables (Psichogios and Ungar, 1992; Balasubramanian et al., 1996; Park et al., 1994; Adhikari and Jindal, 2000; Ramesh et al., 1996). During the network learning procedure, weights are adjusted so that the introduction of a set of inputs produces the desired set of outputs. The most widely used neural network learning method is backpropagation. This method uses an iterative gradient search technique to minimize a performance function, equal to the mean

Hidden 1

Input 1 . . .

Output 1

Input n . . .

1 Threshold value

Hidden n Threshold value

1

FIGURE 13.1 General structure of a multilayered neural network with one output.


difference between the desired and the actual network outputs (Psichogios and Ungar, 1992). The neural network process model should have as few neurons in the hidden layer as possible, respecting accuracy limitations (Huang et al., 1998b). The aims of this work are to model mass transfer during brine salting of Prato cheese by a traditional phenomenological model and an empirical approach and to compare their results.

13.2 MATERIALS AND METHODS 13.2.1 CHEESE PROCESSING The cheese manufacturing trials were carried out in a dairy pilot plant of the Chemistry and Food Engineering Department of the Mauá Institute of Technology. In each experiment, 100 l of milk were used per cheese vat to produce twelve pieces 3 of cheese with a mean weight of 1.0 kg and dimensions of (11.5 × 5.5 × 5.5) cm each. During the cheesemaking process, pasteurised, refrigerated whole milk was first heated to 32°C in a cheese vat. It was then inoculated with 1% of Probat 8/12 WIESBY culture (composed of Lactococcus lactis subsp. lactis + L. lactis subsp. cremoris + L. lactis subsp. lactis biovar. diacetilactis + Leuconostoc mesenteroides subsp. cremoris), a calcium chloride solution (23 g/100 l of milk) was added, the “urucum” coloring (10 ml/100 l of milk) was added, and calf rennet (WIESBY) was added (2 g/100 l of milk). The curd was formed during approximately 35 min and was cut to a size of ca. 0.8 to 1.0 cm. The temperature of the curd and whey mixture was raised slowly and maintained at 40°C for 60 min. Whey was subsequently drained, and the curd was placed in Prato hoops and held at room temperature (25°C) throughout pressing (2 kg weights) for 4 h, during which the cheese pieces were inverted five times, held (2 h), and then brined.

13.2.2 CHEESE SAMPLING The cheese pieces were immersed in nonagitated brine solutions of 15, 20, and 25% NaCl (w/w) at 10°C with a brine:cheese ratio of 2:1 (v/v). These are the usual conditions for Brazilian Prato cheese manufacturing. At each sampling time (30, 60, 120, 240, 360, and 480 min), one sample was removed from the brine vessel. After that, all samples and the control sample were divided into three regions: external, intermediate, and internal, as shown in Figure 13.2. These samples, classified by region, were weighed, ground, and kept at −10°C prior to chemical analysis. Note that in Figure 13.2 only two cut directions are shown; the third is the same as the vertical cut (B), once these two directions have the same dimensions (B = C = 5.5 cm).

13.2.3 ANALYTICAL METHODS In the control cheese samples, acidity, fat content (Atherton and Newlander, 1981), ash content, pH, total solids, and protein (AOAC, 1984) were analyzed. Salt content and moisture of all ground samples were determined by the Volhard method (AOAC, 1984). Analytical assays were performed in triplicate.


External Intermediate Internal

B/6 B/6 B/3 B/6 B/6

y

C A/6

A/6

A/3

z

A/6 A/6 x

FIGURE 13.2 Schematic representation of cheese cut.

13.3 MATHEMATICAL MODELING 13.3.1 FICKIAN DIFFUSION MODEL Considering tridimensional unsteady state mass transfer, the phenomenon of Fickian diffusion for rectangular geometry is described by the following equation:  ∂2C A ∂2C A ∂2C A  ∂C A = Deff  + +  ∂t ∂y 2 ∂z 2   ∂x 2

(13.1)

where CA is the salt concentration in cheese and Deff is the effective diffusivity. It was assumed that the cheese was homogeneous, that brine concentration remained constant during the experiments and was equal to the salt concentration at cheese surfaces, and that no film resistance existed between the brine and cheese faces. These assumptions correspond to an infinite Biot number and unity equilibrium distribution coefficient (K). To solve Equation (13.1), it was necessary to apply the solution suggested by Neumann (Crank, 1975), a product of the solutions of three plane sheets: f (x, y, z, t) = Ψ (x,t)⋅ζ (y, t)⋅ϕ (z, t)

(13.2)

The solution for a single plane sheet is given by: CA − C As = C A0 − C As

∞

∑ n=0

 − Deff (2 n + 1)2 π 2 t  8 exp   (2 n + 1)2 π 2 2a 2  

Initial condition: t = 0,

z > 0,

CA = CA0

t > 0,

z = 0,

CA = CAs

t > 0,

z = a,

∂C A =0 ∂z

Boundary conditions:


(13.3)

where CA0 is the initial salt concentration in cheese, CAs is the salt concentration on the cheese surface, n is the number of terms of the series, and 2a is the plane sheet thickness size in directions x, y, and z. Since all variables of Equation (13.3) are known except Deff, the effective diffu sivity (Deff) was obtained by a least squares estimation using Microsoft Excel (1997) to fit experimental data, with 100 terms of the series considered.

13.3.2 AVERAGE CONCENTRATION The proposed model predicted the average concentration of salt within cheese as a function of time. It was calculated from experimental data measured at three regions (internal, intermediate, and external) of the cheese piece; hence, one divided the total experimental volume V in integrating volumes Vi representing each region of cheese analyzed. The average concentration CA was calculated as follows: 3

∑C V i

CA =

i =1

V

i

(13.4)

13.3.3 NEURAL NETWORK MODEL A neural network model was constructed using a multilayer perceptron architecture. The network had two inputs, brining time and brine concentration, because these variables have a greater effect than other process variables on salt diffusion, and one output, average salt concentration in cheese. The training algorithm used was the well-known backpropagation algorithm, using the experimental data obtained, so as to minimize the differences between desired and calculated values of average salt concentration in cheese. Determination of the number of hidden neurons is commonly done empirically. The procedure used to choose the structure of the artificial neural network was a systematic trial and error procedure using two data sets (training and testing) in an attempt to avoid overfitting and underfitting. A description of these phenomena can be found elsewhere (Huang et al., 1998b; Arteaga and Nagai, 1993). From the 21 vectors available, containing the two inputs and the output, 18 were used to form the training set, while the remaining data formed the testing set. Data were presented to training in a random manner in order to achieve a better training. Error for comparison between structures was calculated by the sum of the squared error between the value in the training or test set and the value calculated by the ANNs. The method for choosing the number of nodes in the hidden layer is presented as follows (Zorzetto et al., 2000). Topology of the network was chosen initially to have a large number of hidden nodes, i.e., 10 nodes. Training was carried out until a minimum error for the test set was observed with this number of hidden nodes. This step is shown in Figure 13.3.


5.0

0.25 testing 0.2

3.0

0.15

2.0

0.1

1.0

0.05

0.0 0

50

100

150

200

Squared error - testing

Squared error - training

training 4.0

0 250

Number of iterations

FIGURE 13.3 Squared error for training and testing sets with 10 hidden nodes.

Brine time

Average concentration

Brine concentration 1 Threshold value Threshold value

1

FIGURE 13.4 Neural network model for Prato cheese salting.

As can be observed from Figure 13.3, minimum squared error for the testing set was reached with 100 iterations. Using the same number of iterations found in this structure, i.e., 100 iterations, nodes in the hidden layer were gradually cut off, until the smallest structure that could provide the same performance was reached. Following this procedure, it was noticed that minimum error was reached using 10 nodes in the hidden layer. However, differences in the testing error among the structures were not high. Thus, it was decided to use a smaller network with five hidden nodes to accelerate the training procedure, while still having an admissible error. From this analysis, the network topology used in this work is shown in Figure 13.4.


13.4 RESULTS AND DISCUSSION 13.4.1 CHEESE COMPOSITION The average pH of Prato cheese before brine salting was 5.20 ± 0.01. Results of the mean centesimal composition analysis are shown in Table 13.1.

13.4.2 SALTING PROCESS It is important to evaluate the changes of salt concentration within the cheese during the salting process to select the best operating conditions (especially salting time and brine concentration). As presented in Figure 13.5, no influence of brine concentration was observed in the internal portion of the cheese, but it is important to notice that although the values of salt concentration increased with time, the amount of salt permeated is almost null. However, at the intermediate position (Figure 13.6), differences of about 75% can be observed after 240 min between 15 and 20–25% NaCl. These variations become clear for the external region (Figure 13.7), where the solution of 15% NaCl yielded a lower salt concentration for all time periods studied. Nevertheless, no considerable differences between 20 and 25% NaCl were found even in this position.

TABLE 13.1 Composition of Control Cheese Samples (%)

Moisture Fat Fat (dry matter) Total protein Ash

Average

Standard Deviation

48.3 27.2 52.6 20.5 4.00

0.4 0.2 0.4 0.1 0.01

Internal Salt Concentration (%)

0.3 15% NaCl 0.2

20% NaCl 25% NaCl

0.1

0 0

100

200

300

400

Time (min)

FIGURE 13.5 Salt concentration at the internal position.


500

600

Salt Concentration (%)

Intermediate 0.3

0.2

15% NaCl 20% NaCl

0.1

25% NaCl

0.0 0

100

200

300

400

500

600

Time (min)

FIGURE 13.6 Salt concentration at the intermediate position.

Salt Concentration (%)

External 4.0 3.0 15% NaCl 2.0

20% NaCl 25% NaCl

1.0 0.0 0

100

200

300

400

500

600

Time (min)

FIGURE 13.7 Salt concentration at the external position.

Since the initial moisture of cheese samples varied for each trial, normalized values M /Mi (moisture wet basis/initial moisture) were plotted against time (Figure 13.8). It is important to note the greater water loss of cheese treated with 25% NaCl solution—more than 10% of the initial value—when compared to the other solutions (maximum of 5% moisture loss). It can be concluded that the concentration of 15% NaCl in the solution did not provide a gradient of activity coefficient high enough to maintain a driving force for efficient mass transfer between the sodium chloride in solution and the liquid phase in cheese. Besides, Furtado (1991) reported that salt concentrations lower than 17% cause a smoothing of the Prato cheese skin, probably due to hydration of the surface. As can be seen, the water loss of the cheese salted in 15% NaCl brine solution was minimal, with a small moisture increase after 360 min. However, as mentioned before, the increase of brine concentration from 20 to 25% NaCl did not raise the sodium chloride concentration within the cheese for all studied conditions. This behavior can be explained by a greater outward water flux of 25% NaCl solution when compared to 20% NaCl (Figure 13.8). This dewatering could lead to the formation of a hard skin on the surface of the cheese, thus avoiding


1.00

M / Mi

0.98 0.95

15% NaCl 20% NaCl

0.93

25% NaCl 0.90 0.88 0

100

200

300

400

500

600

Time (min)

FIGURE 13.8 Normalized average moisture of cheese samples.

salt penetration. These assumptions agreed with the visual observation that the cheese treated in 25% NaCl concentration showed small fissures on the cheese surface, probably due to shrinkage leading to modifications of skin permeability.

13.4.3 PERFORMANCE

OF THE

MODEL APPROACHES

Figure 13.9 allows a general visualization of the performance of the models for average salt concentration prediction at different brine salt concentrations. It can be observed from these plots that the neural network model had a better adjustment to the experimental data than Fick’s model did. Although experimental data did not show the expected behavior from the theoretical analysis due to experimental errors as shown in Figure 13.9(a), the neural network model had a good agreement with these data. Hence, the neural model does not require information regarding the understanding of the process, but on the other hand does not bring any additional comprehension. Theoretical knowledge and analysis of data are of principal importance during the selection of training and testing data in order to avoid discrepancy in the neural model. Fick’s model did not present a good fit to the experimental data, probably due to some model assumptions, especially concerning the unity equilibrium distribution coefficient, since it used a nonagitated solution and a low solution:cheese ratio. These factors could lead to a dilution of the solution with time and a varying concentration at the cheese surface.

13.4.4 STATISTICAL ANALYSIS One-way ANOVA (α = 0.01) tests were performed in order to detect significant differences between experimental average salt concentrations and those predicted 2 by Fickian and neural models. The correlation index (R ) and the sum of squared errors (SSE) between calculated and experimental data were also calculated.


0.8 Exp 15%

0.6

Neural 15% 0.4

Fick 15%

0.2 0.0 0

100

200

300

400

500

3.0

2.5 2 1.5 Exp 20% Neural 20%

1

Fick 20% 0.5 0

600

0

100

200

Time (min)

Time (min)

Exp. Data

Fick

SE F

300

400

500

600

Average Salt Concentration (%)

1.0




1.2

2.5 2.0 1.5

SE N

Time (min)

Exp. Data

Fick

SE F

Neural 25% Fick 25%

0.5 0.0 0

Time (min)

Neural

Exp 25%

1.0

100

200

300

400

500

600

Time (min)

Neural

SE N

Time (min)

Exp. Data

Fick

SE F

Neural

SE N

0

0.000

0.090 0.0081 0.029 0.0009

0

0.000 0.120 0.0144

0.022

0.0005

0

0.000

0.150

0.0226

0.025

0.0006

30

0.306

0.301 0.0000 0.275 0.0009

30

0.373 0.558 0.0343

0.358

0.0002

30

0.433

0.588

0.0243

0.412

0.0004

60

0.344

0.368 0.0006 0.390 0.0020

60

0.656 0.785 0.0166

0.683

0.0007

60

0.704

0.827

0.0152

0.735

0.0010

120

0.470

0.518 0.0023 0.436 0.0012

120

1.129 1.105 0.0006

1.114

0.0002

120

1.205

1.165

0.0016

1.182

0.0005

240

0.670

0.729 0.0035 0.653 0.0003

240

1.728 1.551 0.0315

1.730

0.0000

240

1.829

1.637

0.0367

1.835

0.0000

360

0.876

0.890 0.0002 0.893 0.0003

360

2.211 1.888 0.1040

2.188

0.0005

360

2.256

1.995

0.0680

2.257

0.0000

480

1.086

1.025 0.0037 1.084 0.0000

480

2.349 2.170 0.0323

2.357

0.0001

480

2.565

2.294

0.0732

2.551

SUM

0.2417

SUM

0.0185

(a) 15 % NaCl SE F – Squared error Fick model SE N – Squared error neural model

0.0056

SUM

0.2337

0.0022

(b) 20 % NaCl

FIGURE 13.9 Model performances at brine salting in 15% (a), 20% (b), and 25% (c) NaCl solutions.

(c) 25 % NaCl

0.0002 0.0028

TABLE 13.2 Modeling Results Fickian R 15% NaCl 20% NaCl 25% NaCl

2

0.9893 0.9912 0.9972

Neural 2

FRatio

P

R

1.792E–03 0.002 6.759E–07

0.999 0.932 0.999

0.9933 0.9996 0.9995

FRatio

P

8.387E–06 3.600E–07 6.759E–07

0.995 0.999 0.999

Note: Fratio = Fcalculated/Ftable

TABLE 13.3 Effective Diffusion Coefficients as a Function of the Salt Concentration 7

2

Deff. 10 (cm /sec) 15% NaCl 20% NaCl 25% NaCl

1.64 4.25 3.00

The data analyzed by ANOVA (Table 13.2) showed no significant differences (p ≥ 0.01) between the experimental data and Fickian models or between the experimental data and ANN models. It is known that for a reliable statistical investigation some statistical parameters should be analyzed simultaneously, thus preventing a misinterpreted conclusion. One can observe that the better fit to experimental data was obtained by the neural model, 2 which showed greater R value, lower SSE parameters, and smaller F ratios for all brine concentrations tested. The advantage of this procedure is that it is not necessary to have any previous knowledge about the process to model it. On the other hand, from the neural network it was not possible to understand the physical behavior of the process, once it was modeled from a “black box” approach. Hence, this approach is suitable when it is not necessary to have a detailed understanding of the process, or in process control strategies such as Internal Model Control (IMC). Statistical analysis for the Fickian diffusion model also showed good results, making it possible to calculate the effective diffusivity coefficient, which is related to the mass transfer rate. However, some attention must be paid to Fick’s model because of the previous discussions regarding Figure 13.9.

13.4.5 EFFECTIVE DIFFUSION COEFFICIENTS Effective diffusion values were affected by brine concentration. The brine solution of lowest concentration, 15% NaCl, presented the lowest Deff (Table 13.3). On the other hand, the 20% NaCl solution showed the highest effective diffusion coefficient,


rather than 25% NaCl. Reduced effective diffusivity with increasing salt concentration has been previously reported (Turhan and Kalentuç, 1992) for Turkey white cheese. The authors associated this occurrence with a lower outer water flux in relation to inner salt penetration for the solutions of lower concentration. Regardless, Deff values are similar to those presented by Zorrilla and Rubiolo (1996) for Fynbo cheese and slightly slower than those reported by Turhan and Kalentuç (1992), probably due to compositional differences.

13.5 CONCLUSIONS Although statistical analysis demonstrated that neural network as well as Fickian diffusion models can successfully model salt diffusion in Prato cheese, observations on the experimental data plotted together with the applied models showed a noticeably better fit with the ANN approach, mainly because of the inadequacy of some assumptions on the resolution of Fick’s law. No considerable difference in salt concentration as a function of time was detected between brine concentrations of 20 and 25%. However, cheese treated in 25% NaCl solution had a larger moisture loss than that brined at 20% NaCl. For 15% NaCl solution, the salt penetration and water removal were less intense. Marginal salt uptake was found at the internal and intermediate regions even after 480 min of processing. Effective diffusion coefficients were affected by brine concentration. A maximum value was found at the 20% NaCl solution rather than 25% NaCl, due to greater dehydration of the cheese samples treated with the latter solution.

ACKNOWLEDGMENTS The authors gratefully acknowledge Prof. Walter Borzani and Prof. Marcello Nitz for their helpful discussions and suggestions and students Graziela dos Santos Viola and Rafaela R. C. Bandeira de Mello for their valuable assistance during the experiments.

REFERENCES Adhikari, B. and Jindal, V.K., Artificial neural networks: a new tool for prediction of pressure drop of non-Newtonian fluid foods through tubes, J. Food Eng., 46, 43–51, 2000. AOAC (Association of Official Analytical Chemists), Official Methods of Analysis, 14th ed., Williams, S., Ed., AOAC, Washington, DC, 1984. Arteaga, G.E. and Nagai, S., Predicting protein functionality with artificial neural networks: foaming and emulsifying properties, J. Food Sci., 58, 1152–1156, 1993. Atherton, H. and Newlander, J.A., Chemistry and Testing of Dairy Products, 4th ed., AVI, Westport, CT, 1981. Balasubramanian, A., Panda, R.C., and Ramachandra Rao, V.S., Modelling of a fluidized bed drier using artificial neural network, Drying Technol., 17, 1881–1889, 1996. Benne, M., Grondin-Perez, B., Chabriat, J.P., and Hervé, P., Artificial neural networks for modelling and predictive control of an industrial evaporation process, J. Food Eng., 46, 227–234, 2000. Crank, J., The Mathematics of Diffusion, Oxford University Press, Oxford, 1975.


Furtado, M.M., A Arte e a Ciência do Queijo, Editora Globo S.A., São Paulo, 1991. Furtado, M.M. and Neto, J.P.M.L., Tecnologia de Queijos: Manual Técnico para Produção Industrial de Queijo, Editora Dipemar Ltda, São Paulo, 1994. Furtado, M.M. and Wolfschoon-Pombo, A.F., Fabricação de queijo prato e minas: estudo do rendimento. Parte II. Previsão da gordura no extrato seco, Rev. Instit. Laticínios Cândido Tostes, 34, 3–13, 1979. Geurts, T.J., Walstra, P., and Mulder, H., Transport of salt and water during salting of cheese. 1. Analysis of the processes involved, Neth. Milk Dairy J., 28, 102–129, 1974. Guinee, T.P. and Fox, P.F., Sodium chloride and moisture changes in Romano-type cheese during salting, J. Dairy Res., 50, 511–518, 1983. Guinee, T.P. and Fox, P.F., Salt in cheese: physical, chemical and biological aspects, in Fox, P.F., Cheese: Chemistry, Physics and Microbiology. 1. General Aspects, Elsevier Applied Science, London, 1987. Huang, Y., Whittaker, A.D., and Lacey, R.E., Neural network prediction modeling for a continuous snack food frying process, Trans. ASAE, 41, 1511–1517, 1998a. Huang, Y., Whittaker, A.D., and Lacey, R.E., Internal model control for a continuous snack food frying process using neural networks, Trans. ASAE, 41, 1519–1525, 1998b. Luna, J.A. and Chavez, M.S., Mathematical model for water diffusion during brining of hard and semi-hard cheese, J. Food Sci., 57, 55–58, 1992. Muir, D.D., Hunter, E.A., and Banks, J.M., Aroma of cheese. 2. Contribution of aroma to the flavour of cheddar cheese, Milchwissenschaft, 52, 85–88, 1997. Oliveira, J.S., Queijos: Fundamentos Tecnológicos, 2nd ed., Ícone Editora, São Paulo, 1986. Park, B., Chen, Y.R., Whittaker, A.D., Miller, R.K., and Hale, D.S., Neural network modeling for beef sensory evaluation, Trans. ASAE, 37, 1547–1553, 1994. Payne, M.R. and Morison, K.R., A multi-component approach to salt and water diffusion in cheese, Int. Dairy J., 9, 887–894, 1999. Psichogios, D.C. and Ungar, L.H., A hybrid neural network – first principles approach to process modeling, AIChE J., 38, 1499–1511, 1992. Ramesh, M.N., Kumar, M.A., and Srinivasa Rao, P.N., Application of artificial neural networks to investigate the drying of cooked rice, J. Food Process Eng., 19, 321–329, 1996. Ribeiro, E.P., Aplicação de Ultrafiltração de Leite no Processo de Fabricação de Queijo Prato, thesis, Universidade Estadual de Campinas, Brasil, 1996. Schiftan, T.Z. and Komatsu, I., Estudos sobre a composição do queijo Prato consumido na cidade de São Paulo, Rev. Instit. Laticínios Cândido Tostes, 35, 33–38, 1980. Turhan, M. and Kaletunç, G., Modeling of salt diffusion in white cheese during long-term brining, J. Food Sci., 57, 1082–1085, 1992. Zorrilla, S.E. and Rubiolo, A.C., Modelling NaCl and KCl movement in Fynbo cheese during salting, J. Food Sci., 59, 976–980, 1994. Zorzetto, L.F.M., Maciel Filho, R., and Wolf-Maciel, M.R., Process modelling development through artificial neural networks and hybrid models, Comput. Chem. Eng., 24, 1355–1360, 2000.


14

Influence of Vacuum Pressure on Salt Transport during Brining of Pressed Curd A. Chiralt, P. Fito, C. Gonzalez-Martínez, and A. Andrés

CONTENTS 14.1 Introduction 14.2 Materials and Methods 14.2.1 Cheesemaking 14.2.2 Salting Treatments 14.2.3 Analysis of the Concentration Profiles 14.2.4 Analytical Controls 14.3 Results and Discussion 14.3.1 NaCl Concentration Profiles 14.3.2 Modeling Salt Transport through Profile Development 14.4 Conclusions Nomenclature Acknowledgments References

14.1 INTRODUCTION Salting of cheese is usually carried out by brine immersion (BI) for a lengthy period of time (24–48 h, depending on the cheese size), and this causes technological and quality problems in cheesemaking. The process implies the management of great amounts of brine and the development of sharp salt concentration profiles in the cheese pieces, which may provoke ripening problems in some cases (Guinee and Fox, 1987). Recently, a new salting procedure has been described (vacuum impregnation, VI), which makes the salting process shorter (thus reducing the involved brine volumes) and the salt concentration profile in cheese flatter (Chiralt and Fito, 1997; Andrés et al., 1997). This new process is based on the action of hydrodynamic


mechanisms (HDM) associated with pressure changes. In the VI process, vacuum pressure is applied in the salting tank at the beginning of the process for a time t1, after which the atmospheric pressure is restored for a time t2. During the vacuum period, the occluded gas in the pressed curd is expanded and flows out. When atmospheric pressure is restored, the residual gas is compressed, leading to the entry of the external brine into the curd pores. In this way, a greater salt uptake occurs in the cheese pieces than when using the conventional immersion for the same salting time, and a different salt profile is achieved (Andrés et al., 1997). Salting of cheese by BI has been described as a diffusion process, but capillary mechanisms may also play an important role in the salt uptake (Geurts et al., 1980). The capillary input of the external brine in the curd pores can contribute to quite an extent to the total salt gain when subatmospheric pressure is applied in the salting tank. Capillary penetration in a pore occurs coupled with the compression of the occluded internal gas. The volume fraction of the sample penetrated by the external liquid (Xc) is a function of the capillary pressure (pc), the pressure in the system (p), and the effective porosity of the product (ε) (Fito, 1994; Fito and Pastor, 1994). According to Equation (1), the lower the pressure in the system (p), the greater the capillary penetration. Xc = ε [pc/(p + pc)]

(14.1)

When atmospheric pressure (p2) is restored in the tank at vacuum pressure (p1), the HDM react, leading to a great penetration of external liquid. The volume fraction of the sample penetrated by the external liquid (XHDM) has been modeled by Equations (14.2) and (14.3) in stiff matrices (Fito, 1994). Nevertheless, pressure changes in the system can also promote sample deformations when the solid matrix exhibits a viscoelastic behavior (Fito et al., 1996). The restoring of atmospheric pressure after a vacuum period can promote curd compression and partial collapse of its pores, decreasing the effectiveness of capillary phenomena for salting. XHDM = ε (1 – 1/r)

(14.2)

r = (p2/p1) + (pc/p1)

(14.3)

The aim of this work is to analyze the contribution of the different mass transfer mechanisms in cheese salting in order to minimize the salting time by using subatmospheric pressures. Analysis of the concentration profiles in a product throughout a mass transfer process is a good approach to clarify the mass transfer mechanisms involved in the process. In this sense, NaCl and moisture concentration profiles have been analyzed in cheeses at different times of salting by using: • BI (brine immersion): salting process at atmospheric pressure • VI (vacuum impregnation): salting process at vacuum pressure (p1) • PVI (pulse vacuum impregnation): a two-step salting process; the former (t1) at vacuum pressure (p1), and the latter (t2) at atmospheric pressure (p2).


14.2 MATERIALS AND METHODS 14.2.1 CHEESEMAKING Cheeses were processed at the food technology pilot plant of the Universitat Autónoma of Barcelona. Ewe’s milk from Manchega sheep was supplied by the university dairy farm. Cheeses were manufactured from pasteurized milk (15 sec at 72°C), in a 250 l vat, according to standard protocol. Milk (100 l in each batch) was heated to 31°C, and 2% lactic starter (Lactococcus lactis ssp. lactis plus L. lactis ssp. cremoris; AM Larbus S.A., Barcelona) was added. After 35 min, the milk coagulum was cut into 8–10 mm cubes with a wire knife. The curds were heated at a rate of 1°C per 5 min to reach 36°C and held at this temperature for 10 min prior to wheying. The curd was molded into cylindrical pieces (10 cm high and 18 cm diameter) of about 2.5 kg and pressed following two steps: first, for 1 h at 800 Pa; second, cheeses were turned upside down and pressed again for 3 h at 1600 Pa.

14.2.2 SALTING TREATMENTS Salting treatments were carried out with a 24% (w/w) brine in a stirred tank at 10°C. A pressure of 50 mbar was applied during the vacuum periods. Development of salt and moisture concentration profiles throughout the salting process was analyzed in cylindrical samples of cheese (6 cm diameter, 10 cm high), the lateral surface of which was covered with a film in order to avoid radial mass transfer during salting. The experiments were carried out using two curd batches and applying different salting times (Table 14.1): t1, time at p1 = 50 mbar and t2, time at atmospheric pressure (p2).

14.2.3 ANALYSIS

OF THE

CONCENTRATION PROFILES

A sharp tubular cork borer (20 mm internal diameter) was used to take out cylindrical samples from salted curd pieces in the axial direction at different times during salting. Starting at the surface in direct contact with the brine, each sample cylinder was TM sliced into disks (1.5 mm thick) with a tissue slicer EMS (Fort, Washington) model OTS-3000–04. The distance to the interface (d) of the disks of each cylinder core was identified by the order of slicing. At each distance to the interface, moisture and salt were analyzed in respective disks.

TABLE 14.1 Salting Treatments for the Analysis of the Salt Concentration Profile Development Salting Treatment

t1 (h) p1 = 50 mbar

t2 (h) p2 = 1013 mbar

Curd Batch

BI PVI VI

0 0.5 0.5; 2; 8; 17

0.5, 2, 8, 17 0; 1.5; 7.5; 16.5 0

1 2 1, 2


14.2.4 ANALYTICAL CONTROLS For sodium chloride determination, samples were homogenized in distilled water at 9000 r/min in an ULTRATURRAX T25 for 5 min and centrifuged to remove any fine debris present in the sample. An aliquot of exactly 500 µl of centrifuged sample was taken and titrated in a chloride analyzer (Sherwood, Mod. 926). Moisture content was determined by drying of the samples at 105°C until a constant weight was reached.

14.3 RESULTS AND DISCUSSION 14.3.1 NACL CONCENTRATION PROFILES Figure 14.1 shows the NaCl concentration profiles for each treatment at 8 and 17 h salting times, in terms of the reduced salt concentration Y(t,d) (Equation (14.4)) in the cheese liquid phase (water plus salt), as a function of the distance d to the interface. In Equation (14.4), z is the salt mass fraction in the cheese liquid phase at time t (zt), t = 0 (z0) and at equilibrium (ze). The equilibrium value has been taken to be equal to the brine concentration, according to previous studies (González et al., 1999). By comparison with the concentration profiles in curds from the same production batch, at a determined time of salting, it can be observed that when vacuum pressure was used, salt penetration was deeper than in the traditional BI method because of the capillary mechanisms, especially in the lower part of the cheeses where more pores exist. The effectiveness of the VI process is greater than that of PVI. Y1

Y1

(a)

0.8

0.8

0.6

0.6

0.4

0.4

0.2

d(mm)

0

(b)

0.2

d(mm)

0 0

10

20

30

40

50

0

10

20

30

40

50

Y1

Y1

(c)

0.8 0.6

0.6

0.4

0.4

0.2

d(mm)

0

(d)

0.8

0.2

d(mm)

0 0

10

20

30

40

50

0

10

20

30

40

50

FIGURE 14.1 Experimental concentration profiles in the upper (a and c) and lower (b and d) part of the curd, at 8 h (a and b) and 17 h (c and d) of salting by the different procedures (BI—batch1: ο, PVI—batch2: ◊, VI—batch1: 䊐, VI—batch2: ∆).


This suggests that the pore collapses to quite an extent when atmospheric pressure is restored, inhibiting the HDM penetration. After 17 h of salting, the concentration profile was not fully developed in cheeses salted by BI and PVI. However, 8 and 17 h were enough to reach a fully developed profile, in the lower and upper part respectively, of curd pieces salted by VI. Differences in porosity of the lower and upper part of the curd (in relation to the plug position during pressing) have been explained in terms of the viscoelastic response during pressing (Chiralt and Fito, 1997). This leads to heterogeneous salt transport in the piece, especially if hydrodynamic mechanisms are promoted. Y=

14.3.2 MODELING SALT TRANSPORT DEVELOPMENT

zt − ze z0 − ze

(14.4)

THROUGH

PROFILE

A pseudodiffusional approach was used to model salt transport in the cheese liquid phase. Equation (14.5) (Crank, 1975) was fitted to the experimental profiles at the different salting times by using a nonlinear optimization method to obtain the effective pseudo diffusion coefficient De for each treatment and curd zone. Equation (14.5) has been applied considering a number of terms (value of n) great enough to satisfy Equation (14.6). The obtained values of De for the different samples are shown in Table 14.2. Despite the scarce physical meaning of De when there is a notable action of hydrodynamic mechanisms, De values reflect the differences observed in the profiles. Nevertheless, great deviations in the predicted profiles with respect to the experimental points were observed. Y(l, t ) =

4 π

∞

( − )n

 − De t(2 n + 1)2 π 2  (2 n + 1)πl  cos 4e 2 2e 

∑ 2n1+ 1 exp n=0

Y (l, t ) − Yn −1 (l, t ) Yn (l, t )

< 0.01

(14.6)

TABLE 14.2 Values of the Pseudo Diffusion Coefficient (De × 10 2 10 , m /sec) Obtained from the Modeling of the Concentration Profile for Each Treatment and Curd Zone (Lower and Upper Part) Treatment BI-1 PVI-2 VI-1 VI-2


(14.5)

Upper Part

Lower Part

4.4 6.1 4.6 9.5

5.2 8.4 7.7 12.3

Ypred 1.00

0.95

0.90

0.85

0.80 0.80

0.85

0.90 Yexp (a)

0.95

1.00

Ypred 1.00

0.95

0.90

BI-1 VI-1 VI-2

0.85

PVI-2

0.80 0.80

0.85

0.90 Yexp (b)

0.95

1.00

FIGURE 14.2 Comparison of experimental Y mean values with those predicted from the models: (a) pseudodiffusional model (PDM) for the different salting treatments in the lower (closed symbols) and upper (open symbols) part of the curd, (b) PDM (䊏) and advancing disturbance front (ADF: ◊) models, for all treatments.

Mean values of Y were calculated for each case by integrating the concentration profiles at a determined salting time, and their values were compared with those obtained from the De values applying Equation (14.7) (Crank, 1975). Figure 14.2a shows the comparison between predicted and experimental values, and a greater deviation for treatments carried out under vacuum conditions can be observed. Y = 1− 2

 De t   πl 2 

0.5

(14.7)

On the basis of these results, a model developed to predict concentration profiles was applied (Fito et al., 1998; Salvatori et al., 1998; Salvatori et al., 1999). This model


TABLE 14.3 Location of the Advancing Disturbance Front (df; mm) in the Upper (UP) and Lower (LP) Part of the Curd, as a Function of Salting Treatment and Time BI-1

PVI-1

VI-1

VI-2

Salting Time (h)

UP

LP

UP

LP

UP

LP

UP

LP

0.5 2 8 17

7.7 13.1 24.4 30.5

10.5 13.8 26.7 30.1

7.7 13.5 21.1 31.2

10.5 16.8 26.4 32.2

9.3 19.8 33.9

10.8 30.8

20.1 27.0 37.7

22.6 25.9

a

a

a

a

a

a

a

Fully developed profile.

TABLE 14.4 Parameters df0 and ν Describing the ADF Rate in the Upper (UP) and Lower (LP) Part of the Curd Salted by Different Treatments BI-1

PVI-1

VI-1

VI-2

Parameter

UP

LP

UP

LP

UP

LP

UP

LP

df0 ν

3.62 6.66

7.06 6.13

3.62 6.66

7.06 6.13

2.25 11.4

0 20.5

14.7 8.19

15.6 8.5

is based on the definition of an advancing disturbance front (ADF) in the material where mass transport occurs. ADF has been defined as a virtual plane that separates the zone of the product near the interface where the original concentration status has been disturbed by mass transport phenomena from the rest of the sample where no changes in concentration or mass fluxes are evident. The location of the ADF in the cheese samples has been estimated as the distance (df) to the plane sheet interface where Y(t,d) = 0.99. Table 14.3 gives the values of df for the different treatments at each time. The df vs. time relationships have been modeled by Equation (14.8), where df0 and ν parameters reflect the particular influence of the properties of each curd on salting rate (González, 1999). Table 14.4 gives the obtained values for each treatment. The advanced rate of ADF in the cheese dropped as the salting process progressed and was greatly affected by the curd porosity and water content in VI treatments. d f = d f 0 + ν ⋅ t 0.5

(14.8)

When the values of Y(t,d) were plotted as a function of a reduced distance to the interface z (Equation (14.9)) according to the ADF model, two generalized


1 0.8 0.6 Y 0.4 0.2 0 0

0.2

0.4

z

0.6

0.8

1

0.6

0.8

1

(a) 1 0.8 0.6 Y 0.4 0.2 0 0

0.2

0.4

z (b)

FIGURE 14.3 Generalized profiles in terms of the reduced distance to the interface (z) for VI (a) and BI/PVI (b) treatments (VI-1: ο, VI-2: ∆, BI-1: ◊, PVI-2: +). Line: ADF fitted model.

profiles were obtained. One of these profiles corresponds to the vacuum treatments (VI) and the other to the BI and PVI treatments (Figure 14.3). The ADF model, shown in Equation (14.10), could be fitted to the points of these generalized profiles by a nonlinear procedure for z < 1, thus obtaining the characteristic model parameters (K and z0). In the ADF model, K is a facility coefficient to mass transport of the system, whereas z0 is a reduced distance near the interface immediately equilibrated with the external medium (Fito et al., 1998). The obtained values of z0 in all cases were zero, and K values were 1.075 and 1.145, respectively, for VI and BI/PVI


treatments. The close fitting of this model to experimental points can be seen in Figure 14.3 (lines). At z ≥ 1, Y = 1 for all treatments. z = d / df Y=

K( z − z 0 ) (z − z 0 ) + (1 − z 0 ) ⋅ (K − 1)

(14.9) (14.10)

The ADF model also allows prediction of the mean values of Y(t) in the cases where samples were not completely affected by disturbance (concentration profile not fully developed) (Fito et al., 1998). Equation (14.11) gives the Y(t) mean values as a m function of K and the maximum z value (zt ) of each sample at each time (Salvatori et al., 1999). Figure 14.3b shows the close agreement between experimental and predicted values.  1   1   Y( t ) = 1 − 1 − K 1 + (K − 1) ln1 −    m      z t  K  

(14.11)

It can be observed in addition that the ADF model allows prediction of the mean reduced concentrations better than the pseudodiffusional model (PDM), but only for those situations when the concentration profile is not fully developed. Nevertheless, in practical terms, this situation occurs in industrial salting of this kind of cheese for the usual overall amounts of salt required. The common Y mean values range between 0.88 and 0.84, which correspond to salt levels of 1.5–2.0% in the cheese. These values are lower than the mean value of Y in the sample disturbed zone (z < 1) for the BI and PVI treatments and of the order of those reached in the VI treatments. Therefore, at the end of the usual salting process, the cheese status corresponds to the non-fully developed concentration profile, and the ADF model will be applicable with more accurate predictions than those reached with the SDM.

14.4 CONCLUSIONS Brining at vacuum pressure greatly enhances salting kinetics due to the promotion of hydrodynamic transport coupled with diffusion in the cheese liquid phase. The salt concentration profiles developed during VI salting can be better modeled by an advancing disturbance front model than by diffusional equations. Curd porosity and moisture greatly affect salt transport kinetics, especially when hydrodynamic mechanisms are promoted in the curd pores by vacuum pressure. The greater the curd porosity and moisture, the faster the salting process. Therefore, special control of these variables will be recommended during cheese production, ensuring a homogeneous salt uptake during the brining process.

NOMENCLATURE d Distance in the sample to the interface (m) df Distance of the advancing disturbance front (ADF) to the interface (m) 2 De Pseudodiffusional coefficient (m /sec)


e K l p t Y z zt z0

Sample half-thickness (m) ADF model parameter (dimensionless) Distance from any point in the sample to center (m) Pressure (mbar) (subscript c, capillary; 1, at the vacuum step; 2, atmospheric) Salting time (sec) Reduced driven force Reduced distance to the sample interface (d/df) Mass fraction of salt in the cheese liquid phase (water plus salt) at time t, (subscript e, equilibrium; 0, initial value) ADF model parameter (dimensionless)

ACKNOWLEDGMENTS The authors thank the Comisión Interministerial de Ciencia y Tecnología (Spain) and CYTED program for their financial support.

REFERENCES Andrés, A., Panizzolo, L., Camacho, M.M., Chiralt, A., and Fito, P., Distribution of salt in Manchego type cheese after brining, in Engineering and Food at ICEF 7, Jowitt, R., Ed., Sheffield Academic Press, Sheffield, 1997, pp. 133–136. Chiralt, A. and Fito, P., Salting of Manchego cheese by vacuum impregnation, in Food Engineering 2000, Fito, P., Barbosa-Cánovas, G., and Ortega-Rivas, E., Eds., Chapman and Hall, New York, 1997, pp. 215–230. Crank, J., The Mathematics of Diffusion, Oxford University Press, London, 1975, pp. 44–68. Fito, P., Modelling of vacuum osmotic dehydration of food, J. Food Eng., 22, 313–328, 1994. Fito, P., Andrés, A., Chiralt, A., and Pardo, P., Coupling of hydrodynamic mechanism and deformation relaxation phenomena during vacuum treatments in solid porous foodliquid systems, J. Food Eng., 27, 229–240, 1996. Fito, P., Chiralt, A., Barat, J., Salvatori, D., and Andrés, A., Some advances in osmotic dehydration of fruits, Food Sci. Technol. Int., 4, 329–338, 1998. Fito, P. and Pastor, R., Non-diffusional mechanism occurring during vacuum osmotic dehydration, J. Food Eng., 21, 513–519, 1994. Geurts, T.G., Walstra, P., and Mulder, P., Transport of salt and water during salting of cheese. 2. Quantities of salt taken up and of moisture lost, Neth. Milk Dairy J., 34, 229–254, 1980. González, C., Salado de Quesos Tipo Manchego por Impregnación a Vacío e Influencia en la Maduración, Ph.D. thesis, Universidad Politécnica de Valencia, Spain, 1999. González, C., Fuentes, C., Andrés, A., Fito, P., and Chiralt, A., Effectiveness of vacuum impregnation brining of Manchego-type curd, Int. Dairy J., 9, 143–148, 1999. Guinee, T.P. and Fox, P.F., Salt in cheese: physical, chemical and biological aspects, in Cheese: Physics and Microbiology, Vol. 1., Fox, P.F., Ed., Elsevier Applied Science, London, 1987, pp. 251–297. Salvatori, D., Andrés, A., Chiralt, A., and Fito, P., Osmotic dehydration progression in apple tissue II: Generalised equations for concentration prediction, J. Food Eng., 42, 133–138, 1999. Salvatori, D., Andrés, A., Albors, A., Chiralt, A., and Fito, P., Structural and compositional profiles in osmotically dehydrated apple, J. Food Sci., 63, 606–610, 1998.


15

Effects of Water Concentration and Water Vapor Pressure on the Water Vapor Permeability and Diffusion of Chitosan Films J.L. Wiles, P.J. Vergano, F.H. Barron, J.M. Bunn, and R.F. Testin

CONTENTS 15.1 Introduction 15.2 Materials and Methods 15.2.1 Water Vapor Permeability Coefficients 15.2.2 Solubility Coefficients 15.2.3 Diffusion Coefficients 15.2.4 Statistical Analysis 15.3 Results and Discussion 15.4 Conclusions Nomenclature References

15.1 INTRODUCTION The mechanism for water vapor permeation through hydrophilic films is extremely complex. The complexity is due to nonlinear water sorption isotherms and water content dependent diffusivities (Schwartzberg, 1986). Hydrophilic films are usually used in applications where water vapor migration is not detrimental or water solubility is advantageous (Donhowe and Fennema, 1994). The water vapor transmission of hydrophilic films varies nonlinearly with the water vapor partial pressure gradient. Since chitosan films are cationic and strongly hydrophilic, water interacts with the polymer matrix and increases permeation for water vapor (Pascat, 1986).


TABLE 15.1 Water Vapor Permeability from Prior Studies on Chitosan Films Water Vapor Permeability (g/m/sec/mmHg) 1.30 4.79 3.35 2.65 4.42

× × × × ×

−8

10 −9 10 −9 10 −8 −9 10 to 8.15 × 10 −9 −9 10 to 7.79 × 10

Reference

Conditions

Description of Film

Muzzarelli et al., 1974 Wong et al., 1992 Butler et al., 1996 Caner, et al., 1998 Park, 1998

38.7°C & 90% RH Ambient 25°C & 50% RH 25°C & 50% RH 25°C & 50% RH

Chitosan membranes Chitosan–lipid films Chitosan–plasticizer Chitosan–plasticizer Chitosan–plasticizer

Most polymer plastic films demonstrate ideal or Fickian permeability behavior because little or no interaction between the film and the permeant occurs. Thus, Fick’s first and Henry’s laws apply. Films that are hydrophilic in nature or react to organic vapors will demonstrate non-Fickian behavior because of the interaction between the permeant and the polymer film. The film’s thickness, moisture content, temperature, time, and pressure affect the water vapor permeation rate. For experimental purposes, most of these variables can be controlled during testing. Other influences on the permeation process include the nature of the polymer and the nature of the permeant (Pascat, 1986). Diffusion coefficients (D) for hydrophilic films such as chitosan films may tend to increase with increasing water vapor concentration. Diffusion measured in steady state is a mean value over the range of concentration with flux still being constant through the film. Since D is concentration dependent, the concentration gradient is not linearly dependent on distance through the film (Crank, 1975). Table 15.1 shows the water vapor permeability coefficients of chitosan films from previous work with this biopolymer. The hygroscopicity of chitin and chitosan is comparable to that of cellulose derivatives, but chitin’s surface is less active and permeable to water (Muzzarelli, 1977). Due to chitin’s insolubility in water, it is chemically treated to increase water solubility. As a result of this treatment, films made with chitosan have significant uptake of water and water vapor. At the present, no research has studied the permeability characteristics of chitosan films over a wide range of relative humidity (RH) conditions. The water vapor permeability (WVP) characteristic of chitosan film’s dependency on water concentration is important because of the magnitude of its effects. Hauser and McLaren (1948) developed a model to predict the WVP of hydrophilic films under any RH gradient. Models are used to predict WVP over a large range of relative humidity conditions. McHugh and Krochta (1994) used the Hauser and McLaren method to predict WVPs of whey protein isolate and glycerin films. They concluded that RH had an exponential effect on the WVP of whey protein films. They recommended that all researchers who study edible films should consider this method. The objective of this study was to determine the water vapor permeability, solubility, and diffusion coefficients of chitosan films over a wide range of relative humidity conditions at 25°C. Previously measured water vapor transmission rates


TABLE 15.2 Measured WVTR, Thickness, and Water Concentration at Various Relative ο Humidity Conditions at 25 C for Chitosan Films % Relative Humidity

LSMEANS of 2 WVTR (g/m /day)

11 23 33 43 53 69 75 84

6.65 14.2 36.5 126 278 504 719 1140

LSMEANS of thickness (m) 3.98 3.93 3.98 3.92 3.98 4.42 4.85 5.93

× × × × × × × ×

−5

10 −5 10 −5 10 −5 10 −5 10 −5 10 −5 10 −5 10

LSMEANS of concentration 3 (g/m ) 4.64 6.36 8.36 1.09 1.38 1.99 2.28 2.76

× × × × × × × ×

4

10 4 10 4 10 5 10 5 10 5 10 5 10 5 10

WVTR × Thickness (g/m/day) 2.64 5.58 1.45 4.95 1.11 2.23 3.48 6.76

× × × × × × × ×

−4

10 −4 10 −3 10 −3 10 −2 10 −2 10 −2 10 −2 10

(WVTR) and thickness changes due to water vapor sorption, which are given in Table 15.2, were used to determine the WVP of chitosan films (Wiles, 2000; Wiles et al., 2000). Before determining the permeability characteristics of chitosan films, changes in thickness due to swelling by water vapor sorption were measured (Wiles, 2000) and applied to this study.

15.2 MATERIALS AND METHODS Chitosan films of three levels of deacetylation (91.7%, 84.0%, 73.0%) were prepared by casting a 2% chitosan solution onto a level glass plate. Water sorption isotherms were determined using the method of discontinuous registration of weight changes in a static system (Labuza, 1984). Two standard test methods were used to determine WVTR of chitosan films (Wiles et al., 2000). All film samples were conditioned for 2 weeks at their specified relative humidity and 25°C. Chitosan films were tested according to ASTM E96–80 (ASTM, 1980), the Desiccant Method, and ASTM F 1249–90 (ASTM, 1990) on the Permatran W600 (MOCON, Minneapolis, MN).

15.2.1 WATER VAPOR PERMEABILITY COEFFICIENTS 2

Equation (15.1) expresses water vapor permeability coefficients in units of g⋅m/m / day/mmHg, achieved by measuring the WVTR, multiplying it by the thickness, and dividing by the water vapor partial pressure difference. Thickness measurement takes swelling into account. A mathematical model was used to describe swelling as a function of ambient water vapor pressure (Wiles, 2000). The model was based on data obtained by using the scanning electron microscopy (SEM) and optical microscope to measure swelling. The thickness model was fitted with a polynomial curve 2 with a R of 1.00. WVP = WVTR l/(p1 − p1)


(15.1)

WVP is defined as the steady amount of water vapor flow in unit time through unit area of a film of unit thickness driven by water vapor pressure difference across the film’s parallel surfaces under specified temperature and RH conditions (ASTM, 1980).

15.2.2 SOLUBILITY COEFFICIENTS Solubility coefficient (S) can be derived from the moisture sorption isotherm. From the measurement for determining the moisture sorption isotherm, the amount of water (grams) per unit of water vapor pressure (mmHg) is measured in a volume 3 (m ). The volume measurement takes swelling into account.

15.2.3 DIFFUSION COEFFICIENTS Diffusion coefficients are calculated at each water vapor pressure by using the measured value of WVP and dividing it by the measured value of S at the same water vapor pressure. D=

WVP S

g m 2 m ⋅ sec⋅ mmHg = g sec 3 m ⋅ mmHg

(15.2)

If flux (WVTR) and concentrations are known, another method uses the equation: D = F(l)/∆C

(15.3)

F is WVTR, l is the film’s thickness, and ∆C is the change in water concentration in the film across the gradient. In the calculation of D, some assumptions are made about the permeation process in chitosan films. Diffusion and swelling are limited to one dimension, i.e., perpendicular to the film’s surface, which is parallel to thickness. It is assumed that the water vapor pressure gradient is linear across the film in the range where a particular value of D is calculated.

15.2.4 STATISTICAL ANALYSIS The experimental design used was a randomized block design performed in three replications with 290 permeation and 682 sorption data measurements. Least-squares means (LSMEANS) were calculated for thickness, permeability, diffusion, and solubility coefficients using the General Linear Models (GLM) of SAS (SAS, 1996). It was determined that no significant difference existed in water vapor permeability characteristics among the types of chitosan films made using 91.7%, 84.0%, and 73.0% deacetylated chitosan. Therefore, the data points are all compiled together in this study for further statistical analysis. LSMEANS of measured values were analyzed


and fitted with curves using a nonlinear (NLIN) procedure in SAS. The NLIN procedure uses the Gauss–Newton iterative method to regress the residuals onto the partial derivatives of the model until the parameter estimates converge. These models were used to express the relationships of permeability and diffusion in chitosan films.

15.3 RESULTS AND DISCUSSION

2

(g m /m /day)

WVTR x thickness

In all cases except for solubility of the chitosan films (Figure 15.1), the values increased with an increase in water vapor pressure. Exponential curves were fitted to the LSMEANS of the measured values and plotted using NLIN procedure in SAS. Figures 15.2 and 15.3 show the exponential relationship of WVP and D with increases in water vapor pressure. Water vapor permeability and diffusion coefficients are not constant but vary with water vapor partial pressure and concentration (Figures 15.2 and 15.3). Concentration of sorbed penetrant (water vapor) and temperature are the two main factors affecting polymer chain segmental motion. Temperature was controlled at 25°C for this study. The presence of water increases the free volume of the chitosan film, therefore increasing permeability and diffusion (Rogers, 1985). Deviations from 8.00E-02 7.00E-02 6.00E-02 5.00E-02 4.00E-02 3.00E-02 2.00E-02 1.00E-02 0.00E+00

3.0E+05

2.5E+05

2.0E+05

1.5E+05

1.0E+05

5.0E+04

0.0E+00

3

Water Concentration (g/m )

FIGURE 15.1 A plot of WVTR × thickness vs. concentration can be used as an estimate of the dependence of diffusion on concentration from the slope.

Water Vapor Permeability (g/m/sec/mmHg)

Exponential LSMEANS

5.00E-08 4.00E-08 3.00E-08 2.00E-08 1.00E-08 0.00E+00 0

5 10 15 20 Water Vapor Pressure (mmHg)

25

FIGURE 15.2 The relationship between water vapor permeability of chitosan films and water vapor pressure at 25°C.


Exponential

Water Vapor Diffusion (m2/s)

LSMEANS

4.00E-12 3.00E-12 2.00E-12 1.00E-12 0.00E+00 0

5

10

15

20

25

Water Vapor Pressure (mmHg)

FIGURE 15.3 The relationship between water vapor diffusion of chitosan films and water vapor pressure at 25°C.

typical Fickian behavior are associated with the finite rates at which the polymer structure may change in response to the sorption or desorption of penetrant molecules (Crank, 1975). Chitosan films exhibit non-Fickian diffusion behavior. A plot of WVTR times thickness vs. water concentration of the film quantifies and displays this nonideal behavior (Figure 15.1). The initial dry thicknesses were adjusted according to the change in thickness (swelling) due to water vapor sorption at the higher water vapor pressures (Table 15.2). The average percentages of increase in thickness were 11% from 16.3 (69% RH) to 17.8 (75% RH) mmHg of water vapor pressure and 22% from 17.8 (75% RH) to 19.9 (84% RH) mmHg of water vapor pressure. Frisch (1957) obtained an expression for the time lag in linear diffusion through a membrane with a concentration dependent diffusion coefficient without solving the diffusion equation. Frisch’s method determined the relationship between diffuβC sion and concentration in the form of D = D0 e . Stern and Saxena (1980) also assumed that the concentration dependent diffusion coefficient was an exponential function of concentration. Zhou and Stern (1989) stated that this exponential expression for the dependency of concentration by diffusion describes the overall effect of water concentration on the diffusivity in glassy polymer but cannot explain the individual components of the sorption separately. Systems such as hydrophilic chitosan films have strong interaction between water vapor and the film itself, which deviates from ideal behavior by the swelling and plasticizing effects that vary with water vapor partial pressure. Figure 15.3 agrees with Frisch and others, showing that D is an exponential function of water concentration. Water vapor solubility of chitosan films showed a slight decrease from the initial solubility at intermediate water vapor pressures, with a gradual increase at higher water vapor pressures. This gave a U-shaped curve (Figure 15.4). Water concentration inside the film also increased substantially with an increase in water vapor pressure (Table 15.2). Here again, the effect of water vapor pressure on water concentration in chitosan films increased exponentially. Permeability measurements were made at eight levels of water vapor partial pressure. The concentration gradient was, in every case, 0 mmHg on one side of the chitosan film, with the other having the prescribed test water vapor pressure. By using the equation for determining the permeability constant of a multilayer film or laminate (Equation (15.4)), the individual permeability in each range of water vapor


3

Solubility (g/m /mmHg)

1.80E+04 1.60E+04 1.40E+04 1.20E+04 1.00E+04 8.00E+03 6.00E+03 4.00E+03 2.00E+03 0.00E+00 0

5

10

15

20

25

Water Vapor Pressure (mmHg)

FIGURE 15.4 The relationship between water vapor solubility in chitosan films and water vapor pressure at 25°C. 5.00E-08 4.00E-08

Permeability per Layer (g/m/sec/mmHg)

3.00E-08 2.00E-08 1.00E-08

17.84–19.99

16.32–17.84

12.54–16.32

7.77–10.23

10.23–12.54

5.34–7.77

2.68–5.34

0.00–2.68

0.00E+00

Water Vapor Pressure Range (mmHg)

FIGURE 15.5 The water vapor ranges represent the eight different layers of permeability in the chitosan film.

pressure can be calculated (Figure 15.5) and the range of the possible thicknesses of this layer can be determined. In this case, the multilayer film consists of eight layers, which are the eight different water vapor pressure ranges shown in Figure 15.5. This information, which is calculated and not measured, can be used to give some insight into the concentration gradient inside the chitosan film at various levels of water vapor pressure. It is important to keep the layers in order of low to high water vapor pressures arranged in the direction of diffusion through the film. If layers are rearranged, then the calculation of permeability per layer is not possible because of the water vapor pressure dependence of diffusion. Using the data in Figure 15.5, the calculated permeability of each layer can be plotted over a possible range of layer thickness. The equation to calculate the permeability of a laminate or multilayer film was used to determine the permeability


at each layer according to its water vapor partial pressure range. If the individual thickness (xn) and permeability coefficient (Pn) are known for each layer, the permeability of the layered film (PT) can be calculated from the following equations if individual P values are independent of pressure (Crank, 1975; Rogers et al., 1957). PT =

xT x1 x 2 xn + + ⋅⋅⋅ + P1 P 2 Pn

(15.4)

Water Concentration (g/m3)

The thickness range limits are then used to plot the steady state water concentration gradient in the chitosan films at various %RH. Figures 15.6–15.12 graphically represent the steady state concentration gradients across chitosan films at various water vapor partial pressures or %RH at 25°C. From the concentration–distance curves, it can be determined that the pattern of concentration inside the film has a consistent pattern with increasing relative humidity. Water vapor diffusion occurs in the direction from high concentration to low concentration of water vapor. These plots show graphically that when chitosan films reach steady state diffusion, the majority of the concentration distance through

7.00E+04 6.00E+04 5.00E+04 4.00E+04 3.00E+04 2.00E+04 1.00E+04 0.00E+00 0

0.2

0.4

0.6

0.8

1

Distance (Dimensionless)


FIGURE 15.6 The steady-state concentration distributions across the chitosan film when D = 0 β(c) D e from 23 to 0% relative humidity at 25°C.

1.00E+05 8.00E+04 6.00E+04 4.00E+04 2.00E+04 0.00E+00 0

0.2

0.4

0.6

0.8

1





1.20E+05 1.00E+05 8.00E+04 6.00E+04 4.00E+04 2.00E+04 0.00E+00 0

0.2

0.4

0.6

0.8

1




1.60E+05 1.40E+05 1.20E+05 1.00E+05 8.00E+04 6.00E+04 4.00E+04 2.00E+04 0.00E+00 0

0.2

0.4

0.6

0.8

1


3


FIGURE 15.9 The steady-state concentration distributions across the chitosan film when D = 0 β(c) D e from 54 to 0% relative humidity at 25°C. 2.50E+05 2.00E+05 1.50E+05 1.00E+05 5.00E+04 0.00E+00 0

0.2

0.4

0.6

0.8

1



the film is at the higher concentration with a sharp decrease in concentration toward the low concentration end of the curve. This is not typical of most polymer plastic films, but it is the norm for hydrophilic chitosan films. This pattern may be found in other types of hydrophilic films.


3


2.50E+05 2.00E+05 1.50E+05 1.00E+05 5.00E+04 0.00E+00 0

0.2

0.4

0.6

0.8

1




3.00E+05 2.50E+05 2.00E+05 1.50E+05 1.00E+05 5.00E+04 0.00E+00 0

0.2

0.4

0.6

0.8

1



15.4 CONCLUSIONS Water vapor partial pressure and concentration have a positive effect on the WVP and D of chitosan films. These dependencies are non-Fickian in behavior and can be expressed exponentially except for S. Sorption and diffusion of water vapor are dependent on the sorbed concentration of water. Water vapor swells and plasticizes the chitosan film’s polymer structure to increase the mobilities of both the chitosan polymer segments and water penetrant molecules. Deviations from Fickian behavior are the results of increased penetrant–polymer and penetrant–penetrant interaction. From the permeation models and results, there is an indication of two modes of diffusion in the permeation process from 2.68 to 19.99 mmHg of water vapor partial pressure. The first mode, from 2.68 to 7.77 mmHg of pressure, is linear in behavior. At water vapor pressures above 7.77 mmHg pressure, the mode of diffusion is exponential in behavior. This study has shown the importance of reporting the actual water vapor partial pressure gradient parameters because of changes that occur in the chitosan film’s polymer structure due to the plasticizing effect of water vapor. This concept also applies to most biopolymer films. For example, stating a water vapor pressure gradient of 50% instead of the water vapor partial pressure range is not adequate because of the deviations from ideal sorption or diffusion behavior due to the dependency on the


water concentration in the film, which changes with water vapor pressure. The permeability measured from 0 to 50% RH is different from that measured from 50 to 100% RH due to microstructural changes in the film due to water at different water concentrations, although the apparent pressure gradient is the same. From the graphical representation of the steady state concentration–distance curves, the concentration gradient through the chitosan film is not ideal through the film and therefore does not obey Henry’s law. The concentration–diffusion curves of chitosan films demonstrate the merit of using the saturation/differential method (Demorest and Mayer, 1996) for measuring the steady state permeation of chitosan films. By conditioning films to equilibrium moisture content or water concentration, these films reach steady state diffusion faster than if they were initially void of permeant. This occurs because of the lesser change needed in the concentration gradient, as shown in the concentration–distance curves. Chitosan films that are fully conditioned have fewer stresses in them from the relaxation and swelling of the polymer chains of the film as compared to a film that is dry or void of water vapor. This occurs because the film has already been exposed to water vapor in order to reach equilibrium. Chitosan films that are nearer equilibrium moisture conditions before testing resulted in more precise WVP measurements.

NOMENCLATURE WVP WVTR l p1 p2 p1 − p2 D S F ∆C D0 e β c PT xT xn Pn

Water vapor permeability coefficient (g/m/sec/mmHg) 2 Water vapor transmission rate (g/m /d) Film’s thickness (m) Water vapor partial pressure at film’s surface outside the cup (mmHg) Water vapor partial pressure at the underside of the film towards the desiccant (mmHg) Water vapor partial pressure difference (mmHg) 2 Diffusion coefficient (m /sec) 2 Solubility coefficient (g/m /mmHg) 2 Flux (g/m /d) 3 Change in water concentration across the film (g/m ) 2 Diffusion coefficient when concentration is zero (m /sec) The base of natural logarithms (2.71828) Constant characterizing the concentration dependency 3 Concentration of water in film (g/m ) Permeability coefficient of a multilayer film (g/m/sec/mmHg) Total thickness of multilayer film (m) Layer thickness (m) Layer permeability coefficient (g/m/sec/mmHg)

REFERENCES ASTM, Standard test method for water vapor transmission of materials, in ASTM Book of Standards, E96–80, American Society for Testing and Materials, Philadelphia, 1980. ASTM, Standard test method for water vapor transmission rate through plastic film and sheeting using a modulated infrared sensor, in ASTM Book of Standards, F1249, American Society for Testing and Materials, Philadelphia, 1990.


Butler, B.L., Vergano, P.J., Testin R.F., Bunn, J.M., and Wiles, J.L., Mechanical and barrier properties of edible chitosan films as affected by composition and storage, J. Food Sci., 61, 952–955, 1996. Caner, C., Vergano, P.J., and Wiles, J.L., Chitosan film mechanical and permeation properties as affected by acid, plasticizer, and storage, J. Food Sci., 63, 1049–1053, 1998. Crank, J., The Mathematics of Diffusion, Clarendon Press, Oxford, 1975, pp. 1–10, 44–68, 254–265. Demorest, R.L. and Mayer D.W., Advanced measuring methods enhanced WVTR testing. Part II, J. Packag. Technol. Eng., 5, 20–24, 1996. Donhowe, G. and Fennema, O., Edible films and coating: characteristics, formation, definitions, and testing methods, in Edible Coating and Films to Improve Food Quality, Krochta, J.M., Baldwin, E.A., and Nisperos-Carriedo, M.O., Eds., Technomic, Lancaster, PA, 1994, pp. 1–63. Frisch, H.L., The time lag in diffusion, J. Phys. Chem., 61, 93–95, 1957. Hauser, P.M. and McLaren, A.D., Permeation through and sorption of water vapor by high polymers, Ind. Eng. Chem., 40, 112–117, 1948. Labuza, T.P., Moisture Sorption: Practical Aspects of Isotherm Measurement and Use, American Association of Cereal Chemists, St. Paul, MN, 1984. McHugh, T.H. and Krochta, J.M., Permeability properties of edible films, in Edible Coating and Films to Improve Food Quality, Krochta, J.M., Baldwin, E.A., and NisperosCarriedo, M.O., Eds., Technomic, Lancaster, PA, 1994, pp. 140–149. Muzzarelli, R.A.A., Chitin, Pergamon Press, Oxford, 1977. Muzzarelli, R.A.A., Isolati, A., and Ferrero A., Chitosan Membranes: Ion Exchange and Membranes, Vol. 1, Gordon and Breach Science Publishers, London, 1974, pp. 193–196. Park, S., Barrier and Mechanical Properties of Acetylated Chitosan Films, thesis, Clemson University, 1998. Pascat, B., Study of some factors affecting permeability, in Food Packaging and Preservation: Theory and Practice, Mathlouthi, M., Ed., Elsevier Applied Science, London, 1986, p. 7. Rogers, C.E., Permeation of gases and vapours in polymers, in Polymer Permeability, Comyn, J., Ed., Elsevier Science, New York, 1985, pp. 11–69. Rogers, C.E., Stannett, V., and Szwarc, M., Permeability of gases and vapors through composite membranes, Ind. Eng. Chem., 49, 1933–1936, 1957. SAS, release 6.12 TS020, SAS Institute, Inc. Cary, NC, 1996. Schwartzberg, H.G., Modeling of gas and vapour transport through hydrophilic films, in Food Packaging and Preservation: Theory and Practice, Mathlouthi, M., Ed., Elsevier Applied Science, London, 1986, pp. 115–135. Stern, S.A. and Saxena, V., J. Membr. Sci., 7, 47, 1980. Wiles, J.L., The Effect of Relative Humidity on the Steady State Water Vapor Permeability of Chitosan Films, dissertation, Clemson University, 2000. Wiles, J.L., Vergano, P.J., Barron, F.H., Bunn, J.M., and Testin, R.F., Water vapor transmission rates and sorption behavior of chitosan films, J. Food Sci., 65, 1175–1179, 2000. Wong, D.W., Gastineau, F.A., Gregorski, K.S., Tillin, S.J., and Pavlath, A.E., Chitosan–lipid films: Microstructure and surface energy, J. Agric. Food Chem., 40, 540–544, 1992. Zhou, S. and Stern, S.A., The effect of plasticization on the transport of gases in and through glassy polymers, J. Polym. Sci. B Polym. Phys., 27, 205–222, 1989.


16

Water Vapor Permeability, Water Solubility, and Microstructure of Emulsified Starch–Alginate–Fatty Acid Composite Films Y. Wu, C.L. Weller, F. Hamouz, S.L. Cuppett, and M. Schnepf

CONTENTS 16.1 Introduction 16.2 Materials and Methods 16.2.1 Materials 16.2.2 Preparation of Films 16.2.3 Thickness 16.2.4 Conditioning 16.2.5 Water Vapor Permeability (WVP) 16.2.6 Water Solubility (WS) 16.2.7 Scanning Electron Microscopy (SEM) 16.2.8 Statistical Analysis 16.3 Results and Discussion 16.3.1 Water Vapor Permeability 16.3.2 Water Solubility 16.3.3 Microstructure of Starch–Alginate–Fatty Acid Composite Films 16.4 Conclusions Nomenclature Acknowledgments References


16.1 INTRODUCTION In the formation of edible films, polysaccharides and proteins are generally used for their ability to establish polymer interactions and create a continuous network responsible for the mechanical properties of the films (Krochta, 1992; Callegarin et al., 1997). Edible films produced from polysaccharides and proteins, however, are highly sensitive to water and provide limited resistance to moisture transmission due to the substantial inherent hydrophilicity of the film-forming substances and to the considerable amount of hydrophilic plasticizers incorporated into the films (Guilbert, 1986; Krochta, 1992; Gennadios et al., 1994b; Guilbert et al., 1996; Callegarin et al., 1997). In contrast, lipids, such as neutral lipids of glycerides, long-chain fatty acids, waxes, resins, and oils, provide their moisture transfer barrier property for the film because of their hydrophobic nature (Hernandez, 1994; Callegarin et al., 1997), but films made from lipids alone are usually too brittle (Kester and Fennema, 1986; Guilbert, 1986; Krochta, 1992). Composite films of polysaccharide–lipid (Kamper and Fennema, 1984a, b, 1985; Kester and Fennema, 1989; Greener and Fennema, 1989; Hagenmaier and Shaw, 1990; Koelsch and Labuza, 1992; Wong et al., 1992; Debeaufort et al., 1993; Park et al., 1994; Sapru and Labuza, 1994; Callegarin et al., 1997) or protein–lipid (AvenaBustillos and Krochta, 1993; Gennadios et al., 1993; McHugh and Krochta, 1994a, b; Gontard et al., 1994, 1995; Saravia, 1995; Shellhammer and Krochta, 1997b; Rhim et al., 1999), in laminated or emulsion forms, have been developed and have shown increased moisture resistance and improved functionality for edible packaging application. Starch–alginate coatings have been applied on precooked pork chops (Handley et al., 1996; Hargens-Madsen et al., 1995) and ground beef patties (Ma-Edmonds et al., 1995) to inhibit lipid oxidation and the formation of warmed-over flavor. However, the poor moisture barrier property of starch–alginate films hinders them from being used to control moisture loss in precooked meat products. When freestanding starch–alginate films were applied on precooked meat products in a previous study, moisture absorption and film swelling were observed on these films and the packaged meat became dry over time (unpublished data). Mechanical properties and other barrier properties can be lost upon the hydration of the swollen films. Incorporation of fatty acids into starch–alginate films, therefore, may provide hydrophobicity and increase film resistance to water transmission. In this study, the effect of fatty acids (lauric, palmitic, and stearic acids) on the moisture barrier property of starch–alginate films was evaluated in terms of water vapor permeability and water solubility. Microstructure examination was performed to help elucidate the structure–function relation of starch–alginate–fatty acid composite films.

16.2 MATERIALS AND METHODS 16.2.1 MATERIALS CstarEmCap 0637, a lipophilic modified starch, was obtained from Cerestar USA Inc. (Hommond, IN). Sodium alginate was provided by Kelco Division of Merck and Co. (Rahway, NJ). Glycerin was purchased from Mallinckrodt Baker Chemical (Phillipsburg, NJ). Lauric acid (LA), palmitic acid (PA), and stearic acid (SA) were


purchased from Sigma Chemical Co. (St. Louis, MO). Lecithin (Lecigran 5750), an emulsifier, was provided by Riceland Foods Inc. (Little Rock, AK). Ethyl alcohol (USP dehydrated, absolute-200 proof) was purchased from McCormick Distilling Co., Inc. (Weston, MO).

16.2.2 PREPARATION

OF

FILMS

A control (0% fatty acid) starch–alginate film solution was prepared by slowly dissolving 5 g starch and 2 g alginate in a constantly stirred mixture of distilled water (127.3 ml), ethanol (63.6 ml), and glycerin (2.1 g). Additionally, three other starch–alginate–fatty acid film solutions were prepared using LA, PA, and SA. For each starch–alginate–fatty acid film, four levels of each fatty acid (10, 20, 30, or 40% w/w of starch and alginate) were used. An emulsion with each fatty acid was prepared by adding a portion of the fatty acid into the 200 ml starch–alginate film solution with 30% (w/w of fatty acid) of lecithin. The solution was then heated to o boiling (78 C) with stirring to completely melt the fatty acid. All added fatty acids had melting points lower than the boiling point of the film solution. Each boiled solution was homogenized using an Ultra-Turrax T25 homogenizer (Janke & Kunkel GMBH & Co. KG, Staufen, Germany) at 9500 r/min for 2 min, strained through eight-layered cheesecloth (grade 40, Fisher Scientific), and poured onto a leveled Teflon coated glass plate (21 cm × 35 cm). Film thickness was controlled by casting volumes of solutions having the same amount of solids (8.82 g) onto each plate. Films were allowed to dry at ambient conditions for about 24 h. Then they were peeled from the plate, and samples for property testing were cut. Water vapor permeability (WVP) and water solubility testing samples were squares of 7 × 7 and 2 × 2 cm, respectively.

16.2.3 THICKNESS Film thickness was measured to the nearest 2.54 µm (0.1 mil) with a hand-held micrometer (B.C. Ames Co., Waltham, MA). Five thickness measurements were taken on each WVP sample, one at the center and four around the perimeter, and the mean values were used in WVP calculations.

16.2.4 CONDITIONING All film samples for WVP and WS were conditioned for 2 d in an environmental chamber (Model RC-5492, PGC Parameter Generation & Control, Inc., Black Mountain, NC) set at 50% RH and 25°C before testing (ASTM Standard Method D 618-61, ASTM, 1995a).

16.2.5 WATER VAPOR PERMEABILITY (WVP) .

2

WVP (g mm/m ⋅ h ⋅ KPa) was calculated as: WVP = (WVTR ⋅ L)/ ∆p


(16.1)

WVTR was determined gravimetrically using a modified ASTM Method E 96-95 (ASTM, 1995b). Film specimens were mounted on poly(methyl methacrylate) cups filled with distilled water up to 1 cm from the film underside. The cups were placed in an environmental chamber set at 25°C and 50% RH. A fan was operated within the chamber, creating an air velocity of 198 m/min over the surface of the cups to remove permeating water vapors. Weights of the cups were recorded every h for a period of 8 h. Steady state was reached after about 1 h. Slopes of the steady state (linear) portion of weight loss vs. time curves were determined by linear regression 2 to estimate WVTR. Coefficient of determination (R ) for all reported data was 0.99 or greater. In calculating WVP, the effect of resistance of the stagnant air layer between the film undersides and the surface of water in cups was corrected for (McHugh et al., 1993; Gennadios et al., 1994b). For each type of film, WVP measurements were replicated three times.

16.2.6 WATER SOLUBILITY (WS) WS was calculated as the percentage of soluble matter to initial dry matter in each film sample using the method of Gontard et al. (1992), with some modifications. Three randomly selected samples from each type of film were first dried at 105°C for 24 h to determine initial dry matter. After drying, films were immersed in 30 ml of distilled water in a 50 ml beaker. Beakers were covered with Parafilm “M” wrap (American National Can, Greenwich, CT) and stored in an environmental chamber at 25°C for 12 h with occasional gentle stirring. Since all film samples disintegrated into small pieces after soaking in water, the sample solutions were set for another 10 min to allow the undissolved sample pieces to deposit on the bottom of the beakers. The supernatant of the solution (water) was then discarded and the deposited undissolved sample was poured into an aluminum drying dish with the water residue and oven dried at 105°C for 24 h. The weight of solubilized matter was calculated by subtracting the weight of undissolved dry matter from the weight of initial dry matter and expressed as a percentage of the initial dry matter content. Film samples were weighed to the nearest 0.0001 g before and after drying. WS was determined in triplicate for each type of film.

16.2.7 SCANNING ELECTRON MICROSCOPY (SEM) Microstructural characteristics of starch–alginate film samples were examined using an AMRAY 1000A electronic microscope (AMRAY Inc., Bedford, MA), while those of starch–alginate–fatty acid composite film samples were examined using a JSM6100 scanning microscope (JEOL, Tokyo, Japan). All film samples were prepared using the same methods as for preparing films for WVP and WS testing, but without adding glycerin. Samples were mounted on an aluminum stub with copper tape, first coated with carbon by vacuum evaporation on a Denton DV-502 vacuum evaporator (Denton Inc., Cherry Hill, NJ) and then sputter coated with gold–palladium alloy with a EBTEC sputter coater (EBTEC Co., Agawam, MA). Samples were examined using an accelerating voltage of 20 kV. Images were recorded on Polaroid film (Type o 665) and photographed at a tilt angle of 60 to the electron beam for views in the cross section.


16.2.8 STATISTICAL ANALYSIS Means and standard deviations on a completely randomized design with three replications were determined using the Genera Linear Models procedure in the SAS program (SAS Institute Inc., 1990). Fisher’s protected least significant differences (Fisher’s protected LSD) were calculated to indicate differences among mean values. Significance was accepted at a level of P < 0.05.

16.3 RESULTS AND DISCUSSION 16.3.1 WATER VAPOR PERMEABILITY The WVP values, along with the thickness values and actual RH conditions at the undersides of films during testing, of control starch–alginate film and emulsified composite starch–alginate–fatty acid films are presented in Table 16.1. The WVP of . 2 the control starch–alginate film was 5.14 g mm/m ⋅ h⋅KPa. This value was of the 2 same order of magnitude as mean WVP values of 2.20 g⋅ mm/m ⋅h ⋅ KPa and . 2 2.99–8.78 g mm/m ⋅ h⋅ KPa previously reported for polysaccharide films made from methyl cellulose (Gennadios et al., 1994a) and chitosan (Rhim et al., 1998), respectively, which were tested with the same method and same conditions, i.e., 25°C and 100/50% RH gradient, as in the present study. TABLE 16.1 Water Vapor Permeability (WVP) and Water Solubility (WS) of Starch–Alginate Films with Lauric Acid (LA), Palmitic Acid (PA), or Stearic Acid (SA) Fatty Acid in Film (% of Starch–Alginate) Control LA

PA

SA

0 10 20 30 40 10 20 30 40 10 20 30 40

a

Thickness (µm)

RH Inside Cup (%)

106 148 128 127 126 101 96 103 109 90 106 116 111

77.1 ± 1.06 f 84.4 ± 1.20 de 88.9 ± 1.16 cde 89.9 ± 1.78 bcd 90.5 ± 1.28 f 85.2 ± 0.52 f 84.8 ± 1.22 f 85.5 ± 0.25 ef 87.3 ± 3.80 f 85.0 ± 2.35 de 89.7 ± 2.44 b 93.3 ± 0.46 bc 92.6 ± 1.08

g

a

a

WVP 2 · (g·mm / m · h·KPa) b

5.14 ± 0.05 b 4.74 ± 1.11 cde 2.52 ± 0.07 cde 2.22 ± 0.47 efg 2.04 ± 0.31 c 2.91 ± 0.19 cd 2.86 ± 0.24 c 2.91 ± 0.18 cde 2.60 ± 0.92 cde 2.69 ± 0.58 def 2.07 ± 0.25 g 1.25 ± 0.10 fg 1.30 ± 0.11

a

WS (%) 54.53 52.78 52.84 47.60 46.29 51.40 47.89 44.74 45.47 47.26 43.51 38.09 35.80

± ± ± ± ± ± ± ± ± ± ± ± ±

b

2.62 bc 1.84 bc 0.93 d 0.57 de 2.87 c 0.61 d 0.53 ef 0.92 def 1.11 de 0.98 f 2.52 g 0.32 g 1.70

Mean of three replicates ± standard deviation. Measured at 25°C and 100/50% RH gradient. RH at film undersides inside cups and WVP were corrected to account for resistance of stagnant air layer between film underside and water surface (Gennadios et al., 1994b). b–g Means within a column are significantly (P < 0.05) different.


With the exception of films with 10% LA, WVP of starch–alginate films was 2 significantly reduced by incorporation of LA, PA, or SA to as low as 1.25 g⋅ mm/m ⋅ h⋅ KPa for films with 30% SA. Decreases in WVP of starch–alginate films may be attributed to the increased hydrophobicity and the structural complexity imparted into starch–alginate films by the fatty acids, which is further discussed in the SEM result section in this work. Fatty acids have been successfully used to reduce WVP of emulsified cellulose ether films (Kamper and Fennema, 1984a,b; Hagenmaier and Shaw, 1990; Koelsch and Labuza, 1992; Sapru and Labuza, 1994) and chitosan films (Wong et al., 1992). The reduction of WVP has also been observed with addition of fatty acids into wheat gluten films (Gontard et al., 1994; Derksen et al., 1995), whey protein films (McHugh and Krochta, 1994b), and soy protein isolate films (Rhim et al., 1999) using emulsion technique. Generally, with increasing length of the lipid hydrocarbon chain, as well as concentration of the lipid, resistance of a lipid film to water vapor transmission increases (Fennema et al., 1993; Callegarin et al., 1997; Shellhammer and Krochta, 1997a). As shown in Table 16.1, the effect of chain length on WVP was mainly shown by the significant differences in WVP values found between SA films and LA and PA films in the present study. In general, incorporation of SA into starch–alginate films resulted in the lowest WVP values at all levels when compared with the incorporation of LA and PA. The WVP values of films with PA and LA were similar, except at the level of 10%, where PA was more effective than LA in reducing WVP. WVP decreased significantly as fatty acid concentration increased in films with LA and SA. However, increasing PA concentration did not significantly reduce WVP. For films with LA and PA, a fatty acid content of 40% resulted in the lowest WVP, while the lowest WVP was obtained in films with 30% of SA. The overall WVP change pattern in this study suggests that there is an optimum concentration of fatty acid at which WVP can be lowered most effectively. According to Kamper and Fennema (1984a), increasing the stearic acid concentration in emulsified hydroxypropyl methylcellulose–fatty acid films decreased their WVP until an optimum was reached. Hagenmaier and Shaw (1990) found that hydroxypropyl methylcellulose–fatty acid emulsion films had decreased WVP as the fatty acid concentration increased from 23 to 45%. However, an increase in permeability occurred above a volume fraction of stearic acid of 46%. Similar results have been reported by Sapru and Labuza (1994) in methyl cellulose–stearic acid composite films, by Gontard et al. (1994) in wheat gluten–lipid composite films, and by Rhim et al. (1999) in soy protein isolate films composed with LA and PA. Sapru and Labuza (1994) found that when the stearic acid volume was above 22%, the increase in WVP was attributed to inadequate filling of the void volume within the stearic acid crystallites by the methyl cellulose–polyethylene glycol matrix.

16.3.2 WATER SOLUBILITY Similar to WVP, WS is an indication of the hydrophilicity of a film. Moreover, WS is affected by the chemical nature and the structural integrity of films. As shown in Table 16.1, incorporation of fatty acids significantly reduced WS of starch–alginate films by as much as 34% with 40% stearic acid. The overall WS decreased as the


chain length of fatty acid increased, as indicated by significant differences between film types. WS reduction was more pronounced at increasing levels of fatty acid in the films. It was also observed that films, with or without fatty acids, disintegrated easily after 12 h of soaking. WS, apparently, is more affected by the hydrophilic nature of films rather than by their structural integrity. WS (total soluble matter) of soy protein–fatty acid films has been reported to be less than that of soy protein films without fatty acids (Gennadios et al., 1997). Lai et al. (1997) determined water absorption rate, another property related to hydrophilicity of zein–fatty acid sheets, and reported that the amount of water absorbed deceased as fatty acid levels increased in the sheets. However, a sharp increase in WS beyond a threshold lipid material content has been observed in wheat gluten–lipid composite films (Gontard et al., 1994) and soy protein–fatty acid composite films (Rhim et al., 1999). These discrepancies may be attributed to the types of matrix materials used for film formation, the types and amounts of emulsifiers added into the films, and the differences in film homogeneity and structure.

16.3.3 MICROSTRUCTURE OF STARCH–ALGINATE–FATTY ACID COMPOSITE FILMS To better understand the structure–function relation existing in starch–fatty acid composite films, scanning electron microscopic examination was performed on the control starch–alginate film and films with 30% fatty acids. Variations in microstructure among these films were evident from a comparison of the SEM photomicrographs, as shown in Figure 16.1. The structure of starch–alginate film without the addition of fatty acid was compact and dense, and the film surface had a smooth contour free from pores and cracks (Figure 16.1A). One of the basic physical properties of carbohydrates is that they attract water. Hydrophilicity is expected due to their numerous hydroxyl groups, which form hydrogen bonds with water molecules (Whistler and Daniel, 1985). It is also believed that water molecules can interact with the matrix and increase the permeation rate when they penetrate hydrophilic materials (Pascat, 1986). The hydrophilic nature of the starch–alginate film, therefore, accounts for the water permeability value obtained in the present study (Table 16.1). When fatty acids were incorporated into the starch–alginate film, the film microstructure was changed dramatically (Figure 16.1B, C, and D). This change in the film structure may play a key role in the barrier properties of the composite films, as evidenced by the reduction in WVP and WS of the films. For all fatty acid composite films, there was no obvious migration of the fatty acid to the air–film surface, as noted by other authors using methylcellulose derivatives as the base component (Greener and Fennema, 1989; Vojdani and Torres, 1989, 1990). Microstructures in the cross sections of these films showed some differences. A stratified flaky structure was found in the starch–alginate–LA film (Figure 16.1B). This layer structure could be easily observed visually. A similar, but more compact, stacking layer structure has been noted in a chitosan–LA film (Wong et al., 1992). The arrangement of flaky layers of starch–alginate films provided an extended network of channels lined with hydrophobic molecules. The lower WVP of starch–alginate–LA films



(a)

(b)

(c)

(d)

FIGURE 16.1 Photomicrographs taken by scanning electron microscopy of starch–alginate–fatty acid composite films (fatty acid = 30% starch–alginate, w/w). A: Pure starch–alginate film; B: Starch–alginate–lauric acid film; C: Starch–alginate–palmitic acid film; D: Starch–alginate– stearic acid film (bar = 20.8 µm).

(Table 16.1) suggests that this network hindered the penetration of water molecules through the films. The structure of starch–alginate–PA showed a nonhomogeneous or irregular layered sponge-like structure (Figure 16.1C). Although the network of this structure is more complicated than that of films with LA, which may have contributed to the lower WS in films with PA, the nonhomogeneous distribution of PA may account for a WVP that did not significantly differ from that of films with different contents of PA and from that of LA films. PA has been incorporated into zein sheets by Lai et al. (1997). They reported that layered structures were progressively formed, and that the structures became more compact when the concentration of PA was increased from 0 to 50% of corn zein protein in zein sheets. This change in structure was shown to coincide with the decreased water absorption of the zein sheets. In starch–alginate–SA films, the structure was somewhat similar to that shown by PA films (Figure 16.1D). However, the layered sponge-like structure of SA films was highly channeled, and SA was evenly distributed in the continuous starch–alginate matrix, which may set up a more tortuous path or increased path length for diffusing water molecules than those found in the other films. Consequently, films with SA exhibited the lowest WVP and WS values among all composite films. It is believed that a fine particle size and even distribution of lipids are important to the effectiveness of an emulsion film (Wong et al., 1992; McHugh and Krochta, 1994a; Debeaufort and Voilley, 1995). The existence of stearic hindrance and tortuosity in the film structure also influences the permeability of films (Holton et al., 1994). According to Koelsch (1994), the high degree of inherent tortuosity that resulted from an even interlocking network within methylcellulose–SA emulsified composite films was responsible for the low WVP values of the films. In these films, the interlocking network of SA chains within the support matrix reached a maximum (i.e., maximum tortuosity) at about 30% SA, a concentration similar to our optimum SA level for starch–alginate–fatty acid films. Besides the dispersion of fatty acids in the base matrix, complex formation between starch amylose/amylopectin fractions and fatty acids may also account for the changed microstructures, and thus the lowered WVP and WS values for the fatty acid composite films. Fatty acids can form inclusion complexes with helical amylose and possibly with the longer outer chains of amylopectin. Such complexes resist entry of water (Whistler and Daniel, 1985). Gray and Schoch (1962) demonstrated that the presence of fatty acids decreased the swelling and solubilization of starch by forming a fatty acid–amylopectin complex. Lagendijk and Pennings (1970) found that the amount of fatty acid complexed with amylopectin increased linearly with increasing fatty acid chain length.

16.4 CONCLUSIONS Moisture barrier properties of starch–alginate films can be improved substantially by incorporation of LA, PA, and SA. The reduction in WVP and WS of starch–alginate films was more pronounced as the fatty acid chain length increased and at increasing levels of fatty acids in general. The tortuous network formed by the interaction of fatty acids and the starch–alginate matrix may contribute improved barrier properties.


The more uniform the distribution of the fatty acid molecules, and the higher degree of the tortuosity of the network formed by the fatty acids in the film’s microstructure, the better the moisture barrier property of the film.

NOMENCLATURE WVP WVPR L ∆p

2

Water vapor permeability (g⋅mm/m ⋅ h⋅ KPa) 2 Measured water vapor transmission rate through a film (g/m ⋅ h) Film thickness (m) Partial water vapor pressure difference across the two sides of a film (Pa)

ACKNOWLEDGMENTS We express special appreciation to Tom Bargar in the Department of Veterinary and Biomedical Sciences at the University of Nebraska–Lincoln for assistance in scanning electron microscopy.

REFERENCES ASTM, Standard practice of conditioning plastics and electrical insulating materials for testing (D618-61), in Annual Book of ASTM Standards, Vol. 80, American Society for Testing and Materials, Philadelphia, 1995a, 37–39. ASTM, Standard test methods for water vapor transmission of materials (E 96–95), in Annual Book of ASTM Standards, Vol. 4.06, American Society for Testing and Materials, Philadelphia, 1995b, 697–704. Avena-Bustillos, R.J. and Krochta, J.M., Water vapor permeability of caseinate-based edible films as affected by pH, calcium crosslinking and lipid content, J. Food Sci., 58, 904–907, 1993. Callegarin, F., Gallo, J.-A.Q., Debeaufort, F., and Voilley, A., Lipids and biopackaging, J. Am. Oil Chem. Soc., 74, 1183–1192, 1997. Debeaufort, F., Martin-Polo, M., and Voilley, A., Polarity, homogeneity and structure affect water vapor permeability of model edible films, J. Food Sci., 58, 426–434, 1993. Debeaufort, F. and Voilley, A., Effect of surfactants and drying rate on barrier properties of emulsified edible films, Int. J. Food Sci. Technol., 30, 183–190, 1995. Derksen, J.T.P., Cuperus, F.P., and Kolster, P., Paints and coatings from renewable resources, Industr. Crop. Prod., 3, 225–236, 1995. Fennema, O., Donhowe, I.G., and Kester, J.J., Edible films: barriers to moisture migration in frozen foods, Food Austr., 45, 521–525, 1993. Gennadios, A., McHugh, T.H., Weller, C.L., and Krochta, J.M., Edible coatings and films based on proteins, in Edible Coatings and Films to Improve Food Quality, Krochta, J.M., Baldwin, E.A., and Nisperos-Carriedo, M., Eds., Technomic, Lancaster, PA, 1994a, pp. 201–278. Gennadios, A., Weller, C.L., and Gooding, C.H., Measurement errors in water vapor permeability of highly permeable, hydrophilic edible films, J. Food Eng., 21, 395–409, 1994b. Gennadios, A., Weller, C.L., and Hanna, M.A., Soy protein/fatty acids films and coatings, Inform., 8(622), 624, 1997.


Gennadios, A., Weller, C.L., and Testin, R.F., Modification of physical and barrier properties of edible wheat gluten-based films, Cereal Chem., 70, 426–429, 1993. Gontard, N., Duchez, C., Cuq, J-L., and Guilbert, S., Edible composite films of wheat gluten and lipids: water vapor permeability and other physical properties, Int. J. Food Sci. Technol., 29, 39–50, 1994. Gontard, N., Duchez, C., Cuq, J-L., and Guilbert, S., Water vapor permeability of edible bilayer films of wheat gluten and lipids, Int. J. Food Sci. Technol., 30, 49–56, 1995. Gontard, N., Guilbert, S., and Cuq, J-L., Edible wheat gluten films: influence of the main process variables on film properties using response surface methodology, J. Food Sci., 57, 190–195, 199, 1992. Gray, V.M. and Schoch, T.J., Effects of surfactants and fatty adjuncts on the swelling and solubilization of granular starches, Starch, 14, 239–246, 1962. Greener, I.K. and Fennema, O., Evaluation of edible, bilayer films for use as moisture barriers for food, J. Food Sci., 54, 1400–1406, 1989. Guilbert, S., Technology and application of edible protective films, in Food Packaging and Preservation Theory and Practice, Mathlouthi, M., Ed., Elsevier Applied Science, New York, 1986, pp. 371–394. Guilbert, S., Gontard. N., and Gorris, L.G.M., Prolongation of the shelf-life of perishable food products using biodegradable films and coatings, Lebensm.-Wiss. Technol., 29, 10–17, 1996. Hagenmaier, R.D. and Shaw, P.E., Moisture permeability of edible films made with fatty acid and (hydroxypropyl) methylcellulose, J. Agric. Food Chem., 38, 1799–1803, 1990. Handley, D., Ma-Edmonds, M., Hamouz, F., Cuppett, S., Mandigo, R., and Schenpf, M., Controlling oxidation and warmed-over flavor in precooked pork chops with rosemary oleoresin and edible film, in Natural Antioxidants: Chemistry, Health Effect, and Application, Shahidi, F., Ed., AOCS Press, Champaign, IL, 1996, pp. 311–318. Hargens-Madsen, M., Schnepf, M., Hamouz, F., Weller, C., and Roy, S., Use of edible films and tocopherols in the control of warmed-over flavor, J. Am. Diet. Assoc., 95, A-41, 1995. Hernandez, E., Edible coatings from lipids and resins, in Edible Coatings and Films to Improve Food Quality, Krochta, J.M., Baldwin, E.A., and Nisperos-Carriedo, M., Eds., Technomic, Lancaster, PA, 1994, pp. 279–303. Holton, E.E., Asp, E.H., and Zottola, E.A., Corn-starch-containing polyethylene film used as food packaging, Cereal Foods World, 39, 237–241, 1994. Kamper, S.L. and Fennema, O., Water vapor permeability of an edible, fatty acid, bilayer film, J. Food Sci., 49, 1482–1484, 1984a. Kamper, S.L. and Fennema, O., Water vapor permeability of edible bilayer films, J. Food Sci., 49, 1478–1481, 1984b. Kamper, S.L., and Fennema, O., Use of an edible film to maintain water vapor gradients in foods, J. Food Sci., 50, 382–384, 1985. Kester, J.J. and Fennema, O., Edible films and coatings: a review, Food Technol., 40, 47–49, 1986. Kester, J.J. and Fennema, O., An edible film of lipids and cellulose ethers: barrier properties to moisture vapor transmission and structural evaluation, J. Food Sci., 54, 1383–1389, 1989. Koelsch, C., Edible water vapor barrier: properties and promise, Trends Food Sci. Technol., 51, 76–81, 1994. Koelsch, C.M. and Labuza T.P., Functional, physical and morphological properties of methyl cellulose and fatty acid-based edible barriers, Lebensm.-Wiss. Technol., 25, 404–411, 1992. Krochta, J.M., Control of mass transfer in foods with edible coatings and films, in Advances in Food Engineering, Singh, R.P. and Wifakaratakusumah, M.A., Eds., CRC Press, Boca Raton, FL, 1992, pp. 517–538.


Lagendijk, J. and Pennings, H.J., Relation between complex formation of starch with monoglycerides and the firmness of bread, Cereal Sci. Today, 15, 354–356, 365, 1970. Lai, H-M., Padua, G.W., and Wei, L.S., Properties and microstructure of zein sheets plasticized with palmitic and stearic acids, Cereal Chem., 74, 83–90, 1997. Ma-Edmonds, M., Hamouz, F., Cuppett, S., Mandigo, R., and Schnepf, M., Use of rosemary oleoresin and edible film to control warmed-over flavor in precooked beef patties, abstract, in Book of Abstracts, Institute of Food Technologists, Chicago, 1995, pp. 139. McHugh, T.H., Avena-Bustillos, R., and Krochta, J.M., Hydrophilic edible films: modified procedure for water vapor permeability and explanation of thickness effects, J. Food Sci., 58, 899–903, 1993. McHugh, T.H. and Krochta, J.M., Dispersed phase particle size effects on water vapor permeability of whey protein beeswax edible emulsion films, J. Food Process. Preserv., 18, 173–188, 1994a. McHugh, T.H. and Krochta, J.M., Water vapor permeability properties of edible whey protein–lipid emulsion films, J. Am. Oil Chem. Soc., 71, 307–312, 1994b. Park, J.W., Testin, R.F., Park, H.J., Vergano, P.J., and Weller, C.L., Fatty acid concentration effect on tensile strength, elongation, and water vapor permeability of laminated edible films, J. Food Sci., 59, 916–919, 1994. Pascat, B., Study of some factors affecting permeability, in Food Packaging and Preservation: Theory and Practice, Mathlouthi, M., Ed., Elsevier Applied Science, London, 1986, pp. 7–24. Rhim, J-W, Weller, C.L., and Ham, K-S., Characteristics of chitosan films as affected by the type of solvent, Food. Sci. Biotechnol., 7, 263–268, 1998. Rhim, J-W, Wu, Y., Weller, C.L., and Schnepf, M., Physical characteristics of emulsified soy protein–fatty acid composite films, Sci. Aliment., 19, 57–71, 1999. Sapru, V. and Labuza, T.P., Dispersed phase concentration effect on water vapor permeability in composite methyl cellulose stearic acid edible films, J. Food Process. Preserv., 18, 359–368, 1994. Saravia, R.A., Sorghum wax and selected applications, thesis, University of Nebraska, Lincoln, 1995. SAS Institute, Inc., SAS User Guide, version 6.0, SAS Institute Inc., Cary, NC, 1990. Shellhammer, T.H. and Krochta, J.M., Edible coatings and film barriers, in Lipid Technologies and Applications, Gunstone, F.D. and Padley, F.B., Eds., Marcel Dekker, Inc., New York, 1997a, pp. 453–479. Shellhammer, T.H. and Krochta, J.M., Whey protein emulsion film performance as affected by lipid type and amount, J. Food Sci., 62, 390–394, 1997b. Vojdani, F. and Torres, A., Potassium sorbate permeability of methylcellulose and hydroxypropyl methylcellulose multi-layer films, J. Food Process. Preserv., 13, 417–430, 1989. Vojdani, F. and Torres, A., Potassium sorbate permeability of methylcellulose and hydroxypropyl methylcellulose coatings: effect of fatty acids, J. Food Sci., 55, 841–846, 1990. Whistler, R.L. and Daniel, J.R., Carbohydrates, in Food Chemistry, Fennema, O.R., Ed., Marcel Dekker, Inc., New York, 1985, pp. 69–137. Wong, D.W.S., Gastineau, F.A., Gregorski, K.S., Tillin, S.J., and Pavlath, A.E., Chitosan-lipid films: microstructure and surface energy, J. Agric. Food Chem., 40, 540–544, 1992.


17

Mass Transport Phenomena during the Recovery of Volatile Compounds by Pervaporation T. Schäfer and J.G. Crespo

CONTENTS 17.1 Introduction 17.2 The Principles of Pervaporation 17.2.1 Nonideal Transport Phenomena at the Membrane Surfaces: Concentration Polarization 17.2.2 Nonideal Transport Phenomena in the Membrane Polymer: Swelling and Flux Coupling 17.3 Technical Aspects of Pervaporation 17.4 Advantages and Applications of Pervaporation in the Food Industry 17.4.1 Hydrophilic Pervaporation 17.4.2 Organophilic Pervaporation 17.5 Conclusions References

17.1 INTRODUCTION This chapter presents pervaporation as a membrane separation process for the selective removal and recovery of dilute compounds from a bulk liquid. First, the principle of pervaporation will be described, followed by a discussion of its main transport phenomena and a mathematical description of ideal conditions. Pervaporation is, at first sight, not a complicated process. It is, however, rather versatile. This is on the one hand an advantage, because pervaporation can be adapted and integrated in process lines with considerable flexibility. On the other hand, this versatility has led to an abundance of different individual applications that cannot readily be compared. Recently, attempts have been made in the literature to search for empirical rules on


pervaporation in individual situations. It is, however, the authors’ opinion that instead of empirical rules, a concise but comprehensive understanding of the scientific background of pervaporation fosters the creativity necessary for a flexible and appropriate process design. The essentials of this background will be presented for ideal conditions in the first section, “The Principles of Pervaporation,” in conjunction with a discussion of the most relevant nonideal phenomena. The second part of this chapter, “Technical Aspects of Pervaporation,” will demonstrate how the theory is put into practice. The final section will illustrate the theoretical information provided by selected case studies with special regard to applications in the food industry. The potential, as well as the limits, of pervaporation will be discussed in comparison with other separation techniques, and it will be shown how careful process design allows the application of pervaporation to diverse separation problems.

17.2 THE PRINCIPLES OF PERVAPORATION Pervaporation is a membrane separation process that employs dense, nonporous membranes for the selective separation of dilute solutes from a liquid bulk solvent (Néel, 1991). The separation concept of pervaporation is based on the molecular interaction between the feed components and the dense membrane polymer. This differs from pressure-driven membrane separation processes involving porous membranes, such as microfiltration, where the general separation concept is primarily based on molecular size exclusion. Pervaporation is a process very similar to vapor permeation, the only difference being that in the latter the feed is not liquid, but a vapor. It should, however, be clearly distinguished from membrane distillation (Luque de Castro and GámizGarcía, 2000), in which a porous membrane is in contact with a liquid feed, and feed components evaporate through the membrane pores. In this case, the membrane polymer does not have any selective impact on the solute transport, but merely serves as a mechanical barrier (Lawson and Lloyd, 1997). The principle of pervaporation is illustrated in Figure 17.1. A nonporous membrane separates a liquid feed, commonly close to atmospheric pressure, from a downstream compartment in which a vacuum is applied. When the feed contacts the membrane, the solutes (denoted i in Figure 17.1) sorb in the membrane surface due to solute–polymer interactions (Figure 17.1, bottom, I). These solute–polymer interactions imply that the solvating power of the polymer is higher with regard to some solutes than that of the bulk solvent. Under ideal conditions, a thermodynamic equilibrium will be reached when the chemical potential of the solute i is equal in the membrane surface and the liquid phase adjacent to it. The sorption of these solutes at the membrane surface creates a solute concentration gradient across the membrane, resulting in a diffusive net flux of solute across the membrane polymer (Figure 17.1, bottom, II). In pervaporation, any solute that has diffused toward the membrane downstream surface is ideally instantaneously desorbed and subsequently removed by a vacuum applied on the downstream side of the membrane (Figure 17.1, bottom, III). As a consequence, the solute concentration on the membrane downstream surface remains practically zero, and a maximum concentration gradient between the two membrane surfaces is maintained. As a result, the diffusive net flux across the membrane is maximal.


µi feed

liquid phase

>>

µi permeate

vapor phase (vacuum)

(ambient pressure)

i i

i

pfeed

pperm

i

i

i i highly porous support

selective, nonporous membrane

I

II

i

i i i

i i

i

i

i

i i

i

i

i

i

i

i

i

i i

i

i

i

i

i i

i

III

i i

i

i

i

i

i i

i

i

i

ii i i

vacuum

FIGURE 17.1 The principle of solute transport across the membrane in pervaporation.

If the vacuum is not sufficiently low to desorb all of the solutes reaching the membrane downstream surface, the concentration of solute i at the membrane downstream surface will not be equal to zero. As a consequence, the concentration gradient will decrease and so will the diffusive net flux across the membrane. The driving force in pervaporation is thus the gradient of the concentration, or more precisely of the chemical potential, of a solute across the selective membrane. It should be noted that the role of the vacuum in pervaporation is merely the efficient desorption and removal of solutes from the membrane downstream surface, hence maintaining the driving force of the process. The pressure difference between the two sides of the membrane does not directly affect the transport of an individual component i within the membrane polymer, as is the case in pressure-driven processes involving porous membranes. Ideally, the molecular motion of a component i within the membrane polymer is purely diffusive and thus independent of any operating conditions beyond the membrane surfaces. Of course, these operating conditions contribute to determining the concentration of component i on the membrane surfaces, according to which, in turn, a certain diffusive net flux of this component across the membrane results. To ensure the maximal concentration gradient, the downstream pressure must be low enough to remove any solute at the membrane downstream face. Because the solute i leaves the membrane downstream surface due to the vacuum in the vapor


state, the downstream concentration of solute i is more conveniently expressed in terms of its partial pressure, according to Dalton’s law: p ip = y i ⋅ p p

(17.1)

p

where pi is the partial pressure of i in the permeate, yi is the molar fraction of I, p and p is the total downstream pressure. The solute feed concentration can be expressed in terms of its partial pressure in the feed by p if = x i ⋅ γ i ⋅ p 0i

(17.2) f

Equation (17.2) is a modified Raoult’s law where pi is the partial pressure of a compound i in the feed (Pa), xi is the molar fraction of I, and γi is the activity coefficient of i in the feed. The saturated vapor pressure of i (Pa) at a given 0 temperature is denoted by pi . The activity coefficient γi accounts for nonideal behavior of the solute in the feed solution as, for example, in the case of aqueous salt solutions (increased activity coefficient of the solute due to the salting-out effect) or aqueous ethanol solutions (decreased activity coefficient of the solute due to a co-solvent effect of ethanol). For the partial pressure gradient of solute i across the membrane, it hence follows: ∆p i = p if − p ip = x i ⋅ γ i ⋅ p 0i − y i ⋅ p p

(17.3)

Rearranging Equation (17.3) gives the minimum total downstream pressure p f necessary for ensuring flux of a defined solute i across the membrane, i.e., pi < pi: pp
= βimol pp xi

(17.9)

mol

where βi is defined as the molar enrichment factor of i. If an enrichment of mol compound i is desired in the permeate, βi has to be greater than unity, and hence has the quotient yi/xi. An enrichment of compound i in the permeate is thus obtained by a low downstream pressure, a high saturated vapor pressure of compound i, or its high and positive activity coefficient in the feed solution. It is common to use the enrichment factor βi as the ratio of the mass concentration of i in the permeate, −3 −3 p f ci (kg·m ), to the mass concentration of i in the feed, ci (kg·m ), rather than the mol molar enrichment factor βi : cip = βi cif

(17.10)

The ratio of the enrichment factors of a compound i and a compound j indicates the selectivity αij that a membrane has for compound i in comparison with compound j: cip

βi cf = p i = α ij βj cj c fj

(17.11)

For a given separation problem of a solute i from a bulk solvent j, the partial flux Ji, or more generally the permeability Pi, the enrichment βi, and the selectivity αij, are the parameters that characterize the separation performance of a pervaporation process. When deriving the equations mentioned above, some assumptions were implicitly made with regard to the ideal transport of solute to, across, and away from the membrane. In pervaporation, nonideal transport phenomena can occur in the feed liquid phase adjacent to the membrane upstream surface, within the membrane polymer, and in the vapor phase adjacent to the membrane downstream face. All of these phenomena will be discussed in the following sections.


17.2.1 NONIDEAL TRANSPORT PHENOMENA AT THE MEMBRANE SURFACES: CONCENTRATION POLARIZATION In Equations (17.7) and (17.8), perfect mixing of the feed bulk liquid and instant removal of solutes leaving the membrane downstream face were assumed. If these assumptions are not valid, boundary layers will develop on both sides of the membrane, affecting the solute transport to and away from the respective membrane surface. When feed components sorb in the membrane, a local concentration gradient develops in the liquid phase adjacent to the membrane upstream face. Due to this gradient, transport of components from the bulk liquid into the liquid–membrane interface occurs, replenishing the components sorbed in the membrane. This transport of components across the liquid phase adjacent to the membrane can be either convective or diffusive, depending on the hydrodynamic conditions over the membrane surface. If the flux of a solute i across the liquid phase toward the membrane is lower than that across the membrane, solute i will be depleted in the liquid phase over the membrane upstream surface, resulting in a liquid solute concentration lower than that in the bulk feed (Figure 17.3A). Because the partitioning of the solute into the membrane is determined by the solute concentration in the liquid at the membrane upstream surface, the concentration of i in the membrane upstream surface, Equation (17.5), will be lower than expected according to its bulk feed concentration, as will the concentration gradient over the membrane and hence the trans-membrane flux. This phenomenon is denoted concentration polarization and affects mainly the fluxes of compounds of high sorption coefficient, even under turbulent hydrodynamic conditions over the membrane, as will be shown. Under poor hydrodynamic conditions, a stagnant boundary layer forms over the membrane in which the solute transport is purely diffusive. Similar to the mass transport equation developed above for the solute transport across the membrane,

c mem,f i

c mem,f i c

mem,bl i

c mem,p i

c l,bulk i

c mem,p i

c l,bulk i

c ip

c ip

c l,bl i concentration boundary layer thickness δf

c mem,pbl i

membrane thickness z

(a)

membrane thickness z

concentration boundary layer thickness δp

(b)

FIGURE 17.3 The ideal concentration profile of a solute i in the concentration boundary layer and in the membrane for the case of concentration polarization on (A) the membrane upstream side and (B) the membrane downstream side. The dotted line indicates the ideal concentration profile without any concentration polarization.


the diffusive solute flux across this boundary layer can be described as

J i = Dif ⋅ f

c f − cil,bl dc bl = Dif ⋅ i δ dz

(17.12) l,bl

where ci is the bulk liquid feed concentration of solute i, ci is the concentration of f solute i in the liquid adjacent to the membrane surface, Di is the diffusion coefficient of i in the liquid boundary layer, and δ is the concentration boundary layer thickness. Equation (17.12) implies that no solute–solute interactions occur in the boundary layer and is therefore only valid for dilute solutions. A similar phenomenon as described above can be found on the membrane downstream face. Because in pervaporation one aims at employing selective membranes as thin as possible for obtaining high fluxes (see Equation 17.6), most membranes are composites consisting of a thin selective membrane and a macroporous support for mechanical stability and easier handling. The macroporous support faces the downstream compartment (Figure 17.1) and ideally should not interact with the solutes. During pervaporation, however, the narrow pores of the support constitute a mass transport resistance to the vacuum. This resistance results in a hindered transport of solutes away from the membrane downstream surface, causing an accumulation of solute in the pores of the support and hence a local increase of the solute concentration and thus downstream pressure. Assuming thermodynamic equilibrium at the membrane downstream face, this will cause the solute concentration to increase in the membrane downstream surface (see Equation 17.5), resulting in a lower concentration gradient and hence a reduced flux across the membrane. Especially if the solute possesses a low partial pressure, the pressure drop in the macroporous support can cause the transmembrane flux to cease totally.

17.2.2 NONIDEAL TRANSPORT PHENOMENA IN THE MEMBRANE POLYMER: SWELLING AND FLUX COUPLING In Equation (17.5), a Henry-type sorption of the solute in the membrane polymer was assumed, i.e., the sorption coefficient was assumed linearly dependent on the solute concentration. When sorbing in the membrane polymer, the solutes cause a membrane swelling, which is the more pronounced the higher the degree of solute–polymer interaction or the higher the solute concentration in the feed. At high solute feed concentrations, the membrane polymer can swell to such an extent that its intrinsic properties are significantly altered and strong nonideal interactions occur. These interactions and the swelling of the polymer can be described for complex mixtures by the Flory–Huggins theory (Flory, 1953; Mulder, 1991) and for binary mixtures by the solubility parameter theory if the feed components are hydrophobic (Hildebrand and Scott, 1949; Mulder, 1991). Even if the swelling of the membrane is less pronounced, it can affect the transport of solute across the membrane. Because a vacuum is applied for the removal of the solutes on the membrane downstream face, this side of the membrane is ideally “dry” in comparison to the more swollen and hence more flexible membrane


upstream face resulting from the solute uptake. This anisotropy of the membrane results in a nonuniform diffusivity of solute within the membrane, or in other words, mem Di will be concentration dependent and not constant across the membrane. One way to account for this phenomenon empirically is the relationship Dimem = Dimem,0 ⋅ e

π i ⋅φ

(17.13)

mem,0

where Di is the diffusion coefficient of i in the membrane at zero concentration (under ideal conditions), πi is the so-called plasticizing constant, which is an empirical parameter for the degree of swelling of the membrane polymer caused by the solute, and φI is the volume fraction of solute in the membrane polymer. Equation (17.13) is merely empirical. A more phenomenological approach that considers the mobility of the permeating solute as a function of the free space available within the membrane polymer network is the free volume theory (Fujita, 1961; Fels and Huang, 1971), which has been discussed elsewhere (Huang and Rhim, 1991). In Equations (17.5) through (17.8) and (17.12) it is assumed that only solute– polymer and no solute–solute interactions occur. This excludes the possibility of synergistic effects between membrane permeants, such as flux coupling. Flux coupling takes place when a permeant of low diffusivity (“slower” permeant) is dragged through the membrane polymer by a permeant of higher diffusivity (“faster” permeant), resulting in higher fluxes of the slower permeant than expected. The opposite might also happen, i.e., the slowing down of the diffusion of the faster permeant by the slower one. The Maxwell–Stefan approach is capable of describing such nonideal phenomena (Wesselingh and Krishna, 1990). This theory assumes that the driving force for a solute i within a multicomponent mixture equals the sum of frictional resistances between solutes resulting from their relative motion while diffusing through the membrane polymer, as expressed by: 1 dµ i ⋅ = RT dz

n

∑x⋅ j

j=1

v j − vi D0ji

(17.14)

where dµi/dz is the local chemical potential gradient of i; xj is the mole fraction of solutes j = 1…n, vj is the local velocities of components j = 1…n, vi is the local 0 0 velocity of i, Dji is the Maxwell–Stefan interaction parameter, with 1/Dji indicating a friction coefficient between components j and i, R is the universal gas constant, and T is the absolute temperature. Equation (17.14) represents the general Maxwell–Stefan equation. Applied to the special case of solute flux across a polymer, Equation (17.14) needs to be adapted, which is accounted for in detail elsewhere (Heintz and Stephan, 1994). The Maxwell–Stefan approach is recommended for cases in which a strong nonideal behavior is observed because it is mathematically far more demanding than the equations presented above. It should be stressed, however, that for nonideal systems the Maxwell–Stefan approach might be the most elegant and efficient way to describe the transport phenomena.


The fundamentals of pervaporation were outlined above in the most simplified way, based on the mass conservation principle and Fick’s first law, and discussed with regard to nonideal phenomena that may arise. Nonideal phenomena in pervaporation can be very complex and may demand individual solutions, which explains why until now no general mass transport model has been developed for pervaporation. To describe the mass transport across the dense membrane in a given situation, it might therefore be best to start with the simplest approach assuming Fickian diffusion. If, under the operating conditions of interest, nonideal mass transport phenomena are observed, a choice can still be made between an empirical adjustment of the ideal model or the use of a more complex structured model such as the ones mentioned above. In the latter case, one should be aware that most of the structured models still employ one or more empirical parameters that must be adjusted to experimental data. In addition, the experimental effort of determining all model parameters usually exceeds that needed for empirical models. An empirical model, on the other hand, will only be valid within the range of operating conditions studied and does not allow any extrapolation of the data.

17.3 TECHNICAL ASPECTS OF PERVAPORATION A typical pervaporation set-up is depicted in Figure 17.4. The liquid feed solution is recirculated continuously between the feed reservoir (1) and the pervaporation module (2). Pervaporation modules can be configured in a plate-and-frame, hollow fiber, or spiral wound mode (Rautenbach et al., 1991; Mulder, 1997; Fleming and Slater, 1992). The solutes permeating through the membrane leave the membrane downstream face as a vapor due to the vacuum that is established initially by the vacuum pump (4). The phase transition of the permeating solutes from the liquid to the vapor state goes along with heat consumption. This heat is taken up from the environment, 5

retentate F

P

T

2 feed P

1

T

4

T

permeate

3 fraction 2

fraction 1

FIGURE 17.4 A schematic standard pervaporation set-up. 1: feed tank; 2: pervaporation module; 3: condensation unit; 4: vacuum pump; 5: heat exchanger; F: flow control; P: pressure indicator/ control; T: temperature control.


namely the bulk feed, which consequently cools. In modules with a large membrane area, this causes a temperature drop between the feed and the retentate and must be compensated for. On the industrial scale, heat exchangers (5) are therefore implemented in the feed circuit. In laboratory units, this temperature drop may be neglected because the membrane area is commonly small in comparison to the recirculation rate of the feed, which is maintained at a controlled temperature. Instead of using a vacuum for maintaining the driving force, an inert gas can be swept over the membrane downstream surface for the removal of the permeate solutes (sweeping gas pervaporation). However, this mode of operation is less efficient for maintaining the driving force, and it renders the condensation of the permeate solutes from the sweeping gas stream more difficult. The permeate vapor is recovered in condensers at a temperature that allows a quantitative condensation of the vapor. The condensation can optionally be carried out in a series of condensation stages at different temperatures in order to achieve a permeate fractionation and increased enrichment of target compounds (3). As will be illustrated, a careful design of the condensation strategy is indispensable and most important in pervaporation. Once the vacuum is established, the vacuum pump (4) is shut off and ideally the condensation unit alone maintains the vacuum. Only if noncondensable gases permeate the membrane and enter the downstream compartment is continuous operation of the vacuum pump required for removal of these noncondensable gases, which otherwise would lead to an increased downstream pressure. Theoretically, scaling up is straightforward in pervaporation, and the fluxes obtained on a laboratory scale can serve as a basis for the plant design on the larger scale. However, because it is common and easy to design laboratory-scale experiments close to ideal conditions, which on the larger scale would be uneconomical, care must be taken when up-scaling laboratory experiments with regard to the hydrodynamic conditions, the vacuum pressure, and the condensation strategy. Industrial size plate-and-frame modules, for example, consist of a stack of tightly packed membranes over which the feed solution is recirculated (Mulder, 1997). The membranes are separated by spacers and the permeate withdrawn optionally by a central permeate pipe (Stürken, 1994). Pressure losses on both the feed and the permeate sides of the packed membranes occur and need to be accounted for in the module design. On the feed side, the hydrodynamic conditions over the membrane will be less uniform than on the laboratory scale, resulting in more pronounced concentration polarization. On the permeate side, the packed configuration of the membranes may constitute a mass transport resistance to the vacuum, resulting in a downstream pressure locally higher than that measured in the main vacuum duct. Both aspects may cause solute fluxes lower than expected and a possible shift in selectivities. On the laboratory scale, the permeate vapor is commonly condensed at about 77°K using liquid nitrogen. For economic reasons, the condensation temperature on the industrial scale should be considerably higher. Because condensation efficiency is determined by both the condenser temperature and the residence time of the permeate vapor in the condenser, careful design of the condensation unit becomes crucial. It should be noted that a lower downstream pressure, desirable for high solute fluxes, goes along with a higher permeate vapor velocity in the downstream


compartment, and hence a lower residence time in the condenser. As a consequence, either the heat exchange area or the condenser temperature must be adjusted accordingly. A fractionated condensation at different condensation temperatures can significantly increase the process performance, if the permeate composition allows (Brüschke et al., 1992; Baudot and Marin, 1997), as it enables further product enrichment and a more flexible and defined condensation of the permeate vapor. In the case of coupling a pervaporation unit to an active fermentation, special attention must be paid to noncondensable fermentation off-gases, such as carbon dioxide from a yeast fermentation, which can permeate through the membrane in large quantities. Using liquid nitrogen on the laboratory scale, most of these gases are condensed. On the industrial scale, however, these gases may pass the condensation unit virtually unaffected, lowering both the partial pressure and residence time of other permeants in the condenser. The consequence can be a strongly decreased condenser efficiency resulting in considerable loss of the more volatile permeate vapor compounds. The technical aspects of pervaporation as outlined above illustrate that for process optimization all process parameters must be considered, since they strongly affect each other. It should also be pointed out that although pervaporation is in principle a membrane separation process, its selectivity and effectiveness may be significantly determined by mass and heat transport phenomena occurring beyond the membrane surfaces.

17.4 ADVANTAGES AND APPLICATIONS OF PERVAPORATION IN THE FOOD INDUSTRY As can be seen from the brief description of the pervaporation separation principle, in comparison to mere evaporative techniques the membrane constitutes an additional transport resistance; however, it allows a far more selective recovery of solutes due to solute–polymer interactions. Membrane polymers can be widely tailor made for individual applications. One membrane polymer is barely applicable to every imaginable separation problem, as will be shown in the following section.

17.4.1 HYDROPHILIC PERVAPORATION The most successful application of pervaporation on the industrial scale has been the dehydration of ethanol. Distillations of ethanol/water mixtures reach an azeotropic point at about 95.6 wt% of ethanol. The azeotrope can only be broken and the remaining water removed by adding entrainers such as benzene to the ethanol/water mixture. Entrainers constitute an environmental problem as well as requiring additional downstream processing for their subsequent removal from the final product. Using hydrophilic polyvinylalcohol (PVA) membranes, pervaporation has been successfully coupled to distillation for water removal, yielding a final purity of ethanol of about 99.95 wt% without the need to employ entrainers. PVA is a crystalline or so-called glassy polymer, with the polymer chains constituting a rigid network of little flexibility (Mulder, 1997). Both ethanol and water sorb in PVA; however, due to its larger molecular size, the diffusivity of ethanol through the rigid


PVA network is strongly hindered in comparison to water. Because the use of glassy polymers has been restricted so far mainly because of the selective removal of water from organic solvents, such as dichloroethylene, isopropylalcohol, and tetrahydrofuran, this separation has been widely referred to as hydrophilic pervaporation.

17.4.2 ORGANOPHILIC PERVAPORATION The removal of organic compounds from aqueous solutions, also called organophilic pervaporation, requires the use of a more flexible polymer structure. Almost all organic compounds are of a larger molecular size than water, and their diffusivity is hindered in glassy structures. Membranes used for the separation of organic compounds from water are therefore of a rubbery nature, with a more flexible polymeric network. Evidently, an eased diffusivity for the organic compounds goes along with that of water, resulting in a considerable loss of membrane selectivity. Therefore, special attention is given to modification of rubbery polymers so that strong solute–polymer interactions are promoted, greatly enhancing the sorption of the organic solute in the membrane polymer in comparison with water. For example, using a silicon rubber membrane, a toluene/water separation factor of about 970 has been reported (Baker et al., 1997). Using an ethylene-propylene 1,4-hexadiene copolymer (EPDM) of similar thickness, these researchers obtained a separation factor of 33,400. This illustrates the importance of selecting or developing the appropriate membrane for the individual separation task. Glassy and rubbery membranes differ strongly with regard to the flexibility of their polymeric structure. Within the more rigid polymeric network of a glassy membrane, the diffusivity of components will be strongly related to their molecular volume, whereas in the more flexible structure of a rubbery membrane, it is less the diffusivity that differs strongly for different components, but far more the solute–polymer interaction. This is why the selectivity of glassy membranes is more diffusion controlled, while the selectivity of rubbery membranes is more sorption controlled. In food technology, the main interest in applying pervaporation is in separating organic compounds from an aqueous solution, i.e., the recovery of aroma compounds (so-called bioflavors) formed by microbial fermentations and biotransformations. Bioflavors are preferable to chemically synthesized ones due to their better public acceptance. In addition, bioflavors often exhibit a greater aromatic diversity than their chemically synthesized counterparts, as is the case, for example, with vanillin (Berger, 1995). Aroma concentrates are widely used as food additives to enhance the overall flavor of foods or to compensate for the loss of aromas during food processing. They must hence represent the organoleptic characteristics of the aroma origin as well as be free of any harmful chemical contamination resulting from the aroma recovery process. Traditional aroma recovery processes such as distillation, adsorption, and solvent extraction often are discouraged because they operate at an elevated temperature that deteriorates the aroma quality, are not environmentally friendly due to high energy consumption and the use of toxic solvents, or involve elaborate purification steps to remove solvent residues from the final food product (Fleming, 1992; Schreiber et al., 1997). As a consequence, in recent years separation processes operating at gentler temperatures and avoiding harmful extraction aids have been investigated. These include


steam distillation, air stripping (Le Thanh et al., 1993), the spinning cone column (Wright and Pyle, 1996), supercritical carbon dioxide extraction (Jolly, 1981), and membrane separation processes (Baudot and Marin, 1997). Bioflavors consist of aroma compounds of very differing physicochemical properties (Berger, 1995). Hence, the in situ recovery of these aromas from a microbial fermentation broth requires a process capable of simultaneously extracting aroma compounds of differing chemical nature. In addition, disturbing of or interfering with the ongoing bioconversion process should also be avoided. Thus, the recovery process should be efficient at the bioconversion temperature, usually close to ambient, as well as recover, for example, low and high volatile compounds equally well. Pervaporation can be operated continuously at low temperature, does not require any extraction aid, and does not exert high stress on the active biocatalyst (Schäfer and Crespo, 2000). It is a membrane separation process that has been extensively studied on the laboratory scale for the recovery of flavor compounds (Baudot and Marin, 1997; Karlsson and Träghård, 1993). Membrane fouling is a minor problem in pervaporation because the membranes used are nonporous. Organophilic pervaporation linked to a bioconversion process has until now solely been studied for the recovery of individual aroma compounds or inhibiting metabolic products (Böddeker, 1994; Lamer et al., 1996; Rajagopalan et al., 1994). Most research on aroma recovery by organophilic pervaporation has been conducted using aqueous aroma model solutions (Börjesson et al., 1996; Baudot and Marin, 1997), with the emphasis on the engineering aspects of pervaporation. An evaluation of the actual organoleptic value of an aroma concentrate obtained by pervaporation has only been reported recently (Schäfer et al., 1999). A few examples are presented to illustrate the separation principles outlined above, as well as to demonstrate the potential of pervaporation to respond to very diverse aroma separation problems under well-defined operating conditions. Based on Equation (17.4), it was mentioned that low volatile compounds can still be recovered by pervaporation, given that their activity coefficient in the feed solution is sufficiently high. Two examples of such low volatile compounds are γ-decalactone (boiling point 554°K) and vanillin (boiling point 515°K). Both compounds possess a saturated vapor pressure below 1 Pa at ambient temperature but a high activity in aqueous solution due to their hydrophobicity. TM Using a PDMS GFT 1060 membrane, a silicon membrane supported by a macroporous support, γ-decalactone was successfully recovered by pervaporation from −1 an aqueous feed solution with an initial concentration of 100 mg·kg (Baudot et al., 1999). However, a strong dependence of the γ-decalactone flux on the total downstream pressure was observed (Figure 17.5). While the water flux decreased linearly with increasing downstream pressure, the γ-decalactone flux decreased exponentially. This observation can be explained by the differing capacity of the vacuum to remove both the water and γ-decalactone solutes that have diffused toward the membrane down0 stream surface. Because γ-decalactone is very slightly volatile (Pi of γ-decalactone at 303.15°K is 0.65 Pa while that of water is 4200 Pa), a slight increase of the total downstream pressure can cause an accumulation of γ-decalactone solutes in the macroporous support on the membrane downstream side, and concentration polarization occurs, resulting in a lower γ-decalactone flux. On the contrary, the decrease in


100

1.0

50

0.5

–2

water flux [ g·m ·h

–1

–2

]

–1

1.5

γ-decalactone flux [ g·m ·h ]

150

0

0

500

1000

1500

2000

0.0

total permeate pressure [ Pa ]

FIGURE 17.5 The flux of water (䊊) and γ-decalactone (䊉) as a function of the total downstream pressure (Data from Baudot et al., J. Membrane Sci., 158(1–2), 167, 1999).

water flux with increasing total downstream pressure is solely due to a decreased driving force within the premeate pressure range depicted. The recovery of vanillin (Böddeker et al., 1997) by a homogeneous polyether– block–polyamide membrane is unique in that vanillin crystallizes at ambient temperature in the downstream compartment due to its high melting point, without any need for condensation. Any other more volatile solutes permeating the membrane pass the crystallized vanillin and can either be subsequently condensed or discarded. It should be noted that for such a demanding separation problem composite membranes cannot be used, as the pressure loss within the macroporous support can be sufficient to cause the condensation of vanillin at the membrane downstream face. While most applications of pervaporation deal with the removal of single aroma compounds, little has been reported on the recovery of a distinct aroma profile. An example of such an aroma profile is that of a muscatel wine formed during the vinification process. The aroma produced by yeast during wine must fermentation comprises up to 800 aroma compounds (Rapp, 1990), which makes it a challenge to recover aromas without altering their organoleptic quality. Using a polyoctylmethylsiloxane–polyetherimide (POMS–PEI) composite membrane, a muscatel wine must aroma concentrate can be obtained that is faithful to its origin and of high organoleptic value according to sensory panel evaluation (Schäfer et al., 1999). With the aroma profile of muscatel wine must consisting of a subtle balance among individual aroma compounds of varying physicochemical properties, it is not obvious that their transport across the membrane polymer results in an aroma concentrate organoleptically faithful to the aroma of the feed solution. In fact, it has been observed that concentration polarization phenomena on the feed side strongly affect the permeate composition (Figure 17.6). Based on the equations presented above, the water-free mass fraction of four representative aroma compounds in the permeate, ethyl acetate, ethyl butanoate, ethyl hexanoate and linalool, was simulated as a


permeate mass fraction [ % ]

60

40

20

0

0.01

0.1

1

10

ratio boundary layer/membrane thickness [ ]

FIGURE 17.6 The permeate mass fraction in wt% as a function of the ratio boundary layer/ membrane thickness. The permeate composition is calculated on a water-free basis for a dilute aqueous feed solution of ethyl acetate (solid), ethyl butanoate (dotted), ethyl hexanoate (long dash), and linalool (short dash).

function of the hydrodynamic boundary layer/membrane thickness on the basis of their sorption coefficient Si, as well as their diffusion coefficients in water and the membrane. Of the compounds investigated, ethyl hexanoate showed the highest affinity for the membrane polymer and ethyl acetate the lowest. Diffusion coefficients varied slightly among the compounds, however, to a much lesser extent than their respective sorption coefficients. As can be seen in Figure 17.6, only below a boundary layer/membrane thickness ratio of about 0.01 does the permeate mass fraction remain constant. In other words, if a membrane of 20 µm thickness is used during pervaporation, the hydrodynamic boundary layer thickness should be less than 0.20 µm in order to avoid any shift in the permeate mass fraction due to concentration polarization. Highly turbulent conditions are hence required over the membrane. Less turbulent conditions will lead to a thicker hydrodynamic boundary layer, resulting in a shift in the permeate mass fraction, as is indicated by the dotted vertical lines in Figure 17.6.

17.5 CONCLUSIONS The principles of pervaporation and its advantages for the recovery of volatile compounds from bioconversions in particular have been presented and discussed. It has been shown that pervaporation is a versatile process that should be understood as a whole, and the process components of which can be adapted to individual separation problems. This versatility allows the invention of novel, tailor-made process strategies differing from the standard process design as depicted in Figure 17.4. One might, for example, think of alternative methods for capturing aroma compounds in the vapor permeate. The vapor permeate does not necessarily have to be condensed before the vacuum pump. Using liquid ring vacuum pumps, it may be condensed in the pump itself. Using dry vacuum pumps, it may be captured after the pump under a higher pressure than that of the vacuum. This opens the possibility of conceiving


various alternatives for capturing aromas, such as the recovery of the aromas in an adsorptive matrix resulting immediately in a final product, as for example for applications of controlled release. Likewise, other steps during the pervaporation process may be optimized and adapted for an individual separation problem.

REFERENCES Baker, R.W., Wijmans, J.G., Athayde, A.L., Daniels, R., Ly, J.H., and Le, M., The effect of concentration polarization on the separation of volatile organic compounds from water by pervaporation, J. Membrane Sci., 137, 159–172, 1997. Baudot, A. and Marin, M., Pervaporation of aroma compounds: comparison of membrane performances with vapor–liquid equilibria and engineering aspects of process improvement, Trans. IChemE., 75C, 117–142, 1997. Baudot, A., Souchon, I., and Marin, M., Total permeate pressure influence on the selectivity of the pervaporation of aroma compounds, J. Membrane Sci., 158, 167–185, 1999. Berger, R.G., Aroma Biotechnology, Springer, Berlin, 1995. Böddeker, K.W., Recovery of volatile bioproducts by pervaporation, in Membrane Processes in Separation and Purification, Crespo, J.G. and Böddeker, K.W., Eds., Kluwer, Dordrecht, The Netherlands, 1994, pp. 195–207. Böddeker, K.W., Gatfield, I.L., Jähnig, J., and Schorm, C., Pervaporation at the vapor pressure limit: vanillin, J. Membrane Sci., 137, 155–158, 1997. Börjesson, J., Karlsson, H.O.E., and Trägårdh, G., Pervaporation of a model apple juice solution: comparison of membrane performance, J. Membrane Sci., 119, 229–239, 1996. Brüschke, H.E.A., Schneider, W., and Tusel, G.F., Patent DE 3804236C2, 1992. Fels, M. and Huang, R.Y.M., Theoretical interpretation of the effect of mixture composition on separation of liquids in polymers, in Permselective Membranes, Rogers, C.E., Ed., Dellke Inc., New York, 1971, pp. 89–97. Fleming, H.L., Consider membrane pervaporation, Chem. Eng. Proc., 7, 46–52, 1992. Fleming, H.L. and Slater, C.S., Pervaporation: design, in Membrane Handbook, Ho, W.S. and Sirkar, K.K., Eds., Chapman and Hall, New York, 1992. Flory, P.J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953. Fujita, H., Diffusion in polymer–diluent systems, Fortschr. Hochpolym. Forschung, 3, 1–47, 1961. Heintz, A. and Stephan, W., A generalized solution–diffusion model of the pervaporation process through composite membranes, Part II. Concentration polarization, coupled diffusion and the influence of the porous support layer, J. Membrane Sci., 89, 153–169, 1994. Hildebrand, J.H. and Scott, R.L., The Solubility of Non-Electrolytes, Plenum Press, New York, 1949. Huang, R.Y.M. and Rhim, J.W., Separation characteristics of pervaporation membrane separation processes, in Pervaporation Membrane Separation Processes, Huang, R.Y.M, Ed., Elsevier, Amsterdam, 1991, pp. 111–180. Jolly, D.R.P., Wine Flavour Extraction with Liquid Carbon Dioxide, Process Biochem., Aug/Sept 1981, pp. 36–40. Karlsson, H.O.E., and Träghård, G., Pervaporation of dilute organic–water mixtures. a literature review on modelling studies and applications to aroma compound recovery, J. Membrane Sci., 76, 121–146, 1993.


Lamer, T., Spinnler, H.E., Souchon, I., and Voilley, A., Extraction of benzaldehyde from fermentation broth by pervaporation, Process Biochem., 31, 533–542, 1996. Lawson, K.W. and Lloyd, D.R., Membrane distillation, J. Membrane Sci., 124, 1–25, 1997. Le Thanh, M., Voilley, A., and Tanluu, R.P., The influence of the composition of model liquid culture medium on vapor liquid partition coefficient of aroma substances, Sci. Aliment., 13, 699, 1993. Lee, C.H., Theory of reverse osmosis and some other membrane permeation operations, J. Appl. Polym. Sci., 19, 83–95, 1975. Luque de Castro, M.D. and Gámiz-García, L., Analytical pervaporation: an advantageous alternative to headspace and purge-and-trap techniques, Chromatographia, 52, 265–272, 2000. Mulder, M., Basic Principles of Membrane Technology, Kluwer, Dordrecht, 1997. Mulder, M.H.V., Introduction to pervaporation, in Pervaporation Membrane Separation Processes, Huang, R.Y.M., Ed., Elsevier, Amsterdam, 1991, pp. 1–86. Néel, J., Thermodynamic principles of pervaporation, in Pervaporation Membrane Separation Processes, Huang, R.Y.M., Ed., Elsevier, Amsterdam, 1991, pp. 225–251. Rajagopalan, N., Cheryan, M., and Matsuura, T., Recovery of diacetyl by pervaporation, Biotechnol. Tech., 8, 869–872, 1994. Rapp, A., Natural flavours of wine: correlation between instrumental analysis and sensory perception, Fresenius J. Anal. Chem., 337, 777–785, 1990. Rautenbach, R., Herion, C., and Meyer-Blumenroth, U., Engineering aspects of pervaporation: calculation of transport resistances, module optimization and plant design, in Pervaporation Membrane Separation Processes, Huang, R.Y.M., Ed., Elsevier, Amsterdam, 1991, pp. 225–251. Schäfer, T., Bengtson, G., Pingel, H., Böddeker, K.W., and Crespo, J.P.S.G., The recovery of aroma compounds from a wine-must fermentation by organophilic pervaporation, Biotechnol. Bioeng., 62, 412–421, 1999. Schäfer, T. and Crespo, J.G., Extraction of aromas from active fermentation reactors by pervaporation, in Integration of Membrane Processes into Bioconversions, BélafiBakó, K., Gubicza, L. and Mulder, M., Eds., Kluwer Academic/Plenum Publishers, New York, 2000, pp. 177–186. Schreiber, W.L., Scharpf, L.G., Jr., and Katz, I., Flavors and fragrances: the chemistry challenges, Chemtech, 3, 58–62, 1997. Stürken, K., Organophile Pervaporation: Ein Membranverfahren zur Aufarbeitung verdünnter wäßrig-organischer Lösungen, dissertation, GKSS Forschungszentrum Geesthacht GmbH, 1994. Wesselingh, J.A. and Krishna, R., Mass Transfer, Ellis Horwood, New York, 1990. Wright, A.J. and Pyle, D.L., An investigation into the use of the spinning cone column for in situ ethanol removal from a yeast broth, Process Biochem., 31, 651–658, 1996.


18

Ultrasonic Mass Transfer Enhancement in Food Processing A. Mulet, J. Cárcel, J. Benedito, C. Rosselló, and S. Simal

CONTENTS 18.1 Introduction 18.2 Generation Methods of Power Ultrasound 18.3 Applications 18.3.1 Gas–Solid Mass Transfer Process 18.4 Liquid–Solid Mass Transfer Process 18.4.1 Influence of the Material Treated 18.4.2 Influence of Ultrasound Intensity 18.5 Conclusions Nomenclature References

18.1 INTRODUCTION High intensity ultrasound produces a variety of effects, such as radiation pressure, streaming, cavitation, and interface instabilities. These effects can influence mass transfer processes by producing changes in concentration gradients, diffusion coefficients, or boundary layer (Liang, 1993). Heating produced by absorption of ultrasonic energy (thermoacoustic effect) can also affect mass diffusion. The transducers most commonly used to generate high intensity ultrasound are piezoelectric (Mason, 1998). Based on this technique, ultrasonic cleaning baths are the commercial systems where most experimental results have been obtained and published in the literature. Ultrasonic probe systems with mechanical amplifiers are also used when it is necessary to concentrate the generated acoustic energy. These two methods used to produce ultrasound are suitable for treatments in a liquid environment. Another arrangement of transducers is required for applications in gaseous media, due to their low density, and is responsible for their low specific acoustic impedance.


In order to control a process, it is necessary to measure the main parameters. In ultrasonic applications, the amount and characteristics of the energy applied need to be known. The measurement of power and frequency enables not only a correct evaluation of the importance of these factors in the process, but also the appropriate process control. The use of high intensity ultrasound has been considered to enhance mass transfer for different products and processes such as brining of meat (Sajas and Gorbatow, 1978) and cheese (Sánchez et al., 1999), drying of rice (Muralidhara et al., 1985) and carrots (Gallego-Juárez et al., 1999), osmotic dehydration of apples (Floros and Liang, 1994; Simal et al., 1998), and extraction processes of several products such as pectin from apples (Panchev et al., 1988), chymosin from the fourth stomach of milk-fed calves to manufacture cheese (Kim and Zayas, 1991), bioactive principles from plant materials (Vinaturo et al., 1997), and even extractions with supercritical fluids (Jun et al., 1997). The aim of this work is to examine the application of high intensity ultrasound to mass transfer processes. Four cases are analyzed to show the influence of transmission medium and the material considered. Three are applications in a liquid medium and the fourth is in a gaseous medium. The cases in a liquid medium are: a structured material, microscopically isotropic and with a large quantity of air-filled pores (osmotic dehydration of apple); a low structured, macroscopically isotropic material with a large quantity of liquid-filled pores (pressed cheese curd in brine, an intermediate product in some cheese manufacturing processes); and a highly structured, anisotropic material with low porosity (pork loin in brine). For the mass transfer process in a gaseous medium, a structured vegetal, nonisotropic material with partially air-filled pores will be considered (carrot dried with hot air).

18.2 GENERATION METHODS OF POWER ULTRASOUND Piezoelectric transducers convert a high frequency alternating current into mechanical vibrations. The most common applications of power ultrasound in mass transfer processes take place between a solid and a liquid medium. For these cases, ultrasonic cleaning baths are the most commonly used commercial equipment. In ultrasonic baths, the transducers are attached to the bottom of the bath. When they start to oscillate, vibrations are transmitted into the liquid. Standing waves are created by reflection of the emitted sound waves at the liquid/air interface. High and low intensities are obtained at fixed levels (half wavelength) throughout the bath depth. As a consequence, the ultrasonic field formed inside a bath is irregular and influenced by liquid height and the introduced loading (Figure 18.1). The technology of ultrasonic probe systems is also based on piezoelectric transducers. A “horn” is attached to a piezoelectric sandwich transducer in order to amplify the generated acoustic energy. The length of the horn is generally a half wavelength (or a multiple). The horn shape makes it possible to concentrate the ultrasonic energy. The gain for a given form of the horn is defined by a specific ratio of the diameter of the two faces of the horn. If D is the driven face and d is the emitting face for an exponential horn, the ratio is D/d. This means that for a


2.8 Hydrophone

Transducers

2.6

Acoustic pressure (bar)

2.4

Out

2.2 2.0 1.8 1.6 1.4

Water surface

1.2 1.0 0

5

10

15

20

Distance from the bottom (cm)

FIGURE 18.1 Example of the acoustic pressure variation vs. the distance from the bottom in an ultrasonic bath.

uniform cylinder there is no gain in wave amplitude, but rather the horn acts as an extender for the transference of the acoustic energy. For a stepped horn, the gain 2 is (D/d) . Horns are generally made of titanium due to the high resistance of this material to fatigue and erosion. The erosion of the transmitter surface causes a fall in the power. Variation in horn length can affect the frequency. For applications of power ultrasound in gaseous media, the most important problems are related to the efficient generation and transmission of ultrasonic energy. It is necessary to achieve ultrasonic generators that meet the following requirements: good impedance matching with the air, large amplitude of vibration, high directional radiation, and high power capacity. In addition, for large-scale industrial applications, extensive radiating area would be required in the transducers. Presently, no commercial transducers meeting all the aforementioned conditions are available. A new type of ultrasonic generator has recently been developed (Gallego-Juárez et al., 1989). It is comprised of a transducer with a stepped-plate radiator and an electronic unit for driving the transducer (Figure 18.2). The design of the stepped profile of the radiating plate allows control of the vibration and radiation pattern, thus permitting distribution of the vibration amplitude and, consequently, the acoustic field, according to the needs of the specific application. The signal frequency can slide during operation in association with changes in the mechanical load. Keeping the voltage and current signals in the transducer in phase, a specifically designed electronic circuit can automatically correct the frequency of the generated signal so that the electronic system drives the transducer by producing a signal within the very narrow band corresponding to the resonance frequency of the transducer.


Radiating plate

Piezoelectric ceramics

Mechanical amplifier

Sandwich

FIGURE 18.2 Structure of the stepped-plate transducer. (From Gallego-Juárez, J.A. et al., Drying Technol., 17, 597–608, 1999, courtesy of Marcel Dekker.)

One important point in the use of high intensity ultrasound is the power measurement. Due to problems of sonic wave reflections and the wearing out of transmitters, the power and frequency applied to samples may be modified. Hydrophones can supply local information on the acoustic pressure and frequency (Pugin and Turner, 1990). The signal created by a hydrophone may be captured and treated through an oscilloscope. This allows characterization of the acoustic field and identification of standing waves, high and low intensity zones, etc. A calorimetric method is another way to measure the input of the acoustic power into a medium. The method consists of the measurement of the temperature rise against the time in adiabatic conditions when ultrasound is applied (Margulis and Maximenco, 1990). The average power absorbed is: P = mC p

dT dt

where m is the liquid mass treated, Cp is the heat capacity and dT/dt is the variation of temperature T with time t that can be approximated by ∆T/∆t. As can be seen in Figure 18.3, there is a linear temperature increase. These unpublished data correspond to the results of experiments performed by the authors. A good correlation between acoustic pressure measured by a hydrophone and power estimated through the rate of the temperature increase was found (Figure 18.4). The measurements were carried out in a 600 ml glass beaker with 400 ml saturated brine. The starting temperature was the same in all cases (3°C).

18.3 APPLICATIONS 18.3.1 GAS–SOLID MASS TRANSFER PROCESS Drying carrots is an example of mass transfer from a solid medium to a gas medium. Gallego-Juárez et al. (1989) carried out experiments drying carrots with and without application of airborne ultrasound. A special generator unit and a stepped plate were


Temperature of the medium (°C)

6 5 4 3 Start of the ultrasound application

2 1 0 0

50

100

150

200

250

Time (sec)

FIGURE 18.3 Temperature increase with application of ultrasound with a sound probe system.

Calorimetric measurement (W/cm2)

20 18 16 14 12 10 8 y = 33.922x – 18.22

6

R2 = 0.9958

4 2 0.5

0.7

0.9

1.1

1.3

Hydrophone pressure (bar)

FIGURE 18.4 Correlation between power ultrasound estimation by a calorimetric method and measurement of the acoustic pressure with a hydrophone.

used (Gallego-Juárez, et al. 1989). The experimental tests mainly consisted of measuring the water content of food samples after different times of application of highintensity ultrasonic fields in combination with forced air at various temperatures and flow velocities. In all experiments, the frequency was kept constant at about 20 kHz while different sound pressure levels were applied. The experimental set-up can be seen in Figure 18.5. Different drying temperatures, air velocities, and power ultrasound were used. From data published by Gallego-Juárez et al. (1998), it is possible to calculate an


TABLE 18.1 Diffusivities Calculated for Dehydration with and without Airborne Ultrasound Diffusivity × 10

Experiment T T T T T

= 60°C; Without ultrasound = 60°C; With ultrasound = 90°C; Without ultrasound = 90°C; With ultrasound = 115°C; With and without ultrasound

−6

2

(m /h)

2.67 4.86 6.22 7.74 11.35

Source: Data from Gallego-Juárez et al. (1998). Air velocity (V) = 1.3 m/sec; carrot samples 14 mm × 2 mm

Transducer Electronic generator

Hot air generator

Vibrating plate

Sample

Support plate FIGURE 18.5 Experimental set-up for forced air drying assisted by airborne ultrasound. (From Gallego-Juárez, J.A. et al., Drying Technol., 17, 597–608, 1999, courtesy of Marcel Dekker.)

effective diffusivity of water. For that purpose, a diffusive model for carrots proposed by Mulet (1994) was used. The less complex model was chosen for its simplicity. It was assumed that the solid temperature was constant and equal to drying air temperature and that the diffusivity was constant during drying experiments; the shrinkage was not considered. The model is valid if external resistance to mass transfer is negligible. Mulet et al. (1987) indicated that the air flow had no influence 2 on carrot drying rate when greater than 6000 kg/hm (≈1.29 m/sec). Mitchell and 2 Potts (1958) fixed this flux limit at 4200 kg/hm (≈0.9 m/sec). The analyzed experiments were carried out at constant air velocity of 1.3 m/sec, hence the process was controlled by internal resistance. The diffusivities were calculated from GallegoJuárez et al.’s results and are shown in Table 18.1. The values are similar to those −6 found in the literature. Mulet (1994) reported an effective diffusivity of 6.3 × 10 2 2 m /h for carrot cubes using an air flow of 8000 kg/hm at 60°C through drying. The obtained values showed that under similar conditions the carrot diffusivity in experiments carried out with ultrasound was higher than in experiments without


Transducer Electronic generator Hot air generator

Samples

Vibrating plate

Suction system

Support plate Drain channels Static pressure

FIGURE 18.6 Experimental set-up for dehydration by directly coupled ultrasonic vibration. (From Gallego-Juárez, J.A. et al., Drying Technol., 17, 597–608, 1999, courtesy of Marcel Dekker.)

ultrasound. When the drying air temperature increased, the relative influence of ultrasound decreased. For an air temperature of 115°C, Gallego-Juárez (1998) did not find differences between applying and not applying ultrasound. The ultrasound pressure levels used by these authors were 155 and 163 dB. No important differences in sample water content were found between samples dehydrated under these acoustic pressures. Significant differences were found only in the fast-drying zone (i.e., the first 10 min). The main difficulty in dehydration by airborne ultrasonic radiation is the low penetration of the acoustic energy into the food material due to the mismatch between acoustic impedances. Acoustic impedance is defined as the product of density and sound speed in the material. When ultrasonic waves arrive at the air–material interface, a high percentage of their energy is reflected due to impedance differences between gas and solid. In order to avoid this problem, Gallego-Juárez et al. (1999) carried out another kind of experiment. A stepped-plate transducer was applied to the product directly, and static pressure was simultaneously exerted on the samples (Figure 18.6). To facilitate water removal, an air flow of 1 m/sec at 22°C was applied. Assuming again that internal resistance controls the process, an effective diffusivity using the afore−6 2 mentioned model can be calculated. The result was 21.92 × 10 m /h. This diffusivity was higher than that found in 115°C drying air temperature experiments. In fact, the drying process is more powerful, since the final moisture of the samples could be less than 1% (Gallego-Juárez et al., 1999). These results show that this procedure may be useful for food products with components sensitive to high temperature. The main mechanism involved may be the “sponge effect.” Ultrasound produces contractions and expansions of the material, and water is forced to exit from the vegetal matrix. Additionally, the acoustic stress may create microchannels that assist the water evacuation.


18.4 LIQUID–SOLID MASS TRANSFER PROCESS 18.4.1 INFLUENCE

OF THE

MATERIAL TREATED

In ultrasound application in liquid media, an important parameter that should be kept in mind is the viscosity of the liquid. If this is high, the ultrasonic energy may be absorbed within a few centimeters from the emitters (Mason and Cordemans, 1996). The surface tension is another important factor, since it can prevent the appearance of cavitation phenomena. The solid material processed (in a liquid) is another factor to be considered when examining the effects produced by ultrasound. Simal et al. (1998) studied ultrasound influence (0.81 ± 0.17 bar; 40 kHz) in osmotic dehydration of apples at different temperatures. Water loss and sucrose gain were measured in apple cube samples at different times during the osmotic process. The results showed an acceleration of water loss and sugar gain rate when ultrasound was applied. In Figure 18.7, the differences between experiments with agitation of osmotic solution and with ultrasound are shown. The difference in water content after 3 h of treatment was approximately 25% lower in the ultrasonic experiment. The increase of dry matter in ultrasonic treatment relative to agitation experiments was about 40% at 50°C and 15% at 70°C after 3 h. Simal et al. (1998) found that the solute gain was more important at 40°C with ultrasonic application than at 70°C with agitation. Sánchez et al. (1999) carried out experiments on pressed cheese curd brining, an intermediate process in some types of cheese manufacture. Three kinds of brining

Difference (%) between ultrasonic and agitation brining

100 80 60 40 20 0 -20 -40 0

2000

4000

6000

8000

10000

12000

Time (sec)

FIGURE 18.7 Difference (% d.m.) in water content and solute content between ultrasonic and agitation experience for apple osmotic dehydration. 50°C, solute content; 70°C, solute content; 50°C, water content; 70°C, water content. Difference = 100 × (content with ultrasound − content with agitation)/content with ultrasound.


Difference (%) between ultrasonic and static brining

30 25 20 15 10 5 0 -5

-10 -15 -20 0

2000

4000

6000

8000

10000

12000

Time (sec)

FIGURE 18.8 Difference (% d.m.) in water content and solute content between ultrasonic and static brining of pressed cheese curd. 5°C, solute content; 10°C, solute content; 5°C, water content; 10°C, water content. Difference = 100 × (content with ultrasound– content in static conditions)/content with ultrasound.

were tested: with brine agitation (dynamic conditions); without brine agitation (static conditions); and with applications of ultrasound. An ultrasonic cleaning bath was used as an ultrasonic generator (1.7 ± 0.22 bar; 30 kHz). The difference between ultrasonic and static conditions in 5 and 10°C experiments are shown in Figure 18.8. In this case, the decrease in water content was more important than the increase of solute content. Both processes (osmotic dehydration of apples and pressed cheese curd brining), with and without ultrasonic applications, fitted well to a diffusional model (Simal et al., 1998; Sánchez et al., 1999), thus an effective diffusivity can be calculated. The Arrhenius equation was adequate to model temperature influence on diffusivity in all cases except for solute gain in osmotic dehydration of apple. The fact that apple cubes osmotically treated at different temperatures gained similar amounts of sucrose during the drying process could indicate that the main mechanism involved in sucrose transfer from the osmotic solution into the solid when sonication was used was not of a diffusional nature. A different mechanism for solute gain should be explored. As can be seen in Figures 18.7 and 18.8, the difference of solute and water between ultrasonic and nonultrasonic experiments had distinct behavior. Apple is a porous product with partially air-filled pores. Pressed curd is also a porous product, but its pores are filled with liquid. In osmotic dehydration of apple, the sponge effect, produced by ultrasound, forces the occluded air in the pores to exit. Then the osmotic solution enters the pores. Ultrasound produces a pressure assisted osmotic dehydration. Positive and negative pressures occur with the same frequency as the ultrasound is applied. The entry of solution into the apple matrix


produces an important increase of sugar in ultrasonic assisted osmotic dehydration according to a mechanism other than diffusion. Similar results were obtained by Shi and Fito (1993). These authors found that although an increase in the solution temperature had a favorable effect on the rate of water diffusion, no significant influence of temperature on sugar gain could be observed in apricot and pineapple samples treated through vacuum osmotic dehydration. This could explain the lack of influence of temperature on the sugar gain process. The proportion of air in cheese pores is lower than that in apple pores. Thus, the influence of ultrasonic compressions and expansions is less important. In the first stages of brining, the pore air is “liberated” by ultrasound. Then the differences with regard to static conditions of brining are important. When there is no air in the pores, the effect of ultrasound decreases. Chiralt and Fito (1996) found in vacuum impregnation of Manchego cheese that during vacuum treatment the outflow of gas involved a residual whey release as well, because it was interchanged with brine during the inflow of the external liquid step when atmospheric pressure was restored. This could be the reason that the difference in salt content between ultrasonic and nonultrasonic experiments remains after the first stage of brining in the time span considered. Another factor that may explain the different behavior between apple and cheese could be their different structure. The rigid structure of apples makes them a better ultrasonic vibration conductor than cheese, which presents a “rubbery” structure where the ultrasound waves are more easily absorbed. In both cases, apple and cheese, the water content decreased in experiments with ultrasound more quickly than in experiments without ultrasound. For the time spans considered, the difference in water content between ultrasonic and nonultrasonic experiments increased with the time of treatment. The ultrasonic waves could produce microcurrents in the solid–liquid interface that cause an increase of diffusion coefficients and a decrease of the boundary layer of diffusion. Therefore, the water loss process may be influenced by ultrasound waves due to their effect on external resistance or/and internal resistance.

18.4.2 INFLUENCE

OF

ULTRASOUND INTENSITY

The effects produced by ultrasound in mass transfer processes are influenced by the intensity applied. Several authors have described the existence of an intensity threshold; for lower intensities no ultrasound effects were observed. Arkhangel’skii and Statnikov (1973) found that this threshold in acoustic evaporation was about 140 dB. The effect of acoustic streaming becomes greater than the influence of natural convection. High intensity produces high acoustic vibration, more violent cavitation, and greater microstreaming; thus the influence of ultrasound is intensified. In order to show ultrasound intensity effects, Cárcel et al. (1999) carried out several brining experiments using pork. Loin pork was brined in a NaCl-saturated brine for 45 min. Experiments were performed with and without ultrasound application. In the case of ultrasonic brining, nonexternal agitation was applied to the brine. In nonultrasonic brining, the experiments were carried out with and without brine agitation. For ultrasonic experiments, two types of generators were used: an


TABLE 18.2 Mean Values and Confidence Intervals (95%) for Salt Gain in Pork Loin after 45 Minutes of Brining Experiment Nonultrasonic without agitation Nonultrasonic with agitation Ultrasonic bath Sound probe system

g NaCl/g Initial Dry Matter 0.16 0.18 0.18 0.38

± ± ± ±

0.08 0.08 0.06 0.08

ultrasonic cleaning bath and a probe system. The acoustic pressures applied with the bath and with the probe system were 0.81 ± 0.05 bar (40 kHz) and 1.07 ± 0.20 bar (20 kHz), respectively. During brining, the temperature of the brine was maintained at a constant 2.0 ± 0.5°C. Final moisture and salt gain were measured. For every type of salting, at least three replications were carried out. In experiments with brine agitation, a strong stirring was used to create negligible external resistance to mass transfer. An ANOVA was performed on the final moisture and NaCl gain of samples in the different experiments. Four groups of experiments were considered (nonultrasonic and nonagitation, nonultrasonic and agitation, using an ultrasonic bath, and using a sonic probe system treatment). For final moisture content, at a 95% level of confidence, no differences between different brining groups were observed. However, the experiments carried out with a probe system showed a higher final moisture, and at the 90% level of confidence, this group was significantly different. On the other hand, significant differences were found for salt gain between groups (95%), as can be seen in Table 18.2. The samples treated with the sound probe system gained more NaCl than the others. No differences were found between agitation, nonagitation brine, and ultrasound applied by means of the ultrasonic bath. Treatments in an ultrasonic bath consisted of the application of an ultrasonic energy in a volume of liquid. The fact is that all carcasses vibrate to produce an ultrasonic field more or less irregular in all quantities of liquid contained. The sound probe system can generate a high intensity field in a small zone and then produce an ultrasonic field when focused. Just below the probe tip, the maximum ultrasonic power is generated. If a sample is placed at this point, it receives a large amount of the acoustic pressure emitted. Pork loin is not a porous material, so the effects of ultrasound are less important than for other foods. According to the literature, when the intensity of ultrasound is above the threshold, the salt gain should be increased with respect to the nonultrasonic brining. In that case, the effects created by ultrasound become important when the intensity is sufficient. Only then, microstreaming, contractions and expansions of material, cavitation, etc., force the entry of salt into the meat matrix. However, this high intensity seems to affect the exit of water from the pork loin, an effect different from the one found in apple and cheese osmotic dehydration, where an increase in water loss was observed.


18.5 CONCLUSIONS High intensity ultrasound enhances the mass transfer process, affecting internal as well as external resistance to mass transfer. The propagating medium, material treated, and intensity applied can influence the effects produced by ultrasound. Different technologies are needed for applications in gas and liquid media. Stepped-plate transducers generate the intensity to accelerate the drying of carrots with air. The decrease of the boundary layer and successive contractions and expansions produced by ultrasound aid water exit from carrot matrix. In liquid applications, mass transfer is also improved. Water loss in osmotic dehydration of apple and cheese brining is increased by applications of ultrasound. The material structure also affects the influence of ultrasound. The removal of air from apple pores is assisted by ultrasound, and the air is replaced by osmotic solution. Thus, solute increase in a solid matrix is very important. Cheese pores have less air occluded and, as a consequence, the influence of ultrasound on solute gain is less important. Additionally, apple structure better transmits ultrasound than does cheese. Pork loin has no large pores, and therefore mass transfer is difficult. Nonsignificant differences appear between applications of ultrasound by means of a commercial cleaning bath and agitation or nonagitation brine experiments. Higher ultrasonic intensity generated by a probe system accelerated salt gain. This shows the existence of a threshold where the influence of ultrasound appears. Commercial baths do not usually reach this threshold for some products, and no differences appear, as reported in the literature. High intensities applied to meat increase salt gain and make water loss difficult. These effects could lead to an increase in the process efficiency.

NOMENCLATURE D d P m Cp T t

Diameter of driven face of a transductor; m Diameter of emitting face of a transductor; m Power; W Mass; kg Heat capacity; J/kg°C Temperature; °C Time; sec

REFERENCES Arkhangel’skii, M.E. and Statnikov, Y.G., The action of ultrasonic vibrations on diffusion, in Physical Principles of Ultrasonic Technology, Vol. 2, Rozenberg, L.D., Ed., Plenum Press, New York, 1973. Cárcel, J.A., Benedito, J., Llull, P., Clemente, G., and Mulet, A., Influence of Ultrasound in Meat Brining, Proceedings of AICHE-CoFE 99, Dallas, November 2–5, 1999, pp. 163–170. Chiralt, A. and Fito, P., Salting of Manchego-type cheese by vacuum impregnation, in Food Engineering 2000, Chapman and Hall, New York, 1996, pp. 215–230. Floros, J.D. and Liang, H., Acoustically assisted diffusion through membranes and biomaterials, Food Technol., December 1994, pp. 79–84.


Gallego-Juárez, J.A., Some applications of air-borne power ultrasound to food processing, in Ultrasound in Food Processing, Povey, M.J.W. and Mason, T.J., Eds., Blackie, Glasgow, 1998, pp. 127–143. Gallego-Juárez, J.A., Rodriguez-Corral, G., Gálvez-Moraleda, J.C., and Yang, T.S., A new high-intensity ultrasonic technology for food dehydration, Drying Technol., 17, 597–608, 1999. Gallego-Juárez, J.A., Rodríguez-Corral, G., San Emetano, J.L., and Montoya-Vitini, F., Electroacoustic unit for generating high sonic and ultrasonic intensities in gases and interphases, Spanish patent 8903371, 1989; European patent No. EP 450.030.A1, 1991; U.S. patent No. 5,299,175, 1994. Jun, C., Kedie, Y., Shulai, C., Adschiri, T., and Arai, K., Effects of Ultrasound on Mass Transfer in Supercritical Extraction, The 4th International Symposium on Supercritical Fluids, Sendai, Japan, May 11–14, 1997, pp. 707–710. Kim, S.M. and Zayas, J.K., Effects of ultrasound treatment on the properties of chymosin, J. Food Sci., 56, 926–930, 1991. Liang, H., Modelling of ultrasound assisted and osmotically induced diffusion in plant tissue, thesis, Purdue University, Lafayette, IN, 1993. Margulis, M.A. and Maximenco, N.A., Appendix: measurement of ultrasound-absorbed power (calorimetric methods), in Advances in Sonochemistry, Vol. 2, Mason, T.J., Ed., JAI Press, London, 1990, pp. 288–289. Mason, T.J., Power ultrasound in food processing: the way forward, in Ultrasound in Food Processing, Povey, M.J.W. and Mason, T.J., Eds., Blackie, Glasgow, 1998, pp. 105– 126. Mason, T.J. and Cordemans, E.D., Ultrasonic intensification of chemical processing and related operations: a review, Trans. IchemE., 74, 511–516, 1996. Mitchell, T.J. and Potts, C.S., Through-circulation drying of vegetables. III. Carrots, J. Sci. Food Agric., 9, 93–98, 1958. Mulet, A., Drying modelling and water diffusivity in carrots and potatoes, J. Food Eng., 22, 329–348, 1994. Mulet, A., Berna, A., Borrás, M., and Piñaga, F., Effect of air flow rate on carrot drying, Drying Technol., 5, 245–258, 1987. Muralidhara, H.S., Ensminger, D., and Putnam, A., Acoustic dewatering and drying (low and high frequency): state of the art review, Drying Technol., 3, 529–566, 1985. Panchev, I., Kirchev, N., and Kratchanov, C., Improving pectin technology. II. Extraction using ultrasonic treatment, Int. J. Food Sci. Technol., 23, 337–341, 1988. Pugin, B. and Turner, A.T., Influence of ultrasound on reaction with metals, in Advances in Sonochemistry, Vol. 1, Mason, T.J., Ed., JAI Press, London, 1990, pp. 81–118. Sajas, J.F. and Gorbatow, W.M., Use of ultrasound in meat technology, Fleischwirtschaft, 58, 1009–1021, 1978. Sánchez, E.S., Simal, S., Femenia, A., Benedito, J., and Rosselló C., Influence of ultrasound on mass transport during cheese brining, Eur. Food Res. Technol., 209, 215–219, 1999. Shi, Q.S. and Fito, P. 1993. Vacuum osmotic dehydration of fruits, Drying Technol., 11, 1432–1442, 1993. Simal, S., Benedito, J., Sánchez, E.S., and Rosselló, C., Use of ultrasound to increase mass transport rates during osmotic dehydration, J. Food Eng., 36, 323–336, 1998. Vinatoru, M., Toma, M., Radu, O., Filip, P.I., Lazurca, D., and Mason T.J., The use of ultrasound for the extraction of bioactive principles from plant materials, Ultrason. Sonochem., 4, 135–139, 1997.


19

Mass Transfer and Residence Time Studies in Spinning Cone Columns M.E. Vargas-Ugalde, K. Niranjan, D.L. Pyle, and G.F. Gutiérrez-López

CONTENTS 19.1 Introduction 19.2 Background 19.2.1 Flavor Recovery Technology 19.2.2 The Spinning Cone Column 19.2.3 Mass Transfer Coefficients 19.2.4 Mean Residence Time 19.3 Materials and Methods 19.3.1 Equipment 19.3.1.1 SCC 1000 Flavourtech 19.3.1.2 SCC Perspex Laboratory Model 19.3.2 Methods 19.3.2.1 Determination of Overall Volumetric Mass Transfer Coefficients (KLa) 19.3.2.2 Evaluation of the Mean Residence Time 19.4 Results and Discussion 19.4.1 Mass Transfer 19.4.2 Mean Residence Time 19.5 Conclusions Nomenclature Acknowledgments References

19.1 INTRODUCTION The Spinning Cone Column (SCC) is a multistage liquid–gas contacting unit with increasing applications in food processing. SCC is a type of distillation equipment that operates under vacuum pressures. It presents many advantages because of its mass transfer efficiency, the short processing times involved, and its ability to handle


highly viscous and non-Newtonian liquids under reduced pressure. An understanding of some of the mechanisms of simultaneous transfer of momentum, heat, and mass in this equipment is necessary to save energy and to improve not only the separation efficiency, but also the basis for selection, design, and control of columns and processes involved. Despite the extensive use of this technology, few studies have been conducted concerning the performance and mathematical modeling of this equipment. To study the mass transfer and the residence time distribution in this device, a Perspex model geometrically similar to the pilot plant SCC 1000 Flavourtech Europe LTD installed at the University of Reading, U.K., was employed. This model uses air to simulate the steam current. For these experiments, two spinning and two stationary cones were employed; residence time experiments were also performed in the pilot equipment.

19.2 BACKGROUND 19.2.1 FLAVOR RECOVERY TECHNOLOGY The main objective when processing foods is preservation; the processes used include evaporation, pasteurization, high temperature–short time pasteurization, distillation, drying, and other operations. By applying these processes, it is possible to extend the shelf life of many food materials. However, in most heat treatment procedures, the operating conditions may change the original properties of a product. The use of low temperature and vacuum pressure is a popular way of maintaining flavor and aroma characteristics. In most cases, compounds responsible for flavor and aroma are present in products in small amounts; they are highly volatile and extremely sensitive to heat damage. To minimize losses of aroma and flavor when processing foods, flavor/ aroma recovery technology is widely used. Basically, this technology consists of removing and collecting volatiles before the main process and reincorporating them into the final product at the last stage of processing. In this way, the original sensory characteristics are maintained (Schobinger, 1999). Flavor recovery technique was initially applied to process juice fruits around 1940. There exist different kinds of equipment to perform the separation of volatiles; the use of a particular one depends on the product characteristics, the process costs, and the final quality to be obtained.

19.2.2 THE SPINNING CONE COLUMN Among the uses of the SCC in the food industry are (Menzi and Emch, 1989; Flavourtech, 1998): • • • • • •

Aroma recovery from fruit and vegetable juices, botanicals, herbs and spices Dealcoholization of drinks Desulfiting of grape juice Removal of tannins Coffee, tea, and citrus byproduct processing Removal of off-odors from dairy products


In fruit juice processing, the SCC is employed just before the product enters the concentrator. Through the use of SCC, a substantial aroma recovery can be achieved with a low evaporation rate (from 0.5 to 2.0% depending on the fruit) (Craig, 1987; Schobinger, 1999). The SCC resembles a combination of centrifuge–distillation units, and it is classified in the same group of equipment as the packed, bubble-cap, and plate columns (Menzi and Emch, 1989; Shah, 1995). In this equipment, the inert stripping gas—nitrogen or steam at low temperatures — removes a vapor stream of volatile compounds from liquids or slurries. Internally, the column contains two series of inverted cones. A series of fixed cones is attached to the inside wall of the column; another series of cones, parallel to those fixed, is attached to a rotating shaft. The stationary and rotating cones alternate vertically. The product comes in to the top of the column, flows down the upper surface of the first fixed cone, and drops onto the first rotating cone due to gravity. Then, due to centrifugal force, the liquid is spun into a turbulent thin film that is forced upward to the top of the spinning cone, dropping onto the next stationary cone. The product then goes from cone to cone to the bottom of the column, following a spiral trajectory. The gas for stripping comes into the bottom of the column and flows upward, passing across the surface of the thin film of liquid and collecting volatile compounds along its path. On the underside of the rotating cones are baffles that create a high degree of turbulence in the rising vapor stream. The vapor flows out of the top of the column and passes through a condensing system, which captures the volatile compounds in a concentrated liquid form. The remaining liquid or slurry is pumped out of the bottom of the column (Flavourtech, 1991).

19.2.3 MASS TRANSFER COEFFICIENTS In chemical and food engineering, empirical correlations are commonly used in many situations because real systems exhibit a different behavior from ideal, and it is not easy to determine the contribution of each of the variables involved in a particular case. Although these correlations cannot be applied to general cases, they are useful to predict the behavior of a given phenomenon within the range of conditions studied. The rates of mass transfer between gas and liquid phases are strongly affected by the solubility of the gaseous components in the liquid. Oxygen is among the sparingly soluble gases in aqueous solutions. The solubility of a gas in a liquid depends on the temperature, the equilibrium partial pressure of the solute gas in the gas phase, and the concentration of the solute gas in the liquid phase (Aiba et al., 1973). The total pressure on the system also affects the solubility, but when the value of this parameter is lower than 5 atm, the solubility of gases can be considered independent of the total pressure. In those cases where the concentration of dissolved gas is small and the temperature and pressure are far from the critical temperature and pressure of the gas, Henry’s law can express the solubility of a gas in a liquid: p=Hx


(19.1)

Henry’s constant (H) is affected by temperature, concentration of dissolved salts, and pH. For a sparingly soluble gas, such as oxygen, the overall gas transfer rate (R) is given by the flux equation: *

R = KLa (c − c)

(19.2)

Volumetric R and a are evaluated considering the total liquid volume, since the gas hold-up is considered negligible for calculation purposes. Concentration patterns in the SCC are not those of a perfectly mixed tank, but they resemble those of plug flow. Then, the log mean oxygen concentration applies, and it is possible to calculate KLa from a mass balance, assuming the oxygen concentration in the liquid phase is equal to zero: (V KLa) ∆clm = L ∆c

(19.3)

For cylinders and disks rotating in an infinite fluid, several equations have been developed to predict mass transfer rates. In Sherwood et al. (1975) the following relationship for cylinders is reported: 2/3

0.664

JD = St Sc = (kc/U) Sc

(19.4)

The same authors reported the following equation for oxygen transport to a 5 6 rotating disk in salt water for values of Re between 6 × 10 and 2 × 10 and Sc numbers between 120 and 1200. k cd = 5.6 Re1.1 Sc0.333 D

(19.5)

They reported a semitheoretical relationship for the turbulent region: k cd = α Re 0.4 Sc0.25 D

(19.6)

They also described electrochemical studies by Bagotskaya, who found values of kc proportional to the first power of rotational speed in the case of turbulent flow on a rotating disk. Blanch and Clark (1996) have reported α and β values (see Equation 19.7) to obtain kLa in bubble columns and airlift contactors as a function of the superficial gas velocity and liquid properties for low liquid viscosities. In these systems, α values from 0.4 to 1.09 and β values from 0.89 to 1.23 were reported. β

kLa = α vs

(19.7)

Blanch and Clark (1996) also reported on correlations for kLa in stirred tanks for inviscid systems as follows:


For coalescing (clean) dispersions: −2

0.4

−3

0.7

0.5

(19.8)

0.2

(19.9)

kLa = 2.6 × 10 (Po/V) vs For noncoalescing (dirty) dispersions: kLa = 2.0 × 10 (Po/V)

vs

−3

These equations were obtained for power inputs from 500 to 10,000 watt m and small vessels of up to several thousand liters (Blanch and Clark, 1996) Zivdar (1997) reported results on mass transfer characteristics of structured packed columns 0.1 m in diameter were evaluated for humidification of air and oxygen desorption from water. Gas and liquid transfer mass coefficients were dependent on the corresponding flow rates, and gas phase mass transfer was also influenced by change of liquid hold-up due to gas flow.

19.2.4 MEAN RESIDENCE TIME The measurement of residence time distributions (RTD) is useful to estimate the performance of process equipment or to find how much it deviates from the ideal behavior. Even when the overall flow rate may have no detectable variation, not all particles passing through the system stay there for the same length of time. For a single system, there is a spread of residence times, which can be caused by channelling, recycling, or creation of stagnant regions of fluid in the vessel. The residence time distribution function, E(t), is defined considering that the fraction of product in the outlet stream that has been in the system for a length of time between t and t + dt is equal to Edt (Denbig and Turner, 1984), so that:

∫

∞

E( t )dt = 1

(19.10)

0

The average time spent by a product flowing at flow rate v through a vessel volume V is equal to V/v. If the density of the flowing stream does not change as it passes through the system and for a no-back-mixing situation, the mean residence time (MRT) is equal to V/v and: MRT =

∫

∞

tE( t )dt

(19.11)

0

19.3 MATERIALS AND METHODS Two types of spinning cone columns were used in this work: a pilot plant SCC1000 model (Flavourtech LTD, Europe), and a Perspex laboratory model. To determine the dissolved oxygen concentration in the laboratory column, a JENWAY 9071 dissolved oxygen meter was used. Compressed nitrogen was the degassing agent used to eliminate oxygen from deionized water during the measurements.


An injection gun (APV Co. Ltd.) and a spectrophotometer (CECIL 1000 Series, CE1021 model) were used for the residence time distribution (RTD) studies with cochineal red A spec Pure 70 BAYER AG as a tracer. The carboxymethylcellulose (CMC) working solutions were prepared with Blanose CMC 7HF Sodium Carboxymethylcellulose Aqualon, Hercules Inc.

19.3.1 EQUIPMENT 19.3.1.1 SCC 1000 Flavourtech The SCC column 1000 is composed of a vertical stainless steel cylinder 2.0 m in height and 0.33 m in diameter, with 40 cones and a capacity range of 100–1000 −1 −1 l h . The rotating speed varies from 26 to 160 sec ; nitrogen or low temperature steam can be used as the stripping medium, and the column usually operates at vacuum pressures (Flavourtech, 1991). 19.3.1.2 SCC Perspex Laboratory Model To get information about the performance of the SCC, a Perspex model geometrically similar to the pilot plant column was employed in this research. The model is 0.2 m in height and 0.37 m in diameter; it has three stationary and two spinning cones. It works at atmospheric pressure, and air is used instead of steam or nitrogen for hydrodynamic purposes. The model has a shaft 50 mm in radius, and the column inner radius is 185 mm. Both the spinning and the stationary cones have inner and outer radii of 50 and 145 mm, respectively. Four pressure taps are also available, and it is possible to change shaft velocity, number of cones, and liquid and air flow rates. A diagram of this column is presented in Figure 19.1.

Perspex Column

air

cin

pt Oxygen probe

cones pt

tank

solution

shaft

motor

Recirculation or drain tube

cout

pump water pool pt = pressure tappings

FIGURE 19.1 Schematic diagram of the SCC laboratory Perspex model.


N2

TABLE 19.1 Conditions for Mass Transfer Experiments Water Flow Rate 3 −1 5 (m sec × 10 )

Nitrogen Flow Rate 3 −1 4 (m sec × 10 )

Cone Velocity (sec )

4.2, 5.8, 11.7

8, 1.2, 1.6, 2.0

26, 47, 79, 131

−1

For the RTD determinations, several experiments were performed using the pilot plant SCC.

19.3.2 METHODS 19.3.2.1 Determination of Overall Volumetric Mass Transfer Coefficients (KLa) KLa were calculated by measuring the dissolved oxygen concentration in deionized water at the bottom of the column until the condition of equilibrium was reached, employing a deoxygenating method using nitrogen as the displacing gas (Heineken, 1971; Atkinson and Mavituna, 1991), based on the factorial experimental design presented in Table 19.1. 19.3.2.2 Evaluation of the Mean Residence Time Water and carboxymethylcellulose (CMC) at 0.5 and 1% w/w were used, employing different conditions in both the Perspex model and the pilot plant SCC, as presented in Table 19.2. Mean residence time (MRT) was measured by a pulse method (Levenspiel, −3 1972) using cochineal red dye as a tracer (0.15 g cm for the laboratory model and −3 0.3 g cm for the pilot column) and measuring the concentration as a function of time by reading the absorbance of the samples at the exit of the equipment in a spectrophotometer CECIL 1021 and by applying Equation (19.11).

19.4 RESULTS AND DISCUSSION 19.4.1 MASS TRANSFER KLa values for the conditions presented in Table 19.1 were calculated by applying 3 Equation (19.3) for a tank volume (V) of 0.0215 m . Oxygen concentration values at −3 the inlet of the system (cin) fell within 0.007 to 0.008 kg m for most of the trials. The equilibrium oxygen concentration (cout) was measured with an oxygen probe immersed in the water pool formed at the bottom of the column. Equilibrium was considered to have been reached when no change of this measured value was registered by the probe (see Figure 19.1). In Figure 19.2, calculated KLa values as a function of angular velocity of the −4 3 −1 cones and gas flow rate for a liquid flow rate of 4.2 × 10 m sec are presented. In Figure 19.3, calculated KLa values as a function of gas flow rate and liquid flow


TABLE 19.2 Conditions for Determination of Mean Residence Time Experiments

Water

CMC

Water

CMC 0.5%

SCC Perspex Model −1 Cone velocity (sec ) 3 −1 5 Liquid flow rate (m sec ) × 10 3 −1 4 Gas flow rate (m sec ) × 10 Concentration (%) 3 −1 5 Liquid flow rate (m sec ) × 10 3 −1 4 Gas flow rate (m sec ) × 10 −1 Cone velocity (sec )

26, 47, 79, 131 3.3, 6.7 0, 18.33 0.5 1.6, 3.3 0 79, 47, 131

SCC Pilot Plant −1 Cone velocity (sec ) 3 −1 5 Liquid flow rate (m sec ) × 10 3 −1 Gas flow rate (m sec ) −1 Cone velocity (sec )

26, 61 5.6, 11.1 0 26, 61

3

−1

Liquid flow rate (m sec ) × 10 3 −1 Gas flow rate (m sec )

5

1.0 0.8 0 79

2.8, 5.6 0

0.007

KLa sec-1

0.006 0.005 0.004 0.003

131

0.002

79

0.001

47

0

26 7.88

11.81

15.83

Angular velocity sec-1

19.67

Gas flow rate m3sec-1 × 10-4 −4

3

−1

FIGURE 19.2 KLa for oxygen in water. Liquid flow rate = 4.2 × 10 m sec .


0.009 0.008

KLa sec

-1

0.007 0.006 0.005 0.004

11.67

0.003 0.002

5.83

Liquid flow rate 3 -1 -5 4.17 m sec × 10

0.001 0 7.88

11.81

15.83

19.67

Gas flow rate m3 sec -1 × 10-4

−1

FIGURE 19.3 KLa for oxygen in water. Angular velocity = 47 sec . −1

rate for an angular velocity of 47 sec are presented. From Figures 19.2 and 19.3, it is possible to observe that KLa increases with gas and liquid flow rates and angular velocity; this agrees with findings made by Zivdar (1997) for structured packed columns. Empirical correlations reported in the literature for mass transfer descriptions for rotating disks, cylinders (Sherwood et al., 1975), bubble columns, and stirred tanks (Blanch and Clark, 1996) showed that the dimensionless mass transfer number 0.5 depends on the value of (Re) . An empirical correlation (Equation 19.12) of KLa values and ω, L, and G was obtained by means of the S.A.S. statistical program (S.A.S. Institute, 1989) using an iterative nonlinear least squares methodology based on the Gauss–Newton method. 0.489

KLa = 2.254

(19.12)

It is possible to observe that KLa depends on the dynamic parameters with an exponent close to 0.5. This may indicate that mass transfer in the SCC follows a similar trend as in other situations reported in the literature (Sherwood et al., 1975; Blanch and Clark, 1996; Zivdar, 1997).

19.4.2 MEAN RESIDENCE TIME An example of the E curves obtained using the Perspex laboratory model is presented in Figure 19.4. From Figure 19.4, a plug flow pattern can be observed. Similar findings have been reported by other authors when working with a SCC (Shah, 1995; Menzi and Emch, 1989).


0.09 0.08 0.07

E value

0.06

Fluid: Water Angular velocity = 79 sec-1 Air flow rate = 6.7 x 10-5 m3sec-1 Liquid flow rate= 18.33 x 10-4 m3sec-1

0.05 0.04 0.03 0.02 0.01 0 0

100

200

300

400

time

sec

FIGURE 19.4 Example of an E curve.

TABLE 19.3 Mean Residence Time (MRT) for Water in the SCC Perspex Laboratory Model Conditions Cone Velocity (sec )

Liquid Flow Rate 3 −1 5 (m sec × 10 )

Air Flow Rate 3 −1 (m sec )

MRT (sec)

26 26 26 26 47 47 47 47 79 79 79 79 131 131 131 131

3.3 3.3 6.7 6.7 3.3 3.3 6.7 6.7 3.3 3.3 6.7 6.7 3.3 3.3 6.7 6.7

0 18.33 0 18.33 0 18.33 0 18.33 0 18.33 0 18.33 0 18.33 0 18.33

16 20 13 17 13 16 16 13 14 15 12 12 29 29 18 14

−1

Mean residence times (MRT) for water and CMC were calculated from these E curves (Figure 19.4). Values of MRT for water and CMC are presented in Tables 19.3 and 19.4, respectively. For the conditions studied in the Perspex laboratory model, residence


TABLE 19.4 Mean Residence Time (MRT) for CMC in the SCC Perspex Laboratory Model Conditions Cone Velocity (sec )

Liquid Flow 3 –1 5 (m sec × 10 )

Air Flow 3 –1 (m sec )

Concentration (%)

MRT (sec)

79 79 79 47 131

0.8 3.3 1.6 3.3 3.3

0 0 0 0 0

1.0 0.5 0.5 0.5 0.5

199 30 71 29 35

Mean Residence Time (MRT)

−1

300 Code [ letter] / [value ] / [value] on x-axis represents: [product (W = water, C = CMC0.5%)] / [liquid flow rate m3sec-1 x 10-5] / [cone velocity sec-1]

250 200 150 100 50 0

8/

2.

C

26

8/

C

2.

61

6/

C

5.

26

6

6

W

5.

/2

6/

61

1/

W

5.

W

. 11

26

1/

61

. 11

W

FIGURE 19.5 Mean residence time for water and CMC (0.5%) in the SCC pilot plant.

time for water was not influenced by air nor by liquid flow rate, nor even by cone rotating velocity. The average MRT obtained for this equipment was about 17 seconds. In the case of experiments using 1% CMC solution (apparent viscosity: 960 cP −1 at 10 sec at T = 25°C), MRT was about 11.7 times greater than the average MRT −1 for water. When using 0.5% CMC solution (apparent viscosity: 200 cP at 10 sec ο at T = 25 C), MRT was 1.8 to 4.2 times greater than values for water. The influence of liquid flow rate on MRT was determined using 0.5% CMC solu−5 3 −1 tions. At 3.3 × 10 m sec , the MRT was 2.4 times lower than at a flow rate of 1.6 × −5 3 −1 −1 10 m sec . There was no influence of cone velocity on MRT for 79 and 47 sec . The −1 time for 131 sec was longer; this can be explained by flooding conditions in the first spinning cone (Shah, 1995). Results for the experiments in the SCC pilot plant are presented in Figure 19.5. It is possible to observe that MRT for water depends on flow rate as well as on cone velocity, becoming shorter at higher values of both variables, as expected. The effect of viscosity on MRT can be observed from the experiments with CMC 0.5% at the


same conditions. MRT for CMC is 1.62 times greater than MRT for water. In the −5 3 −1 experiments with water at 5.6 × 10 m sec , MRT is 1.28 times greater for a cone −1 −5 3 −1 speed of 26 than 61 sec . From experiments at a flow rate of 11.1 × 10 m sec −1 and rotating velocity of 26 sec , MRT is 1.14 times greater than that obtained at a −1 rotating speed of 61 sec . In Figure 19.5, the effect of the liquid flow rate using water on MRT is shown, −1 comparing results at a constant angular speed. At 26 sec , MRT is 1.67 times greater −5 3 −1 −5 3 −1 −1 for a flow rate of 5.6 × 10 m sec than for 11.1 × 10 m sec . At 61 sec , MRT −5 3 −1 −5 3 is 1.49 times greater for a flow rate of 5.6 × 10 m sec than for 11.1 × 10 m −1 sec . The effect of liquid flow rate using CMC solution (0.5%) is shown comparing −1 −5 3 −1 data for a constant speed. MRT at 26 sec and 2.8 × 10 m sec is 1.72 times −5 3 −1 greater than MRT for 5.6 × 10 m sec . With a flow rate of CMC solution (0.5%) −5 3 −1 −1 of 2.8 × 10 m sec , MRT obtained at an angular velocity of 26 sec is 1.05 times −1 greater than MRT at 61 sec .

19.5 CONCLUSIONS Mass transfer into the spinning cone column depends on angular cone velocity and liquid and gas flow rate; it is higher at higher values for these variables; their influence can be represented by a power equation. Mean residence time is affected by liquid flow rate, cone speed, and viscosity of the product; the influence of liquid flow rate is greater than the influence of cone speed. The combined effect of high mass transfer rates and short mean residence times for the product into the column brings an advantage for the recovery of aromas and flavors in foods, since it diminishes the damage caused by heat and processing time to these volatile compounds. However, the evidence of a long tail in the residence time distribution must be considered, since this means that some of the volatiles are still present after the mean process and may be affected by having longer processing times. The information given in this work may be used for finding appropriate operating conditions for a particular product, aiming to obtain as much mass transfer as possible within the economic limits of power requirements for pumping the liquid and for rotating the cones so as to obtain low MRT.

NOMENCLATURE a c c* ∆c ∆clm

2

−3

Bubble surface area per unit volume; (m m ) −3 Dissolved oxygen concentration in the bulk liquid; (kg m ) −3 Solubility of oxygen in equilibrium with pG (c* = pG/H); (kg m ) Concentration difference between the inlet and the outlet of the system; (cin − cout) Logarithmic mean of concentration difference between the inlet and the    (c − c )  outlet of the system  inc out  in  ln cout 


D d E(t) G H JD kc

2

−1

Diffusivity coefficient; (cm sec ) Disk diameter; (m) Residence time distribution function 3 −1 Gas flow rate; (m sec ) −1 Henry’s law constant; (atm mole fraction ) Mass transfer dimensionless group Individual mass transfer coefficient in Equations (19.4), (19.5) and −1 (19.6); (cm sec ) −1 KLa Overall volumetric mass transfer coefficient based on the liquid phase; (sec ) −1 kLa Individual volumetric mass transfer coefficient based on the liquid phase; (s ) 3 −1 L Liquid flow rate; (m sec ) l Characteristic length; (cm) MRT Mean residence time; (sec) p Partial pressure of the solute in the gas phase; (atm) pG Gas pressure; (atm) −3 Po/V Gassed power per volume; (watt m ) −3 −1 R Overall oxygen transfer rate; (kg m sec ) Re Reynolds number based on diameter and peripheral velocity of disk in Equation (19.5); (d U ρ/µ) RTD Residence time distribution Sc Schmidt number; (µ/ρ D) Sh Sherwood number; (kc l/D) St Stanton number; (Sh/Re Sc) t Time; (sec) −1 U Peripheral velocity of cylinder in Equation (19.4); (m sec ) 3 V Reactor volume; (m ) −1 v Material flow rate in a reactor; (m sec ) −1 −1 vs Superficial velocity; (cm sec in Equation (19.7); m sec in Equations (19.8) and (19.9)) x Mole fraction of the solute in the liquid phase (molar concentration of the dissolved gas for sparingly soluble gases) α Coefficient in Equation (19.7) β Exponent in Equation (19.7) −1 −1 µ Viscosity; (g cm sec ) −3 ρ Density (g cm ) −1 ω Angular velocity of the cone; (sec )

ACKNOWLEDGMENTS The authors thank the ALFA Network: Food Quality in Food Engineering, coordinated by the University of Reading, U.K., EU Contract Number ALR/B 7–3011/ 94.04–5.0130.9; Flavourtech Europe Ltd. for allowing the use of the Spinning Cone Column model 1000; FES Cuautitlán-UNAM, ENCB-IPN, and CYTED project Xl.13.


REFERENCES Aiba, S., Humphrey, S., and Millis, F., Biochemical Engineering, 2nd ed., Academic Press, New York, 1973, pp. 163–194. Atkinson B. and Mavituna, F., Gas–liquid mass transfer and mixing, in Biochemical Engineering and Biotechnology Handbook, Macmillan Pub. Ltd., U.K., 1991, pp. 703–731. Blanch, H.W. and Clark, D.S., Biochemical Engineering, Marcel Dekker, New York, 1996, pp. 388–392. Craig, A.J.M., Flavor recovery by the Flavortech Recovery System, Food Technol. Aust., 39, 102–104, 1987. Denbig, K.G. and Turner, J.C.R., Chemical Reactor Theory: An Introduction, 3rd ed., Cambridge University Press, Cambridge, 1984. Flavourtech, Ltd., SCC Treatment of UHT Milk, SCC Technical Bulletin TB, February 21, 1998. Flavourtech Ltd., Finally: Processed and Packaged Foods and Beverages Can Taste as Good as the Originals, Commercial Bulletin, 1991. Heineken, F.G., Oxygen mass transfer and oxygen respiration rate measurements utilising fast response oxygen electrodes, Biotech. Bioeng., 13, 599–618, 1971. Levenspiel, O., Chemical Reaction Engineering, 2nd ed., John Wiley and Sons, New York, 1972, pp. 253–325. Menzi, H. and Emch, F., The spinning cone column: An efficient separator of aroma volatiles from a liquid, Lebensmit. Wiss. Technical., 22, 324–328, 1989. S.A.S. Institute Inc., The S.A.S. System for Microsoft Windows, Release 6.10, Cary, N.C., 1989. Schobinger, U., Progress in fruit juice technology during the last 50 years — A survey, Fruit Process., 7, 275–281, 1999. Shah, N.A., Study of the Fluid Dynamics in the Spinning Cone Column, thesis, University of Reading, 1995. Sherwood, T.K., Pigford, R.I., and Wilke, C.H.R., Mass Transfer, McGraw-Hill, New York, 1975, pp. 236–241. Zivdar, M., Distillation for Food Flavour Separations, seminar, July 4, 1997, Department of Chemical Engineering, University of Sydney, Australia, http://www.chem.eng.usyd. edu.au.


Part III Heat Transfer


20

Transport Phenomena during Double-Sided Cooking of Meat Patties S.E. Zorrilla, S. Wichchukit, and R.P. Singh

CONTENTS 20.1 Introduction 20.2 Mathematical Model 20.2.1 The Basic Equation 20.2.2 The Melting Moving Boundary 20.2.3 The Evaporating Moving Boundary 20.3 Thermal Properties 20.3.1 Contact Heat Transfer Coefficient 20.4 Simulation and Model Validation 20.5 Future Considerations 20.6 Conclusions Nomenclature References

20.1 INTRODUCTION In 1998, 5.5 billion hamburgers or cheeseburgers were served in commercial restaurants in the United States (NCBA, 1998). Unfortunately, since 1982 several outbreaks caused by Escherichia coli O157:H7 have been linked to the consumption of beef patties (D’Sa et al., 2000). Consequently, the U.S. Department of Agriculture (USDA) and U.S. Food and Drug Administration (FDA) have recommended cooking ground beef products to 71°C for home preparation and 68°C for 15 sec for food service, respectively (USDA, 1998; FDA, 1997). However, suitable procedures for measuring internal temperature in beef patties are difficult, especially at food service establishments. As a result, patties may be overcooked with undesirable textural and sensory characteristics or undercooked and microbiologically unsafe. Therefore, a fundamental understanding of the heat transfer process is necessary to design equipment and sensors that ensure the desired safety and quality goals during beef patty cooking. One widely used beef patty cooking method in restaurants involves placing frozen patties between two heated plates with the plate surface temperatures generally


greater than 160°C. As heat penetrates the patty, a thawing process begins. As heating continues, higher temperatures cause protein denaturation and considerable lowering in water holding capacity (Lawrie, 1991). Water and fat may be partially squeezed out because of meat shrinkage and external pressure. Near the patty surface, browning reactions occur, and a combination of dehydration and high temperature lead to the formation of a crust (Dagerskog and Bengtsson, 1974). In a simplified model, as heat penetrates the frozen patty, two moving boundaries can be distinguished: the thawing and the evaporation boundaries (Singh, 2000). Predictive mathematical models based on classical transport phenomena equations can be a useful tool to represent this process.

20.2 MATHEMATICAL MODEL Different mathematical models for studying the contact cooking process in hamburger patties have been developed (Dagerskog, 1979a, b; Ikediala et al., 1996; Pan et al., 2000; Zorrilla and Singh, 2000a, b). Dagerskog (1979a) proposed a model of heat transfer based on the heat conduction equation. Internal mass transfer was based on an empirical relation between water retaining capacity and temperature. Dagerskog (1979b) modified this original model for use with the frozen state. In both cases, the models were solved by finite difference methods. However, experimental and theoretical values did not agree well. Ikediala et al. (1996) modeled the heat transfer in unfrozen meat patties during single-sided pan frying with and without turn-over. The main assumptions were that heat was transferred inside the patty by conduction with no heat generation and negligible meat patty shrinkage or swelling. These authors considered cylindrical geometry and incorporated the heat removed due to moisture loss, which was experimentally determined. The model was solved by a finite element method. Pan et al. (2000) developed a model for cooking a frozen hamburger patty based on the enthalpy formulation, considering the effect of mass transfer and variable heating temperature and heat transfer coefficient. Water and fat losses were obtained experimentally and were found to affect the thermal properties. The mathematical prediction taking into account mass transfer did not significantly improve the prediction results. Zorrilla and Singh (2000a) developed a simplified one-dimensional model to predict temperature profiles in a frozen meat patty during double-sided cooking. Conduction was considered the main mechanism for heat transfer, and enthalpy formulation was used to avoid the discontinuity problem of the phase change during melting. The mathematical formulation considered different top and bottom heating temperatures and was validated by comparing predicted and experimental temperature profiles obtained at 163 and 204°C for the bottom plate and 177 and 221°C for the top plate, respectively. Zorrilla and Singh (2000b) extended the model to a twodimensional cylindrical geometry. The results showed no significant difference when a cylindrical geometry or radial shrinkage was considered to predict the temperature at the geometric center of a patty under normal conditions of hamburger cooking. However, the two-dimensional model provides a description of temperature in regions close to the circumferential edge of the patty, which is not possible with the one-dimensional model.


20.2.1 THE BASIC EQUATION The heat conduction equation for a homogeneous, isotropic solid in a cylindrical coordinate system is (Ozisik, 1993): ∂T 1 ∂  ∂T  1 ∂  ∂T  ∂  ∂T   kr  +  k  +  k  + g = ρC p ∂t r ∂r  ∂r  r 2 ∂φ  ∂φ  ∂z  ∂z 

(20.1)

where T is the temperature in the solid; t is the time; r, φ, and z are the cylindrical coordinates; g is the heat generation term; k is the thermal conductivity; ρ is the density; and Cp is the specific heat. When no heat is generated and temperature has no φ dependence, Equation (20.1) becomes: 1 ∂  ∂T  ∂  ∂T  ∂T  kr  +  k  = ρC p r ∂r  ∂r  ∂z  ∂z  ∂t

(20.2)

A meat patty is not a homogeneous solid. However, if the thermal properties are considered as effective properties, and assuming local thermal equilibrium among the different phases present such as ice, fat, and nonfat solids, Equation (20.2) is still valid (Farkas et al., 1996). During cooking of a frozen beef patty, ice melts and water partially evaporates, while solid fat melts. Assuming that the energy involved in fat melting is negligible, only two moving boundaries need to be considered.

20.2.2 THE MELTING MOVING BOUNDARY In problems involving melting or solidification, the location of the moving solid– liquid interface is not known a priori, and it has to be determined as part of the solution (Ozisik, 1994). As a result, phase change boundary problems are nonlinear and their analytical solution is difficult. The situation is more complex when the material is nonhomogeneous. The actual phase change takes place over a wide range of temperatures at which food properties change considerably. Mannapperuma and Singh (1988) proposed a method using the approach of enthalpy formulation to solve this heat conduction problem involving gradual phase change. Therefore, Equation (20.2) can be written in terms of enthalpy (H) as: 1 ∂ ∂T(H)  ∂  ∂T(H)  ∂H  rk(H)  +  k(H) = r ∂r  ∂r  ∂z  ∂z  ∂t

(20.3)

valid for the region where the thawing process takes place.

20.2.3 THE EVAPORATING MOVING BOUNDARY The water boiling at an interface separates two regions of different thermal properties. In immersion frying of potatoes, these regions are core and crust (Farkas et al., 1996).


In cooking of hamburger patties, these regions are better defined as core and sear layer. There are different methods to consider this boundary. Farkas et al. (1996) considered two regions and solved coupled heat and mass transfer equations for each one. Because the crust thickness is thin compared with core dimensions for the usual cooking times, a linear temperature change in the crust region can be assumed (Vijayan and Singh, 1997). Table 20.1 shows the boundary and initial conditions to complete the mathematical formulation.

TABLE 20.1 Summary of Boundary and Initial Conditions to Complete the Mathematical Formulation during Double-Sided Cooking of Meat Patties Heat Transfer in the Core Boundary Conditions T = Tb ∂T =0 ∂r

z = S1(r,t), S2(r,t); 0 < r < a(t); t > 0

(20.4)

r = 0; S1(r,t) < z < S2(r,t); t > 0

(20.5)

∂T = h c (T − Ta ) ∂r T = Tb

r = a(t); S1(r,t) < z < S2(r,t); t > 0

(20.6a)

r = R(z,t); S1(r,t) < z < S2(r,t); t > 0

(20.6b)

Initial Condition T = T0

t = 0; S1(r,t) < z < S2(r,t); 0 < r < a(t)

(20.7)

−k

Heat Balances at Interfaces between Core and Crust − k crust

∂S ( r, t ) ∂T ∂T +k = λ vρx w i ∂t ∂z ∂z

z = Si(r,t); i = 1, 2; t > 0

(20.8)

− k crust

∂R(z, t ) ∂T ∂T +k = λ vρx w ∂t ∂r ∂r

r = R(z,t); t > 0

(20.9)

− k crust

∂T = h(Tp1 ( t ) − T) ∂z

Heat Balances and Approximations at Surface

∂T Tb − T = ∂z S1 ( r, t ) − k crust

∂T = h(T − Tp 2 (t )) ∂z

Tb − T ∂T = ∂z L − S2 ( r, t ) − k crust

∂T = h c (T − Ta ) ∂r

Tb − T ∂T = ∂r a( t ) − R(z, t )


z = 0; t > 0

(20.10)

z = 0; t > 0

(20.11)

z = L; t > 0

(20.12)

z = L; t > 0

(20.13)

r = a(t); t > 0

(20.14)

r = a(t); t > 0

(20.15)

Equations (20.6b), (20.9), (20.14), and (20.15) are valid when the boiling temperature is reached in the radial direction.

20.3 THERMAL PROPERTIES Thermal properties are important to complete the mathematical description. However, while information about the thermal properties of frozen and unfrozen foods at ambient temperature is widely available, data at higher temperatures are scarce. Urbicain and Lozano (1997), Cleland and Valentas (1997), Rahman (1995), and Sweat (1994) reported predictive equations, correlations, and experimental values for thermal properties considering different composition and temperature ranges. Their equations can be used to estimate the properties at lower and ambient temperatures. In Table 20.2, experimental values and correlations for thermal properties of ground beef products during cooking conditions are given.

20.3.1 CONTACT HEAT TRANSFER COEFFICIENT The contact heat transfer coefficient appears between two solids that are not perfectly in contact. In this case, the contact depends on the grill and patty surface characteristics and the characteristics of any other medium between them. Because juice is released as the cooking process takes place, that medium is a thin layer of fat/air/moisture. Generally, a constant value of this coefficient is assumed for

TABLE 20.2 Thermal Properties of Meat Patties during Cooking Property

Source

3

Density (kg/m ) ρ = 1300 − 300 x w − 400 x f − (0.4 x w + 0.7 x f )T 1.025 (30–75°C)

a b

Apparent Specific Heat (J/kg°C) C p = 1600 − 2600 x w + 15T x f

a

Thermal Conductivity (W/m°C) k = 0.5 − 0.92 x f + 0.0024T

a

k = 0.001(668.4 − 704 x f − 305 x w + 1.35T)

c

0.35 to 0.41 (5–70°C)

b

Note: xw = water content in decimal fraction; xf = fat content in decimal fraction; T = temperature in °C. a

Dagerskog, M., Lebensm.-Wiss. Technol., 12, 217–224, 1979. Pan, Z., thesis, University of California, Davis, 1998. c Baghe-Khandan, M.S., Okos, M.R., and Sweat, V.E., Trans. ASAE, 25, 1118–1122, 1982. b


calculations (Dagerskog, 1979a; Ikediala et al., 1996). Most research papers address contact between metals under steady state conditions (Madhusudana, 1996), but little information is available on contact heat transfer coefficient values for solid foods. Houˇsová and Topinka (1985) have shown experimentally that the contact heat transfer coefficient depends on product type, contact plate temperature, contact pressure, and stage in the heat treatment. The heat transfer coefficients measured were in the 2 range of 200 to 1200 W/m °C. Dagerskog and Sörenfors (1978) found contact heat transfer coefficient values for cooking of minced meat patty in the range of 260 ± 2 2 50 W/m °C during contact cooking and 90 ± 20 W/m °C during cooking in a convection oven. Dagerskog (1979a) obtained contact heat transfer coefficient values 2 in the range of 425 ± 33 W/m °C during double-sided contact frying of meat patties. Pan (1998) experimentally determined values of this coefficient by measuring the heat flux involved and the temperatures of the heating surface and the patty surface. The 2 contact heat transfer coefficients obtained were in the range of 200 to 1200 W/m °C. Wichchukit et al. (2000) obtained contact heat transfer coefficient values, h, under various cooking conditions and determined the changes in heat transfer coefficient during the cooking process and the influence of factors for each cooking condition on heat transfer coefficient. Contact heat transfer coefficient values were calculated using Newton’s law of cooling and measuring the heat flux transferring from the grill plate to a hamburger patty, the grill plate temperature, and the patty surface temperature. Different temperatures (177, 191, 204°C), different gap thicknesses between plates (10.0, 10.5, 11.0 mm) and top or bottom plate were considered in a split–split plot design to study the influence of these operating variables. Average h values for the conditions studied are shown in Figure 20.1. The h 2 values varied with cooking time and were in the range of 250 to 650 W/m °C. A maximum value of h was observed in the beginning, and later it reached an asymptotic value. At the beginning of the cooking process, the solid frozen patty offers more resistance to the deformation caused by compression from the top plate, resulting in an increase of the contact area. The decrease in the heat transfer after about 60 sec may be related to the fact that at this point the hamburger is almost completely thawed. Therefore, the hamburger is easily compressed, and the interfacial contact forces and contact area may decrease. When the gap thickness between plates decreases, the patty is more compressed, the contact area increases, and the heat transfer may increase. However, under compression the fat and water discharge also increases, changing the composition of the layer between the hamburger and the plate, thus affecting the heat transfer. A higher peak of h was observed for a 10.5 mm gap thickness compared with the value for a 11.0 mm gap thickness (P < 0.05). The maximum values for h involving the bottom plate were higher than those involving the top plate (P < 0.05). The grill has top heating plates covered with Teflon release sheets to prevent the material from sticking to the plate. Some experiments were carried out placing the release sheet on the bottom plate, and a similar trend as with the top plate was observed. Therefore, it may be concluded that the release sheet offered resistance to heat transfer to the hamburger patties during cooking. The asymptotic values for h involving the bottom plate were also higher than those involving the top plate (P < 0.05). When the plate temperature increased, the


Bottom plate

Top plate

800

600

600

h (W/m2°C)

h (W/m2°C)

Gap thickness = 10.0 mm 800

400

400

200

200

0

0 0

20

40

60

80

100

0

120

20

40

60

80

100

120

Time (sec)

Time (sec)

800

600

600

h (W/m2°C)

800

2

h (W/m °C)

Gap thickness = 10.5 mm

400

200

400

200

0

0

0

20

40

60

80

100

120

0

20

40

Time (sec)

60

80

100

120

100

120

Time (sec)

600

600 2

h (W/m °C)

800

2

h (W/m °C)

Gap thickness = 11.0 mm 800

400

200

400

200

0

0 0

20

40

60

80

Time (sec)

100

120

0

20

40

60

80

Time (sec)

FIGURE 20.1 Average contact heat transfer coefficients involving top and bottom plates for different gap thicknesses and set grill temperatures: (—) 177°C, (—) 191°C, (- - -) 204°C.

asymptotic value of h decreased (P < 0.05). The asymptotic values for 177 and 191 2 or 204°C were 350 and 250 W/m °C, respectively. At high temperatures, more dehydration near the patty surface occurs, increasing the cohesive forces of the protein matrix. Therefore, adhesion to the grill surface may decrease, decreasing the contact area and the contact heat transfer coefficient.


20.4 SIMULATION AND MODEL VALIDATION The solution of the system comprised of Equations (20.3–20.15) was obtained numerically using finite difference methods (Zorrilla and Singh, 2000a, b). A computer program was written in Fortran (Digital Visual Fortran Version 5.0). The input data included gap thickness between plates, radius of the patty, product composition, unfreezable water content, initial freezing point, initial temperature of the product, plate temperature history, surrounding air temperature, contact heat transfer coefficient, combined radiation and convection heat transfer coefficient at the circumferential surface, and total cooking time. The gap thickness, radius of the patty, product composition, initial temperature of product, plate temperature history, and total cooking time were measured during the experiment. Thermal properties varying with temperature were calculated using the procedure developed by Mannapperuma (1988) based on composition, thermal properties for the unfrozen state, unfreezable water content, and initial freezing point. Thermal properties for the unfrozen state were estimated using correlations based on the composition (24% fat content, 60% w.b. moisture, 16% protein content), and 3 the following values were obtained: density, 1026.7 kg/m ; apparent specific heat, 3268 J/kg°C; thermal conductivity, 0.416 W/m°C; unfreezable water, 4%, and initial freezing point, –1°C (Cleland and Valentas, 1997). 2 An average contact heat transfer coefficient of 900 W/m °C, a combined radiation and convection heat transfer coefficient at the circumferential surface of the patty 2 of 60 W/m °C (Geankoplis, 1993), and the average value of the plate temperatures as a surrounding air temperature were assumed. The last assumption was verified experimentally. Typical simulated temperature distributions in two dimensions (radial and axial) are plotted in Figures 20.2 and 20.3 for 50 and 124 sec of cooking time. It is evident that temperature changes in the circumferential region of the patty (region near r = a) are different from the changes in the central region (region near r = 0). Preliminary results showed that this difference affects texture parameters. A one-dimensional model would not be able to find such differences in temperature profiles. The experimental validation is not easy because of the difficulty of appropriately placing the thermocouples. Although thermocouples are placed carefully at the beginning, when the cooking starts, it is possible for them to change position because of the softening of the material. Validation is generally carried out by measuring the temperature at the geometric center of the patty. Typical experimental and predicted temperatures at the patty center are shown in Figure 20.4 (Zorrilla and Singh, 2000a). The predicted values correspond to a simulation for the one-dimensional model. However, Zorrilla and Singh (2000b) observed that the two-dimensional model has no significant effect on the prediction of temperature at the geometric center. For simulation, a constant contact heat transfer coefficient was used. If the experimental h values were used, a lower center temperature would have been predicted because the h values measured are in the range of 2 250–650 W/m °C and are lower than the value assumed for simulations (900 2 W/m °C). However, the calculations carried out with the experimental h values will lead to safer cooking times.


220

Temperature (°C)

180

140

100

60 58.4 46.7 35.1 Radial distance 23.4 from center, r, mm 11.7

20

Distance from top patty surface, z, mm

0.0

11.0

8.8

9.9

7.7

5.5

6.6

3.3

4.4

2.2

0.0

1.1

-20

FIGURE 20.2 Temperature distribution in a hamburger patty at 50 sec of cooking time (L = 2 2 11 mm; a = 58 mm; Tp1 = 221°C; Tp2 = 204°C; h = 900 W/m °C; Ta = 212°C; hc = 60 W/m °C).

220

Temperature (°C)

180

140

100

60 58.4 46.7 35.1 Radial distance 23.4 from center, r, mm 11.7

20

9.9

Distance from top patty surface, z, mm

0.0 11.0

8.8

6.6

7.7

5.5

3.3

4.4

2.2

0.0

1.1

-20

FIGURE 20.3 Temperature distribution in a hamburger patty at 124 sec of cooking time (L = 2 2 11 mm; a = 58 mm; Tp1 = 221°C; Tp2 = 204°C; h = 900 W/m °C; Ta = 212°C; hc = 60 W/m °C).


250 Set top plate temperature = 221°C

200

Temperature (°C)

Set bottom plate temperature = 204°C

150

100

50

0

-50 0

25

50

75

100

125

Time (sec)

FIGURE 20.4 Experimental (ⵧ) and predicted (—) temperatures at patty center (L = 11 mm; 2 h = 900 W/m °C).

20.5 FUTURE CONSIDERATIONS The mathematical model is the first step of an optimization process (Banga et al., 1991, 1997). The results of predictive heat transfer models help to address food safety issues associated with the survival of pathogens. On the other hand, the optimization procedure must take into account information related not only to microbial destruction, but also to textural and sensory characteristics. In recent years, texture has become an important attribute indicative of hamburger patty quality (Berry, 1994; Bigner-George and Berry, 2000; Dreeling et al., 2000; Ju and Mittal, 1999, 2000). Moreover, instrumental methods for textural properties have been successfully correlated with sensory data and are often more precise and reproducible (Beilken et al., 1991). Zorrilla et al. (2000) related the textural and cooking parameters with the plate temperature and gap thickness between plates during double-sided cooking of meat patties as an attempt to relate physical results to any change in the heat transfer mechanism. For correlations, the volume averaged temperature, T, was used:

∫∫∫ T(z, t) dV ∫ T= = dV ∫∫∫ V

L

T(z, t ) dz

0

L

(20.16)

V

where T(z,t) is the temperature profile inside the patty for a one-dimensional model, and L is the gap thickness between plates. An example of the regression surfaces obtained is shown in Figure 20.5. In this case, cooking loss is one of the cooking parameters studied.


Cooking loss (%)

38

34

34

30

30

26

26

22 90

22

11

75

Gap th

0.5

1

icknes s

10

(mm)

65

9.5

70

80

d ge

e tur era

85

Cooking loss (%)

38

) (°C

p

tem

era -av

e lum

Vo

FIGURE 20.5 Cooking loss values correlated with volume averaged temperature and gap thickness between plates.

The optimization problem can be formulated as follows: Find the heating policy of a hamburger contact cooking process in order to obtain minimum cooking loss while ensuring the mandatory level of microbiological destruction and final temperature at the coldest point. At present, the optimization process is a subject of current study.

20.6 CONCLUSIONS Mathematical models based on classical transport phenomena equations satisfactorily predicted the temperature profiles inside hamburger patties during double-sided contact cooking. A model considering a one-dimensional geometry appropriately predicts temperature profiles at the geometric center, while a model for a twodimensional cylindrical geometry provides a temperature history at regions close to the circumferential edge of the patty, which is not possible with a one-dimensional model. The contact heat transfer coefficient is one of the parameters needed to complete the mathematical formulation. Since little information is available on heat transfer coefficient values for solid foods, a detailed discussion was included. The contact heat transfer coefficient changes during the cooking process. The small gap thickness enhances the contact between a patty and the grill surfaces. A Teflon release sheet offers finite resistance to heat transfer to the hamburger patties during cooking. Higher heat transfer coefficient values after reaching maximum value were found with the low set grill temperature and may result from the adhesion phenomenon occurring between the patty surface and the grill surfaces.

NOMENCLATURE a Cp

Radius of the patty, m Apparent specific heat, J/kg°C


3

g h hc H k kcrust L r R S1, S2 t T T

Ta Tb Tp1, Tp2 T0 xf xw z

Heat generation term, J/m sec 2 Contact heat transfer coefficient, W/m °C 2 Combined radiation and convection heat transfer coefficient, W/m °C 3 Enthalpy, J/m Thermal conductivity in the core, W/m°C Crust thermal conductivity, W/m°C Gap thickness, m Cylindrical coordinate, m Position of moving boundary that separates the crust region from the core region in the radial direction, m Positions of moving boundary that separate the crust region from the core region in the axial direction, m Time, sec Temperature, °C Volume averaged temperature, °C Surrounding air temperature, °C Boiling temperature, °C Plate temperatures, °C Initial temperature, °C Fat content, decimal fraction Water content, decimal fraction Cylindrical coordinate, m

GREEK SYMBOLS λv ρ φ

Latent heat of water vaporization, J/kg 3 Hamburger density, kg/m Cylindrical coordinate

REFERENCES Baghe-Khandan, M.S., Okos, M.R., and Sweat, V.E., The thermal conductivity of beef as affected by temperature and composition, Trans. ASAE, 25, 1118–1122, 1982. Banga, J.R., Alonso, A.A., and Singh, R.P., Stochastic dynamic optimization of batch and semi-continuous bioprocesses, Biotechnol. Prog., 13, 326–335, 1997. Banga, J.R., Perez-Martin, R.I., Gallardo, J.M., and Casares, J.J., Optimization of the thermal processing of conduction-heated canned foods: study of several objective functions, J. Food Eng., 14, 25–51, 1991. Beilken, S.L., Eadie, L.M., Griffiths, I., Jones, P.N., and Harris, P.V., Assessment of the textural quality of meat patties: correlation of instrumental and sensory attributes, J. Food Sci., 56, 1465–1469, 1991. Berry, B.W., Fat level, high temperature cooking and degree of doneness affect sensory, chemical and physical properties of beef patties, J. Food Sci., 59, 10–14, 1994. Bigner-George, M.E. and Berry, B.W., Thawing prior to cooking affects sensory, shear force, and cooking properties of beef patties, J. Food Sci., 65, 2–8, 2000.


Cleland, D.J. and Valentas, K.J., Prediction of freezing time and design of food freezers, in Handbook of Food Engineering Practice, Valentas, K.J., Rotstein, E., and Singh, R.P., Eds., CRC Press LLC, Boca Raton, FL, 1997, pp. 71–123. Dagerskog, M., Pan frying of meat patties I. A study of heat and mass transfer, Lebensm.Wiss. Technol., 12, 217–224, 1979a. Dagerskog, M., Pan frying of meat patties II. Influence of processing conditions on heat transfer, crust color formation, cooking losses and sensory quality, Lebensm.-Wiss. Technol., 12, 225–230, 1979b. Dagerskog, M. and Bengtsson, N.E., Pan frying of meat patties: relationship among crust formation, yield, composition, and processing conditions, Lebensm.-Wiss. Technol., 7, 202–207, 1974. Dagerskog, M. and Sörenfors, P., A comparison between four different methods of frying meat patties I. Heat transfer, yield and crust formation, Lebensm.-Wiss. Technol., 11, 306–311, 1978. Dreeling, N., Allen, P., and Butler, F., Effect of cooking method on sensory and instrumental texture attributes of low-fat beefburgers, Lebensm.-Wiss. Technol., 33, 234–238, 2000. D’Sa, E.M., Harrrison, M.A., Williams, S.E., and Broccoli, M.H., Effectiveness of two cooking systems in destroying Escherichia coli O157:H7 and Listeria monocytogenes in ground beef patties, J. Food Protection, 63, 894–899, 2000. Farkas, B.E., Singh, R.P., and Rumsey, T.R., Modeling heat and mass transfer in immersion frying. I. Model development, J. Food Eng., 29, 211–226, 1996. FDA, Food Code, chapter 3, 3–401.11. http://vm.cfsan.fda.gov/~dms/fc-3.html, 1997. Geankoplis, C.J., Transport Processes and Unit Operation, 3rd ed., Prentice Hall, Englewood Cliffs, NJ, 1993. Houˇsová, J. and Topinka, P., Heat transfer during contact cooking of minced meat patties, J. Food Eng., 4, 169–188, 1985. Ikediala, J.N., Correia, L.R., Fenton, G.A., and Ben-Abdallah, N., Finite element modeling of heat transfer in meat patties during single-sided pan-frying, J. Food Sci., 61, 796–802, 1996. Ju, J. and Mittal, G.S., Effects of fat-substitutes, fat levels and cooking methods on the quality of beef patties, J. Food Process. Preserv., 23, 87–107, 1999. Ju, J. and Mittal, G.S., Relationships of physical properties of fat-substitutes, cooking methods and fat levels with quality of ground beef patties, J. Food Process. Preserv., 24, 125–142, 2000. Lawrie, R.A., Meat Science, 5th ed., Pergamon Press, Oxford, 1991. Madhusudana, C.V., Thermal Contact Conductance, Springer-Verlag, New York, 1996. Mannapperuma, J.D., Thawing of foods in humid air, dissertation, University of California, Davis, 1988. Mannapperuma, J.D. and Singh, R.P., Prediction of freezing and thawing times of foods using a numerical method based on enthalpy formulation, J. Food Sci., 53, 626–630, 1988. NCBA, (National Cattlemen’s Beef Association) on the Internet at: http://www. beeffood service.com/working/fs.dll?beef, 1998. Ozisik, M.N., Heat Conduction, John Wiley & Sons, New York, 1993. Ozisik, M.N., Finite Difference Methods in Heat Transfer, CRC Press, Boca Raton, FL, 1994. Pan, Z., Predictive modeling and optimization of hamburger patty contact-cooking process, dissertation, University of California, Davis, 1998. Pan, Z., Singh, R.P., and Rumsey, T.R., Predictive modeling of contact-heating process for cooking a hamburger patty, J. Food Eng., 46, 9–19, 2000. Rahman, S., Food Properties Handbook, CRC Press LLC, Boca Raton, FL, 1995. Singh, R.P., Moving boundaries in food engineering, Food Technol., 54, 44–53, 2000.


Sweat, V.E., Thermal properties of foods, in Engineering Properties of Foods, Rao, M.A. and Rizvi, S.S.H., Eds., Marcel Dekker, New York, 1994, pp. 99–138. Urbicain, M.J. and Lozano, J.E., Thermal and rheological properties of foodstuffs, in Handbook of Food Engineering Practice, Valentas, K.J., Rotstein, E., and Singh, R.P., Eds., CRC Press LLC, Boca Raton, FL, 1997, pp. 425–486. USDA, USDA urges consumers to use food thermometer when cooking ground beef patties, Food Safety and Inspection Service, United States Department of Agriculture, Washington, DC http://www.fsis.usda.gov/OA/news/colorpr.htm, 1998. Vijayan, J. and Singh, R.P., Heat transfer during immersion frying of frozen foods, J. Food Eng., 34, 293–314, 1997. Wichchukit, S., Zorrilla, S.E., and Singh, R.P., Heat transfer coefficients in contact cooking of meat patties as influenced by process conditions, presented at the Institute of Food Technologists Annual Meeting, Dallas, 2000. Zorrilla, S.E., Rovedo, C.O., and Singh, R.P., A new approach to correlate textural and cooking parameters with operating conditions during double-sided cooking of meat patties, J. Texture Stud., 31, 499–523, 2000. Zorrilla, S.E. and Singh, R.P., Heat transfer in meat patties during double-sided cooking, Food Sci. Technol. Res., 6, 130–135, 2000a. Zorrilla, S.E. and Singh, R.P., Heat transfer in double-sided cooking of meat patties considering two-dimensional geometry and radial shrinkage, presented at the Institute of Food Technologists Annual Meeting, Dallas, 2000b.


21

Thermal Processing of Particulate Foods by Steam Injection. Part 1. Heating Rate Index for Diced Vegetables K.S. Pannu, F. Castaigne, and J. Arul

CONTENTS 21.1 Introduction 21.2 Theoretical Considerations 21.3 Materials and Methods 21.3.1 Equipment Design 21.3.2 Data Acquisition 21.3.3 Equipment Operation 21.3.4 Food Samples and Test Sensor Preparation 21.3.5 Data Analysis 21.4 Results and Discussion 21.4.1 Time–Temperature Data 21.4.1.1 Regression for fh and jh Values 21.4.1.2 Temperature at the Food Surface 21.4.2 Heating Rate Index (fh) and the Lag Factor (jh) 21.4.2.1 fhc Values 21.4.2.2 Lag Factor and j Values 21.4.2.3 Errors Associated with fh and jh Values 21.4.2.4 Particle-to-Particle Interaction 21.4.2.5 Steam Flow Direction 21.4.2.6 SMFR and Steam Temperature 21.4.3 Heat Transfer Coefficient (h) and the Biot Number (Bi) 21.4.4 Product Quality 21.4.5 Moisture Loss 21.4.6 Process Implications on Product Blanching/Sterilization


21.5 Conclusions Nomenclature References

21.1 INTRODUCTION Although steam is commercially employed to sterilize and blanch food products, there is a lack of readily available data that can be used with confidence to simulate packed bed heating of diced vegetables by steam injection. Basic parameters and properties required are the food product thermal diffusivity at temperatures over 100°C and the convective surface heat transfer coefficient (h value) for steam. Data on h values for food processing applications range from 400 (Zhang and Cavalieri, 2 2 1991) to 12,000 W/m °C (Tung et al., 1990). An h value of about 5000 W/m °C is commonly employed (Skjöldebrand and Ohlsson, 1993) for steam heating, but for a 12-mm food cube this provides a Biot number (Bi) of about 50, indicating infinite resistance to surface heat transfer. This may not be so in light of data presented by Rumsey et al. (1980), Pannu et al. (1990) and Zhang and Cavalieri (1991), which show that 0.1 ≤ Bi ≤ 40. Also, very little work has been documented in the area of packed bed processing of moist food solids with steam. Thus, the objectives of this study were to investigate the steam injection packed bed heating of potato and carrot cubes/slices. Food center and surface temperatures were measured under a variety of process conditions. The specific objective was to assess the influence of steam temperature, flow rate, flow direction (upward and downward), food particle size/shape, and particle-to-particle interaction on product 1 temperature and the food particle heating rate (fh) during packed bed heating.

21.2 THEORETICAL CONSIDERATIONS The transient heat transfer equation for an object of simple shape, initially at a uniform temperature Ti, exposed to a step-change immersion into a heating medium at temperature Ts, can be expressed as follows (Luikov, 1968): θ object =

Ts − T( x,t ) Ts − Ti

∞

=

∑A B e n

n

− β n2 Fo

(21.1)

n =1

Values for An, Bn, and βn are dependent on object shape (infinite slab, sphere, or infinite cylinder), and T(x,t) (or T without a subscript) represents the time-dependent temperature at a fixed location x within the solid. The Fourier number is defined as 2 Fo = αt/R , with α being the thermal diffusivity, t the time, and R the half thickness. Furthermore, the infinite summation can be approximated by only the first term (n = 1), provided Fo > 0.2 (Heisler, 1947; Singh, 1982). Using first term approximation and by taking logarithms of both sides, Equation (21.1) simplifies to Equation (21.2). 1

The f and j values defined and used here are the true “lumped” form of the analytical transient heat conduction equation, with f as the exponential component and j the first two terms (A and B) of Equation (21.1). The f and j values are used for comparison of products and thermocouple location.


For infinite slabs and cubes, the time-temperature data can be analyzed by making the appropriate substitutions from Equations (21.4) and (21.5), respectively, into Equation (21.2).  Τ − Τi  t = fh ln jh s   Τs − Τ  A1 =

 β1 x  2sin β1 ; B1 = cos  and β1tan β1 = Bi β1 + sin β1cos β1  R 

(21.2)

(21.3)

Infinite slab: R2 and jh = A1B1 (β1 )2 α

(21.4)

R2 and jh = (A1B1 )3 3(β1 )2 α

(21.5)

fh = Cube: fh =

Equation (21.2) is extensively used for evaluating the straight line segment of the heating curve, with fh and jh representing the negative inverse slope (heating rate index) and the intercept, respectively, obtained from a semilogarithmic plot of the 2 temperature ratio at the center (θc) vs. time. A second set of subscripts, c and s, representing the center or surface, respectively, are added to fh and jh. Thus, fhc and fhs represent the f value at the center and at the surface layer, respectively. Equations (21.4) and (21.5) show that f values are position independent (fhc = fhs), but j values are not (jhc ≠ jhs) because B1 is location dependent (Figure 21.1). Care has to be taken in computing the jhs value of a cube because it is location sensitive. However, at the geometric center, jhc = A1, as B1 = 1 since x = 0.

21.3 MATERIALS AND METHODS 21.3.1 EQUIPMENT DESIGN Details of the test apparatus are presented in Figure 21.2. Steam enters from the bottom into the annular jacket, which is equipped with a steam trap and an electric heating coil. A three-way valve is used to direct good quality (dry) steam exiting from the top of the annular jacket into the processing chamber for product treatment. Two three-way valves are required (Figure 21.2, top) to ensure continuous steam flow through the processing chamber or to redirect steam flow via the bypass line. Steam exits as water after passing through a steam condensation unit. Flexible hoses 2

Note that a factor of 2.303 is used with the decimal logarithm scale (log 10). For the log 10 scale: fh = 2.303⋅fh. The natural logarithm (ln) scale was used in this study.


Case 1 Center of X-Y plane Surface of Z plane

-X Food Cube

Case 2 Surface of Y & Z planes Off-center on X plane

-Z

Y

Case 3 Surface of all 3 planes X, Y and Z

Case 4 Surface of Y plane Off-center on X & Z planes

FIGURE 21.1 Location influence for estimating j values.

(hose-1 and hose-2) were used to redirect steam flow through the processing chamber via the three-way valves. Thus, steam flow direction (upward or downward) into the processing chamber can be changed by switching the flexible hoses at locations AA and B-B. Figure 21.2 depicts steam flowing in the upward direction. Each of the four thermocouple locations, T1–T4, is equipped with two thermocouples and a pressure gauge. One thermocouple from each of the four points was connected to the data acquisition unit. The second thermocouple was connected to a digital display unit for visual verification of pressure and temperature.

21.3.2 DATA ACQUISITION ®

The data acquisition system consisted of a Personal Computer (PC), MetraByte ® ® hardware (MetraByte Corporation, Taunton, MA), the Lotus 1–2–3 spreadsheet ® program (Lotus Development Corporation, Cambridge, MA) and Labtec Note® book (Laboratory Technologies Corporation, Wilmington, MA) software packages. ® The MetraByte hardware package included a DASH-8 eight-channel 12-bit high® speed A/D converter and timer/counter board and an EXP-16 expansion board with 16 pairs of screw terminals for the attachment of thermocouple wires. The software package provided real time graphic display of the heating process.

21.3.3 EQUIPMENT OPERATION The pressure was regulated using a Taylor Lin-E-Aire valve (Taylor Instrument Company, Broadview Heights, OH) and controlled pneumatically by an Electromax digital process controller (Leeds and Northrup, North Wales, PA). For system startup, approximate system pressure and temperature were achieved in the bypass mode with the digital process controller. Steam mass flow rate (SMFR) was determined by collecting condensate discharge for 120 sec (Figure 21.2, top). Condensing steam to water is the most reliable method for obtaining data on steam flow rate. Depending on


Personal Computer

Hose-1 T-4 B-B T-2

Food Container

Bypass Line 3-Way Valve

Processing Chamber Needle Valve

Annular Jacket

T-1

PRV: Pressure Regulating Valve

Steam Supply

Cold Water

A-A T-3

Hose-2 Steam Condensation for Mass Flow Rate Measurement

Temp. Pressure Measurement

Steam to Water

Steam Out Cover Plate Thermocouples

Dimensions & Details

Rubber Seals

Process Simulator Electric Heating Element

Material: Aluminum Height = 21 cm I D = 15.2 cm Annular space = 2.5 cm

6-Thermocouple Probe Food Container

Condensation Unit Material: Copper tubing Tube Dia. = 1.9 cm Coil height = 40 cm Coil dia. = 20 cm # of turns = 12

Annular Jacket (Steam Heating) Processing Chamber Steam In

A.C. Supply

Pressure: 275 – 550 kPa Quality: Unknown

Steam Trap Steam from Jacket to Processing Chamber

Steam Supply

Processing Chamber Pressure Release Valve

FIGURE 21.2 Schematics of the laboratory apparatus for heating diced vegetables in steam.

the desired SMFR, flow conditions were established by manually adjusting the needle valve (#SS-1RS8; Whitey Co., Highland Heights, OH) located prior to the condensation unit (Figure 21.2), by restricting steam discharge. The annular jacket performs four tasks: 1. It acts as a buffer tank. 2. It removes condensed water through the steam trap.


3. It acts as a thermal jacket for the processing chamber. 4. It performs steam superheating. For all experiments, a few degrees of superheat (5–15°C) were provided. Detailed information about the equipment and its operation has been presented elsewhere (Pannu, 1992). All test containers were flanged to facilitate their installation into the heating chamber. The container flange fits into a recessed slot with thermocouple wires sandwiched between two silicone rubber rings. This ensures good sealing; no leakage of steam was observed. For each test, the food batch was weighed and transferred to the test container in layers to facilitate positioning of the test sensors and thermocouple wires. Two screws were inserted from the bottom of the flange to attach the test container with the two rubber rings and thermocouple wires to the cover plate. The assembly was placed next to the simulator to facilitate the connection of thermocouple wires to the data logger. Subsequently, the cover plate/food container assembly was installed in place and tightly secured with four butterfly nuts. After commencing data acquisition, steam was diverted to the processing chamber via the three-way valve. The data acquisition system provided real time graphic display of the heating process. Temperatures were recorded until the center temperature of food particles reached that of the heating medium; the steam supply to the processing chamber was then cut off. The product was cooled to 100°C by slow depressurization of the heating chamber. Data acquisition was terminated when the product had cooled to 100°C. The food container was then removed, product quality visually inspected, and thermocouple location/placement verified. SMFR was measured before and during the test and after termination of heating. Temperature data from the steam inlet/exit, preheating chamber, heating chamber, food surface, and center and other locations were used to evaluate the performance of the equipment under selected processing conditions. Time-temperature data were recorded at 1 sec intervals with strategically located fine wire (36-gauge) copper-constantan thermocouples. Special test sensors were fabricated by positioning two thermocouples into the food pieces (Figure 21.3a). These were embedded in the food material to register the temperature history of 36-gauge type-T thermocouple wire

1/2 cube thickness

Food Cube

Wooden toothpick diameter = 1.2 mm

(a)

Fine cord

(b)

FIGURE 21.3 (a) Placement of thermocouple probe in the food cube. (b) Details of the thermocouple probe.


individual solids. Heating time of the food particulates was continually monitored and manually controlled. Steam supply was shut off and redirected through the bypass line when the food particle center temperature reached that of steam. The steam flowing through the processor was condensed to liquid water for determining the system steam mass flow rate (SMFR). Since process come-up time was very short (~5 sec), stable processing conditions were achieved, and heat losses were negligible with a steam jacket surrounding the food treatment chamber.

21.3.4 FOOD SAMPLES

AND

TEST SENSOR PREPARATION

Food cubes were prepared by cutting peeled potatoes and carrots into slices and then passing the slices through a french fry cutter. The material was sorted, washed, surface dried with paper towels, sealed in plastic containers, and set aside for 2 h to allow the product to attain a uniform initial temperature (Ti). Temperatures at the center and the surface layer of a food particle were measured by installing special thermocouple probes into selected food pieces (Figure 21.3b). The probes were constructed with thermocouple junctions positioned half the food piece thickness apart and were secured on opposite sides (180° apart) of a wooden toothpick with a fine cotton thread. A 1.2-mm hole was drilled into the test sample to facilitate the correct positioning of the thermocouples and to prevent particle shape distortion (Figure 21.3a). The wooden toothpick assembly was soaked in water for about 2 h prior to installation into the test piece to avoid migration of water from food to wood and to minimize the thermal insulation effects of wood. Only food cubes of uniform size (measured with a digital caliper) and shape were selected for the construction of the test sensors. Three food sensors were positioned at the top, bottom, and center of the food bed. The top and bottom sensors were placed one food layer away from the top layer and bottom layer of the food bed, respectively. Experiments were conducted with approximately 180 g of food and in a container that resembled a 300 × 400 can (Pannu, 1992). Potato and carrot pieces were processed in two shapes (cubes and slices) and two sizes. Moisture content was determined by vacuum drying in a laboratory oven at 70°C for 48 h. Quality and physical structure of the processed food cubes were examined visually.

21.3.5 DATA ANALYSIS Interactive Lotus 1-2-3 macro programs were developed (Pannu, 1992) to analyze the data and obtain temperature–time graphs. The linear segment of the heating curve was used to obtain an estimate of ƒh and the corresponding intercept jh.

21.4 RESULTS AND DISCUSSION 21.4.1 TIME–TEMPERATURE DATA Typical time–temperature data for steam flowing downward at a rate of 3.8 g/sec through a bed of large (12 mm) potato cubes are shown in Table 21.1. Temperatures of steam from the supply line to the processor are in column 2, and those of steam exiting from the superheater (annular jacket) in column 3. Data for steam entering/exiting


TABLE 21.1 a Typical Time–Temperature Data for Steam Flowing Downward at a Rate of 3.8 g/sec through a Bed of Potato Cubes (12 mm) Sensor Location in the Food Bed 1

2 1

Time V sup (sec) (°C)

3

4

5

6 7 Bottom

8 9 Center

10

11

12

Top

V aj (°C)

V top V bot (°C) (°C)

Fc (°C)

Fs (°C)

Fc (°C)

Fs (°C)

Fc (°C)

Fs (°C)

V pc (°C)

20.0 20.0 20.0 20.0 20.0 19.7 20.0 21.8 24.2 26.8 28.9 34.2 38.6 43.7 50.3 57.4 64.9 71.8 78.7 84.9 90.2 94.7 98.9

17.2 17.2 17.6 17.2 17.6 17.6 61.9 89.9 107.3 114.0 118.6 121.4 121.9 122.1 122.3 122.4 122.5 122.6 122.8 122.9 123.2 123.2 123.2

18.2 17.9 17.9 18.2 18.2 18.2 18.8 21.5 23.6 25.9 26.8 31.0 35.4 40.9 48.3 55.7 63.0 70.2 75.9 81.9 87.3 91.5 95.3

18.2 18.2 18.2 18.2 18.2 18.5 34.8 74.3 85.9 93.4 99.5 109.4 113.7 114.8 116.8 117.3 118.4 118.6 118.6 119.4 119.6 120.2 120.5

20.0 20.0 20.0 20.0 20.3 20.6 21.2 23.0 24.7 25.9 26.8 30.1 33.6 39.4 46.3 54.3 62.4 70.2 77.0 83.5 89.2 94.2 98.4

28.9 29.2 29.2 29.5 29.5 30.1 75.4 103.4 114.3 117.8 120.4 121.7 121.9 122.1 122.2 122.4 122.6 122.4 122.8 123.1 122.9 123.2 123.4

63.0 63.0 63.0 63.3 63.3 63.3 63.0 93.7 110.1 116.3 121.9 122.9 122.7 123.2 123.7 123.9 123.9 123.9 123.9 124.2 123.9 123.9 124.2

122.2 122.2 122.2 121.9 120.6 118.6 101.0 99.5

120.1 120.1 120.4 120.4 120.1 119.1 101.8 100.5

122.4 122.2 121.9 121.7 120.1 118.6 101.3 100.0

123.9 123.4 122.9 122.4 120.6 118.4 101.3 100.0

–4 –3 –2 –1 0 1 2 3 4 5 6 10 15 20 25 30 35 40 45 50 55 60 65

124.2 124.5 124.5 124.2 124.7 125.7 125.2 123.9 123.4 123.9 123.9 125.0 123.9 123.9 123.9 124.2 123.9 123.9 124.2 124.2 124.4 124.2 124.2

132.7 132.7 132.7 132.7 131.2 134.2 131.2 122.2 123.7 126.7 129.2 130.2 130.5 130.7 131.2 131.5 131.7 131.7 131.7 131.7 131.7 132.0 131.7

28.3 28.6 28.6 28.3 28.3 42.9 102.1 115.3 121.7 122.9 123.7 122.4 122.2 122.2 122.7 122.7 123.2 123.2 123.2 123.2 123.2 123.2 123.4

133 134 135 136 138 140 155 160

124.2 124.5 124.2 124.5 124.5 124.5 124.2 124.2

132.5 132.5 132.5 132.5 132.7 132.2 132.2 132.2

122.4 121.9 121.9 121.7 118.9 116.8 103.2 102.9

32.4 32.4 32.4 32.4 32.4 33.0 36.5 115.5 122.2 123.2 123.9 122.7 122.2 122.4 122.7 122.7 123.2 123.2 123.3 123.4 123.4 123.4 123.2

Cooling by Depressurization 122.7 120.1 123.2 118.6 122.2 120.4 122.9 118.9 121.4 120.1 122.4 119.1 120.6 120.4 121.9 119.1 119.4 120.1 119.9 119.1 117.6 119.1 118.1 119.1 100.7 101.5 100.5 101.8 100.5 100.5 99.5 100.2

Note: V: vapor; sup: steam supply; aj: exit at annular jacket; bot: bottom location; pc: processing chamber; Fc: food center; Fs: food surface. a

Time–temperature data recorded at 1 sec intervals have been abbreviated for presentation.


at the top and bottom of the processing chamber, respectively, are in columns 4 and 5. Column 12 provides steam temperatures inside the chamber; the data in columns 6 through 11 are from food sensors. The time scale in column 1 was corrected to reflect the true time zero; this was accomplished by comparing temperature data in columns 4 or 5. A sudden jump in temperature indicated the time when steam first entered the processing chamber and was assigned the value of t = 1 sec. 21.4.1.1 Regression for fh and jh Values Results for fh and jh can vary depending on the values of Ts and Ti and the range selected for regression (Larkin, 1989). To overcome this problem and eliminate bias, Lotus macros were used to carry out the regression analysis. Ts was computed as an average of 93 temperature readings recorded in columns 4, 5, and 12 (Table 21.1). Ti for each food sensor was computed as an average of the food center temperature between t = –5 and 0 sec. The time period selected for the regression analysis was t = 50 to 100 sec for f and j values at the center of the large cube and from 15 to 35 sec for surface temperature data. For the small cubes, the range selected was between 40 and 80 sec for center temperature data and between 10 and 25 sec for surface temperature data. −7 The regression range was selected by calculating te with αpotato = 1.6 × 10 2 m /sec (Kostaropoulos et al., 1975) for Fo = 0.2. An estimate for the isothermal heating time (te), at which the curvilinear portion of a heating curve terminates, can also be obtained with Equation (21.6) (Hayakawa, 1982). For a cube, jhc ≈ 2.0; by assuming that fhc = 35 × 2.303 (log 10 scale), te ≈ 55 sec. A te value of 45 sec was obtained by using Fo = 0.2. Thus, the time periods selected for regression were justifiable. However, these methods cannot be applied for surface temperature data. te = 0.7fh( jh – 1)

(21.6)

Semilogarithmic plots obtained from temperatures registered at the food center and surface layer are presented in Figure 21.4. A value for fhs is only possible 0.5

Temperature Ratio (in-scale)

0

Temp. Ratio (Ta – T)(Ta – Ti)

-0.5 -1 -1.5 -2 -2.5 -3 -3.5

Food center Food surface

-4 -4.5 0

20

40

60

80

100

120

Heating Time (sec)

FIGURE 21.4 Semilogarithmic plots from packed bed heating of large (11.5-mm) carrot cubes at 126°C with steam.


130

130

120

120

Temperature (C)

110

110

(A-1)

100

100

90

90

80

80

70

70

60

60 50

50 Steam F-top F-cen F-bot

40 30 20

Steam F-top F-cen F-bot

40 30 20 10

10 -10


(B-1)

10

30

50 70 90 Heating time (sec)

110

-10

130

10

30

50 70 90 Heating time (sec)

110

130

0.0

0.0 -0.2 -0.4 -0.6

-0.2 Temp Ratio = (Ta – T)/(Ta – Ti)

(A-2)

Temp Ratio = (Ta – T)/(Ta – Ti)

-0.4

(B-2)

-0.6

-0.8 -1.0 -1.2 -1.4 -1.6

-0.8 -1.0 -1.2 -1.4

-1.8 -2.0 -2.2 -2.4

-1.6

Upward Flow

Downward Flow

-1.8

F-top F-cen F-bot

F-top F-cen F-bot

-2.0 -2.2

-2.6

-2.4 0

20

40 60 Heating time (sec)

80

100

0

20

40 60 Heating time (sec)

80

100

FIGURE 21.5 Comparing temperature profiles for upward (A) and downward (B) flow steam trough a bed of 11.7-mm potato cubes processed at 125°C with steam. Bottom plots (A-2 and B-2) compare f values at food bed top, center, and bottom locations.

provided the surface temperature is lower than that of the heating medium. Data 2 for regression analysis provided R values of 0.97 or better for fhc, as opposed to 0.92 or better for fhs. Typical temperature data and the dimensionless temperature ratio results (ln scale) obtained from food cubes placed at the top, center, and bottom locations within the food bed are presented in Figure 21.5. The two cases presented here show steam flow in the upward (A1) and downward (B1) directions through a bed of large (11.7 mm length) potato cubes. Data in plots A2 and B2 were used to obtain the heating rate index (slope: fh) and the intercept (jh). SMFR and steam temperatures for these experiments were approximately 5 g/sec and 124°C, respectively. Figure 21.5 and Tables 21.1 and 21.2 show that the food sensors registered nearly the same initial temperature, but the temperatures changed rapidly as steam was injected into the processing chamber, indicating that food was heated uniformly throughout the bed since the transducers located at the top, center, and bottom positions registered similar temperature increases.


21.4.1.2 Temperature at the Food Surface Food surface temperatures (Figure 21.4) were obtained from a 12-mm carrot cube located at the center of the food bed and processed at 126°C with steam flowing upwards. Temperature recorded at the food surface is highlighted in Table 21.1. These data show that food surface temperatures rose rapidly but took approximately 20–40 sec to reach steam temperature. The fact that temperatures measured at the food surface were slightly lower than the temperature of the steam indicated that Bi was less than 40, suggesting that high temperature short time (HTST) processing of diced vegetables involves finite resistance to surface heat transfer. The h value data evaluated in this work also show that Bi was less than 40.

21.4.2 HEATING RATE INDEX (fh)

AND THE

LAG FACTOR (jh)

21.4.2.1 fhc Values Tables 21.2 and 21.3 provide results for fhc at the top, center, and bottom locations of the food bed and for fhs, jhc and jhs. Data for the latter three are lumped as an average of the data obtained at the top, center, and bottom locations. Additional experimental runs were carried out with the large potato cubes to study the influence of SMFR and steam temperature on product heating. The initiation of the linear heating segment after approximately 45 sec (center) and 10 sec (surface) showed that the previously defined ranges for regression analysis are correct (Figure 21.4). In general, the experimental values for fhc and fhs obtained from the heating of potato cubes varied between 31 and 35 sec for the larger cubes and between 23 and 26 sec for the smaller cubes. Carrot samples provided slightly lower f values, ranging between 28 and 30 sec for the larger cubes and between 21 and 23 sec for the smaller cubes. Carrot cubes heated faster because they have a higher thermal diffusivity. The data also show that f values at the cube surface were similar to those at the food cube center. However, in some instances, larger than general values (underscored) for fhc were obtained. This issue, together with j values, is discussed later. Potatoes and carrots were also processed in the form of cylindrical slices/slabs. The diameter to thickness ratio indicated in Tables 21.2 and 21.3 refers to the fact that the thin slices were cylindrical. Results show that even for a cylinder thickness to diameter ratio of 1:5, a criterion for an infinite slab was not satisfied, since fh slab < 3fh cube. For a ratio of 1:5, the lateral surface of the cylinder represents 30% of its total surface area. Thus, f values for slabs should be less than three times those of cubes of equivalent thickness. It is interesting to note that a 1:5 ratio provided an fh value 2.3 times larger, whereas a 1:4 ratio provided an fh value twice as large as that of a cube of the same thickness. 21.4.2.2 Lag Factor and j Values The jhc values obtained for food cubes were lower than the theoretical value of 2.0. Since j values are location sensitive, small directional errors at the center of food cubes were probably amplified. Average jhc values of 1.69 and 1.63 were obtained for the


TABLE 21.2 fh and jh for Steam-Injection Heating of Diced Potato Samples in Packed Beds f Value at Particle a

Run No.

Steam Temp °C

Center Flow Direction

SMFR (g/sec)

Bottom (sec)

Center (sec)

Surface Top (sec)

j Values at Particle

(sec)

Center

Surface

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16

126.5 124.2 118.2 123.3 124.1 123.9 123.7 131.6 131.5 124.4 118.2 119.0 124.1 123.8 123.8 130.5

Up Up Up Up Up Up Up Up Up Down Down Down Down Down Down Down

Large Cubes (Length = 11.7 mm) 1.15 34.0 34.9 32.4 c c 45.0 32.0 1.32 40.5 1.97 31.5 32.1 31.6 2.33 32.2 33.7 33.3 2.53 34.3 34.9 33.6 3.88 34.5 34.9 34.5 5.25 33.5 33.8 32.7 7.05 35.0 34.7 34.1 c 31.5 7.27 33.3 49.5 1.27 31.0 34.2 31.9 1.95 31.8 32.3 30.7 1.97 32.8 34.2 30.9 2.55 33.8 38.0 32.4 3.83 32.5 37.8 31.8 5.28 34.8 37.0 34.5 7.32 32.1 37.4 32.3

32.9 31.8 33.2 33.1 31.3 34.0 32.2 32.1 34.0 33.3 32.3 32.8 31.6 34.8 31.7 32.6

1.688 1.557 1.747 1.903 1.726 1.765 1.663 1.474 1.845 1.830 1.645 1.552 1.663 1.711 1.651 1.619

0.108 0.113 0.117 0.117 0.121 0.101 0.103 0.108 0.103 0.109 0.108 0.125 0.106 0.117 0.119 0.110

2.1 2.2 2.3 2.4

125.6 123.8 124.1 130.4

Up Up Down Down

Small Cubes (Length = 9.9 mm) 1.32 24.4 25.7 24.1 2.50 24.5 25.9 23.8 1.22 24.0 24.2 23.9 c 2.58 22.7 33.9 23.9

24.4 24.2 23.8 22.8

1.683 1.693 1.487 1.639

0.115 0.118 0.123 0.124

Up Up Down Down

Potato Slabs (Thickness = 1.30 56.1 59.2 2.40 55.8 57.6 1.32 59.0 63.8 c 2.53 58.3 101.2

55.4 53.2 57.2 58.1

1.119 1.158 1.116 1.131

0.096 0.051 0.077 0.081

b

3.2 3.1 3.4 3.3

124.9 124.6 125.2 124.6

9.9 mm) 59.3 52.6 53.2 56.7

Note: SMFR, steam mass flow rate. a b c

f values at particle center are at bed locations: bottom, center, and top. The length to thickness ratio of the slabs was greater than 5. Data out of range.

large and small potato cubes and 1.90 and 1.58 for carrot cubes, respectively. The experimental average jhs values for large/small potato and carrot cubes obtained from the data (Tables 21.2 and 21.3) were 0.112, 0.12, 0.111 and 0.114, respectively, which compare favorably with the theoretical value of 0.15 (1.26 * 1.26 * 0.095 = 0.15).


TABLE 21.3 fh and jh for Steam-Injection Heating of Diced Carrot Samples in Packed Beds f Value at Particle a

Center

Surface

Steam Temp. °C

Flow Direction

1.1 1.2 1.3 1.4 1.5 1.6

126.1 118.6 126.7 117.9 125.9 126.7

Up Up Up Down Down Down

2.1 2.2 2.3 2.4

125.3 125.2 117.6 124.0

Up Up Down Down

126.0 126.6 124.2 125.0 125.4

Large Cylindrical Carrot Slabs (Thickness = 11.5 mm) d Up 1.83 64.0 108.5 63.4 62.6 Up 2.40 60.2 71.2 74.3 68.9 Up 2.53 — 56.7 — 52.2 Up 2.55 68.7 74.3 56.7 70.9 Down 2.47 58.9 56.2 56.4 58.4

118.4 125.5 124.2 125.7 124.2

Cylindrical Potato Slabs Up 1.83 43.6 Up 2.47 45.5 Up 2.53 — Down 1.33 47.6 Down 2.50 43.1

Run No.

SMFR (g/sec)

Bottom (sec)

Center (sec)

Top (sec)

(sec)

j Values at Particle Center

Surface

Large Carrot Cubes (Length = 11.5 mm) 1.32 29.7 28.9 29.7 27.6 1.85 28.5 29.6 28.8 28.3 2.55 29.2 29.3 28.9 29.1 1.92 29.1 32.7 28.6 28.7 2.02 30.2 29.7 29.1 29.4 2.58 29.6 30.2 29.4 28.6

1.841 2.058 1.857 1.908 1.839 1.783

0.117 0.108 0.121 0.111 0.105 0.103

Small Carrot Cubes (Length = 9.8 mm) 1.32 20.3 21.8 21.4 2.53 21.1 21.5 20.9 1.87 21.8 22.5 20.5 2.60 22.5 22.9 22.9

1.823 1.261 1.794 1.441

0.121 0.109 0.111 0.115

1.121 1.163 1.134 1.212 1.115

0.113 0.112 0.101 0.104 0.099

1.218 1.161 1.118 1.213 1.141

0.097 0.118 0.114 0.109 0.102

20.9 20.7 22.2 21.1

b

3.1 3.2 3.3 3.4 3.5

4.1 4.2 4.3 4.4 4.5

c

(Thickness = 9.8 mm) 50.9 40.9 43.1 43.8 39.4 46.1 42.9 — 43.8 44.0 46.7 39.9 43.9 41.1 43.5

Note: SMFR, steam mass flow rate. a b c d

f values at particle center are at bed locations: bottom, center, and top. The diameter to thickness ratio of the cylindrical carrot slabs was between 3:1 and 3.5:1. The diameter to thickness ratio of the cylindrical potato slabs was between 3.5:1 and 4:1. Datum out of range.

21.4.2.3 Errors Associated with fh and jh Values In analyzing the sources of error associated with f and j values, it was observed that a ±1°C change in Ts (steam temperature) altered fhc by less than 5%. However, this was not so for jhc, fhs, and jhs values. The influence of small changes in Ts on f and j values is shown in Figure 21.6, using the surface temperature data with Ts − 1 defined as error 1 and Ts + 1 as error 2. This was done because the temperature ratio (θs)


1


0 Food center -1 -2 -3

ERROR 2

-4 CORRECTED -5 Food surface

-6

ERROR 1

-7 0

20

40 60 Heating Time (sec)

80

100

FIGURE 21.6 Influence of error 1 (Ts – 1) and error 2 (Ts + 1) on the experimental fhs value.

at the surface layer is very sensitive to changes in Ts. Results show that fhs increased with increases in Ts. Furthermore, the value of jhc could be related to that of fhc. During the course of this study, a small increase in steam temperature over time was observed (Figure 21.4). Thus, the use of a slightly lower Ts value would result in a steeper slope. Although such a change had negligible influence on the fhc value, the impact on j value was large since the data were plotted on a logarithmic scale. An in-depth analysis of fhc variations due to factors such as environment and computation procedure was not carried out. 21.4.2.4 Particle-to-Particle Interaction Deviations from general fhc values, presented in Tables 21.2 and 21.3, are due to particle-to-particle interaction. It was indicated earlier that some adhesion of potato cubes was observed due to gelatinization of starch. Particle adhesion resulted in a larger particle mass, and hence the f value increased. However, small variations in the value of fh can be attributed to: (1) edge contacts and (2) experimental errors associated with variability in particle size. Furthermore, a closer inspection of Table 21.2 shows that slightly higher f values were obtained at the center of the food bed. This effect was not as pronounced for carrot cubes (Table 21.3) as for potato cubes. The slightly higher f value for potato cubes, at the center location of the bed, is possibly a result of condensate build-up, which may have slightly depressed the heating rate. 21.4.2.5 Steam Flow Direction It is difficult to define which flow direction is superior from temperature and f value data. However, data recorded on food bed heights showed that 180 g of large potato cubes provided an average bed height of 73 mm before processing. This reduced to 64 and 68 mm after treatment due to bed settling, for downward and upward steam


flow, respectively. Since these data are specific to the simulator used in this study, no general conclusions could be drawn. 21.4.2.6 SMFR and Steam Temperature Steam velocity (SMFR) and processing temperature did not have any influence on particle heating rate within the range tested. The data for large potato cubes show that similar f values were obtained with steam processing temperatures of 118 and 131°C. An increase in temperature of 9°C (from 118 to 127°C) also produced similar results for carrot cubes. Data also show that a sixfold increase in SMFR did not influence the particle heating rate. It can thus be concluded that the steam process temperature and SMFR have negligible influence on the product heating rate.

21.4.3 HEAT TRANSFER COEFFICIENT (h) AND THE BIOT NUMBER (Bi) It was indicated earlier that a finite surface resistance to heat transfer occurs during packed bed HTST processing of diced vegetables in steam. In cases where 0.1 < Bi < 100, the internal dissipation of heat, although slow, maintains its capacity to keep the surface temperature lower than that of the heating medium, as observed. The system h value and the Biot number can be estimated with Equations (21.3) and (21.5), and by using thermal property data (thermal diffusivity, α, and thermal conductivity, k). From results reported by Kostaropoulos et al. (1975) and Pannu −7 2 et al. (1990), the following data are available: αpotato = 1.6 * 10 m /sec, kpotato = −7 2 0.62 W/m°C, αcarrot = 1.7 * 10 m /sec and kcarrot = 0.66 W/m°C. From the data presented in Tables 21.2 and 21.3, we obtain: fpotato = 33.1 sec, fcarrot = 29.3 sec, Rpotato = 1.17/200 m and Rcarrot = 1.15/200 m. For large cubes, the following results were obtained by using Equations (21.2) and (21.5): β1 potato = 1.47, Bipotato = 14.2, 2 2 hpotato = 1500 W/m °C, β1 carrot = 1.49, Bicarrot = 17.8 and hcarrot = 2000 W/m °C. Similar h values were obtained by using f value data for the smaller (9.9 mm) 2 food cubes. These results are higher than the value of 1100 W/m °C reported by 2 Ling et al. (1974) but in the range of 1900–2500 W/m °C reported by Pannu et al. (1990). The latter study was conducted using single particles. However, the above results for the system heat transfer coefficients are lower than the minimum value 2 of 5000 W/m °C for condensing water vapor, as reported by Holman (1976), for processing of food containers and pouches with steam. Thus, there is need for additional work in this area.

21.4.4 PRODUCT QUALITY Product handling and particle identity were observed by placing processed products on a plate. Potato cubes processed at ≤125°C retained their shape and identity. However, potato cubes processed at 130°C were fragile and overcooked, and there was evidence of particle breakup. In a few instances, it was noticed that potato cubes adhered to each other as well as to the processing container wall. On the other hand, carrot cubes showed no clumping, and the particles maintained their shape and


TABLE 21.4 Moisture Content of Raw and Processed Food Particulates and Visual Observations

Batch No.

Treatment

Initial Mass (g)

Final Mass (g)

Moisture Content (%)

1.1

Uncooked

8.28

1.2

Uncooked

8.54

1.71

79.98

1.3 1.4 1.5

Cooked Cooked Cooked

8.34 8.14 8.44

1.58 1.52 1.61

81.06 81.33 80.92

1.6

Cooked

8.12

1.52

81.28

2.1

Uncooked

7.72

2.2

Uncooked

8.47

0.89

89.49

2.3 2.4 2.5 2.6

Cooked Cooked Cooked Cooked

7.25 6.92 7.11 6.31

0.91 0.84 0.85 0.80

87.45 87.86 88.05 87.32

Visual Observation

Potato Cubes 1.65 80.07

Good shape Good, some adhesion Some adhesion to the container Good shape

Carrot Cubes 0.78 89.90

Good, Good, Good, Good,

no no no no

adhesion adhesion adhesion adhesion

integrity. Processed carrots were more intense in color than the raw product and did not adhere to the container wall.

21.4.5 MOISTURE LOSS Table 21.4 shows that there was no significant gain or loss in food moisture content during the processing of potato and carrot cubes in steam. The small increase in moisture of processed potato samples is attributable to higher moisture retention of gelatinized starch. On the other hand, moisture loss by processed carrots probably takes place due to collapse of cellular structure.

21.4.6 PROCESS IMPLICATIONS

ON

PRODUCT BLANCHING/STERILIZATION

To achieve commercial sterility, it is necessary for the center of the food particle to undergo heat treatment to achieve a lethality of 3 min or more (Heldman and Hartel, 1997). The thermal resistance kinetic parameters for Clostridium botulinum are D121 of 0.26 minutes and a z value of 10°C (Perkins et al., 1975). For a 12D process, the particle needs to be held at 121°C for 3 min. Problems such as these need to be resolved through a biological validation study. Furthermore, the definition of the cold point “in the can” may also need to be redefined for packed bed processing.


In this work, it was shown that there was no cold spot in the food container (processing “can”) after 5 to 10 sec of heating during steam injection heating of diced particles. The cold spot effectively is the particle center point, dispersed throughout the “can.” It is generally recognized that HTST processing will provide a product of superior quality in the case of blanching (Heldman and Hartel, 1997). Thus, the role of steam injection heating in providing optimally processed vegetables for readyto-eat meal packages also needs to be evaluated. As blanching/pasteurization requirements are less severe, packed bed heating of particles in batches by steam injection can provide adequate processing. Listeria monocytogenes, which has been of great concern in recent times, has D70 = 0.14 to 0.27 min and a z value between 6 and 7.5°C (Gaze et al., 1989). As other microorganisms of interest present similar trends, using this microorganism as reference and the temperature data presented in Table 21.4, it can be shown that pasteurization requirements for large potato cubes (11.7 mm) are satisfied after approximately 80 sec of steam heating at 125°C. Thus, the data presented in this study can be used to determine thermal process adequacy for various microorganisms of interest.

21.5 CONCLUSIONS The concept of processing food particles in preweighed lots seems promising, and its potential in aseptic processing warrants further investigation. The results of this study show that batch heating of carrot and potato cubes in packed beds with steam provides a processed product that retains shape, identity, and color. Furthermore, steam heating prevents physical losses due to leaching, and under proper conditions the product heats uniformly with negligible gain or loss of moisture. Sensitivity of the temperature data can be appreciated from the fact that carrot and potato cubes produced noticeably different fh values, indicating different heating rates. SMFR (velocity) and steam temperature have only minimal influence on product heating rate. This study also confirms that finite resistance to surface heat transfer occurs during packed bed heating of food particles by steam injection. By using literature values for food thermal properties, the apparent surface heat transfer coefficient 2 value was found to range from 1500 to 2000 W/m °C with Bi on the order of 14–17 for 1.2-cm food cubes. There is need for additional work on h values associated with steam processing, as well as a sensitivity analysis of the influence of process temperature on fh. Process validation, product cook quality, process optimization, and means to aseptically cool the product are areas that need further attention.

NOMENCLATURE α β q Bi Fo

2

−1

Thermal diffusivity (m sec ) Characteristic root, Equation (21.1) Unaccomplished temperature ratio, Equation (21.1) Biot number (= hR/k) 2 Fourier number (= αt/R )


fh H Jh K R T Ts Ti T(x,t)

Heating rate index; negative inverse slope in Equations (21.2), (21.4) and (21.5) (sec) 2 Convective heat transfer coefficient (W/m °C) Thermal lag factor/intercept, Equation (21.2) Thermal conductivity (W/m°C) Characteristic length (m); 1 2 thickness of cube or slice Time (sec) Steam temperature (°C) Initial (food) temperature (°C) Variable internal (particulate) temperature (°C)

REFERENCES Gaze, J.E., Brown, G.D., Gaskell, D.E., and Banks, J.G., Heat resistance of Listeria monocytogenes in homogenates of chicken, beef steak, and carrot, Food Microbiol., 6, 251, 1989. Hayakawa, K.I., Empirical formula for estimating nonlinear survivor curves of thermal vulnerable factors, Can. Inst. Food Sci. Technol., 32, 59, 1982. Heisler, M.P., Temperature charts for induction and constant temperature heating, Trans. ASME, 69, 227, 1947. Heldman, D.R. and Hartel, R.W., Principles of Food Processing, Thomson Publishers, New York, 1997. Holman, J.P., Heat Transfer, 4th ed., McGraw-Hill, New York, 1976. Kostaropoulous, A.E., Spiess, W.E.L., and Wolf, W., Reference values for thermal diffusivity of foodstuffs, Lebensm.-Wiss. Technol., 8, 108, 1975. Larkin, J.W., Use of modified Ball’s formula method to evaluate aseptic processing of foods containing particulates, Food Technol., 43, 124, 1989. Ling, C.C.A., Bomben, J.L., Farakas, D.F., and King, C.J., Heat transfer from condensing steam to vegetable pieces, J. Food Sci., 39, 692, 1974. Luikov, A.V., Analytical Heat Diffusion Theory, Academic Press, New York, 1968. Pannu, K.S., High Temperature Transient Heating of Vegetables by Steam Injection, thesis, Laval University, Ste-Foy, Quebec, 1992. Pannu, K.S., Castaigne, F., Ramaswamy, H., Arul, J., Audet, P., and Desilets, D., Rapid Sterilization of Particulate Food by Steam Injection: Evaluation of Thermal Parameters, ASAE paper #90–6527, presented at the 1990 ASAE winter meeting, Chicago, 1990. Perkins, W.E., Ashton, D.S., and Evancho, G.M., Influence of the z value of Clostridium botulinum on the accuracy of process calculation, J. Food Sci., 40, 1189, 1975. Rumsey, T., Farakas, D.F., and Hudson, J.S., Measuring steam heat transfer coefficient of vegetables, Trans. ASAE, 24, 1048, 1980. Singh, R.P., Thermal diffusivity in food processing, Food Technol., 36, 87, 1982. Skjöldebrand, C. and Ohlsson, T., A computer simulation program for evaluation of the continuous heat treatment of particulate food products. Part 2: Utilization, J. Food Eng., 20, 167–181, 1993. Tung, M.A., Britt, I.J., and Ramaswamy, H.S., Food sterilization in steam/air retorts, Food Technol., 44, 105, 1990. Zhang, Q. and Cavalieri, R.P., Thermal model for steam blanching of green beans and determination of surface heat transfer coefficient, Trans. ASAE, 34, 182, 1991.


22

Thermal Processing of Particulate Foods by Steam Injection. Part 2. Convective Surface Heat Transfer Coefficient for Steam K.S. Pannu, F. Castaigne, and J. Arul

CONTENTS 22.1 Introduction 22.2 Materials and Methods 22.2.1 Computational 22.2.2 Equipment and Experimental 22.2.3 Data Analysis 22.3 Results and Discussion 22.3.1 Temperature Profiles 22.3.2 Impact of Flow Direction 22.3.3 Transducer Location in the Food Bed 22.3.4 Influence of Steam Mass Flow Rate (SMFR) 22.3.5 Steam Temperature Effects 22.3.6 Influence of Particle Size 22.3.7 Inert Material vs. Food 22.3.8 h Values of Food vs. Metal 22.4 Conclusions References

22.1 INTRODUCTION Batch heating of solid foods by steam injection was investigated to determine feasibility and process requirements. This method has potential for high temperature short time (HTST) processing conditions, which are favorable for minimizing nutrient


loss and improving quality of the processed product. Furthermore, product sterilization by steam processing is desirable from the standpoint of minimization of effluent (wastewater) discharge (Poulsen, 1986). The basic premise of this process is that a preweighed quantity of food particulates would be held in specially designed containers and batch heated by flowing steam, which allows HTST processing, with the product flowing continuously. Construction of a laboratory processor and determination of the heating rate of food particulates in a packed bed with steam are described elsewhere (Pannu et al., Chap. 21, this volume). This chapter describes the determination of the surface convective heat transfer coefficient (h value) obtained during packed bed heating with steam. Heat process development involves not only the determination of processing conditions, but also the prediction of product heating time. Basic information requirements are thermal and physical property data for food, thermal property data for steam, steam flow conditions, and the mode of heat transfer. Batch heating of diced carrots and potato cubes showed that the resistance to heat transfer at the food surface was not negligible and that the Biot number was less than 40 when food pieces were heated with steam (Pannu et al., Chap. 21, this volume). These results are in agreement with those reported by Rumsey et al. (1980) and Pannu et al. (1990). In addition, 2 surface heat transfer coefficient values of approximately 2000 W/m °C were obtained with food cubes (Pannu et al., Chap. 21, this volume). These are inferior to the 2 minimum h value of 5000 W/m °C associated with condensing steam (Holman, 1976). This disparity of results requires explanation. To obtain reliable estimates for product heating time, it is imperative to correctly define h and the associated Biot (Bi) number. An assumption that Bi > 40 is valid for the processing of metal cans and pouches in condensing steam because these objects have a characteristically large size. However, heating time estimates of 113 and 103 sec were obtained for steam processing of 12-mm potato and carrot cubes, respectively, where steam temperature was 125°C and the product initial and final temperatures were 25 and 121°C, respectively. Calculations used the first term approximation for the transient heat conduction in the equation of Luikov (1968), assuming infinite Biot −7 number (Bi > 40). The normally accepted thermal diffusivity values of 1.6 × 10 and −7 2 1.7 × 10 m /sec (Kostaropoulos et al., 1975) were used for potato and carrot, respectively. However, with Bi ≈ 15, the heating time increased to 130 and 115 sec for the potato and carrot cubes, respectively. The latter values for product heating time are in agreement with our experimental data (Pannu et al., Chap. 21, this volume) and provide 2 h value estimates for steam on the order of 1500 to 2000 W/m °C. The most common explanation for decrease in h value is the presence of air in the steam. Tung et al. (1990) reported that even a small amount of air present in steam could dramatically decrease the h value. Othmer (1929) reported h values of 2 steam with an air content of 0, 1, and 2% to be 15,900, 7900 and 5200 W/m °C, respectively. On the other hand, Pflug (1964) found that the h value decreased to 2 2 1300 W/m °C (from about 5000 W/m °C) with 10% air in the steam. Tung et al. (1984) obtained h values higher than those of Pflug (1964), but their data support the hypothesis that h values decrease with increasing air content. In addition, they showed that h values were higher during downward steam flow but were not dependent on steam temperature. These differences in results obtained by various researchers


can, at least in part, be attributed to the variations in “come-up time” and/or the “flush time” in different retorts. A second explanation for h value depression, specific to the food processing situation, is the phenomenon described as “gassing out,” where gases present in fruit and vegetable tissue are released during heating. Brown (1974), Ling et al. (1974), and Zhang and Cavalieri (1991) reported that h values decreased to 400–1100 2 W/m °C during the processing of vegetables in condensing steam because of the “gassing out” effect, leading to the formation of a layer of noncondensable gases at the boundary layer of the food particle. Despite its importance, the h value for steam relating to food processing has not 2 been specified by application but is generally said to range from 400 to 16,000 W/m °C (Tung et al., 1984). Holman (1976) indicated that the h value for condensing steam 2 2 could range from 5000 to 100,000 W/m °C. The estimate of 5000 W/m °C reported by Pflug (1964) has gained acceptance in the food industry and is also the value suggested by Skjöldebrand and Ohlsson (1993) as the condensing steam to wall h value. This 2 2 value of 5000 W/m °C is superior to the values of 850 –1700 W/m °C reported by 2 Rumsey et al. (1980) and 1800 –2500 W/m °C reported by Pannu et al. (1990). Thus, it is imperative to understand the factors contributing to the reduction in h value. Packed bed heating with steam injection was investigated primarily to evaluate the h value for steam under a variety of heating conditions and, secondly, to gain an understanding of how steam flow and condensation effect could affect packed bed heating. The convective surface heat transfer coefficient was determined with temperature measurements obtained from metal cubes placed in the bed of food pieces and inert material. The h values were reported for steam heating of packed beds as affected by process variables such as steam flow direction, flow rate, and temperature, as well as bed characteristics such as particle size and bed composition (food vs. inert material).

22.2 MATERIALS AND METHODS 22.2.1 COMPUTATIONAL The internal resistance to heat transfer is small in metal cubes, and hence the lumped heat capacity method is appropriate for estimation of h values, provided the Biot number is small (Bi ≈ 0.1 or lower). Equation (22.1) was used to compute h values from temperatures registered at the geometric center of metal cubes. Ts − T( t ) −(hA/ρCpV)t = θsolid = e Ts − Ti

(22.1)

The terms A, ρ, Cp, V, and L represent the solid surface area, density, specific heat, volume, and characteristic length, respectively. Subscripts for temperature (T), listed as s, i, and t, denote steam, initial (metal cube), and time t (metal cube), respectively. The density and specific heat data are from Holman (1976). ρ and Cp 3 3 for steel are 7830 kg/m and 465 J/kg°C, and those for aluminum are 2707 kg/m and 896 J/kg°C, respectively.


22.2.2 EQUIPMENT

AND

EXPERIMENTAL

Details on apparatus construction and operation have been described elsewhere (Pannu et al., Chap. 21, this volume). The test apparatus had steam superheating capability to provide dry saturated steam to the central treatment chamber. Experiments were conducted with good quality saturated steam, as the outer annular jacket (steam superheater) was always maintained at a slightly higher temperature than the interior treatment chamber, such that heat losses in the central treatment zone were negligible (Pannu, 1992). Two three-way valves and flexible hoses were used for the process setup and to control the steam flow direction through the particle bed. The test apparatus also had provision for regulating the steam velocity through the chamber. Steam was condensed to liquid water, and the condensate collected at intervals was used to determine steam mass flow rate (SMFR). A test container with a diameter of 7.5 cm and a height of 10.3 cm was filled with glass beads of diameter 6 or 14 mm or with rehydrated chickpeas (diameter ≈ 11 mm), to a bed height of 9 cm. For all experiments, three metal cubes (transducers) were positioned in the particle bed at the top, center, and bottom locations to obtain the desired time–temperature data. The top and bottom transducers were positioned approximately 1 cm below and above the bed extremities, respectively. Thermocouples and pressure gauges were also positioned at strategic locations to allow observation and control of steam pressure and the temperature of the process (Pannu et al., Chap. 21, this volume). The metal transducers were fabricated by positioning a fine-wire copper constantan (36 gauge, type-T) thermocouple at the geometric center of each metal cube. To accomplish this task, a 0.7-mm hole was drilled into each metal cube at the central axis up to a depth equivalent to half the cube dimension. An epoxy cement with high heat resistance and high thermal conductivity was used to bond the thermocouples in place (Omegabond 200, Omega Engineering Inc., Stamford, CT). The cube-shaped metal (aluminum and steel) transducers were constructed in three sizes (9.5, 11.1 and 12.7 mm). Most of the experiments were conducted using the 11.1-mm aluminum cubes. The metal transducers and other thermocouples recorded temperatures at 1 reading/sec for 30 sec, followed by 5 readings/sec (fast test cycle) for the next 60 sec. Thereafter, the data logger recorded the temperatures at 1 reading/2 sec until shutdown. Steam was injected into the central processing chamber once the data logger commenced recording data at the fast test cycle.

22.2.3 DATA ANALYSIS Abbreviated time–temperature data acquired for a typical run are given in Table 22.1. This experiment was carried out with steam at a temperature of approximately 140°C at a flow rate of 3.37 g/sec, flowing downwards through a bed of rehydrated chickpeas. Steam temperatures were monitored at three locations (columns 4 to 6). The reference thermocouple (Ref, column 4) was located 2 cm below the test container and measured the temperature of steam inside the processing chamber. Locations Top (column 5) and Bottom (column 6) measured steam temperatures at the entry and exit positions, respectively, of the apparatus. Temperature data from these three


TABLE 22.1 a Abbreviated Time–Temperature Data for a Typical Run 1

2

3 Time (sec)

1 2 3 4 5 6 7 8 9 11 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 29 31 34

4

5

6

Steam Temperature (°C)

7

8

9

Transducer Temperature (°C)

Actual

Corrected

Ref

Top

Bot

Bot

Cen

Top

0.0 10.0 20.0 30.0 31.0 31.2 31.4 31.6 31.8 32.0 32.2 32.4 32.6 32.8 33.0 33.2 33.4 33.6 33.8 34.0 34.2 34.4 34.6 34.8 35.0 37.0 39.0 42.0

–32.0 –22.0 –12.0 –2.0 –1.0 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 5.0 7.0 10.0

22.5 22.7 23.1 23.1 23.1 23.4 23.1 23.4 23.4 23.4 89.8 122.5 126.3 131.1 135.1 136.6 138.0 138.5 138.8 139.3 139.3 139.3 139.8 138.3 138.3 139.3 139.3 139.5

51.0 50.7 50.1 50.1 50.4 49.9 49.9 100.6 107.4 114.1 120.2 124.3 129.3 132.8 136.3 138.0 138.8 140.0 140.3 140.8 140.8 140.8 141.5 140.5 140.0 140.8 141.3 141.3

45.9 45.9 45.6 45.6 45.3 45.3 45.3 45.9 45.6 46.1 46.7 48.7 57.2 84.4 111.0 124.5 130.8 134.8 137.5 138.0 138.8 138.8 137.8 138.0 136.3 139.8 140.3 140.3

23.1 23.4 23.4 23.6 23.4 23.4 23.4 23.6 23.6 27.2 38.4 52.7 64.7 79.0 87.9 96.4 103.2 107.9 112.5 116.6 119.5 121.7 123.8 125.3 127.4 137.8 138.3 139.0

22.8 23.1 22.8 23.1 23.1 22.8 23.1 23.4 25.7 33.1 48.1 64.5 79.8 91.6 101.4 109.7 116.6 122.2 126.3 129.6 131.8 133.8 135.3 136.6 137.0 139.5 139.5 140.0

22.8 22.8 22.5 23.1 22.5 22.5 22.8 23.4 29.6 47.6 65.3 80.4 92.1 102.2 110.0 116.1 121.5 125.0 128.1 130.6 132.1 133.8 134.6 135.3 136.1 139.0 139.3 139.8

Note: Ref, Bot, Cen: Reference, Bottom, and Center, respectively. a

Downward flow of steam (140°C), at the rate of 3.4 g/sec, through a bed of rehydrated chickpeas.

locations facilitated the adjustment of real time (column 3) and calculation of the steam temperature (Ts). The dimensionless temperature ratios (θ) were calculated for each of the metal transducers, and their natural logarithms [ln(θ)] were plotted against heating time. The h value was computed for each metal cube by determining the slope (X coefficient, Table 22.2) of the linear region of the semilogarithmic plot, using standard regression analysis.


TABLE 22.2 Basic Information and Standard Regression Analysis to Determine h Value Material: Mass: Steam Mass Flow Rate: Processing Temperature:

Chickpeas 240 g 3.37 g/m 140.1°C

Transducer: Location Material: Size: 1 (cm) 3 Density: ρ (kg/m ) Specific heat: Cp (J/kg°C)

Bottom Steel 1.111 7830 465

Center Aluminum 1.111 2707 896

Top Aluminum 1.270 2707 896

Regression Analysis: Number of data points Degrees of freedom X coefficient Standard error of coefficient 2 R 2 Estimated h values (W/m °C) Estimated Biot Number (–)

21 19 –0.662 0.0066 99.9 4465 0.459

21 19 –1.214 0.0127 99.8 5455 0.148

21 19 –1.120 0.0070 99.9 5748 0.176

22.3 RESULTS AND DISCUSSION 22.3.1 TEMPERATURE PROFILES Typical temperature profiles obtained from metal transducers under upward and downward steam flow conditions are shown in Figures 22.1a and 22.1b. They indicate that an instant rise in temperature was evident when steam was injected into the processing chamber, and that all metal transducers had a uniform and constant initial temperature just prior to heat treatment. Furthermore, the lower transducer was first to register temperature rise when the flow of steam was upward, and the inverse was true during downward flow. The semilogarithmic plots indicated that linearization was possible after a very short lag period. Biot number values of 0.459 and 0.15 for the 11-mm steel and aluminum test transducers were obtained with the data presented in Table 22.2. The Biot criterion (of Bi ≤ 0.1) was not satisfied by the steel transducer. For this reason, most of the investigation was conducted with aluminum transducers.

22.3.2 IMPACT

OF

FLOW DIRECTION

The h values obtained with steam flowing through a bed of rehydrated chickpeas are presented in Figures 22.2 to 22.4. Comparative data for food and glass beads are presented in Figure 22.5. Figure 22.2 shows the transducer locations and h values


Small glass beads

Upward Flow SMFR = 5.5 g/s Top Center Bottom Steam

0

1

2

3

4

5

6

Large glass beads

130 120 110 100 90 80 70 60 50 40 30 -1

Temperature (°C)

Temperature (°C)

120 110 100 90 80 70 60 50 40 30 20 -1

7

Downward Flow SMFR = 2.1 g/s Top Center Bottom Steam

0

Heating Time (sec)

1

2

3

4

5

Heating Time (sec)

(a)

(b)

FIGURE 22.1 Temperatures registered by the metal cubes in beds of 6 mm (a) and 14 mm (b) glass beads. DOWNWARD

6000

h-value (W/m2°C)

SMFR 5500

DOWNWARD

5.67 g/s

1.58 g/s

7.67 g/s

1.50 g/s

9 cm 8 cm

0 cm 1 cm

5000 4.5 cm

4.5 cm UPWARD

4500

1 cm 0 cm

8 cm 9 cm

4000 0

2

4

6

8

Distance (cm): Bed height = 9 cm

10 UPWARD

FIGURE 22.2 The influence of steam flow rate (SMRF), flow direction, bed height, and location on h values. Experiments were conducted at 120°C with rehydrated chickpeas.

obtained with steam temperatures of ≈120°C in a bed of rehydrated chickpeas under different steam flow rates, with upward and downward flow of steam. In general, all these plots show that h values were higher during downward flow of steam, and the results are in agreement with the general trend reported by Tung et al. (1984). However, the influence of flow direction vanished when live steam at 100°C (atmospheric pressure) was injected (Figure 22.3), and h values were found to be 2 between 4300 and 4500 W/m °C. The h values remained nearly the same for down2 ward flow but decreased to about 4000 W/m °C for upward flow with pressurized steam at 110°C. Although this may appear to be an anomaly, it can be understood when the contribution of phase change or condensation of steam to heat transfer is taken into consideration. With live steam, a quasi-total condensation is expected, whereas it is partial with pressurized steam. At 100°C, the h value is not significantly affected by steam flow direction, presumably because the level of condensation in both directions is nearly the same. With an increase in steam temperature, the h value increases but with a constant disparity between flow directions. This difference has


6000 OPEN HEATING AT 100°C

h-value (W/m2°C)

5500

DOWNWARD

5000 UPWARD 4500 4000 3500 3000 100

110

120

130

140

150

Temperature (°C) FIGURE 22.3 The influence of steam flow direction and temperature on h values. Steam flowed at a rate of ∼3.4 g/sec through a bed of rehydrated chickpeas. 6000

h-value (W/m2°C)

DOWNWARD

Steam Flow Direction

Top Center Bottom

5500

UPWARD DOWNWARD

5000

4500

4000 1

2

3

4

5

6

Steam Mass Flow Rate (g/sec) FIGURE 22.4 The influence of steam flow rate, direction, and location on h at temperature of ∼115°C in a bed of rehydrated chickpeas.

to do with the degree of condensation and/or thickness of the condensate film, which can play a role in the resistance to heat transfer. The results indicate that with downward flow, a higher level of condensation occurs and/or condensate is flushed out more effectively. The results also indicate that there is a lesser degree of condensation with pressurized steam compared with live steam, and that condensation behavior of pressurized steam is affected by its flow direction. Condensation of steam occurs because of temperature and pressure drops through the particulate bed. It is conceivable that


5500

h-value (W/m2°C)

Large glass beads Chickpeas Small glass beads 5000

4500

Steam Flow Direction UPWARD DOWNWARD 4000 0

2

4

6

8

10

Steam Mass Flow Rate (g/sec)

FIGURE 22.5 The influence of bed composition, particle size, steam flow rate, and flow direction at a temperature of ∼120°C on h values in beds of 6 mm and 14 mm glass beads and rehydrated chickpeas. The h values were pooled from all three locations.

the drop in steam temperature due to sensible heat exchange may not be significantly different between the flow directions. But the drop in steam pressure can be different because downward flow tends to compact the packed bed, while upward flow tends to expand it. Consequently, higher pressure drop and reduced void bed volume would take place during downward flow, while the reverse would be true for upward flow. Reduced void space would facilitate retention of a smaller quantity of condensate. Thus, enhanced condensation (latent heat transfer) and reduced condensate film (heat transfer resistance) should lead to a higher h value during downward flow than upward flow under similar heating conditions. Because condensation plays a role, the boundary layer formed by the condensate (liquid water) should be a consideration in heat transfer. Luikov (1968) showed that h is related to the boundary layer thickness (δ) and k, the thermal conductivity of the condensate layer, by equations (2a) and (2b):

h(Ts – Tps) = kf

or δ =

(Ts − Tps ) δ kf h

(22.2a)

(22.2b)

where Ts and Tps are the steam and particle surface temperatures, respectively. An estimate of δ from Equation (22.2b) yields values of 0.3 mm for food materials and 0.12 mm for metal cubes. Larger boundary layer thicknesses increase the resistance to heat transfer and reduce h value.


22.3.3 TRANSDUCER LOCATION

IN THE

FOOD BED

The h values were location sensitive, and more so with steam flowing downward than upward (Figures 22.2 and 22.4). In downward flow, the h values at the uppermost location were highest, and they decreased with bed height; i.e., h values were highest at the point of steam entry into the bed and lowest at the exit end. This was also evident in time–temperature plots for glass beads because the metal transducer at the top location had a steeper slope. In upward steam flow, slightly higher h values were obtained at the top location (Figure 22.2); this is contrary to downward flow, as h values were higher at the exit. Overall, h values were more uniform in upward flow but slightly lower at the bed center.

22.3.4 INFLUENCE

OF

STEAM MASS FLOW RATE (SMFR)

The effect of steam mass flow rate on the value of h is depicted in Figures 22.4 and 22.5. Figure 22.4 shows the changes in h value registered by metal cubes placed in rehydrated chickpeas at steam temperature of 115°C for both flow directions. Figure 22.5 shows the changes in h value for glass beads of diameter 6 and 14 mm and rehydrated chickpeas at 120°C, with steam flow in both upward and downward directions. In the range of SMFR, 1.2 to 5.8 g/sec and Ts ≈ 115°C, (Figure 22.4), h values registered in a bed of chickpeas showed small increases with increase in SMFR at all locations with downward flow. The h values increased at the top, center, and bottom of the food bed 2 2 by about 8% (5000 to 5400 W/m °C), 5% (4650 to 4900 W/m °C), and 14% (4300 to 2 4900 W/m °C), respectively, during downward flow (Figure 22.4). However, the change in h value was negligible when the direction of the steam flow was upward. The pooled data (average for all three locations) presented in Figure 22.5 show that at 120°C the metal transducers experienced small increases in h values, in beds of large glass beads as well as for chickpeas, for both upward and downward flow of steam. However, the increase in h value was negligible for small glass beads for both directions of steam flow (Figure 22.6). Overall, the effect of SMFR on h value was small; the flow direction of steam, as well as particle location, had stronger influences. Nonetheless, the influence of SMFR was noticeable for packed beds with larger void spaces, such as large glass beads or chickpeas. The effect waned when the void volumes became small, as in the case of small glass beads.

22.3.5 STEAM TEMPERATURE EFFECTS The effect of steam temperature on h values for rehydrated chickpeas at SMFR of 3.4 g/sec in both the upward and downward flow of steam is shown in Figure 22.3, pooling data for the three locations. To compare the influence of temperature effects (Figure 22.6) in the three different beds, glass beads of two different sizes and chickpeas, h value data for similar temperatures but different SMFR were pooled for both flow directions and location to provide average values. Whatever the direction or particle size of the bed, h increased with temperature at nearly the same rate. A high h value recorded at 100°C can be attributed to a very high level of condensation of live steam. As steam temperature or pressure was increased, h value remained the same or decreased up to 110°C, depending on steam flow direction; above this


6000

h-value (W/m2°C)

5500 5000 4500 4000 3500

Large glass beads (φ = 14 mm) Chickpeas (φ = 11 mm) Small glass beads (φ = 6 mm)

OPEN HEATING AT 100°C

3000 100

110

120

130

140

Temperature (°C) FIGURE 22.6 The influence of temperature on h in beds of 6 mm and 14 mm glass beads and ∼11 mm rehydrated chickpeas. Data for similar temperatures but different flow rates were pooled for both flow directions and location to obtain mean values.

temperature, h value increased steadily (Figure 22.3). This suggests that the condensation effect is reduced with pressurized steam. However, the temperature effect compensates for the contribution of condensation to heat transfer and hence, the h value increases above 110°C. In effect, quasi-total steam condensation is achieved with pressurized steam at ~108°C in a downward flow and at ~125°C with upward flow (Figure 22.3). The effects of steam flow direction, SMFR, and temperature on h value can also be understood through the condensation effect. When dry steam first enters the packed bed and makes contact with the cold particles, partial condensation occurs and a film of water is deposited on the particle surface, and some of the free void space is filled with the condensate. Since smaller size particles provide more resistance to fluid flow and also have smaller open spaces, it is probable that slightly more water is entrapped because of capillary edge effects, and this increases the boundary layer and consequently reduces the h value. Because condensate is tightly held between the small beads, SMFR and flow direction do not appear to influence the h value (Figure 22.5). However, as particle diameter increases, void spaces are larger, and as the bed opens up, resulting in better condensate flushing and fluid flow, the effects of SMFR and flow direction come into play (Figures 22.4 and 22.5). Furthermore, heat transfer involves not only sensible heat, but also latent heat effects due to condensation. Hence, the differences in h value ultimately depend on the degree of condensation, fluid flow through the bed, and locations depending on the direction of steam flow. In addition, the dry steam provided for heating can, and most likely does, exit as wet steam. The amount of free water that can be readily carried away by the exiting wet steam is probably related to bed restrictions, SMFR, and temperature. Thus, as temperature increases, effects related to the degree of wetness of exiting steam become important, and the h value increases as the condensate boundary layer shrinks.


22.3.6 INFLUENCE

OF

PARTICLE SIZE

The influence of particle size on h value is shown in Figures 22.5 and 22.6. Figure 22.5 shows the effect of particle size on h value at various SMFR, while Figure 22.6 provides data on h values at various temperatures. It is clear from these data that h values are dependent on the particle size of the bed. The value of h is higher in packed beds of larger particle size, attributable to larger void spaces within the bed.

22.3.7 INERT MATERIAL

VS.

FOOD

Figure 22.6 indicates that glass beads (inert) and chickpeas display similar trends in the profile of h value with temperature of steam. Large glass beads (diameter 14 mm) show higher h values at all steam temperatures, and small glass beads (diameter 6 mm) show lower values than chickpeas (~diameter 11 mm), which are intermediate in size. Furthermore, assuming chickpeas to be spheres, and ignoring the end effects, the porosity of all three beds is similar. Hence, differences in h value data are attributable to the void spaces between particles rather than to their composition or to the gassing-out effect, which is a possibility since rehydrated chickpeas contain air dissolved in the rehydrated tissue.

22.3.8 h VALUES

OF

FOOD

VS.

METAL

Pannu et al. (Chap. 21, this volume) determined h values for steam processing of 12-mm potato and carrot cubes. Potato and carrot cubes produced h values of about 2 1500 and 2000 W/m °C, respectively; in comparison, similar size metal cubes 2 produced h values ranging between 4300 and 5500 W/m °C (Figures 22.2 to 22.6). 2 The value of about 5000 W/m °C is in agreement with h values for condensing steam reported by Pflug (1964) and Skjöldebrand and Ohlsson (1993). The results of this study and previous results (Pannu et al., 1990) show that food particles do not produce h values similar to those obtained with metal transducers. The h values obtained with food pieces were about 40% lower than those obtained with metal cubes under similar process conditions. The obvious differences between these two materials are thermal diffusivity and boundary layer thickness of condensed water. The thermal diffusivity of metals is far higher than that of food −5 2 −7 2 materials (~8.4 × 10 m /sec for aluminum vs. ~1.6 and 1.7 × 10 m /sec for potato and carrot, respectively). Furthermore, the water film thickness on metals should be lower because water does not adhere well to metal surfaces, whereas the hydration layer around food facilitates the formation of a thicker boundary layer.

22.4 CONCLUSIONS The h values provided by metal cubes embedded in particle beds heated with steam 2 were closer to the value of 5000 W/m °C of h reported by Pflug (1964). These h values 2 are significantly higher than the value of 2000 W/m °C obtained for food cubes 2 (Pannu et al., Chap. 21, this volume), but lower than the value of 12,000 W/m °C reported by Tung et al. (1984). The h values were not appreciably affected by SMFR over the range tested. Higher h values were obtained by downward steam flow and


higher steam temperature and in beds of larger size particles. It appears that the gassing-out phenomenon is not involved in the reduction of h values, nor in altering the heat transfer characteristics of steam. Excessive condensation of the heating fluid in the particle bed increases the boundary layer thickness and enhances the resistance to heat transfer. The h values obtained from metal cubes during steam heating were about 2.5-fold higher than those obtained from food of similar size and shape. These differences are attributable to thermal diffusivity and the boundary layer thickness of the condensate. Thus, good quality dry steam is necessary for efficient and uniform heating of food beds to avoid build-up of condensate.

REFERENCES Brown, G.E., Importance of surface heat transfer during steam heat and heat/hold processes, J. Food Sci., 39, 1066, 1974. Holman, J.P., Heat Transfer, 4th ed., McGraw-Hill, New York, 1976. Kostaropoulous, A.E., Spiess, W.E.L., and Wolf, W., Reference values for thermal diffusivity of foodstuffs, Lebensm.-Wiss. Technol., 8, 108, 1975. Ling, C.C.A., Bomben, J.L., Farakas, D.F., and King, C.J., Heat transfer from condensing steam to vegetable pieces, J. Food Sci., 39, 692, 1974. Luikov, A.V., Analytical Heat Diffusion Theory, Academic Press, New York, 1968. Othmer, D.F., The condensation of steam, Ind. Eng. Chem., 21, 576, 1929. Pannu, K.S., High Temperature Transient Heating of Vegetables by Steam Injection. Ph.D. thesis, Laval University, Ste-Foy, Quebec, 1992. Pannu, K.S., Castaigne, F., and Arul, J., Chap. 21, this volume. Pannu, K.S., Castaigne, F., Ramaswamy, H., Arul, J., Audet, P., and Desilets, D., Rapid Sterilization of Particulate Food by Steam Injection: Evaluation of Thermal Parameters, ASAE paper # 90–6527, Presented at the 1990 ASAE winter meeting, Chicago, 1990. Pflug, I.J., Evaluation of heating media for processing food in flexible packages. Phase I. Final report, contract DA19-AMC-145(N), U.S. Army Natick Laboratories, Natick, MA, 1964. Poulsen, K.P., Optimization of vegetable blanching, Food Technol., 40, 122–129, 1986. Rumsey, T., Farakas, D.F., and Hudson, J.S., Measuring steam heat transfer coefficient to vegetables, Trans. ASAE, 24, 1048, 1980. Skjöldebrand, C. and Ohlsson, T., A computer simulation program for evaluation of the continuous heat treatment of particulate food products. Part 2: Utilization, J. Food Eng., 20, 167–181, 1993. Tung, M.A., Britt, I.J., and Ramaswamy, H.S., Food sterilization in steam/air retorts, Food Technol., 44, 105, 1990. Tung, M.A., Ramaswamy, H.S., Smith, T., and Stark, R., Surface heat transfer coefficients for steam/air mixtures in two pilot scale retorts, J. Food Sci., 49, 939, 1984. Zhang, Q. and Cavalieri, R.P., Thermal model for steam blanching of green beans and determination of surface heat transfer coefficient, Trans. ASAE, 34, 182, 1991.


23

Modeling of Heat Conduction in Elliptical Cross Sections (Oval Shapes) Using Numerical Finite Difference Models ˇ F. Erdogdu, M.O. Balaban, and K.V. Chau

CONTENTS 23.1 Introduction 23.2 Materials and Methods 23.2.1 Power Curves Method 23.2.2 An Alternate Flexible Method 23.3 Results and Discussion 23.4 Conclusions and Suggestions Nomenclature References

23.1 INTRODUCTION Heat transfer is an important part of several unit operations in food processing. Foods commonly have irregular shapes, and heat transfer analysis is usually avoided because of the difficult mathematical treatment and numerical solutions (Erdoˇgdu et al., 1998b). There has been a wealth of studies, programs and accumulated knowledge of heat transfer in circular cylinders. This is not the case for elliptical cross sections. There have been several studies regarding heat transfer in elliptical cross sections (Erdoˇgdu, 1996). McLahlan (1945) was the first to publish analytical equations governing the heat transfer in an infinite elliptical cylinder with a uniform initial temperature and subjected to an infinite convective heat transfer coefficient. The numerical solution for


McLahlan’s equation was provided by Kirkpatrick and Stokey (1959) for ellipses having eccentricities of 0.6, 0.7, 0.8, and 0.9 against Fourier numbers of 0.1, 0.2, 0.5, and 1.0 at certain locations of the geometry. Manson et al. (1974) predicted the transient temperatures in conduction heating of pear-shaped objects using a finite difference model. Simpson et al. (1989) evaluated thermal processes of foods in oval-shaped containers using a finite difference approximation of the differential equation for transient heat conduction in three dimensions. Eshleman (1976), Califano and Zaritzky (1993), Sheen et al. (1993), Akterian and Fikiin (1994), and Kim and Teixeira (1997) developed finite difference methods for simulating heat conduction in irregularly shaped foods. Erdoˇgdu et al. (1998b) developed a volume element based approach on finite difference models for heat transfer in elliptical cross sections by using power curves. They also reported that the use of power curves, assuming heat transfer is along the curve, can be considered a favored method for heat transfer problems in elliptical cross sections since it reduces the three-dimensional problem into two dimensions in finite cylinders and the two-dimensional problem into one dimension in infinite cylinders. It was also possible to accurately predict the center temperatures in conductively heated elliptical cylinders by substituting them with a right circular cylinder with equal ratio of surface area to volume when the eccentricity of the ellipse was lower than 0.9. However, neither of these methods was flexible enough to accept different boundary conditions at the surface or inside the surface (e.g., different convective heat transfer coefficient and surface temperatures at the different parts of the surface or nonhomogeneous thermophysical properties inside the geometry). Thus, the development of a flexible alternate finite difference solution to heat transfer problems in elliptical cross sections to eliminate the previous methods’ deficiencies would be valuable. Therefore, the objectives of this study were to review the power curves method, to develop an alternate flexible volume element based approach to accept different boundary conditions at the surface and nonhomogeneous thermophysical properties inside, and to compare these methods.

23.2 MATERIALS AND METHODS 23.2.1 POWER CURVES METHOD As explained by Erdoˇgdu (1996) and Erdoˇgdu et al. (1998b), the following procedure was used to generate the numerical network by the power curves method. The equations representing the power curves and an ellipse with half-major axis “a” and half-minor axis “b” (and also elliptical isotherms) are shown in Equations (23.1) and (23.2).


y = c ⋅ xn

(23.1)

x2 y2 + =1 a 2 b2

(23.2)

Since the power curves are orthogonal to the elliptic isotherms, the product of the slope of the power curves and ellipses must be equal to –1 at their intersection. Equidistant points (the number of power curves) on the perimeter of the ellipse with known values of “a” and “b” were found. These points are at the intersections between the ellipse and the power curves, and the arc length of an ellipse between the horizontal axis and angle 2 was given by: θ

p = a⋅

∫

1 − Sin 2 θ ⋅ dθ

(23.3)

0

where e is the eccentricity (the measure of the roundness of ellipse) and is given by: a 2 − b2 a

e=

(23.4)

The arc length between two intersection points was: perimeter/(number of power curves + 1). Equation (23.3) was used to determine the angle necessary for a given arc length; a bisection method was applied to find the intersection point xint, yint. Then, knowing the “a,” “b,” “xint,” and “yint” values, the coefficients of the power curves were calculated using the orthogonality requirement between the ellipse and power curves: n=

y int sp ⋅ x int

(23.5)

n −1 where sp ( = c ⋅ n ⋅ x int ) is the slope of the power curve at the intersection point. Then, the concentric ellipses with major and minor axes decreasing at the same proportions were drawn. The intersection points between the power curves and the concentric ellipses were determined by solving Equation (23.6) using the Newton–Raphson method (Erdoˇgdu, 1996; Erdoˇgdu et al., 1998b).

f=

x 2int c2 ⋅ x 2intn + −1 a2 b2

(23.6)

For an elliptical cross section of a given thickness, a volume element was defined as the volume between two consecutive ellipses and power curves (Figure 23.1). The area bounded by the minor axis and any heat flow line was given by Equation (23.7), and the area of a given volume element was obtained by successive subtraction of areas (Erdoˇgdu, 1996; Erdoˇgdu et al., 1998b). The volume of each element was calculated by multiplying its area by its thickness approximating the curved surfaces by planar surfaces.

A=


x int

x2 b⋅ 1− 2 a

0

c⋅x n

∫ ∫

dy ⋅ dx

(23.7)

a

a a

b

b

a

b b

b

b b

a b

b b

c c

b

b

b b

cc

a

b

b b

b b b b

b b

b b

b

b

a

FIGURE 23.1 Representation of volume element network in an elliptical cross section for the power curves method. (Adapted from Erdoˇgdu, F., Balaban, M.O., and Chau, K.V., J. Food Eng., 38, 223–239, 1998.)

Then, the finite difference equations were generated by writing energy balance equations for each volume element. The outer surface nodes were treated as noncapacitance surface nodes (NCSN) (Chau and Gaffney, 1990; Welt et al., 1997; Erdoˇgdu, 1996; Erdoˇgdu et al., 1998b). The equations generated for an infinite elliptical cylinder (Figure 23.1) are given below for surface volume elements (a), volume elements between surface and center along the heat flow lines (b), and center volume elements (c), respectively. Volume element type a:   k⋅A  ∆t 1 i , j−1 n +1 n n −1 n n   Ti, j = Ti, j + ⋅ ⋅ (Ti, j − Ti, j ) + ⋅ ( T∞ − Ti, j ) ∆ri, j 1  ρ ⋅ c p ⋅ Vi, j  ∆ri, j−1 +   h ⋅ A i, j k ⋅ A i, j  

(23.8)

Volume element type b:

Tin, j+1 = Tin, j +

  k ⋅ A i, j−1 k ⋅ A i, j ∆t ⋅ ⋅ (Tin, j−1 − Tin, j ) + ⋅ (Tin, j+1 − Tin, j ) ρ ⋅ c p ⋅ Vi, j  ∆ri, j−1 ∆ri, j 

(23.9)

Volume element type c:

Tin, j+1 =

∆t ρ ⋅ c p ⋅ Vi, j

  k ⋅ A i, j−1 ⋅ ⋅ (Tin, j−1 − Tin, j )  ∆ri, j−1 

(23.10)

In the case of finite elliptical cylinders, where heat flow is assumed to be in the longitudinal direction and along the heat flow line, nine finite difference equations


were needed. A detailed description for the generation of these equations was given by Erdoˇgdu (1996) and Erdoˇgdu et al. (1998b). An explicit method was used to solve the finite difference equations. At the end of each time step, there were slightly different temperatures for the center volume elements. Therefore, the overall volume average temperature of the center elements was calculated. For the next time step, the initial temperature of all the center volume elements was taken as the average temperature. The computer program to solve these equations was written in Visual Basic V. 6.0 (Microsoft, 1994). The details of the program are given in Erdoˇgdu (1996) and Erdoˇgdu et al. (1998b).

23.2.2 AN ALTERNATE FLEXIBLE METHOD The following procedure was used to generate the numerical network for the alternate flexible method for the numerical evaluation of conduction heat transfer in elliptical cylinders. By using Equations (23.1), (23.2), and (23.3), the equidistant points on the perimeter of the ellipse with known values of “a,” “b,” and “P” were found, and the θ angle values giving this equidistant arc length between the points (Equation (23.3)) were evaluated using a bisection method. The x and y values of the equidistant points were determined using a bisection method with the known θ value. The boundaries of the volume elements were formed by drawing lines parallel to the x and y axes from the equidistant points. The points where the lines parallel to the x axis intersected with the boundary were used as the starting point for the points parallel to the y axis, and all these were applied to the other three quadrants in a cross section (Figure 23.2). Type 5

Quadrant 4

d: Type 4 b e e c

b

Quadrant 1 b

Type 3

e

c

c

e

c

c

c

b

e

c

c

c

c

b: Type 2

g

g a: Type 1

Type 6

Type 7 f

Quadrant 3

g

g

g

Quadrant 2

FIGURE 23.2 Representation of volume element network in an elliptical cross section for the alternate flexible approach.


Vn

yn – 1 yn

xn – 1

xn

FIGURE 23.3 Representation of volume of a surface element in the volume element network for the alternate flexible approach.

Surface volume elements were treated as capacitance surface nodes (CSN) (Welt et al., 1997). The volume and surface area of the volume elements were determined by using those of a rectangle for the inside nodes, and by integration for the surface nodes. The volume of a surface element for a known thickness (∆l) is given in Equation (23.11) (Figure 23.3):  xn  b Vn =  ⋅ a 2 − x 2 ⋅ dx − [( x n − x n −1 ) ⋅ ( y n − y n −1 )] ⋅ ∆ l   a  x n −1 

∫

(23.11)

The heat transfer area of the CSN was taken as the curve bordered by the equidistant points. It was the product of arc length and the given thickness. The finite difference equations were generated developing energy balance equations for each volume element. The thermal and physical properties of each volume element were independent of the others. Figure 23.4 shows the different types of volume elements in quadrant 1 of an elliptical cross section. Equations (23.12) to (23.18) are the generated finite difference equations for these volume elements. Volume element type 1:  k ⋅ YA q1,i, j+1  1  1 − ∆t ⋅ ⋅  q1,i, j (ρ ⋅ c p ⋅ V)q1,i, j  ∆x q1,i, j    Tqn1+1 (i, j) = Tqn1 (i, j) ⋅      + k q 2,i, j ⋅ XA q1,i, j+1 + h q1,i , j ⋅ YA q1,i , j    ∆y q1,i, j    ⋅ XA q1,i, j+1 n k   k q1,i, j ⋅ YA q1,i, j+1 n ⋅ Tq1 (i, j + 1) + q 2,i, j ⋅ Tq1 (i − 1, j + 1)  ∆x q1,i, j ∆y q1,i, j  + ∆t ⋅     + h q1,i, j ⋅ YA q1,i, j−1 ⋅ Tq∞1,n (i, j) (23.12)


Type 3

Type 2

Volume element Type 1

h convection

(i+1,j-1)q1

h convection

(i,j)=(1,2) (i,j+1) q1

(i,j)q1

(i,j+1) q1

(i,j)q2

(i,j) q1

(i,j)q1

(i,j+1)q1

(i,j-1) q1

(i-1,j+1)q1

(i-1,j+1)q1

Type 5 (i+1,j-1)q1

Type 4 h convection (i,j)q4

(i,j)q1 (i,j)q4

(i,j)q1

(i,j-1)q1

(i-1,j+1)q1

(i-1,j+1)q1

Type 7

Type 6

(i+1,j-1) q1

(i+1,j-1) q1

(i,j)q4

(i,j)q1

(i,j)q2

(i,j-1)q1

(i,j+1) q1

(i,j) q1

(i,j-1) q1

(i,j) q2

FIGURE 23.4 Representation of different volume elements in quadrant 1 for the alternate flexible approach (see Figure 23.1).


Volume element type 2:    k q1, i, j + 1 ⋅ YA q1, i, j 1  1 − ∆t ⋅ ⋅ (ρ ⋅ c p ⋅ V) q1, i, j  ∆x q1, i, j    Tqn1+ 1 (i, j) = Tqn1 (i, j) ⋅      k q1, i − 1, j + 1 ⋅ XA q1, i − 1, j + 1 + h q1, i, j ⋅ YA q1, i, j − 1   + ∆y q1, i, j    k q 2, i, j ⋅ XA q1, i − 1, j + 1 n  k q1, i, j + 1 ⋅ YA q1, i, j n  ⋅ Tq1 (i, j + 1) + ⋅ Tq1 (i − 1, j + 1)  ∆ ∆ x y q1, i , j q1, i , j  + ∆t ⋅    + h q1, i, j ⋅ YA q1, i, j − 1 ⋅ Tq∞1, n (i, j)   

(23.13) Volume element type 3:    k q1,i, j+1 ⋅ YA q1,i, j 1  1 − ∆t ⋅ ⋅   (ρ ⋅ c p ⋅ V)q1,i, j  ∆x q1,i, j     k XA k XA ⋅ ⋅ q1,i +1, j−1  q1,i , j−1 Tqn1+1 (i, j) = Tqn1 (i, j) ⋅ + q1,i −1, j−1 + q1,i +1, j−1   ∆y q1,i +1, j−1 ∆y q1,i, j−1       k q1,i −1, j+1 ⋅ XA q1,i, j  +    ∆ y  q1,i , j   k ⋅ XA q1,i, j−1 n   k q1,i, j+1 ⋅ YA q1,i, j n ⋅ Tq1 (i, j + 1) + q1,i, j−1 ⋅ Tq1 (i, j − 1)   ∆x q1,i, j ∆y q1,i, j−1   + ∆t ⋅   k q1,i −1, j+1 ⋅ XA q1,i, j n   k q1,i +1, j−1 ⋅ XA q1,i +1, j−1 n ⋅ Tq1 (i + 1, j − 1) + ⋅ Tq1 (i − 1, j + 1) + ∆y q1,i +1, j−1 ∆y q1,i, j   (23.14) Volume element type 4:   k q 4, i, j ⋅ YA q1, i, j  1  1 − ∆t ⋅ ⋅ ∆x q1, i, j (ρ ⋅ c p ⋅ V) q1, i, j     Tqn1+ 1 (i, j) = Tqn1 (i, j) ⋅      k q1, i − 1, j + 1 ⋅ XA q1, i, j + h q1, i, j ⋅ YA q1, i, j   + ∆y q1, i, j    k q1, i − 1, j + 1 ⋅ XA q1, i, j n   k q 4, i, j ⋅ YA q1, i, j n ⋅ Tq 4 (i, j) + ⋅ Tq1 (i − 1, j + 1)  ∆ x ∆ y q1, i , j q1, i , j  + ∆t ⋅     + h q1, i, j ⋅ YA q1, i, j ⋅ Tq∞1, n (i, j)  


(23.15)

Volume element type 5:  1 1 − ∆t ⋅ (ρ ⋅ c p ⋅ V) q1, i, j  Tqn1+ 1 (i, j) = Tqn1 (i, j) ⋅    k q1, i, j − 1 ⋅ YA q1, i, j − 1 +  ∆x q1, i, j − 1 

 k q 4, i, j ⋅ YA q1, i, j  ⋅ + ∆x q1, i, j     k q1, i − 1, j + 1 ⋅ XA q1, i, j    ∆y q1, i, j  

k q1, i, j − 1 ⋅ YA q1, i, j − 1 n   k q 4, i, j ⋅ YA q1, i, j n ⋅ Tq 4 (i, j) + ⋅ Tq1 (i, j − 1) +   ∆ ∆ x x q1, i , j q1, i , j − 1   + ∆t ⋅     k q1, i − 1, j + 1 ⋅ XA q1, i, j n ⋅ Tq1 (i − 1, j + 1)   ∆ y   q1, i , j

(23.16)

Volume element type 6:   k ⋅ YA q1,i, j 1  1 − ∆t ⋅ ⋅  q 4 ,i , j (ρ ⋅ c p ⋅ V)q1,i, j  ∆x q1,i, j    Tqn1+1 (i, j) = Tqn1 (i, j) ⋅     + k q 2,i, j ⋅ XA q1,i, j + k q1,i, j−1 ⋅ YA q1,i, j−1 + k q1,i +1, j−1 ⋅ XA q1,i +1, j−1    ∆y q1,i, j ∆x q1,i, j−1 ∆y q1,i +1, j−1   ⋅ XA q1,i, j n k  k q 4,i, j ⋅ YA q1,i, j n  ⋅ Tq 4 (i, j) + q 2,i, j ⋅ Tq 2 (i, j)   ∆x q1,i, j ∆yy q1,i, j   + ∆t ⋅   k q1,i +1, j−1 ⋅ XA q1,i +1, j−1 n  k q1,i, j−1 ⋅ YA q1,i, j−1 n  ⋅ Tq1 (i, j − 1) + ⋅ Tq1 (i + 1, j − 1) + x ∆ ∆ y q1,i , j−1 q1,i +1, j−1   (23.17) Volume element type 7:    k q1, i, j + 1 ⋅ YA q1, i, j 1  1 − ∆t ⋅ ⋅ ∆x q1, i, j (ρ ⋅ c p ⋅ V) q1, i, j     Tqn1+ 1 (i, j) = Tqn1 (i, j) ⋅     ⋅ k ⋅ YA k ⋅ XA k XA  q1, i, j − 1 q 2, i , j q1, i , j  q1, i , j − 1 q1, i + 1, j − 1 q1, i + 1, j − 1 + + +   ∆y q1, i + 1, j − 1 ∆y q1, i, j ∆x q1, i, j − 1    k q1, i, j − 1 ⋅ YA q1, i, j − 1 n   k q1, i, j ⋅ YA q1, i, j n ⋅ Tq1 (i, j + 1) + ⋅ Tq1 (i, j − 1)   ∆ ∆ x x q1, i , j q1, i , j − 1   + ∆t ⋅   k q 2, i, j ⋅ XA q1, i, j n   k q1, i + 1, j − 1 ⋅ XA q1, i + 1, j − 1 n ⋅ Tq1 (i + 1, j − 1) + ⋅ Tq 2 (i, j) + y y ∆ ∆   q1, i + 1, j − 1 q1, i , j

(23.18)


An explicit method was used to solve these finite difference equations. All calculations were performed using a computer program written in Visual Basic V. 6.0 (Microsoft, 1994). The developed program calculated the temperature distribution in elliptical cross sections using the power curves method and alternate flexible approach. The flow diagram for the developed program for both approaches is given in Figure 23.5.

A. Volume element network generation Data Input: a: half-major, b: half-minor axis for elliptical cross-section for volume element network generation and all thermophysical properties Calculation of: eccentricity, perimeter, arc length between two intersection points and the coordinates of the intersection points using Eqs. 3.4.1 to 3.4.6

B. Calculation of the physical properties of the volume elements and generation of finite difference equations Defining the volume elements for power curves method of the alternate flexible approach Calculation of the area and volume of each volume element using Eqs. 3.4.7 to 3.4.11 and the distances between the volume elements in each direction

C. Generation of finite difference equations and the solution using an explicit procedure Generation of finite difference equations (using Eqs. 3.4.8 to 3.4.10 for the power curves method and Eqs. 3.4.12 to 3.4.18 for power curves method for the alternate flexible approach) by writing energy balances for each volume element, and using an explicit procedure to solve these equations knowing outside temperatures, product properties, and heat transfer conditions on the surface

FIGURE 23.5 Flow diagram of the developed program for the power curves method and the alternate flexible approach.


23.3 RESULTS AND DISCUSSION The NCSN and CSN volume element finite difference models developed for modeling of heat conduction in elliptical cross sections were validated by experiments. Long plexiglass elliptical cylinders with eccentricities of 0.63 (a = 2.2 cm, b = 1.7 cm, l = 15 cm) and 0.69 (a = 1.25 cm, b = 0.9 cm, l = 8 cm) were used for the validation studies. The thermal conductivity, heat capacity, and density of these 3 cylinders were 0.1875 W/m-K, 1465.4 J/kg-K, and 1190 kg/m , respectively (Polycast Technology, 1996). Heating experiments were performed in a well-agitated water bath (infinite h), 2 and cooling was done in ice slush (h ≈200 W/m °K) (Erdoˇgdu et al., 1998a). Time–temperature data from the experiments were gathered using Type-T, 36 gauge thermocouples (Ecklund, Fort Myers, FL), inserted and stabilized in the cylinders, and a data acquisition system (DAS-TC, Kethley Metrabyte, Taunton, MA). The experimental data were compared with the model results. Ten experiments were conducted for the validation studies at temperatures of 75 and 85°C. Figures 23.6 and 23.7 show the comparison of two of the experimental and predicted temperatures (at center and off-center points) of “infinite” elliptical cylinders of eccentricity of 0.69 and 0.63 by the alternate flexible approach and the power curves method. In both approaches, the model predictions were close to the experimental results. Since a circular cross section can be described as an elliptical cross section with equal “a” and “b” values, both approaches can also be used for simulation studies in circular cylinders. The advantage of the power curves method is that it reduces the heat transfer problem from three- to two-dimensional in finite cylinders, and from two- to onedimensional in infinite cylinders. However, the advantage of the second approach over the power curves method is that it allows simulations with different boundary conditions in different directions and different volume elements, as seen in Figure 23.8.

Temperature (°C)

100 80 60 40 20 0 5

0

10

15

20

25

30

35

Time (min) Model Center

Exp. Center

Exp. Off-center

Medium Temperature

Model Off-center

FIGURE 23.6 Comparison of experimental and model center temperatures in a long elliptical cylinder (a = 2.2 cm, b = 1.7 cm, l = 15 cm, e = 0.63; by alternate flexible approach).


Temperature (°C)

100 80 60 40 20 0 0

5

10

15

20

Time (min) Model Center

Exp. Center

Exp. off-center

Medium Temperature

Model off-center

FIGURE 23.7 Comparison of experimental and model center temperatures in a long elliptical cylinder (a = 1.25 cm, b = 0.9 cm, l = 8 cm, e = 0.69; by power curves method).

Quadrant 4 T1, h1

Quadrant 3 T2, h2

Quadrant 1 T1, h1

Quadrant 2 T2, h2

* Shaded areas show the volume elements with different thermophysical properties.

FIGURE 23.8 Representation of using the alternate flexible approach for different boundary conditions and volume element properties.


For example, the heat transfer problem for an elliptic–cylindrical shaped food with each half having different boundary conditions, such as one half in sauce and the other facing air, can be solved using the developed alternate flexible approach. Since the power curves method assumes the heat transfer along the power curves, not between them, it becomes impossible to use this method for this kind of problem. Other examples showing the efficacy of the second method include the following: Burfoot and James (1988) reported that the heat transfer coefficient is location dependent for thawing and cooking of cylindrical shaped meats. Burfoot et al. (1990) compared the heating and cooling times of meat joints. The developed alternate flexible approach can be easily used for both problems, taking into account the different thermophysical properties of meat joints and including the bones inside the meats. The flexibility of the second method when applied to systems with nonhomogeneous initial temperature distribution can be given as another example of the efficacy of this method.

23.4 CONCLUSIONS AND SUGGESTIONS Both power curves and the developed alternate flexible approach were successfully applied to analyze the heat conduction in elliptical cross sections. Both of these methods can also be used to predict the temperature distribution in response to dynamic surface temperature history. The Windows-based software developed in this study calculated the temperature distribution in infinite elliptical cylinders for both approaches at any given time, knowing outside temperature (constant or variable), product properties (homogeneous or heterogeneous), and heat transfer conditions at the surface (constant throughout the surface or variable). The validation of the second approach for different food systems and modification to simulate the heat transfer in finite cylinders and finite slabs will result in a powerful tool which can be used for the analysis of thermal processing in food systems with nonhomogeneous thermal properties.

NOMENCLATURE a Ai,j b c cp cp qm,i,j ∆r ∆t ∆l e f h k

Half-major axis for ellipse, m 2 Surface area of the volume element i, j, m Half-minor axis for ellipse, m Constant Heat capacity, J/kg-K Heat capacity of volume element i, j in quadrant m, J/kg-K Distance between the volume elements along the heat flow line, m Time step used in explicit solution method, sec Thickness for a volume element Eccentricity of an ellipse Function for the intersection points of ellipses and power curves 2 Convective heat transfer coefficient, W/m -K Thermal conductivity, W/m-K


kqm,i,j n p ρ ρqm,i,j se sp

Thermal conductivity of volume element i, j in quadrant m, W/m-K Constant Perimeter of an ellipse, m 3 Density, kg/m 3 Density of volume element i, j in quadrant m, kg/m Slope of a tangent line at any point on the ellipse Slope of a tangent line at any point on the power curve

Tin,j

Temperature of the volume element i, j at the time step n, °C

n +1 i, j n qm

T

Temperature of the volume element i, j at the next time step, °C

T (i, j)

Temperature of the volume element i, j at the time step n for quadrant m, °C Temperature of the volume element i, j at the next time step for quadrant m, °C Medium temperature of the volume element i, j at the next time step for quadrant m, °C The angle of the line connecting the center of an ellipse and a point on the ellipse circumference, radian 3 Volume of the volume element i, j, m Intersection points between a heat flow line and an ellipse The surface area of the volume element i, j in quadrant m in y direction, ∫ m ∫ The surface area of the volume element i, j in quadrant m in x direction, m

n+1 Tqm (i, j) ∞,n Tqm (i, j)

θ Vi,j xint,yint XAqm,i,j YAqm,i,j

REFERENCES Akterian, S.G. and Fikiin, K.A., Numerical simulation of unsteady heat conduction in arbitrary shaped canned foods during sterilization processes, J. Food Eng., 21, 343–354, 1994. Burfoot, D. and James, S.J., The effects of spatial variations of heat transfer coefficient on meat processing times, J. Food Eng., 7, 41–61, 1998. Burfoot, D., Self, K.P., Hudson, W.R., Wilkins, T.J., and James, S.J., Effect of cooking and cooling method on the processing times, mass losses and bacterial condition of large meat joints, Int. J. Food Sci. Technol., 25, 657–667, 1990. Califano, A.N. and Zaritzky, N.E., A numerical method for simulating heat transfer in heterogeneous and irregularly shaped foodstuffs, J. Food Process Eng., 16, 159–171, 1993. Chau, K.V. and Gaffney, J.J., A finite difference model for heat and mass transfer in products with internal heat generation and transpiration, J. Food Sci., 55, 484–487, 1990. Erdoˇgdu, F., Modeling of temperature distribution in shrimp, and measurement of its effect on texture, shrinkage and yield loss, thesis, University of Florida, Gainesville, 1996. Erdoˇgdu, F., Balaban, M.O., and Chau, K.V., Automation of heat transfer coefficient determination: development of a Windows-based software tool, Food Technol. Turkey, 10, 66–75, 1998a Erdoˇgdu, F., Balaban, M.O., and Chau, K.V., Modeling of heat conduction in elliptical cross-section: I. Development and testing of the model, J. Food Eng., 38, 223–239, 1998b. Eshleman, W.D., A numerical method for predicting heat transfer in axi-symmetrical shaped solids, thesis, University of Florida, Gainesville, 1976.


Kim, K.H. and Teixeira, A.A., Predicting internal temperature response to conduction-heating of odd-shaped solids, J. Food Process Eng., 20, 51–63, 1997. Kirkpatrick, E.T. and Stokey, W.F., Transient heat conduction in elliptical plates and cylinders, J. Heat Transfer, 80, 54–60, 1959. Manson, J.E., Stumbo, C.R., and Zahradnik, J.W., Evaluation of thermal process for conduction heating foods in pear-shaped containers, J. Food Sci., 39, 276–281, 1974. McLahlan, N.W., Heat conduction in elliptical cylinder and an analogous electromagnetic problem, Philosophical Magazine, 36, 600–609, 1945. Microsoft Corporation, Visual Basic Professional, Version 3.0, Microsoft, Redmond, WA, 1994. Polycast Technology, Thermal Property Data for Acrylic Plexiglass, Polycast, Westport, CT, 1996. Sheen, S., Tong, C.H., Fu, Y., and Lund, D.B., Lethality of thermal processes for food in anomalous-shaped plastic containers, J. Food Eng., 20, 199–213, 1993. Simpson, R., Aris, I., and Torres, J.A., Sterilization of conduction-heated foods in oval-shaped containers, J. Food Sci., 54, 1327–1331, 1363, 1989. Welt, B.A., Teixeira, A.A., Chau, K.V., Balaban, M.O., and Hintenlang, D.E., Explicit finite difference methods for heat transfer simulation and thermal process design, J. Food Sci., 62, 230–236, 1997.


24

Heat Transfer Coefficient for Model Cookies in a Turbulent Multiple Jet Impingement System N. Nitin and M.V. Karwe

CONTENTS 24.1 Introduction 24.1.1 Flow Field in a Submerged Turbulent Jet 24.1.1.1 Free Jet 24.1.1.2 Impinging Jet 24.1.1.3 Flow Field in Turbulent Multiple Submerged Jets 24.1.2 Heat Transfer Characteristics of a Single Impinging Jet 24.1.3 Heat Transfer Characteristics for Multiple Jet Configurations 24.2 Materials and Methods 24.2.1 Experiments (Stationary Mode) 24.2.2 Experiments (Rotational Mode) 24.3 Results and Discussion 24.3.1 Maximum Jet Velocity 24.3.2 Heat Transfer Coefficient 24.4 Conclusions Nomenclature Acknowledgment References

24.1 INTRODUCTION Jet impingement technology is used in several industrial applications such as electronic cooling, cooling of hot metal plates and gas turbine blades, high performance low weight heat exchangers, and driers. In the food industry, jet impingement technology has been used in baking and freezing operations for faster and better processing due to high heat transfer rates with better quality retention. Jet impingement ovens are used to bake pizza


shells, crackers, and cookies and to toast ready-to-eat cereals. Prior studies (Walker and Sparman, 1989) showed that jet impingement ovens could not only reduce the time of processing by a factor of about two, but also produce good quality products in terms of better moisture retention. However, little is known about the detailed transport phenomena associated with multiple jet impingement oven systems. Several researchers have experimentally investigated the heat transfer associated with impinging jets. The major studies in this area have been carried out to understand the fluid flow in a single and multiple jet system. A few studies have focused on heat transfer characteristics. Most of the heat transfer studies apply only to a narrow range of Reynolds numbers, with the major focus on cooling applications over a flat surface. Jets can be broadly classified as submerged or nonsubmerged. When the medium of the jet is the same as the surrounding medium, it is classified as a submerged jet. Submerged jets can be further divided into two categories, free jets and impinging jets. The majority of industrial applications involve impinging jets. In impinging jets, a jet of liquid or gas comes from a nozzle and impinges on a desired surface, while in the case of free jets there is no target surface for impingement. Submerged impinging jets can be further classified as laminar or turbulent jets. When the Reynolds number based on the nozzle diameter of the jet exceeds 3000, it is generally classified as a turbulent jet. In this chapter, we focus on turbulent submerged jets because they are used in food processing operations. cvf(Van Dyke, 1982)

FIGURE 24.1 Laminar region near the nozzle in a turbulent free jet with Reynolds number Re = 30,000. (From Van Dyke, M., An Album of Fluid Motion, The Parabolic Press, Stanford, CA, 1982. With permission.)


shows a typical turbulent jet (Re = 30,000) coming out of a tube of diameter 6.35 mm. It shows a laminar region near the nozzle before the jet becomes turbulent. Since the nature of the flow depends upon the axial downstream distance, the heat transfer characteristic in jet impingement will depend upon the distance of the target plate from the nozzle.

24.1.1 FLOW FIELD

IN A

SUBMERGED TURBULENT JET

24.1.1.1 Free Jet Flow field studies (Gardon and Akfirat, 1965) for a submerged single turbulent free jet have shown that as the jet comes out of a nozzle, there is a mixing of the jet air with the surrounding air with an increase in the width of the mixing region. This is shown schematically in Figure 24.2 (Gardon et al., 1965). In the initial region of about 4–5 nozzle diameters, the mixing of the jet air with surrounding air takes place only at jet boundaries. The mixing results in an exchange of momentum of the jet at the jet boundaries with the surrounding medium. This exchange of momentum decreases the jet velocity at the jet boundary, while the centerline velocity does not change significantly over this initial region, characterized as a potential core region. Beyond this potential core region, the centerline velocity decreases as more and more of the surrounding fluid is entrained. This results in an increase in the

@ Z/D= 2

@ Z/D= 12

@ Z/D= 6

U/ UE

0 0

U/ UE

0

U/ UE

1

1

1

VELOCITY VELOCITY

VELOCITY

TURBULENCE TURBULENCE TURBULENCE

MIXING REGION

0

0.2

U’/ UM

POTENTIAL CORE

0.2

0 U’/ UM

0.2

0 ’

U / UM

FIGURE 24.2 Schematic distribution of velocity and turbulence in an axis-symmetric jet. U = jet velocity; UE = jet exit velocity; UM = r.m.s. value of fluctuating component of velocity; U′ = centerline velocity of jet; Z/D = ratio of axial distance to nozzle diameter. (Adapted from Gardon, R. and Akfirat, J.C., Int. J. Heat Mass Transfer, 8, 1261–1272, 1965.)


D

FREE JET

STAGNATION ZONE

LATERAL SPREAD REGION

PLATE FIGURE 24.3 Schematic diagram of three zones in a single impinging jet.

width of the mixing region. The mixing of the surrounding air with the jet air generates turbulence in the free jet (Gardon and Akfirat, 1964; Gardon and Cobonpue, 1961). Figure 24.2 shows that the turbulence level increases with the increase in jet length. The peak of the turbulence shifts from the outer region of the jet to the center of the jet with downstream distance. 24.1.1.2 Impinging Jet Flow studies (Gardon and Akfirat, 1964; Martin, 1977) have indicated that flow in a turbulent single impinging jet can be divided into three regions: the free jet region, the stagnation region, and the lateral spread region. Figure 24.3 shows these three regions in a single impinging jet. In the free jet region, the flow behavior is the same as discussed above for the free submerged turbulent jet. The impinging jet behaves as free jet up to a distance of about 1–1.5 nozzle diameters from the impinging surface. After this, deceleration of the impinging jet begins as the jet enters into the stagnation region. In the stagnation region there is impingement and deflection of the jet. In this region, the jet experiences a deceleration (Gardon and Akfirat, 1965). The axial velocity decreases rapidly, and the static pressure rises. In the lateral spread region, the negative pressure gradient causes a rapid increase in the radial velocity near the deflection region, and it drops away to a region away from it. This drop in velocity is due to wider area available for the same mass of material to flow and also due to the viscous losses at the wall. 24.1.1.3 Flow Field in Turbulent Multiple Submerged Jets The majority of studies in the flow field of multiple jets geometry for turbulent regime (Re ~ 25,000) have been done using nonsubmerged water jets (Viskanta and Huber, 1994b; Saripalli, 1983). Only a few studies on flow field have been reported using a submerged air jet system. One of the major flow field studies (Marcoft, 1999) using a submerged air multiple jets configuration has been done using the technique of LDA (Laser Doppler Anemometry). It was used to map the velocity field for three- and four-jet configurations. The results showed that the axial velocity contours for multiple submerged air jet configurations show similar trends in the


60

Uz (m/sec)

Y(mm)

40

20

23.94 20.16 16.38 12.61 8.83 5.05 1.28 -2.50 -6.27 -10.05

Jet #3 Jet #2

0

Jet #1 -20

-40 -80

-60

-40

-20

0

20

X(mm) -X +Z

10 mm

FIGURE 24.4 Contours of constant axial (uz) velocity for three jets impinging on a flat plate. Measurements made in a plane 10 mm from the flat plate.

free mixing and stagnation region as discussed for single jet configuration. Figure 24.4 and Figure 24.5 show the details of velocity contours at a distance of 10 mm from the impingement plate for three- and four-jet configurations, respectively. In addition to the characteristics of the velocity field for a single jet, the measured velocity profiles also showed negative velocity regions in the lateral flow regime. These negative velocity regions result because of the interaction between the adjacent jets, which causes flow reversal in the lateral flow regime of the multiple jets. This region has been shown to be of great importance in describing the heat transfer characteristics of multiple jets (Marcoft, 1999). For measurements made close to the flat plate, the magnitude of maximum negative velocity in these regions was almost 50% of the maximum jet velocity measured near the nozzle exit. The range of this negative velocity depends on the nozzle-to-nozzle spacing, velocity of the jets, and nozzleto-plate spacing. Similar results of jet interaction resulting in reverse flow have been shown (Viskanta and Huber, 1994b; Saripalli, 1981) using water jets. Both of these studies have shown the formation of an interaction fountain, i.e., the reverse flow in between the adjacent impinging jets. Figure 24.6 shows the details of the reverse flow in the form of an interaction fountain between the adjacent jets for water jets.


Jet #3 40 Jet #4

20

Uz (m/sec) 21.49 18.18 14.87 11.56 8.25 4.94 1.62 -1.69 -5.00 -8.31

0 Jet #2

-20

Y (mm)

Jet #1 -40

-60

-80

-100 -80

-60

-40

-20

0

20

40

X (mm) -X +Z

10 mm

FIGURE 24.5 Contours of constant axial (uz) velocity for four jets impinging on a flat plate. Measurements made in a plane 10 mm from the flat plate.

INTERACTION FOUNTAIN

INTERACTION FOUNTAIN

FIGURE 24.6 Schematic diagram of reverse flow (interaction fountain) in multiple jet configurations.

24.1.2 HEAT TRANSFER CHARACTERISTICS OF A SINGLE IMPINGING JET The heat transfer characteristics for a single jet impinging on a flat surface show maxima in the magnitude of the heat transfer coefficient in the stagnation region of the jet. The magnitude of these maxima is shown to be a function of the Reynolds number and nozzle-to-plate separation (Gardon and Akfirat, 1964; Martin, 1977). With an increase in Reynolds number and in nozzle-to-plate spacing up to 5–6 nozzle diameters, the magnitude of the maxima in heat transfer coefficient increases (Gardon and Akfirat, 1965). Figure 24.7 (Gardon and Akfirat, 1965) shows the variation of heat transfer coefficient at different nozzle-to-plate spacings. The increase in heat transfer coefficient with the increase in nozzle-to-plate spacing is due to the increase


LOCAL HEAT TRANSFER COEFFICIENT, h (Btu/h ft2 degF)

100

B = 1/8 in. Ree = 11000 8 6

80

B = 1/8 in. Ree = 11000

B = 1/8 in. Ree = 11000

8

5 16

2

zn / B

60

zn / B

zn / B = 1/3

32 2

40

5

6

8 8

16

32 80

80

20

(a) 2 ≤ zn / B ≤ 8 0 -20

-10

0

(b) 8 ≤ zn / B ≤ 80 10

20 -20

-10

0

(c) zn / B = 1/3 10

20 -20

-10

0

10

20

DISTANCE FROM STAGNATION POINT, x/B

FIGURE 24.7 Lateral variation of local heat transfer coefficients between a plate and an impinging two-dimensional air jet at different nozzle-to-plate distances. (From Gardon, R. and Akfirat, J.C., Int. Heat Mass Transfer, 8, 1261–1272, 1965. With permission.)

in level of turbulence developed as a result of mixing of the surrounding air while the centerline velocity is nearly constant (Gardon and Akfirat, 1965). For nozzle-toplate spacing greater than eight nozzle diameters, a decrease in the magnitude of maxima for the heat transfer coefficient around the stagnation region is observed, as shown in Figure 24.7. This has been explained by the decrease in the velocity of the jet with no appreciable increase in the turbulence level with an increase in the distance from the jet nozzle beyond seven nozzle diameters. This clearly shows that the magnitude of the maxima in heat transfer coefficient in the stagnation region is due to two parameters, turbulence and jet velocity. The studies of small nozzle-to-plate spacing, i.e., between 0.25 and 1 nozzle diameters, have shown great potential for cooling applications in the electronic packaging industry (Viskanta and Huber, 1994a). This study showed that at a large Reynolds number (Re ~ 15,000) and small nozzle-to-plate spacing (0.25–1 nozzle diameters), the maximum in the heat transfer coefficient is observed not at the stagnation point as for large nozzle-to-plate spacing (5–6 nozzle diameters), but at a distance away from the center. Figure 24.8 (Viskanta et al., 1994a) shows the variation of local heat transfer coefficient (represented in terms of the dimensionless heat transfer number, Nusselt number (Nu)) for small nozzle-to-plate spacing. Similar trends were also observed by others (Goldstein and Seol, 1991; Ichimiya and Okuyama, 1980; Lytle and Webb, 1994; Saad et al., 1980). For nozzle-to-plate spacing of less than one nozzle diameter, the local heat transfer coefficient in the stagnation region increases as the distance between the plate and the nozzle decreases. The existence of secondary maxima in the local heat transfer has also been indicated for small nozzle-to-plate spacing, whereas there is a monotonous


(a)

Local Nusselt number NuD

120

0.25

100

1.0 80 60 40

H/D = 6.0 20 0

0

0.5

1

1.5

2

2.5

3

2.5

3

r/D (b)

Local Nusselt number NuD

140

0.25

120

1.0 100 80 60

H/D = 6.0

40 20 0

0

0.5

1

1.5

2

r/D

FIGURE 24.8 Effect of separation distance (H/D) on local Nusselt number distributions for Reynolds number (a) 10,300 and (b) 17,100. (From Huber, A.M. and Viskanta, R., ASME J. Heat Transfer, 116, 573, 1994a. With permission.)

decrease in heat transfer coefficient in the lateral region for large nozzle-to-plate spacing. This difference in the variation of heat transfer coefficient at very small nozzle-to-plate spacing (less than one nozzle diameter) has been attributed to the difference in the nature of the jet. For very small spacing the shear layer does not develop, as the jet does not get a chance to mix with the surrounding air. The existence of local maxima near the center of the jet has been attributed to the sudden acceleration of the fluid after impinging on the plate (Viskanta and Huber, 1994a). In the lateral spread region of a single jet, the heat transfer coefficient drops monotonously with the radial distance for nozzle-to-plate spacing greater than five nozzle diameters. For nozzle-to-plate spacing between two and five nozzle diameters, the heat transfer does not drop monotonously, as in the earlier case. There are small


secondary maxima in the heat transfer coefficient around the stagnation point. This point is illustrated in Figure 24.7. Beyond these small secondary maxima in the lateral region for the case of nozzle-to-plate spacing of between two and five nozzle diameters, the heat transfer coefficient drops monotonously in the lateral spread region of the jet. The drop in the heat transfer coefficient is steeper in the case of smaller nozzle-toplate spacing (Gardon and Akfirat, 1965; Martin, 1977). Thus, the magnitude of the heat transfer coefficient is more uniform over a wider area for higher nozzle-to-plate spacing; however, the magnitude of the maxima at stagnation point decreases for large nozzle-to-plate spacing (greater than eight nozzle diameters). For nozzle-to-plate spacing of less than one nozzle diameter, there is a sharp increase in the magnitude of the secondary heat transfer coefficient maxima in the lateral jet region (Figures 24.7 and 24.8). The location of these secondary maxima has been found to be a function of the Reynolds number and plate-to-jet spacing. Studies (Martin, 1977; Viskanta and Huber, 1994a) have shown that for smaller nozzle-to-plate spacing, the average heat transfer coefficient is higher due to existence of these secondary maxima.

24.1.3 HEAT TRANSFER CHARACTERISTICS JET CONFIGURATIONS

FOR

MULTIPLE

Relatively few studies have been carried out to characterize heat transfer in multiple jet configurations. The heat transfer characteristics for multiple jets in the stagnation region are expected to be similar to those of a single jet. In the lateral spread region, some difference in heat transfer characteristics is expected due to interaction of adjacent jets. As described earlier, the fluid flow in the case of the multiple jet system shows an interaction between adjacent jets depending upon the spacing of adjacent jets. This interaction, which induces the reverse flow, has been shown to significantly increase the heat transfer for multiple jet configurations (Lytle and Webb, 1994; Viskanta and Huber, 1994b). The existence of secondary maxima in the lateral region due to jet interaction has also been shown in the study by Viskanta and Huber (1994b). The increase in the heat transfer coefficient has been attributed to increased turbulence due to jet interaction. The objective of our study reported in this chapter was to determine the correlations for average surface heat transfer coefficient for a cookie-shaped object in a pilot scale jet impingement oven. The variation of the heat transfer with position of the model object in the oven, size of the model object, rotation of the oven plate, air velocity, and air temperature was investigated.

24.2 MATERIALS AND METHODS During baking or cooking in a jet impingement oven, simultaneous heat and mass transfer occurs. The resistance to heat transfer in an actual baking operation lies both in the hot fluid and the food material. Since our goal was to determine only the surface heat transfer characteristics, we chose a lumped mass approach. In this approach, the resistance offered by the material to heat transfer was kept negligible, which happens when the Biot number is less than 0.1. Aluminum disks were used as model cookies


in this study. Aluminum has a high thermal conductivity value (200 W/mK). Therefore, the Biot number (=hH/ks) based on the thickness of the disk was expected to be much less than 0.1, where h is the average surface heat transfer coefficient and it is supposed 2 100 and d/D = 1/50, a curved pipe had greater resistance than a straight pipe of the same diameter and length. The resistance to flow was 2.9 times that in a straight pipe at NRe = 6000 (~NRe when flow becomes turbulent). When d/D = 1/15, turbulent flow was seen at NRe ~ 9000, and when d/D = 1/2050, turbulence was seen at NRe ~ 2250 to 3200. Several correlations (for example, Equation (7), developed by Kutateladze and Borishanskii (1966)) have been developed to determine the critical Reynolds number: N Rec = 2300 + 12930 ( r / R )0.3


for 0.0417 < d/D < 0.166

(25.7)

25.1.4 RESIDENCE TIME DISTRIBUTION (RTD) Uniformity of heating is important when considering flow in tubes in the chemical, food, pharmaceutical, and various other industries, as it relates to product quality, cost of production, and other factors. As a Newtonian fluid flows through a straight pipe under laminar flow conditions, a parabolic velocity profile develops. Thus, different fluid elements take different times to pass through the pipe, and the distribution in the time spent by various fluid elements in the pipe is referred to as the Residence Time Distribution (RTD). RTD becomes narrower or broader in a tube depending on the type of flow and type of fluid. In order to achieve uniform heating, a narrow RTD is desired. Some methods for narrowing RTD are to induce turbulence or to use helical pipes. The effect of helical pipes in narrowing RTD is described below. Taylor (1953) described the process by which axial dispersion of a solute in a helical tube was reduced by radial diffusion, which is now usually referred to as Taylor dispersion. Radial mixing due to molecular diffusion is complemented by convective secondary motions in a helical tube. Koutsky and Adler (1964) showed that axial dispersion could be greatly reduced as compared to a straight tube when the tube was formed into a helix. They found that this effect was more pronounced in the turbulent region. The study also indicated that in a laminar flow regime, the fluid elements on the secondary flow streamlines of smaller perimeter moved downstream more rapidly than those on larger streamlines. More narrow and peaked tracer distributions were observed as a result of reduced axial dispersion, especially at high Reynolds numbers. These researchers reported that the transition to turbulent flow (3000 < NRe < 7000) was more gradual and occurred at higher Reynolds numbers in helical tubes when compared to straight tubes. The strength of secondary flow at low curvatures was diminished by the onset of turbulent flow, whereas at high curvatures secondary flow was shown to be still effective in reducing the axial dispersion even at higher Reynolds numbers. They also observed that the axial dispersion in helical tubes in the Reynolds number range of 7,000–40,000 was considerably less than that in laminar flow. Saxena and Nigam (1984) studied the flow inversion in helical coils and utilized the centrifugal force concept to develop secondary flow in laminar flow systems. They reported that a 90° bend located midway in the coiled tube was more effective when compared to 30, 45, and 60° bends in inducing secondary flow, thus narrowing the RTD. They also found that increasing the number of bends (more than three) significantly decreased the RTD. The study revealed that up to a Dean number (NDe) of 1.5, the secondary flow was responsible for narrowing the RTD, whereas above 1.5, the change in the direction of centrifugal force caused RTD narrowing by mixing the fluid elements. Mathematical modeling by Liu and Zuritz (1995) and Sandeep et al. (1995) indicated that RTD in helical tubes was always narrower than the RTD in straight tubes of similar lengths and diameters. Parametric analyses were conducted to ascertain the effects of various system and process parameters for the cases of single and two-phase flows. Sandeep et al. (1997) found that the two counter-rotating vortices that developed in the radial direction in the case of helical flow are responsible for mixing the fluid


elements and reducing the velocity gradient across the tube, which in turn results in a decrease in the RTD. Residence time distribution in a helically coiled tube and a chaotic system, where the tube was formed into an array of bends with the plane of curvature of each bend making a 90° angle with that of the adjacent one, was studied by Castelain et al. (1997) within the Reynolds number range of 800–13,500. These authors reported that for Reynolds numbers higher than 2500, a chaotic system was 20% more effective in minimizing the axial dispersion than a helically coiled tube having the same number of bends. They attributed this further improvement to the chaotic trajectories resulting from the change in the plane of Dean vortices due to the bends of the chaotic system. The two configurations were reported to be comparable in terms of their axial dispersion effect for Reynolds numbers less than 2500.

25.1.5 HEAT TRANSFER COEFFICIENTS Heat transfer for laminar flow in a straight tube (heat exchanger or holding tube) is usually low since very little mixing in the radial direction occurs. However, flow in a coiled tube is fully three-dimensional, and the rate of heat transfer can be much higher than that in a straight tube. This is due to the development of secondary flow (in the directions normal to the main direction of flow) due to the pressure gradient imposed as a result of the centrifugal forces present in the curved section. The secondary flow serves as a means of redistributing fluid elements in the radial direction, thereby transferring heat more efficiently between the bulk of the fluid and the fluid elements near the tube wall. Various studies have been conducted in helical tubes to determine Nusselt number correlations under different flow and system conditions in helical tubes, and some of those correlations are shown below. Seban and McLaughlin (1963) developed the following equations to predict the heat transfer coefficient under laminar and turbulent flow conditions: Laminar flow: f  N Nu = 0.13 N1Pr/3  c N 2Re  8  

1/ 3

with fc being determined from White’s (1929) correlation. Turbulent flow: N Nu =

fc N N 0.4 8 Re Pr

with

fc = 0.023 N −Re0.15 ( r / R ) 0.01

2

for NRe (r/R) > 6 The first order approximation for determination of the Nusselt number for flow in a coiled tube developed by Mori and Nakayama (1965) is as follows: N Nuc N Nus


=

0.1979 0.5 N De z

(25.8)

where z is the ratio of the thickness of the thermal and velocity boundary layer and is given by: 2   77  z = 1 + 1 +  11   4 N 2Pr  

0.5

  

for NPr ≥ 1

and (25.9)

z=

 1   10 2 +  2 − 1 5   N Pr  

0.5

  

for NPr ≤ 1

Ozisik and Topakoglu (1968) developed the following expression for Nusselt number in a curved pipe: N Nu

−1 ˙ 2  C1 r 2  48  V c = ˙  1 − R 2  11  V s

(25.10)

where C1 =

C2  C2  6.95513 N 2Pr    1  165.47 + 7 N Pr + −71.6 +  9.96576 +   264  112  2.8  1.1    2

C2 = NRe /36

and

˙  V r 2  1.541 C 22 c = 1 − + 1.1 C 2 − 1 2  ˙ 48 R  67.2 Vs 

(25.11)

˙ and V ˙ being the mass flow rates in the curved and straight tubes, respectively. with V c s Dravid et al. (1971) conducted numerical studies for flow in a helical tube for small aspect ratios and for the case where axial conduction is negligible in comparison with axial convection (NPe > 20). They developed the following correlation for heat transfer coefficient at distances close to the inlet of the tube: N Nuc N Nus

= 0.447N1De/ 6

(25.12)

At NDe = 125, the heat transfer coefficient in the helical tube is the same as that in the straight tube. It should not, however, be inferred that for lower values of the Dean number the heat transfer coefficient is higher in the straight tube, since at short distances from the inlet the temperature field does not significantly penetrate the secondary flow field. Based on their experimental study, Dravid et al. (1971) developed the following correlation for the asymptotic Nusselt numbers applicable for 50 < NDe < 2000 and 5 < NPr < 175:

[

]

N Nu = 0.76 + 0.65 N De N 0Pr.175


(25.13)

The asymptotic Nusselt numbers for the straight tube case are 3.66 for the case of constant wall temperature and 4.36 for the case of constant wall heat flux. These values are significantly higher for the helical tube case. Oliver and Asghar (1976) developed the following Nusselt number correlation for flow in a helical coil: µ  N Nu  w   µb 

0.14

µ  N Nu  w   µb 

0.14

(

)

for 4 < NDe < 60

(25.14)

(

)

for 60 < NDe < 2000

(25.15)

0.13 = 1.75 N Gz 1 + 0.36 N 0De.25

0.13 = 1.75 N Gz 1 + 0.118 N 0De.5

Janssen and Hoogendoorn (1978) developed the following Nusselt number correlations for different ranges of Dean numbers:

(

)

(

)

N Nu = 1.7 N 2De N Pr N Nu = 0.9 N 2Re N Pr

1/ 6

1/ 6

(

for NDe < 20 and N 2De N Pr

)

0.5

> 100

for 20 < NDe < 100

N Nu = 0.7 N 0Re.43 N1Pr/6 ( r / R ) 0.07

(25.16)

(25.17)

for 100 < NDe < 8300

(25.18)

These researchers concluded that the effect of d/D can be neglected for Dean numbers less than 100 in the fully developed thermal region. They also found that for all cases with N 2De N Pr > 100, the Nusselt number in the fully developed thermal region was proportional to N1Pr/6 and that for the thermal entry region, the Nusselt number was proportional to N1Pr/3 . Abul-Hamayel and Bell (1979) conducted experimental studies of heating of ethylene glycol, distilled water, and butyl alcohol in a helical coil with D/d = 20.2 and developed the following Nusselt number correlation:

(

N Nu

)

3.94   N Gr   = 4.36 + 2.84 2   1 + 0.0276 N 0De.75 N 0Pr.197  N Re   

[

2.78 −1.33 N Gr   µ  0.14   N Gr  N 2De  b  × 1 + 0.9348 2  e   µ b    N De 

6

for 92 < NRe < 5500, 2.2 < NPr < 101, and 760 < NGr < 10 .


] (25.19)

In the above equation, the quantities in the first, second, third, and fourth brackets are the expressions for flow in a straight tube, secondary flow contribution, natural convection contribution, and viscosity correction factor, respectively. Yang and Chang (1994) developed the following Nusselt number correlation as a function of Dean number, Prandtl number, Rayleigh number, and curvature: N Nuc N Nus

= 0.42 N 0De.111 N 0Pr.21 N 0Ra.0513 δ − 0.0974

(25.20)

where δ = r/R, for 10 < NDe < 25,000, 0.7 < NPr < 100, 0 < NRa < 320, and 0.01 < δ < 0.8. Thus, it can be seen that different correlations have been developed for different flow conditions and different tube geometries. It is thus important to look into the range of applicability of various correlations before applying them to different situations.

25.2 EXPERIMENTS In order to compare heat transfer coefficients in straight and helical heat exchangers, we conducted studies using both configurations. The helical heat exchanger used was a Multicoil heat exchanger (VRC Co., Inc., Cedar Rapids, IA). The Multicoil heat exchanger (Figure 25.4) consists of a shell within which two coils of tubes (inner coil and outer coil) are situated. The heating medium (pressurized water) flows in the shell at a very high flow rate and recirculates through the system while the product flows inside the coiled tube(s). Baffles present outside the coils increase the turbulence of the heating medium and thus enhance the outside heat transfer coefficient. Either one or both of the coils can be used, and they can be connected in a series or parallel configuration. A shell and tube heat exchanger that utilizes steam and has an automatic control system to maintain a desired temperature is used to provide the hot water to heat the product. The temperature and pressure of both the product and heating medium are monitored at various locations within the system. The outer shell of the Multicoil heat exchanger is insulated to minimize heat loss to the surroundings. City water was used as the test fluid, and the properties of water (thermal conductivity, density, and specific heat) were calculated using correlations (Popiel and Wojtkowiak, 1998). The first set of experiments involved pumping the product using a piston pump (Model 629A, Marlen Research Corp., Overland Park, KS) through the system (Multicoil heat exchanger, holding tube, and four tubular coolers) at different flow rates (0.19–0.63 kg/sec) and at different set point temperatures (100–150°C). The first two tubular cooling units utilized city water as the cooling medium, while the third and fourth coolers utilized a 30% propylene glycol solution. Temperatures of the product and heating medium were monitored at various locations within the system using type-T thermocouples and recorded in a data acquisition system using the software PC208W (Model CR10, Campbell Scientific, Inc., Logan, UT). The volumetric flow rate of the heating medium was measured using an external flowmeter (Model 1010WP1, Controlotron, Inc, Hauppauge, NY), and the volumetric flow rate of the product was determined by noting the time taken by the product to fill a


FIGURE 25.4 Schematic representation of the Multicoil heat exchanger.

container of known volume. The specifications of the experimental set-up are given in Table 25.1. The holding tube and tubular cooling units did not have any insulation on the outside. The Multicoil heat exchanger is similar to a countercurrent heat exchanger and was modeled as a countercurrent heat exchanger for calculation purposes. Based on the


TABLE 25.1 Specifications of the Experimental Set-Up Multicoil Heat Exchanger Total length of outer coil of Multicoil heat exchanger = 39.6 m Diameter of outer coil of Multicoil heat exchanger = 0.3 m a I.D. of pipe of Multicoil heat exchanger = 0.01 m b O.D. of pipe of Multicoil heat exchanger = 0.013 m Thermal conductivity of the material of the pipe = 16 W/m-K Holding Tube Length of holding tube = 48.8 m a I.D. of holding tube = 0.023 Tubular Coolers Length of each of the four coolers = 6.1 m a I.D. of the inner pipe = 0.023 m b O.D. of the inner pipe = 0.025 m a I.D. of the outer pipe = 0.048 m b O.D. of the outer pipe = 0.05 m Thermal conductivity of the material of the pipe = 16 W/m-K a b

I.D. = inside dimension. O.D. = outside dimension.

recorded temperatures and flow rates, the overall heat transfer coefficient in the Multicoil heat exchanger and the tubular coolers were determined using the following equations: ˙ =m ˙ c p ∆T Q (25.21) ˙ = U A ∆T Q lm lm ˙ is the energy gained by the product. where Q The results of the calculations are summarized in Figure 25.5. It can be seen from this figure that the overall heat transfer coefficient was always higher (more than double) in the Multicoil heat exchanger than in the tubular coolers. Based on the limited experimental runs, there does not seem to be any effect of set point temperature on the overall heat transfer coefficient. The next study was conducted with the objective of determining the effect of the diameter of the helical coil on the overall heat transfer coefficient. In order to determine this, we conducted experiments with water as the product and pumped it through both the inner and outer coils separately, but at the same flow rate. A similar analysis to the one performed above was carried out to determine the overall heat


10.0 9.0

U (W/m2-K)

8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 3

4

5

7

6

8

9

10

Flow rate (gpm) Outer, 93 °C

Outer, 121 °C

Outer, 149 °C

Straight tube

FIGURE 25.5 Overall heat transfer coefficient in helical and straight heat exchangers. 11,000 10,000 9,000

U (W/m2-K)

8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

Flow Rate (gpm) Inner 93

Outer 93

FIGURE 25.6 Overall heat transfer coefficient in the inner and outer coils of the Multicoil heat exchanger.

transfer coefficient. The results of the calculations are summarized in Figure 25.6. It can be seen from this figure that the overall heat transfer coefficient was always higher in the coil of smaller radius than in the coil of larger radius. Another study was conducted to directly compare the amount of energy transferred in the Multicoil heat exchanger and in tubular coolers. In order to perform this, the product (water) was pumped through the same system as described above at a specific flow rate (0.26 kg/sec) and a specific set point temperature (121°C) through one of the coils (outer coil), with hot water flowing through the system at 10.25 kg/sec. The mass flow rates of city water in the first and second coolers were


TABLE 25.2 Comparison of Tubular and Multicoil Heat Exchangers (at 0.26 kg/sec) Multicoil Heat Exchanger Energy gained by product = 118,244 W 2 Overall heat transfer coefficient = 3,889 W/m -K Cooler #1 Energy lost by product = 66,989 W 2 Overall heat transfer coefficient = 2,020 W/m -K Cooler #2 Energy lost by product = 24,701 W 2 Overall heat transfer coefficient = 1,603 W/m -K Cooler #3 Energy lost by product = 11,166 W 2 Overall heat transfer coefficient = 901 W/m -K Cooler #4 Energy lost by product = 7950 W 2 Overall heat transfer coefficient = 927 W/m -K a Net gain in energy based on the inlet and exit temperatures of the product = 6441 W a Net gain in energy based on energy gained in heater and energy lost in coolers = 7438 W a

The difference in the above two values is due to experimental errors and the fact that we neglected heat loss in the holding tube.

1.58 and 0.94 kg/sec, respectively, while the mass flow rates of glycol in the third and fourth coolers were 0.74 and 0.47 kg/sec, respectively. The results of the experiments are summarized in Table 25.2. A similar experiment was conducted at a product flow rate of 8.16 g/min. The only other change in the experimental conditions was that the mass flow rate of city water in the first cooler was 1.71 instead of 1.58 kg/sec. The results of this experiment are summarized in Table 25.3. From Tables 25.2 and 25.3 it can be seen that the energy transferred in the Multicoil heat exchanger was much higher than that transferred in each of the coolers. This is further seen from the fact that the overall heat transfer coefficient in the Multicoil heat exchanger was much higher than that in any of the coolers. This is due to two factors: 1. The outside heat transfer coefficient was much higher in the Multicoil heat exchanger due to the high flow rate of the heating medium and the presence of the baffles. 2. The high inside heat transfer coefficient in the Multicoil heat exchanger was due to the development of secondary flow.


TABLE 25.3 Comparison of Tubular and Multicoil Heat Exchangers (at 0.52 kg/sec) Multicoil Heat Exchanger Energy gained by product = 235,857 W 2 Overall heat transfer coefficient = 6,107 W/m -K Cooler #1 Energy lost by product = 97,680 W 2 Overall heat transfer coefficient = 2612 W/m -K Cooler #2 Energy lost by product = 49,268 W 2 Overall heat transfer coefficient = 2195 W/m -K Cooler #3 Energy lost by product = 24,916 W 2 Overall heat transfer coefficient = 1287 W/m -K Cooler #4 Energy lost by product = 18,895 W 2 Overall heat transfer coefficient = 1246 W/m -K a Net gain in energy based on the inlet and exit temperatures of the product = 47,006 W a Net gain in energy based on energy gained in heater and energy lost in coolers = 45,098 W a

The difference in the above two values is due to experimental errors and the fact that we neglected heat loss in the holding tube.

25.3 CONCLUDING REMARKS Numerous studies have been conducted on secondary flow in helical tubes in order to either understand the fundamental principles of flow and heat transfer in coils or to determine friction factors, pressure drops, extent of secondary flow, axial dispersion coefficient, critical Reynolds number, or Nusselt number, and how various system and process parameters affect them. However, there is no standard correlation or law that can be applied to all scenarios of helical tube flow (unlike in straight tubes); hence, readers are cautioned to carefully review the conditions under which the equations are applicable. Based on the experiments conducted in the Multicoil heat exchanger and tubular cooling units, it can be seen that the overall heat transfer coefficient in the Multicoil heat exchanger is much higher than that in the tubular coolers. It was also seen that the smaller the radius of the coil of the heat exchanger, the higher the heat transfer coefficient. However, it should be noted that the pressure drop in a coil of smaller radius is higher than that in a coil of larger radius. In conclusion, it is noted that the use of helical heat exchangers can result in rapid and relatively uniform processing of a product, thereby resulting in a high quality product.


NOMENCLATURE 2

A cp C1, C2 d dp D Dvertical Ef f g h k L ˙ m

Area, m Specific heat, J/kg-K Constants Diameter of tube, m Particle diameter, m Coil diameter, m Vertical dimension, m Energy loss due to friction, J/kg Friction factor 2 Acceleration due to gravity, m/sec 2 Convective heat transfer coefficient, W/m -K Thermal conductivity, W/m-K Length of pipe, m Mass flow rate, kg/sec

NDe

Dean number N De = N Re r / R

NGr

Grashof number N Gr =

NGz

Graetz number N Gz =

NNu

Nusselt number N Nu =

βf g ρf (Tsurface − T∞ ) D3vertical µ 2f

N Re N Pr x/D

NPe

hD kf Peclet number NPe = (NRe)(NPr)

NPr

Prandtl number N Pr =

NRa

Rayleigh number NRa

NRe

Reynolds number N Re =

N Rec

Critical Reynolds number

˙ Q r R T Tsurface T∞ u U ˙ V x z

Energy transferred per unit time, W Radius of tube, m Radius of helical coil, m Temperature, K Surface temperature, K Bulk fluid temperature, K Average velocity, m/sec 2 Overall heat transfer coefficient, W/m -K 3 Volumetric flow rate, m /sec Distance from entrance, m Ratio of thickness of thermal to velocity boundary layer


C Pf µ f kf = (NGr)(NPr) ρf uD µr

SUBSCRIPTS b c f lm p s w

Bulk fluid temperature Coiled (or helical) tube Fluid Logarithmic mean Particle Straight tube Wall temperature

GREEK SYMBOLS β δ ∆Pf ∆T µ ρ

−1

Coefficient of volumetric thermal expansion, K Dimensionless curvature Pressure loss due to friction, Pa Change in temperature, K Viscosity, Pa-sec 3 Density, kg/m

REFERENCES Abul-Hamayel, M.A. and Bell, K.J., Heat Transfer in Helically Coiled Tubes with Laminar Flow, Presented at the annual meeting of the American Society of Mechanical Engineers (Heat Transfer Division), 1979. Bara, B., Nandakumar, K., and Masliyah, J.H., An experimental and numerical study of the Dean problem: flow development towards two-dimensional multiple solutions, J. Fluid Mech., 244, 339–376, 1992. Berger, S.A., Talbot, L., and Yao, L.S., Flow in curved pipes, Annu. Rev. Fluid Mech., 15, 461–512, 1983. Carlson, V.R., Enhancement of Heat Transfer in Heat Exchangers for Aseptic Processing, Presented at the ASAE Annual International Meeting, Paper No. 916608, 1991. Castelain, C., Mokrani, A., Legentilhomme, P., and Peerhossaini, H., Residence time distribution in twisted pipe flows: helically coiled system and chaotic system, Exper. Fluids, 22, 359–368, 1997. Cheng, K.C., Lin, R.C., and Ou, J.W., Fully developed laminar flow in curved rectangular channels, J. Fluids Eng., 98, 41–48, 1976. Choi, U.S., Talbot, L., and Cornet, I., Experimental study of wall shear rates in the entry region of a curved tube, J. Fluid Mech., 93, 465–489, 1979. Colebrook, C.F., Friction factors for pipe flow, Inst. Civil Eng., 11, 133, 1939. Dean, W.R., Note on the motion of fluid in a curved pipe, Philosophical Magazine, Series 7, 4, 208–223, 1927. Dean, W.R., The stream-line motion of fluid in a curved pipe, Philosophical Magazine, Series 7, 5, 673–695, 1928. Dean, W.R. and Hurst, J.M., Note on the motion of fluid in a curved pipe, Mathematika, 6, 77–85, 1959. Dennis, S.C.R. and Ng, M., Dual solutions for steady laminar flow through a curved tube, Q. J. Mech. Appl. Math., 35, 305–324, 1982.


Dravid, A.N., Smith, K.A., Merrill, E.W. and Brian, P.L.T., Effect of secondary fluid motion on laminar flow heat transfer in helically coiled tubes, AIChE J. 17, 1114–1122, 1971. Duh, T.Y. and Shih, Y.D., Fully developed flow in curved channels of square cross sections inclined, ASME J. Fluids Eng., 111, 172–177, 1989. Fiedler, H.E., A note on secondary flow in bends and bend combinations, Exp. Fluids, 23, 262–264, 1997. Futagami, K. and Aoyama, Y., Laminar heat transfer in a helically coiled tube, Int. J. Heat Mass Transfer, 31, 387–396, 1988. Janssen, L.A.M. and Hoogendoorn, C.J., Laminar convective heat transfer in helical coiled tubes, Int. J. Heat Mass Transfer, 21, 1197–1206, 1978. Kao, H.C., Torsion effect on fully developed flow in a helical pipe, J. Fluid Mech., 184, 335–356, 1987. Koutsky, J.A. and Adler, R.J., Minimization of axial dispersion by use of secondary flow in helical tubes, Can. J. Chem. Eng., 42, 239–246, 1964. Kumar, K.R., Rankin, G.W., and Sridhar, K., Fully developed flow of power law fluids in curved ducts with heat transfer, Numerical Heat Transfer Part A, 16, 101–118, 1989. Kutateladze, S.S. and Borishanskii, V.M., A Concise Encyclopaedia of Heat Transfer, Pergamon Press, London, 1966, p. 114. Liu, Y. and Zuritz, C.A., Mathematical modeling of particulate two-phase flow in a helical pipe, J. Food Process Eng., 18, 321–341, 1995. Manlapaz, R.L. and Churchill, S.W., Fully developed laminar flow in a helically coiled tube of finite pitch, Chem. Eng. Commun., 7, 57–78, 1980. Masliyah, J.H., On laminar flow in curved semicircular ducts, J. Fluid Mech., 99, 469–479, 1980. Moody, L.F., Friction factors for pipe flow, ASME Trans., 66, 671–684, 1944. Mori, Y. and Nakayama, W., Study of forced convective heat transfer in curved pipes, Int. J. Heat Mass Transfer, 8, 67–82, 1965. Muguercia, I., Li, W., and Ebadian, M.A., An experimental investigation of the secondary flow inside a helicoidal pipe, Contrib. Papers Fluids Eng., 170, 19–24, 1993. Nandakumar, K. and Masliyah, J.H., Bifurcation in steady laminar flow through curved tubes, J. Fluid Mech., 119, 475–490, 1982. Oliver, D.R. and Asghar, S.M., Heat transfer to Newtonian and viscoelastic liquids during laminar flow in helical coils, Trans. Inst. Chem. Eng., 54, 218–224, 1976. Ozisik, M.N. and Topakoglu, H.C., Heat transfer for laminar flow in a curved pipe, J. Heat Transfer, 90, 313–318, 1968. Patankar, S.V., Prediction of laminar flow and heat transfer in helically coiled pipes, J. Fluid Mech., 62, 539–551, 1974. Popiel, C.O. and Wojtkowiak, J., Simple formulas for thermophysical properties of liquid water for heat transfer calculations (from 0°C to 150°C), Heat Transfer Eng., 19, 87–101, 1998. Sandeep, K.P., Zuritz, C.A., and Puri, V.M., Velocity and Temperature Profiles in Two-Phase Non-Newtonian Flow in Conventional and Helical Holding Tubes, Presented at the ASAE Annual International Meeting, Chicago, June 18–23, 1995, Paper No. 956534. Sandeep, K.P., Zuritz, C.A., and Puri, V.M., Residence time distribution of particles during two-phase non-Newtonian flow in conventional as compared with helical holding tubes, J. Food Sci., 62, 647–652, 1997. Saxena, A.K. and Nigam, K.D.P., Coiled configuration for flow inversion: its effect on residence time distribution, AIChE J., 30, 363–368, 1984. Seban, R.A. and McLaughlin, E.F., Heat transfer in tube coils with laminar turbulent flow, Int. J. Heat Mass Transfer, 6, 387–395, 1963.


Taylor, G., Dispersion of soluble matter flowing slowly through a tube, Proc. R. Soc. London, Series A, 219, 186–203, 1953. Taylor, G.I. and Yarrow, F.R.S., The criterion for turbulence in curved pipes, Proc. R. Soc. London, Series A, 124, 243–249, 1929. Thomson, D.L., Bayazitoglu, Y., and Meade, A.J., Jr., Low Dean number convective heat transfer in helical ducts of rectangular cross-section, Trans. ASME J. Heat Transfer, 120, 84–91, 1998. White, C.M., Streamline flow through curved pipes, Proc. R. Soc. London, Series A, 123, 645–663, 1929. White, C.M., Fluid friction and its relation to heat transfer, Trans. Inst. Chem. Eng., 18, 66–86, 1932. Yang, G. and Ebadian, M.A., Mixed convective flow heat transfer in a vertical helicoidal pipe with finite pitch, Comput. Mech., 14, 503–512, 1994. Yang, G. and Ebadian, M.A., Effect of pitch on fully developed turbulent flow in the helicoidal pipe, ASME Thermal Eng. Conf., 1, 425–432, 1995. Yang, R. and Chang, S.F., Combined free forced convection for developed flow in curved pipes with finite curvature ratio, Int. J. Heat Fluid Flow, 15, 470–476, 1994.


26

Relating Food Frying to Daily Oil Abuse. Part I. Determination of Surface Heat Transfer Coefficients with Metal Balls K.S. Pannu and M.S. Chinnan

CONTENTS 26.1 Introduction 26.2 Materials and Methods 26.3 Results and Discussion 26.4 Conclusions Nomenclature Acknowledgments References

26.1 INTRODUCTION During immersion frying, convective heat transfer occurs at the food–oil interface (Singh, 1995; Sahin et al., 1999; Costa et al., 1999). The mode of heat/mass transfer in deep fat frying is rather complex because it is difficult to precisely define and/or quantify the vigorous movement of water vapor bubbles escaping at the food surface. An added complication is that the thermophysical properties of the food material continually change due to moisture depletion and temperature change (Costa et al., 1999; Hallstrom et al., 1988). Data on the convective surface heat transfer coefficients (h values) obtained during food frying through the use of metal transducers, food, and model foods have been reported in the literature (Sahin et al., 1999; Costa et al., 1999; Pannu and Chinnan, 1999; Dincer, 1996; Miller et al., 1994). Selected results are presented in Table 26.1 to show the large range that exists for these reported results. For the development of predictive models, it is usually assumed that the h value is constant during immersion frying. Comprehensive mathematical models were developed by Farkas et al. (1996a, b) and Farid and Chen (1998) to quantify heat


TABLE 26.1 Some Reported h Value Data Obtained for Food Frying 2 h Value (W/m °C)

~110 90–200 250–300 250–600 300–650 a ~14 a

Temp. (°°C)

Material/Shape

Source

140–160 150–190 170–190 140–180 140–180 180

Potato/finite cylinder Model food/finite slab Metal/sphere Potato/French fry Metal/French fry Potato/long cylinder

Pannu and Chinnan (1999) Sahin et al. (1999) Miller et al. (1994) Costa et al. (1999) Costa et al. (1999) Dincer (1996)

Heat and mass transfer have been decoupled; h-value is only for heating.

and mass transfer during food frying. Large variations in h value data obtained during frying, with food vs. without food, have led researchers to use two h values for the development of predictive models (Farkas, 1996a; Costa et al., 1999). One h value is used in the absence of bubbling and another for the bubbling phase. It is also assumed that a nonbubbling phase is followed by an enhanced bubbling phase, and that at the end there is reduced bubbling as less moisture is evaporated. However, this is not true when commercially prepared frozen foods are deep fried. Under such conditions there is an instantaneous condition of high turbulence or, more specifically, forced convection conditions are created as the surface ice melts, producing a vigorous bubbling action. Costa et al. (1997) also reported that the bubbling action increases to a maximum value prior to decreasing, and this can be related to the water loss rate. Thus, bubbling may be expected to have an influence on h values. In the absence of bubbling, h value data reported in the literature, and obtained through the use of metal trans2 ducers, range between 220 and 300 W/m °C (Miller, et al., 1994; Tseng et al., 1996; Pannu and Chinnan, 1999). Fellows (1996) provided similar data for the nonbubbling 2 phase, along with h values of 800 to 1000 W/m °C for the surface boiling conditions. Based on product surface temperatures, Hubart and Farkas (1998) predicted that for 2 frying at 180°C, the h value started at 300 W/m °C, increased to a maximum value 2 2 of 1100 W/m °C and then decreased to below 200 W/m °C after 900 sec of frying. Pannu and Chinnan (1999) conducted a heat transfer study by heating finite potato cylinders (with a diameter and length of 2.54 cm) in hot oil at 70 and 90°C (no moisture loss) and compared these results with those obtained during frying at 160°C. Their results show that the heating rate index values obtained at the food center and the mid-plane (between center and surface) were identical for the first two cases. In addition, similar values were obtained under frying conditions for the first 120 sec at the mid-plane and for 200 sec at the food center. Based on these and additional data for moisture loss, the h value was computed to range between 100 2 and 130 W/m °C. These values are similar to those reported by Sahin et al. (1999), 2 but lower than the h value of about 250 W/m °C obtained with metal transducers. Additional details on methods that can be employed to obtain a value for the overall


convective surface heat transfer coefficient (U) from moisture loss data are provided in Part 2 of our study (Pannu and Chinnan, Chap. 27, this volume). Sahin et al. (1999) used a model food made from aluminum wool and mashed potatoes to conduct their study. Thin square slabs (0.3 × 5 × 5 cm) were vertically suspended in hot oil to minimize the variability of h values on product geometry. In essence, h values were determined by solving the conduction equation after compiling a database on transducer moisture loss and product temperature as a function of time, and by varying the thermal properties as a function of temperature and moisture. The h values obtained by Sahin et al. (1999) show a stepwise increase 2 from 90 to 200 W/m °C as oil temperature increased from 150 to 190°C. In comparison, Costa et al. (1999) reported that h values are time dependent. Data with metal transducers show that h values peak 10 to 40 sec after frying begins and thereafter start to decline. For potato crisps the peak occurs before that for French fries, and the peak also occurs earlier as the frying oil temperature increases, indicating that h values are tied to the moisture evaporation characteristics. Essentially, the peak is related to the high moisture evaporation regions because higher moisture loss rates induce more bubbling and provide higher h values. Thus, peak h values occur shortly after immersion for the thin potato crisps. This suggests that frozen foods should provide the highest h value immediately after they are immersed into the hot oil. The experimental data of Costa et al. (1999) showed peak h values 2 2 of about 750 W/m °C (after 15 sec of frying) and 650 W/m °C (after 35 sec of frying) for crisps and French fries, respectively, at a frying temperature of 180°C. However, the peak h value measured directly from French fries at a processing 2 temperature of 140°C was about 600 W/m °C and higher than the value of 440 2 W/m °C obtained with the metal transducer. Data in the literature suggest that the oil viscosity, heat capacity, and surface tension are the primary physical properties that may influence the product heating rate (Blumenthal, 1991; Miller et al., 1994). Blumenthal (1991) outlines the surfactant theory and suggests that the heat transfer rate from oil to food can change as the oil quality degrades through use. It is clearly stated that during food frying heat is transferred from the nonaqueous medium (oil) into the mostly aqueous medium (food). Thus, during food frying there is an oil–water interface that does not exist when metal transducers are used to ascertain the h value. Therefore, metal transducers may not register changes in h values due to the buildup of surfactants. This is supported by results presented by Tseng et al. (1996), who showed that the h value is not affected until the oil discard point is reached. Surfactants should be regarded as wetting agents that increase the contact time between oil and food (Blumenthal, 1991) through the oil–water interface. Therefore, excessive darkening and drying at the food surface as a result of oil breakdown may be related to the above phenomenon and not to h values obtained with metal transducers. Parameters commonly used to define discard point are foaming due to buildup of surfactants, oil color, smoke point, dielectric properties, free fatty acid (FFA) levels, and total polar compounds. The first three are visual parameters and probably the ones most commonly used. Miller et al. (1994) used h value data to study the influence of temperature and oil type on oil use (abuse) and found that the major effect was due to viscosity. However, the differences in h values were rather small.


Further, although the transient heating equations indicate that h values for heating and cooling should be similar, no comparative data are available. The objectives related to this part of our research were to estimate h values with metal balls in a range of food frying conditions. Specific objectives were to: determine h value without food; compare h value data for heating vs. cooling in the same temperature range; determine h value during the frying of commercial grade frozen French fries (par-fries) and breaded chicken fillets; observe the influence of loading condition (less food vs. more food); and study the influence of oil abuse on h value data.

26.2 MATERIALS AND METHODS A schematic of the equipment and tools required to conduct this research is presented in Figure 26.1. Although the picture shows one metal ball suspended in the pot fryer, four metal balls were simultaneously immersed into the hot (or cold) oil to obtain the time–temperature data. A fine wire (36 gauge, type K) thermocouple was positioned at the geometric center of each ball. To accomplish this task, a 0.8-mm hole was drilled into each ball in the central axis up to a depth equivalent to half the ball diameter. Omegabond 200 (Omega Engineering Inc., Stamford, CT), a high temperature, high thermal conductivity epoxy cement, was used to bond the thermocouples in place. The diameter of the small aluminum and stainless steel balls used in this study was 1.27 cm (0.5 inch), and the large balls had a diameter of 2.54 cm (1.0 inch). The four balls were suspended on a frame to facilitate their simultaneous immersion into the frying medium and were identified as Al- 1 2 , SS- 1 2 , Al-1 and SS-1 (Figure 26.2), with 1 2 and 1 indicating ball diameters of 1.27 cm and 2.54 cm, respectively. The fifth thermocouple used to measure the oil temperature was also attached to the frame and positioned at the same height as the balls. Kitchen pot fryers (3.5 l capacity) were employed for this study, and commercial grade hydrogenated soybean oil was used as the food frying medium. Commercial grade, frozen (for the food service sector) breaded uncooked chicken fillets (~85 g of breast meat) and par-fries (1-cm thick French fries) were used. A 1:7.5 food-to-oil ratio was used, i.e., ~340 g of food were fried in 2.6 kg

Thermocouple wire: Type K - 36 gauge

Pot Fryer

Metal ball FIGURE 26.1 Schematic of “tools” used to conduct the frying study.


00

200

40

80

120

Oil Al-1/2

150

Al-1

-1

SS-1

SS-1

Ln (T. R.)

Temp. (C)

SS-1/2

Abused Oil FFA > 1.5

50

Al-1

-2

-3 SS-1/2

-4

0 0

20

40 Time (sec) 80

100

120

Al-1/2

Time (sec)

FIGURE 26.2 Typical temperature data registered by the metal balls during heating in oil at 190°C. Data in the right graph were used to obtain h value data. (Al and SS refer to aluminum and stainless steel, respectively; suffix 1 2 or 1 refers to the metal ball diameter in inches.)

of oil. Food batches were fried every 20 min at a temperature of 180°C for 5 min (cook cycle) followed by drain/cool for 5 min (Table 26.2). The 10 min gap between batches provided time for preparation of the next batch and permitted the pot fryers to return to the initial set temperature. Twenty food batches were fried daily over a 7.5 h period, and make-up oil was added after 10 batches had been fried. At the end of each day, the frying oil was filtered to remove debris and charred materials, and make-up oil was added to maintain the 2.6 kg level after a 50 ml oil sample was removed for analysis (Table 26.2). Days 1, 4, and 7 were chosen to conduct the variable food mass study using six batches of French fries and 14 batches of chicken fillets. The frying protocol for the variable food mass study is detailed in Part 2 (Pannu and Chinnan, Chap. 27, this volume). On days 2, 3, 5, 6, and 8, two batches of French fries and 18 batches of chicken fillets were fried, as shown in Table 26.2. The overall mass of food fried each day was approximately 6.7 kg, with the normal loading condition defined as 340 g of French fries or four chicken fillets. For a few experiments, chicken fillet and French fry pieces were instrumented to obtain temperature profiles. Fine wire (36 gauge, type K) thermocouples were positioned at the geometric center of a 7-cm French fry and a chicken fillet. Thermocouples were also inserted close to the food surface. Food surface for the chicken fillet represented the top part of the breading layer. The instrumented French fry was placed on top of the four chicken fillet pieces and subsequently processed in 180°C oil. Experiments with metal balls were conducted at varying intervals to cover the 8-day test period. Temperature data from the metal balls were recorded at 0.5 sec intervals. The exterior food temperature data were recorded prior to the commencement of food frying, i.e., in between batches. The experiments with food were conducted by immersing the metal transducers 60 to 90 sec after commencement of food frying. A time delay was employed for safety reasons because of enhanced bubbling and oil splattering at the commencement of frying due to ice melt conditions.


TABLE 26.2 Example of Experimental Protocol Used in Frying Studies File Name: ABC-2 Frying Cycle 2 of 8 days Weight of Empty Fryer: 1.612 kg Batch # 1 2 3 4 5 6 7 8 9 10

Time

Food (g)

9:00 325.6 9:20 332.8 9:40 348.3 10:00 340.7 10:20 310.2 10:40 319.9 11:00 313.2 11:20 350.4 11:40 350.6 12:00 346.5 TOTAL W1 = 3338.2

Product CH-4 CH-4 CH-4 CH-4 CH-4 CH-4 CH-4 CH-4 FF CH-4

Batch #

Date: April 30, 1999 Weight of Oil: 2.615 kg Time

Food (g)

11 13:20 356.8 12 13:40 308.5 13 14:00 310.7 14 14:20 313.3 15 14:40 313.9 16 15:00 340.5 17 15:20 351.4 18 15:40 324.8 19 16:00 344.4 20 16:20 407.3 TOTAL W2 = 3371.6

Product CH-4 CH-4 CH-4 CH-4 CH-4 CH-4 CH-4 CH-4 FF CH-4

Note: FF: French fries; CH, chicken. 1) Weight of fryer + oil + basket: 2) Weight of fryer + oil + basket: a) Oil loss = 1–2 3) Weight of fryer + oil + basket: 4) Weight of fryer + oil + basket: b) Oil loss = 1–2 c) Total oil loss = a + b d) Total food fried = W1+W2

4440 4412 28 g 4440 4415 25 g 53 g 6710

g at start g at noon g after oil addition g at end

g

Note: Frying basket weight = 4440 – (1612 + 2615) = 213 g

In addition, when 340 g of frozen food are rapidly immersed in hot oil, very unstable processing conditions exist in the beginning, resulting in a rapid drop in oil temperature. Such conditions make it difficult to obtain accurate time–temperature data during the first 60 sec of frying. Further, the metal balls were immersed in cold oil (room temperature) at all times when not in use. This was done to ensure that the balls had a uniform initial temperature. Thus, for the cooling experiments, the balls were returned to the cold oil reservior (also 2.6 kg of oil) after the balls had attained steady temperature in the heating cycle. Time–temperature data for this study were recorded using the HP34970 (Hewlett Packard, Palo Alto, CA) data acquisition unit. The natural logarithm (ln) of the dimensionless temperature ratio was plotted against time to provide a value for the inverse slope and used to determine h value by the lumped capacity method. Equation (26.1) was used to compute h values from temperatures registered at the geometric center of the metal balls. The terms A, ρ, Cp and V represent the solid surface area, density, specific heat, and volume, respectively. Subscripts for


temperature (T) listed as a, i, and (t) denote ambient (oil), initial (metal ball), and the variable time-dependent temperature (metal ball), respectively. Ta − T( t ) −(hA/ρCpV)t = T. R. = θBall = e Ta − Ti

(26.1)

Thermal property data for ρ and Cp used to estimate the h values were (Holman, 1976): 3 ρStainless steel = 7897 kg/m CpStainless steel = 452 J/kg°C 3 ρAluminum = 2707 kg/m Cp Aluminum = 896 J/kg°C

26.3 RESULTS AND DISCUSSION Typical time–temperature data obtained during the heating of metal balls in oil are shown in Figure 26.2. The graph on the right shows that there were no lag factors, and straight lines were obtained when the natural logarithm of the dimensionless temperature ratio was plotted vs. time. The data presented also show that the smaller metal balls heated faster and that aluminum balls registered accelerated heating in comparison to stainless steel balls. Table 26.3 summarizes the data for h values obtained under free convection heating conditions. The results show that transducer material and ball diameter do not appear to appreciably influence the h value data. However, if average values are considered, the 2 large and small balls provide mean h values of 230 and 250 W/m °C, respectively. These results are similar to those reported by Miller et al. (1994). The low Biot numbers (Bi = h R/k) of 0.015 and 0.06 obtained for the large aluminum and stainless steel balls, with thermal conductivity, k, of 204 and 50 W/m°C, respectively, indicate minimal internal resistance to heat transfer and justify the use of the lumped capacity approach. Results presented in Table 26.3 also show that the h values are not influenced by oil abuse, as FFA values increased from a low of 0.05 (fresh oil) to a high of 1.55 on day 8.

TABLE 26.3 Comparison of h Value Data for Fresh vs. Used Frying Oil (Free Convection) 2 h Value (W/m °C)

Day 1 2 4 6 8

FFA

Temp. (°°C)

Al- 1 2

SS- 1 2

Al-1

SS-1

dTl + m p C pp dt dt

(31.15)

The following are assumed for deriving Equation (31.15): uniform initial and transient temperatures for liquid, constant heat transfer coefficient, constant thermal and physical properties for both liquid and particles, and no energy accumulation in the can wall. The second term on the right side of Equation (31.15) can be described as the heat transferred to particles from the liquid through the particle surface, and Equation (31.15) can be rewritten as: UA c (Tr − Tl ) = m l C pl

dTl + h fp A p ( Tl − Tps ) dt

(31.16)

It is also assumed that the particle receives heat only from the liquid and not from the can wall when it impacts, i.e., heat is transferred first from the can wall to the liquid and then to the particles. The transient heat flow in a particle immersed


in fluid can be described by the following partial differential equations (Jaluria and Torrance, 1996): ∂T ∂t

(31.17)

t=0

(31.18)

t > 0 at the surface

(31.19)

∇.k∇T = ρC p The initial and boundary conditions are: T = Ti −k

∂T = h fp (T − Tl ) ∂n

where n is the normal at the surface taken positive outward. Using the finite difference method, the fluid-to-particle heat transfer coefficient, hfp, can be estimated by solving the governing partial differential equations (Equations (31.17–19)) of conduction heat transfer. The inverse heat transfer approach, where surface heat flux is estimated using one or more measured temperature histories inside a heat-conducting body, is used to estimate hfp values. The overall heat transfer coefficient, U, can be determined by integrating Equation (31.16) over the set process time (Ramaswamy and Sablani, 1997a,b). In order to calulate U, it is necessary to have the transient temperatures of the liquid and particle surface. In Equation (31.16), the experimentally measured value of the liquid is used, and the transient temperature of the particle surface is computed from Equations (31.17–19). Once the convective heat transfer coefficients (U and hfp) are known for a range of processing conditions, the equations mentioned above (31.15–31.19) can be used to predict the liquid and particle center temperatures. The predicted time–temperature data can then be used to establish the thermal process schedule. In situations where the associated viscosities of the liquid are high (as in the case of highly viscous Newtonian or non-Newtonian carrier fluids), the liquid portion tends to have only limited convection except at high rotation speeds. The system then moves predominantly towards conduction heating behavior, with characteristically nonuniform temperature distribution throughout the container. Under such a scenario, the concept of U and hfp is difficult to recognize; however, the concept of fh and jch would still be valuable for comparative purposes and for the purpose of establishing thermal processing schedules. Hence, in this chapter, we have compared all conditions of rotational processing involving both Newtonian and non-Newtonian liquids using the fh and jch concept.

31.3 MATERIALS AND METHODS Water and high temperature bath oil (model 100 CST at 38°C, Fisher Scientific Ltd., Montreal), giving different viscosity levels, were used as Newtonian model liquids. Guar gum, a cross-linked waxy starch with commercial applications in several food preparations (soups, gravies, and pudding) was used to prepare non-Newtonian


test samples containing 0–1% (w/v) gum. The gum solutions were made 12 h before the experiment to allow the gum to hydrolyze. The rheological properties of the prepared solutions were determined with a Haake RV2 concentric cylinder rotation viscometer. 3 Nylon spheres (density = 1128 kg/m , heat capacity = 2073 J/kgK, thermal conductivity = 0.369 W/mK) of different diameters (0.01905, 0.02225, and 0.025 m) were used as particles. Can liquid temperature was measured at the geometric center of the cans using CNS needle type thermocouples. For the purpose of measuring particle transient temperatures, a fine hole was drilled to the center of the spherical particle, which was filled with a 1:1 mixture of epoxy resin and hardener. A fine wire (0.0762 mm diameter) copper-constantan thermocouple was inserted into the center. The thermocouple-equipped particle was mounted inside the can using brass connectors (stuffing box for plastic pouches C-5.2, Ecklund Harrison Technologies, Inc., Cape Coral, FL) at half the height of the can. The thermocouple wire attached to the particle was long enough (half the can height) to allow particle movement inside the can, and short enough to avoid entangling with the needle-type thermocouple used for measuring the liquid temperature. In this type of arrangement, the particle was fairly free to move during processing. The thermocouple-equipped particle was positioned near the geometric center of the can. The other ends of both thermocouples were connected to a 32-circuit slip ring assembly through 24 AWG copper-constantan thermocouple wire (Omega Engineering Corp., Stamford, CT). The thermocouple inputs were recorded at 15-sec time intervals using a data acquisition system and LABTECH NOTEBOOK (Laboratory Technologies Corporation, Wilmington, MA) software or an HP datalogger (HP3497/HP34901, Hewlett Packard, Loveland, CO). Cans of size 307 × 409 (0.0873 m diameter × 0.116 m height) containing the test liquid and multiple particles were subjected to end-over-end rotational processing in a pilot scale rotary, single cage, full water immersion retort (Stock RotomatPR900; Hermann Stock Maschinenfabrick, Germany). A common air over pressure of 70 kPa was used for all test runs with the Rotomat. The cans were arranged in vertical orientation (can axis perpendicular to axis of rotation) to give end-over-end rotation to the cans. In some test runs, as opposed to the end-over-end rotation in a continuous fashion, the cage was rotated for a few rounds in one direction, stopped and rotated in the opposite direction giving an oscillatory mode of operation for the end-over-end agitation. With Newtonian liquids (Sablani, 1996), three experimental designs were selected to study the influence of different system and product parameters. The experimental variables used were: three particle concentrations (20, 30, and 40% v/v); three rotational speeds (10, 15, and 20 r/min); and two kinematic viscosities −6 −4 2 (1.0 × 10 and 1.0 × 10 m /sec). Three rotational speeds (10, 15, and 20 r/min), −6 −4 2 two kinematic viscosities (1.0 × 10 and 1.0 × 10 m /sec), and three particle diameters (0.01905, 0.02225, 0.025 m) were selected to study the influence of particle size. A can head space of 0.01 m, a radius of rotation of 0.19 m, and a retort temperature of 120°C were used as fixed parameters. In order to study the influence of mode of end-over-end agitation, the experimental variables were: three radii of rotation (0, 0.13, and 0.26 m); three speeds of rotation (8, 16, and 24 r/min); and


two modes of rotation (continuous and oscillatory). In this case, the particle diameter of 0.025 m, a high temperature bath oil, a can head space of 0.01 m, and a retort temperature of 120°C were used as fixed parameters. A full factorial experimental design was employed in each of the three studies. Four replicate cans were kept at a given radius of rotation, and each processing condition was repeated twice. With non-Newtonian liquids (Krishnamurthy et al., 2000), nine cans at three radii of rotation (0, 0.1, and 0.19 m, three replicates) were loaded into the cage, and the rest of the space was filled with dummy cans containing water. Five retort temperatures (110, 115, 120, 125, and 130°C), five rotational speeds (0, 5, 10, 15, and 20 r/min), five particle concentrations (10, 15, 20, 25, and 30% v/v), and five guar gum concentrations (0.00, 0.25, 0.50, 0.75, and 1.00% w/v guar gum) were employed as system (operating) variables. A central composite rotatable design (CCRD) was used for the experiments.

31.4 DATA ANALYSIS Since small variations in initial temperature and heating medium temperature were unavoidable, liquid and particle transient temperatures were normalized to an initial temperature of 20°C and the respective retort temperature (110, 115, 120, 125, or 130°C). Heat penetration parameters (fh and jch) were evaluated from the plots of logarithms of differences in the retort temperature and the temperature of liquid/ particle vs. time. The heating rate index and heating lag parameters for liquid and particle were analyzed using statistical analysis software available on the McGill University (Montreal) computer network. An analysis of variance (ANOVA) procedure was used to evaluate the level of significance of system and product parameters and their interactions. Process lethality (Fo) and cook value (Co) were calculated for each processing condition by numerical integration of time–temperature data (z = 10°C and reference temperature 121.1°C for Fo, and z = 33°C and reference temperature 100°C for Co):

∫

Fo = 10 ( T −121.1)/ z dt

∫

C o = 10 ( T −100 )/ 33 dt

(31.20)

When the process lethalities of test runs are not constant, as in the present study, the ratio Co/Fo gives a relative parameter for comparing the degree of cooking at various retort temperatures, with higher ratios indicating more severe cooking conditions (Ramaswamy et al., 1993).

31.5 RESULTS AND DISCUSSION 31.5.1 TEMPERATURE PROFILES Typical linear and semilogarithmic time–temperature profiles of the can liquid and particle center are shown in Figure 31.1. The retort come-up times varied from 2 to 4 min and showed logarithmic come-up profiles. Initially, heat is transferred from the heating medium to the can liquid through the can wall and then to the particle.


140

2.5

120

2.0

100

1.5

Log (Tr – Tpih) Log (Tr – Tih)

Log(Tr – T)

Temperature (°C)

Liquid

80 60 40

1.0 fh 0.5 0.0

Retort

0

Liquid

20

Particle center jch = (Tr – Tpih)/(Tr – Tih)

5

10

15

-0.5

Particle center -1.0

0 0

10

5

15

20

Time (min)

Time (min)

FIGURE 31.1 Typical time temperature profiles of liquid and particle.

In the early part of heating, particle center temperature was lower than the liquid temperature; this is due to convective heat transfer resistance at the particle surface and conductive resistance inside the particle. The liquid temperature was lower than the retort heating medium temperature due to conductive resistance of the can wall and to external and internal convective resistance at the can wall surface. In the latter part of heating, the difference between liquid and particle temperatures generally decreased as these temperatures approached the heating medium temperature.

31.5.2 HEATING BEHAVIOR OF NEWTONIAN LIQUID/PARTICLE MIXTURES Analysis of variance of results with Newtonian liquids showed that all factors under study affected fh for both liquid and particle, except the rotation effect on particle fh (Table 31.1). All factors were found to influence (p < 0.05) liquid and particle jch except the effects of radius of rotation and mode of rotation on particle j and radius of rotation effect on liquid jch. Of the six factors considered, only viscosity influenced the liquid Co/Fo (p < 0.005). All factors influenced particle Co/Fo (p < 0.05), except radius of rotation. Although two-way interaction effects were significant in some instances, their overall contribution to the main effects was very small. The effects of different factors considered on the heating rate index of the particle and liquid are shown in Figure 31.2. The fh value of liquid decreases with an increasing rotational speed (17% decrease between 10 and 20 r/min), presumably due to the enhanced mixing resulting in faster heat transfer rates. Increasing rotational speed from 10 to 20 r/min decreased the particle f value by 11%. Moving cans away from the center of the cage to 0.13 or 0.26 m radius of rotation did not influence (p > 0.05) either liquid or particle fh. From a thermal


TABLE 31.1 Analysis of Variance Results (% Sum of Squares and Significance Level) Showing the Influence of System and Product Parameters on Three Variables Considered (fh, jch, and Co/Fo) while Employing Newtonian Liquids A: Parameters for Liquid Portion Rotational speed (RS) Mode of rotation (MR) Radius of rotation (RR) Viscosity (V) Particle concentration (PC) Particle size (PS) B: Parameters For Particles Rotational speed (RS) Mode of rotation (MR) Radius of rotation (RR) Viscosity (V) Particle concentration (PC) Particle size (PS)

Effect of RS, V, PC fh

Effect of RS, V, PS

Co/Fo

fh

a

1.7 — —

3.2 — —

a

a

87.9 —

a

4.8

a

9.4

a

2.7 — — a 18.7 —

a

13.7

jch a

11.0 — —

87.0 a 2.1

a

5.3 a 29.4

8.7 1.9

—

—

—

1.7 — — a 78.6 a 11.7

1.31 — — a 4.7 a 39.4

2.9 — — a 74.0 b 12.4

3.6 — — a 8.0 —

—

—

—

9.1

2.5 — —

a

a

a

Effect of RS, MR, RR

jch

Co/Fo

fh

1.7 — —

1.0 — —

71.9 a 13.8 a 2.0

13.9 a 37 2.0

— —

— —

— —

—

—

—

a

17 —

13.1 —

a

a

a

3.9

b

5.5 — — a 73.4 —

a

9.5

Co/Fo

jch

a

68.4 a 19.0 0.6 — —

a

30.0 2.0 2.8 — —

a

—

—

a

a

2.4 0 2.6

a

66.0 a 21.0 0.5 — — —

Note: RS, rotational speed; MR, mode of rotation; RR, radius of rotation; V, viscosity of fluid; PC, particle concentration; PS, particle size. a b

p < 0.005. p < 0.05.

processing design point of view, this is a desirable situation. The smaller the influence of radius of rotation, the higher the uniformity in terms of lethality achieved in cans placed at different locations in the retort for processing. Shifting from the continuous mode of end-over-end rotation to the oscillatory mode increased both liquid and particle fh values by 10%. The fh values of liquid and particle were higher at all the processing conditions when high viscosity fluid oil was used; this could be due to the thicker boundary layer (Sablani and Ramaswamy, 1997a). The temperature of the water reached the heating medium temperature more rapidly, indicating a faster rate of heat transfer. Increasing the level of particle concentration from 20 to 40% increased the average fh value of the particle by 28%, but increased that of the liquid only by 5%. Increasing particle concentration resulted in a packed bed situation, reducing the free movement of particles, thus increasing the fh value for the particle. Particle motion and mixing have an important effect on the associated heat transfer rates


min 14

Rotation Speed

12 10

10 rpm

8

15 rpm

6

20 rpm

4

min 14 12 10

Continuous

8

Oscillatory

6

2

4 2

0

0

Liquid

min 14

Liquid

Particle

Radius of Rotation

12 10

0.0 m 0.13 m

8

0.26 m

6

Liquid

Particle

min Particle Concentration

Particle

min 14

Particle Size

12

20%

0.0190 m

10

30%

0.0225 m

8

0.0250 m

40%

6

Oil

6

0

8

Water

8

0

12 10

Viscosity

12 10

4 2

14

Particle

min 14

4 2 Liquid

Mode of Rotation

6 4

4 2

2

0

0 Liquid

Particle

Liquid

Particle

FIGURE 31.2 Effect of processing conditions on heating rate index (fh, min) with Newtonian liquid and particle mixture in cans.

(Ramaswamy and Sablani, 1997a). The presence of multiple particles is expected to cause secondary agitation, which will influence the mixing of materials in the cans (Sablani and Ramaswamy, 1998). The small increase in liquid fh indicated that increased particle concentration increased the drag forces exerted by the particle on the liquid.


The fh values of liquid increased by 24% when the particle diameter was increased from 0.01905 to 0.025 m. On the other hand, the particle fh increased 27% as particle size increased from 0.01905 to 0.02225 m, while a further increase in diameter up to 0.025 m decreased the fh by 8%. Lenz and Lund (1978) also observed a decrease in heat transfer rates to liquid and particles (U and hfp) as particle diameter increased from 0.021 to 0.03 m in a 60% aqueous sucrose solution. Deniston et al. (1987), however, did not find a straightforward relationship for heat transfer rates (U) associated with canned liquids. In general, factors affecting fh also affected the jch values, except that the mode of rotation did not influence the particle jch, and neither mode of rotation nor speed of rotation influenced the particle jch. Particle concentration had the greatest influence on the liquid and particle jch (Figure 31.3). Increasing particle concentration reduced the free movement and liquid/particle mixing, which delayed the constant heating rate of the product. The Fo value (process lethality) of a process indicates process severity with respect to microbial destruction and is the key in process time establishments. Co (cook value), on the other hand, indicates the cumulative equivalent of cooking minutes at 100°C. The ratio Co/Fo gives a relative measure of the degree of cooking and could be a suitable multiplier in cases where process time is not constant (Ramaswamy et al., 1993). Liquid viscosity had the greatest influence on liquid Co/Fo (p < 0.005). The liquid ratio Co/Fo was 10% higher for oil than for water. Rotational speed and liquid viscosity effects were more pronounced on particle Co/Fo ratio (Figure 31.4). At any condition, the particle Co/Fo was always higher than the ratio Co/Fo of liquid. This is because in the sterilization of canned liquid–particle mixtures, heat is first transferred from the heating medium to the canned liquid through the wall, then to the particles. In such a situation, some overprocessing of the liquid portion is normal. Any attempt to minimize the overprocessing of the canned liquid portion will be desirable. Temperature is supposed to be the most influential factor on Co/Fo (as will be demonstrated with the non-Newtonian fluids), which is part of the reason for the well-accepted phenomenon that high temperature short time (HTST) techniques provide a quality advantage. The bulk of the above studies, however, were carried out at a constant temperature (120°C), and hence the temperature effect is not illustrated.

31.5.3 HEATING BEHAVIOR OF NON-NEWTONIAN LIQUID/PARTICLE MIXTURES Statistical analysis of results with non-Newtonian liquids (Krishnamurthy et al., 2000) on fh, jch, and Co/Fo during the thermal process are presented in Table 31.2. The results revealed that both guar gum concentration and rotational speed influenced fh significantly (p < 0.05). Further, both gum and particle concentrations influenced the jch (p < 0.05) of liquid, but only particle concentration was important with respect to particle jch. Of the four factors considered, only retort temperature influenced Co/Fo (p < 0.005). An interaction between the factors retort temperature and gum


min 2

min Rotation Speed 10 rpm

1.5

2

Mode of Rotation

1.5

Continuous

1

Oscillatory

15 rpm 1

20 rpm

0.5

0.5

0

0 Liquid

Liquid

Particle

Particle

min

min 2

Radius of Rotation 0.0 m

1.5

2

Viscosity 1.5

Water

0.13 m 1

0.26 m

0.5

Oil

1 0.5

0

0

Liquid

min 2

Particle

Liquid

Particle Concentration 20%

1.5

30% 40%

1 0.5

Particle

min 2

Particle Size 0.0190 m

1.5

0.0225 m 1

0.0250 m

0.5

0

0 Liquid

Particle

Liquid

Particle

FIGURE 31.3 Effect of processing conditions on heating lag factor with Newtonian liquid and particle mixture in cans.

concentration in their effect on Co/Fo was also observed (0.005 < p < 0.05), but was of less significance compared to the main effects. Radius of rotation did not have any significant effect on any of the heat transfer parameters considered. All factors that influenced the heat penetration parameters (fh and jch) with Newtonian liquids also influenced liquid and particle fh and jch with different magnitudes.


min

min 10

10

Mode of Rotation

Rotation Speed

8

10 rpm

8 Continuous

6

15 rpm

6

4

20 rpm

4

2

2

0

0

Liquid

Oscillatory

Liquid

Particle

Particle

min 10

min 10 Radius of Rotation 0.0 m

8

0.13 m

6

0.26 m

4

Water

6

Oil

4

2

2

0

0

Liquid

min 10

Liquid

Particle

Particle Concentration 20%

8

6

30%

6

40% 4

4

2

2

0

0 Particle

Particle

min 10

8

Liquid

Viscosity

8

Particle Size 0.0190 m 0.0225 m 0.0250 m

Liquid

Particle

FIGURE 31.4 Effect of processing conditions on Co /Fo with Newtonian liquid and particle mixture in cans.

In general, fh and jch values with non-Newtonian liquids were higher than those obtained with Newtonian liquids, indicating that rate of heat transfer to both liquid and particles was slower in the former case. This can be attributed to poor mixing of non-Newtonian liquid and particles. Guar gum concentration was the major factor



TABLE 31.2 Analysis of Variance Results (% Sum of Squares and Significance Level) Showing the Influence of Operating Variables on the Three Variables Considered (fh, jch, and Co /Fo) at Three Radii of Rotation (center, 100, and 190 mm) with Non-Newtonian Liquids fh A: Parameters for Liquid Portion Solution concentration Particle concentration Retort temperature Rotational speed B: Parameters for Particles Gum concentration Particle concentration Retort temperature Rotational speed a b

p < 0.005. p < 0.05.

Center a

100 mm

jch 190 mm

Center

Co/Fo

100 mm

190 mm

Center

100 mm

190 mm

46.0 9.00 8.00 b 12.10

51.14 14.20 10.71 b 20.12

a

42.24 18.52 9.96 b 23.08

a

85.50 b 7.64 1.00 1.00

a

74.00 b 15.55 3.00 4.75

a

71.38 b 14.81 3.03 4.71

a

0.58 7.60 a 70.16 6.40

3.00 5.90 a 83.96 1.44

1.32 7.00 a 84.41 4.90

a

51.38 7.89 7.32 b 31.19

a

51.38 7.89 7.32 b 20.87

a

11.19 b 38.19 4.95 10.27

23.87 b 38.75 6.4 9.51

b

10.27 b 25.61 4.95 10.27

2.34 1.40 a 86.03 1.36

2.08 1.23 a 87.08 1.21

1.00 1.4 a 85.12 0.00

58.38 5.86 9.57 b 24.46

min

min

12

Rotation Speed

10

12

Retort Temperature 130°C

20 rpm

10

8

15 rpm

8

125°C

6

10 rpm

6

120°C

5 rpm

4

0 rpm

2 0

115°C

4

110°C

2 0

Liquid

min 12

Particle

Liquid

Gum Concentration

min 12

Particle

Particle Concentration

10

1.0%

10

8

0.75%

8

0.25%

6

0.5%

6

0.2%

4

0.15%

0.25%

4

0.0%

2

0.3%

0.1%

2

0

0

Liquid

Particle

Liquid

Particle

FIGURE 31.5 Effect of processing conditions on heating rate index (fh, min) with nonNewtonian liquid/particle mixture in cans. (From Krishnamurthy, H., Sanchez, G., Ramaswamy, H.S., Sablani, S.S., and Pandey, P.K., poster presentation at ICEF8, Puebla, Mexico, April 10–13, 2000. With permission.)

accounting for nearly 40 to 50% of the total variation in heating rate index, followed by rotational speed accounting for 12 to 24%. Figure 31.5 demonstrates typical variations in fh under different conditions. In general, fh was found to increase with increasing gum concentration and decreasing rotational speed. This was expected and can be explained based on apparent viscosity, which is highest for 1% guar gum and lowest for water. The decreasing effect of rotational speed on fh results from the forced mixing of can contents due to relative motion (turbulence) of liquid and particles in the can. Guar gum concentration was again the major factor accounting for 74% of the total variation in heating lag (jch) of the liquid (Figure 31.6). Particle concentration came next with 15%. The samples with the highest solution and particle concentrations


min 3.5

Rotation Speed

3

20 rpm

2.5

15 rpm

2 1.5

3

5 rpm 0 rpm

1

0.5

0.5

0

0 Liquid Particle

Gum Concentration 1.0%

2 1.5

min 3.5

125°C 120°C 115°C 110°C

2.5

0.5%

2 1.5

0.0%

Particle Concentration

3

0.75%

0.25%

1 0.5

130°C

Liquid Particle

min 3.5

3 2.5

Retort Temperature

2.5 2 1.5

10 rpm

1

min 3.5

0.3% 0.25% 0.2% 0.15%

1

0.1%

0.5

0

0

Liquid Particle

Liquid Particle

FIGURE 31.6 Effect of processing conditions on heating lag factor with non-Newtonian liquid/particle mixture in cans. (From Krishnamurthy, H., Sanchez, G., Ramaswamy, H.S., Sablani, S.S., and Pandey, P.K., poster presentation at ICEF8, Puebla, Mexico, April 10–13, 2000. With permission.)

exhibited a greater lag factor compared to the other four levels, possibly due to the high initial viscosity (strongly non-Newtonian). Contrary to expectation, retort temperature and rotational speed did not significantly affect lag factor. Particle concentration was the only major factor, accounting for 38% of the total variation in particle heating lag. As shown in Figure 31.7, the ratio Co/Fo for non-Newtonian liquids dramatically decreased from about 40 to 1.5 as temperature increased from 110 to 130°C. Overall, the magnitude of Co/Fo for liquid and particle was larger than that observed with Newtonian liquids, suggesting slower heat transfer rates with non-Newtonian solutions. The effect of gum concentration on this ratio was small, but the increasing trend was clear. The cook quality of the product could thus be improved by high temperature short time processing.


Surface Response Graph for Co/Fo (Liquids) 40 35 30

Co/Fo

25 20 15 10 5 0

0.25

0.50

0.75

Gum Concentratio

110 115 (C) 120 ure 1.00 130125 perat

n (%w/v)

ort

Ret

Tem

Surface Response Graph for Co/Fo (Particle) 40 35

Co/Fo

30 25 20 15 10 5 0

110 Gum 0.3 0.5 115 Con 120 cen 0.8 trat ) 1.00 130 125 ion ure (C (%w perat m e T /v) Retort

FIGURE 31.7 Response surface plots of Co /Fo as a function of retort temperature and guar gum concentration. (From Krishnamurthy, H., Sanchez, G., Ramaswamy, H.S., Sablani, S.S., and Pandey, P.K., poster presentation at ICEF8, Puebla, Mexico, April 10–13, 2000. With permission.)

31.6 CONCLUSIONS With Newtonian liquids, liquid viscosity and rotational speed had major effects on fh and Co/Fo, while the effects of particle concentration and rotational speed were more pronounced with jch. The influence of mode of rotation and particle size on the different thermal process parameters was minimal. The effect of radius of rotation was not significant. With non-Newtonian liquids, the gum concentration and rotational speeds had a major effect on the heating rate index (fh) for both liquid and particles, and the effect of particle concentration was especially noticed on lag factor (jch) for


both liquid and particles. Co/Fo was highly influenced by the retort temperature, and a trend was observed when it came to the effect of gum concentration. Both heating rate index and heating lag factor values were higher with non-Newtonian liquid/particle mixtures than those observed with Newtonian liquid/particle mixtures.

NOMENCLATURE a A Bi Co Cp Fo Fo fh hfp jch k m R S T t U U

Radius of sphere, m 2 Total external surface area, m Biot number = hfp a/k Cook value, min Heat capacity, J/kgK 2 Fourier number = αt/a Process lethality, min Heating rate index, sec 2 Fluid-to-particle heat transfer coefficient, W/m K Heating lag factor Thermal conductivity, W/mK Mass, kg A function of Biot number A function of Biot number Temperature, °C Time, sec 2 Overall heat transfer coefficient, W/m K (T – Tr)/(Ti − Tr) Volumetric average

SUBSCRIPTS c i l p r s

Can Initial condition Liquid Particle Retort Surface

GREEK SYMBOLS 2

α Thermal diffusivity, m /sec δ Characteristic root (Equation (31.3))

REFERENCES Anantheswaran, R.C. and Rao, M.A., Heat transfer to model non-Newtonian liquid foods in cans during end-over-end rotation, J. Food Eng., 4, 21–35, 1985. Ball, C.O. and Olson, F.C.W., Sterilization in Food Technology, McGraw-Hill, New York, 1957.


Berry, M.R., Jr. and Bradshaw, J.G., Heating characteristics of condensed cream of celery soup in a steritort: heating penetration and spore count reduction, J. Food Sci., 45, 869–874, 879, 1980. Berry, M.R., Jr. and Bradshaw, J.G., Heat penetration for sliced mushrooms in brine processed in still and agitating retorts with comparisons to spore count reduction, J. Food Sci., 47, 1698–1704, 1982. Berry, M.R., Jr. and Dickerson R.W., Heating characteristics of whole kernel corn processed in a steritort, J. Food Sci., 46, 889–895, 1981. Berry, M.R., Jr., Savage, R.A., and Pflug, I.J., Heating characteristics of cream-style corn processed in a steritort: effect of headspace, reel speed and consistency, J. Food Sci., 44, 831–835, 1979. Deniston, M.F., Hassan, B.H., and Merson, R.L., Heat transfer coefficients to liquids with food particles in axially rotating cans, J.Food Sci., 52, 962–966, 979, 1987. Fernandez, C.L., Rao, M.A., Rajavsireddi, S.P., and Sastry, S.K., Particulate heat transfer to canned snap beans in steritort, J. Food Process Eng., 10, 183–198, 1988. Jaluria, Y. and Torrance, K.E., Computational Heat Transfer, Hemisphere Publishing Corporation, New York, 1996, pp. 403–424. Krishnamurthy, H., Sanchez, G., Ramaswamy, H.S., Sablani, S.S., and Pandey, P.K., Heating Rates of Canned Particulate Non-Newtonian Liquids as Influenced by Product and System Parameters During End-Over-End Agitation Processing in a Rotary Autoclave, Poster #P-203 presented at ICEF8 (International Congress on Engineering and Food), Puebla, Mexico, April 10–13, 2000. Lenz, M.K. and Lund, D.B., The lethality–Fourier number method. Heating rate variations and lethality confidence intervals for forced-convection heated foods in containers, J. Food Process Eng., 2, 227–271, 1978. Naveh, D. and Kopelman, I.J., Effect of some processing parameters on the heat transfer coefficients in a rotating autoclave, J. Food Process. Preserv., 4, 67–77, 1980. Pandey, P.K., Sablani, S.S., and Ramaswamy, H.S., Heat transfer to model non-Newtonian liquid and particle mixtures in cans subjected to end-over-end rotation, Proceedings of the 1997 Annual Conference of the Canadian Society of Agricultural Engineering (CSAE), Norum, D.I. and Savoie, P., Eds., CSCE, Montreal, Volume A, 1997, pp. 235–244. Parchomchuk, P., A simplified method for agitation processing of canned foods, J. Food Sci., 42, 265–268, 1977. Ramaswamy, H.S., Abbatemarco, C., and Sablani, S.S., Heat transfer rates in a canned food model as influenced by processing in an end-over-end rotary steam/air retort, J. Food Process. Preserv., 17, 269–286, 1993. Ramaswamy, H.S., Lo, K.V., and Tung, M.A., Simplified equations for transient temperatures in conductive foods with convective heat transfer at the surface, J. Food Sci., 47(6), 2042–2047, 2065, 1982. Ramaswamy, H.S. and Sablani, S.S., Particle motion and heat transfer in cans during endover-end processing: influence of physical properties and rotational speed, J. Food Process. Preserv., 21, 105–121, 1997a. Ramaswamy, H.S. and Sablani, S.S., Particle shape influence on heat transfer in cans containing liquid particle mixtures subjected to end-over-end rotation, Lebensm.-Wiss. Technol., 30, 525–535, 1997b. Rao, M.A. and Anantheswaran, R.C., Convective heat transfer to fluids in cans, Adv. Food Res., 32, 39–84, 1988. Sablani, S.S., Heat Transfer Studies of Liquid Particle Mixtures in Cans Subjected to EndOver-End Processing, Ph.D. thesis, McGill University, Montreal, 1996.


Sablani, S.S. and Ramaswamy, H.S., Fluid to particle heat transfer coefficient in cans during end-over-end processing, Lebensm.-Wiss. Technol., 28(1), 56–61, 1995. Sablani, S.S. and Ramaswamy, H.S., Particle heat transfer coefficients under various retort operating conditions with end-over-end rotation, J. Food Process Eng., 19, 403–424, 1996. Sablani, S.S. and Ramaswamy, H.S., Heat transfer to particles in cans with end-over-end rotation: influence of particle size and particle concentration (% v/v), J. Food Process Eng., 20, 265–283, 1997. Sablani, S.S. and Ramaswamy, H.S., Multi-particle mixing behavior and its role in heat transfer during end-over-end agitation of cans, J. Food Eng., 37, 141–152, 1998. Sablani, S.S. and Ramaswamy, H.S., End-over-end agitation processing of cans containing liquid particle mixtures. Influence of continuous vs. oscillatory rotation, Food Sci. Technol. Int., 5(5), 385–389, 1999. Stoforos, N.G. and Merson, R.L., Measurement of heat transfer coefficients in rotating liquid particle systems, Biotechnol. Prog., 7, 267–271, 1991. Stoforos, N.G. and Merson, R.L., Physical property and rotational speed effect on heat transfer in axially rotating liquid/particulate canned foods, J. Food Sci., 57, 749–754, 1992.


32

Dimensionless Correlations for Forced Convection Heat Transfer to Spherical Particles under Tube-Flow Heating Conditions H.S. Ramaswamy and M.R. Zareifard

CONTENTS 32.1 Introduction 32.2 Theoretical Background 32.3 Materials and Methods 32.3.1 Calorimetric Method 32.3.2 Particle Oscillatory Motion Method 32.3.3 Data Analysis 32.4 Results and Discussion 32.5 Conclusions Nomenclature References

32.1 INTRODUCTION One of the potential advantages of aseptic processing of particulate foods is the minimum loss of nutrients and overall quality improvement due to the use of a high temperature, short time processing technique. To design a process that ensures the commercial sterility of processed foods, thermal process calculations require accurate time–temperature data. Due to difficulties in measuring the temperature of a particle as it moves through the heat exchanger and holding tube sections of an aseptic processing system, the particle center temperatures are usually predicted by mathematical modeling. Data on residence time distribution (RTD) and associated


fluid-to-particle heat transfer coefficient (hfp) in a continuous flow system are necessary in such models (Dignan et al., 1989). Numerous studies have been carried out to evaluate hfp under a wide range of experimental conditions simulating some aspects of aseptic processing. Several of these studies have made use of particles fixed in a tube for gathering time–temperature data, while others have used novel approaches to monitor transient temperatures of moving particles. Dimensional analysis is a useful technique for generalization of data, as it reduces the number of variables that must be studied and permits the grouping of physical variables that affect the process of heat transfer. In the dimensional analysis of convection heat transfer, the Nusselt number (Nu), a dimensionless measure of convective heat transfer coefficient, is correlated with other dimensionless numbers such as the Reynolds (Re), Prandtl (Pr), and Grashof (Gr) numbers. Kramers (1946), Ranz and Marshal (1952), and Whitaker (1972) were early researchers who presented empirical equations for heat transfer correlations. During the last decade, several researchers have developed different forms of correlations to predict fluid-to-particle heat transfer coefficients for processing of particulate food under a range of tubeflow conditions (Sastry and Zuritz, 1987; Chandarana et al., 1988, 1990; Zuritz et al., 1990; Mwangi et al., 1992; Balasubramaniam, 1993; Zitoun and Sastry, 1994a,b; Astrom and Bark, 1994; Bhamidipati and Singh, 1995; Awuah and Ramaswamy, 1996; Chakrabandhu and Singh, 1998). The objectives of this study were to develop dimensionless correlations for experimental data obtained from two techniques allowing particle motion during the heating process: a calorimetric method in which the particle was free to move and rotate along the length of holding tube (Ramaswamy and Zareifard, 2000), and a particle oscillatory motion method in which the particle was allowed controlled movement in an oscillatory fashion (Zareifard and Ramaswamy, 2001).

32.2 THEORETICAL BACKGROUND Heat transfer from liquid to particles in continuous processing of multi-phase systems is affected by both liquid and particle thermophysical properties as well as system parameters. In the dimensional analysis of forced convection heat transfer, the fluid-to-particle heat transfer coefficient (hfp) is expressed in terms of the Nusselt number (Nu), which is generally described as a function of other dimensionless numbers [Nu = f (Re, Pr, Gr)]. Different forms of dimensionless correlations have been published concerning the flow field and heat transfer to particles in tube-flow conditions. Kramers (1946), Ranz and Marshal (1952), and Whitaker (1972) presented the following equations for forced convection conditions, respectively: 0.15

Kramers (1946): Nu = 2 + 1.3 Pr

0.31

+ 0.66 Pr 0.5

0.5


0.66

+ 0.06 Re

(32.1)

0.33

Ranz and Marshal (1952): Nu = 2 + 0.6 Re Whitaker (1972): Nu = 2 + (0.4 Re

0.5

Re

+ Pr 0.4

) Pr

(32.2) 0.25

(µ∞/µs)

(32.3)

These correlations involve the Reynolds number as a dominant factor influencing hfp. However, some researchers have defined the Reynolds number based on either fluid or particle velocity, while others have used the relative velocity between particle and fluid, also called slip velocity, which is the difference between fluid and particle velocities. The slip velocity appears to be more suitable under the conditions of this study. The Whitaker (1972) model suggests that fluid velocity far removed from the particle should be used to calculate Re number. The particle velocity is easy to measure. However, the fluid velocity in tube-flow conditions of a two-phase system is a source of differences among the researchers. In the case of tube-flow conditions with moving particles, several additional factors can influence particle motion and thus the fluid-to-particle heat transfer coefficient. Particle shape and size, thermophysical properties of both particle and carrier fluid, and system parameters such as tube diameter and fluid flow rate are some influencing factors. These parameters can be grouped into dimensionless forms and included in the correlations. Further, to improve the prediction of the developed model, other groups of dimensionless numbers, such as the Peclet, Froude, and Rayleigh numbers, or dimensionless ratios 2 such as Gr/Re , d/D, αf/αp, and µ∞/µs have also been included in the empirical equations. Some published empirical equations are presented in Table 32.1. In the present study, the Nusselt number and generalized forms of the Reynolds, Prandtl, and Grashof numbers were used and calculated as follows: Nu = hfp dp/kf

(32.4)

GRe = ρ Vs dp/µ

(32.5)

GPr = Cp µ/kf

(32.6)

2

3

GGr = g β ρ ∆T dp /µ

(32.7)

Vs = Vf − Vp

(32.8)

∆T = (Tf − Tip)/2

(32.9)

where:

2

2

Vf = Q/π (Rt − Rp )

(32.10)

Cp = 1.675 + 0.025 (%W)

(32.11)

n−3

µ=2

n

1−n

m (3n + 1/n) (dp/Vs)

(32.12)

2

(32.13)

kf = [326.575 + 1.0412 T − 0.000337 T ] [(0.796 + 0.009346 (%W)]



TABLE 32.1 Some Published Dimensionless Correlations for Estimating Nusselt Number under Forced Convection Heat Transfer Conditions Reference Kramers (1946) Ranz and Marshal (1952) Whitaker (1972) Zuritz and Sastry (1987) Chandarana et al. (1988) Chandarana et al. (1988) Chandarana et al. (1990) Chandarana et al. (1990) Sastry et al. (1990) Sastry et al. (1990) Zuritz et al. (1990) Awuah et al. (1993) Awuah et al. (1993) Astrom and Bark (1994) Astrom and Bark (1994) Bhamidipati and Singh (1995) Kelly et al. (1995) Baptista et al. (1997a) Baptista et al. (1997a)

Fluid Air, oil, water Air Air, oil, water a CMC Starch Water Starch Water Water Water CMC CMC CMC Silicon oil Silicon oil CMC CMC CMC CMC

Sphere (steel) Sphere Sphere Mushroom (aluminum) Cubes (silicone) Cubes (silicone) Cubes (silicone) Cubes (silicone) b Sphere (hollow al ) moving therm. b Sphere (hollow al ) moving therm. b Mushroom (al ) Cylinders (carrot) Cylinders (potato) Spheres (lead) Cubes (lead) Cylinder (temp. sensor) Sphere, cube, cylinder (aluminum) Sphere (hollow aluminum) stationary Sphere (hollow aluminum) rotating

Chakrabandhu and Singh (1998) Chakrabandhu and Singh (1998)

CMC CMC

Cubes (sodium alginate) stationary Cubes (sodium alginate) rotating

a b

CMC = carboxymethyl cellulose. al = aluminum.

Particle

Equation [Nu = f (Re, Pr…..)] 0.15 0.5 0.31 2 + 1.3Pr + 0.66Re Pr 0.5 0.33 2 + 0.6Re Pr 0.5 0.66 0.4 0.25 2 + (0.4Re + 0.06Re ) Pr (µs/µ∞) 0.21 0.14 2 + 2.95Re Pr 0.44 0.35 0.55Re Pr 0.43 0.85Re 1.6 0.89 2 + 0.028Re Pr 1.08 2 + 0.0333Re 0.00455 6.46E–26[Re (d/D)] 0.0125 85.67 –342.5 2.7E–27Re (d/D) Pr 0.23 0.14 1.79 2 + 28.37Re Pr (d/D) 0.108 2.45(Pr Gr) 0.113 2.02(Pr Gr) 0.54 8 + 2.53Re 0.61 8 + 2.25Re 0.2 0.33 0.27Re Pr 0.132 –0.08 2 + 19.36Re Pr 0.59 0.28 Nus + 0.64Re Pr 0.71 0.42 0.28 Nus + 0.17Re Pr (rp/Rt) 0.33 0.5 Nus = 2 + 0.025 Pr Gr 0.29 1.52 4.35 18197Re Pr (d/D) 0.86 0.22 1.67 338Re Pr (d/D)

2

R — — — 0.99 0.92 0.95 0.79 0.89 0.82 0.92 0.96 0.80 0.88 — — 0.85 0.83 — — 0.91 0.89

32.3 MATERIALS AND METHODS Data from two previous reports (Ramaswamy and Zareifard, 2000; Zareifard and Ramaswamy, 2001) were used for developing correlations under tube-flow conditions. The techniques for evaluating hfp have been detailed by Zareifard and Ramaswamy (1997, 1999). A brief description of the two methods is given below for reference:

32.3.1 CALORIMETRIC METHOD The calorimetric method (Zareifard and Ramaswamy, 1999) involves an indirect measurement of particle temperature. Instead of recording the particle center point temperature, the mass average (bulk) temperature of a freely moving particle, which is not attached to any thermocouple, is estimated using a calorimeter. Therefore, the particle is allowed to flow with the carrier fluid along the tube without any restriction. The technique consists of introducing a particle at an upstream location into a holding tube, in which a carrier fluid under prestabilized temperature and flow conditions is circulated, and retrieving it from a downstream location after a known time interval. The particle is transferred immediately into a specially constructed calorimeter, and its bulk temperature is determined upon equilibration. Knowing the medium temperature, particle initial temperature, exposure time (residence time), and thermophysical properties, the associated hfp is computed from the evaluated bulk temperature of the particle using an iterative numerical technique. The procedure involves numerical computation, by trial and error, of bulk temperature (also known as mass average temperature) for a given heating time (residence time), using the available property data and an assumed hfp value, and comparing it with the experimental value. Aluminium spherical particles of three sizes (19.05, 22.22, and 25.4 mm in diameter) were used as test particles in a Pyrex glass holding tube (45 mm inside diameter).

32.3.2 PARTICLE OSCILLATORY MOTION METHOD The particle oscillatory motion method (Zareifard and Ramaswamy, 1997) involves direct measurement of center point temperature of thermocouple-equipped particles fixed at the center of a set of circular holding tubes of different radii cut out in quarter lengths. An assembly of four such tubes is supported by an aluminum shaft and connected to a programmable servo motor equipped with an electronic board for imparting preset oscillatory motions. Spherical nylon particles of three different sizes (12.7, 16, and 17.5 mm in diameter) as well as spherical particles (17.5 mm in diameter) made of aluminum epoxy (Soudotec Inc., Lachine, Quebec) were used as the test particles. The aluminum epoxy particle was formed by mixing aluminum powder and epoxy resin in a soft form. All test particles were drilled to insert thermocouples to their geometric centers. The assembly of holding tubes containing the particles was kept in a cooling tank at 5°C for equilibration of the temperature in test particles, and an appropriate oscillatory motion was programmed to result in particle linear velocity in the range 0.06 to 0.21 m/sec. The assembly was then transferred to the test tank containing a heating medium at a preset temperature. Time–temperature data were gathered at


3-sec time intervals using a data logger connected to a personal computer. Water as well as two concentrations (0.5 and 1% w/w) of commercial carboxymethyl cellulose, CMC (Sigma, St. Louis, MO) were used as the heating media. Thermophysical properties of the different test particles such as density (ρ), thermal diffusivity (α) and heat capacity (Cp) were experimentally evaluated using procedures described elsewhere (Sablani and Ramaswamy, 1996). The thermal conductivity (k) was calculated as k = ρCpα. Rheological data for test fluids at different concentration and temperature were obtained in the form of power law model parameters [consistency coefficient (m) and flow behavior index (n)] using a rotational viscometer Model RV20 (Haake Mess-Technik, Karlsruhe, Germany) as described by Abdelrahim and Ramaswamy (1995). These data are summarized in Tables 32.2 and 32.3. Data obtained on hfp from the two techniques are given in Tables 32.4 and 32.5 and were used for developing the dimensionless correlations.

TABLE 32.2 Thermophysical Properties of Spherical Test Particles Material

Aluminum 3

Density (kg/m ) Heat capacity (J/kg K) Thermal conductivity (W/m K) 2 Thermal diffusivity (m /sec) a b

a

Aluminum Epoxy

2707 896 204 8.42E–5

b

1750 1370 1.13 4.7E–7

Nylon

1128 2073 0.37 1.58E–7

From Holman, G.P., Heat Transfer, McGraw-Hill, New York, 1990. Measured experimentally.

TABLE 32.3 The Power Law Model Parameters of CMC Solutions for Different Concentrations and Temperatures Determined Using a Rotational Viscometer CMC %

Temp. °C

0.25 0.25 0.25 0.50 0.50 0.50 1 1 1

50 60 70 50 60 70 50 60 70

M n Pa.sec 0.150 0.052 0.011 0.510 0.252 0.115 1.715 1.191 0.778

± ± ± ± ± ± ± ± ±

0.031 0.018 0.028 0.020 0.033 0.045 0.151 0.101 0.082

n 0.785 0.821 0.976 0.590 0.681 0.754 0.560 0.510 0.583

± ± ± ± ± ± ± ± ±

0.030 0.015 0.021 0.011 0.026 0.017 0.019 0.029 0.031

Note: Results are means ± standard deviations of three replicates (m: consistency coefficient; n: flow behavior index).


b

TABLE 32.4 2 Values of hfp (W/m ·K) Associated with Aluminum Spherical Particles of Different Sizes and Fluid Flow Rates with Various Fluid Concentrations and Temperatures Using the Calorimetric Approach % CMC Concentration F (l/min)

9

11

13

16

19

P s m l s m l s m l s m l s m l s m l s m l s m l s m l s m l s m l s m l s m l

T (°C)

0.0

0.25

0.5

50

60

70

50

60

70

50

60

70

1430 (3.5) 1730 (3.5) 2200 (2.7)

50

60

70

70

1730 2150 2550 2030 2300 2660

(5.2) (3.7) (2.3) (3.4) (2.6) (3.4)

920 (2.2) 690 (0.1) 980 (0.1) 890 (2.2) 990 (0.2) 1020 (1.0) 880 (2.3) 930 (2.1) 980 (0.1) 1090 (1.8) 1100 (1.8) 1190 (1.8) 1070 (0.9) 1120 (3.6) 1180 (2.5) 1000 (1.0) 1070 (3.7) 1140 (3.5) 1310 (1.5) 1420 (3.5) 1490 (4.0) 1260 (1.6) 1310 (1.5) 1390 (2.2) 1240 (3.2) 1320 (1.5) 1350 (2.2)

790 880 950 770 830 930 750 810 900

(2.5) (1.1) (1.0) (1.3) (1.2) (1.1) (1.3) (2.5) (1.1)

1 740 (2.7) 800 (2.5) 870 (2.3) 680 (1.5) 730 (1.4) 800 (1.2) 650 (1.5) 700 (1.4) 750 (1.3) 750 (4.0) 860 (1.2) 900 (1.1) 710 (1.4) 780 (1.3) 870 (1.1) 680 (1.5) 720 (1.4) 840 (1.2) 890 (2.2) 910 (2.2) 1020 (2.9) 820 (1.2) 870 (1.1) 970 (1.0) 740 (2.7) 810 (1.2) 890 (2.2)

Note: Values of hfp are the means of four replicates, and the coefficient of variation is given in parentheses. F: fluid flow rate; T: fluid temperature; P: particle size: s, small; m, medium; l, large.


TABLE 32.5 2 Values of hfp (W/m ·K) Associated with Different Fluid Concentrations and Temperatures for Spherical Particles of Different Sizes and Materials at Various Particle Velocities Using the Particle Oscillatory Motion Technique 2

C (% CMC)

T (°C)

0

60

60

0.5

70

80

60

1.0

70

80

2

hfp (W/m ·K)

Vp (m/sec)

AL

0.10 0.16 0.21 0.10 0.16 0.21 0.10 0.16 0.21 0.10 0.16 0.21 0.10 0.16 0.21 0.10 0.16 0.21 0.10 0.16 0.21

1460 (11) 1830 (11) 2020 (9) 590 (15) 950 (6) 1220 (9) 750 (5) — 1360 (9) 1140 (6) 1290 (3) 1380 (12) 530 (10) 570 (10) 630 (11) 650 (7) 730 (10) 880 (7) 720 (10) 1010 (8) 1110 (4)

hfp (W/m ·K)

NL

Vp (m/sec)

NS

NM

1120 (7) 1310 (6) 1410 (6) 460 (15) 620 (7) 690 (9) 500 (18) — 730 (7) 640 (16) 740 (14) 800 (4) 430 (17) 520 (12) 620 (19) 490 (8) 530 (18) 660 (10) 500 (18) 560 (17) 740 (8)

0.06 0.09 0.12 0.06 0.09 0.12 0.06 0.09 0.12 0.06 0.09 0.12 0.06 0.09 0.12 0.06 0.09 0.12 0.06 0.09 0.12

1100 (11) 1250 (12) 1410 (8) 520 (15) 720 (15) 750 (18) 530 (3) — 790 (12) 600 (18) 790 (9) 840 (8) 360 (16) 420 (11) 570 (4) 490 (13) 530 (17) 610 (12) 520 (7) 520 (10) 660 (5)

1070 (11) 1120 (10) 1320 (8) 430 (10) 580 (9) 610 (12) 460(9) — 640 (15) 520 (11) 680 (13) 730 (9) 350 (14) 440 (9) 530 (16) 430 (13) 510 (4) 580 (9) 470 (12) 500 (4) 640 (10)

Note: Values of hfp are the means of four replicates, and the coefficient of variation is given in parentheses. Vp: particle velocity; T: medium temperature; C: medium concentration; AL; aluminum epoxy; NL: nylon large; NS: nylon small; NM: nylon medium.

32.3.3 DATA ANALYSIS In the literature, the heat transfer coefficients associated with pure forced convection with unbounded flow are generally modeled in dimensionless forms, as shown in Equation (32.14). c

d

Nu = A + B Re Pr

(32.14)

For tube-flow conditions involving biphasic systems, other dimensionless terms are added to make the correlations better explain the experimental variability. In the present study, regression analyses were performed starting with the simplest form of correlation: Nu = f (GRe). This was progressively expanded to include other dimensionless numbers such as GPr and GGr as well as other dimensionless groups: 2 αf/αp, GGr/GRe , d/D, Vp/Vf, and Vp/Vs. After successful improvement of the model


prediction by including other variables was observed, a multiple regression technique with a backward elimination procedure using the SAS statistical package (SAS Institute, 1994) to obtain the best model was adopted. The models were individually developed with data from the calorimetric method and particle oscillatory motion method and subsequently with the pooled data (600+ experimental values).

32.4 RESULTS AND DISCUSSION Regression analysis between the Nusselt number and other dimensionless groups 2 resulted in a number of correlations with different coefficients of determination (R ) and coefficients of variation (CV). Overall, these coefficients varied over the fol2 lowing ranges: 0.4 < R < 0.99; 1 < CV < 15%. As more dimensionless groups were 2 added to the correlation, the associated R and CV values demonstrated a progressive improvement of the model. The following is the sequence of the regression analysis with data obtained from the calorimetric method: 0.147

Nu = 2 + 1.84 E–1 GRe

0.601

GPr

0.440

GPr

Nu = 2 + 2.49 E–1 GRe Nu = 2 + 3.51 E–2 GRe

0.448 0.681

0.25

(32.15)

2

(32.16)

2

(32.17)

2

(32.18)

2

(32.19)

(R = 0.71; CV = 6.1) 0.191

GGr

(R = 0.73; CV = 5.8)

–2.18

Nu = 2 + 1.15 E11 GRe GPr −1.125 4.508 × GGr (d/D) 0.479

2

(R = 0.60; CV = 7.1)

−0.655

Nu = 2 + 3.80 E4 GRe GPr 2.293 0.514 × (d/D) (Vp/Vs)

(R = 0.91; CV = 3.2) −0.526

GGr

(R = 0.97; CV = 1.8)

The simplistic form of the correlation relating Nu to just GRe (Equation (32.15)) yielded a fairly poor fit and largely scattered data points (not shown). This was expected because GRe alone cannot be expected to explain all the variability associated with Nu. The second correlation in the form of Nu as a function of Re and Pr (Equation (32.16)) is commonly employed in many published reports including the well known Ranz and Marshal correlation. Logarithmic plots of experimental vs. predicted Nu based on Equations (32.16–19) are presented in Figure 32.1. The 2 R improved somewhat (from 0.6 to 0.7) with the inclusion of Pr in the model (Equation (32.16), Figure 32.1a). Adding Gr to the model (Equation (32.17), 2 2 Figure 32.1b) did not improve R much since the range of Gr/Re employed was less than 1.0, indicating that experimental conditions involved forced convection rather than natural convection. Adding the ratio of particle-to-tube diameter d/D (Equation (32.18), Figure 32.1c) resulted in a good improvement of the model, and 2 R increased from 0.7 to 0.9. Including the ratio of particle-to-slip velocity, Vp/Vs (Equation (32.19), Figure 32.1d), also contributed to improving model prediction. 2 This combination model had an R greater than 0.97. Because additional different forms of dimensionless numbers could have the potential to improve the model performance, a multiple regression technique with backward elimination of statistically insignificant (p > 0.05) terms was employed to achieve the optimal model configuration. The regression analysis with the backward elimination procedure resulted in the following model equation with the best


2

R2 = 0.71 Predicted Log Nu

Predicted Log Nu

2

1.8

1.6

1.4

1.8

1.6

1.4

1.2

1 1

R2 = 0.73

1.2

1.2

1.4

1.6

1.8

2

1 1

2.2

1.2

1.4

Experimental Log Nu

1.6

1.8

2

2.2

Experimental Log Nu

(a)

(b) 2.2

2

R2 = 0.91 Predicted Log Nu

Predicted Log Nu

2

1.8

1.6

1.4

1.2

1 1

R2 = 0.97

1.8

1.6

1.4

1.2

1.2

1.4

1.6

1.8

2

1 1

2.2

1.2

1.4

Experimental Log Nu

1.6

1.8

2

2.2

Experimental Log Nu

(c)

(d)

FIGURE 32.1 Logarithmic plot of experimental vs. predicted Nusselt numbers using different models: (a) Equation (32.16); (b) Equation (32.17); (c) Equation (32.18); (d) Equation (32.19). 2

fit (R = 0.98), employing about 300 experimental data points obtained from the calorimetric method under different conditions: 0.589

Nu = 2 + 5.89 E24 GRe

0.977

GPr

0.231

GGr

−0.607

(Vp/Vf)

−8.402

0.731

(Vp/Vs)

(αf/αp) (32.20)

Figure 32.2 shows the plot of Nusselt numbers calculated from Equation (32.20) vs. experimental values of Nusselt numbers obtained with the calorimetric method. The same procedure, backward elimination, was also used for data obtained 2 from the particle oscillatory motion method, and the best model (R = 0.88) was found to be: 2.032

Nu = 2 + 2.27 E–8 GPr


1.120

GGr

−1.843

(d/D)

2 −0.316

(GGr/GRe )

0.305

(αf/αp)

(32.21)

2.2

R2 = 0.98

Predicted Log Nu

2

1.8

1.6

1.4

1.2

1 1

1.2

1.4

1.6

1.8

2

2.2

Experimental Log Nu

FIGURE 32.2 Logarithmic plot of experimental vs. predicted Nusselt number for the best model for data from the calorimetric method (Equation (32.20)).

Contributions from natural convection were accounted for by introducing both 2 2 GGr and the GGr/GRe ratio in Equation (32.21). The magnitude of Gr/Re describes the ratio of buoyant to inertia forces indicating free, forced, and mixed convection regimes (Chapman, 1989). Yuge (1960) indicated that forced convection dominates 2 for flow over a sphere when condition Gr/Re < 0.01 is satisfied. According to Johnson et al. (1988), mixed convection occurs when the ratio falls between 0.08 and 5.1. Figure 32.3 shows the logarithmic plot of Nusselt numbers calculated from Equation (32.21) vs. experimental values of Nusselt numbers obtained from the particle oscillatory motion method. Model prediction of Nu using Equation (32.20) 2 (developed from the calorimetric approach) resulted in a better fit with an R of 0.98 than model Equation (32.21) (developed from oscillatory motion technique) with an 2 2 2 R of 0.88. The model R represents a combination of partial R associated with individual parameters included in the model. Obviously, it is important to include meaningful dimensionless numbers in the model. In this regard, the variables in Equation (32.20) appear to do a better job (with calorimetric approach data). The same variables did not give similar results with the data obtained from the oscillatory technique. The lack of rotational motion of the particles (because of the attachment of the thermocouple to the particle) could be a reason for the discrepancy. The overall 2 lower R associated with Equation (32.21) also indicates that perhaps some additional parameters may be necessary to describe the rather unusual oscillatory motion for the particles.


2

Predicted Log Nu

R2 = 0.88

1.5

1

0.5 0.5

1

1.5

2

Experimental Log Nu

FIGURE 32.3 Logarithmic plot of experimental vs. predicted Nusselt number for the best model for data from the oscillatory motion technique (Equation (32.21)).

Multiple regression and backward elimination for all of the 600 experimental 2 data values obtained from both methods resulted in the following model (R = 0.91) after including all the possible influencing factors associated with hfp: 0.555

Nu = 2 + 4.45 E2 GRe 0.227

× (αf/αp)

−0.294

GPr

−0.352

GGr

1.628

(d/D)

1.748

(Vp/Vf)

−0.533

(Vp/Vs)

(32.22)

The validity of the above equation is limited to the following ranges of the dimensionless group numbers: 2.9 < GRe < 7660; 2.5 < GPr < 3077; 4.15 < GGr < 1.5 E7; 0.2 < d/D < 0.6; 0.30 < Vp/Vf < 1; 0.43 < Vp/Vs < 3.37; and 0.33 < αf/αp < 1892. Figure 32.4 shows the logarithmic plot of the Nusselt number calculated from Equation (32.22) vs. experimental values of Nusselt numbers obtained from the pooled data including both calorimetric and particle oscillatory motion methods. It is worth mentioning that in the heat transfer literature the convective heat transfer coefficient in the form of the Nusselt number is correlated to the Reynolds and Prandtl numbers due to the influence of properties associated with the carrier fluid, which can affect the boundary conditions. The constant value of two has been reported in the developed correlations to be considered as a limiting Nusselt number value for the steady state heat conduction between fluid and sphere. This number may vary for different particle geometry (Whitaker, 1976). In the present study, particle-to-tube diameter ratio (d/D) and thermal diffusivity ratio (αf/αp), as well as velocity ratios (Vp/Vf or Vp/Vs) were found to be appropriate for hfp correlations under tube-flow situations investigated.


2.2

R2 = 0.91

2

Predicted Log Nu

1.8 1.6 1.4 1.2 1 0.8 0.6 0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Experimental Log Nu

FIGURE 32.4 Logarithmic plot of experimental vs. predicted Nusselt number for the overall model from the combined data (Equation (32.22)).

Dimensionless numbers as well as the associated coefficients in the above models are different from those reported in the literature. Attempts to compare the above model with others in the literature resulted in either overestimation or underestimation of the Nusselt number. Comparison with well-known correlations such as those of Ranz and Marshal (1952) or Whitaker (1972) is not possible because these equations were derived for an unbounded flow situation. In general, it is difficult to compare the empirical equations in the literature because of differences in experimental procedures, and most importantly, differences in the range of Re number and other dimensionless numbers which might differ from the experimental condition for the present study.

32.5 CONCLUSIONS Dimensionless correlations between Nusselt number and other dimensionless numbers were developed for data obtained from a particle oscillatory motion method and a calorimetric approach separately as well as for the combination of data from both. A multiple regression analysis with backward elimination procedure was used to obtain the best model with statistically significant parameters associated with hfp. Heat transfer to the particle from the carrier medium was modelled using Reynolds, Prandtl, and Grashof numbers. Their relationships were attributed to the carrier fluid and various parameters affecting the boundary condition. In addition, particle-totube diameter ratio (d/D) and thermal diffusivity ratio (αf/αp) as well as velocity ratio (Vp/Vf) were found to be appropriate for hfp correlations under tube-flow and


force convection situations investigated in the present study. Nusselt numbers estimated from the developed models showed good agreement with experimental data 2 (R = 0.88–0.98).

NOMENCLATURE a Cp d D g hfp k m n Rt Rp Tf Ti Tp Vf Vp Vs W

Radius of sphere (m) Heat capacity (J/Kg °C) Particle diameter (m) Inside tube diameter (m) 2 Local acceleration due to gravity (m/sec ) 2 Fluid-to-particle heat transfer coefficient (W/m K) Thermal conductivity (W/m K) n Consistency coefficient (Pa.sec ) Flow behavior index Radius of the tube Radius of the particle Fluid or medium temperature Initial temperature Particle temperature Fluid velocity (m/sec) Particle velocity (m/sec) Slip velocity (m/sec) Moisture content (%)

GREEK LETTERS α β ρ µ µ∞

2

Thermal diffusivity (m /sec) Root of characteristic equation or thermal expansion coefficient (1/K) 3 Density (kg/m ) 2 Dynamic or apparent viscosity (µ = ρυ) (N.sec/m ) 2 Fluid apparent viscosity at bulk fluid temperature (N.sec/m )

DIMENSIONLESS NUMBERS Bi Gr Pr Re Nu

Biot number (hfp a/kp) 2 3 2 Grashof number (g β ρ ∆T d /µ ) Prandtl number (µ Cp/kf or υ/α) Reynolds number (ρ V d/µ) Nusselt number (hfp a/kf)

SUBSCRIPTS f fp i

Fluid Fluid-to-particle Initial


p s ∞

Particle Surface or slip Ambient

REFERENCES Abdelrahim, K.A. and Ramaswamy, H.S., High temperature/pressure rheology of carboxymethyl cellulose (CMC), Food Res. Int., 28(3), 285–290, 1995. Astrom, A. and Bark, G., Heat transfer between fluid and particles in aseptic processing, J. Food Eng., 21, 97–125, 1994. Awuah, G.B. and Ramaswamy, H.S., Dimensionless correlations for mixed and forced convection heat transfer to spherical and finite cylindrical particles in an aseptic processing holding tube simulator, J. Food Process Eng., 19, 241–267, 1996. Awuah, G.B., Ramaswamy, H.S., and Simpson, B.K., Surface heat transfer coefficients associated with heating of food particles in CMC solutions, J. Food Process Eng., 16, 39–57, 1993. Balasubramaniam, V.M., Liquid-to-Particle Convective Heat Transfer in Aseptic Processing Systems, Ph.D. dissertation, The Ohio State University, Columbus, 1993. Baptista, P.N., Oliveira, F.A.R., Oliveira, J.C., and Sastry, S.K., Dimensionless analysis of fluid-to-particle heat transfer coefficients, J. Food Eng., 31, 199–218, 1997a. Baptista, P.N., Oliveira, F.A.R., Oliveira, J.C., and Sastry, S.K., The effect of transitional and rotational relative velocity components on fluid-to-particle heat transfer coefficients in continuous tube flow, Food Res. Int., 30(1), 21–27, 1997b. Bhamidipati, S. and Singh, R.K., Determination of fluid–particle convective heat transfer coefficient, Trans. ASAE, 38(3), 857–862, 1995. Chakrabandhu, K. and Singh, R.K., Determination of fluid-to-particle heat transfer coefficients for rotating particles, J. Food Process Eng., 21, 327–350, 1998. Chandarana, D.I., Gavin, A., and Wheaton, F.W., Particle/Fluid Interface Heat Transfer During Aseptic Processing of Food, ASAE Paper No. 88–6599, American Society of Agricultural Engineers, St Joseph, MI., 1988. Chandarana, D.I., Gavin, A., and Wheaton, F.W., Particle/fluid interface heat transfer under UHT conditions at low particle/fluid relative velocity, J. Food Process Eng., 13, 191– 206, 1990. Chapman, A.J., Heat Transfer, 4th ed., Macmillan Publishing Co., New York, 1989. Dignan, D.M., Berry, M.R., Pflug, I.J., and Gardine, T.D., Safety considerations in establishing aseptic processing for low-acid foods containing particulates, Food Technol., 43, 118–121, 1989. Holman, G.P., Heat Transfer, McGraw-Hill, Inc., New York, 1990. Johnson, A.G., Kirk, G., and Shin, T., Numerical and experimental analysis of mixed forced and natural convection about a sphere, Trans. ASAE, 31(1), 293–299, 1988. Kelly, B.P., Megee, T.R.A., and Ahmed, M.N., Convective heat transfer in open channel flow: effect of geometric shape and flow characteristics, Trans. Inst. Chem. Eng., U.K., 73 (part C), 171–182, 1995. Kramers, H., Heat transfer from spheres to flowing media, Physica, 12, 61, 1946; as cited in Zenz, F.A. and Othmer, D.F., Fluidization and Fluid Particle Systems, Reinhold Pub. Corp., New York, 1960. Mwangi, J.M, Datta, A.K, and Rizvi, S.S., Heat transfer in aseptic processing of particulate foods, 1992, as cited in Singh, R.K. and Nelson, P.E., Advances in Aseptic Processing Technologies, Elsevier, London, 1995, pp. 73–102.


Ramaswamy, H.S., and Zareifard, M.R., Evaluation of factors influencing tube-flow fluid-toparticle heat transfer coefficient using a calorimetric technique, J. Food Eng., 45, 127–138, 2000. Ranz, W.E. and Marshal, W.R., Evaporation from drop, Chem. Eng. Prog., 48, 141–147, 173–180, 1952. Sablani, S.S. and Ramaswamy, H.S., Particle heat transfer coefficient under various retort operating conditions with end-over-end rotation, J. Food Process Eng., 19, 403–424, 1996. SAS Institute Inc., SAS User’s Guide, version 6, SAS Institute, Cary, NC, 1994. Sastry, S.K., Lima, M., Brim, J., Brunn, T., and Heskitt, B.F., Liquid-to-particle heat transfer during continuous tube flow: influence of flow rate and particle to tube diameter ratio, J. Food Process Eng., 13, 239–253, 1990. Sastry, S.K. and Zuritz, C.A., A review of particle behavior in tube flow: applications to aseptic processing, J. Food Process Eng., 10, 27–52, 1987. Whitaker, S., Force convection heat transfer calculations for flow in pipes, past flat plates, single cylinder, single sphere and for flow in packed beds and tube bundles, J. AIChE., 18, 361–371, 1972. Whitaker, S., Elementary Heat Transfer Analysis, Pergamon Press, New York, 1976. Yuge, T., Experiments on heat transfer from spheres including combined and forced convection, J. Heat Transfer, 82, 214–220, 1960. Zareifard, M.R. and Ramaswamy, H.S., A new technique for evaluating fluid-to-particle heat transfer coefficients under tube-flow conditions involving particle oscillatory motion, J. Food Process Eng., 20(6), 453–475, 1997. Zareifard, M.R. and Ramaswamy, H.S., A calorimetric approach for evaluation of fluid to particle heat transfer coefficient under tube-flow condition, Lebensm.-Wiss. Technol., 32, 495–502, 1999. Zareifard, M.R. and Ramaswamy, H.S., Evaluation of tube-flow fluid-to-particle heat transfer coefficient under controlled particle oscillatory motion, Food Res. Int., 34(4), 289– 249, 2001. Zitoun, K.B. and Sastry, S.K., Convection heat transfer coefficient for cubic particles in continuous tube flow using the moving thermocouple method, J. Food Process Eng., 17, 229–241, 1994a. Zitoun, K.B. and Sastry, S.K., Determination of convective heat transfer coefficient between fluid and cubic particles in continuous tube flow using noninvasive experimental techniques, J. Food Process Eng., 17, 209–228, 1994b. Zuritz, C.A., McCoy, S., and Sastry, S.K., Convection heat transfer coefficients for irregular particles immersed in non-Newtonian fluids during tube flow, J. Food Eng., 11, 159–174, 1990. Zuritz, C.A. and Sastry, S.K., Convective Heat Transfer Coefficients for Non-Newtonian Flow Past Food-shaped Particulates, ASAE Paper No. 87–6538, 1987.


33

Heat and Mass Transfer Modeling in Microwave and Spouted Bed Combined Drying of Particulate Food Products H. Feng and J. Tang

CONTENTS 33.1 33.2

Introduction Model Development 33.2.1 Transport Relations 33.2.2 Mass Balance 33.2.3 Energy Balance 33.2.4 Governing Equations 33.2.5 Initial and Boundary Conditions 33.3 Model Reduction 33.4 Effective Moisture Diffusivity 33.5 Permeability of Apple Tissues 33.6 Dielectric Properties and Microwave Power Absorption 33.7 Numerical Analysis 33.8 Experimental 33.9 Model Validation 33.10 Conclusions Nomenclature Acknowledgments References Appendix


33.1 INTRODUCTION Microwave drying involves simultaneous heat, mass, and momentum transfer, as well as volumetric heat generation. The internal heat generation results in the development of a gas/vapor pressure gradient from interior to product surface that distinguishes microwave drying from other drying methods (Turner and Jolly, 1991). Internal heat generation also produces a positive temperature gradient (from the interior to the surface), in contrast to the negative temperature gradient produced during drying by conventional hot air heating. A comprehensive heat and mass transfer analysis that takes into account the contribution of the pressure gradient to moisture migration is desirable in order to understand the underlying physics involved in microwave drying and to predict temperature and moisture changes during drying. Moisture exists in porous food material in the forms of free water, bound water, and vapor. The migration of moisture through the food can be caused by the individual or combined contributions of moisture, temperature, and pressure gradients. In microwave drying, free water transport has been attributed to diffusion (Torringa et al., 1996; Adu and Otten, 1996) or capillary flow (Wei et al., 1985; Constant et al., 1996; Lian et al., 1997; Turner et al., 1998; Ni et al., 1999). For vapor migration, diffusion (Lian et al., 1997), capillary flow (Turner et al., 1998; Ni et al., 1999) and a combination of capillary flow and diffusion (Chen and Schmidt, 1990; Constant et al., 1996) have been considered as transport mechanisms. Few studies have considered the migration of bound water. In studies regarding the mechanism of bound water transfer, Turner et al. (1998) assumed that bound water transfer was caused by diffusion, while Chen and Pei (1989) employed a capillary mechanism to characterize bound water flow. Knowledge of thermophysical and dielectric properties is essential to a heat and mass transfer analysis in microwave drying. There is a lack of understanding of the thermophysical and dielectric properties needed in such an analysis, especially for food materials. Moisture diffusivity, permeability, and dielectric properties were used in this study to account for moisture gradient driven flow, pressure gradient driven flow, and microwave heat generation. It is important to understand the effect of moisture content and product temperature on these properties. In order to understand more about microwaves and their application to drying, it is necessary to develop a comprehensive heat and mass transfer model for microwave and spouted bed combined drying of hygroscopic porous food products; to determine moisture diffusivity, permeability, and dielectric properties for a model food, apple tissue; and to experimentally validate the model developed in this study.

33.2 MODEL DEVELOPMENT In this study, drying of a particulate product was conducted in a spouted bed placed in a microwave cavity. Microwave energy was applied to the cavity at a constant and adjustable rate. The hot air velocity was set such that the aerodynamic conditions needed to sustain stable fluidization in the spouted bed could be realized. The pneumatic agitation in the spouted bed helped to improve the heating uniformity among particles during microwave drying. The hot air flow was at constant temperature.


The following assumptions were used in model development: the material is homogeneous and isotropic on the macro scale; transport of free water is governed by the generalized Darcy’s law; bound water transport is driven by chemical potential difference (Stanish et al., 1986; Gong, 1992); and transport in the gas phase is determined by the contribution of convection and diffusion. Other assumptions included: 1. Local thermodynamic equilibrium exists, that is, the solid, liquid, and gas phases are at the same average temperature at any moment in a control volume. 2. Solid, liquid, and gas phases are continuous. 3. The binary gas mixture of air and vapor obeys the ideal gas law. 4. Vapor pressure as a function of moisture content and temperature can be estimated using sorption isotherms. 5. The model material used in this study, diced apples, can be treated as equivalent spheres. 6. The diced apples are exposed to a uniform microwave field. 7. Electromagnetic field intensity is uniform throughout diced apples. The governing equations for microwave drying were derived at a macroscopic level. Variables examined in this study are averaged values over a control volume. This approach was first proposed by Whitaker (1977) and has been widely used in multiphase heat and mass transfer studies related to drying of porous materials (Bories, 1991).

33.2.1 TRANSPORT RELATIONS Fluid velocities in a multiphase porous food system are given by the generalized Darcy’s law: r r Kk rf uf = − (∇Pg − ∇Pc − ρf g) µf

(33.1)

Kk rg r r ug = − (∇Pg − ρg g) µg

(33.2)

where the capillary pressure Pc = Pg – Pf. Diffusion of vapor and air is governed by Fick’s law: r r ρ  jv = − ja = − ρg D av ∇ v   ρg 

(33.3)

In many food materials, bound water migration is important. It has been found that migration of the bound water cannot be simply treated as a diffusion process (Chen and Pei, 1989). In the present study, a universal driving force, the chemical


potential gradient, was employed as the driving force for the bound water flow (Gong, 1992). The bound water flux is then given by: n b = D b (1 − ε ′)

∂ µb ∂z

(33.4)

From the local thermodynamic equilibrium assumption, the chemical potential of bound water, µ′b, equals the chemical potential of vapor, µ′v. Hence thermodynamic relations for vapor can be used to express the bound water flux. A detailed derivation can be found elsewhere (Stanish et al., 1986), and the resulting equation is given in Equation (33.7). Thermal diffusion through a food material is controlled by Fourier’s law: r q = − λ eff ∇T

(33.5)

33.2.2 MASS BALANCE The mass balance equations for free water, bound water, vapor, and air can be written as:

Free water:

(1 − ε )ρs

∂Xf r ˙ + ∇ ⋅n f = − m ∂t

(33.6)

Bound water:

(1 − ε )ρs

∂X b r ˙b + ∇⋅ n b = − m ∂t

(33.7)

∂X v r ˙ +m ˙b + ∇⋅ n v = m ∂t

(33.8)

∂Xa r + ∇⋅ n a = 0 ∂t

(33.9)

(1 − ε )ρs

Vapor:

(1 − ε )ρs

Air:

In Equations (33.6–9), the mass fluxes are specified by: Free water:

Bound water:

Vapor:

Kk rf Kk rf r r n f = ρf u f = − ρf ∇Pl = − ρf ∇( Pg − Pc ) µf µf

(33.10)

  ε ∇Pv Sv r r n b = ρb u b = −ρb D b (1 − ε ′) − ∇T Mv   ρv

(33.11)

r ρ  Kk rg r r n v = ρv u v + jv = − ρv ∇Pg − ρg Dav ∇ v  µg  ρg 

(33.12)


Air:

r ρ  Kk rg r r n a = ρa u a + ja = −ρa ∇Pg + ρg Dav ∇ v  µg  ρg 

(33.13)

Gas:

Kk rg r r v n g = n a + n v = − ρg ∇Pg µg

(33.14)

In Equation (33.10), the free water transfer caused by gravity is ignored, and in Equation (33.14), ρg = ρa + ρv.

33.2.3 ENERGY BALANCE The total energy balance for a representative elementary volume is written as (Bird et al., 1960): r r ∂ DP +Φ (ρ h ) + ∇ ⋅(ρ uh ) = − (∇ q ) − ( τ: ∇ u) + ∂t Dt

(33.15)

r The viscous dissipation term ( τ:∇µ ) and pressure work term (DP/Dt) are usually negligible. Hence, Equation (33.15) reduces to an enthalpy balance equation: r ∂ (ρ h ) + ∇ ⋅(ρ uh ) = − (∇ q ) + Φ ∂t

(33.16)

ρ h = ρs h s + ρv h v + ρa h a + (ρf + ρb ) h l

(33.17)

r r r r r ρ uh = ρv u v h v + ρa u a h a + (ρf u f + ρb u b ) h l

(33.18)

where

33.2.4 GOVERNING EQUATIONS Due to the geometric symmetry, moisture transfer was assumed to take place only in the radial direction. Hence, a one-dimensional heat and mass transfer problem in spherical coordinates was considered. The total moisture transport equation can be obtained by adding Equations (33.6) to (33.8); the temperature equation was obtained by substituting Equations (33.5), (33.17), and (33.18) into the enthalpy balance equation (Equation (33.16)), and the total pressure equation was obtained from the air balance equation (Equation (33.9)). The resulting drying equations are as follows: ∂ Pg  ∂Xl 1 ∂  ∂T 2 ∂Xl = 2 + DT r 2 + DP r 2 D X r  ∂t ∂r ∂r ∂r  r ∂r 


(33.19)

C TX

∂ Pg  ∂Xl ∂T ∂T 1 ∂  2 ∂Xl + C TT = 2 + D TT r 2 + D TP r 2 D r +Φ r ∂ r  TX ∂t ∂t ∂r ∂r ∂r 

C PX

(33.20)

∂ Pg ∂ Pg  ∂Xl ∂T ∂T 1 ∂  a 2 ∂Xl + C PT + C PP = 2 + D aT r 2 + D aP r 2 D X r  ∂t ∂t r ∂r  ∂t ∂r ∂r ∂r  (33.21) k

In Equations (33.19) to (33.21), Dij and Cij are kinetic and capacity coefficients, respectively. The subscripts i and j can be temperature, moisture, or pressure, while the superscript k denotes air, free water, bound water, or vapor. The expressions for these coefficients are detailed in this chapter’s Appendix

33.2.5 INITIAL

AND

BOUNDARY CONDITIONS

Initial conditions can be written as: Xl

t =0

= X0

T t =0 = T0

Pg

t =0

= Patm

(33.22)

Because of the pneumatic agitation in a spouted bed, it is reasonable to assume that moisture arriving at a sample surface from the interior evaporates immediately and is carried away by the hot air stream. The moisture transfer boundary condition in spherical coordinates is then given by: ∂ Pg   ∂Xl ∂T + DT + DP −(1 − ε )ρs  D X ∂r ∂r ∂ r  

= ε h m (ρ vs − ρ v∞ )

(33.23)

r =R0

An energy balance over the interface can be set up to obtain the temperature boundary condition: − λ eff

∂T ∂r

r =R0

∂ Pg   ∂Xl ∂T = h ( T∞ − Ts ) − ∆h v (1 − ε )ρs  D fX + D Xb + D fT + D Tb + D fP  ∂r ∂r ∂r   (33.24)

(

)

(

)

Pressure at the surface of a spherical particle can be written as: P

r =R0

= Patm

(33.25)

The symmetry condition at the center of the sphere must be satisfied: ∂ Xl ∂r


= r=0

∂T ∂r

= r=0

∂ Pg ∂r

=0 r=0

(33.26)

33.3 MODEL REDUCTION To simplify the microwave drying equations given in Equations (33.19–33.21) and (33.22–33.26) for numerical analysis and further elucidation of the mechanisms, a scaling method was used to analyze the magnitude of each term in the equations. The following scaling groups were used (Plumb et al., 1986): r∗ =

r ; R0

X∗l = tc =

t∗ =

Xl ; X0

t ; tc

X∗b =

T∗ =

Xb − Xe ; X0 − Xe

µ a∞ R 20 R0 ; = u c K 0 ( Pmax − Patm )

T − T0 ; Tmax − T0 ρ∗vs = uc =

Pg∗ =

Pg − Patm Pmax − Patm

ρvs P T = vs ∞ ; ρ v∞ Pv∞ Ts

K 0 Pmax − Patm ; µ a∞ R0

(33.27) Φ∗ =

Φ Φ0

In the scaling analysis, a comparison of the magnitude of each dimensionless group helped to determine which terms were insignificant and hence negligible. In the analysis, a term with a coefficient at least one magnitude smaller than the other terms was considered negligible and was eliminated from the drying equations. The magnitude for each term was assessed by using literature values of the physical, thermal, and transport parameters in the capacity and kinetic coefficients. For moisturedependent parameters, two extreme moistures, X0 = 7.0 (db) and Xl = 0.1 (db), were used to estimate the range of the coefficients. For temperature-dependent parameters, an average drying temperature of 343°K was used. Detailed analyses can be found elsewhere (Feng, 2000). Simplified drying equations can be written as: ∗ Tc ∂ X∗l ( Pmax − Patm ) ρf Kk rf ∂ Pg   ∂ X1∗ ∂  ∗2  = r D X +  (33.28)  ∂ t∗ R 20 X 0 r ∗2 ∂ r ∗   eff 0 ∂ r ∗ 1− ε ρs µ f ∂ r ∗  

(ρC p )eff

∗ Tmax − T0 ∂ T∗ Tmax − T0 ∂  ∗2 ∂ T  ˙ + Φ Φ∗ = λ r   + ∆h v M 0 ∂ t∗ ∂ r∗  tc R 20 r ∗2 ∂ r ∗  eff

(1 − ε ) =

(33.29)

 ε M a Pmax − Patm ∂ Pg∗ ρaρs X 0 ∂ X∗l  1 − ε ρs + 1 − Xl  ∗ ρf t c ∂ t ε ρf + ρ b  R ′ T tc ∂ t∗  1 R r

2 ∗2 0

∂ ∂ r∗

 ∗2 Kk rg ∂ Pg∗   r ( Pmax − Patm )ρa µ g ∂ r ∗  

(33.30)

The simplified boundary conditions are given by: ∗ ∂ X∗ P − Patm ρf Kk rf ∂ Pg  1  Deff X 0 ∗l + max ∗   1 − ε ρs µ f ∂ r  R0  ∂r


=− r ∗ =1

ερv∞ h m ∗ (ρ − 1) (1 − ε )ρs vs

(33.31)

−

λ eff (Tmax − T0 ) ∂ T∗ ∂ r∗ R0 −

∆h v (1 − ε )ρs R0

= h (T∞ − T0 ) − hTs∗ (Tmax − T0 )

∗

r =1

∗  ∂ X∗ ( P − P ) ρ Kk rf ∂ Pg  Deff X 0 ∗l + max atm f  1− ε ρs µ f ∂ r ∗  ∂r 

P*| ∂ X ∗l ∂r∗

= r∗ = 0

r*=1

=0

∂T∗

=

∂r∗

∗

r =0

(33.32) (33.33)

∂ Pg∗ ∂r∗

=0

(33.34)

∗

r =0

It can be seen from Equations (33.28–33.34) that terms with capillary pressure were eliminated through the scaling analysis. The remaining unknown parameters in Equations (33.28–33.34) include effective moisture diffusivity, intrinsic and relative permeabilities, and dielectric properties. Experiments were designed and conducted in this study to measure these properties as function of moisture and temperature. All other thermal and transport relations for apples are from the literature and are listed in Table 33.1. TABLE 33.1 Correlations for Physical, Thermal, Thermodynamic, Dielectric, and Mass Transfer Parameters Parameter

Correlation

Reference

1.282 + 1.65(1.899 + X l )X l (1 + 1.65X l )(1.899 + X l )

Porosity, ε'

ε′ =

Viscosity of free water, µf

µ f (T) = µ f 0 exp(a − bT + cT 2 + dT 3 − eT 4 )

Feng, 2000 Turner, 1991

−4

µf0 = 1 × 10 ; a = 29.619; −4 b = 0.152; c = 0.648 × 10 ; −6 −8 d = 0.815 × 10 ; e = 0.120 × 10 Viscosity of gas, µg

µ g (T) = µ g 0 {T 1/ 2 / (a + b / T − c / T 2 + d / T 3 )}

Turner, 1991

a = 0.672; b = 85.229; c = 2111.475; d = 106417.0; µg0 = 1 × 10 Latent heat of water, ∆hv Effective thermal conductivity of apple, λeff Thermal conductivity of air, λa Effective specific heat of apple, Cpeff Air–vapor binary diffusivity, Dav


−6

6

∆hv = 2.792 × 10 − 160 T − 3.43 T λeff = 0.12631 + 0.0595 Xl

2

λ a = 0.0035 + 7.67 × 10 −5 ∗ T C peff = 1415 +

Turner, 1991 Niesterus, 1996

27.21 X l 1 + Xl

D av = 2.20 × 10 −5

101325  T  Pg  273.15 

Stanish et al., 1986 Donsì et al., 1996

1.75

Stanish et al., 1986

33.4 EFFECTIVE MOISTURE DIFFUSIVITY The effective moisture diffusivity Deff combines the contributions of both bound water and free water. From Equation (33.28), it can be seen that Deff can be determined with a hot air–only drying test at moderate temperatures, in which the internal vapor generation is negligible. Drying tests with apple tissues were conducted, and Deff was obtained using the slope method (Feng et al., 2000a). Deff as function of moisture content and temperature is given in the following equation:  a + a2 ∗ Xl  D eff = a 0 ∗ exp − 1    T −4

3

(33.35) 2

where a0 = 6.273 × 10 ; a1 = 5.843 × 10 ; and a2 = –2.038 × 10 .

33.5 PERMEABILITY OF APPLE TISSUES Intrinsic and relative permeabilities of apple tissues were measured with a set-up designed based on the following relation:

Q g = ug A = − A

K (ε ) k rg ∆Pg µg

H

(33.36)

From Equation (33.36), when the gas phase flow rate Qg and pressure drop over a specimen ∆Pg are determined, the product of intrinsic permeability K and gas relative permeability krg can be obtained. The separation of K with krg was achieved by conducting a dry sample test in which krg = 1 and thus K as function of porosity could be attained. Details of the set-up and experimental procedures can be found in Feng et al. (2002b). The intrinsic permeability K(ε) of apple tissue as a function of porosity was determined by the above method and can be fitted to a Kozeny–Carman equation: K(ε ) = 5.578 × 10 −12

ε3 (1 − ε ) 2

(33.37)

The gas and liquid relative permeabilities for apple tissue were also determined (Feng et al., 2002b) and can be correlated to saturation level S using the following empirical equations: k rg = 1.01 e −10.86 S k rl = S3


(33.38) (33.39)

33.6 DIELECTRIC PROPERTIES AND MICROWAVE POWER ABSORPTION Dielectric constant ε* and loss factor ε″ are important parameters in the determination of microwave penetration depth and volumetric heat generation. The open-ended coaxial probe technique was used to measure ε* and ε″ for apple tissues over a moisture content range of 4 to 87.5% (wb) and at 22 and 60°C. From measured results with apple tissues, a correlation of loss factor ε″ with moisture and temperature changes was obtained (Feng et al., 2000): ε ′′ = a 1 + a 2 T + a 3 T 2 + a 4 X + a 5 X T + a 6 X T 2 + a 7 X 2 + a 8 X 2 T + a 9 X 3 (33.40) where a1 = −23.5999; a2 = 0.158233; a3 = −0.000256978; a4 = −1.87998; a5 = −6 −5 0.00768435; a6 = −5.6363 × 10 ; a7 = 0.0289568; a8 = −7.66337 × 10 ; a9 = −5 −4.09947 × 10 . Microwave power absorption by the sample during drying can be obtained using: Φ = Φ0

ε ′′ ε ′′0

(33.41)

where Φ0 is the absorbed microwave power at the beginning of a microwave and spouted bed drying (MWSB) drying test. It is given by Φ0 = Φinput − Φreflect, the difference between measured input and reflected power. ε ′′0 is the loss factor corresponding to the value at the beginning of a drying test.

33.7 NUMERICAL ANALYSIS The simplified drying equations (Equations (33.28) to (33.34)) are coupled and highly nonlinear. The finite difference method was used to find numerical solutions. The analytical domain was a sphere with a diameter of 5 mm divided into 15 layers, each with a thickness δ = 1/3 mm. A time step of 1 sec was selected in the simulation. The Crank–Nicolson scheme was used to discretize the partial differential equations for the internal nodes. The boundary conditions at nodes 0 and 16 were discretized with a three-point formula to obtain the same accuracy as with internal nodes. The moisture, temperature, and gas pressure equations were solved simultaneously. A program was written using Matlab to implement the simulation. Details of the finite difference formulation can be found elsewhere (Feng, 2000).

33.8 EXPERIMENTAL A microwave and spouted bed combined drying system was developed to conduct drying tests to validate the model simulation. This system consisted of a 2.45 GHz microwave supply system and a hot air system (Figure 33.1). In the microwave supply system, a magnetron generated the microwaves, a wave guide transmitted


1

2

3

4

5

10

11

6

12

7 Air

9

8

FIGURE 33.1 Schematic of 2.45 GHz microwave and spouted bed drying system. 1. Direction coupler; 2. circulator; 3. magnetron; 4. power controller; 5. power meter; 6. temperature controller; 7. HP network analyzer; 8. air pump; 9. heater; 10. three-stub tuner; 11. microwave cavity; 12. spouted bed.

the waves to the cavity, and a directional coupler with power meters measured power components. In the system, a circulator with a water load was installed to absorb the reflected power and a three-stub tuner was used to adjust the matching impedance of the drying cavity. The power generated from the magnetron could be continuously adjusted using the power controller. Both the incident and the reflected power were measured using HP power meters so that the power absorbed by the drying sample could be accurately determined. Diced Red Delicious apples (Malus domestica Borkh) with average initial moisture content of 20.2% (wb) were used as the model food in the validation. The spouted bed superficial air velocity was 2.1 m/sec in all experiments, the velocity at which a stable particle circulation can be achieved during drying to ensure uniform heating. Forty grams of diced apples were used in each drying test. Moisture loss was monitored by periodically weighing the sample on an electric balance. The average moisture content of samples was determined using the vacuum oven method (AOAC, 1990). The drying temperature of the dice was measured at the core of ten randomly chosen apple pieces with a type T thermocouple (response time 0.8 sec) at predesignated time intervals. The pressure increase during microwave heating was measured using fresh apples with a fiber optical pressure probe, which has a resolution of 1 kPa. The pressure probe was led through a hole opened on the wall of the spouted bed and inserted into the center of a fresh-cut apple sample sealed with


vacuum grease to prevent vapor leakage. A data acquisition system was used to record the pressure data.

33.9 MODEL VALIDATION Figure 33.2 compares the predicted and experimentally determined moisture contents for MWSB and spouted bed (SB) drying. In MWSB drying, the data reported were the averages of four replicates, and drying tests were conducted at microwave power density of 4 W/g (wb) and hot air temperature of 70°C. For SB alone drying, the same air conditions as for MWSB drying were used, and experiments were performed in triplicate. Relatively small error bars in Figure 33.2 indicate good repeatability in drying tests. The simulated moisture content followed an exponential decay curve that characterizes drying in the falling rate period for hygroscopic materials. At the beginning of drying, the simulation slightly underestimated moisture loss. The assumption of spherical geometry for apple dice might be a major reason for this discrepancy due to an underestimation of the surface-to-volume ratio. Generally, model predictions were in good agreement with experiments for both drying methods. The temperatures measured from diced apples and predicted with the model are compared in Figure 33.3 for MWSB (MW power 4 W/g and air temperature 70°C) and SB (air temperature 70°C) drying. Temperature prediction in MWSB drying agreed well with the measured values. At the beginning of drying, the hot air heated the apple dice from outside, and the microwaves heated volumetrically. This led to a rapid increase in product temperature. When the apple dice surface temperature surpassed the air temperature, the air started to cool the apple dice. The center temperature, however, continued to increase as a result of microwave heating until it reached about 83°C, when a balance between the energy supplied by microwaves and the heat losses due to surface convective cooling and evaporation was established.

Moisture Content (wb), %

25 Model prediction SB drying test MWSB drying test

20 15 10 5 0 0

10

20

30

40

50

60

70

80

90

Drying Time, min

FIGURE 33.2 Moisture content comparison between model predictions and experimental results for microwave and spouted bed (MWSB) drying and spouted bed (SB) drying. (From Feng, H. et al., AICHE J., 47(7), 1499–1509, 2001. With permission.)


90 80

Bed temperature

Temperature, °C

70 60 50 40 30

Model prediction SB drying test MWSB drying test

20 10 0 0

10

20

30

40

50

60

70

80

90

Drying Time, min

FIGURE 33.3 Temperature comparison between model predictions and experimental results for MWSB and SB drying. (From Feng, H. et al., AICHE J., 47(7), 1499–1509, 2001. With permission.) 10

Over-pressure, kPa

8 6 4 Experiment #1 Experiment #2 Experiment #3 Model

2 0 0

1

2

3

4

5

Drying Time, min

FIGURE 33.4 Comparison of predicted and measured pressures for MWSB drying with microwave power 10 W/g, hot-air temperature 70°C and initial moisture content 84% (wb). (From Feng, H. et al., AICHE J., 47(7), 1499–1509, 2001. With permission.)

The product temperature remained nearly constant throughout the rest of drying. This is a unique feature associated with MWSB drying. The high surface heat transfer in the spouted bed helped to maintain a nearly constant drying temperature. This temperature leveling effect prevents the product from overheating and charring. Therefore, MWSB drying has potential application in drying of heat-sensitive food products. In SB drying, the sample temperature approached air temperature at a slightly lower rate compared to that of MWSB drying, and the predicted temperature agreed well with the bed temperature. The over-pressure readings measured with the fiber optical probe are compared to model prediction in Figure 33.4. Experiments to validate model prediction of pressure were conducted at conditions different from those for moisture and


temperature verification. This is because at the moisture content of commercial evaporated diced apple (20.2%), and the power level used in moisture and temperature validation tests, the over-pressure generated is lower than the threshold of the pressure probe (1 kPa). To generate a vapor pressure above the threshold of the pressure probe (1 kPa), fresh apple samples in a cylindrical shape (8 mm in diameter and 15 mm in length), and microwave power of 10 W/g and sample moisture of 84% (wb) were used in the tests. The measured pressure increased during the first two minutes of drying to reach a peak and started to decrease thereafter. At this stage, the moisture content of the sample remained high, and hence volumetric heating was still intensive enough to maintain the pressure. Therefore, the measured pressure decrease might have been caused by a gradual failure of the seal to the pressure probe.

33.10 CONCLUSIONS The comprehensive heat and mass transfer drying model developed in this study was demonstrated to be an effective tool to predict moisture, temperature, and pressure history and distribution for MWSB of particulate foods. The model prediction of moisture, temperature, and pressure agreed with experimental results. A temperature leveling effect was predicted and in agreement with experiment results. This unique temperature leveling effect in MWSB drying makes it possible to use this method for drying heat-sensitive food products.

NOMENCLATURE C Cp D Dav Db g h ∆hv j kr K m ˙ M n P Pc q R R′ S t

Capacity coefficient −1 −1 Specific heat, J·kg K Kinetic coefficient; material derivative 2 −1 Binary air–vapor diffusivity, m ·sec 2 −1 Bound water diffusivity, m ·sec −2 Gravitational acceleration, m·sec −1 −2 −1 Enthalpy, J·kg ; surface heat transfer coefficient, W·m K −1 Latent heat of free water, J·kg −2 −1 Diffusive mass flux, kg·m ·sec Relative permeability 2 Intrinsic permeability, m −3 −1 Moisture evaporation rate, kg·m ·sec −1 Molar mass, kg·mol −2 −1 Mass flux, kg·m sec Pressure, Pa Capillary pressure, Pa −2 −1 Heat flux, J·m ·sec −1 Sample radius, m −1 −1 Universal gas constant, J·mol K Saturation Time, sec


T u X

Average absolute temperature, K −1 Spouted bed air superficial velocity, m·sec −1 Moisture content (dry basis), kg H2O·(kg solid)

GREEK SYMBOLS ε ε′ ε* ε″ Φ λ µ µ′ ρ ρs τ

Porosity [(gas + liquid)/total volume], tolerance limit on iteration scheme Porosity defined in Table I (gas volume/total volume); dielectric constant Dielectric constant Dielectric loss factor −3 Heat source, W·m −1 −1 Thermal conductivity, W·m ·K ; wave length, m −1 −1 Dynamic viscosity, kg·m ·sec Chemical potential −3 Density, kg·m −3 Solid density (solid mass/solid volume), kg·m −2 Shear stress tensor, kg·m

SUBSCRIPTS 0 a atm b eff f g l max P s T v w X ∞

AND

SUPERSCRIPTS

At saturated condition or free space; initial condition (refers to fresh sample) Air Atmospheric pressure Bound water, bulk density Effective Free water Gas = air + vapor Liquid = free water + bound water Maximum values Pressure Solid, or relating to surface Temperature Vapor Total moisture = free water + bound water + vapor Moisture Surrounding

ACKNOWLEDGMENTS Support for this research has been provided by Washington State University IMPACT Center and the Northwest Center for Small Fruits Research. Special thanks go to Dr. Ovid Plumb and Dr. Ralph Cavalieri for their valuable input to this work. The authors would like to thank Timothy Wig and Wayne DeWitt for assistance in developing the 2.45 GHz MWSB drying system, and TreeTop, Inc., Selah, WA, for donating evaporated apples.


REFERENCES Adu, B. and Otten, L., Modeling microwave heating characteristics of granular hygroscopic solids, J. Microwave Power Electromag. Energy, 31, 35–42, 1996. AOAC, Official Methods of Analysis, 15th ed, Association of Official Analytical Chemists, Washington, DC, 1990. Bird, R.B., Stewart, W.E., and Lightfoot, N.E., Transport Phenomena, John Wiley and Sons, New York, 1960. Bories, S.A., Fundamentals of drying of capillary-porous bodies, in Convective Heat and Mass Transfer in Porous Media, Kakac, S. et al. (Eds.), Kluwer Academic Publishers, Dordrecht, 1991. Chen, P. and Pei, D.C.T., A mathematical model of drying processes, Int. J. Heat Mass Transfer, 32, 297–310, 1989. Chen, P. and Schmidt, P.S., An integral model for drying of hygroscopic and nonhygroscopic materials with dielectric heating, Drying Technol., 8, 907–930, 1990. Constant, T., Moyne, C., and Perré, P., Drying with internal heat generation: theoretical aspects and application to microwave heating, AIChE J., 42, 359–368, 1996. Donsì, G., Ferrari, G., and Nigro, R., Experimental determination of thermal conductivity of apple and potato at different moisture contents, J. Food Eng., 30, 263–268, 1996. Feng, H., Microwave Drying of Particulate Foods in a Spouted Bed. Ph.D. dissertation, Washington State University, 2000. Feng, H., Tang, J., and Cavalieri, R., Dielectric properties of dehydrated apples as affected by moisture and temperature, Trans. ASAE, 45(1), 129–135, 2002a. Feng, H., Tang, J., Cavalieri, R.P., and Plumb, D.A., Heat and mass transport in microwave drying of porous materials in a spouted bed, AICHE J., 47(7), 1499–1509, 2001. Feng, H., Tang, J., and Dixon-Warren, St.J., Determination of moisture diffusivity of Red Delicious apple tissues by thermogravimetric analysis, Drying Technol., 18(6), 1183–1199, 2000. Feng, H., Tang, J., and Plumb, G., Intrinsic and relative permeability for flow of humid air in unsaturated apple tissues, J. Food Eng., accepted, 2002b. Gong, L., A Theoretical, Numerical and Experimental Study of Heat and Mass Transfer in Wood during Drying, Ph.D. dissertation, Washington State University, 1992. Lian, G., Harris, R.E., and Warboys, M., Coupled heat and moisture transfer during microwave vacuum drying, J. Microwave Power Electromag. Energy, 32, 34–44, 1997. Ni, H., Datta, A.K., and Torrance, K.W., Moisture transport in intensive microwave heating of biomaterials: a multiphase porous media model, Int. J. Heat Mass Transfer, 42, 1501–1512, 1999. Niesterus, R., Changes of thermal properties of fruits and vegetables during drying, Drying Technol., 14, 415–422, 1996. Plumb, O.A., Couey, L.M., and Shearer, D., Contact drying of wood veneer, Drying Technol., 4, 387–413, 1986. Stanish, M.A., Schajer, G.S., and Kayihan, F., A mathematical model of drying for hygroscopic porous media, AIChE J., 32, 1301–1311, 1986. Torringa, E.M., van Dijk, E.J., and Bartels, P.S., Microwave puffing of vegetables: Modeling and measurements, in Proceedings of 31st Microwave Power Symposium, Int. Microwave Power Inst., Manassas, VA, 1996. Turner, I.W., The Modeling of Combined Microwave and Convective Drying of a Wet Porous Material, Ph.D. thesis, University of Queensland, Australia, 1991. Turner, I.W. and Jolly, P.G., Combined microwave and conventional drying of a porous material, Drying Technol., 9, 1209–1269, 1991.


Turner, I.W., Puiggali, J.R., and Jomaa, W., A numerical investigation of combined microwave and convective drying of a hygroscopic porous material: a study based on pine wood, Trans. IChemE, 76, Part A, 193–209, 1998. Wei, C.K., Davis, H.T., Davis, E.A., and Gorden, J., Heat and mass transfer in water-laden sandstone: microwave heating, AIChE J., 31, 842–848, 1985. Whitaker, S., Simultaneous heat, mass, momentum transfer in porous media: a theory of drying, Adv. Heat Transfer, 13, 119–203, 1977.

APPENDIX D fX = −

1 ρf Kk rf  ∂ Pc  ; 1 − ε ρs µ f  ∂ X w  T

D fP =

1 ρf Kk rf 1 − ε ρs µ f

D Xb =

1 − ε ′ D b ε R ′ T  ∂ Pv  1 − ε ρs Pv M v  ∂ X w  T

D Tb = −

D fT = −

1 ρf Kk rf  ∂ Pc    1 − ε ρs µ f  ∂ T  X

ε R ′ T  ∂ Pv   1 − ε ′ D b  Sv b  −   ; D P = 0 1 − ε ρs  M v Pv M v  ∂ T  X  w 

D Xv =

 ∂ Pv  Ma M v (1 − ε )ρs R ′ T( Pg M a + (M v − M a )Pv )  ∂ X w  T

D Tv =

 ∂ Pv  Ma M v   (1 − ε )ρs R ′ T( Pg M a + (M v − M a )Pv )  ∂ T  X

D Pv =

Dav Pg

Dav Pg

1 (1 − ε )ρs

C TX = ∆h v

Mv R′

 P M Kk rg  D av M a M v Pv −  v v  R ′ T( Pg M a + (M v − M a )Pv )   R ′ T µ g  (1 − ε )ρs Pv  (1 − ε )ρs  1 ∂Pv  + ε − X −  ρf + ρ b l  T ∂ X l   ρf + ρ b T 

C TT = (ρC p ) eff + ∆h v


w

(1 − ε )ρs  ∂( Pv / T) Mv  X ε − ρf + ρb l  ∂ T R′ 

w

D TX = ∆h v (1 − ε )ρsD Xv ;

D TT = λ eff + ∆h v (1 − ε )ρsD Tv

D TP = ∆h v (1 − ε )ρsD Pv

C PX =

 1 ∂Pv  ε M a  Pg − Pv  1 − ε ρs   1 − ε ρs − 1 − Xl  −    ε ρf + ρ b  T ∂ X l  R ′  T  ε ρf + ρb  

C PT =

εMa R′

 1 − ε ρs   Pg ∂( Pv / T)  1 − ε ρ + ρ X l   T 2 − ∂ T    f b

C PP =

εMa R′

 1 − ε ρs  1 1 − ε ρ + ρ X l  T   f b  ∂ Pv   ∂X   wT

D aX = − (1 − ε )ρs D Xv = − D av

Pg M a Mv R ′ T Pg M a + (M v − M a )Pv

D aT = − (1 − ε )ρs D Tv = − D av

Pg M a Mv  ∂ Pv    R ′ T Pg M a + (M v − M a )Pv  ∂ T  X

D aP =

( Pg − Pv )M a Kk rg R′T


µg

+ D av

Mv Pv M a R ′ T Pg M a + (M v − M a )Pv

w

Transport Phenomena in Food Processing (Food Preservation Technology)

Progress in Food Preservation

Food Preservation Techniques (Woodhead Publishing in Food Science and Technology)

Food Preservation Techniques (Woodhead Publishing in Food Science and Technology)

Unit Operations in Food Engineering (Food Preservation Technology)

Fats in Food Technology (Sheffield Food Technology)

Handbook of Vegetable Preservation and Processing (Food Science and Technology)

Food Preservation Process Design (Food Science and Technology)

Handbook of Food Preservation

Food Preservation Techniques

Food preservation techniques

Food biochemistry and food processing

Food Biodeterioration and Preservation

Proteins in Food Processing

Food Biochemistry and Food Processing

Food Biochemistry And Food Processing

Hydrocolloids in Food Processing

Proteins in Food Processing

Ozone in Food Processing

Enzymes in Food Technology

Enzymes in Food Technology

Emulsifiers in Food Technology

Organic Acids and Food Preservation

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Cereals Processing Technology (Woodhead Publishing in Food Science and Technology)

Thermal Technologies in Food Processing

Improving Traceability in Food Processing

Fundamentals of Food Processing and Technology

Food Safety in Shrimp Processing

Food Processing Operations Modeling

Food Processing Technology: Principles and Practice, 2nd Edition (Woodhead Publishing in Food Science and Technology)

Transport Phenomena in Food Processing (Food Preservation Technology)

Progress in Food Preservation

Food Preservation Techniques (Woodhead Publishing in Food Science and Technology)

Food Preservation Techniques (Woodhead Publishing in Food Science and Technology)

Unit Operations in Food Engineering (Food Preservation Technology)

Fats in Food Technology (Sheffield Food Technology)

Handbook of Vegetable Preservation and Processing (Food Science and Technology)

Food Preservation Process Design (Food Science and Technology)

Handbook of Food Preservation

Food Preservation Techniques

Food preservation techniques

Food biochemistry and food processing

Food Biodeterioration and Preservation

Proteins in Food Processing

Food Biochemistry and Food Processing

Food Biochemistry And Food Processing

Hydrocolloids in Food Processing

Proteins in Food Processing

Ozone in Food Processing

Enzymes in Food Technology

Enzymes in Food Technology

Emulsifiers in Food Technology

Organic Acids and Food Preservation

Organic Acids and Food Preservation

Cereals Processing Technology (Woodhead Publishing in Food Science and Technology)

Thermal Technologies in Food Processing

Improving Traceability in Food Processing

Fundamentals of Food Processing and Technology

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Food Processing Technology: Principles and Practice, 2nd Edition (Woodhead Publishing in Food Science and Technology)

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