Food Properties Handbook, Second Edition (Contemporary Food Science)

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Food Properties Handbook Second Edition

ß 2008 by Taylor & Francis Group, LLC.


Food Properties Handbook Second Edition

Edited by

M. Shafiur Rahman


CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-0-8493-5005-4 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Food properties handbook / edited by M. Shafiur Rahman. -- 2nd ed. p. cm. Previous edition has main entry under Rahman, Shafiur. Includes bibliographical references and index. ISBN-13: 978-0-8493-5005-4 ISBN-10: 0-8493-5005-0 1. Food--Analysis. 2. Food industry and trade. I. Rahman, Shafiur. II. Title. TX541.R37 2009 664--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com


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Contents Preface Acknowledgments Editor Contributors Chapter 1 Food Properties: An Overview Mohammad Shafiur Rahman Chapter 2 Water Activity Measurement Methods of Foods Mohammad Shafiur Rahman and Shyam S. Sablani Chapter 3 Data and Models of Water Activity. I: Solutions and Liquid Foods Piotr P. Lewicki Chapter 4 Data and Models of Water Activity. II: Solid Foods Piotr P. Lewicki Chapter 5 Freezing Point: Measurement, Data, and Prediction Mohammad Shafiur Rahman, K.M. Machado-Velasco, M.E. Sosa-Morales, and Jorge F. Velez-Ruiz Chapter 6 Prediction of Ice Content in Frozen Foods Mohammad Shafiur Rahman Chapter 7 Glass Transitions in Foodstuffs and Biomaterials: Theory and Measurements Stefan Kasapis Chapter 8 Glass Transition Data and Models of Foods Mohammad Shafiur Rahman


Chapter 9 Gelatinization of Starch Shyam S. Sablani Chapter 10 Crystallization: Measurements, Data, and Prediction Kirsi Jouppila and Yrjö H. Roos Chapter 11 Sticky and Collapse Temperature: Measurements, Data, and Predictions Benu P. Adhikari and Bhesh R. Bhandari Chapter 12 State Diagrams of Foods Didem Z. Icoz and Jozef L. Kokini Chapter 13 Measurement of Density, Shrinkage, and Porosity Panagiotis A. Michailidis, Magdalini K. Krokida, G.I. Bisharat, Dimitris Marinos-Kouris, and Mohammad Shafiur Rahman Chapter 14 Data and Models of Density, Shrinkage, and Porosity Panagiotis A. Michailidis, Magdalini K. Krokida, and Mohammad Shafiur Rahman Chapter 15 Shape, Volume, and Surface Area Mohammad Shafiur Rahman Chapter 16 Specific Heat and Enthalpy of Foods R. Paul Singh, Ferruh Erdo gdu, and Mohammad Shafiur Rahman Chapter 17 Thermal Conductivity Measurement of Foods Jasim Ahmed and Mohammad Shafiur Rahman Chapter 18 Thermal Conductivity Data of Foods Jasim Ahmed and Mohammad Shafiur Rahman


Chapter 19 Thermal Conductivity Prediction of Foods Mohammad Shafiur Rahman and Ghalib Said Al-Saidi Chapter 20 Thermal Diffusivity of Foods: Measurement, Data, and Prediction Mohammad Shafiur Rahman and Ghalib Said Al-Saidi Chapter 21 Measurement of Surface Heat Transfer Coefficient Shyam S. Sablani Chapter 22 Surface Heat Transfer Coefficients with and without Phase Change Liyun Zheng, Adriana Delgado, and Da-Wen Sun Chapter 23 Surface Heat Transfer Coefficient in Food Processing Panagiotis A. Michailidis, Magdalini K. Krokida, and Mohammad Shafiur Rahman Chapter 24 Acoustic Properties of Foods Piotr P. Lewicki, Agata Marzec, and Zbigniew Ranachowski Appendix A Appendix B Appendix C



Preface A food property is a particular measure of a food’s behavior as a matter or its behavior with respect to energy, or its interaction with the human senses, or its efficacy in promoting human health and well-being. An understanding of food properties is essential for scientists and engineers who have to solve the problems in food preservation, processing, storage, marketing, consumption, and even after consumption. Current methods of food processing and preservation require accurate data on food properties; simple, accurate, and low-cost measurement techniques; prediction models based on fundamentals; and links between different properties. The first edition was a well received bestseller, and it received an award. Appreciation from scientists, academics, and industry professionals around the globe encouraged me to produce an updated version. This edition has been expanded with the addition of some new chapters and by updating the contents of the first edition. The seven chapters in the first edition have now been expanded to 24 chapters. In this edition, the definition of the terminology and measurement techniques are clearly presented. The theory behind the measurement techniques is described with the applications and limitations of the methods. Also, the sources of errors in measurement techniques are compiled. A compilation of the experimental data from the literature is presented in graphical or tabular form, which should be very useful for food engineers and scientists. Models can reduce the number of experiments, thereby reducing time and expenses of measurements. The empirical and theoretical prediction models are compiled for different foods with processing conditions. The applications of the properties are also described, mentioning where and how to use the data and models in food processing. Chapter 1 provides an overview of food properties, including their definition, classification, and predictions. Chapters 2 through 4 present water activity and sorption isotherm and include terminology, measurement techniques, data for different foods, and prediction models. Chapters 5 through 12 present thermodynamic and structural characteristics including freezing point, glass transition, gelatinization, crystallization, collapse, stickiness, ice content, and state diagram. Chapters 13 through 15 discuss the density, porosity, shrinkage, size, and shape of foods. Chapters 15 through 23 present the thermophysical properties including specific heat, enthalpy, thermal conductivity, thermal diffusivity, and heat transfer coefficient. Chapter 24 provides the acoustic properties of foods. This second edition will be an invaluable resource for practicing and research food technologists, engineers, and scientists, and a valuable text for upper-level undergraduate and graduate students in food, agriculture=biological science, and engineering. Writing such a book is a challenge, and any comments to assist in future compilations will be appreciated. Any errors that remain are entirely mine. I am confident that this edition will prove to be interesting, informative, and enlightening. Mohammad Shafiur Rahman Sultan Qaboos University Muscat, Sultanate of Oman



Acknowledgments I would like to thank Almighty Allah for giving me life and blessing to gain knowledge to update this book. I wish to express my sincere gratitude to the Sultan Qaboos University (SQU) for giving me the opportunity and facilities to initiate such an exciting project to develop the second edition, and supporting me toward my research and other intellectual activities. I would also like to thank all my earlier employers, Bangladesh University of Engineering and Technology, University of New South Wales (UNSW), and HortResearch, from whom I built my knowledge and expertise through their encouragement, support, and resources. I wish to express my appreciation to the UNSW, SQU, and HortResearch library staffs, who assisted me patiently with online literature searches and interlibrary loans. I sincerely acknowledge the sacrifices made by my parents, Asadullah Mondal and Saleha Khatun, during my early education. Appreciation is due to all my teachers, especially Professors Nooruddin Ahmed, Iqbal Mahmud, Khaliqur Rahman, Jasim Zaman, Ken Buckle, Drs. Prakash Lal Potluri and Robert Driscoll, and Habibur Rahman, for their encouragement and help in all aspects of pursuing higher education and research. I would like to express my appreciation to Professor Anton McLachlan, Drs. Saud Al-Jufaily, Yasen Al-Mula, Nadya A-Saadi, and S. Prathapar for their support toward my teaching, research, and extension activities at the SQU. Special thanks to my colleagues Dr. Conrad Perera, Professor Dong Chen, Drs. Nejib Guizani, Ahmed Al-Alawi, Shyam Sablani, Bhesh Bhandar, and Mushtaque Ahmed, and my other research team members, especially Mohd Hamad Al-Ruzeiki, Rashid Hamed Al-Belushi, Salha Al-Maskari, Mohd Khalfan Al-Khusaibi, Nasser Abdulla Al-Habsi, Insaaf Mohd Al-Marhubi, Intisar Mohd Al-Zakwani, and Zahra Sulaiman Al-Kharousi. I owe many thanks to my graduate students for their hard work in their projects related to food properties and building my knowledge base. Special thanks for the contributing authors; it was a great pleasure working with them. I would also like to appreciate the enthusiasm, patience, and support provided by the publisher. I wish to thank my relatives and friends, especially Professor Md. Mohar Ali and Dr. Md. Moazzem Hossain, Dr. Iqbal Mujtaba, and Arshadul Haque for their continued inspiration. I am grateful to my wife, Sabina Akhter (Shilpi), for her patience and support during this work, and to my daughter, Rubaba Rahman (Deya), and my son, Salman Rahman (Radhin), for allowing me to work at home. It would have been very hard for me to write this book without my family’s cooperation and support.



Editor Mohammad Shafiur Rahman is an associate professor at the Sultan Qaboos University, Sultanate of Oman. He has authored or coauthored more than 200 technical articles including 81 refereed journal papers, 71 conference papers, 40 book chapters, 33 reports, 8 popular articles, and 4 books. He is the editor of the internationally acclaimed 2003 bestseller, Handbook of Food Preservation published by CRC Press. He was invited to serve as one of the associate editors for the Handbook of Food Science, Engineering and Technology, and he is one of the editors for the Handbook of Food and Bioprocess Modeling Techniques, also published by CRC Press. Dr. Rahman has initiated the International Journal of Food Properties (Marcel Dekker, Inc.) and has served as its founding editor for more than 10 years. He is a member of the Food Engineering Series editorial board of Springer Science, New York. Presently, he is serving as a section editor for the Sultan Qaboos University journal, Agricultural Sciences. In 1998, he was invited to serve as a food science adviser for the International Foundation for Science (IFS) in Sweden. Dr. Rahman is a professional member of the New Zealand Institute of Food Science and Technology and the Institute of Food Technologists; a member of the American Society of Agricultural Engineers and the American Institute of Chemical Engineers; and a member of the executive committee for International Society of Food Engineering (ISFE). He received his BSc Eng (chemical) (1983) and MSc Eng (chemical) (1984) from Bangladesh University of Engineering and Technology, Dhaka; his MSc (1985) in food engineering from Leeds University, England; and his PhD (1992) in food engineering from the University of New South Wales, Sydney, Australia. Dr. Rahman has received numerous awards and fellowships in recognition of research=teaching achievements, including the HortResearch Chairman’s Award, the Bilateral Research Activities Program (BRAP) Award, CAMS Outstanding Researcher Award 2003, SQU Distinction in Research Award 2008, and the British Council Fellowship. The Organization of Islamic Countries has named Rahman as the fourth ranked agroscientist in a survey of the leading scientists and engineers in its 57 member states.



Contributors Benu P. Adhikari School of Science and Engineering The University of Ballarat Mount Helen, Victoria, Australia

Stefan Kasapis Department of Chemistry National University of Singapore Singapore

Jasim Ahmed Polymer Source, Inc. Dorval, Quebec, Canada

Jozef L. Kokini Department of Food Science Rutgers, The State University of New Jersey New Brunswick, New Jersey

Ghalib Said Al-Saidi Department of Food Science and Nutrition Sultan Qaboos University Muscat, Sultanate of Oman Bhesh R. Bhandari School of Land, Crop and Food Sciences The University of Queensland Brisbane, Queensland, Australia G.I. Bisharat Department of Chemical Engineering National Technical University of Athens Athens, Greece Adriana Delgado School of Agriculture, Food Science and Veterinary Medicine University College Dublin Dublin, Ireland Ferruh Erdogdu Department of Food Engineering University of Mersin Mersin, Turkey Didem Z. Icoz Department of Food Science Rutgers, The State University of New Jersey New Brunswick, New Jersey Kirsi Jouppila Department of Food Technology University of Helsinki Helsinki, Finland


Magdalini K. Krokida Department of Chemical Engineering National Technical University of Athens Athens, Greece Piotr P. Lewicki Department of Food Engineering and Process Management Warsaw University of Life Sciences Warsaw, Poland K.M. Machado-Velasco Chemical and Food Engineering Department University of the Americas–Puebla Cholula, Puebla, Mexico Dimitris Marinos-Kouris Department of Chemical Engineering National Technical University of Athens Athens, Greece Agata Marzec Department of Food Engineering and Process Management Warsaw University of Life Sciences Warsaw, Poland Panagiotis A. Michailidis Department of Chemical Engineering National Technical University of Athens Athens, Greece

Mohammad Shafiur Rahman Department of Food Science and Nutrition Sultan Qaboos University Muscat, Sultanate of Oman

M.E. Sosa-Morales Chemical and Food Engineering Department University of the Americas–Puebla Cholula, Puebla, Mexico

Zbigniew Ranachowski Institute of Fundamental Technological Research Polish Academy of Sciences Warsaw, Poland

Da-Wen Sun School of Agriculture, Food Science and Veterinary Medicine University College Dublin Dublin, Ireland

Yrjö H. Roos Department of Food Science and Technology University College Cork Cork, Ireland Shyam S. Sablani Department of Biological Systems Engineering Washington State University Pullman, Washington R. Paul Singh Department of Biological and Agricultural Engineering University of California, Davis Davis, California


Jorge F. Velez-Ruiz Chemical and Food Engineering Department University of the Americas–Puebla Cholula, Puebla, Mexico Liyun Zheng School of Agriculture, Food Science and Veterinary Medicine University College Dublin Dublin, Ireland

CHAPTER

1

Food Properties: An Overview

Mohammad Shafiur Rahman CONTENTS 1.1 1.2

Definition of Food Property..................................................................................................... 1 Classification of Food Property ............................................................................................... 2 1.2.1 Physical and Physicochemical Properties .................................................................... 3 1.2.2 Kinetic Properties......................................................................................................... 4 1.2.3 Sensory Properties........................................................................................................ 4 1.2.4 Health Properties .......................................................................................................... 5 1.3 Applications of Food Properties .............................................................................................. 5 1.3.1 Process Design and Simulation.................................................................................... 5 1.3.1.1 Process Design .............................................................................................. 6 1.3.1.2 Process Simulation ........................................................................................ 6 1.3.1.3 Continuous Need........................................................................................... 6 1.3.2 Quality and Safety ....................................................................................................... 6 1.3.3 Packaging Design......................................................................................................... 7 1.4 Prediction of Food Properties .................................................................................................. 7 1.5 Conclusion ............................................................................................................................... 7 References ......................................................................................................................................... 8

1.1 DEFINITION OF FOOD PROPERTY A property of a system or material is any observable attribute or characteristic of that system or material. The state of a system or material can be defined by listing its properties (ASHRAE, 1993). A food property is a particular measure of the food’s behavior as a matter, its behavior with respect to energy, its interaction with the human senses, or its efficacy in promoting human health and well-being (McCarthy, 1997; Rahman and McCarthy, 1999). It is always attempted to preserve product characteristics at a desirable level for as long as possible. Food properties, in turn, define the functionality of foods (Karel, 1999). Food functionality, as defined by Karel, refers to the control of food properties that provides a desired set of organoleptic properties, wholesomeness (including health-related functions), as well as properties related to processing and engineering, in particular, ease of processing, storage stability, and minimum environmental impact.


In general, food preservation and processing affects the properties of foods in a positive or negative manner. During food processing, attempts to achieve the desired characteristics can be grouped as (1) controlling food characteristics by adding of ingredients=preservatives or removing components detrimental to quality, (2) applying different forms of energy, such as heat, light, electricity, and physical forces, and (3) controlling or avoiding recontamination. In the investigation of foods in the temperature range between 808C and 3508C many effects can be observed during processing, preservation, and storage. These phenomena may either be endothermic (such as melting, denaturation, gelatinization, and evaporation) or exothermic processes (such as freezing, crystallization, and oxidation). Through precise knowledge of such phase transitions, optimum conditions for safe storage or processing of foods can be defined. In addition to thermal energy, other forms of energy, such as electricity, light, electromagnetism, and pressure are also used in food processing. 1.2 CLASSIFICATION OF FOOD PROPERTY Classifying food properties is a difficult task, and any attempt to do so is likely to be controversial. However, it is necessary to develop a well-defined terminology and classification of food properties (Rahman, 1998). Rahman and McCarthy (1999) attempted to develop a widely accepted classification terminology for food properties. A good classification could facilitate sound interdisciplinary approaches to the understanding of food properties, and of the measurement and use of food property data, leading to better process design and food product characterization. Jowitt (1974) proposed a classification of foodstuffs and their physical properties. Rahman (IJFP, 1998) settled on the list that appears at the end of the first issue of the International Journal of Food Properties, after several revisions based on discussions with many academics and scientists around the world (Table 1.1). The classification now proposed contains four major classes (Rahman and Table 1.1 Food Properties Grouped in the First Issue of International Journal of Food Properties Acoustical properties Colorimetric properties Electrical properties Functional properties Mass transfer properties Mass–volume–area-related properties Mechanical properties Medical properties Microbial death–growth-related properties Morphometric properties Optical properties Physico-chemical constants Radiative properties Respiratory properties Rheological properties Sensory properties Surface properties Thermodynamic properties Textural properties Thermal properties Quality kinetics parameters Source: From Int. J. Food Prop., 1, 78, 1998.


Table 1.2 List of Four Classes of Food Properties Physical and physicochemical properties a. Mechanical properties 1. Acoustic properties 2. Mass–volume–area-related properties 3. Morphometric properties 4. Rheological properties 5. Structural characteristics 6. Surface properties b. Thermal properties c. Thermodynamic properties d. Mass transfer properties e. Electromagnetic properties f. Physicochemical constants Kinetic properties a. Quality kinetic constants b. Microbial growth, decline, and death kinetic constants Sensory properties a. Tactile properties b. Textural properties c. Color and appearance d. Taste e. Odor f. Sound Health properties a. Positive health properties 1. Nutritional composition 2. Medical properties 3. Functional properties b. Negative health properties 1. Toxic at any concentration 2. Toxic after critical concentration level 3. Excessive or unbalanced intake Source: Rahman, M.S. and McCarthy, O.J., Int. J. Food Prop., 2, 1, 1999.

McCarthy, 1999): (1) physical and physicochemical properties, (2) kinetic properties, (3) sensory properties, and (4) health properties (Table 1.2). 1.2.1

Physical and Physicochemical Properties

Physical and physicochemical properties are properties defined, measured, and expressed in physical and physicochemical ways. However, there is no clear dividing line between these two types of properties. Paulus (1989) classified physical properties as mechanical, thermal, transport, and other electrical and optical properties. It is considered misleading to use transport as a subclass of physical properties, since many mechanical, thermal, and electrical properties are considered transport properties, e.g., electrical conductivity and thermal conductivity. Moreover, among thermal properties, specific heat is a constitutive property, whereas thermal conductivity and diffusivity are transport


properties. The classification proposed here is similar to the classification of physical properties proposed by Jowitt (1974): first, two new subclasses, thermodynamic and mass transfer properties, replace Jowitt’s subclass of diffusion-related properties; most of the properties included in Jowitt’s subclass are in fact thermodynamic ones. The new mass transfer properties subclass now proposed includes mass transfer by both diffusion and other mechanisms, and is thus more generic. Second, a new subclass of physicochemical constants has been added. Mechanical properties are related to food’s structure and its behavior when physical force is applied. Structure is the form of building or construction, the arrangement of parts or elements of something constructed or of a natural organism to give an organization of foods. In addition to natural structure, man-made structured foods use assembly or structuring processes to build product microstructure. Examples of essential tools to create microstructure are crystallization, phase inversion, phase transition, glass transition, emulsification (e.g., margarine, ice cream, sauces, and mayonnaise), freeze alignment, foaming (e.g., whipped cream), extrusion, puffing, drying, kneading of dough, and baking. In these products, a complicated multiphase microstructure is held together by binding forces between the various phases. This microstructure leads to acceptance of desired product texture and mouth feel during mastication, which is the key to final product quality and is appreciated by the consumer. The control of the microstructure of man-made structured foods is the key qualitydetermining factor apart from requirements on microbial stability and safety. A detailed review on structuring processes since the past 25 years as well as the challenges that lie ahead has been presented by Bruin and Jongen (2003). In the past 25 years, substantial progress has been made in the understanding and control of product microstructure, and new ways of achieving them have been developed. The mechanical properties based on structure are further classified into six subclasses: acoustic properties, mass–volume–area-related properties, morphometric properties, rheological properties, structural characteristics, and surface properties. Thermal properties are related to heat transfer in food, and thermodynamic properties are related to the characteristics indicating phase or state changes in food. Mass transfer properties are related to the transport or flow of components in food. Electromagnetic properties are related to the food’s behavior with the interaction of electromagnetic energy (e.g., dielectric constant, dielectric loss, and electrical resistance). 1.2.2

Kinetic Properties

Kinetic properties are kinetic constants characterizing the rates of changes in foods. These can be divided into two groups. The first comprises kinetic constants characterizing the rates of biological, biochemical, chemical, physicochemical, and physical changes in food. It could include respiratory constants, rate constant, decimal reduction time, half-life, Arrhenius equation constants, temperature quotient (Q10), and D and z values. The second comprises kinetic constants characterizing the rates of growth, decline, and death of microorganisms in food. It could include properties such as specific growth rate, the parameters of the logistic and Gompertz equations (mathematical models of microbial growth), generation time, square root (Ratkowsky) equation constants, and decimal-reduction time. It should be noted that these properties are not actually properties of food, but properties of microorganisms as moderated by the food they are in (Rahman and McCarthy, 1999). 1.2.3

Sensory Properties

A sensory property can be defined as the human physiological–psychological perception of a number of physical and other properties of food and their interactions. The physiological apparatus (fingers, mouth, eyes, taste and aroma receptors, and ears) examines the food and reacts to the food’s properties. Signals are sent to the brain, which interprets the signals and comes to a decision about the food’s sensory quality; this is the psychological bit. Sensory properties are measured


subjectively using trained and untrained panels, and individuals or consumers. Sensory properties can be subdivided into tactile properties, textural properties, color and appearance, taste, odor, and sound. Tactile properties are perceived as touch, i.e., by the fingers. For example, the surface roughness and softness of a food can be evaluated by touch. The main difference between texture and other sensory attributes is that texture is perceived mainly by biting and masticating, i.e., by the mouth. Many of the sensory properties are related to physical and physicochemical properties as measured objectively with instruments. However, this does not mean that instrumentally measured characteristics are sensory properties. The following discussion could help to highlight the difference. The rheological nature of a food and the food’s texture are two different things. Rheological properties are measured objectively using suitable instruments that allow controlled deformation of the food. Texture, however, has to be measured subjectively. It depends partly, of course, on the food’s rheological properties, but also, potentially, on a number of other properties (e.g., shape, size, porosity, and thermal properties) and on the expectations and prior experience of the person(s) assessing the texture. In many cases, texture can be correlated quite well with an instrumentally measured rheological property (often an empirical or imitative one), but texture as such can be measured only by subjective means (Rahman and McCarthy, 1999). Both subjective and objective methods have their own advantages and limitations. However, food properties measured by subjective methods could be correlated with properties measured by objective methods and this could make the quality control process easy during processing, preservation, and storage. 1.2.4

Health Properties

Health properties relate to the efficacy of foods in promoting human health and well-being. Not all foods consumed are safe; thus foods have positive or negative impacts on health. Positive effects can be subdivided into nutritional composition (as defined in nutritional composition tables), medical properties, and functional properties. Functional properties are those that impact on an individual’s general health, physical well-being, and mental health, and slow the aging process; medical properties are those that prevent and treat diseases. It is not easy to make a clear-cut distinction between functional and medical properties. For example, the antioxidant character of a food has effects both in controlling heart disease (a medical effect) and in slowing down the aging process (a functional effect). Some components of foods, such as pesticides and fungicides are toxic at any level, or when some critical level is exceeded. It is not safe to consume unlimited quantities of some foods. Some components (e.g., sugar, salt, fat, fat soluble vitamins, and alcohol) have negative effects if intake is excessive, or if the diet as a whole is unbalanced. Thus negative health properties are grouped as toxic at any concentration, toxic above a critical concentration level, and excessive or unbalanced intake.

1.3 APPLICATIONS OF FOOD PROPERTIES An understanding of food properties is essential for scientists and engineers to solve the problems in food preservation, processing, storage, marketing, consumption, and even after consumption. It would be very difficult to find a branch of food science and engineering that does not need the knowledge of food properties. The application of food properties are discussed in the following sections. 1.3.1

Process Design and Simulation

Processing causes many changes in the biological, chemical, and physical properties of foods. A basic understanding of these properties of food ingredients, products, processes, and packages is


essential for the design of efficient processes and minimization of undesirable changes due to processing. The present lack of sufficient data on physical properties (such as rheological, thermal, mass, and surface properties) of basic components under real conditions has limited the application of many well-established engineering principles (IFT, 1993). 1.3.1.1

Process Design

Food properties are used in the engineering design, installation, optimization, and operation of food processing equipments including a complete plant. For example, during canning, foods need to be heated for sterilization. The duration and the temperature at which heating needs to be carried out can be based on quality, safety, nutrition content, and process efficiency. In this case, thermal properties such as thermal conductivity and diffusivity as well as microbial lethality and nutrition loss are required for all heat transfer calculations and to predict the end point of heating process. 1.3.1.2

Process Simulation

Process simulation is an important tool for food engineers to develop concept, design, operation, and improvement of food processes. For example, flow modeling can investigate more alternatives of better products in less time at a lower cost. Dhanasekharan et al. (2004) described examples in which flow modeling was used to overcome challenging design problems in extrusion, mixing, and food safety by incorporating HACCP. Schad (1998) warned not to gamble with physical properties when making the most of process simulation benefits. Physical properties are critical in simulating a process. Thus, it is important to know from where pure-component properties have come, what basic property models are being used, and from where the basic equation has originated. It is important to be careful in interpreting the results of the simulation based on the quality and source of critical physical property data. The missing or inadequate physical properties undermine the accuracy of simulation. The problem is that the simulation software is not likely to tell us whether answer is erroneous. The results may appear to be correct, but they may be totally wrong. It is our responsibility alone to ensure that we are using the right property models and have inputted or accessed correct and sufficient data to describe our physical properties. There are no shortcuts (Schad, 1998). 1.3.1.3

Continuous Need

We may think that food properties are important only for the initial design of a plant or process, thus only those who are building new equipment need food properties. It is misleading to think that after the plant has been commissioned, food properties are not required for process design, process operation, and product development. In many instances, the existing equipment need to be updated for new product lines or when some units are not operating efficiently. In this case, it is very expensive to replace whole processing lines or equipment. Some modifications in the process need the applications of process design. Thus, the use of food properties in process design is necessary during the entire life of a processing plant. 1.3.2

Quality and Safety

Quality is an illusive, ever-changing concept. It is a relative perception and is always pegged to expectations based on past experiences. It may have different dimensions or attributes, which could be rotated based on the types of users. Several authorities have defined quality in various ways, but the term generally appears to be associated with the degree of fitness for use or the satisfaction level of consumers (ITC, 1993); the absence of defects or a degree of excellence (Shewfelt, 1999); the degree of conformance to the desired functionality (Karel, 1999); or the degree of acceptability of a


Processing conditions

Properties of raw materials (input)

Processing

Properties of finished product (output)

Figure 1.1 Interaction of processing variables with input and output materials variables.

product to users. Every food product has characteristics measurable by sensory evaluation methods or physicochemical tests. Some characteristics or properties are physical and are easily perceived; others are unseen. The applications of food properties can also be described as characterizing the defects in a food product. Understanding these quality characteristics and emotional factors and familiarity with the appropriate measuring tools are vital to the quality control of food products. Quality loss can be minimized at any stage thus quality retaining depends on the overall control of the processing chain. When preservation fails, the consequences range broadly from being extremely hazardous to the loss of color. Automatic control of food-processing systems helps to improve final product quality, increase process efficiency, and reduce waste of raw materials. Food processes are generally multiple-input, multiple-output systems involving complex interactions between process inputs and outputs (Figure 1.1). 1.3.3

Packaging Design

It is important to know product and packaging characteristics, food-packaging interaction, and stability of packaging during storage and distribution (Petersen et al., 1999). Information on food properties is needed in the selection of packaging materials, and in the design of packages, packaging operations, and packaging machines (McCarthy, 1997). It is important to know how food materials interact with packaging materials, and deterioration kinetics of food during storage and distribution. 1.4 PREDICTION OF FOOD PROPERTIES The experimental measurement is very costly, labor intensive, and may require specialist knowledge. Computer models can be run very quickly, and in many cases do not require a lot of detailed technical knowledge. They can be used to predict what might happen in the process, handling, storage, and consumption. One of the best features of computer models is that they can be used to explore any number of ‘‘what if ’’ scenarios. In many instances simulation refers to ‘‘what if ’’ scenarios and optimization refers to ‘‘best way to do it.’’ This can be useful as tools, since they can be used to investigate the possible effects before undertaking detailed and time-consuming experimental work. There is a need for models that can predict complicated phenomena such as taste development or the effect of complex food processing events on product properties.

1.5 CONCLUSION A clear definition of food properties is presented followed by well-defined classifications. The needs of understanding food properties are clearly identified with the different applications in food processing, preservation, storage, and quality control. Food properties can be measured


experimentally when needed. This task could be achieved by developing prediction models, which would save money for costly instruments or methods, reduce labor costs, and avoid hiring skilled operators for complex methods. However, prediction models will not be able to replace the needs of developing measurement techniques.

REFERENCES ASHRAE. 1993. ASHRAE Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York. Bruin, S. and Jongen, T.R.G. 2003. Food process engineering: The last 25 years and challenges ahead. Comprehensive Reviews in Food Science and Food Safety, 2: 42–80. Dhanasekharan, K.M., Grald, E.W., and Mathur, R. 2004. How flow modeling benefits the food industry. Food Technology, 58(3): 32–35. IFT. 1993. IFT special report: America’s food research needs into the 21st century. Food Technology, 47(3): 1S–39S. IJFP. 1998. Instructions for preparation of manuscript. International Journal of Food Properties, 1: 95–99. ITC. 1993. Quality Control for the Food Industry: An Introductory Handbook. International Trade Centre UNCTAD=GATT, Geneva. Jowitt, R. 1974. Classification of foodstuffs and physical properties. Lebensmittel-Wissenschaft und Technologie, 7(6): 358–378. Karel, M. 1999. Food research tasks at the beginning of the new Millennium—a personal vision. In: Water Management in the Design and Distribution of Quality of Foods, Roos, Y.H., Leslie, R.B., and Lillford, P.J. (eds.). Technomic Publishing, Lancaster, Pennsylvania, pp. 535–559. McCarthy, O.J. 1997. Physical properties of foods and packaging materials—an introduction. In: Food and Packaging Engineering I Course Material. Department of Food Technology, Massey University, Plmerston North. Paulus, K. 1989. Nutritional and sensory properties of processed foods. In: Food Properties and ComputerAided Engineering of Food Processing Systems, Singh, R.P. and Medina, A.G. (eds.). Kluwer Academic Publishers, New York, pp. 177–200. Petersen, K., Nielsen, P., Bertelsen, G., Lawther, M., Olsen, M.B., Nilsson, N.H., and Morthensen, G. 1999. Potential biobased materials for food packaging. Trends in Food Science and Technology, 10: 52–68. Rahman, M.S. 1998. Editorial. International Journal of Food Properties, 1(1): v–vi. Rahman, M.S. and McCarthy, O.J. 1999. Classification of food properties. International Journal of Food Properties, 2(2): 1–6. Schad, R.C. 1998. Make the most of process simulation. Chemical Engineering Progress, 94(1): 21–27. Shewfelt, R.L. 1999. What is quality? Postharvest Biology and Technology, 15: 197–200.


CHAPTER

2

Water Activity Measurement Methods of Foods

Mohammad Shafiur Rahman and Shyam S. Sablani CONTENTS 2.1 2.2

Introduction .............................................................................................................................. 9 Water Activity Measurement ................................................................................................. 10 2.2.1 Colligative Properties Methods.................................................................................. 10 2.2.1.1 Vapor Pressure Measurement ..................................................................... 10 2.2.1.2 Water Activity above Boiling ..................................................................... 13 2.2.1.3 Water Activity by Freezing Point Measurements....................................... 14 2.2.2 Gavimetric Methods Based on Equilibrium Sorption Rate ....................................... 14 2.2.2.1 Discontinuous Registration of Mass Changes ............................................ 15 2.2.2.2 Methods with Continuous Registration of Mass Changes ......................... 20 2.2.3 Hygrometric Methods ................................................................................................ 25 2.2.3.1 Mechanical Hygrometer.............................................................................. 25 2.2.3.2 Wet and Dry Bulb Hygrometer .................................................................. 25 2.2.3.3 Dew Point Hygrometer ............................................................................... 26 2.2.3.4 Hygroscopicity of Salts............................................................................... 26 2.2.3.5 Electronic Sensor Hygrometer .................................................................... 27 2.2.4 Other Methods ........................................................................................................... 29 2.3 Selection of a Suitable Method.............................................................................................. 29 2.4 Conclusion ............................................................................................................................. 29 References ....................................................................................................................................... 30

2.1 INTRODUCTION Water is an important constituent of all foods. In the middle of the twentieth century, scientists began to discover the existence of a relationship between the water contained in a food and its relative tendency to spoil. They also began to realize that the chemical potential of water is related to its vapor pressure relative to that of pure water was more important. This relative vapor pressure (RVP) is termed as water activity or aw. Scott (1957) clearly stated that the water activity of a medium correlated with the deterioration of food stability due to the growth of microorganisms. Thus, it is possible to develop generalized rules or limits for the stability of foods using water


activity. This was the main reason why food scientists started to emphasize water activity along with water content. Since then, the scientific community has explored the great significance of water activity in determining the physical characteristics, processes, shelf life, and sensory properties of foods. Recently, Rahman and Labuza (2007) have presented a detailed review on this aspect of water activity. Details of the various measurement techniques are presented by Labuza et al. (1976), Rizvi (1995), Rahman (1995), and Bell and Labuza (2000). Water activity, a thermodynamic property, is defined as the ratio of the vapor pressure of water in a system to the vapor pressure of pure water at the same temperature, or the equilibrium relative humidity (ERH) of the air surrounding the system at the same temperature. Thus, water activity can be expressed as follows: Pvw sy aw ¼ ¼ ERH (2:1) Pvw where aw is the water activity (fraction) at t (8C) (Pvw )sy and Pvw are the vapor pressures of water in the system and pure water, respectively, at t 8C (Pa) ERH is the equilibrium relative humidity of air at t 8C

2.2 WATER ACTIVITY MEASUREMENT Wiederhold (1987), Labuza et al. (1976), Rizvi (1995), Smith (1971), and Stoloff (1978) studied the accuracy and precision of various water activity measuring devices and found considerable variations. The accuracy of most of the methods lies in the range of 0.01–0.02 water activity units (Rizvi, 1995). The choice of one technique over another depends on the range, accuracy, cost, response time (speed), suitability, portability, simplicity, precision, maintenance and calibration requirements, and types of foods to be measured (Wiederhold, 1987; Rizvi, 1995; Rahman and Al-Belushi, 2006). The required accuracy of the routine and reference methods is given in Table 2.1. More details of the measurement techniques are presented by Rizvi (1995), Wiederhold (1987), Gal (1981), and Smith (1971). The water activity measurement methods can be classified as given in Table 2.2. 2.2.1 2.2.1.1

Colligative Properties Methods Vapor Pressure Measurement

The water activity of food samples can be estimated by direct measurement of vapor pressure using a manometer (Sood and Heldman, 1974; Lewicki et al., 1978; Lewicki, 1987, 1989). A simple Table 2.1 Precision Requirements for Temperature, Water Activity, and Moisture Measurement Equipment Variable Temperature (8C) Relative humidity (%) Xwe (%)

Routine Method

Reference Method

0.2 1.0

0.02 0.10

0.1

0.01

Source: Spiess, W.E.L. and Wolf, W. in Water Activity: Theory and Applications to Food, Rockland, L.B. and Beuchat, L.R. (eds.), Marcel Dekker, Inc., New York, 1987.


Table 2.2 Methods for the Determination of Sorption Isotherm Colligative properties methods 1. Vapor pressure measurement 2. Freezing point measurement 3. Boiling point measurement Gravimetric methods 1. Methods with discontinuous registration of mass changes a. Static systems (isopiestic method) b. Evacuated systems c. Dynamic systems 2. Methods with continuous registration of mass changes a. Static chamber b. Dynamic systems c. Evacuated system Hygrometric systems 1. Mechanical hygrometers 2. Wet and dry bulb hygrometers 3. Dew point hygrometers 4. Hygroscopicity of salts 5. Electronic sensor hygrometers Other methods

schematic diagram is shown in Figure 2.1. A sample of mass 10–50 g of unknown water activity is placed in the sample flask and sealed on to the apparatus. The airspace in the apparatus is evacuated with the sample flask excluded from the system. The sample flask is connected with the evacuated airspace and the space in the sample flask is evacuated to less than 200 mmHg, which is followed by the evacuation of sample for 1–2 min. After isolating the vacuum source and equilibration for 30–50 min the pressure exerted by the sample is recorded (Dh1). The sample flask is subsequently excluded from the system, and the desiccant flask is opened. Water vapor is removed by sorption onto CaSO4, and the pressures exerted by volatiles and gases are indicated by Dh2 after equilibrium. The water activity of the sample is calculated as (Labuza et al., 1976): aw ¼

[h1 h2 ]rg Pvw

(2:2)

where Pvw is the vapor pressure of pure water at t 8C (Pa) r is the density of manometric fluid (kg=m3) h1 and h2 are the manometer readings (m) Rizvi (1995) mentioned that for precise results it is necessary to maintain the following conditions: (1) the whole system should be maintained at a constant temperature, (2) ratio of the sample volume to vapor space volume should be large enough to minimize changes in water activity due to loss of water by vaporization, and (3) a low-density and low-vapor-pressure oil should be used as the manometric fluid. Apiezon B manometric oil (density ¼ 866 kg=m3) is generally used as manometric fluid. If Tsa (sample temperature) and Tme (medium temperature) are different, then water activity is corrected as (Rizvi, 1995):


Temp. control Vacuum gauge

Sensor 1

2 Tm 3

Voltmeter

Fan 4

5 Ts

Sample flask Desiccant flask

Baffle

Vacuum

Manometer Liquid N2 trap

Sensor Heaters

Figure 2.1 Schematic diagram of a thermostatized vapor pressure manometer apparatus. Numbers 1–5 indicate the locations of the stopcocks used for performing an experiment. (From Rizvi, S.S.H., in Engineering Properties of Foods, 2nd edn., Rao, M.A., Rizvi, S.S.H., and Datta, A. (eds.), CRC Press, Boca Raton, FL, 1995.)

Dh1 Dh2 Tsa rg aw ¼ Pvw Tme

(2:3)

The capacitance manometer can be used for more compactness of the large setup and better temperature control (Troller, 1983). In order to incorporate the change in volume that occurs when water vapor is eliminated from the air–water mixture during desiccation, Nunes et al. (1985) presented the following corrections: [h1 Ch2 ]rg Vd aw ¼ (2:4) and C ¼ 1 þ Pvw Vs where C is the correction factor Vd and Vs are the volumes of vapor space and sample, respectively The additional step performed for the correction of volume requires initially placing 1 g of P2O5 in both sample and desiccant flasks. With stopcocks 1, 3, and 5 in open position and stopcock 4 in


close position the sample flask is evacuated. Manometric reading (h1) is taken by closing stopcock 3, and the manometric reading (h2) is obtained with stopcock 5 in close position and stopcock 4 in open position. The void volumes Vs and Vd corresponding to the sample and the desiccant flasks, respectively are also measured. The details were provided by Rahman et al. (2001). Although vapour pressure manometer (VPM) is considered a standard method, it is not suitable for materials either containing large amounts of volatiles and bacteria or mold, or undergoing respiration processes. This method can be used only in the laboratory and is limited for field applications. Stamp et al. (1984) measured the water activity of salt solutions and foods by several electronic methods as compared to direct vapor pressure measurement. An error of approximately 0.01 water activity units was found using the 1 h data but no significantly better regression line was found using the 24 h data. Measurement of the aw of five foods, however, gave values differing by an average of 0.051 aw units as compared to the VPM readings. Their study demonstrates justification of the food and drug administration (FDA) cutoff aw values of 0.85 for low-acid foods as a margin of safety. 2.2.1.2

Water Activity above Boiling

Loncin (1988) proposed a method to determine the water activity above 1008C. If any substance initially containing free water is heated in a closed vessel at a temperature above 1008C (say 1108C) and if the pressure is released in order to reach the atmospheric pressure (1.0133 105 Pa), then water activity is only a function of temperature (Loncin, 1988). The vapor pressure of water in the product is 1.0133 105 Pa because it is in equilibrium with the atmosphere. The vapor pressure of pure water is 1.43 105 Pa at 1108C (Table B.1 in Appendix B). Thus, water activity of the product at 1108C is aw ¼

1:0133 105 ¼ 0:70 1:43 105

(2:5)

In this case, water activity does not depend on the binding forces between water and solutes or solids and composition. This fact is very important for extrusion, where water activity at the outlet is a function of the temperature only (Loncin, 1988). Bassal et al. (1993) proposed a method based on the equilibration of food samples with an atmosphere of pure water vapor at constant pressure. The equilibration cell consisted of 100 mL glass bottle with a Teflon stopper through which a capillary tube (1.3 mm diameter and 20 cm long) was inserted. The bottle with the sample was placed in a temperature-regulated oven (air circulated) and the total pressure inside the bottle was measured by a barometer. An in situ weight-measuring device was also attached to the system. Boiling equilibrium (TB) was assumed to be reached when the variation of the sample mass was less than 0.01 g for 1 h. The test duration depends on the set temperature and air circulation rate in the oven. At equilibrium, the vapor pressure of the sample must be equal to the steam surrounding it and the water activity can be written as aw ¼

PSS PvST

(2:6)

where PSS is the pressure of the surrounding steam, and PvST is the vapor pressure of water at temperature TB from steam tables. The moisture content of the equilibrated sample can be determined by air drying of the equilibrated sample in a conventional dryer. At the end of the test, the capillary tube was damped to prevent any loss of steam from the bottle during cooling. A correction for the partial condensation of water vapor on the sample can be estimated from the known temperature and volume. Bassal et al. (1993) used the above procedure for measuring desorption isotherms of microcrystalline cellulose (MCC) and potato starch at temperatures from 1008C to


1508C at 1 atm and recommended the suitability of this method to measure water activity of foods at higher pressure or vacuum. Boiling point elevation of solution can also be used to predict the water activity by the equation given by Fontan and Chirife (1981) as ln aw ¼ 1:1195 104 (Tbs Tbw )2 35:127 103 (Tbs Tbw )

(2:7)

where Tbs and Tbw are the boiling points of sample and pure water, respectively. 2.2.1.3

Water Activity by Freezing Point Measurements

The determination of water activity by cryoscopy or freezing point depression is very accurate at water activity above 0.85 as mentioned by Wodzinski and Frazier (1960), Strong et al. (1970), Fontan and Chirife (1981), Rey and Labuza (1981), and Lerici et al. (1983). This method is applicable only to liquid foods and provides the water activity at freezing point instead of at room temperature. In the case of solution, the difference is not larger than 0.01 water activity unit (Fontan and Chirife, 1981). Rahman (1991) measured the water activity and freezing point of fresh seafood independently, and found that water activity prediction from freezing point data was 0.02–0.03 units higher than the actual water activity data. This method has advantages at high water activity and for the materials having large quantities of volatile substances which may create error in vapor pressure measurement and in electric hygrometer due to contamination of the sensor. In a two-phase system (ice and solution) at equilibrium, the vapor pressure of solid water as ice crystals and the interstitial concentrated solution are identical; thus water activity depends only on the temperature, and not on the nature and initial concentration of solutes, present in the third or fourth phase (i.e., with respective kind of food). This creates a basis to estimate the water activity of foods below the freezing point using the equation: aw ¼

Vapor pressure of solid water (ice) Vapor pressure of liquid water

(2:8)

At 108C, water activity in an aqueous system at equilibrium containing ice crystals is equal to 260.0=286.6 ¼ 0.907 (data Table A.3) and is independent of nature and initial concentration of solutes, presence of third or fourth phase as in the case of ice cream (Loncin, 1988). Fennema (1981) concluded that changes in properties could occur below freezing point without any change in water activity. These include changes in diffusion properties, addition of additives or preservatives, and disruption of cellular systems. The water activity data of ice from 08C to 508C are correlated with an exponential function as (Rahman and Labuza, 2007):

595:1 aw ¼ 8:727 exp T

(2:9)

where T is measured in kelvin. The maximum error in prediction is 0.012 unit water activity and the average is 0.0066. Other colligative properties such as osmotic pressure and boiling point elevation have not yet been used for food systems (Rizvi, 1995). 2.2.2

Gavimetric Methods Based on Equilibrium Sorption Rate

The gravimetric method is based on the equilibration of samples with its atmosphere of known humidity. In this method, it is important to achieve both hygroscopic and thermal equilibrium (Gal, 1981).


2.2.2.1

Discontinuous Registration of Mass Changes

In this method, the sample in the controlled atmosphere needs to be taken out for weighing and is then placed back in the atmosphere chamber for equilibration. The balance is not a fixed part of the apparatus and samples must be conditioned to different ERH values and conditioning can be carried out in a static or dynamic way. With these methods it is possible to visually examine the samples to detect immediate physical changes, like caking, shrinkage, discoloration, and loss of free-flowing properties (Gal, 1975).

2.2.2.1.1

Static Systems (Isopiestic Method)

The static method is the most simple and common method of measuring water activity of food. This method is also known as isopiestic method. In this method, a weighted sample of known mass (around 2–3 g) is stored in an enclosure and allowed to reach equilibrium with an atmosphere of known ERH (or aw), for example, by a saturated salt solution, and reweighed at regular intervals until constant weight is established. The condition of equilibrium is thus determined in this manner. The moisture content of the sample is then determined, either directly or by calculation from the original moisture content and the known change in weight. A desiccator is commonly used as a chamber to generate controlled atmosphere (Figure 2.2). The details of measuring water activity using isopiestic methods are presented in Rahman and Al-Belushi (2006), Lewicki and Pomaranska-Lazuka (2003), and Sablani et al. (2001). Several days, or even weeks, may be required to establish equilibrium under static air conditions, but results can be obtained for all relative humidity values simultaneously with little effort if the apparatus is replaced with different salt solutions (Smith, 1971). The main advantages of this method are its simplicity, low cost, ability to handle many samples simultaneously, and easy operability (Lewicki and Pomaranska-Lazuka, 2003; Rahman and Al-Belushi, 2006). The main disadvantages of this simple method are

Sample A Sample B Sample C Sample container

Saturated salt solution

Figure 2.2 Humidity control chamber using desiccator.


Excess salt

.

.

.

. .

Slowness of the equilibrium process, which usually takes from 3 to 6 weeks. In certain instances, it could take a few months to equilibrate. It is therefore doubtful whether the microbial and physicochemical stability remain valid in the sample during long experimental periods, especially at higher water activity. At high relative humidity values, the delay in equilibration can lead to mould or bacterial growth on the samples and consequent invalidation of the results. Although it is recommended to place toluene or thymol in the chamber for slowing the microbial growth, there is no option available to ensure physical and chemical stability in the course of the equilibration period. In addition, care should be taken not to inhale these toxic chemicals from the desiccator chamber while preparing the sample and performing the weighing process. Condition of equilibrium is determined by reweighing the sample at regular intervals until constant weight is established. The equilibrium can also be hastened by evacuating the conditioning chamber. The loss of conditioned atmosphere each time the chamber is opened to remove the sample for weighing delays the equilibrium process. Lewicki and Pomaranska-Lazuka (2003) studied the effects of individual operations on this process such as opening the desiccator, and transferring samples to the balance for checking mass. It was shown that opening the desiccator, taking the sample, and closing it again caused the most disturbance. The error depends on the water activity and number of times the desiccator was opened. At low water activity (aw < 0.6) a maximum of about 20% overestimation in sample mass was observed, while at high water activity (0.6 < aw < 0.8) an underestimation of 20% was observed. The process of equilibration can be enhanced in desiccators with or without vacuum. It was shown that equilibration of samples with or without vacuum yields different water contents (Laaksonen et al., 2001). All materials showed clearly that water contents were higher at the high water activities (>0.60) and lower at the low water activities ( 0.70) it showed underestimation of the moisture content. One of the reasons could be due to the higher relative humidity of the room atmosphere, which caused condensation or adsorption of humidity resulting in overestimation at low water activity. In the case of desorption, evaporation could occur for the high water activity sample. In addition, high water activity equilibration for long periods of time may also cause continuous physical and chemical deterioration of the sample. 2.2.2.2.3

Other Applications of DVS

Rahman and Al-Belushi (2006) discussed other applications of the dynamic method. Dynamic method has high potential to be used for microlevel drying studies. Microlevel drying kinetics could be performed as a function of air composition (modified atmosphere drying with varied oxygen, carbon dioxide, and nitrogen levels), relative humidity, temperature, and varied sample geometry and structure. Roques et al. (1983) used dynamic systems to study sorption and diffusion in rapeseed


within the temperature range of 408C–1058C. May et al. (1997) studied the isothermal drying kinetics of foods using a thermogravimetric analyzer (Perkin-Elmer 7 series), which was automated and could handle samples between 10 and 100 mg. It recorded mass automatically at set intervals with a precision of 0.1 mg. They used a sample size of about 30 mg and recorded mass against time at a set temperature of 408C with an accuracy of 0.18C. The purge gas used was nitrogen at a constant velocity of 1.25 103 m=s. Drying curves of five different foods (apple, potato, carrot, asparagus, and garlic) were analyzed and found different drying characteristics with varied constant and falling rate periods. Lin and Chen (2005) discussed a prototype setup for controlled air humidity and temperature chamber for isothermal drying kinetics and moisture sorption isotherm. Teoh et al. (2001) used DVS (DVS-2000, Surface Measurement Systems, London) analysis to measure cornmeal snack moisture isotherm and compared the values by equilibrating the sample in PEC (Lang et al., 1981) and measured the water activity using Aqualab CX-2 (Pullman, Washington). They identified that DVS analysis could produce rapid isotherms and could accurately condition the individual model systems to the same moisture content. With such rapid equilibration times, it was possible to obtain isotherm points at up to water activity of 0.95, without having to worry about microbial and other physicochemical degradation, while increasing the accuracy of the isotherm modeling at higher water activities. Rahman et al. (2005) studied the quality of dried lamb meat produced by simulating modified atmosphere (nitrogen gas) drying using dynamic system SGA-100. The dried lamb meat was evaluated for their microbial and physicochemical characteristics. Sannino et al. (2005) used the DVS-1000 system to study the drying process of lasagna pasta at controlled humidity and temperature with a sensing device to measure the electrical conductivity of pasta during the drying process. An anomalous diffusion mechanism has been observed, typical of the formation of a layered sample structure: a glassy shell on the surface of the pasta slice, which inhibits a fast diffusion from the humidity, rubbery internal portion. Internal stresses at the interface of the glassy–rubbery surfaces are responsible for the formation and propagation of cracks and thus lasagna sample delamination and breakage. The kinetics of adsorption and desorption isotherms could be used to explore other structural characteristics, such as glass transition. A break in the mass transfer rate constant or moisture diffusivity at the glass–rubber transition is expected to occur. The mass transfer rate constant or diffusivity could be estimated from the adsorption or desorption kinetics of samples having the same initial moisture content and can be plotted as a function of temperature in order to identify a break in the plot at glass transition. Rahman et al. (2007) measured moisture diffusivity of spaghetti (moisture content: 9.77 kg water=100 kg spaghetti) within the temperature range 108C–808C and plotted as a function of temperature. They found a clear break at 508C, which is close to the glass–rubber transition of the spaghetti measured by differential scanning calorimetry (DSC). Garcia and Pilosof (2000) attempted to correlate water sorption kinetics with the glass transition temperature. They measured adsorption and desorption kinetics by placing samples in a desiccator at 308C with different relative humidity values maintained by saturated salts. They plotted rate constant as a function of relative humidity and observed a change in slope at the relative humidity when the sample was transformed to a glassy state. In general, the dynamic method used at microlevel has high potential to be used in foods. Del-Nobile et al. (2004) developed a new approach based on the use of oscillatory sorption tests, which was proposed to determine the water-transport properties of chitosan-based edible films. Oscillatory sorption tests as well as stepwise sorption tests were conducted at 258C on chitosan films. Two different models were fitted to the experimental data to determine the relationship between the water-diffusion coefficient and the local water concentration. One of the two tested models accounted only for stochastic diffusion while the other accounted also for the superposition of polymer relaxation to stochastic diffusion. A comparison between experimental and predicted water permeability indicated that stepwise sorption tests cannot be used to determine the dependence of water-diffusion coefficient on local water concentration when the diffusion process has characteristic time much smaller than that of polymer relaxation. In fact, in these cases the diffusion process controls only the very early stage of sorption kinetics, whereas the remaining part of the


transient is controlled by polymer relaxation (Del-Nobile et al., 2004). In the future, more other potential applications of the dynamic methods could emerge from the literature. 2.2.2.2.4

Evacuated Systems

Evacuated or vacuum systems are also used in the dynamic system with continuous registering of the mass of the sample during the equilibration period. 2.2.3

Hygrometric Methods

One of the most commonly used methods of measuring ERH is to equilibrate the sample with air in a closed vessel and then to determine the relative humidity of the air with a hygrometer. Any instrument capable of measuring the humidity or psychrometric state of air is a hygrometer. Hygrometric devices are based on many different scientific principles such as dew point, frost point, wet and dry bulb temperature, expansion of a material, and electric resistance and capacitance of salt. The details of measurement steps are provided in Rahman and Sablani (2001). In a hygrometer there are three zones: testing enclosure, sample environment, and sensor environment. The temperatures within the three zones need to be in equilibrium. Unless the temperature of the sample is known, the relative humidity in this region cannot be determined. Unless the temperature of the sensor is known, the relative humidity indicated the sensor cannot be converted into a reliable estimate of the vapor pressure throughout the system. Without careful control and measurement of temperatures in the sample and the sensor, no meaningful data can be collected (Reid, 2001). 2.2.3.1

Mechanical Hygrometer

This method is based on the dimensional changes in natural or synthetic materials suitably amplified by mechanical linkage to indicate atmospheric humidity. The most common one is the hair type hygrometer, in which relative humidity is indicated on a dial by the change in length of a bundle of other fibers. Smith (1971) mentioned that these instruments are comparatively slow to react to changes in the ambient atmosphere and are difficult to maintain in calibration. They are not suitable for use below 0.25 relative humidity and above 508C, as the element may undergo a permanent change in length under these conditions. Prolonged exposure to high humidity involves the risk of deterioration due to mould growth on the hairs. So hair hygrometers have limited use in ERH measurement but may be useful for approximate ERH. 2.2.3.2

Wet and Dry Bulb Hygrometer

The wet bulb temperature depends on the amount of moisture present in the air–vapor mixture. The atmospheric humidity can be estimated from the water vapor pressures at saturation (dew point) and dry bulb temperatures. The vapor pressures can be related to wet and dry bulb temperatures as (Smith, 1971) Pvw ¼ Pvws PP(Tsy Twb ) where Pvw is the vapor pressure of water Pvws is the vapor pressure of water on saturation at the wet bulb temperature P is the barometric pressure Tsy and Twb are the ambient and wet bulb temperatures P is a constant that depends on the instruments and certain conditions


(2:10)

The constant P is predominantly affected by the airflow rate over the wet bulb and to a lesser extent by the dimensions of the thermometer. Results become independent of flow rate at speeds above 3 m=s for all practical purposes. This can be achieved by physically shifting the instrument or more commonly by circulating the air by means of a fan. The dimensions of the thermometer affect constant P mainly through conduction of heat along the uncooled stem to the cooled mercury bulb. This effect is considerably reduced by substituting smaller temperature sensors such as platinum resistance thermometers, thermocouples, or thermistors. These have added advantages of greater sensitivity and the ability to provide remote reading facilities (Smith, 1971). This method is not suitable for small volumes of air and has been primarily used to determine the relative humidity of large storage atmospheres and commercial dehydrators (Rizvi, 1995). Small hygrometers especially designed for use in foods are now commercially available. Major limitations of this method are condensation of volatile materials, heat transfer by conduction and radiation, and the minimum wind velocity requirement of at least 3 m=s (Rizvi, 1995). 2.2.3.3

Dew Point Hygrometer

The dew point is the temperature at which the air under investigation will reach saturation point due to the moisture conent and below which a dew or frost will form. A typical simple dew point apparatus consists of a polished silver thimble projecting into the enclosed atmosphere. The thimble is cooled by the forced evaporation of a solvent in the thimble. The temperatures at which dew is just formed and disappears on rewarming are averaged to give the actual dew point. The visual detection of the deposition and disappearance of the dew requires considerable care, and various attempts have been made to improve accuracy by substituting the human eye with detection devices (Smith, 1971). Modern instruments, based on Peltier effect cooling of mirrors and the photoelectric determination of condensation on the reflecting surface via a null-point type of circuit, give very precise values of dew point temperatures (Rizvi, 1995). Dew point measuring devices are reported to have an accuracy of 0.003 water activity unit in the range of 0.75–0.99 (Prior, 1979). At lower water activity levels there is not sufficient vapor in the headspace to cover the reflecting surface and the accuracy of these instruments is therefore diminished. Thus, the measurement based on psychrometry has the highest accuracy near relative humidity 1.0 (Wiederhold, 1987). 2.2.3.4

Hygroscopicity of Salts

The micromethod based on the hygroscopicity of salt does not yield an accurate result but gives a useful indication of the ERH of a sample between fairly narrow limits within about an hour (Smith, 1971). It is based on the fact that water vapor will condense on a salt crystal only from an atmosphere which has a higher relative humidity than the critical relative humidity of the salt. Each salt has its own characteristic or critical relative humidity transition point. A salt crystal will remain dry if the surrounding air relative humidity is lower than the critical relative humidity and will show a wet short line when inspected under a lens if air relative humidity is higher than the critical relative humidity. An approximate estimate of ERH can be obtained by observing the change in color of a salt enclosed in the headspace over the sample as silica gel which changes color from blue at low humidities, through shades of lilac, to pink at high humidities. Solomon (1945) used papers impregnated with cobalt chloride, potassium thiocyanate, or sodium thiosulfate which is blue at low and pale at high humidities, with a series of lilac colors in between. The error in relative humidity measurement may be up to 5% from color matching against a suitably calibrated series of standard colors. Solomon (1957) mentioned that the measuring of relative humidity by matching the colors of tissue paper impregnated with cobalt thiocyanate becomes convenient due to the


commercial availability of impregnated paper and colored glass standards (commercial kits). However, color correction for temperature and up to 2 h exposure for equilibration is necessary. This method is suitable for general use when elaborate equipment cannot be employed. 2.2.3.5

Electronic Sensor Hygrometer

The hygrometers are more sensitive compared to the static and dynamic methods at water activity above 0.9 (Pezzutti and Crapiste, 1997). Several hygrometers are commercially available for direct determination of water activity. These types of hygrometers are based on the measurement of the conductivity of salt solution (usually LiCl) which is in equilibriums with the air (Figure 2.8). The electrolytic sensor consists of a small hollow cylinder covered with a glass fiber tape, which is impregnated with saturated lithium chloride solution. A spiral bifilar electrode is wound over the tape and a temperature sensor is mounted at the center of the cylinder. An alternating voltage is applied to the electrodes and a current is allowed to pass through the inverting electrolyte. The resulting rise in the temperature opposes the absorption of moisture by the lithium chloride and the sensor rapidly reaches an equilibrium temperature at which the vapor pressure of the salt solution equals that of the air. Temperature is determined by a sensor at the core of the cylinder and a calibration chart is used to convert this relative humidity. Hygroscopic organic polymer films are also used instead of lithium chloride salt. Another type is the anodized aluminum sensor. This sensor consists of an aluminum strip that is anodized by a process that forms a porous oxide layer. A very thin coating of gold is then evaporated over this structure. The aluminum base and the gold layer form the two electrodes of what is essentially an aluminum oxide capacitor (Smith, 1971; ASHRAE, 1993; Rahman, 1995). Different variations in sensor construction are available. The disadvantages due to the nonlinear response to humidity, large temperature coefficient for conductivity values, and contamination of the lithium chloride can cause erroneous conductivity readings. The range of operation is restricted to 0.15–0.90 ERH and each sensor exhibits individual characteristics and must be calibrated against reference atmospheres (Smith, 1971). The performance of the sensor is subject to change on aging

Sensor

Water activity display aw = 0.453 T = 25.1⬚C

Chamber

Sample Electric heaters for temperature control

Sample holder

Clamps Clamps

Insulated box

Figure 2.8 Schematic diagram of setup for water activity measurement by electronic sensor. (From Rahman, M.S. and Sablani, S.S., in Current Protocols in Food Analytical Chemistry, John-Wiley & Sons, Inc., New York, 2001, A2.5.1–A2.5.4.)


and with contamination of the element by foreign particles, but the probe can be readily calibrated with saturated salt solutions. Equilibrium times obtained in measurement on various food products ranged from a few seconds to a minute or more. Readings can be obtained in small areas and the sensors’ physical shape make them particularly suitable for insertion through the small packs and pouches. A small probe-type hygrometer is based upon the effect of humidity on the capacitance and resistance of a condenser. Different commercial probes are available. This type of instrument provides rapid and reliable means of measuring water activity provided some precautions are taken. The specific problems that arise from the use of an electric hygrometer for measurement of water activity are the equilibration period, the calibration of sensors, and the influence of temperature on the values measured in saturated salt solutions and food products. Stekelenburg and Labots (1991) found that the equilibration time required for various products to reach a constant water activity value increased at higher water activity and mentioned that the following precautions must be taken for reliable measurement: (1) the water activity value should be taken when the reading (0.001 unit) has been constant for 10 min, (2) the humidity sensors should be calibrated regularly to compensate for drift, (3) a separate calibration curve should be made for each sensor, and (4) sensors should be calibrated at the same temperature at which the samples are measured and differences in temperature between sample and sensor should be avoided because of possible formation of condensate on the sensor. Labuza et al. (1976) mentioned that the equilibration time varied from 20 min to 24 h depending on the humidity range and food materials. A mechanical or chemical filter is available to protect the sensor from contamination. The Sina-scope sensor was equipped with a mechanical filter to protect it from dust, oil, and water vapor condensation. A chemical filter can also be used to protect the sensor from chlorine, formaldehyde, ammonia, sulfur dioxide, hydrogen sulfite, amino acids, hydrocarbons, and oil droplets (Labuza et al., 1976). It is difficult to find a sensor that will operate at very low temperature and above 1008C (Wiederhold, 1987). The sensor consists of a small hollow cylinder covered with a glass fiber tape which is impregnated with saturated lithium chloride solution. A spiral bifilar electrode is wound over the tape and a temperature sensor is mounted at the center of the cylinder. An alternating voltage is applied to the electrodes and a current is allowed to pass through the intervening electrolyte. The resulting rise in temperature opposes the absorption of moisture by the lithium chloride and the sensor rapidly reaches an equilibrium temperature at which the vapor pressure of the salt solution equals that of the air. This temperature is determined by the sensor at the core of the cylinder and a calibration chart is used to convert this relative humidity. Stoloff (1978) presented collaborative studies of using instruments with immobilized salt solution sensors in different laboratories. A multilaboratory and multi-instrument study of the method, applied to a cross section of commodities and reference standards, demonstrated that measurements of water activity by the described method with instruments using immobilized salt solution sensors can be made with an accuracy and precision within 0.01, provided there are no commodity–instrument interactions. Their study showed that commodity–instrument interactions did exist with lupine beans in brine, soy sauce, and cheese spread, but not with fudge sauce. A sampling error is distinctly evident with walnuts and is probably a component of variance with rice. The sensitivity of the sample or the sensor to vapor transfer must also be considered. Here, the quantity of material represented by the sample or by the sensor is important. Vapor pressure is established by the presence of a particular number of molecules in a defined volume of space. The transfer of water molecules into the vapor phase may cause a measurable change in the gravimetric water contents of the sample and sensor. It is necessary that the sample water content (or the initial sample weight) be known. It is also necessary to know the sensor water content, depending upon the operating principle of the sensor. The amount of moisture transfer from the sample does not significantly change its moisture content (Reid, 2001).


2.2.4

Other Methods

Another method involves preparation of a standard curve by equilibration of a specific amount of dry standard material over different saturated salt solutions. The standard must be stable during reuse of the material. The standard curve is a plot of water activity versus water content of the standard material. The standards that could be used are soy albumin, MCC, glycerol, and sulfuric acid solution (Stoloff, 1978). The jar or desiccator used should be exactly the same as will be used later. For the measurement, the sample size of the standard should be in a controlled narrow range, e.g., 1.6 0.1 g in duplicate or triplicate. Once the standard is made, dry samples of the same weight as in the standard curve are equilibrated over a large quantity of the food material (10–20 g). Once equilibrated, the moisture content of the standard material is measured (e.g., by weight gain) and then aw is estimated from the standard curve (Vos and Labuza, 1974). This method avoids preparation or storing of saturated salts for each determination. Moreover, use of a standard shortens the equilibration, thus requiring less time for measurement. The sensitivity of this method is highly dependent on the accuracy of the calibration curve (Rizvi, 1995). This technique is not accurate below 0.50 and above 0.90 water activity (Troller, 1983). Gilbert (1993) described the application of inverse gas chromatography (IGC) to determine sorption isotherms in different sorbents. IGC is used to measure sorption as a function of retention volume, but the short transit time of normal chromatography which requires near ideal conditions hindered the application of IGC in complex food systems exhibiting slow or complex changes with water sorption resulting in hysteresis. Chromatography offers another dimension—time, removing the requirement for equilibrium. This makes possible the measurement of limited rates of change approaching zero as in slow reactions or restricted diffusivity in the so-called lag phase.

2.3 SELECTION OF A SUITABLE METHOD The selection of an appropriate method or instrument for a specific food material and the purpose of measurement is very important for more valuable and meaningful results. It is clear that many of the methods discussed above fail to meet the present requirements of portability, speed, cost, and simplicity. The manometric, volumetric, sorption isotherm, and sorption rate methods are all excluded by their dependence on fixed laboratory equipment. The methods based upon the wetting of hygroscopic salts or the color changes of cobalt salts are free from this limitation and might be adapted to give cheap, semirapid, and approximate procedure suitable for some applications (Smith, 1971). For more accurate measurements, the various electrical hygrometers now commercially available appear to meet the present requirements. Among these, the most promising instruments are anodized aluminum sensors (Smith, 1971). The characteristics of a sensor depend upon the conditions of manufacture, particularly on the nature of the anodized film, and therefore each manufacturer’s instrument must be calibrated separately. The anodized sensors are advantageous because of their ruggedness and small dimensions, fast response and freedom from large temperature coefficients, and susceptibility to contamination of lithium chloride conductivity sensors (Smith, 1971). However, there is no single humidity instrument which is suitable for every application (Wiederhold, 1987). It is necessary to develop instruments to measure water activity of food at higher temperatures and pressures in the future.

2.4 CONCLUSION Different types of water activity measurement equipments are presented in the literature. However, the static isopiestic method is the most widely used due to its simplicity and low cost;


electronic sensor types are also popular due to their simplicity, speed, and portability. In recent times, DVS systems are also being used due to their multidimensional applications for sorption, drying rate, controlled atmosphere, and other structural changes. REFERENCES ASHRAE. 1993. Measurement and instruments. In: ASHRAE Handbook Fundamentals. American Society of Heating, Refrigeration and Air-Conditioning Engineers, Atlanta, GA, pp. 13.1–13.23. Bassal, A., Vasseur, J., and Lebert, A. 1993. Measurement of water activity above 1008C. Journal of Food Science, 58(2): 449–452. Bell, L.N. and Labuza, T.P. 2000. Moisture Sorptions: Practical Aspects of Isotherm Measurement and Use. Egan Press AACC, Egan, MN. Del-Nobile, M.A., Buonocore, G.G., and Conte, A. 2004. Oscillatory sorption tests for determining the watertransport properties of chitosan-based edible films. Journal of Food Science, 69(1): 44–49. Fennema, O. 1981. Water activity at subfreezing temperatures. In: Water Activity: Influences on Food Quality, Rockland, L.B. and Stewart, G.F. (eds.). Academic Press, New York, pp. 713–732. Fontan, C.F. and Chirife, J. 1981. The evaluation of water activity in aqueous solutions from freezing point depression. Journal of Food Technology, 16: 21–30. Gal, S. 1975. Recent advances in techniques for the determination of sorption isotherms. In: Water Relations of Foods, Duckkworth, R.B. (ed.). Academic Press, London, pp. 139–154. Gal, S. 1981. Recent developments in techniques for obtaining complete sorption isotherms. In: Water Activity Influences and Food Quality, Rockland, L.B., and Stewart, G.F. (eds.). Academic Press, London, pp. 89–111. Garcia, L.H. and Pilosof, A.M.R. 2000. Kinetics of water sorption in okara and its relationship to the glass transition temperature. Drying Technology, 18(9): 2105–2116. Gilbert, S.G. 1993. Applications of IGC for research in kinetic and thermodynamic problems in food science. In: Shelf Lie Studies of Foods and Beverages. Charalambous, G. (ed.). Elsevier Science Publishers B.V., London, pp. 1071–1079. Greenspan, L. 1977. Humidity fixed points of binary saturated aqueous solutions. Journal of Research and National Bureau of Standards [A]: Physics and Chemistry, 81A(1): 89–96. Igbeka, J.C. and Blaisdell, J.L. 1982. Moisture isotherms of a processed meat product—bologna. Journal of Food Technology, 17:37–46. Kanade, P.B. and Pai, J.S. 1988. Moisture sorption method for hygroscopic samples using a modified proximity equilibrium cell. Journal of Food Science, 53(4): 1218–1219. Laaksonen, T.J., Roos, V.H., and Labuza, T.P. 2001. Comparisons of the use of desiccators with or without vacuum for water sorption and glass transition studies. International Journal of Food Properties, 4(3): 545–563. Labuza, T.P. 1984. Moisture Sorptions: Practical Aspects of Isotherm Measurement and Use. American Association of Cereal Chemists, St. Paul, MN. Labuza, T.P., Acott, K., Tatini, S.R., Lee, R.Y., Flink, J., and McCall, W. 1976. Water activity determination: A collaborative study of different methods. Journal of Food Science, 41: 910–917. Lang, K.W., McCune, T.D., and Steinberg, M.P. 1981. A proximity equilibration cell for rapid determination of sorption isotherms. Journal of Food Science, 46: 936–938. Lerici, C.R., Piva, M., and Rosa, M.D. 1983. Water activity and freezing point depression of aqueous solutions and liquid foods. Journal of Food Science, 48: 1667–1669. Lewicki, P.P. 1987. Design of water activity vapor pressure manometer. Journal of Food Engineering, 6: 405–422. Lewicki, P.P. 1989. Measurement of water activity of saturated salt solutions with the vapor pressure manometer. Journal of Food Engineering, 10: 39–55. Lewicki, P.P. and Pomaranska-Lazuka, W. 2003. Errors in static desiccator method of water sorption isotherms estimation. International Journal of Food Properties, 6(3): 557–563. Lewicki, P.P., Busk, G.C., Peterson, P.L., and Labuza, T.P. 1978. Determination of factors controlling accurate measurement of aw by the vapor pressure manometric technique. Journal of Food Science, 43: 244–246.


Lin, S.X.Q. and Chen, X.D. 2005. An effective laboratory air humidity generator for drying research. Journal of Food Engineering, 68: 125–131. Loncin, M. 1988. Activity of water and its importance in preconcentration and drying of foods. In: Preconcentration and Drying of Food Materials, Bruin, S. (ed.). Elsevier Science Publishers B.V., Amsterdam, pp. 15–34. Lowe, E., Durkee, E.L., and Farkas, D.F. and Silverman, G.J. 1974. Anidea for precisely controlling the water activity in testing chambers. Journal of Food Science, 39: 1072–1073. May, B.K., Shanks, R.A., Sinclair, A.J., Halmos, A.L., and Tran, V.N. 1997. A study of drying characteristics of foods using thermogravimetric analyzer. Food Australia, 49(5): 218–220. Nunes, R.V., Urbicain, M.J., and Rotstein, E. 1985. Improving accuracy and precision of water activity measurements with a water vapor pressure manometer. Journal of Food Science, 50: 148–149. Pezzutti, A. and Crapiste, G.H. 1997. Sorption equilibrium and drying characteristics of garlic. Journal of Food Engineering, 31: 113–123. Prior, B.A. 1979. Measurement of water activity in foods: a review. Journal of Food Protection, 42, 668–674. Rahman, M.S. 1991. Thermophysical properties of seafoods. PhD thesis, University of New South Wales, Sydney. Rahman, M.S. 1995. Handbook of Food Properties. CRC Press, Boca Raton, FL. Rahman, M.S. and Al-Belushi, R.H. 2006. Dynamic isopiestic method (DIM): Measuring moisture sorption isotherm of freeze-dried garlic powder and other potential uses of DIM. International Journal of Food Properties, 9(3): 421–437. Rahman, M.S. and Labuza, T.P. 2007. Water activity and food preservation. In: Handbook of Food Preservation, 2nd edn., Rahman, M.S. (ed.), CRC Press, Boca Raton, FL, pp. 447–476. Rahman, M.S. and Sablani, S.S. 2001. Measurement of water activity using electronic sensors. In Current Protocols in Food Analytical Chemistry, Eds, Wrolstad, R.E., Acree, T.E., Am, H., Decker, E.A., Penner, M.H., Reid, D.S., Schwartz, S.J., Shoemaker, C.F., and Sporms, P., John-Wiley & Sons, Inc., New York, pp. A2.5.1–A2.5.4. Rahman, M.S., Sablani, S.S., Guizani, N., Labuza, T.P., and Lewicki, P.P. 2001. Direct manometric measurement of vapor pressure. In: Current Protocols in Food Analytical Chemistry, John-Wiley & Sons, Inc., New York, pp. A2.4.1–A2.4.7. Rahamn, M.S., Salman, Z., Kadim, I.T., Mothershaw, A., Al-Riziqi, M.H., Guizani, N., Mahgoub, O., and Ali, A. 2005. Microbial and physico-chemical characteristics of dried meat processed by different methods. International Journal of Food Engineering, 1(2): 1–14. Rahman, M.S., Al-Marhubi, I.M., and Al-Mahrouqi, A. 2007. Measurement of glass transition temperature by mechanical (DMTA), thermal (DSC and MDSC), water diffusion and density methods: A comparison study. Chemical Physics Letters, 440: 372–377. Reid, D. 2001. Factors to consider when estimating water vapor pressure. In: Current Protocols in Food Analytical Chemistry, Eds, Wrolstad, R.E., Acree, T.E., Am, H., Decker, E.A., Penner, M.H., Reid, D.S., Schwartz, S.J., Shoemaker, C.F., and Sporms, P., John Wiley and Sons, New York, p. A2.1.1–A2.1.3. Rey, D.K. and Labuza, T.P. 1981. Characterization of the effect of solute in water-binding and gel strength properties of carrageenan. Journal of Food Science, 46: 786. Rizvi, S.S.H. 1995. Thermodynamic properties of foods in dehydration. In: Engineering Properties of Foods, 2nd edn., Rao, M.A., Rizvi, S.S.H., and Datta, A. (eds.). CRC Press, Boca Raton, FL. Roques, M., Naiha, M., and Briffaud, J. 1983. Hexane sorption and diffusion in rapeseed meals. In: Food Process Engineering, Vol. 1, Food Processing Systems, Linko, P., Malkki, Y., Olkku, J., and Larinkari, J. (eds.). Applied Science Publishers, London, pp. 13–21. Sannino, A., Capone, S., Siciliano, P., Ficarella, A., Vasanelli, L., Meffezzoli, A. 2005. Monitoring the drying process of lasagna pasta through a novel sensing divice-based method. Journal of Food Engineering, 69: 51–59. Sablani, S.S., Rahman, M.S., and Labuza, T.P. 2001. Measurement of water activity using isopiestic method. In Current Protocols in Food Analytical Chemistry, Eds, Wrolstad, R.E., Acree, T.E., Am, H., Decker, E.A., Penner, M.H., Reid, D.S., Schwartz, S.J., Shoemaker, C.F., and Sporms, P., John-Wiley & Sons, Inc., New York, pp. A2.3.1–A2.3.10. Scott, W.J. 1957. Water relations of food spoilage microorganisms. Advances in Food Research, 72: 83. Smith, P.R. 1971. The Determination of Equilibrium Relative Humidity or Water Activity in Foods— A Literature Review. The British Food Manufacturing Industries Research Association, England.


Solomon, M.E. 1945. The use of cobalt salts as indicators of humidity and moisture. Annals of Applied Biology, 32: 75–85. Solomon, M.E. 1957. Estimation of humidity with cobalt thiocyanate papers and permanent colour standards. Bulletin Entomological Research, 48(3): 489–507. Sood, V.C. and Heldman, D.R. 1974. Analysis of a vapor pressure manometer for measurement of water activity in nofat dry milk. Journal of Food Science, 39: 1011–1013. Spiess, W.E.L. and Wolf, W. 1987. Critical evaluation of methods to determine moisture sorption isotherms, In: Water Activity: Theory and Applications to Food, Rockland, L.B. and Beuchat, L.R. (eds.). Marcel Dekker, New York, pp. 215–233. Stamp, J.A., Linscott, S., Lomauro, C., and Labuza, T.P. 1984. Measurement of water activity of salt solutions and foods by several electronic methods as compared to direct vapor pressure measurement. Journal of Food Science, 49: 1139–1142. Stekelenburg, F.K. and Labots, H. 1991. Measurement of water activity with an electric hygrometer. International Journal of Food Science and Technology, 26(1): 111–116. Stoloff, L. 1978. Calibration of water activity measuring instruments and devices: Collaborative study. Journal of the Association of Official Analytical Chemists, 61(5): 1166–1178. Strong, D.H., Foster, E.M., and Duncan, C.L. 1970. Influence of water activity on the growth of Clostridium perfrigens. Applied Microbiology, 19: 980. Teoh, H.M., Schmidt, S.J., Day, G.A., and Faller, J.F. 2001. Investigation of cornmeal components using dynamic vapor sorption and differential scanning calorimetry. Journal of Food Science, 66(3): 434–440. Troller, J.A. 1983. Methods to measure water activity. Journal of Food Protection, 46: 129. Vos, P. and Labuza, T.P. 1974. A technique for measurement of water activity in the high aw range. Journal of Agricultural Food Chemistry, 22: 326–327. Wiederhold, P. 1987. Humidity measurements. In: Handbook of Industrial Drying, Mujumdar, A.S. (ed.). Marcel Dekker, New York. Wodzinski, R.J. and Frazier, W.C. 1960. Moisture requirement of bacteria. 1. Influence of temperature and pH on requirements of Pseudomonas fluorescens. Journal of Bacteriology, 79: 572.


CHAPTER

3

Data and Models of Water Activity. I: Solutions and Liquid Foods

Piotr P. Lewicki CONTENTS 3.1 Introduction ............................................................................................................................ 33 3.2 Water Activity of Solutions and Liquid Foods ..................................................................... 37 3.3 Semiempirical Equations ....................................................................................................... 48 3.4 Empirical Equations............................................................................................................... 54 References ....................................................................................................................................... 62

3.1 INTRODUCTION Food is a multicomponent and multiphase system that is usually not in a thermodynamic equilibrium. This lack of equilibrium causes many chemical and physical changes, which are passed within the material during storage. Processing creates domains within the food with higher and lower concentration of constituents. Concentration gradients result in diffusion, which in turn enable contact between substrates and can facilitate chemical reactions. Redistribution of constituents, especially water, affects rheological properties of the material, hence its texture, structure, ability to creep and relax all change during storage. Interactions between polymers strongly influence properties of the material and depend on rotational and translational diffusion of polymer chains. In materials with interphase boundaries, mass and surface forces cause changes of structure and constituents, spatial concentration. Destabilization of foams and coalescence of emulsions is a macroscopic result of those changes. All the above-mentioned examples show that food is a dynamic system, far from the equilibrium state, which undergoes many changes during storage. Research conducted over the years prove that the course and the dynamics of all those changes are related to the thermodynamic state of water in the material (Lewicki, 2004). The thermodynamic state of water in food arises firstly from unusual properties of water and its ability to form strong hydrogen bonds. Interactions with hydrophilic solute lead to the formation of structure water and hydration water. On the other hand, an interaction with hydrophobic solute affects the structure of the solvent water. A clathrate-like structure is formed. Water molecule mobility in both structure and hydration state as well as in clathrate-like structure is reduced. In the presence of electrolytes, ionic interactions also occur and induce some structuring of water


molecules. Finally, porosity of the material, which is quite common for foods, also affects the thermodynamic state of water. Curvature of the solvent surface in capillaries, in drops or interstices, reduces the ability of water molecules to do the work. The thermodynamic state of water is expressed by its Gibbs free energy, which is a criterion of feasibility of chemical or physical transformation. The Gibbs free energy is quantitatively expressed by the equation: G ¼ H TS

(3:1)

where H is the enthalpy (J) T is the temperature (K) S is the entropy (J=K) and can be understood as total energy (H) diminished by unavailable energy (TS) Gibbs free energy in differential form: dG ¼ dH TdS SdT

(3:2)

Substituting dH ¼ dE þ PdV þ VdP and dE ¼ TdS – PdV yield the differential change in the Gibbs free energy: dG ¼ VdP SdT

(3:3)

where E is the internal energy (J) P is the pressure (Pa) V is the volume (m3) The above equation applies to a homogeneous system of constant composition in which only work of expansion takes place. In an open multicomponent system, the Gibbs free energy will depend not only on the temperature and pressure, but also on the amount of each component present in the system. Hence G ¼ f (T, P, n1 , n2 , . . . , nn )

(3:4)

where n1, n2, . . . , nn is the number of moles of component 1, 2, . . . , n. Under this situation, change of Gibbs free energy is described by the equation: @G dG ¼ @P

T,nj

@G dP þ @T

dT þ P,nj

nj X @G ni

@n

dni

(3:5)

T,P,nj

in which i denotes that component where concentration changes and j denotes all those components where concentration remains constant. The partial molar Gibbs free energy @G @ni

(3:6) T,P,nj

expressed by Equation 3.6 is called chemical potential of component i and is denoted as mi. It represents the change in Gibbs free energy of the system caused by addition of one mole of


component i keeping temperature, total pressure, and number of moles of all other components constant. Then Equation 3.3 can be written as follows: X dG ¼ VdP SdT þ mi dni (3:7) Gibbs has shown that for any system the necessary and sufficient condition for equilibrium is the equality of chemical potential of component i in all phases. That is mIi ¼ mIIi ¼ mIII i

(3:8)

where the superscripts refer to different phases. Dividing Equation 3.3 by the number of moles of component i, the following can be written: dG V S ¼ dP dT dni dni dni

(3:9)

Denoting V=dni ¼ vi and S=dni ¼ si, Equation 3.9 becomes dmi ¼ vi dP si dT

(3:10)

dmi ¼ vi dP

(3:11)

where vi is the molar volume (m3=mol) si is the molar entropy J=(mol K) At constant temperature

Taking the substance as an ideal gas, Lewis obtained chemical potential in terms of easily measurable properties. vi ¼

RT P

and dmi ¼

RT dP P

(3:12)

where R is the gas constant (J=mol K). Integrating Equation 3.12 the following formula is obtained: mi mi ¼ RT ln

p p

(3:13)

where mi is the chemical potential of component i at standard conditions p is the initial pressure in the system In a real system properties of gases deviate from the ideal one and to account for the deviation Lewis proposed a new function called fugacity, f. Hence, Equation 3.13 for a real gas takes the following form: mi mi ¼ RT ln

fi fi

(3:14)

In a pure ideal gas fi ¼ pi, the partial pressure of the gas. The no ideality of the gas follows from the existence of intermolecular forces (Prausnitz, 1969). The ratio between fugacities is called activity, a. Hence, Equation 3.14 becomes mi mi ¼ RT ln ai


(3:15)

Table 3.1 Fugacity and Activity of Water Vapor in Equilibrium with the Liquid at Saturation and at Pressure 0.01 MPa Temperature (8C)

Pressure (kPa)

Fugacity (kPa)

Activity

0.01 10

0.611 1.227

0.611 1.226

0.9995 0.9992

20

2.337

2.334

0.9988

40 60 80

7.376 19.920 47.362

7.357 19.821 46.945

0.9974 0.9950 0.9912

99.856

0.9855

100

101.325

120 140

198.53 361.35

194.07 349.43

0.9775 0.9670

160 180

618.04 1002.7

589.40 939.93

0.9537 0.9374

Source: Adapted from Hass, J.L., Geochim. Cosmochim. Acta, 34, 929, 1970.

Analysis of data presented in Table 3.1 shows that in the range of temperatures important for food processing, the water vapor deviates from the ideal gas by not more than by 6%, and at ambient temperature and pressure, the deviation is less than 0.2%. Thus, under the conditions experienced in food processing and storage activity, water vapor at saturation can be assumed to be equal to 1 and it can be written as fw pw ¼ fw pw

(3:16)

where pw and pw are the vapor pressure of water in the system and of pure water at the same temperature and total pressure, respectively. Equation 3.16 is used to calculate water activity in food when partial pressure of water vapor over that food is known. Assuming that food is in equilibrium with the gas phase, the activity of gas phase calculated from Equation 3.16 is taken as the activity of water in solid or liquid food according to Equation 3.8. Taking a system consisting of an ideal solution and an ideal gas, the equilibrium state can be described as follows: ¼ mvapor msolution i i

(3:17)

An ideal solution behaves analogously to an ideal gas and follows Raoult’s law. Fugacity of a component i in an ideal solution is proportional to the concentration of that component. Thus fi ¼ kxi

(3:18)

fisolution ¼ xi ¼ asolution i fi

(3:19)

where k is the constant x is the mole fraction of component i For xi ¼ 1 fi ¼ k ¼ fi , hence


In liquids, deviations from ideality are strong and arise from electrostatic forces, induction forces between permanent dipoles, forces of attraction, and repulsion between nonpolar molecules and specific interactions such as hydrogen bonds. To account for these interactions, an activity coefficient g was introduced. Thus ¼ g i xi asolution i

(3:20)

msolution mi solution ¼ RT ln g i xi i

(3:21)

and

Activity of water in solution can be calculated from Equation 3.20 but the activity coefficient gw must be known. However, this is a function of temperature and composition and is usually undetermined.

3.2 WATER ACTIVITY OF SOLUTIONS AND LIQUID FOODS Water activity of solutions and liquid foods depends on concentration, chemical nature of solutes, and temperature. In most fresh foods, it is close to 1 and its measurement presents some difficulties. Water activity of some solutions and liquid foods is presented in Tables 3.2 through 3.6. Table 3.2 Water Activity of Some Liquid Foods Product

Concentration (%)

Apple juice Apple juice concentrate Apple juice concentrate

Aronia juice concentrate Black currant juice concentrate

Temperature (8C)

0.986 40–42 66.0 65.4

0.91–0.93 0.792 0.798

Reference Chirife and Ferro Fontan (1982)

24.2 24.3

Chen (1987a) Jarczyk et al. (1995) Jarczyk et al. (1995)

66.2

0.795

23.7

Jarczyk et al. (1995)

66.2 70.1

0.791 0.739

23.5 23.4

Jarczyk et al. (1995) Jarczyk et al. (1995)

62.4

0.821

24.1


71.2 60.7

0.734 0.823

23.8 23.7


64.5

0.810

24.4


65.0 65.0

0.812 0.822

24.3 23.3


60.6 63.9

0.846 0.803

23.8 24.2


66.6

0.815

23.3


Cherry juice Coffee beverage, freeze-dried

aw

0.986

Chirife and Ferro Fontan (1982)

5

0.996

Chen (1987b)

10 20

0.991 0.990

Chen (1987a) Chen (1987b)

30 40

0.978 0.964

Chen (1987a) Chen (1987a) (continued)


Table 3.2 (continued) Water Activity of Some Liquid Foods Product Corn syrup Cream, 40% fat

Concentration (%) 17% water

aw

Temperature (8C)

0.62 0.979

Reference

Flavored milks

0.9899 0.0007

Chirife and Ferro Fontan (1982) Esteban et al. (1990a)

Fruit juices

0.9901 0.0017

Esteban et al. (1990a)

Glucose, 66% Grape juice

0.78 0.983

Richardson (1986) Chirife and Ferro Fontan (1982)

Grape juice Grape juice concentrate

10 20 30

0.986 0.976 0.962

Chen (1987a) Chen (1987b) Chen (1987b)

Grapefruit juice concentrate Honey

59

0.844 – 0.860

Chen (1987b)

17% water

0.54

Honey, buckwheat

17.6%–20.9% water

aw ¼ 0.203 þ 0.02x

Bakier (2006)

Honey, rape

17.4%–19.5% water

aw ¼ 0.060 þ 0.026x

Bakier (2006)

Lemon juice concentrate Maracuja juice concentrate Milk, 1.5% fat

40–42

0.90 – 0.932

Chen (1987a)

54

0.88

Chen (1987a)

0.995


Milk, pasteurized Milk, whole

0.995 0.001 0.995

Esteban et al. (1989) Chirife and Ferro Fontan (1982)

Milk, whole

0.994 – 0.995


Milk condensed sweetened

0.833 0.0017

Favetto et al. (1983)

Molasses

26% water

0.76

NaCl solution

2.18% salt 10.16% salt

0.977 0.927

Chuang and Toledo (1976) Chuang and Toledo (1976)

15.45% salt 23.41% salt

0.877 0.799

Chuang and Toledo (1976) Chuang and Toledo (1976)

0.988

Orange juice Orange juice concentrate

Pineapple juice concentrate

15

0.982

Chirife and Ferro Fontan (1982) Chen (1987a)

40 50 55

0.908 0.892 – 0.905 0.88 – 0.90

Chen (1987a) Chen (1987a) Chen (1987a)

60

0.84 – 0.87

65.0 65

0.824 0.802 – 0.835

68

0.79

Chen (1987a) 23.2

Jarczyk et al. (1995) Chen (1987a)

72

0.735

Chen (1987a)

61

0.84

Chen (1987a)


Table 3.2 (continued) Water Activity of Some Liquid Foods Product

Concentration (%)

Raspberry juice

aw

Temperature (8C)

0.988

Reference Chirife and Ferro Fontan (1982)

Raspberry juice concentrate

53.6

0.903

23.5


Red currant juice concentrate

61.0

0.828

23.2


65.6

0.811

23.6


62.3

0.825

23.3


10 20

0.997 0.990

Chen (1987b) Chen (1987b)

30 40

0.982 0.971

Chen (1987b) Chen (1987b)

67.0

0.788

24.8


64.2 66.2

0.796 0.756

24.0 24.1


67.4

0.796

23.4


64.3 65.3

0.816 0.793

24.3 24.0


0.810 0.018

Stoloff (1978)

5.87 15.29 40.78

0.993 0.982 0.944

Chuang and Toledo (1976) Chuang and Toledo (1976) Chuang and Toledo (1976)

59

0.90

Favetto et al. (1983)

66 66.29

0.86 0.839

Richardson (1986) Chuang and Toledo (1976)

0.991


Skim milk solution

Sour cherry juice concentrate

Soy sauce Sucrose

Strawberry juice Strawberry juice concentrate Tomato ketchup

56.0

0.838

22.4


0.959 0.012

20

Jakobsen (1983)

Tomato paste, triple concentrated

0.934 0.0013

Tomato purée

0.978 0.029

Whey cheese Yogurt beverages

0.996 0.000 0.9876 0.0008

Favetto et al. (1983) 20

Jakobsen (1983) Esteban et al. (1989) Esteban et al. (1990a)

Culture media used in food microbiology Nutrient broth (Difco)

0.997

20

Esteban et al. (1990b)

Brain heart infusion broth (Oxoid)

0.995

20


Staphyloccocus medium no. 110 (Difco) Malt yeast extract 50% glucose broth Halophylic broth (25% NaCl)

0.936

20


0.876

20


0.816

20


Malt yeast extract 70% glucose fructose broth

0.740

20



Table 3.3 Water Activity of Solutions of Low Molecular Weight Compounds Concentration (%)

Lactose

1

0.999

3 5

0.997 0.995

Fructose

Sucrose

Glucose

Reference Chen (1987b) Chen (1987b) Chen (1987b)

0.996

5 5 5.87 8

Glycerol

0.96 0.98 0.993

Chuang and Toledo (1976) Chen (1987b) Chen (1987b)

0.992

10

0.991

0.992

15 15

0.985

0.988

Chen (1987b) Grover and Nicol (1940)

0.93

15 15.29

0.96 0.982

20 25.0

0.983

30 35.0

0.961

40

0.935

40.78 50.0

Chen and Karmas (1980) Chuang and Toledo (1976)

0.923


0.887


0.971 0.955

Chen (1987b)

0.944 0.814

60.0 66.29

Grover and Nicol (1940) Chen and Karmas (1980)

Chuang and Toledo (1976) Grover and Nicol (1940)

0.737

Grover and Nicol (1940) Chuang and Toledo (1976)

75.0 83.0

0.587 0.446

Grover and Nicol (1940) Grover and Nicol (1940)

0.92

0.275

Grover and Nicol (1940)

0.839

Concentration (%) Solute

5

15

Arabinose

0.97 0.98

0.96

Chen and Karmas (1980) Chen and Karmas (1980)

Galactose

0.98

0.96

Chen and Karmas (1980)

Gluconic acid Glucuronic acid

0.97 0.97

0.96 0.96


Glycine Lactic acid Malonic acid

0.97 0.96 0.98

0.94 0.94 0.95

Chen and Karmas (1980) Chen and Karmas (1980) Chen and Karmas (1980)

Maltose

0.98

0.96


Mannitol Mannose

0.97 0.98

0.96 0.96


Oxalic acid

0.98

0.96


Ribose Sorbitol Succinic acid

0.98 0.96 0.99

0.96 0.95

Chen and Karmas (1980) Chen and Karmas (1980) Chen and Karmas (1980)

Tartaric acid

0.97

0.96


Xylose

0.98

0.95


L-Alanine


Table 3.4 Water Activity of Glucose Solutions Water Activity at Temperature (8C) Concentration (%)

20

25

30

35

30.0

0.954

0.955

0.956

0.957

35.0 40.0

0.942 0.927

0.943 0.929

0.944 0.930

0.945 0.930

45.0 50.0

0.910 0.891

0.912 0.891

0.913 0.893

0.914 0.895

55.0 60.0

0.865 0.835

0.867 0.837

0.869 0.839

0.870 0.841

Glucose

Source: Adapted from Viet Bui, A., Minh Nguyen, H., and Muller, J., J. Food Eng., 57, 243, 2003.

Nonideality of the liquid solution can be characterized by the thermodynamic excess function. The excess functions are thermodynamic properties of solutions, which are in excess of those of an ideal solution at the same temperature, pressure, and concentration. Thus msolution mideal ¼ RT ln i i

ai ¼ RT ln g i xi

(3:22)

Hence, the excess function, the deviation of properties of a real solution from those of an ideal solution, is expressed by the activity coefficient. The problem resolves to finding the activity coefficient of component i in the solution. Activity coefficients for some sugars are given in Table 3.7. Several approaches have been used to calculate activity coefficients of liquid solutions. This includes empirical equations based on solution composition, equations derived by the thermodynamic approach, and equations based on solution theories (Sereno et al., 2001). Table 3.5 Water Activity of Sugar Solutions Glucose

Fructose

Sucrose

Maltose

aw

W

aw

W

aw

W

aw

0.049

0.995

0.049

0.995

0.049

0.998

0.046

0.998

0.096 0.142

0.990 0.983

0.096 0.142

0.989 0.985

0.096 0.142

0.995 0.991

0.091 0.134

0.995 0.990

0.186 0.229

0.977 0.970

0.186 0.229

0.978 0.971

0.186 0.228

0.989 0.985

0.176 0.217

0.988 0.984

0.272

0.962

0.274

0.965

0.270 0.292

0.980 0.974

0.258

0.979

0.349

0.967

0.399 0.445

0.958 0.948

W

Source: Adapted from Valezmoro, C., Meirelles, A.J., and Vitali, A., in Engineering & Food at ICEF 7, Jowitt, R. (Ed.), Sheffield Academic Press, Sheffield, UK, 1997, pp. A145– A148. Note: W, weight fraction for solute.


Table 3.6 Water Activity of Sugars Solutions aw Sucrose (8C) Molality 0.1 0.5

25

50

25

50

Fructose (8C) 25

0.9982 0.9907 0.9735 0.9806

1.0727 1.4

0.9719

1.5212 1.926

0.9599

1.9976 2.0

0.9581

0.9851

Ross (1975) 0.985 0.005

Correa et al. (1994b) Robinson and Stokes (1965)

0.979 0.005


0.971 0.005

Correa et al. (1994b) Ross (1975)

0.963 0.005


0.9617

2.379

0.9400

0.9517

Ross (1975) 0.954 0.005

0.9457 0.9442

0.9477

2.7466 3.0 3.1491

0.9328

3.5

0.9193

Reference Robinson and Stokes (1965) Robinson and Stokes (1965)

0.617 0.7513 1.0

2.3865 2.5 2.563

Glucose (8C)

Correa et al. (1994b) Robinson and Stokes (1965) Ross (1975)

0.947 0.005

Correa et al. (1994b)

0.938 0.005

Robinson and Stokes (1965) Correa et al. (1994b) Robinson and Stokes (1965)

0.924 0.005

3.9199 4.0

0.9057


4.5 5.0

0.8917 0.8776

Robinson and Stokes (1965) Robinson and Stokes (1965)

5.0339 5.5

0.8634

0.897 0.005

5.508 5.551

0.8631

6.0

0.8493

0.8758 0.8722

6.0778 7.2271

Correa et al. (1994b) Robinson and Stokes (1965) Ross (1975) Ross (1975)

0.8878

Robinson and Stokes (1965) 0.870 0.005 0.841 0.005

Correa et al. (1994b) Correa et al. (1994b)

For a simple two-component mixture, the Wohl equations for both components are obtained by combining the total excess Gibbs energy with the Gibbs–Duhem equation. qw 2 ln g w ¼ zs A þ 2zw B A qs (3:23) q s 2 ln g s ¼ zw B þ 2zs A B qw where z is the effective volume fraction of component in solution q is the effective molar volume of component in solution A, B are constants subscripts w and s denote water and solute, respectively


Table 3.7 Activity Coefficients for Saccharides at 258C Maltose (Miyajima et al., 1983a)

Maltotriose (Miyajima et al., 1983a)

D-Glucose

D-Mannose

D-Galactose

(Miyajima et al., 1983b)



0.1

1.003

1.003

1.002

1.000

1.001

0.2 0.3

1.007 1.010

1.010 1.016

1.004 1.006

1.000 1.000

1.001 1.002

0.4

1.014

1.023

1.009

1.000

1.002

0.5 0.6 0.7

1.020 1.027 1.036

1.030 1.040 1.049

1.013 1.017 1.021

1.001 1.001 1.002

1.003 1.004 1.005

0.8

1.044

1.061

1.026

1.003

1.006

0.9 1.0

1.052 1.061

1.074 1.088

1.031 1.036

1.005 1.006

1.007 1.009

1.1

1.070

1.103

1.2 1.3

1.081 1.091

1.120 1.138

1.047

1.010

1.013

1.4 1.5

1.103 1.114

1.158 1.179

1.059

1.014

1.018

1.6 1.7

1.126 1.138

1.199 1.220

1.071

1.019

1.025

1.8 1.9

1.150 1.163

1.242 1.265

1.084

1.025

1.032

2.0

1.176

1.288

Molality

1.097

1.031

1.039

2.5 3.0 3.5

1.131 1.166 1.200

1.048 1.068 1.088

1.061 1.083 1.104

4.0

1.233

1.109

4.5 5.0

1.266 1.298

1.130 1.150

5.5

1.329

1.169

6.0 Activity coefficients of xylose in water solution Activity coefficients of water in xylose solution Activity coefficients of maltose in water solution Activity coefficients of water in maltose solution

1.306 1.189 ln gs ¼ 0.02814m þ 0.003432m2 0.0006427m3, where m is the molality

Uedaira and Uedaira (1969)

ln gw ¼ 1.282x2 9.093x3 þ 65.61x4, where x is the mole fraction of the solute ln gs ¼ 0.007270m þ 0.02727m2 0.003207m3, where m is the molality ln gw ¼ 2.518x2 60.71x3 þ 302.5x4, where x is the mole fraction of the solute

Uedaira and Uedaira (1969) Uedaira and Uedaira (1969) Uedaira and Uedaira (1969)

The Wohl equation for activity coefficient of water is simplified to the following form: ln gw ¼ x2s (A þ kxw )

(3:24)

Assuming that molecules are similar in size, shape and chemical nature (q1 ¼ q2) and that A and B are equal, the two-suffix Margules equations are obtained (Prausnitz et al., 1999)


A 2 x RT s A 2 x ln g s ¼ RT w

ln g w ¼

(3:25)

Further expansions of these derivations are as follows (Poling et al., 2001). Three-suffix Margules equations: ln g w ¼ (2B A) x2s þ 2(A B) x3s

(3:26)

van Laar equations: A ln g w ¼ A 1 þ B B ln g s ¼ B 1 þ A

xw xs xs xw

2 2

(3:27)

The selected parameters for the above equations for sugar solutions are collected in Table 3.8. Solution theories were used to develop molar excess Gibbs energy and activity coefficients. Wilson (1964) applied local-composition models and developed the following equation, which can be written for a binary mixture as Lws Lsw ln g w ¼ ln (xw þ Lws xs ) þ xs xw þ Lws xs xs þ Lsw xw (3:28) vs vw L12 ¼ e(Ews =RT) ; L21 ¼ e(Esw =RT) vw vs The molar volume vw and vs refer to pure liquids with an activity of 1. A pure liquid phase does not exist for dissolved components, hence the ratio of their molar volumes is used and the equation is simplified to the following form: ! n n X X xL Pnk wk ln g w ¼ ln xk Lwk þ 1 (3:29) i¼1 xi Lik k¼1 k¼1 Table 3.8 Parameters of Equations 3.24 through 3.27 for Sugar Solutions (de Cindio et al., 1995) Equation

A

B

k

Glucose Margules (Equation 3.25) Margules (Equation 3.26)

2.217 4.240

2.977

van Laar (Equation 3.27)

8.494

3.866

Wohl (Equation 3.24)

4.240

2.526

Sucrose Margules (Equation 3.25) Margules (Equation 3.26)

5.998 2.1819

2.064

van Laar (Equation 3.27)

2.064

3.948

Wohl (Equation 3.24)

2.585

9.916

Source: Adapted from de Cindio, B., Correra, S., and Hoff, V., J. Food Eng., 24, 405, 1995.


Renon and Prausnitz’s (1968) equations known as the non-random two liquid equations (NRTL) are also based on the concept of local composition. The NRTL equation for a binary mixture is expressed as " ln g w ¼

x2s

t sw

Gsw xw þ xs Gsw

Gws ¼ exp (a12 t ws ) t ws

2

t ws Gws þ (xs þ xw Gws )2

# (3:30)

Aws ¼ RT

Parameters of the NRTL equation are presented for some food volatiles in Table 3.9. The NRTL local composition model was extended to electrolyte solutions (Chen et al., 1982). Recently, attempts have been made to develop equations of state for polymer systems. The segment-based polymer NRTL model was developed by Chen (1993) and the perturbed hard-sphere-chain model was proposed by Song et al. (1994). Mixing of high molecular weight polymer with solvent causes the entropy change. This reduction in the entropy of mixing was first formulated independently by Flory (1942) and Huggins (1942) using a lattice theory. The developed equation describes a free energy change during mixing of a solvent and amorphous polymer.

1 2 m ¼ m0 þ RT ln (1 n2 ) þ 1 n2 þ xn2 b n2 b¼ n1

(3:31)

where n1 is the molar volume of solvent n2 is the molar volume of polymer x is the Flory interaction parameter denoting intermolecular interactions The above equation is valid for such polymer concentrations at which polymer chains interlace in the solution. When the solution is sufficiently diluted and the chains are not interacting each with other, Equation 3.31 simplifies to the form: m ¼ m0 þ RT ln (1 n2 ) þ n2 þ xn22

(3:32)

Table 3.9 Parameters of the NRTL Equation for Binary Mixtures Substance in Water

Temperature (8C)

Acetic acid

100–119

Aws 183.82

Ethanol

40

1443.6

Ethyl acetate Ethyl acetate

50 71–76

2241.3 1881.15

Asw 47.032 391.8 618.1 727.01

a12

Reference

0.39688

Faúndez and Valderrama (2004)

0.1803

Lee and Kim (1995)

0.2834 0.39304

Lee and Kim (1995) Faúndez and Valderrama (2004)

Methanol

25

699.9

280.6

0.2442

Lee and Kim (1995)

Methanol

65–100

573.90

208.12

0.29426


1-Propanol

88–95

0.30282



1094.4

12.175

Equation 3.32 rewritten for water activity in the polymer solution is as follows: n1 f þ xf2p ln aw ¼ ln fw þ 1 n2 p

(3:33)

where f is the volume fraction subscripts w and p are for water and polymer, respectively The Flory–Huggins model was applied to solutions of simple molecules, and the results of calculation are presented in Table 3.10. The method, based on the group contribution, views molecules as made of a certain number of standard functional groups. The Analytical Solution of Groups (ASOG) was developed by Wilson and Deal (1962), Derr and Deal (1969), and Kojima and Tochigi (1979). According to this method the activity coefficient of component i in a liquid mixture is a result of differences both in molecular size and shape and in intermolecular forces. The ASOG method was used by Correa et al. (1994a) to calculate water activities of solutions of sugars and urea at 258C. An average deviation of 0.4% was obtained between experimental and predicted values of aw. In another publication, Correa et al. (1994b) showed that water activity of binary and ternary solutions containing sugars, glycerol, or urea can be predicted with the ASOG method with the deviation between the measured and predicted values not exceeding 1%. The method was applied to electrolytes solutions (Kawaguchi et al., 1981) and it was shown by Correa et al. (1997) that calculated values of water activity for binary, ternary, and quaternary systems deviated from measured values by 0.21%, 0.28%, and 0.20%, respectively. The concept of group contribution developed by Abrams and Prausnitz (1975) known as Universal Quasi Chemical method was used by Peres and Macedo (1996) to predict thermodynamic properties of sugar aqueous solutions. Predicted water activities of binary systems water=D-glucose, water=D-fructose, and water=sucrose differed from the measured values by 0.28%, 0.44%, and 0.96%, respectively. Another approach based on the group contribution concept known as universal functional activity coefficient (UNIFAC) was proposed by Fredenslund et al. (1975). This has received more attention and a continued development of this method has been observed. In this method, a molecule is pictured as an aggregate of functional groups, and its physical properties are the sum of Table 3.10 Parameters of the Flory–Huggins Equation for Glucose Solution Molality

fs

x

0.14

0.011

77.0

0.28

0.021

27.6

0.56 0.83

0.041 0.060

7.1 4.3

1.11 1.39

0.079 0.096

2.6 1.55

2.22

0.146

0.40

2.56 3.33

0.176 0.204

0.07 þ0.11

Source: Adapted from Napierala, D., Popenda, M., Surma, S., and Plenzler, G., in Properties of Water in Foods, Lewicki, P.P. (Ed.), Warsaw Agricultural University Press, Warsaw, 1998, pp. 7–13.


contributions made by the groups. The contributions are independent and additive. Contributions are due to differences in molecular size and shape as well as molecular interactions. A solution is treated as a mixture of groups, which contribute to the partial molal excess free energy independently and additively. The contributions are associated with differences in molecular size and with interactions of the structural groups (Wilson and Deal, 1962). The UNIFAC model assumes that water activity results from the combinatorial and residual contributions. Hence ln aw ¼ ln aw )c þ ln aw )r

(3:34)

The combinatorial part is derived from the pure component properties such as group volume and area constants. This part is described by equations: ! n z Qw Fw X þ lw xj l j ln aw ) ¼ ln Fw þ qw ln 2 Fw xw j¼i X X qw xw rw xw ; Fw ¼ Pn ; ri ¼ vik Rk ; qi ¼ vik Qk Qw ¼ Pn j qj x j j rj xj k k c

(3:35)

The residual part of the activity is a function of group area fractions and their interactions in pure components and in mixtures. It is given by ln aw )r ¼

X

w vw k ln Gk ln Gk

(3:36)

k

where Gk is a group residual activity and Gw k is the group residual activity of group k in a reference solution containing only molecules of water (w): " ln Gk ¼ Qk 1 ln Qm Xm Vm ¼ P ; n Qn Xn

!

# X Vm Ckm P Vm Cmk n Vn Cnm m m P j a j vm xj mn Xm ¼ P P j ; Cmn ¼ exp T j xj m vm

X

(3:37)

where amn is the group interaction constant; li ¼ z=2(ri qi) (ri 1); Ak is the van der Waals area of group k; Qk is the group area constant, Qk ¼ (Ak=2.5 109); Rk is the group volume constant, Rk ¼ (Vk=15.17); vk is the number of groups of type k in molecule i; Vk is the van der Waals volume of group k; xi are mole fraction of component i; z is coordination number, z ¼ 10. Qi is the component area fraction, Fi is the component volume fraction, Vm is the area fraction of group m, Xm is the mole fraction of group m. Subscripts denote i component i; k, m, n are groups, w is water. The UNIFAC model was used by Choundry and Le Maguer (1986) to predict water activities of glucose solutions. Catté et al. (1995) used the UNIFAC equations to model aqueous solutions of sugars. In this model, conformational equilibrium was taken into account, thus isomers and anomers can be distinguished. Spiliotis and Tassios (2000) modified a UNIFAC model in order to predict phase equilibriums in aqueous and nonaqueous sugar solutions. The model allowed satisfactory prediction of water activity in sugar solutions. Further modification of a UNIFAC model was done by Peres and Macedo (1997) who introduced a new group in a sugar molecule. The ‘‘OH-ring’’ group behaves in a different way from the alcohol OH-group; hence, the two cyclic structures ‘‘PYR’’ and ‘‘FUR’’ can be treated as single groups. In systems, D-glucose–sucrose–water, water


Table 3.11 Water Activities for the System D-Glucose–Sucrose–Water at 258C aw

Molality D-Glucose

Predicted from UNIFAC Model (Peres and Macedo, 1997)

Measured (Lilley and Sutton, 1991)

2.8039

0.5542

0.92098

0.9260

2.3504 1.2166

1.0996 2.4489

0.92189 0.92385

0.9260 0.9260

1.7391 0.8286

1.8737 2.9503

0.92207 0.92350

0.9251 0.9251

0.3648

3.4947

0.92406

0.9251

Sucrose

activity was predicted with relative deviation lower than 0.6%. Some results of the UNIFAC model application are presented in Table 3.11. The modified UNIFAC model proposed by Peres and Macedo (1997) was used to predict the water activities of fruit juice concentrates and synthetic honey (Peres and Macedo, 1999) assuming that they are mixtures of sugars and water. For a total sugar concentration between 10% and 60% the water activities were predicted with an average deviation of 0.34% for apple juice and 0.37% for grape juice. The UNIFAC model with a new assignment of groups was used to describe the activity coefficients in binary systems of amino acids and peptides in water and other biochemicals. The new assignment of groups was needed because biochemicals have several asymmetric carbons and form electrolyte complexes with ions. Activity coefficients for solutions of DL-proline, xylose, D-glucose, D-mannose, D-galactose, maltose, sucrose, raffinose, and KCl were predicted with root mean square deviation (RMSD) lower than 1%. For L-hydroxyproline, sodium and potassium glutamate, Lagrinine HCl, L-histidine HCl, NaCl, and sodium glucuronate the RMSD was between 1% and 2.69%. For other studied compounds, the RMSD was larger than 5% (Kuramochi et al., 1997). Attempts to predict thermodynamic properties of solution of complex molecules such as proteins have been made (Chen et al., 1995; Curtis et al., 2001). New methods such as cosmo-Rs (Eckert and Klamt, 2002) and group and segment contribution solvation models (Lin and Sandler, 1999, 2002) using quantum chemistry and molecular modeling are under development. The predictions based on group contribution models are good although they consider binary and in some cases ternary solutions. However, liquid foods are multicomponent mixtures and the group contribution models are not very suitable unless there are some dominant components, and the solution can be treated as binary or ternary mixture. For practical application empirical and semiempirical equations are developed.

3.3 SEMIEMPIRICAL EQUATIONS The semiempirical equations are based on Raoult’s law for an ideal solution. Water activity is described by the following equation: aw ¼

nw nw þ ns

where n is the number of moles subscripts w and s denote water and solid, respectively


(3:38)

Interaction of water molecules with solute molecule leads to formation of hydrated moieties. At solvation equilibrium, average hydration number denotes the average number of molecules of bound solvent per solute molecule. For real solutions the hydration number is given by the following equation: 55:51 aw h¼ m 1 aw

(3:39)

where m is the molality aw is the water activity The above approach was taken up by Stokes and Robinson (1966) and equation predicting activity of solution of several solutes based on known data for single solutes was developed 55:51 aw s ¼ þ m 1 aw S

(3:40)

where ¼ 1 þ Kaw þ (Kaw )2 þ þ (Kaw )n s ¼ Kaw þ þ n(Kaw )n and K is the solvation equilibrium constant For sucrose solutions, n ¼ 11 and K ¼ 0.994. For glucose, n ¼ 6 and K ¼ 0.786 and for glycerol n ¼ 3 and K ¼ 0.720. The n value was assumed to be equal to oxygen sites in a molecule able to interact with water molecules. For a mixture of two solutes the equation is 55:51 aw mA hA þ mB hB ¼ þ m A þ m B 1 aw mA þ mB

(3:41)

with the assumption that there is no interaction between solutes A and B. For sucrose and sorbitol a very good agreement was obtained for measured and calculated values. Sucrose and glucose, sucrose and arabinose, and sucrose and glycerol deviation not exceeding 1.5% was noticed. Poliszko and coworkers (2001) developed an equation accounting for hydration water and its effect on water activity of sugar solutions. The equation is as follows:

Dm hc RT (1 c) Dmh Dm 2pc ¼ h 1 þ A cos RT RT D 1 cs h¼ cs aw ¼ exp

where Dm is the average excess chemical potential of hydration water h is the average hydration degree A is the fluctuation amplitude c is the concentration cs is the saturation concentration


(3:42)

Table 3.12 Parameters of Equation 3.42 Sugar

cs (g=g)

A

h

Dm (J=mol)

Fructose

0.824

0.0979

0.214

1226

Glucose Maltose

0.470 0.380

0.1755 0.2520

1.128 1.632

285 144

Xilitol

0.600

0.1680

0.670

406

Source: From Baranowska, H.M., Klimek-Poliszko, D., and Poliszko, S., in Properties of Water in Foods, Lewicki, P.P. (Ed.), Warsaw Agricultural University Press, Warsaw, 2001, pp. 12–20.

Parameters of Equation 3.42 for some sugars are presented in Table 3.12. Raoult’s law written in mass fractions yields the following equation: aw ¼

1x 1 x þ Ex

(3:43)

where x is the solute content in the solution (kg=kg solution) E is the ratio of the molecular weight of water to molecular weight of the solute E¼

Mw Ms

(3:44)

Assuming that some amount of water is bound with the solute, Schwartzberg (1976) proposed to modify Raoult’s law and gave the expression: aw ¼

1 x bx 1 x bx þ Ex

(3:45)

where b is the amount of water bound by unit weight of solid (kg=kg solids). Modification of Raoult’s law was presented by Palnitkar and Heldman (1970) in which effective molecular weight of the solute was introduced aw ¼

(xw =M) (xw =M) þ (xs =EMW)

(3:46)

where xw and xs are the mass of water and solute, respectively (kg=kg solution) M is the molecular weight of water (kg=mol) EMW is the effective molecular weight of solute (kg=mol) The above equation was used to calculate activity coefficients for sugars, polyols, amines and amino acids, organic acids, inorganic salts, and isolated soy proteins (Chen and Karmas, 1980). It was found that EMW for Promine (Central Soya, Chicago) was between 821.2 and 950.1, depending on its composition. Caurie (1983) derived an equation based on Raoult’s law in which water activity of a solution is expressed as the difference between the values calculated for an ideal solution and the amount of overestimation arising from interactions between solvent and solutes aw ¼


55:5 ms (1 aw ) ms þ 55:5 ms þ 55:5

(3:47)

Table 3.13 Parameters of Equation 3.48 Sugar

E

b

D-Fructose

0.100

0.10–0.18

D-Glucose

Lactose

0.100 0.053

0.15–0.25 0.21–0.46

Maltose Sucrose

0.053 0.053

0.21–0.26 0.26–0.30

Freeze-dried skim milk (10%–40% concentration)

0.054

0.11–0.13

Freeze-dried coffee beverage (5%–40% concentration)

0.052

0.00–0.02

Concentrated orange juice

0.075

0.15–0.20

Fruit juice (average)

0.089

0.20

Source: Adapted from Chen, C.S., LWT, 20, 64, 1987a.

where ms is the molal concentration of the solute. It was concluded that water activity is a measure of free, available, and unbound fraction of water in a solution. Chen (1987a) proposed to derive activity coefficient by comparison of ideal and real solution activities. Taking Equations 3.44 and 3.45 the following is obtained: g ¼1

Eby2 1 þ y(E b)

(3:48)

where y ¼ x=(1 x). Calculated water activities from the above equation up to sugar concentration of 40% agreed with the experimental data within 0.01 unit of aw. Equation parameters for sugars are collected in Table 3.13. Chen (1989) developed a simple equation to predict water activity of single solutions. The equation predicts water activity of a wide range of solutes with an accuracy of 0.001 aw. The equation is as follows: aw ¼

1 1 þ 0:018(b þ Bmn )m

(3:49)

where m is the molality b and n are constants Values of constants for some solutes are given in Table 3.14. The above equation in combination with the Ross (1975) equation was used to calculate water activity in ternary and quaternary electrolyte and nonelectrolyte solutions (Chen, 1990). In electrolyte solutions such as NaCl þ KCl, NaCl þ KNO3, NaCl þ KCl þ LiCl, water activity was predicted with an accuracy higher than 0.01 aw. For sucroseþglucose and sucrose þ glycerol, water activity can be predicted with an accuracy of at least 0.005 aw. The mixture of sucrose and NaCl yielded water activities, which differed from the predicted values by some 0.002 aw. Water activities of some multicomponent mixtures are presented in Table 3.15. Prediction of water activity in a mixed solution of interacting components is of particular interest, since equilibrium measurement would be difficult if not impossible. The development is based on the experimental fact of linear or near linear relationship between the molalities of most


Table 3.14 Parameters of Equation 3.49 b

Solution

B

n

Molality Range 0, below which the overall conductivity of a sufficiently large system is zero. In terms


of a binary composite with k2 k1, the prediction of percolation theory is that for a sufficiently large system ke ¼ 0 (k2) if « > «cr, and ke ¼ 0 (kd if « < «cr). In this case, «cr is referred to as the percolation threshold or simply the threshold volume fraction. Gurland (1966) studied compacted mixtures of silver balls and kelite powder and found a conductivity threshold «cr ¼ 0.3. Malliars and Turner (1971) showed that the threshold value of «cr for a compact of metal particles (nickel) and insulator (polyvinyl chloride) depended on the size and shape of metal particles. Shante and Kirkpatrick (1971) found «cr ¼ 0.29 for a continuum percolation model, in which the conducting material consisted of identical spheres that were permitted to overlap and whose centers were randomly distributed. By computer studies on a network of conductors randomly placed on a single lattice, Kirkpatrick (1973) determined that the threshold fraction was 0.25. Finally, EMT (described below) yields «cr ¼ 1=3. Thus, the threshold volume theoretically found varied from 1=4 to 1=3.

19.5 EFFECTIVE MEDIUM THEORY Landauer (1952) successfully described the conductivity of several binary metallic mixtures, and Davis et al. (1975), by different reasoning, arrived at the same expression for a medium with n constituents: n X i¼1

«i

ke ki ¼0 ki þ 2ke

(19:34)

where ke is the effective thermal conductivity of the medium (W=m2 K). Kirkpatrick (1973) extended EMT to regular networks of resistors distributed at random, obtaining a similar equation: n X i¼1

ke ki ¼0 ki þ ððz=2Þ 1Þke

(19:35)

where z is the coordination number of the system (i.e., number of conductors with a site in common). Kirkpatrick (1973) assumed that a continuous medium can be represented by a cubic arrangement, for which z ¼ 6. For a system of two components, with k1 and k2 not equal to zero, Equation 19.33 can be written as (Mattea et al., 1986) ke 2p 2 ¼vþ v þ z2 k2

(19:36)

where k1 k2 ðz«2 =2Þ 1 þ p½ðz=2Þð1 «2 Þ 1 v¼ z2 p¼

Carson et al. (2001) used finite element numerical models to examine how the effective thermal conductivity of a material containing voids is affected by its structure. Simulated results were fitted by two commonly used effective thermal conductivity models with adjustable parameters: Krischer’s model and the EMT. The sizes of the individual voids were found to have a minor


influence on the effective thermal conductivity, whereas a more significant influence on thermal conductivity was the void shape. There are many other theoretical models available in the literature (Yagi and Kunii, 1957; Kunii and Smith, 1960; Tsao, 1961; Cheng and Vachon, 1969; Chen and Heldman, 1972; Okazaki et al., 1977). The above theoretical models were developed assuming a simple structure and heat transfer mechanisms were involved. This theoretical model has limited applications in food materials due to its complex structure or arrangement of the phases. The parameters in theoretical models vary not only with different foods but also with the variety, harvesting location and time, and also the measurement location in the food materials. Due to this limitation of the theoretical models, semiempirical and empirical models are popular and widely used for food materials. The semiempirical models, which include the weighting structure factor, have prospects in food applications due to their theoretical basis and only a single parameter is necessary to represent the structure. Theoretically, the structure factor varied from 0 to 1 if component phase thermal conductivity does not change after mixing or heating. In the case of food materials at a higher temperature, chemical and physical structure changes of the component phases may occur due to the interaction of the phases. Hence, in some cases the factor may be higher than one. For example, in the case of potato, the experimental values were much higher than the maximum limit predicted by the parallel model at a higher temperature (t > 708C). Califano and Calvelo (1991) proposed two reasons for higher thermal conductivities: (1) intercellular air may be considered as air trapped in a capillary porous body, where vapor was transferred through the capillary due to temperature difference between its wall, and (2) the structural changes produced by starch gelatinization in the samples might contribute to increase in thermal conductivity at a higher temperature. A similar increase in thermal conductivity was detected by Drouzas and Saravacos (1988) for corn starch at water contents above 11.1% and temperature above 508C.

19.6 EMPIRICAL MODELS 19.6.1

Fitting of Data for a Specific System

Thermal conductivity varies with chemical composition, distribution of the phases (i.e., geometric factor), density or porosity, temperature, and pressure. For solids and liquids, pressure has negligible effect on thermal conductivity. Thermal conductivity prediction equations of the major food components are given in Table 19.2. Water content was found to be the most important factor in determining thermal conductivity while the nonaqueous part of the food was of lesser importance. This may be due to the relative magnitude of the conductivities of water and other food constituents (Cuevas and Cheryan, 1978). Thermal conductivity of foods decreases with a decrease in moisture content. It is common to find a linear relation between thermal conductivity and moisture content or temperature. Mohsenin (1980), Miles et al. (1983), and Sweat (1986) reviewed this type of correlation for different types of food materials (Tables 19.3 and 19.4). A number of authors used the linear multiple regression equation to relate the thermal conductivity (Table 19.5). The linear correlation of thermal conductivity with moisture content is limited to small changes in moisture and the constants of the correlation vary not only with the types of food materials but also with the variety and growing or harvesting location and time. Hence, a nonlinear correlation is necessary to cover the whole range of moisture contents. Quadratic and multiple form correlations are also used for food materials (Table 19.5). Baghe-khandan et al. (1982) presented correlations to predict the thermal conductivity of fresh whole and ground beef meat as h i ke ¼ 303:7 454Xfa 219Xw þ 306ðt Þ0:039 103 (whole)


(19:37)

TABLE 19.2 Thermal Conductivity of Major Food Components as a Function of Temperature Material

Equation

Reference

Air (dry air)

k ¼ 0.0184 þ (1.225 104)t

Luikov (1964)

Air (moist air: 208C–608C) Air

k ¼ 0.0076 þ (7.85 104)t þ 0.0156f k ¼ 3.24 103 þ (5.31 104)t

Luikov (1964) Luikov (1964)

Air (08C–10008C)

k ¼ 2.43 102 þ (7.98 105)t(1.79 108)t2 (8.57 1012)t3

Maroulis et al. (2002)

Air (P: mm Hg, P 2 mm Hg) Air (P: mm Hg, P 2 mm Hg)

k ¼ 0.0042P þ 0.01 k760=k ¼ 1 þ 1.436 (1=P)

Fito et al. (1984) Fito et al. (1984)

Protein (t ¼ 408C to 1508C) Gelatin

k ¼ 1.79 101 þ (1.20 103)t (2.72 106)t2 k ¼ 3.03 101 þ (1.20 103)t (2.72 106)t2

Choi and Okos (1986) Renaud et al. (1992)

Ovalbumin

k ¼ 2.68 101 þ (2.50 103)t

Renaud et al. (1992)

Carbohydrate (t ¼ 408C to 1508C)

k ¼ 2.01 101 þ (1.39 103)t (4.33 106)t2

Choi and Okos (1986)

Starch Gelatinized starch

k ¼ 4.78 101 þ (6.90 103)t k ¼ 2.10 101 þ (0.41 103)t

Renaud et al. (1992) Maroulis et al. (1991)

Sucrose

k ¼ 3.04 101 þ (9.93 103)t

Renaud et al. (1992)

Fat (t ¼ 408C to 1508C) Fiber (t ¼ 408C to 1508C) Ash (t ¼ 408C to 1508C)

k ¼ 1.81 101 (2.76 103)t (1.77 107)t2 k ¼ 1.83 101 þ (1.25 103)t (3.17 106)t2 k ¼ 3.30 101 þ (1.40 103)t (2.91 106)t2

Choi and Okos (1986) Choi and Okos (1986) Choi and Okos (1986)

Water (t ¼ 408C to 1508C)

k ¼ 5.71 101 þ (1.76 103)t (6.70 106)t2


Water Water

k ¼ 5.87 101 þ [2.80 103(t 20)] k ¼ 5.62 101 þ (2.01 103)t (8.49 106)t2

Water (08C–3508C)

k ¼ 5.70 101 þ (1.78 103)t(6.94 106)t2 þ (2.20 109)t3

Renaud et al. (1992) Nagaska and Nagashima (1980, Cited in Hori, 1983) Maroulis et al. (2002)

Ice (t: 408C to 1508C)

k ¼ 2.22 (6.25 103)t þ (1.02 104)t2


ke ¼ ½668:4 7:4Xfa 305Xw þ 1:35t 103 (ground)

(19:38)

The limits were 0.50 < Xw < 0.78 and 308C < t < 908C. Wang and Hayakawa (1993) developed a correlation for thermal conductivity of gelatinized starch gel (Xw ¼ 0.396–0.713; t ¼ 808C–1208C): k ¼ 4:1676 101 þ (6:6375 104 )t þ (3:6 107 )Xw3 (9:5 106 )t Xw

(19:39)

Wang and Hayakawa (1993) developed a correlation for thermal conductivity of gelatinized starch gel with dissolved sucrose (Xw ¼ 0.492–0.750; t ¼ 808C–1208C): ke ¼ 4:1830 101 þ (1:1312 104 )t þ (1:009 105 )Xw3 (4:3 107 )t Xw

(19:40)

Mattea et al. (1989) and Lozano et al. (1979) used a nonlinear correlation, which was in exponential form, for correlating their thermal conductivity data of apple and pears during drying. The correlation was as follows:


ke ¼ 0:490 0:443½exp (0:206Mw )(apple)

(19:41)

ke ¼ 0:513 0:301½exp (1:107Mw (pear)

(19:42)

TABLE 19.3 Numerical Values for the Coefficients A and B in the Equations for the Thermal Conductivity of an Unfrozen Food (ke ¼ A þ BXw) Xw Range

t Range (8C)

Fish Mince meat

— —

— —

0.032 0.096

0.329 0.340

Miles et al. (1983) Miles et al. (1983)

Sorghum

—

—

0.564

0.086

Miles et al. (1983)

0.140 0.080 0.139

0.420 0.520 0.119

Miles et al. (1983) Sweat (1975) Chandra and Muir (1971)

0.139 0.144

0.141 0.095

Chandra and Muir (1971) Chandra and Muir (1971)

Material

— 0 to 60 27 to 20

A

B

Reference

Fruit juice Fresh meat Spring wheat (hard red)

— 0.600–0.800 0.044–0.225

Spring wheat (hard red) Spring wheat (hard red)

0.044–0.225 0.044–0.225

Spring wheat (hard red)

0.044–0.225

1

0.136

0.136

Chandra and Muir (1971)

Spring wheat (hard red) Spring wheat (hard red)

0.044–0.225 0.044–0.225

6 17

0.133 0.141

0.154 0.094

Chandra and Muir (1971) Chandra and Muir (1971)

27

20 5


0.044–0.225

0.144

0.095

Chandra and Muir (1971)


0.014–0.148

32–60

0.129

0.274

Moote (1953, cited in Chandra and Muir, 1971)

Soft wheat (white)

0.000–0.203

21–44

0.117

0.113

Kazarian and Hall (1965, Cited in Chandra and Muir, 1971)

Bulk corn (four varieties) Tomato paste Tomato paste

0.000–0.400 0.538–0.708 0.538–0.708

40 30 40

0.108 0.029 0.066

0.180 0.793 0.978

Kustermann (1984) Drusas and Saravacos (1985) Drusas and Saravacos (1985

50

0.079

1.035

Drusas and Saravacos (1985

20

0.050 0.156

0.077 0.404

Mohsenin (1980) Sweat and Parmelee (1978)

Tomato paste

0.538–0.708

Bulk rice (rough) Dairy product

0.090–0.190 0.160–0.822

Dairy product

0.160–0.822

0

0.166

0.381

Sweat and Parmelee (1978)

Dairy product

0.167–0.822

40

0.169

0.395

Sweat and Parmelee (1978)

—

TABLE 19.4 Numerical Values for the Coefficients c and d in the Equations for the Thermal Conductivity of a Food (ke ¼ A þ Bt) Material

Xw

t Range (8C)

A

Mince meat

—

—

0.295

2.00 104

B

Sorenfors (1973)

Reference

Chicken meat (dark)

0.763

0 to 20

0.481

8.65 104

Sweat et al. (1973)

Chicken meat (white) Chicken meat (dark) Milk powder (skim) (rb ¼ 293 kg=m3)

0.744 0.763 0.014

0 to 20 0 to 20 10 to 40

0.476 0.481 0.028

6.05 104 8.65 104 7.51 104

Sweat et al. (1973) Sweat et al. (1973) MacCarthy (1984)

Milk powder (skim) (rb ¼ 577 kg=m3) Milk powder (skim) (rb ¼ 724 kg=m3) Apple ‘‘Golden Delicious’’

0.042

15 to 50

0.052

5.80 104

MacCarthy (1984)

0.040

15 to 50

0.073

7.28 104

MacCarthy (1984)

0.873

0 to 25

0.394

2.12 103

Ramaswamy and Tung (1984)

Apple ‘‘Granny Smith’’ Apple ‘‘Golden Delicious’’

0.858 0.873

0 to 25 25 to 0

0.367 1.290

2.50 103 9.50 103

Ramaswamy and Tung (1984) Ramaswamy and Tung (1984)

Apple ‘‘Granny Smith’’

0.858

25 to 0

1.070

1.11 102

Ramaswamy and Tung (1984)

Potato

0.798

20 to 85

0.624

1.19 103

Lamberg and Hallstrom (1986)

Tomato paste

0.618

30 to 50

0.482

1.50 103

Drusas and Saravacos (1985)

0.087

9.36 104

Maroulis et al. (1990)

Soy flour (defatted)


—

—


TABLE 19.5 Multiple and Quadratic Form of Thermal Conductivity Prediction Equations for Food Materials (ke ¼ a þ bXw þ ct) Material

Xw

t Range (8C)

Equation

References 3

Mince meat Coconut

0.40–0.80 0.02–0.51

25–50 25–50

ke ¼ 0.049 þ 0.340Xw þ (2.40 10 )t ke ¼ 0.058 þ 0.181Xw þ (0.90 103)t

Sorenfors (1973)

Fresh meat Apple

0.65–0.85 0.2–1.6 (Mw)

40 to 5 5–45

ke ¼ 0.280 þ 1.900Xw (9.20 103)t ke ¼ 0.051 þ 0.072Mw þ (0.67 103)t

Sweat (1975) Singh and Lund (1984)

20–90

ke ¼ 0.310 0.820Xw þ (1.30 103)t

Metel et al. (1986)

Dough wheat

—

Jindal and Murakami (1984)

Chicken meat (dark) Chicken meat (white) Milk powder (rb ¼ 512 kg=m3)

0.763 0.744 0.031

75 to 10 75 to 10 12 to 43

ke ¼ 1.140 1.460t (9.86 105)t2 ke ¼ 1.070 1.490t (10.4 105)t2 ke ¼ 0.0076 þ 0.268t (2.30 105)t2

Sweat et al. (1973) Sweat et al. (1973) MacCarthy (1985)

Milk powder (rb ¼ 605 kg=m3)

0.031

17–43

ke ¼ 0.0003 þ 0.646t (9.58 105)t2

MacCarthy (1985)

Potato

0.800

50–100

ke ¼ 1.05 – (1.960 102)t þ (19.00 105)t2

Califano and Calvelo (1991)

Mattea et al. (1986) used other correlations for potato and pear: ke ¼ 0:4875

0:1931 þ 0:0227 ln (Mw )(potato) Mw

(19:43)

0:1931 0:0301 þ (pear) Mw Mw2

(19:44)

ke ¼ 0:5963

Rahman and Potluri (1990) presented a more general form of correlation than Mattea et al. (1989) and Lozano et al. (1979) for correlating their data on squid meat during air-drying by normalizing both sides of the correlation with initial water content before drying or cooking. The correlation was ke Xw ¼ A B exp C o keo Xw

(19:45)

The parameters are given in Table 19.6 for different food materials. The above form of correlation can predict the thermal conductivity with less than 10% error (Rahman, 1992). There was always a tendency to make general correlations to predict any property of food materials for use in process design equations. For example, Baroncini et al. (1980) compiled a number of general correlations to predict the thermal conductivity of a liquid. They then suggested an improved correlation for calculation of liquid thermal conductivity. Again, Kubaitis et al. (1990) studied the possibility of investigating saturation processes using a generalized expression of process rate constant. For fruits and vegetables, Lozano et al. (1983) developed a general correlation to predict the apparent shrinkage during a drying process. Miles et al. (1983) compiled the thermophysical properties prediction models of foods and gave ke ¼ 0:344Xw 0:0644Xpr 0:133Xfa þ 0:0008t

(t > tF )

(19:46)

Riedel (1949, cited in Ramaswamy and Tung, 1981) proposed a correlation to predict the thermal conductivity of fruit juices, sugar solutions, and milk:

ke ¼ 0:566 þ (1:799 103 )t (5:882 106 )t 2 (7:598 104 ) þ 9:342Xw

(19:47)

It was estimated that between 08C and 1808C there was an error of 1% when this model (Equation 19.45) was used (Singh, 1992), but it is very limited in use, since it can be applied only to the indicated products. Salvadori and Mascheroni (1991) proposed a general correlation for meat products. When heat transfer is parallel to fiber, ke ¼ 0:1075 þ 0:501Xw þ (5:052 104 )Xw t

(t tF )

(19:48)

TABLE 19.6 Parameters of Equation for Different Foods Material

Process

n

A

B

C

Apple

Air-drying

15

0.155

0.021

3.713

Beef Pear

Cooking Air-drying

41 15

1.832 1.120

1.737 1.166

0.814 2.368

Potato

Air-drying

10

1.245

1.279

1.654

Squid

Air-drying

39

1.200

1.350

1.750

Source: Rahman, M.S., J. Food Eng., 15, 261, 1992. With permission.


ke ¼ 0:398 þ 1:448Xwo þ

0:985 t

(t < tF )

(19:49)

When heat transfer is perpendicular to fiber, ke ¼ 0:0866 þ (0:052 104 )Xw t ke ¼ 0:378 þ 1:376Xwo þ

0:930 t

(t tF ) (t < tF )

(19:50) (19:51)

Vagenas et al. (1990) correlated the thermal conductivity of raisins (t ¼ tF to 1008C; Xw ¼ 0–1.0; and «b ¼ 0.37–0.49) as kb ¼ 0:118 þ 0:422Xw þ 0:002t þ 0:120«b

(19:52)

Morita and Singh (1979) and Jindal and Murakami (1984) also used this form of equation. In the case of food materials, thermal conductivity empirical correlation is limited to specific materials and varieties. 19.6.2

Data Mining Approach

There has been consistent effort spent in developing generalized correlations to predict properties of food materials for use in process design and optimization (Rahman, 1992). Sweat (1986) developed a correlation using a varied data set of about 430 points for liquid and solid foods: ke ¼ 0:58X w þ 0:155X pr þ 0:25X ca þ 0:135X as þ 0:16X fa

(19:53)

Again the drawback of the above general correlation is that it does not include temperature, porosity terms, and geometric factors. Hence, it is not applicable in porous solid foods. Rahman et al. (1991) tested Sweat’s (1986) model to predict the thermal conductivities of fresh seafood and found that the error varied from 2.5% to 7.5%. However, Sweat (1974) presented a linear model for predicting the thermal conductivity of fresh fruits and vegetables giving predictions within 15% of most experimental values. The model was ke ¼ 0:148 þ 0:493Xwo

(19:54)

The above model was limited to Xwo > 0:60 and does not include temperature. According to Sweat, there was a strong relation between water content and thermal conductivity of all fruits and vegetables tested except for apples, which were highly porous. Hence, the general correlation should include a porosity term. For nonporous materials, the conductivity can be correlated with only water or proximate composition. In the case of porous materials, the porosity term must be included in the correlation because air has a thermal conductivity much lower than food components. A more general correlation was developed by Rahman (1992) using data of five foods (apple, beef, pear, potato, and squid). The correlation was ke 1 Xw ¼ 1:82 1:66 exp 0:85 1 «a keo Xwo


(19:55)

Squid had a maximum error of 12% and potato had a minimum error of 4%. The above correlation was developed by varying the water content from 5% to 88% (wet basis), porosity from 0 to 0.5, and temperature from 208C to 258C. The plot of experimental data and predicted line is given in Figure 19.2D. The above correlation had a lower error than Sweat’s (1974) correlation (Equation 19.54), was not limited to fruit or vegetables, and had a higher moisture range. When «a ¼ 1, it was later identified by Rahman and Chen (1995) that the left-hand side of the above correlation becomes

1.2

Bulk density >900 kg/m3

1.0

1.0

0.8

0.8 kep /keo

kep /keo

1.2

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Xω /Xωo

(A)

0.0

0.2

0.4

0.6

0.8

1.0

Xω /Xωo

(B) 1.4 1.2

kep

0.6

1.0 0.8

keo

1 kep /keo

0.8

1⫺eα

1.0

0.6

0.4

0.4

0.2

0.2

0.0

0.0 0.0

(C) FIGURE 19.2

0.2

0.4 Xω /Xωo

0.6

0.8

0.0

1.0 (D)

0.2

0.4

0.6

0.8

1.0

Xω /Xωo

Dimensionless thermal conductivity as a function of dimensionless water content. (A) Squid meat (o, test 1; D, test 2; ., test 3). (From Rahman, M.S. and Potluri, P.L., J. Food Sci., 56, 582, 1990. With permission.) (B) Beef. (From Rahman, M.S., J. Food Eng., 15, 261, 1992. With permission.) (C) Fruits and vegetables (., apple; D, potato; o, pear). (From Rahman, M.S., J. Food Eng., 15, 261, 1992. With permission.) (D) general correlation (o, apple; D, pear; r, beef; !, potato; ., squid meat). (From Rahman, M.S., J. Food Eng., 15, 261, 1992. With permission.)


infinity, which is physically incorrect. The other disadvantages of the above correlation are that it does not include any temperature effects on thermal conductivity and it also needs the conductivity values of the fresh foods (i.e., before processing). Thus, a more general equation should be developed using the temperature term and more experimental data points. Rahman et al. (1997) developed another improved model for fruits and vegetables during drying based on structural factors: ke ¼ «a ka þ fr ½(1 «a «w )kso þ «w kw

(19:56)

A general correlation of structural factors for fruits and vegetables was developed based on 164 data points: 0:731 ka T fr ¼ 0:996 1 «a þ Xw0:285 Tr (kw )r

(19:57)

where Tr indicates the reference temperature 273 K T is the absolute temperature in K Xw is the mass fraction of water (wet basis) The above model was developed for Xw varying from 0.14 to 0.94, temperature from 58C to 1008C, and apparent porosity from 0.0 to 0.56. The prediction error varied from 7% to 15% based on the type of product tested. The model, however, did not consider temperatures below 08C and above 1008C. Considering the wide variations in materials and processing conditions makes it difficult to develop an analytical model for the prediction of thermal conductivity. Thus, Hussain and Rahman (1999) used the artificial neural network technique for predicting thermal conductivity using a data set of 164 points. Artificial neural networks are optimization algorithms, which attempt to mathematically model the learning process. The model is a simple approximation of such a complex process, but it utilizes the basic foundations and concepts inherent in the learning processes of humans and animals. One of the major advantages of an artificial neural network (ANN) is its efficient handling of highly nonlinear relationships in data. Neural network modeling has generated increasing acceptance and is an interesting method in the estimation and prediction of food properties and process-related parameters. Similarly, ANN modeling has been used in the prediction of the thermal conductivity of bakery products (Sablani et al., 2002). An ANN model was presented for the prediction of the thermal conductivity of foods as a function of moisture content, temperature, and apparent porosity (Sablani and Rahman, 2003). The food products considered in the study by Sablani and Rahman (2003) were apple, pear, cornstarch, raisin, potato, ovalbumin, sucrose, starch, carrot, and rice (676 data points). Thermal conductivity data of food products (0.012–2.350 W=m K) were obtained from the literature for a wide range of moisture contents (Xw ¼ 0.040.98), temperature (428C–1308C), and apparent porosity (0.0– 0.70). Several configurations were evaluated while developing the optimal ANN model. The optimal ANN model consists of two hidden layers with four neurons in each hidden layer. This model was able to predict thermal conductivity with a mean relative error of 13% and a mean absolute error of 0.081 W=m K. Neural network-based equations for estimation of effective thermal conductivity, ke (W=m K) for known moisture content (Xw), temperature ratio (T=Tr, T in K), and apparent porosity («a) are


X2 ¼ Xw (2:13) þ (1:09) X3 ¼ (T=Tr )(3:18) þ (3:69) X4 ¼ «a (2:86) þ (1) X5 ¼ tanh [(0:66) þ (0:99)X2 þ (3:28)X3 þ (0:38)X4] X6 ¼ tanh [(1:42) þ (0:82)X2 þ (0:54)X3 þ (2:11)X4] X7 ¼ tanh [(3:49) þ (2:11)X2 þ (3:66)X3 þ (1:00)X4] X8 ¼ tanh [(0:18) þ (0:92)X2 þ (0:36)X3 þ (0:042)X4] X9 ¼ tanh [(0:21) þ (0:62)X5 þ (0:72)X6 þ (0:01)X7 þ (0:73)X8]

(19:58)

X10 ¼ tanh [(0:025) þ (0:058)X5 þ (0:043)X6 þ (0:17)X7 þ (0:189)X8] X11 ¼ tanh [(0:54) þ (0:55)X5 þ (0:69)X6 þ (1:13)X7 þ (0:19)X8] X12 ¼ tanh [(0:15) þ (2:76)X5 þ (0:53)X6 þ (2:92)X7 þ (0:021)X8] X13 ¼ tanh [(0:078) þ (0:14)X9 þ (0:032)X10 þ (0:31)X11 þ (0:29)X12] ke ¼ X13(1:95) þ (1:18) Although Equations 19.58 are difficult to use in a normal calculator, it could be easily used in computer programming for process design and optimization in a generic approach considering the main factors of moisture content, porosity, and temperature.

19.7 CONCLUSION Thermal conductivity can be predicted from empirical correlations if they are available for the specific product at the desired processing conditions. Alternatively, if structures or types of distribution in the component phases are known, then parallel, series, and Kopelman models could be used. If the distribution factor is known, models based on the distribution concepts, such as Krischer’s models, should be used. If the above approach is difficult to use, then complex neural network-based models could be used. However, there is still a need to develop a more generic user-friendly model to predict the thermal conductivity of foods in the future.

NOMENCLATURE A a B C D d f g h i j k L M

model parameter (Equation 19.45) activity model parameter (Equation 19.45) model parameter (Equation 19.45) diffusivity (m2=s) differentiation distribution factor model parameter (Equation 19.18) model parameter (Equation 19.18) model parameter (Equation 19.18) model parameter (Equation 19.18) thermal conductivity (W=m K) latent heat of vaporization (J=kg) mass fraction (dry basis, kg=kg solids)


n P p q R T t X X1–X13 y z

number of components in a mixture pressure (Pa) thermal conductivity ratio (Equation 19.36) parameter of Equation 19.17 ideal gas constant temperature (K) temperature (8C) mass fraction (wet basis, kg=kg sample) functions in Equation 19.58 parameter in Equation 19.16 coordination number

Greek Symbols « d n j g h v w c

volume fraction thermal conductivity contrast [(kd kc)=kc] parameter in Equation 19.7 model parameter of Equation 19.9 distribution factor in Equation 19.25 distribution factor in Equation 19.25 model parameter of (EMT) Equation 19.36 parameter in Equation 19.12 sphericity

Subscripts a as b c ca cn cr d e ec F fa i k L m max min n on p pa pr r

air or apparent ash bulk continuous phase carbohydrate Carson’s model threshold or critical dispersed effective evaporation–condensation freezing point fat ith component Krischer model liquid phase Maxwell maximum minimum nth term Torquato–Sen pore parallel protein Rahman et al.’s model or reference


ra re rv sat se so w 1, 2, 3

random Renaud model Rahman–Potluri–Varamit model saturation series model solids water component 1, 2, 3

Superscript o

initial or before processing (treatment)

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CHAPTER

20

Thermal Diffusivity of Foods: Measurement, Data, and Prediction

Mohammad Shafiur Rahman and Ghalib Said Al-Saidi CONTENTS 20.1 20.2

Definition ........................................................................................................................... 649 Measurement Techniques .................................................................................................. 650 20.2.1 Direct Measurement ............................................................................................. 650 20.2.1.1 Methods Based on Analytical Solution of Heat Conduction Equation........................................................................... 653 20.2.1.2 Method Based on Analytical Solution, j and f Factors....................... 661 20.2.1.3 Nesvadba (1982) Method.................................................................... 668 20.2.1.4 Numerical Method .............................................................................. 669 20.2.1.5 Pulse Method ...................................................................................... 669 20.2.1.6 Probe Method...................................................................................... 674 20.2.2 Indirect Prediction ................................................................................................ 674 20.3 Thermal Diffusivity Data of Foods.................................................................................... 674 20.4 Thermal Diffusivity Prediction of Foods........................................................................... 675 Nomenclature ................................................................................................................................ 690 References ..................................................................................................................................... 692

20.1 DEFINITION Thermal diffusivity indicates how fast heat propagates through a sample while heating or cooling. Thermal diffusivity is a lumped parameter used in the heat transfer calculation by conduction. The rate at which heat diffuses by conduction through a material depends on the thermal diffusivity and can be defined as a¼


k rCp

(20:1)

where a is the thermal diffusivity (m2=s) r is the density (kg=m3) Cp is the specific heat (J=kg K) at constant pressure k is the thermal conductivity (W=mK)

20.2 MEASUREMENT TECHNIQUES Thermal diffusivity can be determined either by direct experiment or estimated from the thermal conductivity, specific heat, and density data. The estimation of thermal diffusivity can be roughly divided into two groups: (a) direct measurement and (b) indirect prediction. 20.2.1

Direct Measurement

Direct measurement of thermal diffusivity is usually determined from the solution of onedimensional unsteady-state heat transport equation with the appropriate boundary conditions for finite or infinite bodies by analytical method and for irregular bodies by numerical technique. The analytical solution of unsteady-state heat conduction depends on the relative importance of internal and external heat transfer resistance. The internal and external heat transfer resistance depends on the dimensionless Biot number and can be defined as Bi ¼

Internal resistance hR ¼ External resistance k

(20:2)

High Biot number greater than 40 indicates that external resistance is negligible and Biot number less than 0.2 indicates that internal resistance is negligible. Between Biot number 0.2 and 40, both internal and external resistances to heat transfer are important (Singh, 1992). The mathematical expressions for temperature ratio are summarized below for different conditions (Mohsenin, 1980; Singh, 1992): Negligible internal resistance to heat transfer T Tm hAs t (20:3) ¼ exp T0 Tm rCV Negligible surface resistance to heat transfer For infinite slab 1 h X T Tm 2 br at i ¼ (1)nþ1 cos n exp b2n 2 R R T0 Tm n¼1 bn

(20:4)

where R is the half thickness of a slab and r is the variable distance from the center axis and bn ¼ (2n 1)

p 2

(20:5)

For infinite cylinder, 1 h X T Tm 2 br at i J0 n exp b2n 2 ¼ R R T0 Tm n¼1 bn J1 (bn ) where R is the radius of a cylinder and r is the variable distance from the center axis and


(20:6)

J0 (bn ) ¼ 0

(20:7)

1 h X T Tm br at i nþ1 bn r sin n exp b2n 2 ¼ 2(1) R R R T0 Tm n¼1

(20:8)

For infinite sphere

where R is the radius of a sphere and r is the variable distance from the center axis. bn ¼ np

(20:9)

Finite surface and internal resistance to heat transfer For infinite slab 1 h X T Tm 2 sin bn br at i ¼ cos n exp b2n 2 R R T0 Tm n¼1 bn þ sin bn cos bn

(20:10)

where R is the half thickness of a slab and r is the variable distance from the center axis. bn is the root of the equation below bn tan bn ¼

hR k

(20:11)

For infinite cylinder 1 h X T Tm 2(sin bn bn cos bn ) br at i ¼ J0 n exp b2n 2 R R T0 Tm n¼1 bn sin bn cos bn

(20:12)

where R is the radius of a cylinder and r is the variable distance from the center axis. J0 (bn ) ¼ 0

(20:13)

1 h X T Tm 2(sin bn bn cos bn ) bn r at i exp b2n 2 ¼ R R T0 Tm n¼1 bn sin bn cos bn

(20:14)

For infinite sphere

where R is the radius of a sphere and r is the variable distance from the center axis. bn is the root of the equation below: bn cot bn ¼ 1

hR k

(20:15)

The temperature history charts are also used to analyze the data due to the complexity of the series solutions. These charts are presented in Figures 20.1 through 20.3 for infinite slab, cylinder, and sphere.


1.0

m =∞ x d

m=6

d

Slab

m= m=

n = 1.0 0.8 0.6 0.4 0.2 0.0

2

2

0.10

m m

=1 n = 1.0 0.8 0.6 0.4 0.2 0.0

=1

T − Tm m

T0 − Tm

.5

=0

m .5

=0

m= 0

m=

0.010

0

n = 1.0 0.8 0.6 0.4 0.2 0.0

n=

x d

m = x hd

n = 0.8 0.6 0.4 0.2 0.0

m=0 n=1

0.0010 0

1.0

2.0

3.0

4.0

5.0

6.0

at/d 2 FIGURE 20.1 Unsteady-state temperature distribution in an infinite slab. (From Foust, A.S., Wenzel, L.A., Clump, C.W., Maus, L., and Andersen, L.B., Principles of Unit Operations, John Wiley & Sons, New York, 1960.)


1.0

m =∞ m=6

m= m=

m

R

2

2

m 0.10

r

n = 1.0 0.8 0.6 0.4 0.2 0.0

Cylinder

=1

=1

m= 0.5

m =0 .5

T −Tm

0

m=0

m=

T0−Tm

n = 1.0 0.8 0.6 0.4 0.2 0.0

0.010 n = 0.8 0.6 0.4 0.2 0.0

k hR n = r R m=

n = 1.0 0.8 0.6 0.4 0.2 0.0

m=0 n=1

0.0010 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

t /R 2 FIGURE 20.2 Unsteady-state temperature distribution in an infinite cylinder. (From Foust, A.S., Wenzel, L.A., Clump, C.W., Maus, L., and Andersen, L.B., Principles of Unit Operations, John Wiley & Sons, New York, 1960.)

20.2.1.1 20.2.1.1.1

Methods Based on Analytical Solution of Heat Conduction Equation Dickerson (1965) Method

Dickerson (1965) described an apparatus to measure thermal diffusivity based on transient heat transfer. The heat transport equation can be written as V d2 T 1 dT ¼ 2þ a dr r dr


(20:16)

1.0

m =∞ r

m=

R

6

Sphere

m m 0.10

=2

=2

m =1

m =1

0.5

n = 1.0 0.8 0.6 0.4 0.2 0.0

0.5

T0−Tm

m=

m=

T −Tm

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

0.010

m=

n = 0.8 0.6 0.4 0.2 0.0

k hR

n = r R

m=0 n=1

0.0010

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

at /R 2 FIGURE 20.3 Unsteady-state temperature distribution in a sphere. (From Foust, A.S., Wenzel, L.A., Clump, C.W., Maus, L., and Andersen, L.B., Principles of Unit Operations, John Wiley & Sons, New York, 1960.)


where V is the constant rate of temperature rise at all points in the cylinder (8C=s). Equation 20.16 can be solved when the temperature gradient is no longer time dependent as T¼

Vr 2 þ v ln r þ w 4a

(20:17)

Boundary conditions are T þ Vt ¼ Ts ,

t > 0, r ¼ Rsa

(20:18)

Boundary conditions are dT ¼ 0, dr

t > 0, r ¼ 0

(20:19)

The solution of Equation 20.17 with the above boundary conditions is (Ts T) ¼

V 2 Rsa r 2 4a

(20:20)

At r ¼ 0 and T ¼ Tc, thermal diffusivity can be expressed as a¼

VR2sa 4(Ts Tc )

(20:21)

Time (s)

A plot of sample temperature versus time is shown in Figure 20.4. Dickerson (1965) showed that 95% or more of the maximum temperature difference (Ts Tc) or the establishment of steady state takes place when (at=R2sa ) > 0:55. Thus, the appropriate radius for the sample cylinder can be determined from the above inequality. Olson and Schultz (1942) have shown that a simulation of the mathematical model of infinite cylinder is possible if the length to diameter ratio (L=D) is greater than 4. The length of the sample holder cylinder must be four times greater than the diameter. The Dickerson (1965) apparatus consisted of an agitated water bath and a sample holder metal cylinder (Figure 20.5). One thermocouple is soldered to the outside surface of the cylinder to monitor the surface temperature of the sample and another thermocouple probe is inserted at the center of the sample. The sample holder contains two caps at the bottom and a top made of Teflon.

Surface temperature

Ω e=

Ts

–T

c

op

Sl

Center temperature

Temperature (C) FIGURE 20.4 Temperature versus time plot of Dickerson (1965) method.


Thermocouple Sample holder Stirrer

Heater

Water bath

FIGURE 20.5 Thermal diffusivity measurement apparatus as used by Dickerson (1965) method.

The cylinder is then placed in the agitated water bath, and the time and temperature are recorded until a constant rate of temperature rise is obtained for both inner and outer thermocouples (Figure 20.4). 20.2.1.1.2

Hayakawa (1973) Method

If one surface of an infinite slab is insulated and the other surface is subjected to a step change in its surface temperature at the zero time of heating or cooling, then the temperature at any location can be calculated by the following equation: T ¼1

1 4 X (1)n a(2n þ 1)2 p2 t (2n þ 1)px exp cos p n¼0 (2n þ 1) 4l2 2l

(20:22)

where l is the thickness of the sample. If the temperature of the uninsulated surface Ts changes with time, Hayakawa (1973) derived the following equation for the temperature distribution in the sample using Duhamel theorem: " # g 1 1 X X 16l2 X T ¼ Ts þ am Qgn(m1) Qgnm ap3 m¼1 m¼1 m1

(20:23)

where Qgnm ¼ (1)nþ1

cos (2n 1)px 1 exp½Yn (g m)Dt 2l (2n 1)3 Yn ¼

a(2n 1)2 p2 2l2

(20:24) (20:25)

where am is the slope of mth line segments (8C=s) g is the number of line segments with which a temperature history curve on the uninsulated surface of sample material is approximated


Temperatures of the sample material are monitored experimentally at its internal locations and on its exposed surface. These temperatures are recorded at uniform time intervals (mDt: m ¼ 1,2, . . . , g). Equation 20.23 is used to calculate theoretical temperatures at the same time intervals to compare them with recorded temperatures. For this calculation, experimental temperatures on the exposed surface are used with various assumed thermal diffusivity values. For each value assumed, the sum of squares of differences between experimental and theoretical temperatures is calculated. The diffusivity value of the sample material is determined by calculating a value at which the sum of squares of differences becomes minimal. The advantage of the procedure is that a food sample can be exposed to any time-variable heating or cooling temperatures during the experimentation. 20.2.1.1.3

Nordon and Bainbridge (1979) Method

When a cylindrical sample is transferred from a bath at temperature (T0) to another temperature (Tm), the temperature at the center of the cylinder changes according to the equation (Nordon and Bainbridge, 1979): 2 1 X 1 h at i T T0 8 X (1)2 b2m þ p(2nLþ 1)R exp 2 ¼1 p n¼0 m¼1 2n þ 1 R Tm T0 bm J1 (bm )

(20:26)

Where bm is the root of the following equation: J0 (bm ) ¼ 0

(20:27)

Theoretical values of temperature ratio can be plotted as a dimensional variable at=R2. Nordon and Bainbridge (1979) mentioned that when the shape of the theoretical and experimental temperature ratio curves is identical, then a single point from the theoretical curve can be considered to calculate the thermal diffusivity. A typical theoretical plot is shown in Figure 20.6. At temperature ratio 0.5, the value of at=R2 is known. When the experimental cooling time reaches the temperature ratio at 0.5, then thermal diffusivity can be calculated from at=R2 by knowing the time, t.

1.0

Temperature ratio

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1.0

at /R 2 FIGURE 20.6 Computer-generated plot of normalized temperature against cd=R2. (From Jones, J.C., Wootton, M., and Green,S., Food Austr., 44, 501, 1992.)


20.2.1.1.4

Olivares, Guzman, and Solar (1986) Method

Olivares et al. (1986) used the heat conduction equation for a finite cylinder. The series equation is rapidly convergent for a Fourier number greater than 0.2 (Singh, 1982), and it is possible to consider only the first two terms with accurate results (Yanez, 1983) after some time has elapsed. The resultant equation is 2 2 X 2 X Ts T 2(1)mþ1 2z 2J0 (bn r=R) b 4b2 exp 2n þ 2m at ¼ cos bm Ts T0 m¼1 n¼1 L bn J1 (bn ) bm R L

(20:28)

where bn and bm are the roots of the cosine and Bessel functions. The values of T and Ts can be obtained experimentally by placing thermocouples at the center and outside of the can. To obtain an average thermal diffusivity, the experimental values and predicted ones for T are compared for some arbitrarily selected value of thermal diffusivity. Using an iterative technique, the values of thermal diffusivity is changed until the sum of the difference is minimized. 20.2.1.1.5

Gordon and Thorne (1990) Method

The differential equation for heat conduction in a sphere suddenly transferred to a different constant temperature surrounding is d2 u 2 du du ¼ þ dh2 h dh dF0 u¼

T Tm ; T0 Tm

h ¼ r=R;

(20:29)

F0 ¼ (at=R2 )

For 0 h 1 and F0 0, the solution is u¼

1 X n¼1

"

# 2 2Bi sin (bn h) exp bn F0 h b2n þ Bi sin bn

(20:30)

where Bi is the Biot number (hR=k) and bn is the nth root of the transcendental equation as below: bn cot bn ¼ 1 Bi

(20:31)

The center temperature (h ¼ 0) can be written as uc ¼

1 X n¼1

"

2Bi bn 2 exp b2n F0 2 bn þ Bi Bi sin bn

# (20:32)

Gordon and Thorne (1990) provided two methods for estimating the thermal diffusivity: (i) slope method and (ii) lag method. 20.2.1.1.5.1

Slope Method—Considering a single term, Equation 20.32 can be transformed to ln uc ¼ ln I


b21 at R2

(20:33)

where I¼

b21

2Bib1 þ Bi2 Bi sin b1

(20:34)

Thus, the logarithm of uc is a linear function of time and thermal diffusivity can be determined from the slope of the Equation 20.33 by knowing the first root of Equation 20.31 from the intercept. When u 0.4, the first root was found to be the only significant contributing root of Equation 20.31 and use of a single root is justified. From the plot of ln uc versus t, the portion of the curve satisfying the above criterion is used to determine the intercept I. Then, the value of b1 can be determined from Equation 20.34 by binary chop iteration. 20.2.1.1.5.2 Lag Method—If a sphere at uniform temperature is transferred to a different, lower constant temperature, and temperatures at two points, the center (h ¼ 0) and half way from the center to the surface (h ¼ 0.5), are recorded, the temperature difference between the two points would increase with time from an initial value of zero to a maximum and then tend to zero again as the temperature within the solid tended toward that of the cooling medium. Figure 20.7 shows typical temperature profiles for the two points. The time tmax corresponding to the maximum temperature difference (Figure 20.7) is that period when the tangents to the two curves are parallel. The time lag (tL) shown in Figure 20.7 is inversely related to thermal diffusivity. If a is large, heat will pass rapidly through the food and tL will be small, and vice versa. A regression correlation was developed using random data (Gordon and Thorne, 1990): tL ¼

3:89 105 5:3471 a

(20:35)

A vigorously stirred water bath of 22 L capacity at 48C 18C was used by Gordon and Thorne (1990) to effect a high heat transfer rate (h ¼ 300 W=m2 K) (Figure 20.8). The ratio of water bath volume to sphere volume (200:1) satisfies the boundary conditions of a constant ambient temperature (0.28C). A jig can be used to locate thermocouples within the samples, which are suspended in a water bath while temperature and time are recorded. To submerge the fruits and vegetables, a piece of glass capillary tubing is inserted through the base and a piece of nylon thread tied to a lead weight is passed through this (Figure 20.8). The experimental conditions are h ¼ 300 W=(m2 K), T0 ¼ 208C, and Tm ¼ 408C. For values of thermal diffusivity between 0.85 107 and 3.23 107 m2=s, a range that includes almost 1

0.5 q

tL Δqmax

qc 0

FIGURE 20.7

tc

Time

Estimation ohime lag from u versus time plot. (From Gordon, C. and Thorne, S., J. Food Eng., 11, 133, 1990. With permission.)


Support

Nylon line and thermocouple anchored to rubber septum Water bath

Test material Glass capillary to hold Nylon line holding Weight

FIGURE 20.8 Sample arrangement to measure thermal diffusivity as used by Gordon and Thorne (1990). (From Gordon, C. and Thorne, S., J. Food Eng., 11, 133, 1990. With permission.)

all foodstuffs, the mean and maximum errors were 1.3% and 4% by the slope method and 1.2% and 5.3% by the lag method. The slope method offers the advantage that it requires only temperature measurements at a single point at the center. For typical foods, an error of 3 mm at the center results in a maximum error of 0.8% in temperature measurement. For a thermocouple located at h ¼ 0.5, the effect of location errors are much greater, particularly if the location error is toward the center; for example, the error in temperature measurement is 5.4% if the thermocouple is inserted 2 mm too far in. 20.2.1.1.6

Moore and Bilanski (1992) Method

The time–temperature history of a sample was measured using the plane heat source method as described by Mohsenin (1980). The construction of the device is shown in Figure 20.9. In this Heater stirrer

Water vessel

Thermocouple wires Sample Copper base plate Insulated box FIGURE 20.9 Thermal diffusivity measurement apparatus as used by Moore and Bilanski (1992). (From Moore, G.A. and Bilanski, W.K., Appl. Eng. Agric., 8, 61, 1992.)


method, a slab of material is insulated on one plane and is subjected to an isothermal heat source on a parallel plane. To simulate a semi-infinite slab, the edges of the slab are insulated. One thermocouple is attached centrally to the underside of the water vessel so that it is in direct contact with the heat source and the sample. A second thermocouple is soldered to the underside of the 0.2 mm thick copper base plate. The heat source is maintained at approximately 808C while the sample is placed in the cell at ambient temperature (approximately 208C). Bulk density is controlled by placing a known mass evenly in the cell and maintaining the heat source at a known distance above the base of the cell. The solution of the heat conduction equation to give the temperature at any time and position within the material undergoing heating or cooling depends on the boundary conditions. When a slab of material initially with uniform temperature T0 and the heat source are placed on the face of the sample, the temperature in the slab can be written as (considering only the first term of the series solution) Tm T 4 h pxi p2 at sin exp 2 þ Tm T0 p 2l 4l

(20:36)

The assumptions made in the derivation of this equation are (1) the material is homogeneous and of constant thermal properties, (2) there is no surface thermal resistance, i.e., an infinite heat transfer coefficient at the surface. An error of 0.5 mm in distance would lead to an error in calculation of thermal diffusivity of about 12% at a thickness of sample of 8 mm and 5% when the sample is 20 mm thick. The temperature rise was recorded until the base of the sample reached 608C. For times greater than 100 s, the error occurred by ignoring higher terms less than 1% (Moore and Bilanski, 1992). In this case, thermal diffusivity can be calculated from the slope of the plot ln(Tm T)=(Tm T0) versus time. 20.2.1.2 20.2.1.2.1

Method Based on Analytical Solution, j and f Factors Bhowmik and Hayakawa (1979) Method

The heat conduction equation for an infinite cylinder whose surface is subject to convective heat exchange is obtained from an equation given by Carslaw and Jaeger (1959) as follows: " # 1 X Tm T J0 (bn r=R) 2 2 ¼ 2Bi exp abn t=R 2 Tm T0 Bi þ b2n J1 (bn ) n¼1

(20:37)

where bn is the nth positive root of the equation. Bi J0 (bn ) bn J1 (bn ) ¼ 0

(20:38)

The summation series may be approximated with the first term as " # 1 X Tm T J0 (bn r=R) 2 2 ¼ 2Bi exp ab1 t=R 2 Tm T0 Bi þ b21 J1 (bn ) n¼1

(20:39)

The temperatures at the center and surface of the cylinder can be estimated from the following equations:


" # 1 X Tm Tc 1 2 2 ¼ 2Bi exp ab1 t=R 2 Tm T0 Bi þ b21 J1 (bn ) n¼1 " # 1 X Tm Ts J0 (b1 ) 2 2 ¼ 2Bi exp ab1 t=R 2 Tm T0 Bi þ b21 J1 (bn ) n¼1

(20:40)

(20:41)

By taking the ratio of Equations 20.40 and 20.41, one can express Gcs ¼

Tm Tc 1 ¼ J Tm Ts 0 (b1 )

(20:42)

The constant value of Equation 20.42 can be obtained by plotting (Tm Tc)=(Tm Ts) against the heating or cooling time of food on a rectangular graph paper (Figure 20.10). The values of b1 may be easily determined from this constant value. According to Pflug et al. (1965), there exists a relationship between b1 and thermal diffusivity a given by f a 2:303 ¼ R2 b21

(20:43)

where f is the slope index of the heating or cooling curve of the sample food in seconds. The values of f can be obtained from the semilogarithmic plotting of (T0 Tc) versus t. A typical plot is shown in Figure 20.11 from experimental heat penetration data. Then a can be determined using Equation 20.43 provided f value is given. Bhowmik and Hayakawa (1979) used this method to measure the thermal diffusivity of different materials and found that the error varied from 1% to 7%. Later Hayakawa and Succar (1983) used this technique for spherical food materials. When F0 is moderately large, the analytical solution for transient state heat conduction in a sphere is (Carslaw and Jaeger, 1959) Tm T 2Bi 2 b21 þ (b1 1)2 b1 r b1 F0 (sin b ¼ ) 1 sin 2 2 Tm T0 r=R R b1 b1 þ Bi(Bi 1)

(20:44)

(Tm Tc)/(Tm Ts)

10

5

1 0

50

100

150

Time (s) FIGURE 20.10 Graph of (Tm Tc)=(Tm Ts) versus time for cherry tomato pulp. (From Bhowmik, S.R. and Hayakawa, K., J. Food Sci., 44, 469, 1979.)


Tm Tc

End of heating

1

10 f

T0

j= Tei

Tm Tei Tm T 0

100 0

10

20

30

Time (min) FIGURE 20.11 Typical heat penetration curve describing f and j.

where b1 is the first positive root of the following equation: b1 cot b1 ¼ 1 Bi

(20:45)

The values of (Tm T)=(Tm T0) can be related to Ball’s f and j values as (Ball and Olson, 1957) Tm T ¼ j½10t=f , t > t1 Tm T0

(20:46)

Combining Equations 20.44 and 20.46, one can express a¼

R2 ln 10 b21 f

(20:47)

The ratio of temperatures at two locations in the sample can be written from Equation 20.44 as G12 ¼

Tm T1 (r2 =R) sin (b1 r1 =R) ¼ Tm T2 (r1 =R) sin (b2 r2 =R)

(20:48)

The constant value of r can be determined from the plot of G versus t (Figure 20.12). The values of b1 can be estimated from the Equation 20.48. When b1 and f are known, thermal diffusivity can be calculated from Equation 20.47. 20.2.1.2.2

Uno and Hayakawa (1980) Method

Uno and Hayakawa (1980) used an analytical solution of heat conduction in a finite cylinder for the experimental determination of thermal diffusivity. This solution was derived by assuming that surface heat transfer conductances at its top, bottom, and side surfaces were all finite and different from each other. A first-term approximation of the series solution and an empirical f value were used


Γ

0.8

0.4

0 0

3000

6000

Time (s) FIGURE 20.12 Plot of temperature ratio against time for a 26.78C fresh tomato heated in a 42.98C chamber. (From Hayakawa, K. and Succar, J., Food Sci. Technol., 16, 373, 1983.)

to develop this method. Temperature distributions in a cylindrical can of heat conduction food may be estimated by the following equation (Carslaw and Jaeger, 1959): u¼

1 X 1 T Tm X ¼ Gm Hn exp (Bmn t) T Tm m¼1 n¼1

(20:49)

where 2Bisi J0 (pm h) Gm ¼ 2 Bisi þ p2m J0 (pm ) 2 q2n þ Bib ½sin qn ¼ ðBit ( cos qn )=qn Þ þ ðBit =qn Þ Hn ¼ qn cos (qn j) þ Bib sin (qn j) Bmn ¼ p2m S2 þ q2n

(20:50)

(20:51) (20:52)

where t t=L2, L is the height of the cylinder (m) Bisi is the side Biot number (hsiR=k) Bit is the top Biot number (htL=k) Bib is the bottom Biot number (hbL=k) h is r=R j is z=L pm and qn are the positive roots of the following characteristic equations for an infinite cylinder and an infinite slab


Bisi J0 (p1 ) ¼ p1 J1 (p1 )

(20:53)

(q21 Bit Bib ) sin q1 ¼ (Bit þ Bib )q1 cos q1

(20:54)

z

L/4

r

L /2

1

2

4

3 R

r

q

FIGURE 20.13 Locations of the thermocouples in the can for Uno and Hayakawa (1980) method. Points 1, 2, 3, and 4 are the locations of the thermocouples.

For a moderately long heat exchange time, Equation 20.49 may be approximated by the first term of the infinite series. Temperatures at four locations in the can are measured (Figure 20.13): (1) T1 along the central axis of the cylinder and also are apart about one-quarter of the total height of the can content, (2) T2 at the center of the cylinder, (3) T3 below T2 along the central axis, and (4) T4 is positioned away from the central axis of the cylinder and its axial distance measured from the top surface of the cylinder is identical to that of location T2. The ratios of dimensionless temperatures at two locations may be given by the following expressions after a moderately long heat exchange time: G12 ¼

Tm T1 q1 cos (q1 j1 ) þ Bit sin (q1 j 1 )J0 (p1 h1 ) ¼ Tm T2 q1 cos (q1 j2 ) þ Bit sin (q1 j 2 )J0 (p1 h2 )

(20:55)

G23 ¼

Tm T2 q1 cos (q1 j2 ) þ Bit sin (q1 j 2 )J0 (p1 h2 ) ¼ Tm T3 q1 cos (q1 j3 ) þ Bit sin (q1 j 3 )J0 (p1 h3 )

(20:56)

Tm T2 J0 (p1 h2 ) ¼ Tm T4 J0 (p1 h4 )

(20:57)

G34 ¼

where p1 and q1 are the first positive roots of Equations 20.53 and 20.54, respectively. By plotting G12, G23, and G34 against time, reasonably stable constant values of G12, G23, and G34 can be obtained (Figure 20.14). The values of p1 may be obtained by trial and error by substituting G34 into Equation 20.57. By substituting this p1 value into Equation 20.53, Bisi can be obtained. From Equations 20.55 and 20.56, an equation is derived by eliminating Bit. This equation is used to determine a q1 value through iterative calculations. Experimental parameters f and j are used in temperature estimation after a relatively long exposure time: Tm T ¼ j½10t=f Tm T0


(20:58)

1.5

Γ

Γ34

1.0

Γ23

0.6

Γ12

0

10

20

Time (min) FIGURE 20.14 Temperature ratios as a function of time in a can. (From Uno, J. and Hayakawa, K., J. Food Sci., 45, 692, 1980.)

Combining Equation 20.58 and first-term approximation of Equation 20.49, the value of thermal diffusivity may be obtained by 2 L ln 10 a¼ 2 2 f p1 S þ q21

(20:59)

where S is the shape factor of the can (L=R). 20.2.1.2.3

Singh (1982) Method

For a finite cylinder at a uniform initial temperature, exposed to a constant temperature environment and having negligible surface convective resistance, the solution of the heat conduction equation is as follows (Carslaw and Jaeger, 1959): 2 1 X 1 X Ts T 2(1)mþ1 2J0 (bn r=R) b 4b2 exp n2 þ 2m at ¼ cos (bm 2z=L) Ts T0 m¼1 n¼1 bn J1 (bn ) bm R L where J0 and h are the Bessel function of the first kind of order zero and one R and L are the radius and length of the cylinder (m), respectively Ts and T0 are the surrounding and initial temperature (8C), respectively bm and bn are the root of cosine and Bessel functions, respectively z is the axis from the center along the length of the cylinder


(20:60)

For a long time, the first term of the series solution alone may be sufficient. For a finite cylindrical object, the solution expressed by Equation 20.60 can be simplified to retain only the terms with m ¼ n ¼ 1; i.e., bm ¼ p=2, bn ¼ 2.4048, and J1 (2.4048) ¼ 0.5191 (Appendix C). At the center of the cylindrical object, z ¼ 0, r ¼ 0, and J0(0) ¼ 1.0. Thus, the approximation for long times can be expressed as Ts T (2:4048)2 p2 ¼ 2:0396 exp þ 2 at Ts T0 R2 L

(20:61)

Ball and Olson (1957) plotted experimental heat penetration curves on a semilogarithmic expression: Ts T0 t ¼ f log j Ts T

(20:62)

where j¼

Ts Tei Ts T0

(20:63)

where Tei is the extrapolated initial temperature obtained by linearizing the entire heating curve (8C). Combining Equations 20.61 and 20.62, thermal diffusivity can be written as a ¼

2:303 (2:4048) =R2 þ ðp2 =L2 Þ f 2

(20:64)

Equation 20.64 allows the determination of thermal diffusivity if the heating rate parameter f (slope index of heating or cooling curve) is obtained from a heat penetration study (Figure 20.11). 20.2.1.2.4

Poulsen (1982) Method

Ball and Olson (1957) developed the relation between temperature, time, thermal diffusivity, and geometry of the bodies during heating or cooling. Time–temperature relation for an infinite cylinder with high surface heat transfer coefficient can be written as R2 Tm T0 t ¼ 0:398 log 1:6 a Tm T

(20:65)

Thermal diffusivity can be calculated from Equations 20.62 and 20.65 as a¼

0:398R2 f

(20:66)

In the graphical solutions of the heat equation by Gurney and Lurie (1923), straight lines give the relation between log[(Tm T)=(Tm T0)] and at=R2. The straight lines lie approximately between the following range: 0
0, k dx t > 0, k

(20:74) (20:75)

where l and f are the sample thickness (m) and penetration depth, respectively, of pulse energy in the front face at t e is the pulse energy by unit surface area (J=m2) The analytical solutions for different conditions are given by Andrieu et al. (1986): 1. Bi1 ¼ Bi2 ¼ 0 (i.e., no heat losses). The temperature evolution on the back face (x ¼ l) is given by " T(l, T) ¼ Ta 1 þ

1 X

# cos (np) exp (n2 p2 t )

(20:76)

n¼1

where Ta ¼ e=(Cprl) is the adiabatic temperature, t* ¼ t=m is the reduced time, and m is the time constant (l2=a) in seconds. 2. Bi1 ¼ Bi2 ¼ Bi (i.e., equal losses on each face). The solution is now given by the relation T(l, T) ¼ 2Ta

1 X bn cos bn þ Bi sin bn exp b2n t (b þ Bi þ 2) ð Bib Þ n n n¼1

(20:77)

where bn are the roots of the transcendental equation: tbn ¼0 2Bi b2n Bi2 bn

(20:78)

3. Bi16¼ Bi2 (i.e., different losses on each face). The thermogram equation is " T(l, T) ¼ 2Ta

1 X

# (bn cos bn þ Bi1 sin bn ) En exp (b2n t*)

(20:79)

n¼1

where b n En ¼ 2 bn þ Bi21 1 þ Bi2 =b2n þ Bi22 þ Bi1


(20:80)

where bn are the roots of the transcendental equation: 2 bn Bi1 Bi2 tbn ¼ bn (Bi1 þ Bi2 )

(20:81)

The above analytical solutions are difficult to use due to calculation or estimation of heat losses (Bi1 or Bi2) because the maximum value and shape of the thermogram are sensitive to these parameters. A typical thermogram is shown in Figure 20.17. Degiovanni (1977) proposed a satisfactory analysis in the case of a cylindrical sample. This method is based on the use of experimental and theoretical normalized temperatures. Theoretical reduced temperature T* is defined by the relation

Q

0

x=e x T Ta

H=0

Tmax T5/6 H≠0 T1/6

t1/6

(a) T∗=

t

t5/6

Tr

Te =

Tr,max

1

1

tz

b

y

a

0

ty∗

(b)

t z∗

Simulated

t∗

T Tmax

0 ty

tz

t

Experimental

FIGURE 20.17 Schematic of reduced thermograms by pulse method. (a) Sample heating with pulse energy, shape, and peak in thermogram, (b) reduced temperature thermogram. (From Andrieu, J., Gonnet, E., and Laurent, M., Food Engineering and Process Applications, Le Maguer, M. and Jelen, P. (eds.), Elsevier Applied Science, London, 1986.)


T* ¼

Tr Tr, max

Tr ¼

and

T(l,t) Ta

(20:82)

The experimental reduced temperature Ter is given by the relation Ter ¼

T Tmax

(20:83)

where Tmax is the maximum temperature (Figure 20.17). After determination of the points (c, tc) and (z, tz) on the experimental reduced thermograms (Figure 20.17), Andrieu et al. (1986) calculated the diffusivity with the relation l2 * tz t§

a¼

(20:84)

where the reduced time tz, characteristic of heat losses, is calculated by the following empirical relation: * ¼ 0:954 1:58 1:58 t5=6

2 t1=2 t1=2 þ 0:558 t5=6 t5=6

(20:85)

where c and z are 1=2 and 5=6. The main disadvantage of this method is that it uses few points to calculate thermal diffusivity. The moment theory is also usually used to estimate the parameter. Andrieu et al. (1986) provided details of the moment method to estimate the thermal diffusivity. The fth order partial moments are defined as ðtz wf ¼

t f T(l,t)dt

(20:86)

tc

If f ¼ 0, W0 the zeroth moment, and if f ¼ l, then W1 minus one moment (independent of timescale) (Figure 20.18). Andrieu et al. (1986) derived an equation to calculate the thermal diffusivity from partial moment: a¼

* l2 W1 W0

(20:87)

t5=6

ð

* ¼ W1 t1=6

T* t*

(20:88)

A schematic diagram of the equipment is shown in Figure 20.19. A pulse generator delivers a radiant energy pulse of about 800 J during 102 s. Figure 20.19 shows an apparatus for pulse method with a semiconductor thermocouple (bismuth telluride, Bi2Te3) with high thermoelectric power (360 mV=8C at 208C) and short delay time vegetable oil (20%) > pork fat

(20:93)

Table 20.9 Thermal Diffusivity of Potato as a Function of Temperaturea

Tmax,s (8C)

Apparent Density (kg=m3)

72.2 81.7 90.6

57.2 63.3 68.3

90.6 100.0 98.9 100.0

Tmax,c (8C)

Thermal Diffusivity (m2=s) Raw

Heated

1038 1051 1048

1.332 107 1.411 107 0.975 107

1.371 107 1.488 107 1.063 107

72.8

1049

0.961 107

1.044 107

61.1 73.9

1052 1046

1.251 107 1.239 107

1.335 l07 1.444 107

77.2

1048

1.007 107

1.043 107

Source: Matthews, F.V. and Hall, C.W., Trans. ASAE, 11, 558, 1968. a Tmax,c, temperature at the center of the sample; Tmax,s, temperature at the surface of the sample.


Table 20.10 Thermal Diffusivity of Sucrose Solution Thermal Diffusivity 107 (m2=s) T (8C) Xwo

39

36

33

30

28

25

22

19

16

14

11

8

5

0.98 0.96

12.97 12.95

12.50 12.50

12.03 12.03

11.49 10.93

11.27 10.68

10.67 9.74

10.38 9.69

9.72 8.69

9.25 8.37

8.68 7.59

7.62 6.48

6.83 5.41

5.31 3.68

0.94

12.51

11.74

11.08

10.73

10.28

9.61

9.55

8.81

7.98

7.39

6.30

5.22

3.38

0.92 0.90

12.34 11.46

11.47 10.83

10.83 10.83

10.07 9.27

9.80 9.09

9.05 8.23

8.69 7.98

7.90 7.04

7.34 6.44

6.58 5.69

5.41 4.54

4.26 3.43

2.52 1.94

0.88 0.86

10.92 10.58

10.18 10.18

9.26 8.81

8.34 8.05

8.17 7.83

7.44 6.99

6.99 6.67

6.22 5.8

5.64 5.15

4.82 4.39

3.80 3.41

2.76 2.41

1.52 1.29

0.84 0.82

10.49 9.84

9.76 9.21

8.54 7.75

7.67 7.03

7.17 6.57

6.58 6.03

6.06 5.58

5.26 4.79

4.69 4.19

3.81 3.43

2.94 2.60

2.03 1.79

1.03 0.86

0.80 0.75

9.86 8.9

9.11 8.28

7.64 7.05

6.85 6.11

6.34 5.62

5.75 4.87

5.29 4.49

4.59 3.81

3.94 3.28

3.26 2.6

2.42 1.88

1.58 1.23

0.76 0.57

0.70

7.81

7.32

6.03

5.15

4.64

4.02

3.73

3.11

2.53

2.01

1.41

0.87

0.40

0.65

6.65

6.32

4.94

4.05

3.75

3.16

2.95

2.42

2.01

1.53

1.05

0.63

0.29

Source: Keppler, R.A. and Boose, J.R., Trans. ASAE, 13, 335, 1970.

Martens (1980) performed multiple regression analysis on 246 published values on thermal diffusivity of a variety of food products and obtained the following regression equation: a ¼ ½0:057363Xw þ 0:000288(T þ 273) 106

(20:94)

The standard error of estimate was 0.14 107 m2=s. Metel et al. (1986) provided the thermal diffusivity of dough wheat as a function of moisture content, density, and temperature as (208C < T < 908C)

Table 20.11 Thermal Diffusivity of Fish at Different Temperatures Thermal Diffusivity 107 (m2=s) T (8C) Xwo

XPA

Xp

20

10

0

10

20

30

40

Saman

0.773

0.0046

0.202

8.330

5.094

1.670

1.678

1.969

2.426

2.61

Sol

0.776

0.0046

0.205

7.988

5.263

1.904

2.153

2.372

2.650

3.175

Black bhitki Black pomphret

0.777 0.750

0.0060 0.0476

0.195 0.179

9.763 8.726

5.706 4.381

1.872 1.804

2.113 1.966

2.459 2.038

2.556 2.329

2.696 2.542

Mackerel

0.774

0.0058

0.187

6.984

4.122

1.408

1.519

1.793

2.070

—

Red bhitki Singara

0.795 0.779

0.0042 0.0057

0.169 0.201

7.232 6.836

5.274 3.827

1.634 1.130

1.843 1.508

2.084 1.717

2.430 1.894

2.779 2.131

Hilsa Surama White Pomphret

0.747 0.780 0.748

0.0513 0.0077 0.0410

0.172 0.188 0.186

8.460 6.689 9.846

5.569 3.920 4.385

2.156 1.469 1.753

2.131 1.609 2.034

2.75 1.994 2.434

2.768 2.092 2.815

2.866 2.304 2.938

4.262

1.595

1.775

2.016

2.498

—

5.083

1.760

1.904

2.016

2.318

2.491

Fish

Malli

0.782

0.0045

0.194

Rohu

0.752

0.0057

0.215

7.038

Source: Kumbhar, B.K., Agarwal, R.S., and Das, K., Int. J. Refrig., 4, 1981.


Table 20.12 Thermal Properties of Dough and Bread, Compiled by Rask (1989) a 107 (m2=s)

T (8C)

Xaw

rap (kg=m3)

Dough Bread solids

— —

— —

— —

— 1.13

0.386 0.309b

— —

Bakshi and Yoon (1984) Bakshi and Yoon (1984)

Bread (5 min) Bread (10 min)

— —

0.334b 0.269b

202b 181b

2.151c 1.952c

0.085b 0.093b

— —

Bakshi and Yoon (1984) Bakshi and Yoon (1984)

Bread crumb

—

0.340a

—

—

—

4.07

Bread crumb

30.0

0.41

—

2.560b

—

—

Bread crumb

100.0

0.41

—

2.626b

—

—

Bread crust

100.0

—

—

—

—

0.367

Johnsson and Skjoldebrand (1984)

Bread crust

150.0

—

1.656b

—

—

Johnsson and Skjoldebrand (1984)

0.550c

—

Kafiev et al. (1987)

Product

0

Cp (kJ=kg k)

k (W=m K)

Reference

Johnsson and Skjoldebrand (1984) Johnsson and Skjoldebrand (1984) Johnsson and Skjoldebrand (1984)

Dough

35.0

—

—

1.700c,d

Bread (rye, 10 min)

—

—

—

—

—

0.24

Kriems and Reinhold (1980)

Bread (rye, 40 min) Bread (wheat, 10 min) Bread (wheat, 40 min)

—

—

430c

—

—

0.52

Kriems and Reinhold (1980)

—

—

—

—

—

0.43

—

—

290c

—

—

1.38

Kriems and Reinhold (1980) Kriems and Reinhold (1980)

Dough Dough Dough

43.5 28.5 22.0

0.435 0.435 0.435

1100 1100 1100

1.76 1.94 —

— 0.88

4.78 3.95 —

Lind (1988) Lind (1988) Lind (1988)

Dough Dough

16.5 23.0

0.435 0.435

1100 1100

2.76 —

0.46

— 1.45

Lind (1988) Lind (1988)

1.03

5.3

Lind (1988)

0.98

— 4.35

Lind (1988) Lind (1988)

0.92

—

Dough

38.0

0.461

1100

1.760

Dough Dough

28.0 16.0

0.461 0.461

1100 1100

1.88 —

Dough

19.0

0.461

1100

—

0.5

1.63

Lind (1988)

Dough Dough (wheat) Bread crumb (loaf)

21.0 28.0 18.0

0.461 0.420 0.418

1100 623 402

2.81 2.883 3.19

— 0.414 0.298

— 1.770 1.915

Lind (1988) Lind (1988) Makljukow and Makljukow (1983)

Bread crumb (tin loaf)

18.0

0.428

340

2.975

0.244

2.42

Makljukow and Makljukow (1983)

Bread crumb (crust) Bread solids

140.0

0

300

1.575

0.066–0.43

0.268

0.366

—

—

—

—

Makljukow and Makljukow (1983) Makljukow and Makljukow (1983) Mannheim et al. (1957)

12.5

—

Bread solids

0

1 x þ1 and

t>0

and

x¼0 x ¼ 1

The solution from Carslaw and Jaeger (1959) is 1 X 2Bi cos (hi x=l) (hi ) u¼ exp h2i F0 2 Bi(Bi þ 1) þ hi i¼1

(21:13) (21:14) (21:15)

(21:16)

where hi is the ith root of the following equation: hi cot hi ¼ Bi where F0 ¼

at hl ; Bi ¼ a2 k

(21:17) (21:18)

where l is half of the slab thickness (m). The solutions for other simple geometries also follow a similar form. For other geometries, the solutions are available in Carslaw and Jaeger (1959). The use of digital computers is necessary to evaluate h from the above equation. But one term approximation of the infinite series solution can also be used assuming the higher terms in the summation equation are negligible, especially after some time has elapsed. In this case, Equation 21.16 reduces to u¼

2Bi[cos (h=l) sec (h) exp h2 F0 Bi(Bi þ 1) þ h2

(21:19)

The values of h can be determined from the slope of the line ln u versus F0. Then h can be calculated from Equation 7.17 by knowing the value of Biot number (Bi). This method unfortunately only uses a part of the data obtained in cooling=heating curves, where the temperature of the body changes slowly, thus magnifying the measurement error (Arce and Sweat, 1980). If the body is nonisotropic or nonhomogeneous with regard to their thermal properties, this will introduce another source of error. The above authors mentioned that this scheme to calculate h gave reasonable accuracy if properly used. This method is also not suitable for irregular materials. Chang and Toledo (1990) used the first four terms of the exact series solution for an infinite slab to calculate the heat transfer coefficient of cubes immersed in an isothermal fluid. The multiplicative technique was employed in simplifying the infinite series solution for the finite cube. They calculated the time–temperature data iteratively at three locations until the h value that gave the best fit between experimental and predicted data was found. 21.2.3.2

Numerical Solution

The numerical techniques such as finite differences or finite elements are also used to calculate the heat transfer coefficient heat conductance. In the case of heterogeneous complex geometries or variable boundary conditions, only the numerical method is available to solve the heat conduction equations. This technique is the most accurate if the numerical solution uses small space and time increments to calculate the solution (Arce and Sweat, 1980).


21.2.3.3

Plank’s Method

Plank’s (1932) equation can also be used to predict the heat transfer coefficient during the freezing process. The equation is t¼

l l l2 þ DT 2h 8k

(21:20)

where t is the freezing time (s) l is the thickness of the slab (m) k is the thermal conductivity of the frozen layer (W=mK) l is the latent heat effusion (J=kg) DT is the temperature difference between the ambient and the slab (8C) 21.2.3.4

Surface Heat Flux Method

Federov et al. (1972) used heat flux pickup to measure the heat flux and the surface temperature as a function of time. Arce and Sweat (1980) mentioned that this method gave approximate values due to the presence of a device at the surface of the body.

21.3 MEASUREMENT OF HEAT TRANSFER COEFFICIENT IN DIFFERENT FOOD PROCESSES The surface heat transfer coefficient is influenced by the composition of the fluid, the nature and geometry of particle surface, and the hydrodynamics of the fluid moving past the surface. Hence, it is important to measure the surface heat transfer coefficient under the conditions simulating the actual process. The estimation of the heat transfer coefficient falls under the category of an inverse heat conduction problem. This approach requires experimental measurement of the transient temperatures inside a body of known geometry at a specified location and estimation of transient temperatures, at the same location, by solving the governing heat conduction equations with an assumed convective boundary condition (i.e. the Biot number, Bi). In doing so, Bi is varied systematically to produce computed temperature–time histories closely matching the experimentally measured temperature histories. The following section describes the experimental protocols used for measurement of surface heat transfer coefficient under various food processing situations. 21.3.1

Canning

Accurate prediction of particle center temperature through mathematical models depends on how well overall heat transfer coefficient (U) and surface heat transfer coefficient (h) data are estimated under simulated process conditions, where both system and product parameters are expected to be influencing factors. Traditionally, U and h are determined from the measurement of the temperature responses of the particle and liquid under well-characterized initial and boundary conditions (Maesmans et al., 1992). Particle and liquid temperatures are generally measured using thermocouples. In real processing, situations involving rotational retorts, the particle motions are expected to be influenced by centrifugal, gravitational, drag, and buoyancy forces and hence, the associated U and h. The particle, attached to a rigid thermocouple, will not simulate the particle motion during agitation processing, which in turn causes some deviations in the measured heat transfer coefficients. Recently, attempts have been made to monitor the temperature of the particle


Table 21.1 Methods Employed for Determining U and h, with Canned Liquid=Particle Mixtures, Subjected to Agitation Processing Motion of Test Particle

References

Location of Particle Temperature Measurement

Mathematical Procedure

Lenz and Lund (1978)

Restricted

Center

Lumped capacity, Duhamel’s theorem

Hassan (1984)

Restricted

Surface

Lekwauwa and Hayakawa (1986)

Restricted

Center

Integration of overall energy balance equation Empirical formulae containing heating rate index and factor, Duhamel’s theorem

Deniston et al. (1987)

Restricted

Surface

Integration of overall energy balance equation

Fernandez et al. (1988)

Restricted

Center

Lumped capacity, Duhamel’s theorem

Stoforos and Merson (1991) Stoforos and Merson (1991) Weng et al. (1992)

Free

Only liquid temperature needed Surface (liquid crystal)

Analytical solution in the Laplace transform plane Integration of overall energy balance equation Numerical solution of heat conduction equation with convective boundary conditions Integration of overall energy balance equation and numerical solution of heat conduction equation with convective boundary conditions

Sablani and Ramaswamy (1995)

Free Free

Free

Indirect, accumulated process lethality at center Center

without inhibiting particle motion during agitation processing. Apart from difficulties in measuring the temperature of the moving particle, the mathematical solution, of the governing equation of the energy balance on the can (containing liquid and particles), is also complex due to the time variant temperature of the can liquid. Table 21.1 summarizes published methods to determine U and h in particulate liquids in cans subjected to agitation processing together with a description of the experimental procedure and the mathematical solution. The procedures used for the determination of U and h are classified into two groups, based on the motion of the experimental particle whose temperatures are monitored, (1) fixed particle and (2) moving particle, during agitation processing. 21.3.1.1

Mathematical Procedures to Determine U and h

The surface heat transfer coefficient is determined using an inverse heat transfer approach, in which the boundary condition is determined using measured transient temperatures. The governing partial differential equations have to be solved to describe the conduction heat flow inside the particle with appropriate initial and boundary conditions. Experimental data needed for this analysis are the transient temperatures of both liquid and particle and thermophysical properties of the particle. The liquid bulk temperatures are measured using a needle-type copper-constantan thermocouple, with its tip located at an appropriate location of the can, and a thermocouple, embedded into a predefined location in the particle, is used to monitor particle transient temperatures. The overall thermal energy balance on a particulate–liquid food system is used to calculate the associated convective heat transfer coefficients. The governing equation for heat transfer in such systems can be written as (all symbols are detailed in nomenclature) UAc (TR Tf ) ¼ mf Cpf


d < Tp > dTf þ mp Cpp dt dt

(21:21)

where Ac is surface area of can Tf is fluid temperature Cpf is specific of heat of fluid is average particle temperature mf is mass of fluid mp is mass of particles Cpp is specific heat of particle TR is retort temperature U is overall heat transfer coefficient The following are assumed in the solution of Equation 21.21: uniform initial temperature for the particle, uniform initial and transient temperatures for the liquid, constant heat transfer coefficients, constant physical and thermal properties for both liquid and particles, and no energy accumulation in the can wall. The second term on the right side of Equation 21.21 is equal to the heat transferred to particles from the liquid through the particle surface: mp Cpp

d < Tp > ¼ hAs Tf Tps dt

(21:22)

where Tps is particle surface temperature. 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 particle. For example, the heat flow in a spherical particle immersed in liquid can be described by the following partial differential equation: 2 @T @ T 2 @T ¼ ap þ @t @r2 r @r

(21:23)

where T is temperature r is radial coordinate system (m) ap is particle thermal diffusivity The initial and boundary conditions are T(r, 0) ¼ Ti , at

t¼0

(21:24)

where Ti is initial temperature dT(0, t) ¼ 0, at t > 0 dr dT ¼ h(Tf Tps ) kp dr

(21:25) (21:26)

where Kp is particle thermal conductivity The solution of Equations 21.23 through 21.26 is complex because of time-dependent liquid temperatures.


21.3.1.2 21.3.1.2.1

Experimental Procedure Restricted Particle Motion

Lenz and Lund (1978) developed a numerical solution using the fourth-order Runge–Kutta method and Duhamel’s theorem to determine U and h, for the low Biot number (Bi < 0.1) situation. They measured the transient temperatures of a lead particle fixed at the geometric center of the can with liquid moving around it and verified that the lead particle quickly approached to one temperature at all the points. Assuming that the retort temperature instantly reached its operating condition, they proposed the following solution of Equation 21.27 for the temperature at any position in the particle:

TR T 2Bi X b2n þ (Bi 1) sin bn ¼ sin bn (r=a) TR Ti r=a b2n þ Bi(Bi 1) b2 n tp exp (t f t) exp (t p t) exp (t p t þ tp tf

(21:27)

where TR is retort temperature tp and tf are time constants for particle are fluid (t is ab2=ap; where a is thermal diffusivity, b is root of the equation: b cot(b) þ Bi1 ¼ 0, Bi is Biot number: hap=k; h is heat transfer coefficient (W=m2k2), ap is charateristic dimension of a particle, k is thermal conductivity). They estimated h by minimizing the sum of the squared deviations between measured and predicted particle temperature profile. They also obtained the equation for the particle average temperature from Equation 21.27 and used it in Equation 21.21 to calculate the overall heat transfer coefficient. Hassan (1984) derived the following equations for U and h by integrating Equations 21.21 and 21.22, respectively, allowing the heating time to approach infinity ( ¼ Tf(1) ¼ TR): TR Tpi mf Cpf mp Cpp TR Tfi U¼ þ Ac 1 Ð Ð Ac 1 TR Tf dt TR Tf dt 0

(21:28)

0

where Tfi and Tpi are initial temperatures of fluid and particle. mp Cpp final initial h¼ 1 Ð Tf Tps dt Ap

(21:29)

0

where and are average temperature of particle and fluid. For cans subjected to end-over-end rotation, Lekwauwa and Hayakawa (1986) developed a model using an overall heat balance equation in combination with an equation for transient heat conduction in a particle. They considered the probability function representing the statistical particle volume distribution. The temperature distribution for individual particles was obtained using Duhamel’s theorem and empirical formulae describing the heat transfer to spherical, cylindrical, or oblate spheroid-shaped particles in a constant temperature liquid. In their numerical solution, they assumed that within each time step the liquid temperature was a linear function of time and the coefficients of these functions were determined iteratively such that the resulting particle and liquid temperatures satisfied the overall heat balance equation.


Deniston et al. (1987) used Equations 21.28 and 21.29 to determine U and h in axially rotating cans. In their experiments, the heating time was sufficiently long to allow liquid and particle average temperatures to reach the heating media temperature to satisfy infinite time limits in the above equations. They measured the transient temperature at the particle surface using rigid-type thermocouples. They recognized the difficulties and errors associated with the measurement of surface temperatures. Fernandez et al. (1988) determined the convective heat transfer coefficients for bean-shaped particles, in cans processed in an agitated retort. They preferred a high conductivity material such as aluminum to give a low Biot number (Bi < 0.01) condition and used the lumped capacity method for U and h evaluation. They measured time–temperature data for both the liquid and particle using rigid-type thermocouples and used the scheme developed by Lenz and Lund (1978) to solve the heat balance equations. Recently, Stoforos and Merson (1995) proposed a solution to the differential equations governing heat transfer to axially rotating particulate liquids in cans. They used an analytical solution, Duhamel’s theorem for particle temperature, and a numerical solution based on the fourth-order Runge–Kutta scheme for the liquid temperature. The solution avoided the need for empirical formulae or a constant heating medium within short time intervals. Their comparison between predicted values and experimental data from Hassan (1984) showed good correlation for liquid temperature; however, it deviated for particle surface temperatures. 21.3.1.2.2

Allowing Particle Motion

From liquid temperature only: Stoforos and Merson (1990) used a mathematical procedure requiring only the measurement of liquid temperature to estimate U and h in axially rotating cans. They solved an overall energy balance equation (Equation 21.21) for a can and the differential equation for a spherical particle with appropriate initial and boundary conditions (Equations 21.23 through 21.26). Since the can liquid temperature depends on both U and h, by systematically varying these coefficients and minimizing the error between experimental and predicted liquid temperatures (in Laplace plane), they estimated the U and h. This method allows for free movement of particles. The authors reported that calculated h values did not always coincide with those determined from particle surface temperature measurements. Liquid crystal: Stoforos and Merson (1991) extended the solution procedure for the overall energy balance of the can for finite heating time, previously used by Hassan (1984) to determine U and h. They used a liquid crystal, which changes color with temperature, as a sensor to monitor the particle surface temperature. The method involved coating the particle surface with an aqueous solution of liquid crystals, videotaping of the color changes on the particle surface as a function of temperature, and comparing it with standard color charts after calibration. This method does not impose any restriction on particle motion, while monitoring the particle temperature during can rotation. They measured the liquid temperatures using rigid thermocouples for finite heating time and exponentially extrapolated them to obtain liquid temperature data for ‘‘long time’’ approximation of heating time in the solution equation (Equation 21.27). They used an iterative procedure to calculate the average particle temperature, required in the solution equation for hip calculation. By using the first term approximation to the series solution, for average particle temperature () as a function of particle surface temperature and Biot number (Stoforos, 1988), they obtained an analytical expression. Initially, as a first approximation, h was estimated by assuming ¼ Tps (t) in Equation 21.28; the particle average temperature was obtained from analytical expression. This value for average particle temperature was substituted into Equation 21.28 to find an improved value for h. The authors reported that one or two iterations were usually enough for convergence. However, they cautioned that the first term approximation in the expression of particle average temperature was limited to tow particle thermal diffusivity materials and suggested that for a high thermal diffusivity


particle ¼ T (t) may be a good assumption. The experiments were carried out between the temperature range of 208C to 508C. Time–temperature integrator (TTI): The combined use of a TTI has been proposed in the form of microorganism, chemicals or enzymes, and a mathematical model to determine the convective heat transfer coefficient. In this approach, a particle loaded with an indicator at the center can be processed without affecting its motion in real processing conditions. The process lethality received by the particle during processing can be calculated from TTI’s initial (N0) and final (N) status: N0 (21:30) FTTI ¼ Dref log N FTTI is integrated process lethality recorded by time temperature indicator. By using heating liquid temperature and assuming a constant h, a time–temperature profile at the particle center can be generated using a mathematical model and Fmodel could be calculated: ðt Fmodel ¼ 10ðTTref =zÞ dt

(21:31)

0

The value of h is estimated by minimizing the difference between FTTI and Fmodel. Hunter (1972) and Heppel (1985) were the first to use this approach and used microorganisms suspended in beads to back calculate the convective heat transfer coefficients, during continuous sterilization. Weng et al. (1992) used a TTI in the form of immobilized peroxidase, and determined the heat transfer coefficients in cans at pasteurization temperatures. A polyacetal sphere loaded with the indicator at the center was hooked onto a thermocouple and placed at the geometric center of a can. They calculated the time–temperature history and associated lethality from the equation of the heat conduction with assumed h and known thermophysical properties, using an explicit finite difference method. The h was modified and lethality recalculated until the difference between the predicted and actual lethalities fell within a tolerance limit. The authors named this approach as the least absolute lethality difference (LALD) method. They also gathered the transient temperature data for liquid and particle during heating and cooling in the same experiments and estimated h by minimizing the sum of the square of the temperature difference (LSTD) between measured particle center temperature and predicted center temperature, using the mathematical model. Maesmans et al. (1994) studied the feasibility of this method and the factors that can affect the choice of this methodology. Since these factors can influence the measurement of h, care is necessary in the design of experiments to obtain accurate results with this method (Maesmans et al., 1994). Moving thermocouple method: A methodology was developed to measure the heat transfer coefficients, while allowing the movement of the particle inside the can, by attaching it to a flexible fine wire thermocouple (Sablani and Ramaswamy, 1995, 1996, 1997; Sablani, 1996) (Figure 21.1). U was calculated from the thermal energy balance on the can (Equation 21.21) and h was evaluated using a finite difference computer program, by matching the particle time–temperature data with those obtained by solving the governing partial differential Equations 21.23 through 21.26 of conduction heat transfer in different geometries, with appropriate initial and boundary conditions. 21.3.2

Aseptic Processing

Evaluating the surface heat transfer coefficient under aseptic processing conditions is an important step in the process design of particulate liquid. The conditions that can influence the heat transfer coefficient are particle=fluid relative velocities, the shape and size of particles, pipe diameter, temperature, and concentration of carrier fluid (Ramaswamy et al., 1997). In general, h is


Needle-type thermocouple

Brass connector stuffing box

To data logger via slip ring

Test particle

Flexible thin-wire thermocouple Figure 21.1 A schematic showing thermocouple-equipped particle mounted inside the can using a brass connector.

determined from the physical measurement of temperature history within a particle or particle surface under well-defined experimental conditions and by solving the transient heat transfer equation for particle geometry along with appropriate initial and boundary conditions (Table 21.2). Table 21.2 Methods Employed for Determining h in Aseptic Processing Situation

References

Method

Particle Temperature= Lethality Measurement

Mathematical Procedure

Hunter (1972) and Heppel (1985)

TTI

Microbial population reduction

Weng et al. (1992)

Free TTI

Peroxidase activity

Sastry et al. (1989)

Moving thermocouple Melting point

Center Surface

Stoforos and Merson (1990) and Moffat (1990)

Liquid crystal

Surface

Whitaker (1972)

Relative velocity

Balasubramaniam and Sastry (1994a,b)

Transmitter method

Measure relative velocity Center

Using correlation of Ranz and Marshall (1952) Numerical solution of heat conduction equation with convective boundary conditions

Chang and Toledo (1989); Alhamdan and Sastry (1990); Zuritz et al. (1990); Chandarana et al. (1990); Awuah et al. (1993); Ramaswamy et al. (1996a,b); Awuah et al. (1996)

Restricted

Center

Numerical=analytical solution of heat conduction equation with convective boundary conditions

Mwangi et al. (1993)


Numerical solution of heat conduction equation with convective boundary conditions Numerical solution of heat conduction equation with convective boundary conditions Lumped capacity Numerical solution of heat conduction equation with convective boundary conditions Numerical solution of heat conduction equation with convective boundary conditions

Ramaswamy et al. (1997) presented an excellent review of techniques used for the measurement of heat transfer coefficients in aseptic processing situations. 21.3.2.1

Time–Temperature Integrator Method

In the time–temperature integrator method, instead of comparing the experimentally measured and estimated transient temperatures, the cumulated effect of time–temperature is compared for estimation of heat transfer coefficient. Hunter (1972) and Heppel (1985) used the microbial population reduction approach by heating spore-impregnated alginate beads to estimate heat transfer coefficient. This approach led to the use of a chemical indicator. Weng et al. (1992) used a time– temperature integrator in the form of immobilized peroxidase to calculate heat transfer coefficients at pasteurization temperatures. In this method, a particle of regular geometry is loaded with the indicator (known microbial population or enzyme of known activity) at the center whose kinetic data are known. After the heating=cooling, the particle is recovered and actual destruction of microbes or inactivation of enzymes is measured for estimation of actual lethality. Assumed heat transfer coefficient and known thermophysical properties are used to compute transient temperature profile using any numerical method. The estimated lethality is then calculated from predicted time– temperature profile and compared with actual=experimental lethality. The heat transfer coefficient is then modified and lethality is recalculated until the difference between the predicted and actual lethalities is minimized within a tolerable limit. This approach of estimation of heat transfer coefficient was termed as LALD. In this approach, the weight of lethal temperatures is more pronounced than the weight of lower temperatures. However, in the approach where the least sum of squared temperature differences (LSTD) is used, each measured temperature is considered in minimizing the difference between experimental and predicted temperatures. In this method, loading of enzymes=microbes may alter the thermophysical properties due to materials used in holding the indicator in place. Although this method is noninvasive in relation to particle trajectory as well as suitable to high-temperature applications, sample-to-sample variation in microbial population for instance can cause major problems and discrepancies in replicated data (Ramaswamy et al., 1997). 21.3.2.2

Moving Thermocouple Method

In aseptic processing, the particles move in the holding tube along with the carrier fluid. To simulate particle motion during measurement of heat transfer coefficient, Sastry et al. (1989) developed the moving thermocouple technique that involved a thermocouple hooked particle moving in a tube. The heat transfer coefficient was calculated from the experimentally measured time–temperature data using the lumped capacity approach. Sastry et al. (1990) modified the above setup by using a motor-driven setup at the downstream end of the tube to withdraw the thermocouple at the predetermined velocity and using a magnetic flowmeter to measure the fluid flow rate (Figure 21.2). Sastry (1992) indicated that the moving thermocouple method predicts conservative values. The main advantage of this technique was the measurement of transient temperatures of the moving particle. However, this approach cannot be adopted for high temperatures and pressures owing to difficulties in equipment design. Also, the fluid motion and profiles are constrained when particles are withdrawn from the tube (Ramaswamy et al., 1997). 21.3.2.3

Melting Point Method

The melting point method uses polymers that change color at specific temperatures. The polymer with calibrated color=temperature is transplanted as an indicator at the center of transparent particles. Mwangi et al. (1993) used transparent polymethylmetacrylate spheres with diameters


Particle Thermocouple

Photosensor

Particle N

Thermocouple

Units Process fluid Heat exchanger Motorized pinch rollers

Magnetic flow meter Figure 21.2 Schematic diagram of experimental setup for particle to fluid heat transfer coefficient measurement. (From Sastry, S.K., Lima, M., Bruin, J., Brunn, T., and Heskitt, B.F., J. Food Process Eng., 13, 239, 1990.)

ranging from 8 to 12.7 mm. The particles were introduced through a venture into a transparent holding tube containing glycerin=water mixtures. The time at which the color change occurred was recorded in addition to time temperature data of the fluid. The temperature at the surface of the indicator within the particle was predicted with a finite difference program using an assumed value of heat transfer coefficient. An iterative procedure for minimizing the difference between time needed to reach the melting temperature was used to calculate the heat transfer coefficient. The method is noninvasive but limited to transparent tubes, and particles are not usable since the color change is irreversible (Mwangi et al., 1993). 21.3.2.4

Liquid Crystal Method

Liquid crystal method involves coating of the surface of a particle with a liquid crystal that changes color with a change in temperature. The change in color is videotaped as crystal-coated particles travel through the transparent tube. The surface transient temperatures are obtained using a color temperature-calibrated chart. This method is noninvasive and rapid. The accuracy of measured temperature thus estimated by heat transfer coefficient is limited by (1) the range of temperatures over which color changes and (2) the resolution of the video image. The method is not suitable for high temperature=pressure applications involving opaque carrier liquids. 21.3.2.5

Relative Velocity Method

Relative velocity method requires estimation of relative velocity from experimentally measured velocities of a liquid and particle. The velocities of the particle and liquid are measured using a video camera. The carrier liquid velocity is essentially measured by introducing a tracer particle into the fluid stream. Ramaswamy et al. (1992) presented an apparatus for particle fluid relative velocity measurement in tube flow at various temperatures under nonpressurized flow conditions (Figure 21.3). The relative velocity along with flow properties and particle properties is used to calculate the heat transfer coefficient from established Nusselt number correlations presented by Kramers (1960),


Medium (water or starch)

Steam kettle

3-way valve

Pump A Port for particle entry

By-pass line

B

Rubber connector

Glass tube (ID, 41.2 mm; Length, 1.2 m)

By-pass system to measure flow rate

C

Port for particle exit

Figure 21.3 Equipment setup for the particle to fluid relative velocity measurement. (From Ramaswamy, H.S., Pannu, K., Simpson, B.K., and Smith, J.P., Food Res. Int., 25, 277, 1992. With permission.)

Ranz and Marshall (1952), and Whitaker (1972). The accuracy of the heat transfer coefficient estimated using this method will depend on the reliability of the adopted correlation (Ramaswamy et al., 1997). However, the technique can be useful in characterizing the radial variations in the heat transfer coefficient (Sastry, 1992). 21.3.2.6

Transmitter Method

The transmitter technique involves placement of a miniature sensor in the particle that can transmit the signals to an external receiver. The purpose is to gather time–temperature data from a particle in motion with no obstruction such as in the case of the moving thermocouple method. Balasubramaniam and Sastry (1994a,b) used a hollow cylindrical capsule made of boron nitride with a transmitter. The magnetic signals transmitted were converted to temperature reading using a calibrated scale and an iterative procedure was used to back calculate the heat transfer coefficient. Ramaswamy et al. (1997) noted that this method has all the potentials for aseptic studies but little information has been reported in the literature. One of the reasons may be that the placement of the transmitter may cause considerable variation in the density of the particle in which it is embedded. 21.3.2.7

Stationary Particle Method

In this method, a particle is placed in a flowing stream and transient temperatures are measured within the particle at a known location and fluid. An iterative procedure is used to estimate the heat transfer coefficient by solving appropriate governing equations of transient heat transfer with convective boundary condition using the analytical or numerical method. This method may not reflect conditions in real situations since both translational rotational motions are restricted; thus it can lead to deviation in the estimated value of the heat transfer coefficient. However, several researchers (Chang and Toledo, 1989; Alhamdan and Sastry, 1990; Chandarana et al., 1990;


Zuritz et al., 1990; Awuah et al., 1993, 1996; Ramaswamy et al., 1996a,b) have used this method to study the influence of various processing parameters on the heat transfer coefficient under low- or high-temperature processing conditions. This method has been useful in providing considerable insight on the influence of various parameters on the heat transfer coefficient (Ramaswamy et al., 1997). 21.3.3

Baking

Energy balance during baking in an oven will require data on surface heat transfer coefficient on baked products in addition to thermal properties of product, construction, size of the oven and operating conditions such as the velocity, temperature, and humidity of the hot air. The direct-fired oven is the most common type for commercial baking. However, heat transfer studies in such ovens are still limited. In particular, a nonuniform temperature distribution between heating elements and product surface is an important factor as well as variable temperature profiles along the continuous oven, which makes it difficult to estimate surface heat transfer coefficients (Baik et al., 1999). There have been some studies reported on the measurement of heat transfer coefficients of the dough and the internal oven environment. In most such studies, the heat flux or heat transfer coefficient was measured directly in the oven using a heat flux sensor (typically a metal cylinder or sphere) or a commercially available h-monitor (Krist-Spit and Sluimer, 1987; Sato et al., 1987; Huang and Mittal, 1995; Li and Walker, 1996; Baik et al., 1999). These studies assumed that heat transfer between the oven and dough is the same as that between the oven and the model food (heat flux sensor). Baik et al. (1999) described an experimental and mathematical procedure to estimate the surface heat transfer coefficient on cakes baked in a tunnel type industrial oven using h-monitor (Figure 21.4). They represented convective flux and radiative flux from wall and heating element in the form of convective and radiative heat transfer coefficients: Convective heat flux Qc ¼ hc (Ta Ts )

(21:32)

Radiative heat flux from wall Qrw ¼ hc (Ta Ts )

(21:33)

Radiative heat flux from heating element Qew ¼ hc (Ta Ts )

(21:34)

5 2 1

3

4

6

Upper zone Heating element

4 5

Measurement position

6 2 1

Plate

3

Band (A)

4

(B)

Figure 21.4 (A) Basic description of h-monitor and (B) measurement position inside the baking chamber of the tunnel type multizone industrial oven. (1) SMOLE, (2) aluminum plate, (3) thermocouple for plate temperature, (4) thermocouple position for air–mass temperature, (5) insulation, (6) thermocouple holder for air temperature measurement.


This method is useful when direct measurement of radiation heat transfer is not available. These radiant heat transfer coefficients can be treated similarly to the convective heat transfer coefficient (Kreith, 1973); in that case the total surface heat flux, Qt, can be expressed as Qt ¼ (hc þ hrw þ hre )(Ta Ts ) ¼ ht (Ta Ts )

(21:35)

Hence ht is defined as effective surface heat transfer coefficient. The total heat flux was measured by a commercial h-monitor and a moving temperature recorder, super multichannel occurrent logger evaluator (SMOLE) equipped with K-type thermocouples. The aluminum plate of h-monitor had emissivity similar to that of the cake (0.90–0.95). Other properties such as mass (m), specific heat (Cp), and the surface area (A) of the aluminum plate were known. By measuring the temperature for both the air and the plate at selected intervals during the baking process, the total heat flux and effective surface heat transfer coefficient were determined as ht ¼

mCp DT A(Ta Ts )Dt

(21:36)

The Bi of aluminum plate of the h-monitor was 200

(22:7b)

Other relations for heated surface facing upward (or cooled surfaces facing downward) are (Gebhart, 1971) Nu ¼ 0:54Ra1=4 , Nu ¼ 0:14Ra1=3 ,

22.2.1.2

105 < Ra < 107

(22:7c)

107 < Ra < 3 1010

(22:7d)

Heat Transfer against the Direction of Gravitational Force

Fujii and Imura (1972) also proposed the following correlation for heat transfer to or from horizontal plates in the opposite direction of gravitational force: Nu ¼ 0:16Ra1=3 ,

Ra < 2 108

(22:8a)

Nu ¼ 0:13Ra1=3 ,

5 108 < Ra

(22:8b)

In Equations 22.6 and 22.8 b is evaluated at T1 þ 0.25(Ts T1) and all other properties at Ts 0.25(Ts T1), and Ts is the average wall temperature related to the heat flux (Holman, 1986; Suryanarayana, 1995). Alternatively, the following relation is recommended by McAdams (1954) (Gebhart, 1971): Nu ¼ 0:27Ra1=4 ,

3 105 < Ra < 3 1010

(22:9)

For circular plates of diameter D in the stable horizontal configurations, the data of Kadambi and Drake (1959) suggest that (Lienhard and Lienhard, 2004) Nu ¼ 0:82Ra1=5 Pr0:034

22.2.2

(22:10)

Vertical Plates

Churchill and Chu (1975a) proposed the following equation for the mean value of Nu number for vertical plates:


Nu ¼

8 > < > :

92 > =

1=6

0:387Ra 0:825 þ h i8=27 > ; 1 þ ð0:492=Pr Þ9=16

(22:11a)

where the characteristic length L is the height of the plate. b is evaluated at T1 if the fluid is a gas and all other properties at the film temperature Tf ¼ (Ts þ T1)=2. Equation 22.11a provides a good representation for the mean heat transfer over a complete range of Ra and Pr from 0 to 1 even though it fails to indicate a discrete transition from laminar to turbulent flow (Churchill and Chu, 1975a). For Ra < 109, the following equation gives more accurate prediction of h: Nu ¼ 0:68 þ h

0:67Ra1=4 1 þ ð0:492=Pr Þ9=16

i4=9

(22:11b)

Equations 22.11a and b are suitable for vertical plates with uniform wall temperature. However, if uniform heat flux is applied to the vertical plates, a small adjustment to Equations 22.11a and b is needed by replacing the constant 0.492 in the denominator by 0.437. In such a case, the fluid properties are evaluated at the temperature at the mid-height of the plate, i.e., Tf ¼ (TL=2 þ T1)=2 and TL=2 T1 is used to form the Raleigh number. Alternatively, for vertical plates with uniform heat flux, the following correlations proposed by Sparrow and Gregg (1956) and Vliet and Liu (1969) can be used to calculate the local Nusselt number as a function of the length from the plate edge (Suryanarayana, 1995): Nux ¼ 0:6(Gr x*Pr)0:2 , Nux ¼ 0:568(Gr x*Pr)0:22 ,

105 < Gr x*Pr < 1013 1013 < Gr x*Pr < 1016

(22:12a) (22:12b)

where all the properties are evaluated at Tf ¼ (Ts þ T1)=2 and x is the coordinate from the leading edge along the plate. Free convection with uniform heating is often correlated in terms of the modified Raleigh number (Ra* ¼ Gr x* Pr), based on the local heat flux q, to avoid explicit inclusion of the surface temperature (Churchill and Chu, 1975b). The modified Grashof number in Equation 22.12 and the modified Ra number are defined as gbr2 qx4 m2 k gbqx4 Ra* ¼ yak

Gr x* ¼

where q ¼ hx(Ts T1) 22.2.3

Inclined Plates

For heat transfer in the general direction of gravitational force either from the plate or to the plate, the above correlations may be used for inclined plates, if the component of the gravity vector along the surface of the plate is used in the calculation of the Grashof number. In other words, for downward-facing heated inclined plates or upward-facing cooled inclined plates,


Equation 22.11 for vertical plates can be used by taking a modified Rayleigh number (Rau) where g is replaced by g cos u: Rau ¼

g cos ubL3 DT ya

and u is the angle of inclination of the plate to the vertical. However if the angle is greater than 888, it is suggested to use the correlations for horizontal plates. If the inclined plates are supplied with a uniform heat flux, Fussey and Warneford (1978) suggested the following correlations for the local heat transfer coefficients: Nux ¼ 0:592(Rax* cos u)0:2 , Rax* < 6:31 1012 e0:0705u , 0 < u < 86:5 Nux ¼ 0:889(Rax* cos u)0:205 , Rax* > 6:31 1012 e0:0705u , 0 < u < 31

(22:13a) (22:13b)

All the fluid properties are evaluated at the average temperature (Ts þ T1)=2. For heat transfer against the general direction of gravitational force, i.e., for an upward-facing heated inclined plate, the situation is more complicated, and the reader is advised to consult a book on specialized convection heat transfer for relevant correlations. 22.2.4

Long Horizontal Cylinders

Churchill and Chu (1975b) proposed a simple empirical expression for the mean value of the Nusselt number over the cylinder of diameter D for uniform surface temperature for all Pr and Ra values: 91=6 12 > = C B Ra C B Nu ¼ @0:60 þ 0:387 h i A 16=9 > > : 1 þ ð0:559=Pr Þ9=16 ; 0

8 >

=

Ra Nu ¼ 0:36 þ 0:518 h i16=9 > , > : 1 þ ð0:559=PrÞ9=16 ;

106 < Ra < 109

(22:14b)

Equation 22.14a is probably a good approximation also for uniform heat flux; however, Equation 22.14c provides a possibly better correlation than Equations 22.14a and b for uniform heat flux in the laminar regime for small Pr: 8 91=4 > > < = Ra Nu ¼ 0:36 þ 0:521 h i16=9> > ; : 1 þ ð0:442=Pr Þ9=16

(22:14c)

For large temperature differences such that the variation of physical properties is significant, the properties may be evaluated at the average of the bulk and surface temperatures as a first


approximation; b at Tf ¼ (Ts þ T1)=2 for liquids, and b ¼ 1=T1 for gases. For a vertical cylinder, the same equations can be used as those for a vertical plate if the following can be satisfied (Gebhart, 1971): D 35 1=4 L Gr where the Grashof number is based on L. In general, the average natural convection heat transfer coefficients for isothermal surfaces can be expressed by the following general equation for a variety of circumstances: Nu ¼ cRam

(22:14d)

where c and m are constants and are given in Table 22.1 for different geometries. All the physical properties are evaluated at the film temperature, Tf ¼ (Ts þ T1)=2. Simplified equations for heat

Table 22.1 Constants for Equation 22.14d for Natural Convection Physical Geometry

Ra

c

m

109

0.59 0.13

1=4a 1=3a

109–1013

0.021

2=5b

10 1010–102 102–102

0.13 0.675 1.02

1=3a 0.058b 0.148b

102–104

0.85

0.188b

10 –10 107–1012

0.48 0.125

1=4b 1=3b

105–(2 107)

0.54

1=4a

(2 107)–(3 1010)

0.14

1=3a

105–1011

0.58

1=5a

105–1011

0.27

1=4b

Vertical cylinder (height ¼ diameter, characteristic length ¼ D)

104–106

0.775

0.21b

Irregular solids (characteristic length ¼ distance fluid particle travels in boundary layer)

104–109

0.52

1=4b

Vertical planes and cylinders (vertical height L < 1 m)a

Horizontal cylinders (diameter D used for L and D < 0.20 m)b

9

4

Horizontal plates (upper surface of heated plates or lower surface of cooled plates)

Horizontal plates (lower surface of heated plates or upper surface of cooled plates)

Sources:

a

7

Geankoplis, C.J., in Transport Processes and Unit Operations, Prentice-Hall International Inc, New Jersey, 1993; bHolman, J.P., in Heat Transfer, 6th edn., McGraw-Hill, New York, 1986.


transfer coefficients from various surfaces to air and water at atmospheric pressure are given in Rahman (1995), Geankoplis (1993), and Holman (1986). The relations for air may be extended to higher or lower pressures by multiplying by the following factors (Holman, 1986):

p 1=2 , for laminar cases 101:32 p 2=3 , for turbulent cases 101:32 where p is the pressure in kPa. 22.2.5

Spheres

Yuge (1960) recommends the following empirical expression for natural convection from spheres to air (Holman, 1986): Nu ¼ 2 þ 0:392Gr1=4 ,

1 < Gr < 105

(22:15a)

Equation 22.15a can be rearranged by the introduction of the Prandtl number to give Nu ¼ 2 þ 0:43Ra1=4

(22:15b)

It is expected that this relation would be applicable for free convection in gases and Prandtl numbers in the vicinity of 1. However, in the absence of specific information it may also be used for liquids. Properties are evaluated at the film temperature Tf ¼ (Ts þ T1)=2, and b at T1. A more complex expression (Raithby and Hollands, 1998) encompasses other Prandtl numbers (Lienhard and Lienhard, 2004): 0:589Ra0:25 12 Nu ¼ 2 þ , Ra < 10 9=16 4=9 1 þ (0:492=Pr)

(22:15c)

Equation 22.15c has an estimated uncertainty of 5% for air and a root-mean-square error of approximately 10% at higher Prandtl numbers. For higher ranges of the Rayleigh number the experiments of Amato and Tien (1972) with water suggest the following correlation (Holman, 1986): Nu ¼ 2 þ 0:50Ra0:25 ,

22.2.6

3 105 < Ra < 8 108 ,

10 Nu 90

(22:15d)

Natural Convection in Enclosed Spaces

Free convection inside enclosed spaces occurs in a number of processing applications. The flow phenomena inside spaces are examples of complex fluid systems, which are important in energy conservation processes in buildings (e.g., in multiply glazed windows, uninsulated walls, attics), in crystal growth and solidification processes, and in hot or cold liquid storage systems (Lienhard and Lienhard, 2004). Different types of flow patterns can occur inside enclosed spaces; at low Grashof numbers heat is transferred mainly by conduction across the fluid layer, while different flow regimes are encountered when the Grashof number increases (Geankoplis, 1993). Empirical correlations and review papers on natural convection in enclosures can be found in Geankoplis (1993), Holman (1986), Yang (1987), Raithby and Hollands (1998), and Catton (1978).


22.3 FORCED CONVECTION CORRELATIONS In the food industry, most of the cooling or heating processes occur under forced convection. Forced convection is defined as that because the heat transfer from or to the foods is due to the motion of the heating or cooling fluid caused by external means such as a fan or a pump. Therefore unlike natural convection, the movement of the fluid is not dependent on the temperature difference, and fluid properties and flow velocity play an important part in the convective heat transfer coefficients. Hence Equation 22.5 can be simplified as Nu ¼ f (Re, Pr)

(22:16)

Therefore, finding the above relation is the main task in convection heat transfer research. Depending on the fluid flow position, studies on the forced convective heat transfer can be classified into two cases: external flow and internal flow. 22.3.1

External Flow

According to the shape of the solid object, a fluid can flow externally over a plate, a cylinder, or a sphere among other shapes. Heat transfer correlations have therefore been developed accordingly. In the flow over the external surface of an object, there exists a boundary layer. Within the boundary layer, the fluid velocity changes from zero at the surface to a uniform velocity V1 (the free stream velocity), and the flow can be laminar or turbulent. If the flow is laminar, the layer is defined as the laminar boundary layer. Likewise if turbulent flow exists in the layer, the layer is the turbulent boundary layer. As a result, different correlations are developed for predicting convective heat transfer coefficients under different boundary layer conditions. In order to identify the boundary layer conditions, for simplicity, abrupt transition from laminar to turbulent boundary is normally assumed. Therefore, a critical Reynolds number Recr is used to indicate the abrupt point, which is commonly defined as Recr ¼ 5 105. 22.3.1.1 22.3.1.1.1

Flat Plates Laminar Boundary Layers (Re < Recr)

For simple flow problems where a fluid flows parallel to the plate with uniform temperature, if all the properties are evaluated at the film temperature, Tf ¼ (Ts þ T1)=2, i.e., average of wall and free stream temperatures, and the Reynolds number is determined at the local position, i.e., Rex ¼ rV1x=m, the local Nusselt number can be correlated by the following equations (Suryanarayana, 2000; Lienhard and Lienhard, 2004): Nux ¼

hx x 1=3 ¼ 0:332Re1=2 , x Pr k

1=2 Nux ¼ 0:564Re1=2 , x Pr

Rex Pr 100 and

1=3 , Nux ¼ 0:339Re1=2 x Pr

0:6 Pr 50 Pr 0:01 or Rex 104 Pr ! 1

(22:17a) (22:17b) (22:17c)

For all Prandtl numbers, Churchill and Ozoe (1973a) and Rose (1979) recommended the following empirical correlations (Holman, 1986; Suryanarayana, 2000):


Nux ¼

0:3387Pr 1=3 Re1=2 x , 2=3 1=4 1 þ (0:0468=Pr)

Nux ¼

Rex Pr > 100

(22:18a)

1=6

(22:18b)

1=2 Re1=2 x Pr

27:8 þ 75:9Pr 0:306 þ 657Pr

In the range of 0.001 < Pr < 2000, Equation 22.18a is within 1.4% and Equation 22.18b is within 0.4% of the exact numerical solution to the boundary layer energy equation. If the fluid is flowing parallel to a flat plate and heat transfer is occurring between the whole plate of length L (m) and the fluid, the average convective heat transfer coefficient h (either q=DT in the uniform wall temperature problem or q=DT in the uniform heat flux problem) and thus the Nu number (Nu ¼ 2Nux ¼ L) are as follows (Geankoplis, 1993): Nu ¼ 0:664Re1=2 Pr 1=3 ,

Pr > 0:7

(22:19)

Likewise for liquid metal flows (Lienhard and Lienhard, 2004), Nu ¼ 1:13Re1=2 Pr 1=2 ,

Pr 1

(22:20)

Alternatively for all Prandtl numbers (Churchill, 1976), Nu ¼

0:6774Pr 1=3 Re1=2 1=4 , Re < Recr 1 þ (0:0468=Pr)2=3

(22:21)

Correspondingly, if the plate is supplied with a uniform heat flux, the following correlations should be used instead (Lienhard and Lienhard, 2004; Suryanarayana, 1995): 1=3 Nux ¼ 0:453Re1=2 , x Pr 1=2 , Nux ¼ 0:886Re1=2 x Pr

Pr > 0:1

(22:22)

Pr < 0:05

(22:23)

For all Pr numbers and uniform heat flux Churchill and Ozoe (1973b) recommended the following single correlation: Nux ¼

22.3.1.1.2

0:4637Pr 1=3 Re1=2 x 1 þ (0:02052=Pr)2=3

1=4 ,

Rex Pr > 100

(22:24)

Turbulent Boundary Layer (Re > Recr)

When the flow in the boundary becomes turbulent (Re > Recr), Equations 22.17 through 22.24 are no longer valid. For a flat plate with uniform surface temperature, the local convective heat transfer coefficients can be predicted by the following equations (Suryanarayana, 1995, 2000): Nux ¼

hx x 1=3 ¼ 0:0296Re4=5 , x Pr k

Nux ¼ 1:596Rex (ln Rex )2:584 Pr 1=3 ,


0:6 Pr 60, Recr < Rex < 107 0:6 Pr 60,

107 < Rex < 109

(22:25a) (22:25b)

0:43 Nux ¼ 0:0296Re4=5 x Pr

m1 ms

1=4 , 0:6 Pr 60,

Rex > Recr

(22:25c)

For average convective heat transfer coefficient (Janna, 2000), h i 4=5 4=5 Nu ¼ 0:664Re1=2 Pr1=3 , þ 0:0359 Re Re cr cr

0:6 < Pr < 60

(22:26)

If Recr ¼ 5 105 is inserted, Equation 22.26 can be reduced to Nu ¼ (0:0359Re4=5 830)Pr1=3 ,

0:6 < Pr < 60, Recr < Re < 108

(22:27)

If Re > 107 and Recr ¼ 5 105 (Suryanarayana, 2000), Nu ¼ 1:963Re(ln Re)2:584 871 Pr 1=3 , 0:7 < Pr < 60,

107 < Re < 109

(22:28)

An alternative expression to Equation 22.27 is the Zhukauskas–Whitaker equation, which accounts for the temperature dependence of the fluid viscosity (Whitaker, 1972): Nu ¼ 0:036(Re4=5 9200)Pr0:43 (m=ms )1=4

(22:29)

where ms is evaluated at the uniform surface temperature Ts, while all other properties are obtained based on the free stream temperature T1. In addition, the equation is valid only for the following conditions: 0.7 < Pr < 380, 105 < Re < 5.5 106, and 0.26 < (m=ms) < 3.5. If Equation 22.29 is used to predict heat transfer to a gaseous flow, the viscosity-ratio correction term should not be used and properties should be evaluated at the film temperature (Lienhard and Lienhard, 2004). A problem with Equation 22.29 is that it does not deal with the question of heat transfer in the rather lengthy transition region, since it is based on the assumption that the flow abruptly passes from laminar to turbulent at a critical value of x (Lienhard and Lienhard, 2004). Churchill (1976) suggested the following equations to predict heat transfer for laminar, transitional, and turbulent flow (Lienhard and Lienhard, 2004): (

Nux ¼ 0:45 þ (0:3387f

1=2

(f=2,600)3=5 ) 1þ [1 þ (fu =f)7=2 ]2=5

)1=2 (22:30a)

where " f ¼ Rex Pr

2=3

#1=2 0:0468 2=3 1þ Pr

(22:30b)

and fu is a number between 105 and 107. If the Reynolds number at the end of the turbulent transition region is Reu, an estimate is fu f(Rex ¼ Reu). For the average heat transfer coefficient, Churchill (1976) also proposed (Lienhard and Lienhard, 2004) (

Nu ¼ 0:45 þ (0:6774f


1=2

(f=12,500)3=5 ) 1þ [1 þ (fum =f)7=2 ]2=5

)1=2 (22:30c)

where f is defined as in Equation 22.30b using Re over the length L of the plate in place of Rex, and fum f(Re ¼ Reu). This equation may be used for either uniform heat flux or uniform surface temperature. The following equation can also be used for turbulent boundary layer along the whole plate and constant wall temperature (Petukhov and Popov 1963; Schlichting 1979): Nu ¼

0:037Re0:8 Pr 1 þ 2:443Re0:1 (Pr 2=3 1)

(22:31)

Another simplified empirical relationship for estimating the average heat transfer coefficient is as follows (Geankoplis, 1993): Nu ¼ 0:0366Re0:8 Pr 1=3 ,

Pr > 0:7,

Re > Recr

(22:32)

If the plate is supplied with a uniform heat flux, the local heat transfer coefficient is higher. Kays and Crawford (1980) suggested the following: 0:6 Nux ¼ 0:03Re0:8 x Pr

(22:33)

Thomas and Al-Sharifi (1981) also recommended the following alternatives: Nux ¼

pffiffiffiffiffiffiffiffiffiffi cfx =2Rex Pr

pffiffiffiffiffiffiffiffiffiffi , 0:5 < Pr < 10 2:21 ln Rex cfx =2 0:232 ln Pr þ 14:9Pr0:623 15:6

pffiffiffiffiffiffiffiffiffiffi c =2Rex Pr pffiffiffiffiffiffiffiffiffiffi fx Nux ¼ , 2:21 ln Rex cfx =2 0:232 ln Pr þ 10Pr0:741 6:21

10 < Pr < 500

(22:34a)

(22:34b)

where the local friction factor is cfx 0:0592Rex0:2 . Equation 22.30 is valid for uniform surface temperature, but it may be used for uniform heat flow if the constants 0.3387 and 0.0468 are replaced by 0.4637 and 0.02052, respectively. In laminar boundary layers, the convective heat transfer coefficient with uniform heat flux is approximately 36% higher than that with uniform surface temperature. With turbulent boundary layers, the difference is small and the correlations for the local heat transfer coefficient can be used for both uniform surface temperature and uniform heat flux (Suryanarayana, 2000). 22.3.1.2

Cylinders

Many interesting practical problems on forced convection heat transfer deal with the bodies of curvature complex shape such as cylinders or spheres. The special features of heat transfer on a curved surface manifest themselves in a boundary layer, which can no longer be simply classified as laminar or turbulent boundary layer in the same way as flows over a flat plate. The boundary layer for flows over a cylinder can be laminar or partly laminar and partly turbulent, and then followed by a turbulent wake. Therefore, correlations are not developed based on the condition of the boundary layer, but rather based on a range of Reynolds numbers. For flows over a cylinder, the characteristic dimension is the diameter of the cylinder, Reynolds number is evaluated by using the uniform velocity V1, i.e., the free stream velocity, and all the properties of the fluid are calculated at Tf ¼ (Ts þ T1)=2; the correlation is as follows: Nu ¼


hD ¼ cRem Prn k

(22:35)

Table 22.2 Constants for Equation 22.35 for Air at 0.5 < Pr < 10 Re 4–35 35–5,000

cPr1=3 (Pr ¼ 0.7)

c

m

0.795 0.583

0.895 0.657

0.384 0.471

5,000–50,000

0.148

0.167

0.633

50,000–230,000

0.0208

0.0234

0.814

The values of c, m, and n in Equation 22.35 are given in Table 22.2 (Hilpert, 1933; Morgan, 1975). For Re Pr > 0.2, Churchill and Bernstein (1977) recommend the following correlations (Suryanarayana, 2000): " 5=8 #4=5 0:62Re1=2 Pr 1=3 Re Nu ¼ 0:3 þ , Re > 400,000 1=4 1 þ 282,000 1 þ (0:4=Pr)2=3 " 1=2 # 0:62Re1=2 Pr 1=3 Re , 100,00 < Re < 400,000 Nu ¼ 0:3 þ 1=4 1 þ 282,000 1 þ (0:4=Pr)2=3 Nu ¼ 0:3 þ

0:62Re1=2 Pr1=3 1=4 , Re < 10,000 1 þ (0:4=Pr)2=3

(22:36a)

(22:36b)

(22:36c)

For flow of liquid metals, Ishiguro et al. (1979) suggested the following correlation (Suryanarayana, 2000): Nu ¼ 1:125(RePr)0:413 ,

1 < RePr < 100

(22:37)

Another correlation equation is given by Whitaker (1972, cited by Holman, 1986) as 0:25 m Nu ¼ 0:4Re0:5 þ 0:06Re2=3 Pr0:4 1 ms

(22:38)

Equation 22.38 is valid for 40 < Re < 105, 0.65 < Pr < 300, and 0.25 < m1=ms < 5.2; all properties are evaluated at the free stream temperature except that ms is at the wall temperature. For RePr < 0.2, Nakai and Okazaki (1975) present the following relation (Holman, 1986): h i1 Nu ¼ 0:8237 ln (Re1=2 Pr1=2 )

(22:39)

Properties in Equations 22.36, 22.37, and 22.39 are evaluated at the film temperature. A distinction was made earlier between constant wall temperature and constant surface flux problems. However, in the flow past a cylinder, the equations apply to either case, and the distinction is not so significant (Janna, 2000). 22.3.1.3

Spheres

For flows over spheres, if the characteristic dimension is the diameter of the spheres, all properties are evaluated at T1, except ms at Ts, and if 3.5 < Re < 76,000, 0.71 < Pr < 380, and


1 < m1=ms < 3.2, where m1 is the uniform viscosity, i.e., the free stream viscosity, Whitaker (1972) suggests the following equation for gases and liquids flowing past spheres: Nu ¼ 2:0 þ (0:4Re1=2 þ 0:06Re2=3 )Pr 2=5

m1 ms

0:25 (22:40)

Achenbach (1978) proposed the following equations for Pr ¼ 0.71, where properties are evaluated at (Ts þ T1)=2 (Suryanarayana, 2000): Nu ¼ 2 þ (0:25Re þ 3 104 Re1:6 )1=2 , 3

9

100 < Re < 2 105

Nu ¼ 430 þ 5 10 Re þ 0:25 10 Re 3:1 10 2

17

Re ,

4 10 < Re < 5 10 5

(22:41a)

3

6

(22:41b)

Witte (1968a) from experimental results with liquid sodium recommends the following for liquid metals (Suryanarayana, 2000): Nu ¼ 2 þ 0:386(RePr)1=2 ,

22.3.2

3:6 104 < Re < 1:5 105

(22:42)

Internal Flows

Internal flows refer to flows inside tubes or ducts. In such a flow, a boundary layer also forms near the inside surface of a tube. In the boundary layer, the flow can be laminar or turbulent. However, unlike the boundary layer in external flows, the development of the boundary layer is restricted by the radius of the tube. Furthermore, in the entrance region of the tube, the thickness of the boundary layer increases rapidly, which also causes significant changes in the local convective heat transfer coefficients. When the thickness of the boundary layer becomes the maximum, the thickness then remains unchanged for the rest of the tube. Therefore, the entrance region where the flow velocity profile varies is known as the hydrodynamically developing region, while the remaining region where the velocity profile is invariant is defined as the hydrodynamically fully developed region. Correlations relating the Nusselt number, Reynolds number, and Prandtl number are therefore developed in these two regions, respectively. In these correlations, the inside diameter is taken as the characteristic dimension. 22.3.2.1

Laminar Flows

For flows inside circular pipes, if the pipes are in uniform temperature, and all fluid properties (except ms at Ts) are evaluated at the bulk temperature Tb ¼ (Ti þ Te)=2, i.e., the average temperature between the inlet and exit; Sieder and Tate (1936) proposed the following equation for the average heat transfer coefficient over a length L of the tube in the entrance region: 1=3 0:14 D m Nu ¼ 1:86 RePr L ms

(22:43)

Equation 22.43 is valid for the following conditions: 0.48 < Pr < 16,700, 0.0044 < m=ms < 9.75, (L=D) < (8=RePr)(ms=m)0.42, and Gz > 100, where Gz is the Graetz number ¼ RePr D=L. Equation 22.43 is satisfactory for small diameters and temperature differences. A more general expression


covering all diameters and temperature differences is obtained by including an additional factor 0.87 (1 þ 0.015Gr1=3) on the right-hand side of Equation 22.43 (Knudsen et al., 1999). For 0.1 < Gz < 104, the following equation is recommended (Knudsen et al., 1999): Nu ¼ 3:66 þ

0:14 0:19Gz0:8 m 1 þ 0:117Gz0:467 ms

(22:44)

Stephan (1961, 1962) also proposes the following correlations for the entrance region for uniform surface temperature and uniform heat flux, respectively: Nu ¼ 3:657 þ Nu ¼ 4:364 þ

0:0677ððD=LÞRePr Þ1:33 1 þ 0:1Pr ððD=LÞReÞ0:83 0:086ððD=LÞRePr Þ1:33 1 þ 0:1Pr ððD=LÞReÞ0:83

(22:45a)

(22:45b)

which are valid over the range 0.7 < Pr < 7 or if RePr D=L < 33 also for Pr > 7. For fully developed flow, the convective heat transfer coefficient remains a constant as given below (Janna, 2000): For uniform surface temperature Nu ¼ 3:658

(22:45c)

Nu ¼ 4:364

(22:45d)

For uniform heat flux

For flows inside concentric annular ducts, the annular duct is formed by two concentric tubes, and each tube has a finite wall thickness, and the annular flow area is bounded by the outer diameter of the inner tube Di and the inner diameter of the outer tube Do. Approximate heat transfer coefficients for laminar flow in annuli may be predicted by the following equation (Knudsen et al., 1999): Nu ¼ 1:02Re0:45 Pr 0:5

0:4 0:8 0:14 Dh Do m Gr 0:05 mi L Di

(22:46)

where mi is the viscosity at the inner wall of annulus Dh is the hydraulic mean diameter defined as the ratio of four times the area of cross section of the flow perpendicular to the direction of the flow Ac to the wetted perimeter of the duct Pw, that is: Dh ¼ 4

22.3.2.2

Ac Pw

Turbulent Flows

If flows in the pipes are turbulent, for both uniform surface temperature and uniform heat flux, the following equation can be used, which is valid for 0.7 Pr 16,700, Re 10,000, and L=D 60 (Sieder and Tate, 1936):


Nu ¼ 0:027Re4=5 Pr 1=3

0:14 m ms

(22:47)

For better predicted results, Dittus and Boelter (1930) recommend the following (Janna, 2000): Nu ¼ 0:023Re4=5 Prn , 0:7 Pr 160, Re 10,000, L=D 60

(22:48)

where n ¼ 0.4 for heating (Ts > Tb) and n ¼ 0.3 for cooling (Ts < Tb); properties are evaluated at the fluid bulk temperature. Equation 22.48 is useful when the wall temperature is unknown. For even better prediction accuracy, Gnielinski (1976, 1990) proposes the following equations for 2300 < Re < 106 and 0 < D=L < 1 to cover both the entrance region and the fully developed region: "

2=3 # D , 0:5 < Pr < 1:5 Nu ¼ 0:0214(Re 100)Pr 1þ L " 2=3 # D 0:87 2=5 Nu ¼ 0:012(Re , 1:5 < Pr < 500 280)Pr 1þ L 4=5

2=5

(22:49a)

(22:49b)

For describing the fully developed region, ratio D=L should be set to zero in Equation 22.49. If there exists a large variation in fluid properties due to temperature, Gnielinsky (1990) suggests multiplying Equation 22.49 by (Tb=Ts)0.45 for gases and by (Pr=Prs)0.11 for liquids, where Prs is evaluated at the surface temperature Ts. Equations 22.47 through 22.49 are developed for uniform surface temperature of the tube, but they can also be used for uniform heat flux in turbulent flows. If heat transfers with uniform heat flux, Petukhov (1970) recommends the following correlation for better accuracy with the conditions of 104 < Re < 5106 and 0.08 < m=ms < 40: n ( f =8)RePr m Nu ¼ 1=2 2=3 m 1:07 þ 12:7( f =8) (Pr 1) s

(22:50a)

where f ¼ [0.79 ln(Re) 1.64]2 n ¼ 0.11 for liquids heating n ¼ 0.25 for liquids cooling n ¼ 0 for constant heat flux or for gases (Holman, 1986) Alternative to Equation 22.50a, for uniform heat flux, the following Gnielinski correlation is suggested for 2300 < Re < 5 106 (Suryanarayana, 1995): " 2=3 # f =8(Re 1000)Pr D pffiffiffiffiffiffiffi Nu ¼ 1þ L 1 þ 12:7 f =8(Pr 2=3 1)

(22:50b)

The Petukhov correlation (Equation 22.50a) is similar to the Gnielinksi correlation (Equation 22.50b) and is generally accurate at 104 < Re < 5 106, but Equation 22.50a generally provides slightly higher heat transfer coefficients. For uniform surface temperature, Equation 22.50 can be also used with negligible error for fluids with Pr > 0.7. For turbulent flows in noncircular ducts,


Equations 22.47 through 22.50 are also valid except that the Reynolds number and Nusselt number should be calculated based on the hydraulic mean diameter as the characteristic dimension. For turbulent flows (Re > 2300), and for flows inside concentric annular ducts, Petukhov and Roizen (1964) suggest the following: For heat transfer at the inner tube with outer tube insulated Nu ¼ 0:86ðDi =Do Þ0:16 Nutube

(22:51a)

For heat transfer at the outer tube with inner tube insulated Nu ¼ 1 0:14ðDi =Do Þ0:6 Nutube

(22:51b)

where the characteristic dimension Dh ¼ Do Di is used to determine the Reynolds number and Nusselt number, and all properties are evaluated at the fluid bulk mean temperature (arithmetic mean of inlet and outlet temperatures). If heat transfer occurs at both tubes, which have the same wall temperatures, Stephan (1962) recommends the use of the following correlation (Knudsen et al., 1999): Nu ¼ Nutube

h i 0:86ðDi =Do Þ0:16 þ 1 0:14ðDi =Do Þ0:6 1 þ ðDi =Do Þ

(22:52)

For diameter ratios Di=D0 > 0.2, Monrad and Pelton’s (1942) equation is recommended for either or both the inner and outer tubes (Knudsen et al., 1999): Nu ¼ 0:020Re0:8 Pr1=3

22.3.3

Do Di

0:53 (22:53)

Combined Forced and Natural Convection

In any forced convection, natural convection always plays a part, and therefore the heat transfer is actually a combination of forced convection and natural convection. However, in most of the forced convections, the fluid velocity is high enough so that the contribution of the natural convection is negligible in formulating the correlations. On the other hand, if the velocity is not sufficiently high in forced convection, the contribution from the natural convection becomes significant. In this case, contributions from both the forced convection and natural convection should be considered and consequently different correlations are developed. The magnitude of the dimensionless group Gr=Re2 describes the ratio of buoyant to inertia forces and thus governs the relative importance of free to forced convection (Awuah and Ramaswamy, 1996). The combined free and forced convection regime is generally the one for which (Gr=Re2) 1 (Incropera and DeWitt, 1996). For flows in horizontal tubes, if the tubes have uniform surface temperature, Depew and August (1971) propose the following correlation to account for similar contributions from both natural and forced convections (Suryanarayana, 1995): h i1=3 m 0:14 1=3 0:36 0:88 Nu ¼ 1:75 Gz þ 0:12(GzGr Pr ) , ms


L < 28:4 D

(22:54)

where the Graetz number Gz ¼ mcp=(kL), and all the properties are evaluated at the average bulk temperature, except ms at Ts. If the tubes are supplied with a uniform heat flux, Morcos and Bergles (1975) suggest using the following equation to evaluate the average convective heat transfer coefficient for the following conditions: 3104 < Ra < 106, 4 < Pr < 175, and 2 < hD2=(kwt) < 66 (Suryanarayana, 1995): 8 9 " #2 0:5 < 1:35 0:265 = Gr Pr Nu ¼ 4:362 þ 0:145 ; : P0:25

(22:55)

where P ¼ (kD=kwt) t is the tube wall thickness kw is the tube wall thermal conductivity All the fluid properties are evaluated at (Tw þ Tb)=2 with Tw and Tb being the inside wall temperature and bulk temperature, respectively. 22.4 CORRELATIONS FOR CONVECTION WITH PHASE CHANGE Heat transfer with phase change occurs regularly in food processing. Examples of phase change include boiling, condensation, melting, freezing, and drying processes. Heat transfer to a boiling liquid is very important in different kinds of chemical and biological processing, such as control of the temperature of chemical reactions, evaporation of liquid foods, etc. 22.4.1

Boiling

Boiling takes place on a solid–liquid interface when the temperature at the solid surface is higher than the saturation temperature of the liquid. During boiling, vapor is generated at the interface. Depending on the state of the fluid, boiling can be divided into pool boiling if the fluid is stationary and forced convection boiling if the fluid is in motion. 22.4.1.1

Pool Boiling

In pool boiling, there exist several distinct regimes. In nucleate boiling regime, bubbles are formed in microcavities adjacent to the solid surface. They then grow until they reach some critical size, at which point they separate from the surface and enter the fluid stream. In this regime, high heat flux is achieved at low values of temperature difference between the surface and the saturation temperature of the liquid, i.e., DT ¼ Ts Tsat. The maximum heat flux achieved in this regime is defined as the critical heat flux (CHF), i.e., qmax. In the film boiling regime, as the temperature at the solid surface is much higher than the saturation temperature of the liquid, the layer of liquid closest to the surface turns to vapor; as a result, the surface is completely blanketed by the vapor. Due to the mass exchange occurring at the vapor liquid interface, bubbles of vapor periodically form and migrate upward. There also exists a transition boiling regime where both nucleate boiling and film boiling take place. In this transition regime, with the increase in the temperature at the solid surface, the area of the surface covered by film boiling increases while that with nucleate boiling decreases, with a net decrease in the average heat flux. In the above three regimes, food processing equipment are normally designed to operate in the nucleate boiling regime below CHF to achieve a high heat transfer rate at low DT values. Therefore, most of the correlations developed are focused on this regime and the prediction of CHF.


It is worth noting that pool boiling correlations are generally regarded as being valid for both subcooled (the liquid temperature is below the saturation temperature) and saturated nucleate boiling (the liquid temperature is equal to the saturation temperature); and that at moderate-tohigh heat flux levels, a pool boiling heat transfer correlation developed for one heated surface geometry in one specific orientation often works reasonably well for other geometries and other orientations (Carey, 2000). 22.4.1.1.1

Nucleate Boiling

Rohsenow (1952) developed the following correlation for predicting the heat transfer rate in the nucleate boiling regime (for horizontal wires): q ¼ h(Ts Tsat ) ¼ mLv

gDr s

1=2

cp DT Cs Lv Pr n

3 (22:56)

where all properties are those of liquid except rv, which is the density of the vapor, Dr ¼ r rv, i.e., the density difference from the liquid and the vapor, s is the surface tension of the liquid–vapor interface, Lv is the enthalpy of vaporization of the liquid. Constant n generally equals 1.7, but for water it is equal to 1, while constant Cs is dependent on the combination of the liquid and the material of the surface and its surface finish, which can cause significant deviations from the values predicted by Equation 22.56. Table 22.3 lists the values of the constant Cs for water (n ¼ 1) as suggested by Rohsenow (1952) and Vachon et al. (1968). Equation 22.56 may be used for geometries other than horizontal wires, since it is found that heat transfer for pool boiling is primarily dependent on bubble formation and agitation, which is dependent on surface area and not surface shape (Holman, 1986). For the critical heat flux, Kutateladze (1963) recommends the following for boiling from an infinite horizontal plate: 0:25 CHF ¼ qmax ¼ Cr0:5 v Lv ðsgDr Þ

(22:57)

where the constant C is between 0.12 and 0.18; e.g., Zuber (1958) theoretically estimated C ¼ p=24, Kutateladze (1963) correlated data for C ¼ 0.13, and Lienhard and Dhir (1973) and Lienhard et al. (1973) correlated data for C ¼ 0.15. Stephan and Preuer (1979) also suggest the following equation for nucleate boiling regime:

Table 22.3 Constants Cs and n for Equation 22.56 Surface

Cs

Brass, nickel

0.006

Copper, polished Copper, lapped

0.0128 0.0147

Copper, scored Stainless steel, ground and polished

0.0068 0.008

Stainless steel, Teflon pitted

0.0058

Stainless steel, chemically etched

0.0133

Stainless steel, mechanically polished

0.0132

Platinum

0.013


Nu ¼

0:371 2 0:350 hdA qdA 0:674 rv 0:156 Lv dA2 ar ¼ 0:0871 ðPrÞ0:162 sdA k kTs r a2

(22:58)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where dA is the bubble departure diameter ¼ 0:851bo ð2s=gDrÞ with bo ¼ p=4 rad for water; 0.0175 rad for low-boiling liquids; and 0.611 rad for other liquids. Simplified empirical equations to estimate heat transfer coefficients for water boiling on the outside of submerged surfaces at 1.0 atm absolute pressure are developed (Geankoplis, 1993): For a horizontal surface q < 16 A q 16 < < 240 A

h ¼ 1043DT 1=3 , h ¼ 5:56DT 3 ,

(22:59a) (22:59b)

For a vertical surface q 60DT

14. Non-Newtonian (power law) fluids in turbulent flow (Wilkinson, 1960): 0:14 0:4 Nub ¼ 0:023Re0:8 b Prb ðhb =hw Þ

Reb ¼

DnT u2n av r b , K

h¼K

n1 uav , DT

Prb ¼

Cp Kun1 av ^ exp DE=T ^ , K¼Y n1 DT

^ is a parameter of the equation. where Y 15. Suspension through tube (Orr and Valle, 1954): DT uav rsu 0:8 Cp hsu 1=3 hi 0:14 Nub ¼ 0:027 hsu ksu hiw hsu ¼

hi ð1 «sb =«se Þ1:8

104 < Re < 3 105 where «sb and «se are the volume fraction of solids in the suspension and in the sedimented suspension. 16. Slurries in turbulent flow (Salamone and Newman, 1955): hDT DT uav rsu 0:62 Cl hsu 0:72 kpa 0:05 DT 0:05 Cpa 0:35 ¼ 0:0131 kl hsu kl kl Dpa Cl where subscript l refers to liquid properties. 17. Particles in bundle (Schlunder, 1972): Nubundle ¼ JNusingle particle J ¼ 1 þ 1:5ð1 «FL Þ where «FL is the volume fraction of fluid ranging between 0.4 and 1. 18. Fluids flowing past a staggered tube bank (Whitaker, 1976): 2=3 Nub ¼ 0:4Re0:5 Prb0:4 ðhb =hw Þ0:14 b þ 0:2Reb 80 < Reb < 200, 0:42 < «FL < 0:65 This equation has been applied for air and oil; it can also be applied in packed beds for variable velocity near the surface and bulk. 19. Falling films on the exterior of a vertical tube without evaporation, turbulent regime (Wilke, 1962): h¼

n2l gkl3

1=3

¼ 8:7 103

0:4 0:344 4G nl al hl

This equation has been applied for water and mixtures of water–ethylene glycol.


Table 23.6 Heat Transfer Correlations for Heat Exchangers 1. Plate heat exchanger (Auth and Loiano, 1978, cited by Jackson and Lamb, 1981): ^

^

^ Rebb Prbc^ ðhb =hs Þd Nub ¼ a ^ < 0:85, 0:30 < c ^ < 0:2 ^ < 0:4, 0:65 < b ^ < 0:45, 0:05 < d 0:15 < a 0:33 ðhb =hs Þ0:14 , for laminar flow Nub ¼ a_ Reb Prb Deq =L 1:86 < a_ < 4:5 where L is the plate length. 2. Plate heat exchanger (Dutta and Chanda, 1991): Nub Prb0:33

hb hw

0:14

AD ¼ 0:244 Apr

Reb ¼

1:32

lri 1þ lga

1:216

!1:55 u^ 1þ Re0:30 b 180

2m hDeq Cb hb , Nub ¼ , Prb ¼ lhb kb kb

120 < Reb < 200 where AD is the developed area Apr is the projected area lri is the rise of ribs above the plate surface lga is the gap between plates u^ is the angle of ribs with the flow direction (degree) l is the plate width Deq is the equivalent diameter of the channel between plates These equations have been developed in plate exchangers of length 0.69 m, width 0.21 m, thickness 0.001–0.0023, projected plate area 0.1349 m2, developed plate area 0.1349–0.1415 m2, gap between adjacent plates 0.00176 m, equivalent diameter 0.0024–0.0035 m, ribs to flow direction angle 08–1808, and rise of ribs 0–0.0027 m for water and water–glycerin solution 3. Scraped surface heat exchanger Product side (Skelland et al., 1962): ðDT Dsa Þuav rp Cp hp k DT N 0:62 Dsa 0:55 0:53 hp DT ¼Y n kp hp kp DT uav where Y and k are 0.014 and 0.96, respectively, for cooling viscous liquids; 0.039 and 0.70, respectively, for cooling thin mobile liquids Dsa is the diameter of the shaft; and y is the number of rows of scraper blades Trommelen (1970):

0:5 ^f hp ¼ 1:13 kp rp Cp Ny

where ^f is an empirical correction factor (values are given by Van Boxtel and De Fielliettaz Goethart for different food systems, cited by Rahman, 1995). 4. Jacket side (water cooled system) (Van Boxtel and De Fielliettaz Goethart, 1983, 1984): 0:40 Nub ¼ 18:2 þ 0:0158Re0:8 b Prb

This equation has been developed for flow rate 0.171–0.417 kg=s and jacket side heat transfer coefficient 3220–5572 W=m2 8C.


Table 23.7 Heat Transfer Correlations for Extruders 1. Single-screw extruder (fed with hard wheat flour dough) (Levine and Rockwood, 1986): Nub ¼ 2:2Brb0:79 hou H r Cb pNDsr H K ðpNDsr Þnþ1 , Peb ¼ b , Brb ¼ kb ðTi Ts ÞHn1 kb L 15 < Nub < 50, 15 < Brb < 58

Nub ¼

where H is the tread or screw channel depth Dsr is the diameter of the screw N is the screw rotational rate Ti is the initial temperature Ts is the barrel temperature L is the length of the screw. This correlation characterizes the specific extruder 2. Twin-screw extruder, corotating (Todd, 1988): 2 0:28 D Nr hb Cb 0:33 nb 0:14 Nub ¼ 0:94 sr b hb kb hw hin Dsr Nub ¼ kb 3. Twin-screw extruder (Mohamed and Ofoli, 1989): Nub ¼ 0:0042Gz1:406 Brb0:851 b Nub ¼

hin Dsr , kb

Gzb ¼

mCb Ki L2 g_ anþ1 ^ exp DE=RT , Brb ¼ i kb kb ðTi Tw Þ

where Ki is the reference consistency coefficient Ti is the reference temperature g_ is the average shear rate DÊ is the activation energy

Table 23.8 Heat Transfer Correlations for Freezing 1. Flat plate surface during air freezing (Heldman, 1980): Nux ¼ 0:665Re0:571 , for acrylic x Nux ¼ 0:579Re0:582 , for ground beef x 1 104 < Rex < 5 105 air speed: 0–14 m=s, temperature: 28.48C to 17.88C, plate thickness: 0.945 and 1.89 cm. 2. Belt freezer (air to disk face heat transfer coefficient) (Flores and Mascheroni, 1988): Nub ¼ 10:998Re0:277 b Parallel to belt airflow, material: copper, both surfaces Nub ¼ 7:891Re0:328 b Parallel to belt airflow, material: unpacked hamburger, both surfaces Nub ¼ 4:293Re0:395 b Parallel to belt airflow, material: packed hamburger, both surfaces Nub ¼ 0:412Re0:578 b Transverse to belt airflow (downward), material: copper, both surfaces Nub ¼ 0:326Re0:640 b


Table 23.8 (continued) Heat Transfer Correlations for Freezing Transverse to belt airflow (downward), material: hamburger, both surfaces Nub ¼ 4:190Re0:355 b Transverse to belt airflow (upward), material: hamburger, upper surface only Nub ¼ 1:312Re0:496 b Transverse to belt airflow (upward), material: hamburger, lower surface only Reb ¼

Duav rb hD , Nub ¼ hb kb

20 103 < Reb < 70 103 where D is the disk diameter. 3. Heat transfer due to temperature gradient without vapor condensation in the case of refrigerated food products (Leichter et al., 1976): Nub ¼ 0:860Grb0:25 Nub ¼

gbr2b H3 DT hwoc HT , Grb ¼ kb h2b

where HT is the height of the cylinder and the subscript woc indicates the process without condensation of water vapor. The effect of temperature and relative humidity of atmosphere in the heat transfer coefficient on a cold surface (at 08C) is given by the expressions hwic =hwoc ¼ 0:584 þ 0:0113 %RH, Tb ¼ 15 C hwic =hwoc ¼ 0:760 þ 0:0104 %RH, Tb ¼ 20 C hwic =hwoc ¼ 0:813 þ 0:0112 %RH, Tb ¼ 25 C hwic =hwoc ¼ 0:879 þ 0:0123 %RH, Tb ¼ 30 C where %RH is the percent relative humidity, ranging from 40% to 100%, and the subscript wic indicates the process with condensation of water vapor.

Table 23.9 Heat Transfer Correlations for Sterilization—Retort Canning 1. Retort canning (sterilization) (Merson et al., 1980) 1=3

1=3 Nub ¼ 0:17Re0:52 b Prb ðHc =eÞ

Reb ¼

Dr pDc Nrb hov Dc Cb hb , Nub ¼ , Prb ¼ hb kb kb

where hov is the overall heat transfer coefficient Nc is the can rotational speed Hc is the can height Dc is the can diameter e is the head space height with can vertical; bulk fluid properties are evaluated at the average bulk temperature between the start and the end of the process. 2. Steam heated steritort (container–fluid interface for fluids without particles) (Lenz and Lund, 1978): 0:08 Nub ¼ 115 þ 15Re0:3 b Prb

Reb ¼

ðDr =2Þ2 Nrb hin Dr C b hb , Nub ¼ , Prb ¼ hb 2kb kb

This equation has been applied for reel speed: 3.5–8 rpm, «l: 0.32–0.45, steam temperature 1218C, Dp: 0.97–3.81 cm, container size: 303 406 and 608 700. (continued)


Table 23.9 (continued) Heat Transfer Correlations for Sterilization—Retort Canning 3. Steam heated steritort, container–fluid interface for fluids with spherical particles (Lenz and Lund, 1978): Dp 0:14 Nub ¼ 33 þ 53Re0:28 b Prb Dr =2ð1 «l Þ Reb ¼

ðDr =2Þ2 Nrp hDr Cp hb , Nub ¼ , Prb ¼ hb 2kp kp

This equation has been applied in conditions similar to the previous case. 4. Steam heated retort and axially rotated cans (Soule and Merson, 1985): 0:356 0:154 hb 0:278 Hc Nuw ¼ 0:434Re0:571 Pr w w Dc hw hin ¼ 1:07hov Rew ¼

D2c Nc rw , hw

Nuw ¼

hov Dc Cw hw , Prw ¼ kw kw

12 < Rew < 4:4 104 , 2:2 < Prw < 2,300, 1:11 < Hc =Dc < 1:61, 1:22 < hb =hw < 1:79 where hin and hov are the internal film and overall coefficient, respectively Nc is the can rotational speed; wall fluid properties are evaluated at the arithmetic average of the initial and final temperatures of the can wall; head space 1.0 cm, vacuum 20 in Hg, Nc: 0–2.5 rps 5. Steam heated, axially rotating cans (Deniston et al., 1987): !0:126 ap B0:53 F0:171 Nub ¼ 1:87 104 Re1:69 p vD2p rp rb v2 Dcx þ 2g Dp _ b 6cr Dcx v2 Dcx Hce vD2cx F ¼ 1 «p Dci ab Dp up r b hov Dcx Arb Nr Dr Rep ¼ , Nub ¼ , c_ ¼ , Nc ¼ hb kb 4Re2p Dcx

3 2 4Dp rb rp rb g þ v ðDcx =2Þ Arb ¼ 3h2b B¼

where c_ is the drag coefficient Arb is the Archimedes number Hce is the effective can length (internal can length: head space) Dci and Dcx are the internal and external can diameter, respectively v is the can angular velocity (2pNc) Nc is the can rotational speed Nr is the reel rotation 6. Steritort (concentrated tomato 7.28Brix) (Deniston et al., 1992): 0:069 Hcx Pss 0:806 Nub ¼ 0:703 þ 0:417Re1:103 Prb0:324 b Dcx PTO C p hp D2cx Nc rb hov Dcx Reb ¼ , Nub ¼ , Prb ¼ hp kp kp 1:99 < Reb < 5:38, 23,500 < Prb < 34,800, 1:12 < Hcx =Dcx < 1:51, 0:75 < Pss =PTO < 1:00 where Hcx is the can height Dcx is the external can diameter Pss is the absolute steam pressure PTO is the absolute total pressure hov is the overall heat transfer coefficient based on total external surface (heating medium [can wall] internal fluid); reel rotation speed: 4–10 rpm, air: 0%–25%, can size: 211 300, 300 407, 307 503, Tre ¼ 121.18C, F: 60.9–84.9


Table 23.9 (continued) Heat Transfer Correlations for Sterilization—Retort Canning 7. Steritort (particle–fluid heat transfer coefficient for canned snap beans) (Fernandez et al., 1988): Prb0:33 C6:98 Nub ¼ 2:69 104 Re0:294 b 2=3 6V p ppa C¼ Apa Dc Nc rb hpa Dpa Cb hb Reb ¼ , Nub ¼ , Prb ¼ hb kb kb Beans mass: 164–244 g, reel speed: 2–8 rpm, steam temperature 115.68C, can size: 303 406, head space: 0.64 cm. 8. Steritort (cans containing non-Newtonian liquids) (Rao et al., 1985): Nub ¼ 2:60ðGrb Prb Þ0:205 þ 7:15 107 ½Reb Prb ðDc =Hc Þ1:837 Reb ¼

D2c N2n c rb n , 8n1 K 3nþ1 4n

Nub ¼

n gD3c r2b bDT hov Dc C b hb K8n1 3n þ 1 , Prb ¼ , Grb ¼ , hb ¼ 1n hb 4n kb kb Nc

0:4 < Reb < 96, 205 < Prb < 5100, 24 < Nub < 160, 100 < Grb < 5:2 105 , 1:13 < Hc =Dc < 1:37 This model has been applied for aqueous guar gum solutions (i. 0.3%, K ¼ 0.041, n ¼ 0.58, ii. 0.4%, K ¼ 0.077, n ¼ 0.69, iii. 0.5%, K ¼ 0.178, n ¼ 0.63, iv. 0.75%, K ¼ 0.922, n ¼ 0.58) and glycerin (K ¼ 0.082, n ¼ 1.00); Tsteam: 1108C–115.68C, e ¼ 6.35 mm, steam heated retort, axially rotated can. 9. Steretort (cans containing Newtonian liquids) (Anantheswaran and Rao, 1985a): Prb0:287 Nub ¼ 2:9Re0:436 b Reb ¼

D2r Nr rb hin Dr Cb hb , Nub ¼ , Prb ¼ hb kb kb

11 < Reb < 2:1 105 , 2:8 < Prb < 498, 0:73 < Hcx =Dcx < 1:37 where Nr is the speed of rotation Dr is the diameter of rotation or reel diameter; end-over-end rotation in steam at atmospheric pressure steretort, can size: 303 406, Nr: 0–38.6 rpm, Dr: 0–29.8 m, headspace volume: 3%–9%. 10. Steretort, cans containing non-Newtonian liquids (Anantheswaran and Rao, 1985b): Prb0:355 Nub ¼ 1:41Re0:482 b Reb ¼

D2r N2n rb r , 8n1 KVn

Nub ¼

hin Dr Cb K8n1 Vn , Prb ¼ , 0:73 < Hcx =Dcx < 1:37 kb kb N1n r

70 < Reb < 1:2 104 , Prb ¼ 48, for aqueous guar gum solution 0.3% 4 < Reb < 1:17 103 , 508 < Prb < 953, for aqueous guar gum solution 0.4% 1 < Reb < 478, 1250 < Prb < 2800, for aqueous guar gum solution 0.5% 0:1 < Reb < 46:2, 1:66 104 < Prb < 5:7 104 , for aqueous guar gum solution 0.75% end-over-end rotation in steam retort, can size: 303 406, Nr: 0–38.6 rpm, Dr: 0–29.8 m 11. Retort-canned gelatinized starch (Ramaswamy et al., 1993): ln hov ¼ 5:99 2:72 103 Tre þ 3:76 102 ðNr =60Þ 3:55hini ln Nu ¼ 4:604 þ 152:26Fr 3:29 104 hinsp n1 n 1n K8 V =Nr hov Dr Dr N2r Nu ¼ , Fr ¼ , hinsp ¼ kb g hwater where Fr is the rotational Froude number hini is the initial apparent viscosity of the product hinsp is the specific apparent viscosity; end-over-end rotation of can, product concentration: 3%–4%, reel rotation speed: 10–20 rpm, can headspace: 6.4–12.8 mm, Tre: 1108C–1308C, heating medium: steam (75%)–air (25%) mixture (continued)


Table 23.9 (continued) Heat Transfer Correlations for Sterilization—Retort Canning 12. Flame sterilization (flame heating of CMC solutions) (Teixeira Neto, 1982): 0:60

Pr Re0:68 Nu ¼ 0:433Re0:56 ci co ci

c rai Dr pDc Nrci Dc u hov Dc Cci hci Reci ¼ , Reco ¼ , Nu ¼ , Prci ¼ hci hai kci kci where c is the cylinder surface peripheral velocity u rai is the air density, hai is the air viscosity Dc is the can diameter Dr is the reel diameter, and subscripts ci and co refer to internal and outside of the can, respectively 13. Liquids heating in a can with different modes of outer heating (Duquenoy, 1980): Nu ¼ 17:0 105 Re1:449 Pr 1:190 We0:551 Fr 0:932 «0:628 p Re ¼

C p hp vDc 2Dc Hc hin Dc pv2 H2 Dc , Nu ¼ , Pr ¼ , We ¼ , 2hp Dc =2 þ Hc 2kp kp sp

Fr ¼

Dr N2r , v ¼ 2pN g

2:07 < Re < 9:91, 0:8 < Pr < 8:06, 2:73 < Nu < 5:05 where hin is the inside heat transfer coefficient v is the speed of rotation (rad=s), «p is the product volume fraction compared to total volume of can, We is the Weber number, and sp is the surface tension of liquid product (N=m); Ti: 208C–608C, v: 0.21–4.19 rad=s, Dc: 13.75–24.25 mm, H: 45.4–55.7 mm, kp: 0.118–0.670 W=m, rp: 792–1150 kg=m3, Cp: 1448–4332 J=kg, Vp: 0.75–0.99.

Table 23.10 Heat Transfer Correlations for Sterilization—Aseptic Canning 1. Aseptic canning based on slip velocity (Kramers, 1946): 0:31 Nub ¼ 2:0 þ 1:3Prb0:15 þ 0:66Re0:5 s Prb

2. Aseptic canning based on slip velocity (Ranz and Marshall, 1952): 0:33 Nub ¼ 2:0 þ 0:6Re0:5 s Prb

3. Aseptic canning based on slip velocity (Whitaker, 1972): 2=3 Nub ¼ 2:0 þ 0:4Re0:5 Prb0:4 ðhb =hs Þ0:25 s þ 0:06Res

upa ul Dpa r l Res ¼ hl where upa and ul are the particle and liquid fluid velocity, respectively. 4. Aseptic canning (Sastry et al., 1990): Nu ¼ 26:81 þ 0:00455Repa Dpa =DT 1:71 0:64 Dpa =DT Frpa Nu ¼ 6:023 106 Re1:79 pa Nu ¼ 0:0046Repa þ 41:54 Dpa =DT 35:65Frpa 5:24 Repa ¼

Dpa uav rb hpa Dpa u2 h av i , Nu ¼ , Frpa ¼ hb kb gDpa rp =rb 1

3600 < Repa < 27300, 0:39 < Frpa < 14:83, 0:2618 < Dpa =DT < 0:6273 Dpa: 1.33–2.39 cm, flow rate: 1.33 104 – 7.98 104 m3=s.


Table 23.10 (continued) Heat Transfer Correlations for Sterilization— Aseptic Canning 5. Aseptic canning (Chandarana et al., 1990): 1:94 hpa ¼ 1:14 104 Re0:07 pa F

where F is the surface area to volume ratio of the particle. 6. Aseptic processing (shear flow) (Mwangi et al., 1993): Nub ¼ 0:184Re0:58 b , laminar flow, solid fraction: 2.12% Nub ¼ 0:195Re0:74 b , laminar flow, solid fraction: 3.22% Nub ¼ 0:0078Re0:95 b , turbulent flow, solid fraction: 1.22% Nub ¼ 0:034Re0:80 b , turbulent flow, solid fraction: 3.22% 0:33 Nub ¼ 0:100Re0:58 , laminar flow, Reb < 563, 3.68 < Prb < 5.52 b Prb 0:32 Nub ¼ 0:0336Re0:80 , turbulent flow, 563 < Reb < 6000, 3.68 < Prb < 5.52 b Prb

Dpa: 8.0–12.7 mm, Dpa=DT: 0.156–0.25

Table 23.11 Heat Transfer Correlations for Flow through Packed Beds 1. Forced convection through packed bed (Yoshida et al., 1962): Reb < 50 j ¼ 0:91Re0:51 b Reb > 50 j ¼ 0:61Re0:41 b h Cb hb 2=3 G , Reav ¼ j¼ kb Cb G Asp hav where G is the superficial mass velocity (kg=m2=s) is the particle shape factor (Gamson presented the following values of this factor for packed bed correlations: 1.00 for sphere, 0.91 for cylinder, 0.86 for flake, 0.80 for berl saddle, 0.79 for rasching ring, 0.67 for partition ring) Properties’ values are calculated in temperature, which is the arithmetic mean of inlet T0 and outlet Te temperature. It should be noted that the heat transfer rate from particle to fluid in a packed bed can be calculated by the following equation: q_ ¼ hAsp Acs HðT0 Te Þ where Acs is the sectional area of the bed (m2) Asp is the total particle surface area per unit volume of the bed (m2=m3) H is the bed height (m) 2. Packed bed of sphere and liquid (Rohsenow and Choi, 1961): 0:33 Nub ¼ 0:8Re0:7 b Prb

3. Packed bed during drying (Hallstrom et al., 1988): Nupa ¼ 0:3Re1:3 pa Repa ¼

Dpa uap rb hpa Dpa , Nupa ¼ hb c

where c is the mass transfer coefficient. 4. Packed bed (particle-to-fluid heat transfer coefficient) (Whitaker, 1976): 2=3 Nub ¼ 0:4Re0:5 Prb0:4 b þ 0:2Reb Reb ¼

Dpa G , hb ð1 «FL Þ

Nub ¼

hDpa «FL 6Vpa , Dpa ¼ kb 1 «FL As

3:7 < Reb < 8000, Prb ¼ 0:7, hb =hw ¼ 10:34 < «FL < 0:74 where As is the particle surface.


Table 23.12 Heat Transfer Correlations for Flow through Fluidized Beds 1. Fluidized bed (bed-to-wall heat transfer) (Levenspiel and Walton, 1954): hw Dec Dpa Gai Dpa 1:16 ¼ 0:0018 kai hai DT 0:65 hw Dec Gec ¼ 10 Cai Gai hai hw Dpa Gai 0:7 ¼ 0:6 Cai Gai hai Dec ¼

pDpa , 6ð1 «ai Þ

Gec ¼

Gai «ai

where «ai is the air volume fraction and the subscript ec refers to effective; Dp: 0.35–4.33 mm. 2. Fluidized bed, bed-to-cylindrical surface heat transfer coefficient during freezing (Sheen and Whitney, 1990): Nub ¼ 37:47Re0:222 ðHi =Hs Þ0:436 Dp =Hs b Reb ¼

Hs uap rb hw H s , Nub ¼ hb kb 32,000 < Reb < 83,000, 0:46 < Hi =Hs < 0:92, 0:05 < Dpa =Hs < 0:07

where uap is the superficial air velocity Hs is the height of the cylindrical heat transfer surface Hi is the initial height of the bed and the physical properties are those of air 3. Fluidized bed (fluid-to-particle heat transfer coefficient) (Wen and Chang, 1967): Ar 0:198 j ¼ 0:097Re0:502 pa 2=3 hpa Cg hg j¼ , Cg Gg kg

Repa

Dpa Gg ¼ at minimum fluidization, hg

Ar ¼

2 gD3so rso rg h2g

where subscripts so and g refer to solid and gas, respectively. 4. Fluidized bed drying, fluid-to-particle heat transfer coefficient in the case of rotary stream (Shu-De and FangZhen, 1993): ^j 0:790 0:42Re1:285 Nu ¼ 0:937 þ 0:133Re0:622 T T ReT ¼

DT uap rg , hg

Nu ¼

hpa Dpa kg

12 < ReT < 189, 0:0863 < ^j < 0:17 where ^j is the mass flow ratio of solid to gas DT is the inside diameter of the drying chamber uap is the gas velocity; inlet gas temperature: 1008C, DT: 15.0 cm, chamber height: 1.5 m 5. Fluidized bed drying, air-to-particle heat transfer coefficient (Ramirez et al., 1981): Dpa 0:71 Dpa 0:94 0:33 Nu ¼ 4:948 103 Re1:14 Pr pa H DT Dpa uav rb hDpa Repa ¼ , Nu ¼ hb kb 0:28 < Repa < 3:0, 0:007 < Dpa =DT < 0:012, 0:0016 < Dpa =H < 0:0152, 0:39 < H=DT < 1:49 Dpa: 0.0125–0.035 cm, minimum fluidization velocity: 0.4113–1.67 cm=s, H: 2–8 cm, airflow rate: 5.3–20.29 L=min.


Table 23.12 (continued) Heat Transfer Correlations for Flow through Fluidized Beds 6. Fluidized bed, total heat transfer coefficient, particle-to-fluid and particle-to-particle (Wen and Chang, 1967): 0:038 0:0529Re0:552 Cr 0:236 pa Ar «g Dpa Gg hpa Dpa Cso rso ¼ , Nu ¼ , Cr ¼ hg 2kso Cg rg

Nu ¼ Repa

300 < Repa < 5,000, 5:86 106 < Ar < 2:55 107 , 30:6 < Nu < 89:4 where rg and rso are the apparent density of air and grain, respectively ua is the superficial air velocity Gg is the mass flow rate of air «g is the porosity of fluidized bed; Dpa: 4.76–6.35 mm, «g: 0.425–0.898, Gg: 1.557–16.637 kg=(m2 s), air temperature: 528C–1508C 7. Fluidized bed: total heat transfer coefficient (Shirai et al., 1965): 0:36 u h ¼ 7:4 D0:06 D0:27 pa T umin where h is the particle-to-fluid and particle-to-particle heat transfer coefficient umin is the minimum fluidizing velocity; Dpa: 4.50–8.20 mm, DT: 7.99–28.01 cm, umin: 7.99 103– 1.05 101 m=s, u=umin: 2–8

Table 23.13 Heat Transfer Correlations for Mixed Convection 1. Mixed convection from an immersed solid (fluid-to-cylindrical particles heat transfer coefficient during heating in non-Newtonian fluid) (Awuah et al., 1993): Nub ¼ 2:45ðGrb Prb Þ0:108 ,

for carrot

Nub ¼ 2:02ðGrb Prb Þ0:113 ,

for potato

Nub ¼

hDpa , Reb ¼ kb

Dnpa u2n av rb 8n1 KVn

, Prb ¼

gbr2b D3pa ðTb Ti Þ Cb KVn 2n3 n1 , Grb ¼ n 2 KV 2n1 kb Dpa =uav 1n n1 4uav D

3 103 < Grb Prb < 6 104 where Tb is the temperature of the bulk fluid Ti is the initial temperature of the solid; Tb: 508C–808C, fluid: CMC (0%–1%), fluid flow parallel to the length of the cylinder (upward and downward), uav: 0.2 103–0.7 103, Dpa: 0.016–0.023 m 2. Mixed convection in a horizontal tube (Colburn, 1933): h 0:14 1=3 b Nub ¼ 1:75Gzb 1 þ 0:015Gra hw 3:7 103 < Gra < 3:0 108 , 0:76 < Pra < 160 This equation has been applied for water, air, and light oil; L=DT: 24–400, Gr number is calculated for the average conditions between surface and bulk. 3. Flow in horizontal tubes where natural convection is significant (Eubank and Proctor, 1951; McAdams, 1954): 8 9 " 0:4 #1=3 = 0:14 < DT hb Nub ¼ 1:75 Gzb þ 12:6 Prw Grw : ; hw L 3:3 105 < Grw Prw < 8:6 108 , 12 < Gzb < 4,900, 140 < Prw < 15,200 where subscript w refers to wall; L=DT: 61–235. This equation has been applied in petroleum oil. (continued)


Table 23.13 (continued) Heat Transfer Correlations for Mixed Convection 4. Mixed convection in pseudoplastic fluids in horizontal tubes of circular cross section (Metzner and Gluck, 1960): 8 9 " 0:14 0:4 #1=3 = < DT Kb Nub ¼ 1:75 Gzb þ 12:6 Prw Grw V1=3 : ; L Kw 0:32 < n < 0:75, Gzb < 1,000 This equation has been applied for applesauce, banana puree, ammonium alginate, and carbopol. 5. Mixed convection for non-Newtonian fluid in a horizontal tube (Oliver and Jenson, 1964: Rodriguez-Luna et al., 1987): h i1=3 K 0:14 b Nub ¼ 1:75 Gzb þ 0:0083ðPrw Grw Þ0:75 Kw 6. Mixed convection for Newtonian fluid in a horizontal tube (Jackson et al., 1961; Oliver, 1962): h i1=6 Nub ¼ 2:67 Gz2b þ 0:00872 ðPrw Grw Þ1:5 based on logarithmic mean temperature between wall and bulk h i1=3 h 0:14 b Nub ¼ 1:75 Gzb þ 0:0083ðPrw Grw Þ0:75 hw based on arithmetic mean temperature between wall and bulk 1:57 106 < Grw < 3:14 106 , 33 < Gzb < 1,300, Prw < 0:71, L=DT ¼ 31 7. Mixed convection in horizontal tube (combined heat transfer) (Oliver, 1962): h i1=3 h 0:14 b Nub ¼ 1:75 Gzb þ 5:6 104 ðPrb Grb L=DT Þ0:70 hw based on arithmetic temperature difference between fluid and wall 1:0 104 < Grw < 1:1 105 , 7 < Gzb < 11, 1:9 < Prw < 3:7 (water) 4:9 104 < Grw < 1:6 105 , 24 < Gzb < 187, 4:8 < Prw < 7:0 (ethyl alcohol) 29 < Grw < 64, 20 < Gzb < 176, 62 < Prw < 326 (glycerol–water) Fluid properties are calculated at the average inlet and outlet bulk temperature. 8. Mixed convection in a horizontal tube (Brown and Tomas, 1965): h i1=3 Nub ¼ 1:75 Gzb þ 0:012ðGzb Grb Þ1=3 based on arithmetic mean temperature 7:1 104 < Grb < 8:9 105 , 19 < Gzb < 112, 3:5 < Prb < 7:4, 36 < L=DT < 108 This equation has been applied for water. 9. Mixed convection in a horizontal isothermal tube (Depew and August, 1971): 0:88 1=3 h 0:14 1=3 b Nub ¼ 1:75 Gzb þ 0:12 Gzb Grb Prb0:36 hw based on arithmetic mean temperature 0:7 105 < Grw < 5:8 105 , 25 < Gzb < 338, 5:7 < Prw < 8:0 (water) 2:7 105 < Grw < 9:9 105 , 36 < Gzb < 712, 14:2 < Prw < 16:1 (ethyl alcohol) 510 < Grw < 900, 53 < Gzb < 188, 328 < Prw < 391 (glycerol–water) L=DT ¼ 28:4 10. Mixed convection for air flowing in a horizontal isothermal tube (Yousef and Tarasuk, 1982): 0:882 1=3 h 0:14 1=3 b Gr Region I: Nub ¼ 1:75 Gzb þ 0:245 Gz1:5 b b hw 1 104 < Grb < 8:7 104 , 20 < Gzb < 110, 0:0073 < j < 0:040


Table 23.13 (continued) Heat Transfer Correlations for Mixed Convection 0:14 hb Region II: Nub ¼ 0:969Gz0:82 b hw 0:8 104 < Grb < 4 104 , 3:2 < Gzb < 20, 0:04 < j < 0:25 Region III: Nub ¼ 2:0 0:8 104 < Grb < 4 104 , Gzb < 3:2, j < 0:25 where j ¼ x=(Reb Prb DT ) and x is the distance in x-axis Region I corresponds to the area near the tube inlet Region II is further downstream Region III is the tube area far from its inlet Heat transfer coefficient and Grashof number are calculated on logarithmic mean temperature difference. 11. Laminar mixed convection for non-Newtonian foods in a horizontal tube (Rodriguez-Luna et al., 1987): 0:14 h i1=3 Kb V1=3 Nub ¼ 1:75 Gzb þ 0:0083ðPrw Grw Þ0:75 Kw Nub ¼

hDT hDT , Nub ¼ , kb kb

Prw ¼

Cw hw gbr2w D3T DT , Grw ¼ kw hw

0:8 104 < Grb < 4 104 , 500 < Gzb < 7000, 500 < Gzb < 7,000, 0:18 < n < 0:63, Tw ¼ 708C, 3n þ 1 1=3 L < 1:21, 18:8 < < 126:6 1:01 < ðKb =Kw Þ0:14 < 1:08, 1:06 < V1=3 ¼ 4n DT where L is the tube length and subscripts b and w refer to bulk and wall conditions, respectively. 12. Combined forced and natural convection in a horizontal cylinder (water flow parallel) (Kitamura et al., 1992): 0:14 0:07 Num Grm Num Grm ¼ 1:18 and ¼ 1:62 Nuf Num Re2 Nun Num Re2 0:55 0:37 Nun ¼ 0:586Ra0:2 , Nu ¼ 0:364Re Pr f m a a hDT gbqs r2a D4T Ca ha Nu ¼ , Grm ¼ , Pra ¼ , Ram ¼ Pra Grm kb ka h2a ka Grm 105 < Ram < 109 , 100 < Rea < 2,000, 0:3 < < 1,000 Num Re2a 13. Mixed convection from a rectangular box to a cold fluid (laminar flow range) (Hanzawa et al., 1991): Hbo 0:60 Nub ¼ 9:2Re0:088 Grb0:14 b l luav rb hl gbr2b l3 ðTs T0 Þ Reb ¼ , Nub ¼ , Grb ¼ hb kb h2b 100 < Reb < 500, 4 105 < Grb < 2 107 where l is a characteristic length (i.e., space between heated surface and box) Hbo is the box height; Ts: 408C–808C, Ti: 138C–218C, Hbo=l: 6.25–15.0, w ¼ 1808 Table 23.14 Heat Transfer Correlations for Convection with Phase Change 1. Film condensation outside a horizontal tube (Bird et al., 1960): 3 2 1=3 gkcv rcv L or h ¼ 0:954 mhcv m Re ¼ < 1,000 Lhcv

1=4 gkcv lv r2cv h ¼ 0:725 hcv DT ðTd Tw Þ

where Td is the dew point m=L is the mass rate of condensation per unit length of tube lv is the latent heat of vaporization, and the subscript cv refers to condensing vapor. Physical properties are calculated at the average film temperature of the condensed vapor (continued)


Table 23.14 (continued) Heat Transfer Correlations for Convection with Phase Change 2. Film condensation outside a vertical tube or wall (Bird et al., 1960): pffiffiffi 3 2 1=3 1=4 3 4 gkcv rcv 2 2 gkcv lv r2cv or h ¼ h¼ hcv LðTd Tw Þ 3 3Ghcv 3 m G¼ pDT where G is the total rate of condensate flow from the bottom of the condensing surface per unit width of the surface. This equation is applied for relatively short tubes (up to 15 cm long). 3. Film condensation outside a vertical tube or wall (turbulent flow) (Bird et al., 1960): 3 1=2 gkcv rcv HðTd Tw Þ h ¼ 0:033 lv h3cv

3 2 1=3 gkcv rcv G or h ¼ 0:021 , for small DT h3cv

4. Nucleate boiling of liquid from a hot surface (case of pool boiling) (Rohsenow, 1951): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!0:33 q_ Cl ðTs Tsv Þ 1 q_ s Cl hl 1:7 ¼Q kl lv hl lv As As gðrl rv Þ where q_ is the heat flow rate As is the surface area Ts is the surface temperature Tsv is the saturation temperature of the liquid s is the interfacial tension between liquid and vapor Q is the Rohsenow constant dependent on fluid and heating surface and is given in Table 24.17. Fluid properties are evaluated at the saturation temperature corresponding to local pressure It should be noted that the heat flow rate is related to heat transfer coefficient through the equation q_ ¼ hAs ðTs Tsv Þ Also, the maximum flux in nucleate boiling is given by the equation (Zuber, 1958) q_ p r 1=2 1=4 lv r1=2 ¼ 1þ v v ½gs ðrl rv Þ As max 24 rl or by the equation (Rohsenow and Griffith, 1956) 0:25 ^ q_ g rl rv 0:6 ¼ 0:0121rv lv As max g rv ^ is the induced gravitational field such as in the case of centrifuge. where g 5. Nucleate boiling of liquid from a hot surface (case of pool boiling) (Kutateladze, cited by Jackson and Lamb, 1981): 1=2 0:7 q_ h s P s Cl hl 0:35 ¼ 0:0007 al lv rv s gðrl rv Þ kl kl gðrl rv Þ where P is the absolute pressure. 6. Nucleate boiling both inside and outside of tubes (Gilmour, 1959): ! ^ 0:3 h Cl hl 0:6 gsrl 0:425 DT G ¼ 0:001 ^ kl P2 hl Cl G m r v l ^ ¼ G As rv where mv is the vapor mass flow rate. 7. Nucleate boiling of liquid from a hot surface (Palen and Taborek, 1962): _ T 0:69 PDT 0:31 rl rv 0:31 hDT Cl hl 0:69 qD ¼ 0:225 kl kl hl lv s rv


Table 23.14 (continued) Heat Transfer Correlations for Convection with Phase Change 8. Stable film boiling on a submerged horizontal tube (Breen and Westwater, 1962): 1=4 xhv DT x ¼ 0:59 þ 0:069 h gqrv kv3 ðrl rv Þ DT x ¼ 2p

2 1=2 s Cv , q ¼ lv 1 þ 0:34 DTv lv gðrl rv Þ

where DTv is the temperature drop across vapor film x is the wavelength of the smallest wave growth on a flat horizontal surface 9. Film boiling of liquid nitrogen droplet on the food surface (cryogenic freezing) (Baumeister et al., 1966): _ 1=12 for small spherical drops hav ¼ 1:2KV i € 1=4 þ 1:2KV _ 11=12 3:89 108 K i for large drops Vi ! 1=2 1=2 1=2 3 € ¼ g rl rv sl l kv , l* ¼ lv 1 þ 7 Cv DT K hv DT 20 lv

€ 1=4 V 1:29K i

5=6

hav ¼ K_ ¼

grl rv l kv3 hv DT

1=4 ,

where DT is the mean temperature difference between droplet and food surface Di is the initial average diameter (mean of major and minor axes of spheroid) of droplet Vi is the initial volume hav is the average heat transfer coefficient between droplet and food surface 10. Single nitrogen droplet film boiling on a food surface (Awonorin and Lamb, 1988): 0:25 0:33 1:75 0:17 grl rv Cv D3i kv h2v Cv lv rv hv Cv hav ¼ 13:5 Di kv2 DT hv kv rl kv 2 kv DT < 2:2 0:1 < Di < 2:5 mm, 360 < rl =rv < 668, 0:9 < 2 hv Cv lv

Table 23.15 Heat Transfer Correlations for Evaporation 1. Evaporating liquid films from a surface of water falling films flowing along the outside surface of a vertical tube (Chun and Seban, 1971): 1=3 3 1=3 1=3 gkl 4 4G , laminar region 3 hl n2l 3 1=3 0:22 gkl G , laminar region when surface ripples exist h ¼ 0:606 hl n2l 2 1=3 0:4 0:65 nl nl 3 4G h¼ ¼ 3:8 10 , turbulent regime al hl gkl3 0:4 G h ¼ 11:4 103 , for Prandtl number of 5 and a high hl

h¼

Reynolds number G¼

grl d3 nl 3

where d is the film thickness nl is the kinematic viscosity of the liquid al is the thermal diffusivity of the liquid These properties are given by the equations (continued)


Table 23.15 (continued) Heat Transfer Correlations for Evaporation nl ¼ hl =rl , al ¼ kl =rl Cl G expresses the mass flow rate per unit width of the wall. Laminar-layer surface presents capillary waves when (Kapitza, cited by Dukler, 1960) 4 1=11 4G gh ¼ 2:43 3 l s rl hl The separation of the laminar and turbulent regime starts at the wavy laminar regime at Reynolds number 1:06 4 1=3 gh 4G nl ¼ 5800 ¼ 0:215 3 l al s rl hl 2. Climbing film evaporator (Jackson and Lamb, 1981): n_ hmix 1 € ¼c hl X 0:5 0:1 rv hl 1 x0 0:90 X¼ rl hv x0 € < 7:55, 0:328 < n_ < 0:75 2:17 < c where hl is the heat transfer coefficient of a single liquid flow at the same condition (given by Dittus–Boelter equation for single flow of liquid) c_ and n_ are constants X is the Lockhart–Martinnelli parameter, and x0 is the mass fraction of vapor in the mixture 3. Climbing film evaporator (Bourgois and Le Maguer, 1983a,b, 1987): Pei ^ ^ (B þ 2CZ), liquid zone 4 1=3 2=3 Nuz ¼ 8:5Re0:2 , boiling zone a Pra S

Nuz ¼

hz ¼ 615:5 112:0Z þ 41:55Z 2 , boiling zone ^ ¼ 0:458 þ 9,445 þ 5:22 10 B Pei Pe2i

6

^ ¼ 0:238 þ 755 þ 1:350 10 C Pei Pe2i

9

Z ¼ z=HT hz HT 4G h Ci Nuz ¼ , Pei ¼ Rei Pri , Rei ¼ , Pri ¼ i , liquid zone ki ki phi DT hz z zul rl hl Cl uv Nuz ¼ , Rea ¼ , Pra ¼ , S ¼ , boiling zone hl kl ul kl where HT is the height and DT is the diameter of the tube z is the local distance in the tube S is the slip ratio, and subscripts i and l refer to inlet conditions of liquid and liquid, respectively System at atmospheric pressure, steam at 1.357 105 Pa absolute, liquid flow rate ranging from 0.692 102 to 1.30 102 m=s. All properties of the liquid are calculated at the average of bulk and wall temperatures at the local position z. Nuz expression for the boiling zone is valid when pffiffiffiffiffi uv r v uv ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2:5 gDT ðrl rv Þ where uv is the volumetric vapor flux uv is the modified volumetric vapor flux These equations have been applied for sucrose solutions and tomato juice.


Table 23.16 Heat Transfer Correlations for Dryers 1. Thin layer air-drying (McCabe and Smith, 1976): h ¼ 0:0204G0:8 ai , parallel flow h ¼ 1:17G0:37 ai , perpendicular flow where Gai is the air mass velocity. 2. Drum drying (evaporating vapor film heat transfer coefficient) (Hougen, 1940): hv ¼ 8:484 108 u0:8 dr lv DP=DT where udr is the drum speed DP is the difference between saturation vapor pressure and vapor pressure of air DT is the vapor film temperature difference 3. Spray drying (air film heat transfer coefficient of the droplet in hot air) (Ranz and Marshall, 1952): Nu ¼ 1:6 1 þ 0:3Pr 1=3 Re1=2 Re ¼

Dl ure rai , hai

Nu ¼

hai Dl Cai hai , Pr ¼ kai kai

where Dl is the droplet diameter ure is the relative velocity of droplet and air 4. Tunnel-based convection oven (Mureau and Barreteau, 1981): Nu ¼ 4:5Re0:27 ,

Re < 40

Nu ¼ 0:70Re0:61 , Re > 40 Luav rb hL , Nu ¼ Re ¼ hb kb where L is the width of the tunnel uav is the average air velocity

Table 23.17 Values of Rohsenow Constant Liquid

Surface

Q

Reference

Water

Platinum wire

0.0130

Rohsenow (1952)

Carbon tetrachloride Water

Emery-polished copper Brass tube

0.0070 0.0060

Vachon et al. (1968) Rohsenow (1952)

Water Water

Paraffin-treated copper Emery-polished copper

0.0147 0.0128

Vachon et al. (1968) Vachon et al. (1968)

Water

Scored copper

0.0068

Vachon et al. (1968)

Water Water

Ground and polished stainless steel Teflon-pitted stainless steel

0.0080 0.0058

Vachon et al. (1968) Vachon et al. (1968)

Water

Chemically etched stainless steel

0.0133


Water Ethyl alcohol Benzene

Mechanically polished stainless steel Polished-plated chromium Polished-plated chromium

0.0132 0.0027 0.0100

Vachon et al. (1968) Rohsenow (1952) Rohsenow (1952)

n-Pentane

Lapped copper

0.0049


n-Pentane

Emery rubber copper

0.0074


n-Pentane

Polished-plated chromium

0.0150

Rohsenow (1952)

n-Pentane

Emery-polished copper

0.0154


n-Pentane

Emery-polished nickel

0.0127



h (W/m2. K)

10,000

S 1,000 A

100 0

50

100

RPM FIGURE 23.2 Heat transfer coefficients in agitated kettle. S, sucrose solution 408Brix; A, applesauce; rpm, 1=min. (From Saravacos, G.D. and Maroulis, Z.B., Transport Properties of Foods, Marcel Dekker, New York, 2001.)

The shear rate g_ for the pilot-scale-agitated kettle, described in this reference (0.40 m diameter, anchor agitator), was calculated from the empirical relation g_ ¼ 13N. The heat transfer coefficients h at the internal interface of the vessel for a sugar solution and for applesauce increased linearly with the speed of agitation (rpm), as shown in Figure 23.2. Besides the correlations presented above, there exist a few simplified dimensional equations, applicable to specific equipment geometries and system conditions, for the estimation of heat transfer coefficient of air and water in some important operations (Perry and Green, 1984; Geankoplis, 1993). These are presented in Table 23.18. In Table 23.18, DT is the temperature difference (8C), D0 is the outside diameter (m), L is the length (m), G is the mass flow rate (kg=m2s), G is the irrigation flow rate of the films (kg=ms), and N is the number of horizontal tubes in a vertical plane. In the case of evaporation of fluid foods, heat transfer controls the evaporation rate and high heat transfer coefficients are essential in the various types of equipment. Prediction of the heat transfer coefficients in evaporators is difficult, and experimental values of the overall heat transfer coefficient U are used in practical applications. Equation 23.4 can be written as the following expression: 1 1 x 1 ¼ þ þ þ FR U hi k h 0 Table 23.18 Simplified Heat Transfer Coefficient Equations for Air and Water Natural convection of air: Horizontal tubes, h ¼ 1.42 (DT=D0)1=4 Vertical tubes, h ¼ 1.42 (DT=L)1=4 Falling films of water: h ¼ 9150 G1=3 Condensing water vapors: Horizontal tubes, h ¼ 10,800=[(NbD0)1=4(DT)1=3] Vertical tubes, h ¼ 13,900=[L1=4 (DT)1=4]


(23:7)

U (W/m 2· K)

1,600

1,300

1,000 0

5

10

K (Pa · s n ) FIGURE 23.3 Overall heat transfer coefficient U of fruit purees in agitated kettle. (From Saravacos, G.D. and Maroulis, Z.B., Transport Properties of Foods, Marcel Dekker, New York, 2001.)

where FR is the fouling resistance that becomes important in the evaporation of liquid foods containing colloids and suspensions, which tend to deposit on the evaporator walls, reducing significantly the heat transfer rate. Falling film evaporators are used extensively in the concentration of fruit juices and other liquid foods because they are simple in construction and they achieve high heat transfer coefficients. Figure 23.3 shows the overall heat transfer coefficient for an agitated vessel. Figure 23.4 illustrates overall heat transfer coefficients U for apple juices in a pilot plant falling film evaporator, 3 m long and 5 cm diameter tube (Saravacos and Moyer, 1970). Higher U values were obtained in the evaporation of depectinized (clarified) apple juice (1200–2000 W=m2 K) than the unfiltered (cloudy) juice, which tended to foul the heat transfer surface as the concentration was increased. The U value for water, under the same conditions, was

U (W/m2· K)

10,000

CL 1,000 UFT

100 0

50

100

⬚Brix FIGURE 23.4 Overall heat transfer coefficients U in evaporation of clarified (CL) and unfiltered (UFT) apple juice at 558C. (From Saravacos, G.D. and Maroulis, Z.B., Transport Properties of Foods, Marcel Dekker, New York, 2001.)


higher as expected and equal to 2300 W=(m2 K). Jet impingement ovens and freezers operate at high heat transfer rates due to the high air velocities at the air–food interface. Heat transfer coefficients of 250–350 W=m2 K can be obtained in ovens, baking cookies, crackers, and cereals. Ultrasounds can substantially improve the air-drying rate of porous foods like apples (acoustically assisted drying). Ultrasound of 155–163 db increased the moisture diffusivity at 608C from 7 1010 to 14 1010 m2=s (Mulet et al., 1999).

23.5 HEAT TRANSFER FACTOR IN FOOD PROCESSES Saravacos and Maroulis (2001) and Krokida et al. (2001) retrieved a significant number of recently reported heat transfer coefficient data in food processing from the literature (including the following journals: Drying Technology, Journal of Food Science, International Journal for Food Science and Technology, Journal of Food Engineering, Transactions of the ASAE, and International Journal of Food Properties). The collected data refer to seven different processes presented in Table 23.19 and include about 40 food materials in Table 23.20. Most of the data were available in the form of empirical equations using dimensionless numbers. All available empirical equations were transformed in the form of heat transfer factor versus Reynolds number given by the equation jH ¼ aRen

0

(23:8)

The thermophysical properties of the materials and characteristics of the systems were taken into account to achieve this transformation (Krokida et al., 2002b). This equation was also fitted to all data for each process and the resulting equations characterize the process, since they are based on the data from all available materials. The results are classified based on the process and material and are Table 23.19 Number of Available Equations for Each Food Process Process Baking Forced convection

No. of Equations

1

Blanching Steam

1

Cooling Forced convection

9

Drying Convective Fluidized bed Rotary

16 1 4

Freezing Forced convection

6

Storage Forced convection

4

Sterilization Aseptic

9

Retort

3

Total no. of equations


54

Table 23.20 Number of Available Equations for Each Food Material Material

No. of Equations

Apples Apricots

1 1

Barley

2

Beef Cakes Calcium alginate gel

1 1 1

Canola seeds

1

Carrot Corn

1 2

Corn starch Figs

1 1

Fish

1

Grapes Green beans Hamburger

3 1 2

Maize

1

Malt Meat carcass

1 1

Model food

4

Newtonian liquids Nonfood material Particulate liquid foods

1 3 3

Peaches

1

Potatoes Raspberries

2 1

Rice Soya

1 2

Soybean

1

Strawberries Sugar Wheat

1 1 3

Spherical particles

1

Tomatoes Corn cream

1 1

Rapeseed

1

Meatballs

1

Total no. of equations

54

presented in Table 23.21. Heat transfer coefficient values for process design can be obtained easily from the proposed equations and graphs. The range of variation in this uncertain coefficient can also be obtained to carry out valuable process sensitivity analysis. Estimations for materials, not included in the data, can also be made using similar materials or average values. It is expected that the resulting equations are more representative and predict the heat transfer coefficients more accurately. The results of fitting the equation to all data, for each process, are summarized in Table 23.22.


0

Table 23.21 Parameters of the Equation jH ¼ aRen for Each Process and Each Material A

n0

Cakes (Baik et al., 1999)

0.801

0.390

40

3,000

Blanching Green beans (Zhang and Cavalieri, 1991)

0.00850

0.443

150

1,500

Apples (Fikiin et al., 1999) Apricots (Fikiin et al., 1999) Figs (Dincer, 1995)

0.0304 0.114 8.39

0.286 0.440 0.492

4,000 2,000 3,500

48,000 25,000 9,000

Grapes (Fikiin et al., 1999)

0.472

0.516

1,300

17,000

Model food (Alvarez and Flick, 1999) Peaches (Fikiin et al., 1999)

2.93 0.186

0.569 0.500

2,000 3,700

12,000 43,000

Raspberries (Fikiin et al., 1999)

0.0293

0.320

1,300

16,000

Strawberries (Fikiin et al., 1999) Tomatoes (Dincer, 1997)

0.136 0.267

0.440 0.550

1,900 1,000

25,000 24,000

Barley (Sokhansanj, 1987) Canola seeds (Lang et al., 1996)

3.26 0.458

0.650 0.241

20 30

1,000 50

Carrot (Mulet et al., 1989)

0.692

0.486

500

5,000

Corn (Fortes and Okos, 1981) Corn (Torrez et al., 1998) Grapes (Ghiaus et al., 1997)

1.06 4.12 0.665

0.566 0.650 0.500

400 20 8

1,100 1,000 50

Process=Product (Reference)

Min Re

Max Re

Baking

Cooling

Drying Convective

0.741

0.430

1,000

3,000

11.9 0.196

0.901 0.185

150 60

1,500 80

0.224 4.12

0.200 0.650

2,000 20

11,000 1,000

2.48

0.523

200

1,500

149 3.26

0.340 0.650

50 20

100 1,000

0.101

0.355

3,200

13,000

Fish (Shene et al., 1996) Soya (Alvarez and Shene, 1994)

0.00160 0.00960

0.258 0.587

80 10

300 100

Soya (Shene et al., 1996)

0.000300

0.258

20

80

Sugar (Wang et al., 1993)

0.805

0.528

1,500

17,000

Grapes (Vagenas et al., 1990) Maize (Mourad et al., 1997) Malt (Lopez et al., 1997) Potatoes (Wang and Brennan, 1995) Rice (Torrez et al., 1998) Soybean (Taranto et al., 1997) Wheat (Lang et al., 1996) Wheat (Sokhansanj, 1987) Fluidized bed Corn starch (Shu-De et al., 1993) Rotary

Freezing Beef (Heldman, 1980) Calcium alginate gel (Sheng, 1994) Hamburger (Flores et al., 1988)

0.650 48.6 8.87

0.418

80

25,000

0.535 0.672

300 7,500

600 150,000

Hamburger (Tocci and Mascheroni, 1995) Meat carcass (Mallikarjunan and Mittal, 1994)

4.67 0.228

0.645 0.269

9,000 1,800

73,000 20,000

Meatballs (Tocci et al., 1995)

0.536

0.485

3,400

28,000


0

Table 23.21 (continued) Parameters of the Equation jH ¼ aRen for Each Process and Each Material A

n0

0.658 0.0136

0.425 0.196

70 1,500

90 10,000

Model food (Balasubramaniam and Sastry, 1994) Model food (Sastry et al., 1990)

0.500 0.448

0.507 0.519

5,000 2,400

20,000 45,000

Model food (Zuritz et al., 1990)

3.42

0.687

2,000

11,000

Non-food material (Kramers, 1946) Non-food material (Ranz et al., 1952) Non-food material (Whitaker, 1972)

0.748 0.662 0.517

0.512 0.508 0.441

3,000 3,000 3,000

85,000 85,000 85,000

Process=Product (Reference)

Min Re

Max Re

Storage Potatoes (Xu and Burfoot, 1999) Wheat (Chang et al., 1993) Sterilization Aseptic

Particulate liquid foods (Mankad et al., 1997)

0.225

0.400

140

1,500

Particulate liquid foods (Sannervik et al., 1996) Spherical particles (Astrom and Bark, 1994)

0.0493 2.26

0.199 0.474

1,800 4,300

5,200 13,000

Retort Newtonian liquids (Anantheswaran et al., 1985a)

2.74

0.562

11,000

400,000

Particulate liquid foods (Sablani et al., 1997) Corn cream (Zaman et al., 1991)

0.564 0.108

0.403 0.343

30 130,000

1,600 1,100,000

Source: Adapted from Saravacos, G.D. and Maroulis, Z.B., Transport Properties of Foods, Marcel Dekker, New York, 2001.

All the equations referred to in Table 23.19 are presented in Figure 23.5 to define the range of variation in the heat transfer factor jH versus Reynolds number (Re). The range of variation by process is sketched in Figure 23.6. The data of Tables 23.21 and 23.22 demonstrate the importance of the flow conditions (Reynolds number, Re) and the type of food process and product on the heat transfer characteristics (heat transfer factor, jH). As expected from theoretical considerations and experience in other fields, the heat transfer factor, jH, decreases with a negative exponent of about 0.5 of the Re. The highest heat transfer factor values are obtained in drying and baking operations,

0

Table 23.22 Parameters of the Equation jH ¼ aRen for Each Process Process

A

n0

Min Re

Max Re

Baking

0.80

0.390

40

3,000

Blanching Cooling

0.0085 0.143

0.443 0.455

150 1,000

1,500 48,000

Drying=convective Drying=fluidized bed

1.04 0.10

0.455 0.354

8 3,200

11,000 13,000

Drying=rotary

0.001

0.161

10

300

Freezing Storage Sterilization=aseptic

1.00 0.259 0.357

0.486 0.387 0.450

80 70 140

150,000 10,000 45,000

Sterilization=retort

1.034

0.499

30

110,000

Source: Saravacos, G.D. and Maroulis, Z.B., Transport Properties of Foods, Marcel Dekker, New York, 2001.


100

10

1

jH

0.1

jH=0.344 Re⫺0.423

0.01

0.001

0.0001

0.00001 1

10

100

1,000

10,000 100,000 1,000,000 10,000,000 Re

FIGURE 23.5 Heat transfer factor versus Reynolds number for all the examined processes and materials. (From Saravacos, G.D. and Maroulis, Z.B., Transport Properties of Foods, Marcel Dekker, New York, 2001.)

while the lowest values are in cooling and blanching. Granular food materials, such as corn and wheat, appear to have better heat transfer characteristics than large fruits, e.g., apples (Table 23.21). In drying, heat transfer takes place in a variety of mechanisms, and the heat transfer factor has been calculated for different dryer types, separately (Krokida et al., 2002a). The general presentation of Figure 23.6 for convective drying has been analyzed for the following drying methods: . . . .

Flat plate drying Fluidized bed drying Packed bed drying Rotary drying

Figures 23.7 through 23.10 illustrate the heat transfer factor for each of the above drying methods. The effect of food material is obvious in these diagrams. The range of variation per dryer type is sketched in Figure 23.11.


1

Baking Drying convective

0.1

jH

Storage

0.01

Sterilization aseptic

Freezing Drying fluidized bed

Cooling

0.001

Sterilization retort Blanching

0.0001 10

100

1,000

Re

10,000

100,000

1,000,000

FIGURE 23.6 Ranges of variation in the heat transfer factor versus Reynolds number for all the examined processes. (From Saravacos, G.D. and Maroulis, Z.B., Transport Properties of Foods, Marcel Dekker, New York, 2001.)

23.6 PREDICTION OF HEAT TRANSFER COEFFICIENT USING MASS TRANSFER DATA One of the fundamental theories in transport phenomena is the analogy in momentum, heat, and mass transfer. Due to this analogy, data of any type of transfer are equally useful in the design and operation of thermal processes, as they may be used for the evaluation of the heat=mass transfer mechanisms and the estimation of heat transfer coefficient (or vice versa). Chilton and Colburn quantified this analogy proposing the well-established Chilton–Colburn (or Colburn) analogy, expressed by the following equation: jH ¼ jM ¼ where jH is the heat transfer factor jM is the mass transfer factor f is the Fanning friction factor


f 2

(23:9)

10 Flat plate drying

1 Barley, wheat Soybean

jH

Grapes 0.1

Malt Potatoes Corn Carrot

0.01

0.001 10

100

1,000

10,000

100,000

Re FIGURE 23.7 Heat transfer factor versus Reynolds number for flat plate drying and various materials.

10 Fluidized bed drying

1

jH

Rice

Nonfood material

0.1

Maize

0.01

Corn starch

0.001 1

10

100

1,000

10,000

100,000

Re FIGURE 23.8 Heat transfer factor versus Reynolds number for fluidized bed drying and various materials.


10 Packed bed drying Wheat 1

Canola seeds

jH

0.1

Nonfood materials

0.01

0.001 1

10

100

1,000

10,000

100,000

Re FIGURE 23.9 Heat transfer factor versus Reynolds number for packed bed drying and various materials.

1 Rotary drying

0.1

jH

Nonfood material

0.01

Sugar

0.001 1

10

100

1,000

10,000

100,000

Re FIGURE 23.10 Heat transfer factor versus Reynolds number for rotary drying and various materials.


10

Pneumatic 1

jH

Spray

0.1 Fluidized bed

Packed bed

0.01

Rotary Flat plate

0.001

1

10

100

Re

1,000

10,000

100,000

FIGURE 23.11 Ranges of variation in the heat transfer factor versus Reynolds number for all the examined drying methods.

Analytically, heat and mass transfer factors are given by the equations jH ¼ StH Pr2=3

(23:10)

jM ¼ StM Sc2=3

(23:11)

h urC

(23:12)

hM k C ¼ ur u

(23:13)

hC k

(23:14)

StH ¼ StM ¼

Pr ¼

where h is the heat transfer coefficient hM is the mass transfer coefficient kC is the mass transfer coefficient expressed in concentration units u is the average (bulk) fluid velocity r is the density of the fluid h is the dynamic viscosity of the fluid C is the specific heat of the fluid k is the thermal conductivity of the fluid


All fluid properties refer to bulk temperature. The Chilton–Colburn analogy in air–water mixtures (applications in drying and air conditioning) is simplified when the Pr and Sc are approximately equal (Pr ffi Sc ffi 0.8). Therefore, Equation 23.9, by substituting Equations 23.10 and 23.11, becomes StH ¼ StM

or

h k C hM ¼ ¼ urC u ur

or

h ¼ kC rC

(23:15)

In terms of the mass transfer coefficient hM, the above relationship gives hM ¼

h C

(23:16)

The specific heat of atmospheric air at ambient conditions is approximately C ¼ 1000 J=(kg K). Therefore, Equation 16 yields h ¼ 1000 hM, where h is in W=(m2 K) and hM in kg=(m2 s). If the units of hM are taken as g=(m2 s), the above relationship is written as (Saravacos, 1997) h(W=m2 K) ffi hM(g=m2 s),

for atmospheric air

A similar relationship is obtained between the coefficients h and kC h(W=m2 K) ffi kC (mm=s),

for atmospheric air

Krokida et al. (2001) retrieved recently reported mass transfer coefficient data in food processing from the literature (which include materials such as corn, grapes, maize, meat, model food, potatoes, rice, carrots, and milk) and, following the same procedure as for heat transfer coefficient data, transformed them in the form of mass transfer factor versus Reynolds number given by the equation jM ¼ aRen

0

(23:17)

The results are classified based on the process and material, which are presented in Tables 23.23 and 23.24. All the equations used are presented in Figure 23.12 to define the range of variation in 0

Table 23.23 Parameters of the Equation jM ¼ aRen for Each Process and Each Material Process=Product (Reference)

A

n0

5.15 0.004

0.575 0.462

20 10

1,000 40

0.741

Min Re

Max Re

Drying Convective Corn (Torrez et al., 1998) Grapes (Ghiaus et al., 1997) Grapes (Vagenas et al., 1990) Maize (Mourad et al., 1997) Rice (Torrez et al., 1998) Carrot (Mulet et al., 1987)

0.430

900

3,000

34.6 5.15

1.000 0.575

5 20

15 1,000

0.69

0.486

500

5,000

2.947

0.890

1

2

2.496

0.495

2500

70,000

0.667

0.428

50

55

11.220

1.039

6500

26,000

Spray Milk (Straatsma et al., 1999) Freezing Meat (Tocci et al., 1995) Storage Potatoes (Xu and Burfoot, 1999) Sterilization Model food (Fu et al., 1998)



0

Table 23.24 Parameters of the Equation jM ¼ aRen for Each Process Process

n0

A

Drying=convective

Storage Sterilization

Max Re

0.882

5

5,000

2.95 0.10

0.889 0.268

1 2,500

2 70,000

0.67 11.2

0.427 1.039

50 6,500

55 26,000

23.5

Drying=spray Freezing

Min Re


the mass transfer factor jM versus Reynolds number, Re. The range of variation by process is sketched in Figure 23.13. Especially for drying processes, the available mass transfer data have been classified according to the drying method and the corresponding equipment (Krokida et al., 2002a,b). Figure 23.14 presents the mass transfer factor versus Reynolds number for all the examined drying types and materials, while Figure 23.15 illustrates the ranges of variation in the mass transfer factor versus Reynolds number for all the examined drying types. Figure 23.16 shows the effect of Reynolds number to mass transfer factor in the case of fluidized bed drying for all the examined materials. Regression analysis of published mass transfer data shows the similarity between the heat transfer factor jH and the mass transfer factor jM. Mass transfer coefficient values for process design can be obtained from the proposed equations and graphs. The range of variation in both heat and mass transfer coefficients, which generally presents a degree of uncertainty, can also be obtained from the presented data to carry out valuable process sensitivity analysis. Estimations for materials not included in the data can also be made using similar materials or average values.

10

1

0.1 jM

j M =1.11 Re⫺0.54

0.01

0.001

0.0001 1

10

100

1,000

10,000

100,000

Re

FIGURE 23.12 Mass transfer factor versus Reynolds number for all the examined processes and materials. (From Saravacos, G.D. and Maroulis, Z.B., Transport Properties of Foods, Marcel Dekker, New York, 2001.)


10 Drying convective Spray drying

1

0.1

Storage

jM

Freezing

0.01

0.001

Sterilization 0.0001 1

10

100

1,000

10,000

100,000

Re

FIGURE 23.13

Ranges of variation in the mass transfer factor versus Reynolds number for all the examined processes. (From Saravacos, G.D. and Maroulis, Z.B., Transport Properties of Foods, Marcel Dekker, New York, 2001.)

10

jM

1

j M = 4.78 Re⫺0.68

0.1

0.01

0.001 1

10

100

1,000

10,000

100,000

Re FIGURE 23.14 Mass transfer factor versus Reynolds number for all the examined drying types and materials.


10

1 Fluidized bed

jM

Spray 0.1 Packed bed Pneumatic

Flat plate

0.01

0.001 1

10

100

1,000

10,000

100,000

Re FIGURE 23.15 Ranges of variation in the mass transfer factor versus Reynolds number for all the examined drying types.

10 Fluidized bed drying

Maize

1 Rice

jM

Corn Nonfood materials

0.1

0.01

0.001 1

10

100

1,000

10,000

100,000

Re FIGURE 23.16 Mass transfer factor versus Reynolds number for fluidized bed drying and various materials.


ACKNOWLEDGMENT The authors would like to thank Dr. Zacharias Maroulis for his guidance and comments for the writing of this chapter and for his contribution in preparing the tables and figures.

NOMENCLATURE a a_ â A Ar ^b B ^ B Br c_ €c ^c C C ^ C Cr D d^ e e0 E ^ E f ^f F ^ F Fr g ^g G ^ G Gr Gz h H hd hM j ^j k K K_

equation parameter equation parameter equation parameter heat transfer surface area (m2) Archimedes number equation parameter equation parameter equation parameter Brinkman number drag coefficient equation parameter equation parameter specific heat at constant pressure (kJ=kg=8C) equation parameter equation parameter equation parameter (dimensionless) diameter (m) equation parameter can headspace height (m) equation parameter material, fluid characteristic activation energy (J) friction factor correction factor equation parameter correction function Froude number (dimensionless) acceleration due to gravity (m=s2) induced gravitational field air mass velocity (kg=m2=s) equation parameter Grashof number (dimensionless) Graetz number (dimensionless) heat transfer coefficient (W=m2=8C) height or screw channel depth (m) fouling coefficient of the wall=liquid interface (W=m2=8C) mass transfer coefficient Colburn transfer factor mass flow ratio of solid to gas thermal conductivity (W=m=K) consistency coefficient of the fluid (Pa sn) equation parameter


€ K kC l L m M n n0 n_ N Nb Nu P Pe Pr q_ qs r Ra Re S Sc St T Tu u uv uv uc U V We x x0 x X Y^ Z

equation parameter mass transfer coefficient expressed in concentration units characteristic length (m) film thickness or length (m) mass flow rate (kg=s) equation parameter flow behavior index of the fluid equation parameter equation parameter rotational or agitation speed (s1 or rps) number of horizontal tubes in a vertical plane Nusselt number (dimensionless) pressure (Pa) Peclet number (dimensionless) Prandtl number (dimensionless) heat flux (kW) parameter equation radius (m) Rayleigh number (dimensionless) Reynolds number (dimensionless) vapor velocity to liquid film velocity ratio Schmidt number (dimensionless) Stanton number (dimensionless) temperature (8C) turbulence intensity fluid velocity (m=s) volumetric vapor flux (m3=s) modified volumetric vapor flux (dimensionless) cylinder surface peripheral velocity (m=s) overall heat transfer coefficient (W=m2=8C) volume (m3) Weber number wall thickness (m) mass fraction of vapor in mixture distance in x-axis Lockhart–Martinnelli parameter equation parameter equation parameter

Greek Symbols a b g G g_ d D « h

thermal diffusivity (m2=s) volumetric thermal expansion coefficient of the fluid major to minor axis ratio condensation rate per unit length of perimeter (kg=s=m) shear rate (s1) film or layer thickness (m) difference volume fraction viscosity (Pa s)


q u^ Q k L n l l* j J p r s t^ y Y F x c v V %RH

equation parameter orientation of ribs in plate heat exchanger Rohsenow constant equation parameter non-Newtonian to Newtonian heat transfer coefficient ratio kinematic viscosity (m2=s) latent heat of vaporization (kJ=kg) modified latent heat of vaporization (kJ=kg) equation parameter correction function constant ( ¼ 3.14) density (kg=m3) surface or interfacial tension (N=m) shear stress number of rows of scraper blades equation parameter surface area to volume ratio wavelength mass transfer coefficient particle shape factor can angular velocity non-Newtonian to Newtonian heat transfer rates percent relative humidity

Subscripts ai ap av b bo c ca ce ci co cs cv cx d D dr e ec eg eq er f FL ga

air approach average bulk fluid box can mushroom cup effective can length can internal can outside can surface or cross sectional condensing vapor can external due point developed area drum exit or outlet effective equilibrium at the end of heating or cooling process equivalent ellipsoid of revolution forced convection fluid gap between plates


h H i ini insp iw l m M ma me mi min mix n ou ov p pa pr r re ri s sa sb sc se so sp sr ss su sv T TO v w woc wic x y z

hydraulic heat inlet or initial inside initial inside specific liquid at wall liquid mixed convection mass major axis semiaxis minor axis minimum mixture natural convection outside overall product particle (or sphere) projected reel retort or relative rise of ribs surface shaft bulk suspension center of solid sedimented suspension solid specific screw absolute steam pressure suspension saturation tube or cylinder total vapor wall without condensation of water vapor with condensation of water vapor x-axis y-axis local distance in z-direction REFERENCES

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CHAPTER

24

Acoustic Properties of Foods

Piotr P. Lewicki, Agata Marzec, and Zbigniew Ranachowski CONTENTS 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8

Introduction ........................................................................................................................ 811 Basics of Acoustic Emission ............................................................................................. 812 Measurement of Acoustic Emission .................................................................................. 815 Analysis of Recorded Signal of Acoustic Emission.......................................................... 817 Measurement of Acoustic Emission in Food..................................................................... 820 Calculation of Acoustic Emission Descriptors in Food .................................................... 823 Factors Affecting Acoustic Emission in Food................................................................... 827 Correlations between Acoustic Emission Descriptors and Sensory Assessment of Quality ....................................................................................................... 836 24.9 Recapitulation .................................................................................................................... 838 Acknowledgment .......................................................................................................................... 838 References ..................................................................................................................................... 838

24.1 INTRODUCTION Acoustic emission (AE) generated during biting and mastication of food is an important attribute of food texture. Acoustic properties of snack-type food as well as fresh fruits and vegetables are important from the quality point of view. Crunchiness and crispness are the sensory attributes preferred by consumers and are regarded as signs of freshness and propriety of processing. Lack of these attributes suggests low quality and is at least partly responsible for the unacceptance of the product by consumers. Crunchiness=crispness is a subjective assessment of quality and until now, there are divergences in the description and definition of this sensory attribute. From the point of research, analyzing the crunchiness=crispness sound percept by a human ear is considered as the main physical index influencing the consumers’ assessment of quality. Elastic waves generated during the disintegration of food can be used to identify the acoustic properties of the product. Moreover, analysis of the emitted sound can be used to follow changes in the texture of food, caused during storage or by modifications of recipes or processing parameters.


And lastly, the acoustic properties of food can be used to numerically express one of its important quality attributes, namely the crunchiness=crispness.

24.2 BASICS OF ACOUSTIC EMISSION

Amplitude Time

(A) Figure 24.1

(B)

Direction of motion

Time

Amplitude

An initial inhomogeneity of internal energy distribution can appear in any solid, gaseous, or liquid body. In actual materials, inhomogeneity characteristics are due to structural defects, impurities, internal stresses, and other factors and can be micro- or macroscopic [1]. The following structure elements are examples of body inhomogeneities: (1) surface between different phases of the material, (2) cellular walls and intercellular objects, and (3) gaseous pores in starch polymers, forming the cereal food products. Inhomogeneities have no influence on the state of global mechanical equilibrium of that body. Introduction of external stimuli, such as external force, chemical reaction, or thermal excitation, disturbs the equilibrium state and causes the onset of vibrations of particles in the body volume. The vibration can remain active and propagate through the medium even after decay of the stimuli. The reason for the extended reaction of the medium can be explained by the elastic and inertia properties of the majority of bodies. The elastic properties are responsible for the reaction against the disturbance of the equilibrium. The inertia properties, however, are the source of forces opposing to any change in state of the matter, i.e., opposing to the decrease in pending kinetic activity even when the local equilibrium is achieved. Therefore, the movement of body particles results in a series of transformations, from the kinetic energy supported by inertial forces to the potential energy supported by elastic forces. This process can continue infinitely even after the decay of the stimulus, but is not another property of the real matter—exhibiting the ability to attenuate the internal vibrations. However, the kinetic energy of the movement gradually gets converted into heat and changes the microstructure of the body (dislocation movements, creation of micro- and macro-cracks). In this way, the attenuated vibrations gradually fade. The disturbance of the equilibrium that has appeared in time t0, in a certain spot of a body, would cover the entire body after a sufficient period of time. This effect occurs because every particle of the body interacts with its neighbors and passes a part of its kinetic energy to them. This phenomenon is called wave propagation [2,3]. Two basic modes of wave propagation can occur: (1) a longitudinal wave, propagation of which causes the elements of the medium to move in the direction of the wave propagation; (2) a transverse wave, with the vibrating motion perpendicular to the direction of the wave propagation (Figure 24.1).

Vibrations of an element in (A) longitudinal wave and (B) and transverse wave. (From Sliwinski, A., Ultrasound and Its Application, 2nd ed, WNT, Warszawa, 2001 (in Polish); Malecki, I., Physical Foundations of Technical Acoustics, Pergamon Press, Oxford, 1969. With permission.)


Vibration period

Amplitude

+u0

Time

⫺u0

Figure 24.2 Elastic wave.

In liquids and gases, only longitudinal wave can be excited. Two other modes of wave propagation, namely Rayleigh–Lamb and Love waves, occur near the surface of the solids. The number of times a complete motion cycle takes place in every point of the body during a period of 1 s is called the frequency and is measured in hertz (Hz) [4]. The distance between the equilibrium location and the maximum displacement during the motion cycle is the amplitude [5] (Figure 24.2). Generally, disturbance of mechanical and thermodynamic equilibrium in solids, liquids, and gases leads to the release of local deformation processes and invokes the dissipation of energy. The major part of this emerging energy gets converted into the work done by stresses, heat, and a relatively small portion radiates in the form of elastic waves. The resulting waves propagate through the material and reach its outer surface. When a sensor of elastic waves is placed on the body surface, it can detect the onset of the described phenomenon [6]. The particles of the air surrounding the vibrating body can also transmit the waves to the microphone situated at a certain distance from the body. The explanation of the terminology used has its roots in the past—the elastic waves arising in loaded mechanical elements were situated in the audible range of frequencies (16–16000 Hz), and hence, scientists who first observed these phenomena introduced the terms noise, ultrasonic emission, or AE. It can be stated that when a sudden release of internal energy in the loaded matter takes place, and if an elastic wave’s propagation pattern is recorded, then the research is inclined toward the measurement of AE. An AE source is defined as a certain location or element of the structure that emits the AE signal. Usually, an electric waveform u(t) recorded on the output of the sensor prepared for storage or processing is considered as the AE signal. The AE sources can be divided into groups with respect to the processes involved in their activities [7]. These include (1) crack formation and propagation, (2) relocation of defects in crystalline lattice, (3) phase transitions and chemical reactions, (4) creation of transitions between the energy states of the atoms, (5) body movements with the accompanying friction mechanisms, and (6) biological processes running in live tissues and organisms. The highest AE amplitude levels are detected during the crack formations and destruction processes in solids under mechanical stresses, and the lowest are generated when transitions between the energy states of the atoms occur. The AE signal arising in the processes stated earlier varies with respect to its amplitude and frequency. The nature of waveforms of different frequencies is identical. However, different effects related to the interactions between the propagating elastic wave and surrounding material can occur. The microscopic structure of a solid medium is mostly inhomogeneous (i.e., polycrystalline or composite-like) and anisotropic,


and this decisively influences the mechanism of energy absorption of the wave. This phenomenon depends on the mutual relation between the dimensions of the material particles and the length of the propagating wave [1]. The relation between the wave frequency f (s1), wavelength l (m), and wave velocity c (m s1) in a body is as follows: l¼

c f

(24:1)

A typical AE source is characterized by small dimensions with respect to the element or the volume of its occurrence. Therefore, it can be treated as a source of a spherical wave. The AE propagation mechanism in air can be explained as the spread of harmonic changes in pressure. In solids and liquids, the oscillations of particles are also transferred in all directions from the source. However, the propagation of acoustic waves in vacuum is not possible. The direction of the propagation pattern may vary from its symmetrical form when structural inhomogeneities, inclusions, or particle migrations occur. An acoustic wave traveling through the boundary between two bodies undergoes reflection, refraction, and scattering, which cause problems in tracing the origin of the AE activity. Furthermore, according to Doppler’s effect, a moving AE source is subject to variations in produced wave frequency [8]. The deformation processes can occur in a micro- and macro scale and, therefore, cause generation of pulse trains of AE signals (bursts) differing in amplitude and intensity. Time dependence of the AE signals reflects the type of the process, i.e., the source of emission and kinetics of the process. The American Standard E 1316 91b [9] deals with the basic term definitions for AE, listing two types of AE: continuous and discrete. The AE signal is identified as discrete when the time delays between pulses having relatively large amplitudes are greater than or equal to the duration of the pulses itself. This condition is related to the phenomena of micro- and macro crack formation and their growth. Referring to the above-mentioned standard, continuous AE is caused by ‘‘rapidly occurring’’ AE sources, since the signals they generate are mixed. A good example of this kind of AE source is a steel ball sliding across the smooth surface, causing abrasive wear of it in the bearings [10]. The relaxation processes in the investigated body cause the origin of the AE signal. By assuming an isotropic solid with loss-less structure and analyzing where a local change of stress field was induced, the propagating elastic wave can be described by the following equation [2]: r

@2U ¼ (d þ h)grad div U þ ddiv grad U @t 2

(24:2)

where r is the medium density (kg m3) U is the displacement vector (m) d, h are Lame constants, reflecting the mechanical properties of the investigated medium (N m2) t is the time (s) The analytical solution of Equation 24.2 for a particular body is extremely complex, owing to the body’s geometrical configuration, along with the reflection, refraction, inhomogeneities, or scattering phenomena occurring at its boundaries. Generally, it can be stated that the AE signal registered on the tested body surface is a product of AE-source parameters and properties of a body in which AE wave propagates. To construct a mathematical model of a system comprising an AE source, its environment, and wave sensor, some simplifications are applied that result in imperfections in the description of the real effects. The main problems are related to the negligence of the source dimensions and wave


transformations in the medium. The recorded AE signal is also distorted by the receiving sensor, because this device is capable of detecting the limited frequency bandwidth. Owing to these problems, usually the determination of dependence between a registered AE signal and its origin is impossible [11]. Researchers investigating on AE signal have avoided these limitations by constructing certain signal parameters, called AE signal descriptors, that can be measured or calculated in practice.

24.3 MEASUREMENT OF ACOUSTIC EMISSION The AE method has now found wide application in testing the mechanical properties, phase transitions, and in identifying the physical and chemical processes in different materials. As explained before, the AE signal propagates from its source, across the material under test to the sensors placed on the surface of that material [12]. To gain the maximum benefit from the application of the AE method, the use of the proper sensor is important. There are different contact and noncontact sensors available in the market. Noncontact sensors are made as laser Doppler interferometers. Generally, they can be divided into groups with respect to the operating frequency band, and the typical parameters are listed below: 0.1 Hz–15 kHz 1 Hz–40 kHz 1 kHz–150 kHz 100 kHz–1 MHz

It is important to achieve proper contact between the AE sensor and the body under test. Any air slot or inclusion of a material resistant to elastic waves may cause AE signal attenuation and reflection, introducing severe errors to the measuring procedure. The AE signal amplifiers and registering devices must operate at a suitable speed and the lowest internal noise level to process the data properly. It is also important to suppress the noises from the environment (including the machine loading the tested object) [13]. The typical arrangement to measure AE signals consists of (1) AE sensor, (2) low-noise preamplifier, matched to the signal level produced by the AE sensor, (3) filter to cut off the undesired frequency bands, (4) main amplifier to establish the desired signal voltage for further processing, (5) high-speed analog-to-digital converter with an interface to the computer, and (6) computer for data storing and processing (Figure 24.3). Traditionally, acoustic measurements of food were carried out with microphones as the sensors of AE signal; however, for this study, the contact method was preferred, i.e., using the sensor that remains in mechanical contact with the product sample. In recent years, many food technology

Preamplifier

AE sensor

Main amplifier

Band-pass filter

Specimen Analog-to-digital converter

Computer Figure 24.3 Arrangement of a stand for AE measurement and analysis.


laboratories are carrying out the process of registration of AE signal during the crushing of food products. The instrumentation for such tests consists of a texturometer equipped with a microphone probe, acoustic signal amplifier, and a computer to control the loading process and to register the experimental results, as presented in Figure 24.4A through C. The acoustic waves travel across the body under test, then across the air, and finally excite the thin membrane placed in the microphone probe. The microphone converts the vibrations into electric waveforms ready for further computer processing. The sensitivity of the microphone probe is proportional to the membrane area and the spatial direction of the microphone probe. The place and direction of the probe with respect to the specimen also has a significant influence on the registered signal level. It should be stated that the application of sensitive microphone probes requires accurate calibrations of the instrumentation [14]. The method presented in Figure 24.4A through C cannot be assumed as an effective method when applied under industrial conditions. The sounds generated by crushed-food specimen are relatively weak, and the microphone records the unwanted sounds coming from the environment, echoes, etc. Elastic wave traveling across the bulk of the specimen is subject to high reflection phenomenon on the body–air interface owing to the significant difference in the specific density of both the media. Therefore, the way the signal is propagated differs from actual conditions of human sensing, where the bone sound-transmission plays the most significant role. However, the results of the application of a microphone probe for a sound registration in testing of food products can be found in many research papers [15–20].

F(t)

F(t)

Penetrating tool

Penetrating tool AE sensor

(A) Sample under test

(D) Sample under test and AE sensor F(t)

F(t)

(E)

(B)

F(t)

F(t) AE sensor sends a signal to the recording system

AE sensor sends a signal to the recording system (C)

(F)

Figure 24.4 The schematic diagram of testing a product sample with the application of loading machine and a sensor of AE. (A and D) penetrating tool moves toward a product; (B) penetrating tool induces a mechanical stress in the product volume; the released stress wave travels across the product body, and then a part of it propagates across the air toward a microphone; (C) a microphone sends AE signal as an electric waveform to the recording system; (E) penetrating tool induces a mechanical stress in the product volume; the released stress wave travels across the product body; (F) contact AE sensor sends AE signal as an electric waveform to the recording system.


A contact method of registering AE signals of food specimens, in good agreement with the idea presented in Figure 24.4D through F, has been designed [21]. A contact sensor capable of measuring an acceleration evoked in the investigated product was mounted between the lower end of the moving traverse and penetrating tool of the loading machine [22]. The sensor was placed on a conical adapter to prevent it from the damage caused by the static load produced by the loading machine. Its sensitivity within the operating frequency band (1–15,000 Hz) was set to 8 mV per 1 m s2. It is possible to determine the amplitude of the measured vibration (in micrometer) by double integration of the current–voltage registered during the investigation. Since the accelerometer receives the vibration directly from the penetrating tool through a metal joint, its mechanism of sensing the processes accompanying the deformation of the food specimen remains in close relation to the conditions existing in the human mouth. The electric signal from the sensor is fed to the low-noise amplifier, where it is magnified hundred times (40 dB). A 12-bit analog-to-digital converter of type ADLINK 9112 was applied to convert the current values of the AE signal and to convert the stress level into digital form, which can be stored and processed in a computer. The converter operates at a rate of 44,100 sampling processes per second. The custom-designed software helps to produce two types of records as a result of the performed test. The AE signal is stored in popular Windows multimedia .wav format, which makes it possible to play and edit the signal with the application of standard software. The signal, proportional to the force loading the specimen, is stored in the text format compatible with the requirements of popular worksheet programs. A similar instrumentation was applied to investigate the AE signal generated during the wear processes in bearings and to investigate the mechanical durability of wood and fiberglass specimens. Alchakra et al. [23] applied both an accelerometer operating at 0.2–20 kHz range and a microphone probe operating at 0–40 kHz range in tests of destruction of concrete, graphite, and polystyrene. They have also observed that the loading machine used in the investigation had not emitted measurable background noise within the used frequency range. From the point of sensory assessment, the acoustic investigation of food should be made in the frequency range from 1 to 16 kHz, because the waves of those frequencies evoke audible sensations in humans. This effect was described by Dacremont [16], who had registered the bone transmission of sound from the mouth to the microphone placed in the mastoid. The effectiveness of bone soundtransmission was compared with the air propagation during biting of crisp-bread specimens. Dacremont [16] stated that the sounds of frequency 5–10 kHz propagate through a bone tract, and the sounds of frequency 1–2 kHz propagate through air tract. There are numerous literatures confirming that the frequency range stated here is carefully chosen for the sensory investigation of food [24–27]. Lee et al. [28] has stated that the proper frequency range should reach a minimum of 12 kHz for crispy products. The AE frequencies generated by typical crispy products are in the 5–12.8 kHz range, while those generated during chewing of potato chips are in 3–6 kHz range, and when specimens of chips are compressed in the loading machine, the AE lies in the 1.9–3.3 kHz range [29].

24.4 ANALYSIS OF RECORDED SIGNAL OF ACOUSTIC EMISSION A typical record of the time dependence of AE signal is presented in Figure 24.5. A section of the signal where measurable oscillations are detected is called the AE event. Within a time of an event, the consecutive AE signal amplitudes exceed the preset threshold, called the AE discrimination level [30]. Modern computerized AE processing equipment is capable of producing different AE descriptors during the investigation. The most commonly used descriptors are listed in Table 24.1 [31] and are discussed in the later sections. To determine the time-domain derivatives listed in Table 24.1, some assumptions were made with respect to the shape of the AE waveform. The basic assumption made after practical observations is the approximation of the registered AE signal produced by the impulse AE source to the


Signal amplitude

AE event

AE count

Discrimination level

Time (s)

Figure 24.5 Typical record of AE signal. The AE event is the duration of an effect, when the AE signal crosses a certain preset signal threshold called the discrimination level. The AE count is registered at every moment when the current AE amplitude crosses that threshold.

shape of a damped sinusoid. It is possible to preset a certain signal threshold and register (in other words, count) every moment when current amplitude of the AE signal would cross that threshold. This strategy of tracing the activity of AE source is called AE count processing. The time when the consecutive AE signal amplitudes, belonging to the single AE impulse, exceed the preset threshold is called AE event (Figure 24.5). The AE waveform of the single event can then be approximated using the following formula: n(t) ¼ Am exp (a1 t) sin (2pf0 t)

Table 24.1 Most Commonly Used AE Signal Descriptors Time-domain derivatives Counts number processing Events number processing Zero crossing Peak amplitude Average amplitude Area of waveform above mean ‘‘Half life,’’ i.e., time to reach half area (see the above point) Energy derivatives Peak root-mean-square (rms) Average rms Sum of rms Energy of single count of event Frequency-domain derivatives Frequency in signal spectrum with maximum intensity Median frequency, dividing signal spectrum for parts with equal energies Maximum of the signal spectrum or local maximum Average frequency of registered waveform train Frequency band of AE signal crossing the preset energy level Signal power or area of signal spectrum above the preset level in defined frequency bands


(24:3)

where n(t) is the transient value of amplitude (V) Am is the maximum signal amplitude registered in the event (V) f0 is the frequency in the signal spectrum with maximum intensity (s1) a1 is the decrement of the AE signal damping in the system, including the tested body and the AE sensor (s1) t is the time (s) Several AE analyzers have the ability of registering the number of counts or number of events in the preset time unit. This method of tracing the activity of the AE sources is called AE count=event number measuring. It is possible to determine the number of counts, Nc, within a single AE event when a certain signal threshold, At, is preset using the following formula [7]: f0 Am Nc ¼ ln a1 At

(24:4)

The registered AE signal, presented in Figure 24.5, with amplitude–time coordinates can be characterized by global quantitative descriptor E, the area of waveform above the mean. This parameter is often colloquially called AE signal energy, because it is to some extent proportional to the exact value of the energy. Assuming that N signal samples were recorded during the investigation, the AE signal energy can be calculated (in volts) from the following equation: E¼

N X

n(m T1 )

(24:5)

m¼1

where m is an index of a stored-signal sample T1 is an inverse of a sampling frequency (s)

Sound intensity (dB)

The third part in Table 24.1 lists the AE descriptors as the frequency-domain derivatives. They deal with differences existing in the AE signal, by analyzing its spectral characteristics (Figure 24.6). This method, contrary to the procedure of the AE signal energy determination, does not depend on a specimen volume under the test. If the AE signal, n(t), is absolutely integrable, it can be associated

6 0 ⫺6 ⫺12 ⫺18

a w = 0.037

0

3

6

9

12

15

9

12

15

a w = 0.750 6 0 ⫺6 ⫺12 ⫺18

0

3

6

Frequency (kHz) Figure 24.6 AE signal spectral characteristics of the rye-extruded crisp bread as a function of water activity.


with its spectral density function, A(v), where v is a linear analogue of frequency f, v ¼ 2p f, using a Fourier transform: 1 n(t) ¼ p

1 ð

A(v) sin [vt þ w(v)]dv

(24:6)

0

where w is an argument representing a phase of transformed signal. There are numerous computer procedures to derive a discrete image of A(v) that consists of a set of coefficients, cn. In this study, the procedure was designed to carry out this task by organizing the recorded AE signal sample in time windows at a length of 0.25 s. To reject the influence of background noise, one dominant AE burst was detected (if any present) in each section. All the bursts were processed to obtain their power-spectrum function, keeping the same phase of each burst at the transformation process. This algorithm is sometimes called event filtering and enables to suppress the random noise accompanying the recorded signal. As a result, for each time window, the procedure produces a series of coefficients, cn, and each of them represents AE signal power in a frequency of about 11 Hz. The whole series of cn cover the desired spectral range of 11–15,000 Hz. The recorded time-dependent AE signal, n(t), of each recording is converted into a vector of digital samples. The algorithm performing the n(mT1) ! cn transform is based on the standard approximated formula: cn

N1 1 X jn2pm y(m T1 ) mod e N m¼0 N

(24:7)

where j denotes square root of 1 mod denotes the modulus of a complex number In the analysis of acoustic properties of foods, descriptors such as loudness, pitch, energy of the emitted sound, and the number of peaks are mostly used. Much less attention is paid to the analysis of frequency of emitted sound [32]. 24.5 MEASUREMENT OF ACOUSTIC EMISSION IN FOOD During the early decades of acoustic testing of food, sounds emitted by the products were registered with the use of microphones operating in the acoustic frequency band (16–16,000 Hz). After a desired amplification, the collected data were recorded on a tape recorder [33]. In 1980s, a computer equipped with a soundcard or other analog-to-digital converter was introduced in these investigations. An important part of food sounds registration procedure is a kind of applied method of product crushing. Numerous investigators [33,34] examined the sounds during food mastication, biting with their incisors, or breaking by hand. From the beginning of 1970s, machines for the mechanical loading of the products have been applied for tests. The machines (of universal application like Instron or specialized texturometers) were equipped with the operating tool for mechanical loading of the sample, moving with a controlled speed, and an arrangement enabling the registration of applied force [17]. The experiments carried out by Drake [35] revealed that the sounds emitted during the process of food crushing differ from each other with respect to amplitude, frequencies found in signal spectrum representing dominant intensity, and in characteristics of intensity changes in the registered signal amplitude. In 1976, Vickers and Bourne [33] concluded that biting crisp foods generate specific sounds of sharp, short, and noisy type. Spectral analysis on these sounds revealed that the investigated signal


included the components within 0–10 kHz frequency band. These researchers hypothesized that the perceived crispness level is proportional to the sound amplitude of the signal of the product registered by the instrumentation. In 1979, Vickers and Wasserman [36] attempted to verify the hypothesis that the sound of crushed product includes the pattern specific to the structure of the tested sample. The probability of right product recognition by listening to the prepared records was determined, and about 18 products were examined. Two of them were recognized successfully at a rate greater than 50%, while the success of other product recognitions varied from 0% to 44% of the cases. The typical process in food technology, in which raw material is processed into a final product, usually produces a complex and multiphase structure. The thermodynamic state of the ingredients and phases in the products containing large amounts of water reaches uniformity in short time, and therefore, the physicochemical equilibrium is established. This is caused mostly by the existing possibilities of molecular movements and molecule mobility in water solution. These movements are remarkably reduced or blocked in low-moisture products, resulting in an internal stress. The magnitude of this stress depends on the parameters of the technological process. In other words, the stresses formed in the product by a technological process cannot relax. The example of blocked relaxation is the process of drying. In extruded products, the internal stresses are created during kneading and flow through a high-temperature nozzle, and later, they are preserved in the material owing to immediate evaporation of water. Generation of AE signal in dry food products is caused by the sudden release of accumulated elastic energy. This effect can be invoked by crack formation and propagation, as well as by the destruction processes due to application of external stress, for example, during biting. Crack formation and propagation are accompanied by high-level AE, especially in more crunchy, i.e., less deformable and more resistive-to-stress materials. The nature of this phenomenon is not entirely recognized. Its understanding is complicated by the AE signal transformation, where the signal travels across its path from the crack, through the matter under test, toward the AE sensor. Dry food products often have a cellular structure that highly influences the nature of the emitted signal. The interior of the cells is filled with air, and the AE source is formed by cracking the cell walls [29]. The destruction process, including the greater thickness of dry food products causes the generation of a series of sharp electron-acoustic (EA) waveforms, and each AE pulse is invoked by destroying the single cell wall. When the large destruction processes run simultaneously, a specific sound is emitted. The sound amplitude generated by the product of that kind is proportional to that produced by a single cell break and to the number of cell walls per dimension unit of the investigated product [16]. To explain the anomalies in acoustic properties of such two-phase material, complicated formulas and models are proposed, such as the approach described by Malecki [2]. The consequence of two different phases existing in a medium is that the two bands of wavelengths have privileged conditions to propagate, one band in the first phase and the other in the second phase. Spectral characteristics are valuable sources of parameters of the generated AE signal. Signal differences presented in the averaged spectral characteristics of crisp bread baked by the traditional method, biscuits, crackers, and cornflakes are presented in Figure 24.7. Each product shown in Figure 24.7 emits a signal with unique spectral characteristics. Both low- and high-frequency components are present in these characteristics. The chemical composition influences fewer characteristics than the way the product is produced. Regions in the frequency domain where the high level of power-spectrum function is observed for different food products are as follows [37]: . . . .

Rye-crisp-bread baked by traditional method: 2–8 kHz and 13–14 kHz Crackers: 2–3 kHz and 14 kHz Low-fat biscuits: 2–6 kHz and 12–14 kHz Rich-in-fat biscuits: 2–4 kHz and 13 kHz


Crisp bread baked by traditional methods

6 0 ⫺6 ⫺12 ⫺18


0

3

6

12

15

9

12

15

12

15

12

15

Crackers

6 0 ⫺6 ⫺12 ⫺18 0 6 0 ⫺6 ⫺12 ⫺18

9

3

6

Biscuits with 8% fat

0

3

6

9

Biscuits with 11% fat

6 0 ⫺6 ⫺12 ⫺18 0

3

6

9

Frequency (kHz) Figure 24.7 Spectral characteristics of different food products recorded during a bending test.

Food technologists attribute great importance to water content and its migration when testing the texture changes of food. Interesting experimental results were obtained by Tesch et al. [17] who attempted to equilibrate the water activities (aw) of cheese balls and croutons stored in desiccators. In their study, food samples with water activity ranging from 0.11 to 0.75 were investigated. During specimens’ compression process using the Instron machine, acoustic signatures of the crushing process were recorded using a microphone probe. The following AE signal descriptors were used for product characterization: mean (averaged) value of the time signal, peak amplitude and amplitude standard deviation of time signal, and mean magnitude of its Fourier spectrum. Although the scatter of all four descriptors was ‘‘fairly large,’’ the researchers concluded that all the descriptors clearly illustrated the fact that at aw above 0.65, the AE signal practically vanished, owing to the plasticizing effect of water. Maximum product crispness was observed at aw ¼ 0.23. For higher water activity levels, AE intensity loss was observed. Other investigations [16,25], made with different texturometers for crushing the products with different moisture levels, generally agreed with those obtained by Tesch et al. [17]. However, no distinct maximum limit of the AE signal intensity descriptors was observed in some products, but AE descriptors measured in all of them demonstrated a remarkable variation of the AE intensity in the range of aw from 0.4 to 0.6. This was confirmed by sensory tests for crispness done by the trained assessors. There are several studies on the sound generated by extruded crisp bread [16,24,25] that describe the results obtained using a microphone probe. In the experiment done by Daceremont [16], a sound transmitted by a bone tract was registered. To perform this, the probe was placed on the human mastoid. The other probe was capable of registering a sound transmitted by an air tract. The conclusion of the investigation was that the sounds in the frequency band of 5–10 kHz propagate by both the air and the bone tract, while those in the frequency band of 1–2 kHz propagate only by air.


The number of AE counts in biscuits was also determined by Chen et al. [20] to determine their crunchiness. Registered with a microphone probe, the AE signal was processed in the acoustic detector of the acoustic emission detector (AED) type. The procedure was designed to analyze the sound intensity, and on that basis, to recognize the acoustic events of 250 ms duration. The instrumentation was capable of altering the event duration gate to fit the different food products: chips and products with crunchiness lower than that of chips. They concluded that the method used in their experiment was capable of detecting the variations of the crunchiness in the materials they examined. The above-presented data show that the AE of food can be analyzed in multiple ways and expressed by a number of physical indices. Descriptors used in the research of acoustic properties of food are listed in Table 24.2, which shows the complexity of the AE analysis. Volume stress can exist not just in dry products. Its presence was also confirmed by Konstankiewicz and Zdunek [13] in hydrated bodies under conditions where they contain liquid in the cell interior, which exerts a pressure onto the surrounding cell wall. In a tissue that is under turgor pressure, cell walls are stressed and are perceived as firmness by the consumers. Fruits and vegetable tissues have a very complex structure, in which the basic elements are cell walls, responsible for the mechanical strength. This strength varies among cell walls, and hence, the natural dispersion of internal stress and stored energy can be observed. A part of that energy remains in stressed cell membranes (i.e., in wall and plasmalemma) and in the intercellular layers of the lamellae. Application of the AE method to investigate moist plant tissues requires the identification of possible AE sources. Changes in the tissue structure, evoked by external forces or internal stress due to moisture variations, are one of the possible origins of AE [38]. Sound emitted by fresh fruits and vegetables is caused by a rapid loss of intercellular pressure [33]. Zdunek and Konstankiewicz [39] presented a description of the processes, believed to be the possible reasons for the generation of AE signal. Deformation of the plant tissue by external force causes changes in both the state of internal stress and internal energy. If the external force reaches a critical level, cellular walls would break and a new configuration of stresses would be formed. Evolution of the destruction process of such a structure would depend on the structural parameters of the investigated material, cell turgidity level, and properties of external factors (e.g., load increase speed) [13,40]. The phenomenon described earlier, with respect to the crack formation and propagation in plant tissue, becomes a source of AE signal (Figure 24.8). The AE method enables the investigation of the influence of different factors on the plant tissuedestruction processes. The analysis of the dependence of AE descriptors on the mechanical factors (e.g., stress level) helps in determining the mechanical conditions leading to the onset of the planttissue breaking processes. Moreover, the assessment of the amount of the breaking processes is also possible [39]. The effectiveness of the AE method in recognizing and investigating the plant-tissue breaking processes was observed to be high [40]. It was concluded that there is a possibility of applying the described method in the investigation of potato tubers, and the frequency spectrum of the registered AE signal has specific maxima at 60, 75, 115, and 135 kHz. The other result described was that the initial AE activity occurs at product elongation, equaled to approximately 65% of the critical one.

24.6 CALCULATION OF ACOUSTIC EMISSION DESCRIPTORS IN FOOD The acoustic activity of food can be expressed by several descriptors. These include AE signal energy, number of AE events, energy of the AE event, partition-power spectrum slope, and acoustograms. The AE signal energy is calculated based on the recorded time-dependent AE signal. The calculation is carried out according to Equation 24.5.


Table 24.2 Descriptors of AE Measured Instrumentally Acoustic Parameters Amplitude

Number of sound bursts Number of acoustic events

Energy

Mean sound pressure level (N=m2)

Mean sound pressure level (dB) Acoustic intensity (W=m2) Partition-power spectrum

Fast Fourier analysis


Products

Analyzed Frequency Range

References

Carrot, almond, extruded bread, dry biscuit

10 Hz–10 kHz, 20 Hz–20 kHz

[16]

Snack food Dry apple, dry carrot Cereal flakes

1 Hz–10 kHz 1 Hz–15 kHz 0–10 kHz

[59] [61] [19]

Biscuits

1 Hz–12 kHz

[20]

Crisp bread Cereal flakes

1 Hz–15 kHz 1 Hz–15 kHz

[38,44,48] [63]

Apple, potato

100 Hz–1 MHz

[57]

Cornflakes Celery fresh, turnips, wafer

1 Hz–15 kHz

[62] [60]

Biscuits Crisp bread


[20] [44,48]

Cereal flakes Apple, potato

1 Hz–15 kHz 100 Hz–1 MHz

[63] [57]

Cornflakes Sponge fingers, wafer biscuit, ice cream wafers


[62] [15]

Crisp bread Bread, crackers, cake


[21,45] [43,48]

Cornflakes

1 Hz–15 kHz

[63]

Cornflakes Sponge fingers, wafer biscuit, ice cream wafers

6.3 Hz–20 kHz 0.5–3.3 kHz

[62] [64]

Extruded bread, dry biscuits, toasted rust roll Biscuits, apples

6.3 Hz–20 kHz

[58]

25 Hz–20 kHz

[15]

One food among several Crisp bread, crackers Cereal flakes

0.5–3.3 kHz 1 Hz–15 kHz 1 Hz–15 kHz

[64] [44,48] [63]

Dry apple, dry carrot One food among several

1 Hz–15 kHz 0.5–3.3 kHz

[61] [64]

Apples

25 Hz–20 kHz

[15]

Potato chips Carrot, almond, extruded bread, dry biscuit

0–20 kHz 10 Hz–10 kHz, 20 Hz–20 kHz

[28] [16]

Crisp bread, crackers Cereal flakes Dry apple, dry carrot

1 Hz–15 kHz 1 Hz–15 kHz 1 Hz–15 kHz

[43,48] [63] [61]

Extruded bread, dry biscuits, toasted rusk roll

6.3 Hz–20 kHz

[58]

External force

Internal energy distribution

Modified distribution of internal energy

Types of energy generated in the process

Mechanical work

Heat

AE signal

Figure 24.8 The scheme of AE signal creation in plant tissue. (From Konstankiewicz, K. and Zdunek, A., Acta Agrophis., 24, 87, 1999 (in Polish). With permission.)

Registered AE presented in the amplitude–time coordinates exhibited a number of visible AE bursts caused by the matrix-break processes. More precise signal investigation revealed a typical pattern of bursts among all the records. Within that pattern, a series of AE impulses of similar amplitude and duration occurred. A characteristic noisy and crispy sound of food products was formed by these impulses. Therefore, they can be treated as the integral elements of the analyzed signal and are called AE events. A procedure to calculate the parameters of a single AE event, designed by Ranachowski [41], is capable of analyzing the records of 10 s of AE signal converted into a set of digital samples. The recognition of the occurrence of the AE event is done in the following way: AE event duration is measured as a time when the signal energy, calculated using Equation 24.5, exceeds a preset level. Both the start and the end events were determined on the basis of the energetic criterion mentioned earlier. Under practical conditions, the AE signal energy does not fade to reach a zero level of energy, owing to the accompanying background noise. Because of this, the end of the event is recognized when the energy of the three signal samples is lower than half of the preset level applied to detect the start of the event. When the end of the event is reached, the next two signal samples are omitted to prevent the procedure from the false recognition of the event already processed. Practical examination of the procedure determined the optimal detection threshold for cornflakes to be 2 V and for crisp bread as 3 V. Application of higher thresholds could result in losing of the weaker AE events, while low threshold causes false recognition due to background noise. A software procedure described earlier enables determination of the following parameters of the detected AE events: event duration and peak amplitude of the AE signal within the event, number of events per second, and energy of a single event. The elastic wave propagates in a different way with respect to the frequency. In the 1 kHz and 10 kHz regions, acoustic wavelength, according to Equation 24.1, differs 10 times. Since food texture presents a resonance character to the propagating elastic wave, its wavelength plays a significant role in sound generation. Moreover, low-frequency waves propagate mostly across the air cavities of a product, while the high-frequency components prefer solid-state phases.


The shape of the acoustic-spectral characteristics (Figure 24.3) reveals that, in most of the investigated foods, there are two regions in the frequency domain, where the high level of powerspectrum function is observed. Taking into account this observation, a dimensionless AE descriptor, independent of the sample volume, was proposed. This coefficient is called partition-power spectrum slope (b). According to Equation 24.8, it is calculated as a ratio of AE signal power spectrum registered in the high-frequency range, labeled as Phigh, and AE signal power registered in the low-frequency range, labeled as Plow. Phigh ¼

n7!high X2, kHz

cn ,

Plow ¼

n7!low 2, kHz X

cn ,

b¼

n7!low 1, kHz

n7!high 1, kHz

Phigh Plow

(24:8)

The best measuring conditions for measuring b coefficient exist during the compression test. In this test, elastic waves propagate from the destructed region of the product toward the compressing plate across the larger volume than during the bending process. The researchers of this study suggest that determination of a single AE signal descriptor to characterize complex food products of multilayered composition results in poor effects. Acoustic investigation of an inhomogeneous material can be carried out with the application of an experimental results presentation technique called acoustogram. An acoustogram presents collected data of the process of a product deformation and is a three-dimensional graph. The horizontal axis represents time, the vertical axis indicates the spectral characteristic of the measured signal, and the signal intensity is visualized using color coding. A single graph ‘‘slice’’ is performed by applying Equation 24.7. The acoustograms presented here are constructed with the horizontal resolution of 0.25 s and vertical resolution 22 Hz, in the frequency band of 1–15 kHz. The construction of color scale refers to the reference signal level A0 (equal to 10 mV rms in 22 Hz band), by assuming that signals weaker than 16 dB in relation to A0 would be presented in dark blue. The other signal intensities are presented in different colors. The software designed to create acoustograms also produces a digital image of the data, which enables precise comparison of the product properties. Figures 24.9 and 24.10 demonstrate the application of the described procedure to present experimental data of bending of four samples of flat rye-bread, differing in water activity level and the method of their production. The bands of high level of power-spectrum function are clearly visible in the figure. The effect of signal-energy loss caused by the water sorption process can also be observed. In the latter case, high-frequency components increase with respect to the low-frequency components. The acoustograms were also applied for testing the bread staling process [42]. a w = 0.037

12 9 6 3 1

0

1

2

3

Time (s)

4

a w = 0.750

15 Frequency (kHz)

Frequency (kHz)

15

5

12 9 6 3 1

0

1

2

3

4

5

Time (s)

Figure 24.9 Acoustograms of bending of rye-extruded crisp bread as affected by water activity.


a w = 0.750 15

12

12

Frequency (kHz)

Frequency (kHz)

a w = 0.253 15

9 6 3 1

9 6 3 1

0

2

4

6

8

10

0

2

Time (s)

4

6

8

10

Time (s)

Figure 24.10 Acoustograms of bending of rye-crisp-bread baked by the traditional method, affected by water activity.

The AE method is capable of delivering a useful tool to determine the existing product crispness level. It was proposed by the authors of this chapter to introduce the crispness index x, combining both mechanical and acoustic properties of a tested product. The coefficient can be determined as a ratio of the registered number of AE events and the mechanical work applied to destruct the product under normalized conditions specific to the product type. Hence, x¼

AE number of events W

(24:9)

where Ðt W ¼ u 0 F(t)dt is the work of mechanical destruction of the product (mJ) F(t) is the current level of the force registered by a loading machine during a test (N) u is the velocity of loading cross-head (m s1) t is the test duration (s)

24.7 FACTORS AFFECTING ACOUSTIC EMISSION IN FOOD Investigations have been made to determine the influence of the mechanical-loading speed and the kind of product manufacturing method on the parameters of the generated AE signal [43]. The influence of the mechanical-loading method at the identical loading speed of 20 mm=min on the AE signal energy is presented in Figure 24.11. The results showed that the generated AE signal energy has a significantly greater level when compression is applied in comparison with bending. This effect is caused by the varied size of the contact of loading tool-tested product areas. In addition, the process of product destruction differed during both the tests, since during bending, the contact occupies a rather small specimen area that evokes only a single process of product cell-breaking. The other difference is that at the upper specimen surface, the loading tool evokes compression stresses, while at the lower surface stretching stresses exist. When compression as a method of loading was used, the area of loading tool-tested product was large and caused higher AE energy level. Contrasting effects were observed during the investigation of a product containing more fat, i.e., crackers [44]. It is worth mentioning that the spectral characteristics derived for both the destruction methods, presented in Figures 24.12 and 24.13, are different. Both the destruction methods evoke AE signal in low- and high-frequency regions. The amount of low-frequency band energy dominates during bending, while high-frequency band dominates during compression test.


600

Partition power spectrum

Bending 20 mm/min Compression 20 mm/min 400

200

0

1

2

3

4

5

6 7 8 9 10 11 12 13 14 15 Frequency (kHz)

Figure 24.11 Effect of two mechanical-loading methods on the spectral characteristics (presented in 1 kHz bands) of the generated AE signal in low-fat cakes.

When loading speeds of bending processes were set to 20 and 50 mm min1, the spectral characteristics of AE signal presented the activity in the 13 kHz region. The same speeds applied during the compression test resulted in the shift of the significant activity level to the band of 1–7 kHz and 12–14 kHz (Figures 24.12 through 24.15). Increase in mechanical-loading speed led to the increase in AE signal energy, regardless of the kind of loading method applied (Figures 24.11 and 24.14). The reason of that effect can be that when lower speed of mechanical loading is used, the process of energy dissipation in the product becomes more stable, and low number of micro-cracks arising evokes low level of the AE signal. On the contrary, high rate of the release of accumulated energy causes high intensity of AE. The analysis of the spectral characteristics, presented in Figures 24.12 and 24.13, leads to the conclusion that the compression test should be a recommended method of mechanical testing of the cereal products. Application of the compression test enables increased contact between the


Bending, 20 mm/min 6 0 ⫺6 ⫺12 ⫺18 0

3

6

9

12

15

12

15

Compression, 20 mm/min 6 0 ⫺6 ⫺12 ⫺18 0

3

6

9

Frequency (kHz) Figure 24.12 Comparison of the two mechanical-loading methods on the spectral characteristics of the generated AE signal in low-fat cakes. The upper graph illustrates the bending process and the lower graph illustrates the compression process at 20 mm=min.



Bending, 50 mm/min

6 0 ⫺6 ⫺12 ⫺18 0

3

6

9

12

15

12

15

Compression, 50 mm/min 6 0 ⫺6 ⫺12 ⫺18 0

3

6 9 Frequency (kHz)

Figure 24.13 Comparison of the two mechanical-loading methods on the spectral characteristics of the generated AE signal in low-fat cakes. The upper graph illustrates the bending process and the lower graph illustrates the compression process at 50 mm=min.

elastic waves propagating in the specimen and the flat plate of the loading tool—guiding the waves to the sensor. A similar comparison of the methods of the specimen loading was made with respect to the crisp bread. During the compression or the bending tests, the spectral characteristic of the AE signal presented the activity in 1–3 kHz and 12–15 kHz regions. Acoustic energy was generated at a higher level during the compression test than that during the bending test of the crisp bread. The dependence of the partition-power spectrum slope, b, on water activity measured in extruded ryeand wheat-crisp-bread specimens is presented in Figure 24.16. Till the level of aw ffi 0.4, the b coefficient remained unchanged, whereas after crossing of this level a significant increase in b was _ and observed. A similar trend in b changes was observed in extruded wheat-crisp-bread specimens crisp bread baked by the traditional method [21,37,45]. The other character of spectral characteristics versus water activity was observed in products compressed in bulk, such as cornflakes [26,46]. Regions of a high level of power spectrum in this 600

Partition-power spectrum

Bending 20 mm/min Bending 50 mm/min 400

200

0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Frequency (kHz) Figure 24.14 Effect of the two mechanical-loading velocities on the spectral characteristics (presented in 1 kHz bands) of the generated AE signal in low-fat cakes during bending test.


900

Partition power spectrum

Compression 20 mm/min Compression 50 mm/min 600

300

0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Frequency (kHz) Figure 24.15 Effect of the two mechanical-loading velocities on the spectral characteristics (presented in 1 kHz bands) of the generated AE signal in low-fat cakes during compression test.

product were situated in the frequency bands of 4–9 kHz and 14–15 kHz [47]. In the range of products of water activity from 0.05 to 0.45, the partition-power spectrum slope was constant (Figure 24.17) and at higher water activities b decreased. Application of different manufacturing technologies (backing with traditional method and extrusion) results in the formation of different product structures. The dependence of the detected number of events per second on water activity—calculated for three food products: extruded wheat, rye bread, and rye bread baked by traditional method—is presented in Figure 24.18. However, from the statistical point of view, no valid difference was found when the events rate for the extruded wheat and rye bread at the same water activity level was compared. Spectral characteristics of these products are presented in Figure 24.19. The experimental results presented in Figures 24.18 and 24.19 lead to the conclusion that the manufacturing technology affects the parameters of the emitted AE more significantly than the chemical composition of the investigated products. The influence of the applied manufacturing technology on the acoustic properties of the

Bending

Compression

120

40 Wheat Rye

90

30

60

20

b

b

Wheat Rye

30

0 (A)

10

0

0.2

0.4 Water activity

0.6

0

0.8 (B)

0

0.2

0.4

0.6

0.8

Water activity

Figure 24.16 Influence of the mode of deformation and water activity of extruded rye- and wheat-crisp-bread on partition-power spectrum slope b. (A) Bending and (B) compression.


Partition-power spectrum slope, b

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

0.2

0.4 Water activity

0.6

0.8

Figure 24.17 Influence of water activity on the partition-power spectrum slope b for cornflakes. (From Kuropatwa, M., Influence of water activity on mechanical and acoustic properties of corn flakes. M.Sc. thesis, Warsaw Agricultural University (SGGW), Warszawa, 2003 (in Polish).)

product can be most efficiently demonstrated by a comparison of a spectral characteristics of the product [43]. The influence of water activity on acoustic properties of dry foods is very pronounced. The resulting 3 s long records registered in the region of maximal AE signal activity for five levels of water activity are presented in Figure 24.20. The graphical resolution of Figure 24.20 enables the detection of short AE signal bursts of 5 ms duration. When a recording speed of 44,100 signal samples per second was used, the real AE signal record, stored in the memory of a computer consisting of a file of samples registered with a time delay T1, equaled 22.7 ms, successively. The large number of discrete signal samples can be treated as a real image of the effective value (rms) of the emitted AE signal. The discrete form of the signal, stored and processed in the computer, is denoted as n(m T1) and using Equation 24.5, the AE signal energy can be calculated. A remarkable influence of specimen water activity on the AE signal energy is presented in Figure 24.21.

AE events rate per second

500 Wheat extruded Rye extruded Rye traditional

400

300

200

100

0 0

0.2

0.4

0.6

0.8

Water activity Figure 24.18 Influence of the mode of processing and water activity on the number of acoustic events generated in crisp breads.


Wheat-extruded crisp bread, a w = 0.376

6 0 ⫺6 ⫺12 ⫺18 Sound intensity (dB)

0

3

6

9

12

15

Rye-extruded crisp bread, a w = 0.380 6 0 ⫺6 ⫺12 ⫺18 0

3

6

9

12

15

Rye traditional crisp bread, a w = 0.360

6 0 ⫺6 ⫺12 ⫺18 0

3

6

9

12

15

Frequency (kHz)

Figure 24.19 AE signal spectral characteristics of crisp breads. a w = 0.041 ⫺2 ⫺1 0 1 2

Amplitude (V)

0

1

2

3

2

3

2

3

2

3

a w = 0.330

⫺2 ⫺1 0 1 2 0

1 a w = 0.530

⫺2 ⫺1 0 1 2 0

1 a w = 0.750

⫺2 ⫺1 0 1 2 0

1 Time (s)

Figure 24.20 Test results of bending process of extruded wheat-crisp-bread specimens for different water activity levels. Signal amplitude is presented in volts, as is registered at the output of AE sensor with added 100 times amplification. ß 2008 by Taylor & Francis Group, LLC.

AE signal energy (arbitrary units)

4000

3000

2000

1000

0 0

0.2

0.4 Water activity

0.6

0.8

Figure 24.21 Influence of water activity on AE signal energy (Equation 24.5) in flat, extruded wheat bread.

Water activity also affects the number of acoustic events [48], which decrease with increasing water activity (Figure 24.18). This is caused by the altered stress distribution occurring in the moisturized structures [49]. The increase in water content causes dissipation of the elastic energy stored in the material and in turn, reduces the possibility of crack occurrence [50]. It is interesting to note that the energy of a single acoustic event is a little dependent on the water activity of dry foods (Figure 24.22). In crisp-extruded breads, the energy of a single acoustic event was constant in water activities range of 0.03–0.5, and thereafter, it decreased by about 15% [48]. This suggests that even at high water activities, there are domains capable of breaking and generating vibrations with such energy as that emitted in the dry material. The probability of occurrence of such domains decreases with increasing water activity; hence, the number of acoustic events is affected by the water activity and is lower when the wetness of the material is higher.

Energy of single acoustic event

450

400

350

300

250

200

0.0

0.2

0.4 Water activity

0.6

Figure 24.22 Influence of water activity on the energy of a single acoustic event.


0.8


6 0 ⫺6 ⫺12 ⫺18 6 0 ⫺6 ⫺12 ⫺18 6 0 ⫺6 ⫺12 ⫺18 6 0 ⫺6 ⫺12 ⫺18

a w = 0.041

0

3

6

9

12

15

12

15

12

15

12

15

a w = 0.330

0

3

6

9 a w = 0.530

0

3

6

9 a w = 0.750

0

3

6 9 Frequency (kHz)

Figure 24.23 Spectral characteristics of AE signal generated during the bending process of wheat-extruded crisp bread with different water activity levels.

Spectral characteristics of the AE signal generated in the bending process of extruded wheatcrisp-bread specimens with different water activity levels are presented in Figure 24.23. It can be observed that in the investigated material, there are two regions in the frequency domain where the high level of power-spectrum function is observed. For different kinds of extruded crisp bread, these regions are 1–3 kHz and 12–15 kHz. The AE generated during crushing of cereal products constitutes a significant factor of the features, described as crispness and crunchiness. The consumers are capable of distinguishing between crispy and crunchy sounds. A crispy sound is sharp such as that produced while walking on snow or icy ground, whereas the crunchy ones sound longer and are more firm as that produced while walking on gravel or dry leaves [32]. Recent investigations made by the authors of this work were focused on the phenomena of sound generation with respect to the perceived intensity of crispness and crunchiness. The main reason for the loss of these features is the increase in water content in cereals caused by sorption processes from the environment or its migration from the product interior [51]. The investigations done by Marzec et al. [27] and Lewicki et al. [26,37] proved that food remains crispy within a narrow range of water activity, which is a specific value for a given product. American researchers have proposed a theory stating that crispness is a product texture feature mostly of acoustic nature [33,35]. Mohamed and Jowitt [15], and Liu and Tan [52] have stated that crispy sounds are related to the structure breaking processes and these effects appear when the energy accumulated in the product is instantaneously released in the form of elastic waves with acoustic frequencies. Sound generation is also related to crack increase and fast material destruction. The AE signals can be used to detect the changes in the quality of food. An example is shown in Figure 24.24, which presents the influence of the staling process of bread on the AE signal [53]. It is


600 Fresh bread Bread stored for 7 days at RH = 20%

AE signal energy

500 400 300 200

Format

100 0 0

100

200 300 Penetrating tool position (µm)

400

500

Figure 24.24 Changes in acoustic properties of bread crust during storage.

evident that fresh bread shows a crisp crust of about 200 mm thickness, which emits a strong acoustic signal during penetration. The moist crumb is acoustically inactive. During storage, the moisture moving from the interior of the loaf moistens the surface and the crust loses its attractive property. There is no difference in acoustic activity between the crust and the crumb after 7 days of storage of bread. The AE method is capable of delivering a useful tool to determine the existing product crispness level. It is proposed to introduce the crispness index, x, combining both mechanical and acoustic properties of the tested product. The coefficient is calculated according to Equation 24.9. Figure 24.25 presents the influence of water activity on the crispness index, x, measured in two kinds of crisp breads. The products were baked by the traditional method. They were bended with the speed of 20 mm min1 and the system amplification was set to 20 dB. As can be observed, the x coefficient decreases with respect to the increasing water activity, although there are no statistically measurable variations of the coefficient in the region of water activity from 0.2 to 0.4. The sensory tests have also revealed that in that region of water activity, the product is assessed as

Crispness index, number of acoustic events/mJ

180

Rye “Three cereal”

150 120 90 60 30 0 0

0.1

0.2

0.3

0.4

0.5

Water activity Figure 24.25 Effect of water activity on the crispness index of crisp breads.


0.6

0.7

Crispness index, number of acoustic events/mJ

200 150 100 50 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Water activity Figure 24.26 Effect of water activity on the crispness index of crackers.

crispy and acceptable [54]. Hence, the proposed coefficient can be applied as a useful tool to characterize the textural properties of cereal products [44]. The analogical trend of the dependence of crispness coefficient on aw was determined in crackers (Figure 24.26) by performing the bending test at the speed of 60 mm=min and with the system amplification set to 40 dB. It was found that the product crispness decreases exponentially with an increase in water content. Again, in the region of water activity from 0.2 to 0.4, the statistically measurable variations of the crispness index could be neglected. The profile assessment of the texture (quantitative descriptive analysis, QDA) and general sensory assessment of the crackers as a function of water activity were undertaken by Gondek and Marzec [54]. The researchers were focused on the assessment of mechanical (kinetics) and acoustic features of the texture, and these features were considered as optimal means to describe the textural properties of the crunchy and crispy products [29,55]. The investigation proved that the acoustic impression is a more sensitive indicator of the quality changes occurring in the product due to water sorption than mechanical impressions [54].

24.8 CORRELATIONS BETWEEN ACOUSTIC EMISSION DESCRIPTORS AND SENSORY ASSESSMENT OF QUALITY The comparison between the measured number of AE events and the sensory assessment of the sound intensity as a function of water activity is presented in Figure 24.27. Sensory assessment of crackers with water activity from 0.225 to 0.670 was carried out by trained persons using QDA, according to Stone and Sidel [56]. The procedure used in the assessment is described in the Standard ISO 13299:2300 (E). In the preliminary tests, acoustic descriptors accounting for sensory-texture profile were chosen. The loudness of the sound was defined as the intensity of emission generated during the mastication of the material with teeth. A general texture quality was defined as the sensory impression obtained from mechanical and acoustic senses and their harmonization [54]. The investigation revealed that crackers in aw region from 0.225 to 0.490 were acceptable by the consumers, and there were no significant changes in both the investigated parameters in that region. Figures 24.28 and 24.29 present the relationships between acoustic properties assessed by the sensory impression and those measured instrumentally. It is evident that the general quality of the sensory impression is related to the AE energy and to the crispness index, having a logarithmictype relationship. Although both AE energy and crispness index change with water activity in the range from 0.225 to 0.490, the total sensory quality changes insignificantly within this aw range.


Sensory acoustic intensity

8

a w = 0.225

Product not accepted by panelists

a w = 0.330

6

a w = 0.490 4

a w = 0.570

2

Product accepted by panelists

a w = 0.670 0 0

20

40 60 80 Number of acoustic events

100

120

Figure 24.27 Relationship between the measured number of acoustic events and the sensory assessment of sound intensity of crackers as a function of water activity.

a w = 0.490

a w = 0.330

8 Sensory general quality

a w = 0.225 6

4

a w = 0.570

2

a w = 0.670

0 0

20

40 60 Acoustic energy

80

100

Figure 24.28 Relationship between the measured acoustic energy and the sensory assessment on the general quality of crackers as a function of water activity.

10 Sensory general quality

a w = 0.490 8 a w = 0.225

6

a w = 0.330

4 a w = 0.570 2

a w = 0.670

0 0

20

40 Crispress index

60

80

Figure 24.29 Relationship between crispness index and the general sensory quality of crackers as a function of water activity.


24.9 RECAPITULATION The AE method can be applied to determine the quality characteristics and microstructure changes in the investigated material subjected to mechanical stress. The analysis of the emitted sound delivers the information about the nature of the destruction processes and enables detection of the onset of these phenomena that appears before the occurrence of the maximum material stress. The determined AE signal energy contains information about the scale of the destruction processes. Frequency of the emitted sound is closely related to the kind of destruction process and the type of the tested material, and therefore, it can be used as a descriptor for the quality assessment of fruits and vegetables and for the assessment of the crispness index for snacks. Examples of the AE signal processing described in this work are the evidence for the possibility of determining the AE descriptors in food products. These descriptors can be applied in quality testing and determination of crispness in cereals. To perform a proper analysis of the registered AE signal, sufficient knowledge on the influence of product texture on the image of emitted signal is needed. It is also worth mentioning that a variety of software tools to process AE signals, both in frequency and time domain, have been recently prepared to improve the research work.

ACKNOWLEDGMENT The authors gratefully acknowledge the financial support of the State Committee for Scientific Research (Grant No. 3PO6T 040 25).

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12.

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Appendix A Table A.1 Vapor Pressure, Saturated Liquid Density, Thermal Expansion Coefficient, Compressibility Coefficient, and Refractive Index of Water

t (8C)

Vapor Pressure (kPa)

Saturated Liquid Density (kg=m3)

Thermal Expansion Coefficient (K 1 10 3)

Compressibility Coefficient (atm 1 10 6)

Refractive Na0.5893

0 10

0.6102 1.2259

999.84 999.70

0.07 0.088

50.6 48.6

1.33464 1.33389

20

2.3349

998.20

0.207

47.0

1.33299

25 30 40

3.1634 4.2370 7.3685

997.05 995.65 992.22

0.255 0.303 0.385

46.5 46.0 45.3

1.33287 1.33192 1.33051

50 60

12.3234 19.8984

988.05 983.21

0.457 0.523

45.0 45.0

1.32894 1.32718

70

31.1282

977.79

0.585

45.2

1.32511

80 90

47.3117 70.0485

971.83 965.32

0.643 0.698

45.7 46.5

1.32287 1.32050

100

101.2300

958.35

0.752

48.0

1.31783

t (8C)

Heat Capacity (kJ=kg K)

Latent Heat of Vaporization (kJ=kg)

Enthalpy (kJ=kg)

Entropy (kJ=kg K)

Free Energy (kJ=kg)

Velocity of Sound (m=s)

0 10

4.2174 4.1919

2500.5 2476.9

0.000 42.03

0.0000 0.1511

0.000 42.03

1402.74 1447.59

20 25

4.1816 4.1703

2453.4 2441.7

30

4.1782

40 50

4.1783 4.1804

60

4.1841

2357.9

250.91

70 80 90

4.1893 4.1961 4.2048

2333.1 2307.8 2281.9

292.78 334.72 376.75

100

4.2156

2255.5

418.88

1.3063

83.86 104.74

0.2963 0.3669

83.86 104.74

1482.66 1497.00

2429.9

125.61

0.4364

146.46

1509.44

2406.2 2382.2

167.34 209.11

0.5718 0.7031

167.33 209.10

1529.18 1542.87

0.8304

250.89

1551.30

0.9542 1.0747 1.1920

292.75 334.67 376.68

1555.12 1554.81 1550.79

418.77

1543.41 (continued)


Table A.1 (continued) Vapor Pressure, Saturated Liquid Density, Thermal Expansion Coefficient, Compressibility Coefficient, and Refractive Index of Water

t (8C) 0 10

Dielectric Constant

Viscosity (Pa s) 10

87.69 83.82

Thermal Conductivity (W=m K)

3

1.788 1.305

Surface Tension 10 (N=m)

0.550 0.576

2

Electrical Conductivity

Enthalpy of Ionization (kJ=mol)

1.61 2.85

62.81 59.64

75.62 74.20

20

80.08

1.004

0.598

72.75

4.94

57.00

25 30

78.25 76.49

0.903 0.801

0.608 0.617

71.96 71.15

6.34 8.04

55.84 54.75

40

73.02

0.653

0.633

69.55

12.53

52.75

50 60 70

69.70 66.51 63.45

0.550 0.470 0.406

0.647 0.658 0.667

67.90 66.17 64.41

18.90 27.58 38.93

50.90 49.13 47.39

80

60.54

0.355

0.675

62.60

53.03

45.64

90 100

57.77 55.15

0.315 0.282

0.680 0.683

60.74 58.84

69.65 88.10

43.86 42.05

Source: Horvath, A.L. Handbook of Aqueous Electrolyte Solutions, Ellis Horwood, England, 1985. Table A.2 Physical Constants of Water Property

Values

Molecular weight

18.01534

Melting point at 101.32 kPa (1 atm) Boiling point at 101.32 kPa (1 atm)

08C 1008C

Critical temperature

374.158C

Triple point Heat of fusion at 08C

0.00998C and 610.4 kPa 334.0 kJ=kg

Heat of vaporization at 1008C

2257.2 kJ=kg

Heat of sublimation at 08C

2828.3 kJ=kg

Source: Fennema, O.R., Food Chemistry, 2nd ed., Fennema, O.R., ed., Marcel Dekker, New York, 1985, pp. 23–67. Table A.3 Vapor Pressures of Water and Ice at Subfreezing Temperatures Vapor Pressure (kPa) Iceb

Water Activity

b

0.6104

1.000c

5 10

b

0.4216 0.2865b

0.4016 0.2599

0.953 0.907

15 20

0.1914b 0.1254d

0.1654 0.1034

0.864 0.820

25

0.0806d

0.0635

0.790

30 40

0.0509d 0.0189d

0.0381 0.1290

0.750 0.680

50

0.0064d

0.0039

0.620

t (8C) 0

Liquid Watera 0.6104

Source: Fennema, O.R., Food Chemistry, 2nd ed., Fennema, O.R., ed., Marcel Dekker, New York, 1985, pp. 23–67. Supercooled at all temperatures except degree centigrade. b Observed data. c Applies only to pure water. d Calculated data. a


Appendix B Table B.1 Properties of Saturated Steam

t (8C)


Specific Volume (m3=kg)

Enthalpy (kJ=kg)

Entropy (kJ=kg K)

Liquid

Saturated Vapor

Liquid (Hc)

Saturated Vapor (Hv)

Saturated Liquid

Vapor

0.01 3

0.6113 0.7575

0.0010002 0.0010001

206.1360 168.1320

0.00 12.57

2501.4 2506.9

0.0000 0.0457

9.1562 9.0773

6 9

0.9349 1.1477

0.0010001 0.0010003

137.7340 113.3680

25.20 37.80

2512.4 2517.9

0.0912 0.1362

9.0003 8.9253

12 15

1.4022 1.7051

0.0010005 0.0010009

93.7840 77.9260

50.41 62.99

2523.4 2528.9

0.1806 0.2245

8.8524 8.7814

18

2.064

0.0010014

65.0380

75.58

2534.4

0.2679

8.7123

21 24

2.487 2.985

0.0010020 0.0010027

54.5140 45.8830

88.14 100.70

2539.9 2545.4

0.3109 0.3534

8.6450 8.5794

27

3.567

0.0010035

38.7740

113.25

2550.8

0.3954

8.5156

30 33 36

4.246 5.034 5.947

0.0010043 0.0010053 0.0010063

32.8940 28.0110 23.9400

125.79 138.33 150.86

2556.3 2561.7 2567.1

0.4369 0.4781 0.5188

8.4533 8.3927 8.3336

40

7.384

0.0010078

19.5230

167.57

2574.3

0.5725

8.2570

45 50

9.593 12.349

0.0010099 0.0010121

15.2580 12.0320

188.45 209.33

2583.2 2592.1

0.6387 0.7038

8.1648 8.0763

55 60

15.758 19.940

0.0010146 0.0010172

9.56800 7.67100

230.23 251.13

2600.9 2609.6

0.7679 0.8312

7.9913 7.9096

65 70

25.030 31.190

0.0010199 0.0010228

6.19700 5.04200

272.06 292.98

2618.3 2626.8

0.8935 0.9549

7.8310 7.7553

75 80

38.580 47.390

0.0010259 0.0010291

4.13100 3.40700

313.93 334.91

2635.3 2643.7

1.0155 1.0753

7.6824 7.6122

85

57.830

0.0010325

2.82800

355.90

2651.9

1.1343

7.5445

90 95

70.140 84.550

0.0010360 0.0010397

2.36100 1.98190

376.92 397.96

2660.1 2668.1

1.1925 1.2500

7.4791 7.4159

100

101.350

0.0010435

1.67290

419.04

2676.1

1.3069

7.3549

105 110 115

120.820 143.270 169.060

0.0010475 0.0010516 0.0010559

1.41940 1.21020 1.03660

440.15 461.30 482.48

2683.8 2691.5 2699.0

1.3630 1.4185 1.4734

7.2958 7.2387 7.1833

120

198.530

0.0010603

0.89190

503.71

2706.3

1.5276

7.1296

125 130

232.100 270.100

0.0010649 0.0010697

0.77060 0.66850

524.99 546.31

2713.5 2720.5

1.5813 1.6344

7.0775 7.0269

135 140

313.000 316.300

0.0010746 0.0010797

0.58220 0.50890

567.69 589.13

2727.3 2733.9

1.6870 1.7394

6.9777 6.9299

145

415.400

0.0010850

0.44630

610.63

2740.3

1.7907

6.8833 (continued)


Table B.1 (continued) Properties of Saturated Steam

t (8C)


Specific Volume (m3=kg)

Entropy (kJ=kg K)

Enthalpy (kJ=kg)

Liquid

Saturated Vapor

Liquid (Hc)

Saturated Vapor (Hv)

Saturated Liquid

Vapor

150

475.800

0.0010905

0.39280

632.20

2746.5

1.8418

6.8379

155 160

543.100 617.800

0.0010961 0.0011020

0.34680 0.30710

653.84 675.55

2752.4 2758.1

1.8925 1.9427

6.7935 6.7502

165 170

700.500 791.700

0.0011080 0.0011143

0.27270 0.24280

697.34 719.21

2763.5 2768.7

1.9925 1.0419

6.7078 6.6663

175 180

892.000 1002.100

0.0011207 0.0011274

0.21680 0.19405

741.17 763.22

2773.6 2778.2

2.0909 2.1396

6.6256 6.5857

190 200

1254.400 1553.800

0.0011444 0.0011565

0.15654 0.12736

807.62 852.45

2786.4 2793.2

2.2359 2.3309

6.5079 6.4323

225

2548.000

0.0011992

0.07849

966.78

2803.3

2.5639

6.2503

250 275

3973.000 5942.000

0.0012512 0.0013168

0.05013 0.03279

1085.36 1210.07

2801.5 2785.0

2.7927 3.0208

6.0730 5.8938

300

8581.000

0.0010436

0.02167

1344.00

2749.0

3.2534

5.7045

Source: Keenan, J.H., Keyes, F.G., Hill, P.G., and Moore, J.G., Steam Tables: Thermodynamic Properties of Water Including Vapor, Liquid, and Solid Phases (Metric Measurements), Wiley, New York, 1969.

Table B.2 Specific Volume of Superheated Steam Specific Volume (m3=kg) Temperature (8C) Ts (8C)

100

150

200

250

300

360

420

500

10 50

45.81 81.33

17.196 3.4180

19.512 3.8890

21.825 4.3560

24.136 4.8200

26.445 5.2840

29.216 5.8390

31.986 6.3940

35.679 7.1340

75

91.78

2.2700

2.5870

2.9000

3.2110

3.5200

3.8910

4.2620

4.7550

100 150 400

99.63 111.37 143.63

1.6958

1.9364 1.2853 0.4708

2.1720 1.4443 0.5342

2.4060 1.6012 0.5951

2.6390 1.7570 0.6458

2.9170 1.9432 0.7257

3.1950 2.1290 0.7960

3.5650 2.3760 0.8893

700

164.97

0.2999

0.3363

0.3714

0.4126

0.4533

0.5070

1000 1500

179.91 198.32

0.2060 0.1325

0.2327 0.1520

0.2579 0.1697

0.2873 0.1899

0.3162 0.2095

0.3541 0.2352

P (kPa)

2000

212.42

0.1114

0.1255

0.1411

0.1562

0.1757

2500 3000

223.99 233.90

0.0870 0.0706

0.0989 0.0811

0.1119 0.0923

0.1241 0.1028

0.1399 0.1162



Table B.3 Enthalpy Volume of Superheated Steam Enthalpy (kJ=kg) Temperature (8C) Ts (8C)

100

150

200

250

300

360

420

500

10 50

45.81 81.33

2687.5 2682.5

2783.0 2780.1

2879.5 2877.7

2977.3 2976.0

3076.5 3075.5

3197.6 3196.8

3320.9 3320.4

3489.1 3488.7

75

91.78

2679.4

100 150

99.63 111.37

2676.2

400 700

143.63 164.97

1000 1500

179.91 198.32

2000 2500 3000

P (kPa)

2876.5

2975.2

3074.9

3196.4

3320.0

3488.4

2776.4 2772.6

277.82

2875.3 2872.9

2974.3 2972.7

3074.3 3073.1

3195.9 3195.0

3319.6 3318.9

3488.1 3487.6

2752.8

2860.5 2844.8

2964.2 2953.6

3066.8 3059.1

3190.3 3184.7

3315.3 3310.9

3484.9 3481.7

2827.9 2796.8

2942.6 2923.3

3051.2 3037.6

3178.9 3169.2

3306.5 3299.1

3478.5 3473.1

212.42 223.99

2902.5 2880.1

3023.5 3008.8

3159.3 3149.1

3291.6 3284.0

3467.6 3462.1

233.90

2855.8

2993.5

3138.7

3276.3

3456.5




Appendix C Table C.1 Bessel Functions of First Kind x

J0(x)

J1(x)

x

J0(x)

J1(x)

0.0 0.1

1.000 0.988

0.000 0.050

2.50 2.60

0.048 0.097

0.497 0.471

0.2 0.3

0.990 0.978

0.100 0.148

2.70 2.80

0.142 0.185

0.442 0.410

0.4 0.5 0.6

0.960

0.196

2.90

0.224

0.375

0.938 0.912

0.242 0.287

3.00 3.10

0.260 0.292

0.339 0.301

0.7 0.8 0.9

0.881 0.846 0.808

0.329 0.369 0.406

3.20 3.30 3.40

0.320 0.344 0.364

0.261 0.221 0.179

1.0

0.765

0.440

3.50

0.380

0.137

1.1 1.2

0.720 0.671

0.471 0.498

3.60 3.70

0.392 0.399

0.095 0.054

1.3

0.646

0.511

3.80

0.402

0.013

1.4 1.5

0.567 0.512

0.542 0.558

3.90 4.00

0.402 0.397

0.027 0.066

1.6 1.7

0.455 0.398

0.570 0.578

4.10 4.20

0.389 0.376

0.103 0.139

1.8 1.9

0.340 0.282

0.582 0.581

4.30 4.40

0.361 0.342

0.172 0.203

2.0

0.224

0.577

4.50

0.320

0.231

2.1 2.2

0.167 0.110

0.568 0.556

4.60 4.70

0.296 0.269

0.256 0.279

2.3 2.4

0.055 0.002

0.540 0.520

4.80 4.90 5.00

0.240 0.210 0.178

0.298 0.315 0.328

Source: Tabulated values are available in the Handbook of Mathematical Functions, U.S. Department of Commerce, Applied Mathematics Series No. 55 (cited by Whitaker, S., Elementary Heat Transfer Analysis, Pergamon Press, New York, 1976).



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