Modelling Drying Processes a Reaction Engineering Approach

more information - www.cambridge.org/9781107012103

Modelling Drying Processes This comprehensive summary of the state-of-the-art and the ideas behind the reaction engineering approach (REA) to drying processes is an ideal resource for researchers, academics and industry practitioners. Starting with the formulation, modelling and applications of the lumped-REA, it goes on to detail the use of the REA to describe local evaporation and condensation, and its coupling with equations of conservation of heat and mass transfer, called the spatial-REA, to model non-equilibrium multiphase drying. Finally, it summarises other established drying models, discussing their features, limitations and comparisons with the REA. Application examples featured throughout help fine-tune the models and implement them for process design, and the evaluation of existing drying processes and product quality during drying. Further uses of the principles of REA are demonstrated, including computational fluid dynamics-based modelling, and further expanded to model other simultaneous heat and mass transfer processes. Xiao Dong Chen is currently the 1000-talent Chair Professor of Chemical Engineering at Xiamen University in China, and the Head of Department of Chemical and Biochemical Engineering. He held previously Chair Professorships of Chemical Engineering at Auckland University, New Zealand, and Monash University, Australia, respectively from 2001 to 2010. He is now a fractional Professor of Chemical Engineering and the CoDirector of the Biotechnology and Food Engineering Research Laboratory at Monash University, Australia. He is an Elected Fellow of Royal Society of NZ, Australian Academy of Technological Sciences and Engineering, and IChemE. Aditya Putranto holds a BE of Chemical Engineering from Bandung Institute of Technology, Indonesia and a Master of Food Engineering from University of New South Wales, Australia. He has a Ph.D. in Chemical Engineering from Monash University, Australia. He has worked in Indonesia as lecturer in Parahyangan Catholic University. His research area is heat and mass transfer. He has published a dozen journal papers in peer-reviewed hard-core chemical engineering journals.

‘The Reaction Engineering Approach (REA), which captures basic drying physics, is a simple yet effective mathematical model for practical applications of diverse drying processes. The intrinsic “fingerprint” of the drying phenomena can, in principle, be obtained through just one accurate drying experiment. The REA is easy to use with the guidance of featured application examples given in this book. This book is highly recommended for both academics and industry practitioners involved in any aspect of thermal drying.’ Zhanyong Li, Tianjin University of Science and Technology, China ‘An interesting book on a novel approach to mathematical modelling of an important process. Modelling Drying Processes: A Reaction Engineering Approach is the first attempt to summarize the REA to modelling in a single comprehensive reference source.’ Sakamon Devahastin, King Mongkut’s University of Technology Thonburi, Thailand

Modelling Drying Processes A Reaction Engineering Approach XIAO DONG CHEN Monash University, Australia

ADITY A PUTRANTO Monash University, Australia

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107012103 C

Xiao Dong Chen and Aditya Putranto 2013

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2013 Printed and bound in the United Kingdom by the MPG Books Group A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Chen, Xiao Dong. Modelling drying processes : a reaction engineering approach / Xiao Dong Chen, Monash University, Australia, Aditya Putranto, Monash University, Australia. pages cm Includes bibliographical references and index. ISBN 978-1-107-01210-3 (hardback) 1. Drying. 2. Food – Drying. 3. Porous materials – Drying. 4. Polymers – Curing. 5. Lumber – Drying. I. Putranto, Aditya. II. Title. TP363.C528 2013 2013003983 664 .0284 – dc23 ISBN 978-1-107-01210-3 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

List of figures List of tables Preface Historical background 1

2

page ix xxvi xxvii xxx

Introduction

1

1.1 1.2

1

Practical background A ‘microstructural’ discussion of the phenomena of drying moist, porous materials 1.3 The REA to modelling drying 1.3.1 The relevant classical knowledge of physical chemistry 1.3.2 General modelling approaches 1.3.3 Outline of the REA 1.4 Summary References

6 15 15 17 18 29 30

Reaction engineering approach I: Lumped-REA (L-REA)

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2.1 2.2 2.3 2.4

34 36 40 43 43 47 50

2.5

The REA formulation Determination of REA model parameters Coupling the momentum, heat and mass balances Mass or heat transfer limiting 2.4.1 Biot number analysis 2.4.2 Lewis number analysis 2.4.3 Combination of Biot and Lewis numbers Convective drying of particulates or thin layer products modelled using the L-REA 2.5.1 Mathematical modelling of convective drying of droplets of whey protein concentrate (WPC) using the L-REA 2.5.2 Mathematical modelling of convective drying of a mixture of polymer solutions using the L-REA 2.5.3 Results of modelling convective drying of droplets of WPC using the L-REA

50 51 53 55

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2.5.4

Results of modelling convective drying of a thin layer of a mixture of polymer solutions using the L-REA 2.6 Convective drying of thick samples modelled using the L-REA 2.6.1 Formulation of the L-REA for convective drying of thick samples 2.6.2 Prediction of surface sample temperature 2.6.3 Modelling convective drying thick samples of mango tissues using the L-REA 2.6.4 Results of convective drying thick samples of mango tissues using the L-REA 2.7 The intermittent drying of food materials modelled using the L-REA 2.7.1 Mathematical modelling of intermittent drying of food materials using the L-REA 2.7.2 The results of modelling of intermittent drying of food materials using the L-REA 2.7.3 Analysis of surface temperature, surface relative humidity, saturated and surface vapour concentration during intermittent drying 2.8 The intermittent drying of non-food materials under time-varying temperature and humidity modelled using the L-REA 2.8.1 Mathematical modelling using the L-REA 2.8.2 Results of intermittent drying under time-varying temperature and humidity modelled using the L-REA 2.9 The heating of wood under linearly increased gas temperature modelled using the L-REA 2.9.1 Mathematical modelling using the L-REA 2.9.2 Results of modelling wood heating under linearly increased gas temperatures using the L-REA 2.10 The baking of cake modelled using the L-REA 2.10.1 Mathematical modelling of the baking of cake using the L-REA 2.10.2 Results of modelling of the baking of cake using the L-REA 2.11 The infrared-heat drying of a mixture of polymer solutions modelled using the L-REA 2.11.1 Mathematical modelling of the infrared-heat drying of a mixture of polymer solutions using the L-REA 2.11.2 The results of mathematical modelling of infrared-heat drying of a mixture of polymer solutions using the L-REA 2.12 The intermittent drying of a mixture of polymer solutions under time-varying infrared-heat intensity modelled using the L-REA 2.12.1 Mathematical modelling of the intermittent drying of a mixture of polymer solutions under time-varying infrared-heat intensity using the L-REA

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73 80 81 82 88 89 91 95 96 97 100 101 103 104

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2.12.2 Results of modelling the intermittent drying of a mixture of polymer solutions under time-varying infrared heat intensity using the L-REA 2.13 Summary References

106 116 117

Reaction engineering approach II: Spatial-REA (S-REA)

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3.1 3.2 3.3

121 125 127

The S-REA formulation Determination of the S-REA parameters The S-REA for convective drying 3.3.1 Mathematical modelling of convective drying of mango tissues using the S-REA 3.3.2 Mathematical modelling of convective drying of potato tissues using the S-REA 3.3.3 Results of modelling of convective drying of mango tissues using the S-REA 3.3.4 Results of modelling of convective drying of potato tissues using the S-REA 3.4 The S-REA for intermittent drying 3.4.1 The mathematical modelling of intermittent drying using the S-REA 3.4.2 Results of modelling intermittent drying using the S-REA 3.5 The S-REA to wood heating under a constant heating rate 3.5.1 The mathematical modelling of wood heating using the S-REA 3.5.2 The results of modelling wood heating using the S-REA 3.6 The S-REA for the baking of bread 3.6.1 Mathematical modelling of the baking of bread using the S-REA 3.6.2 The results of modelling of the baking of bread using the S-REA 3.7 Summary References 4

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Comparisons of the REA with Fickian-type drying theories, Luikov’s and Whitaker’s approaches 4.1

4.2 4.3

Model formulation 4.1.1 Crank’s effective diffusion 4.1.2 The formulation of effective diffusivity to represent complex drying mechanisms 4.1.3 Several diffusion-based models Boundary conditions’ controversies A diffusion-based model with local evaporation rate 4.3.1 Problems in determining the local evaporation rate 4.3.2 The equilibrium and non-equilibrium multiphase drying models

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169 169 171 172 173 177 179 180 182

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4.4

Comparison of the diffusion-based model and the L-REA on convective drying 4.5 Comparison of the diffusion-based model and the S-REA on convective drying 4.6 Model formulation of Luikov’s approach 4.7 Model formulation of Whitaker’s approach 4.8 Comparison of the L-REA, Luikov’s and Whitaker’s approaches for modelling heat treatment of wood under constant heating rates 4.9 Comparison of the S-REA, Luikov’s and Whitaker’s approaches for modelling heat treatment of wood under constant heating rates 4.10 Summary References Index

185 188 190 195 200 203 206 207 212

Figures

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1.2 1.3

1.4 1.5 1.6 1.7

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1.9

Some traditional dried products. (a) Broccoli-steam blanched and air dried (kindly provided by Ms Xin Jin, Wageningen University, The Netherlands), (b) air-dried Chinese tea leaves (taken at Xiamen University laboratory), (c) spray dried aqueous herbal extract (particle size is about 80 µm) (taken at Xiamen University laboratory), (d) timber stacked for kiln drying (kindly provided by Professor Shusheng Pang (Canterbury University, New Zealand). page 2 Chemical structures of some chemicals: (a) 1, caffeic acid; 2, gallic acid; 3, vanillic acid; (b) 1, cellulose; 2, starch; 3, pectin; (c) human insulin. 4 ‘Air drying’ of a capillary assembly (a bundle) which consists of identical capillaries (diameter and wall material) – a scenario of symmetrical hot air drying of an infinitely large slab filled with the capillaries (modified from Chen, 2007); the air flows along both sides of the symmetrical material. 7 Schematic showing a common scenario of air drying of a moist solid. 9 Packed particulate material. 10 Cellular structures in plant material. 10 (a) Generation of computational domains of corn geometry for the hybrid mixture theory of corn kernels (adapted from Takhar et al. (2011)). (b) The simulated results (isosurface plots of corn moisture content) for a variety of drying conditions. [Reprinted from Journal of Food Engineering, 106, P.S. Takhar, D.E. Maier, O.H. Campanella and G. Chen, Hybrid mixture theory based moisture transport and stress development in corn kernels during drying: Validation and simulation results, 275–282, Copyright (2012), with permission from Elsevier.] 13 Wood cellular structures employed in pore-network modelling of drying of wood. [Reprinted from Drying Technology, 29, P. Perre, A review of modern computational and experimental tools relevant to the field of drying, 1529–1541, Copyright (2012), with permission from Taylor & Francis.] 15 Schematic illustration of the effect of temperature on final liquid water content (qualitatively derived from Equation 1.3.6). 20

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(a) Drying flux versus average water content X¯ ; (b) the CDRC (characteristic drying rate curve). [Reprinted from Chemical Engineering Science, 9, D.A. van Meel, Adiabatic convection batch drying with recirculation of air, 36–44, Copyright (2012), reprinted with permission from Elsevier.] 1.11 Saturated water vapour concentration in air under 1 atm (Equation 1.3.21). 1.12 Schematic diagram showing the heat of drying as a function of water content of a porous solid of concern (when the water content is beyond the point where the heat of drying becomes the latent heat of pure water evaporation, the water content may be called free water). 2.1 Equipment setup of convective drying of milk droplets (a) measuring droplet shrinkage; (b) measuring droplet temperature; (c) measuring mass change. [Reprinted from Chemical Engineering Science, 66, N. Fu, M.W. Woo, S.X.Q. Lin et al., 1738–1747, Copyright (2012), with permission from Elsevier.] (Adapted from Fu et al. (2011) Chemical Engineering Science 66, 1738–1747). 2.2 The deflection of glass filament and a typical standard curve (a) measuring displacement to measure weight loss; (b) correlation between the displacement and the weight. [Reprinted from Chemical Engineering Science, 66, N. Fu, M.W. Woo, S.X.Q. Lin et al., 1738–1747, Copyright (2012), with permission from Elsevier.] 2.3 The relative activation energy of convective drying of 20%wt. skim milk powder at a drying air temperature of 67.5 °C, velocity of 0.45 m s−1 and humidity of 0.0001 kg H2 O kg dry air−1 . [Reprinted from AIChE Journal, 51, X.D Chen and S.X.Q. Lin, Air drying of milk droplet under constant and time-dependent conditions, 1790–1799, Copyright (2012), with permission from John Wiley & Sons, Inc.] 2.4 Schematic diagram showing the plug-flow spray dryer. 2.5 The schematic diagram showing the parameters for the definition of the classical Biot number. [Reprinted from Drying Technology, 23, X.D. Chen, Air drying of food and biological materials – Modified Biot and Lewis number analysis, 2239–2248, Copyright (2012), with permission from Taylor & Francis.] 2.6 The schematic diagram showing the parameters for the definition of the modified Biot number) (Chen–Biot number). [Reprinted from Drying Technology, 23, X.D. Chen, Air drying of food and biological materials – Modified Biot and Lewis number analysis, 2239–2248, Copyright (2012), with permission from Taylor & Francis.] 2.7 The relative activation energy of convective drying of WPC at different drying air temperatures. [Reprinted from Chemical Engineering and Processing, 46, S.X.Q. Lin and X.D. Chen, The reaction engineering approach to modelling the cream and whey protein concentrate droplet drying, 437–443, Copyright (2012), with permission from Elsevier.] 1.10

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List of figures

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2.10

2.11

2.12

2.13

2.14

The droplet diameter changes during convective drying of WPC. [Reprinted from Chemical Engineering and Processing, 46, S.X.Q. Lin and X.D. Chen, The reaction engineering approach to modelling the cream and whey protein concentrate droplet drying, 437–443, Copyright (2012), with permission from Elsevier.] Heat transfer mechanisms of the convective drying of a mixture of polymer solutions. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2012), with permission from Elsevier.] Normalised activation energy and fitted curve of polyvinyl alcohol/glycerol/water under convective drying at an air temperature of 35 °C and relative humidity of 30%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2012), with permission from Elsevier.] The comparison between experimental and model prediction using the L-REA of convective drying of WPC at drying air temperatures of (a) 67.5 °C (b) 87.1 °C (c) 106.6 °C. [Reprinted from Chemical Engineering and Processing, 46, S.X.Q. Lin and X.D. Chen, The reaction engineering approach to modelling the cream and whey protein concentrate droplet drying, 437–443, Copyright (2012), with permission from Elsevier]. Moisture content profile of convective drying at an air temperature of 55 °C, air velocity of 2.8 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.] Product temperature profile of convective drying at an air temperature of 55 °C, air velocity of 2.8 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.] Moisture content profile of convective drying at an air temperature of 35 °C, air velocity of 1 m s−1 and air relative humidity of 30%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol

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2.15

2.16

2.17

2.18

2.19

2.20

(PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.] Product temperature profile of convective drying at an air temperature of 35 °C, air velocity of 1 m s−1 and air relative humidity of 30%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.] Product temperature profile of convective drying at an air temperature of 55 °C, air velocity of 1 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.] Product temperature profile of convective drying at an air temperature of 55 °C, air velocity of 1 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.] The relative activation energy (Ev /Ev,b ) of convective drying of mango tissues at an air velocity of 4 m s−1 , drying air temperature of 55 °C, and air humidity of 0.0134 kg H2 O kg dry air−1 . [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.] Moisture content profile of convective mango tissues at air temperatures of 45, 55, and 65 °C (modelled using the L-REA which incorporates the temperature distribution inside the sample). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.] Temperature profile of convective mango tissues at air temperatures of 45, 55, and 65 °C (modelled using the L-REA which incorporates the temperature distribution inside the sample). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.]

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List of figures

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2.22

2.23

2.24

2.25

2.26

2.27

Moisture content profile of convective mango tissues at air temperatures of 45, 55, and 65 °C (modelled using the L-REA without approximation of temperature distribution inside the sample). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.] Temperature profile of convective mango tissues at air temperatures of 45, 55, and 65 °C (modelled using the L-REA without approximation of temperature distribution inside the sample). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.] Moisture content profile of mango tissues during intermittent drying at a drying air temperature of 45 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Temperature profile of mango tissues during intermittent drying at a drying air temperature of 45 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Moisture content profile of mango tissues during intermittent drying at a drying air temperature of 55 °C and resting at 27 °C [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Temperature profile of mango tissues during intermittent drying at a drying air temperature of 55 °C and resting at 27 °C [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Moisture content profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from

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Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Temperature profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Relative activation energy profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Surface relative humidity profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Saturated vapour concentration and surface temperature profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Surface and saturated vapour concentration profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Surface vapour concentration and surface temperature profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley,

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Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society]. Moisture content profile of intermittent drying of mango tissues with heating (at a drying air temperature of 45 °C) and resting periods of 4000 s each. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Saturated vapour concentration and surface temperature profile of intermittent drying of mango tissues with heating (at a drying air temperature of 45 °C) and resting periods of 4000 s each. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Surface vapour concentration and surface temperature profile of intermittent drying of mango tissues with heating (at a drying air temperature of 45 °C) and resting periods of 4000 s each. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Surface and saturated vapour concentration profile of intermittent drying of mango tissues with heating (at a drying air temperature of 45 °C) and resting periods of 4000 s each. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] Surface vapour concentration and surface relative humidity profile of intermittent drying of mango tissues with heating (at a drying air temperature of 45 °C) and resting periods of 4000 s each. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] The relative activation energy (Ev /Ev,b ) of the convective drying of kaolin. [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering

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approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.] Moisture content profile of intermittent drying in Case 1 (periodically changed drying air temperatures between 65–43 °C). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.] Temperature profile of intermittent drying in Case 1 (periodically changed drying air temperatures between 65–43 °C). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.] Moisture content profile of intermittent drying in Case 2 (periodically changed drying air temperatures between 100–50 °C). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.] Temperature profile of intermittent drying in Case 2 (periodically changed drying air temperatures between 100–50 °C). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.] Moisture content profile of intermittent drying in Case 3 (periodically changed relative humidity between 4–12%). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.] Temperature profile of intermittent drying in Case 3 (periodically changed relative humidity between 4–12%). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.]

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Moisture content profile of intermittent drying in Case 4 (periodically changed relative humidity between 4–80%). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.] Temperature profile of intermittent drying in Case 4 (periodically changed relative humidity between 4–80%). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.] Relative activation energy (Ev /Ev,b ) of the dehydration of wood during heat treatment generated from the experimental data in Case 2 (refer to Table 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.] Moisture content profiles during the heat treatment of Cases 1 to 3 (refer to Table 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.] Temperature profiles during the heat treatment of Cases 1 to 3 (refer to Table 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.] Moisture content profiles during the heat treatment of Cases 4 and 5 (refer to Table 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.] Temperature profiles during the heat treatment of Cases 4 and 5 (refer to Table 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.] The relative activation energy (Ev /Ev,b ) of baking of thin layer of cake at an oven temperature of 100 °C. [Reprinted from Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of

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2.54

2.55

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2.57

2.58

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baking of cake using the reaction engineering approach (REA), 306–311, Copyright (2012), with permission from Elsevier.] Moisture content profiles at baking temperatures of 100, 140 and 160 °C. [Reprinted from Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of baking of cake using the reaction engineering approach (REA), 306–311, Copyright (2012), with permission from Elsevier.] Moisture content profiles at baking temperatures of 50 and 80 °C. [Reprinted from Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of baking of cake using the reaction engineering approach (REA), 306–311, Copyright (2012), with permission from Elsevier.] Temperature profiles at baking temperatures of 100, 140 and 160 °C. [Reprinted from Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of baking of cake using the reaction engineering approach (REA), 306–311, Copyright (2012), with permission from Elsevier.] Temperature profiles at baking temperatures of 50 and 80 °C. [Reprinted from Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of baking of cake using the reaction engineering approach (REA), 306–311, Copyright (2012), with permission from Elsevier.] Heat transfer mechanisms of convective and infrared-heat drying. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/ water mixture – The reaction engineering approach (REA), 348–357, Copyright (2012), with permission from Elsevier.] Moisture content profile of convective and infrared drying at an air temperature of 35 °C, air velocity of 1 m s−1 , air relative humidity of 18% and intensity of infrared drying of 3700 W m−2 . [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2012), with permission from Elsevier.] Product temperature profile of convective and infrared drying at an air temperature of 35 °C, air velocity of 1 m s−1 , air relative humidity of 18% and intensity of infrared drying of 3700 W m−2 . [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2012), with permission from Elsevier.]

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2.61

2.62

2.63

2.64

2.65

2.66

2.67

Sensitivity of the moisture content profile of cyclic drying, Case 1 (refer to Table 2.12) towards n (on Equation 2.12.1). [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Sensitivity of the temperature profile of cyclic drying, Case 1 (refer to Table 2.12) towards n (on Equation 2.12.1). [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Moisture content profile of cyclic drying, Case 1 (refer to Table 2.12) using the first scheme (T* as function of infrared intensity) with n = 1.8. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Temperature profile of cyclic drying, Case 1 (refer to Table 2.12) using the first scheme (T* as function of infrared intensity) with n = 1.8. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Sensitivity of the moisture content profile of cyclic drying, Case 1 (refer to Table 2.12) towards q (on Equation 2.12.3). [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Sensitivity of the temperature profile of cyclic drying, Case 1 (refer to Table 2.12) towards q (on Equation 2.12.3). [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Moisture content profile of cyclic drying, Case 1 (refer to Table 2.12) using the second scheme (Ev ,b as function of infrared intensity) with q = 1.8. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of

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2.68

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2.71

2.72

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polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Temperature profile of cyclic drying, Case 1 (refer to Table 2.12) using the second scheme (Ev,b as function of infrared intensity) with q = 1.8. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Moisture content profile of cyclic drying, Case 2 (refer to Table 2.12) using the first scheme (T* as function of infrared intensity) with n = 1.5. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Temperature profile of cyclic drying, Case 2 (refer to Table 2.12) using the first scheme (T* as function of infrared intensity) with n = 1.5. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Moisture content profile of cyclic drying, Case 2 (refer to Table 2.12) using the second scheme (Ev ,b as function of infrared intensity) with q = 1.5. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Temperature profile of cyclic drying, Case 2 (refer to Table 2.12) using the second scheme (Ev ,b as function of infrared intensity) with q = 1.5. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] Moisture content profile of cyclic drying, Case 3 (refer to Table 2.12) using the first scheme (T* as function of infrared intensity) with n = 1.6. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]

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Temperature profile of cyclic drying, Case 3 (refer to Table 2.12) using the first scheme (T* as function of infrared intensity) with n = 1.6. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] 2.75 Moisture content profile of cyclic drying, Case 3 (refer to Table 2.12) using the second scheme (Ev,b as function of infrared intensity) with q = 1.6. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] 2.76 Temperature profile of cyclic drying, Case 3 (refer to Table 2.12) using the second scheme (Ev,b as function of infrared intensity) with q = 1.6. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] 3.1 Schematic diagram of a cube dried in a uniform convective environment. 3.2 Moisture content profiles of the convective drying of mango tissues at a drying air temperature of 45 °C solved by the method of lines with 10 and 200 spatial increments. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.] 3.3 Average moisture content profiles of mango tissues during convective drying at different drying air temperatures. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.] 3.4 Centre temperature profiles of mango tissues during convective drying at different drying air temperatures. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.] 3.5 Spatial moisture content profiles of mango tissues during convective drying at drying air temperatures of 45 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction

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3.6

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3.9

3.10

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3.12 3.13

engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.] Spatial water vapour concentration profiles of mango tissues during convective drying at drying air temperatures of 45 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.] Spatial temperature profiles of mango tissues during convective drying at drying air temperatures of 45 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.] Profiles of evaporation rates inside mango tissues during convective drying at a drying air temperature of 55 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.] Moisture content profiles in the core and cortex during convective drying of potato tissues with a diameter of 1.4 cm. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.] Moisture content profiles in the core and cortex during convective drying of potato tissues with a diameter of 2.8 cm. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.] Core temperature profiles during convective drying of potato tissues with a diameter of 1.4 cm. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.] Average moisture content profiles of mango tissues during intermittent drying at different drying air temperatures. Spatial moisture content profiles of mango tissues during intermittent drying at a drying air temperature of 55 °C.

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3.14 3.15 3.16 3.17 3.18

3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 4.1

Spatial water vapour concentration profiles of mango tissues during intermittent drying at a drying air temperature of 55 °C. Centre temperature profiles of mango tissues during intermittent drying at different drying air temperatures. Spatial temperature profiles of mango tissues during intermittent drying at a drying air temperature of 55 °C. Profiles of evaporation rate inside mango tissues during intermittent drying at a drying air temperature of 55 °C. Profiles of average moisture content during heat treatment in Case 2 (refer to Table 3.5) solved by the method of lines using 10 and 100 increments. Effect of liquid diffusivity on profiles of the moisture content during heat treatment in Case 1 (refer to Table 3.5). Effect of liquid diffusivity on profiles of temperature during heat treatment in Case 1 (refer to Table 3.5). Profiles of average moisture content during heat treatment in Case 1 (refer to Table 3.5). Profiles of temperature during heat treatment in Case 1 (refer to Table 3.5). Profiles of average moisture content during heat treatment in Case 2 (refer to Table 3.5). Profiles of temperature during heat treatment in Case 2 (refer to Table 3.5). Profiles of spatial moisture content during heat treatment in Case 2 (refer to Table 3.5). Profiles of spatial water vapour concentration during heat treatment in Case 2 (refer to Table 3.5). Profiles of spatial temperature during heat treatment in Case 2 (refer to Table 3.5). Profiles of average moisture content during the baking of bread at a baking temperature of 150 °C. Spatial profiles of moisture content during the baking of bread at a baking temperature of 150 °C and air velocity of 10 m s−1 . Spatial profiles of concentration of water vapour during the baking of bread at a baking temperature of 150 °C and air velocity of 10 m s−1 . Profiles of top and bottom surface temperatures during the baking of bread at a baking temperature of 150 °C and air velocity of 1 m s−1 . Spatial profiles of temperature during the baking of bread at a baking temperature of 150 °C and air velocity of 10 m s−1 . Experimental setup for convective drying of porcine skin. [Reprinted from Chemical Engineering Research and Design, 87, S. Kar, X.D. Chen, B.P. Adhikari and S.X.Q. Lin, The impact of various drying kinetics models on the prediction of sample temperature–time and

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151 152 152 153 153 154 155 156 156 157 161 162 162 163 163

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moisture content–time profiles during moisture removal from stratum corneum, 739–755, Copyright (2012), with permission from Elsevier.] (a) Overview of a sample/plate assembly for convective drying of porcine skin. (b) Detailed of layering structure of sample support. [Reprinted from Chemical Engineering Research and Design, 87, S. Kar, X.D. Chen, B.P. Adhikari and S.X.Q. Lin, The impact of various drying kinetics models on the prediction of sample temperature–time and moisture content–time profiles during moisture removal from stratum corneum, 739–755, Copyright (2012), with permission from Elsevier.] Moisture content profiles from the convective drying of mango tissues modelled using the L-REA and diffusion-based model (Vaquiro et al., 2009). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of Drying of Food Materials with Thickness of Several Centimeters by the Reaction Engineering Approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.] Temperature profiles from convective drying of mango tissues modelled using the L-REA and diffusion-based model (Vaquiro et al., 2009). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.] Moisture content profiles from the convective drying of mango tissues modelled using the S-REA and diffusion-based model (Vaquiro et al., 2009). [Reprinted from AIChE Journal, A. Putranto and X.D. Chen, Spatial reaction engineering approach as an alternative for non-equilibrium multiphase mass-transfer model for drying of food and biological materials, DOI 10.1002/aic.13808, Copyright (2012), with permission from John Wiley & Sons, Inc.] Temperature profiles from the convective drying of mango tissues modelled using the S-REA and diffusion-based model (Vaquiro et al., 2009). [Reprinted from AIChE Journal, A. Putranto and X.D. Chen, Spatial reaction engineering approach as an alternative for non-equilibrium multiphase mass-transfer model for drying of food and biological materials, DOI 10.1002/aic.13808, Copyright (2012), with permission from John Wiley & Sons, Inc.] Moisture content profiles from the heat treatment of wood modelled using the L-REA and Luikov’s approach. [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.] Temperature profiles from the heat treatment of wood modelled using the L-REA and Luikov’s approach. [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley,

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4.10

4.11 4.12 4.13 4.14

Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.] Moisture content profiles from the heat treatment of wood (refer to Table 4.1) modelled using the L-REA and Whitaker’s approach. [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.] Temperature profiles from the heat treatment of wood (refer to Table 4.1) modelled using the L-REA and Whitaker’s approach. [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.] Moisture content profile from the heat treatment of wood modelled using the S-REA and Luikov’s approach. Temperature profile from the heat treatment of wood modelled using the S-REA and Luikov’s approach. Moisture content profiles from the heat treatment of wood (refer to Table 4.1) modelled using the S-REA and Whitaker’s approach. Temperature profiles from the heat treatment of wood (refer to Table 4.1) modelled using the S-REA and Whitaker’s approach.

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203 204 204 205 205

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2.1

Experimental conditions of convective drying of a mixture of polymer solutions (Allanic et al., 2009). page 51 2.2 R2 and RMSE of modelling of a mixture of polymer solutions using the L-REA. 58 2.3 Experimental conditions of convective drying of mango tissues (Vaquiro et al., 2009). 61 2.4 R2 and RMSE of modelling of convective drying of mango tissues using the L-REA. 66 2.5 Schemes of intermittent drying of mango tissues (Vaquiro et al., 2009). 69 2.6 R2 and RMSE of modelling of intermittent drying of mango tissues using the L-REA. 73 2.7 Settings of intermittent drying of kaolin (Kowalski and Pawlowski, 2010). 81 2 2.8 R , RMSE, average absolute deviation and maximum absolute deviation of profiles of moisture content predicted by and Kowalski and Pawlowski’s model (2010b). 85 2.9 R2 , RMSE, average absolute deviation and maximum absolute deviation of profiles of temperature predicted by Kowalski and Pawlowski’s model (2010b). 85 2.10 Settings of heat treatment of wood samples (Younsi et al., 2006a; 2007). 89 99 2.11 R2 of modelling using the REA. 2.12 The experimental conditions of intermittent drying of a mixture of polymer solutions. 105 3.1 Experimental conditions of convective drying of mango tissues (Vaquiro et al., 2009). 128 2 3.2 R and RMSE of convective drying of mango tissues using the S-REA. 135 3.3 Scheme of intermittent drying of mango tissues (Vaquiro et al., 2009). 141 142 3.4 R2 and RMSE of intermittent drying of mango tissues. 3.5 Experimental settings of wood heating under a constant heating rate (Younsi et al., 2007). 148 3.6 R2 and RMSE of modelling of heat treatment of wood under a constant heating rate using the S-REA. 155 4.1 Experimental settings of the heat treatment of wood (Younsi et al., 2007). 201

Preface

Drying is one of the oldest and most effective methods for preserving food and biological materials. Low moisture content in foods prevents the growth of bacteria responsible for their deterioration so foods can have extended shelf-lives. When foods became abundant, trade became possible. Today, dried products are the main materials trading round the world but this is not limited to food products. Construction materials, textiles, electronic parts and appliances, biomass-based fuels, pharmaceutics and many other materials important to our daily lives and the business world are all included. Essentially over 80% of the products on Earth require drying as one of the steps in their production. Product quality and process parameters are interactive. Industrial drying is energy hungry; a process involving simultaneous heat, mass transfer and momentum transfer. Product quality is determined through compositional and structural rearrangements, as well as chemical reactions in some circumstances. For existing drying facilities, optimisation is often needed to achieve new goals such as energy reduction, quality improvements and development of new materials. There are also opportunities in designing dryer modifications or even brand new dryers that are superior in performance over conventional devices. Modelling of drying processes is very useful for these purposes. A number of drying models have been proposed, which are conveniently classified into empirical and mechanistic models. The empirical models give advantages of being simple in their mathematical formulation. However, these models most often cannot explain the physics of drying and their application is limited since they are valid only for a particular set of drying conditions. On the other hand, the mechanistic models are derived based on fundamental phenomena that occur during drying. These phenomena are crucial in material science (and materials processing) though material scientists themselves may not have yet come to appreciate the process engineering aspects which impact on the product microstructure. Some of these models can capture the physics well. These models are, however, often mathematically complex and sometimes contain too many parameters, which need to be determined experimentally (prior to model predictions). For some decades now, a comprehensive set of macroscopic equations has been developed and used to address heat and mass transfer and mechanical aspects related to drying. The application of macroscopic descriptions of drying (temperature, moisture and sometimes pressure) has been perfected over the past two decades, and relevance has been confirmed in many drying configurations. Some of these involve irreversible thermodynamics formulations, which are lengthy and have many model coefficients. These have

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become the ‘classical’ approach. However, this classical approach has serious limitations. The concept of multi-scale and multi-physics addresses some of these limitations, e.g. coupled meso-scale and equipment scale problems. When a local thermodynamic equilibrium is not attained, however, the time scales usually overlap. This is a real multiscale configuration and challenging in terms of the great demand in computational power and handling of mathematics. Several scales can be considered simultaneously, ranging from simple exchanges between macroscopic phases to comprehensive formulations in which time evolution of microscopic values and microscopic gradients is considered over a representative elementary volume, according to a recent review by Patrick Perre (for a review of modern computational and experimental tools relevant to the field of drying, see Drying Technology, 29, 1529–1541, 2011). While exploring the detailed physics involved in drying using these multi-scale and multi-physics approaches, it is, from an engineering viewpoint, also important to develop new ideas establishing simpler models. In general, today industrial drying applications require mathematical models that are simple and easy to use. For practical purposes, an effective drying model should be simple, accurate, and able to capture the major physics of drying and its application should be robust. This model should also favour short computation time and it should be easy to establish parameters needed (experimentally) to help quicker decision-making in an industry environment (and with the lowest cost). The reaction engineering approach (REA), which is a ‘middle path’ approach, perhaps between the empirical and the mechanistic models, was first thought about by the first author of this book, Chen, in 1996. Through much of the research on its possible applications, it has been revealed that the REA is indeed simple, accurate and robust enough to model many cases of drying, i.e., drying in a constant or variable environment. The REA has also been implemented in industry for prediction of spray dryer performance and shows good agreement with plant data for different scales in the dairy industry. It has also been extended to various other challenging systems of drying, such as polymer drying, intermittent drying, thermal-thick materials, infrared heating and microwave heating. The model is significantly easier to implement and requires less experimentation effort to establish the parameters needed, compared with the more fundamental models. The REA was first taken as a lumped model which does not need us to resolve the spatial distribution of water content, etc.; the lumped-REA (or L-REA), but in recent times, we have also extended the approach to describe spatially distributed systems; spatial-REA (or S-REA). The REA approach has been initiated and exercised over the past 12 years and there is a significant amount of successful applications already illustrated. As mentioned earlier, it is a middle path between the rigorous theory that requires high-level mathematics and the empirical models that do not represent much physics. We can see, through our own practices and from other colleagues in the same area who have used the REA concept, it is a really straightforward approach to modelling some rather complex drying processes; hence, it is simple and cost-effective to establish accurate REA models to use in industry. This book is the most fundamental and comprehensive description of the REA approach to drying modelling – the basic idea, rationale, mathematical description and implementation procedures – for various systems. This approach has been extended,

Preface

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and experimented with, by several quality Ph.D. graduates, in particular, the second author, Aditya Putranto. Regarding the other more established theories, this book not only provides essential details so the readers can refer to them but also illustrates, by comparison, the physics involved in REA concepts. The disadvantages and advantages between theories are also briefly introduced. The book should benefit both academics in drying research and practicing engineers in industry. Undergraduate students in process engineering may also find it useful for quickly setting up a drying model for design purposes. The main emphasis of this book is how to apply the REA to reality. The book will also elaborate on potential applications of similar thinking to more complex reactive systems that couple with drying processes, hopefully to foster their future development. Here, the modern ideas of microstructure development and product qualities created by drying processes, and in turn their impacts on moisture transfer, will be introduced. This should make the book more relevant in years to come. Xiao Dong Chen and Aditya Putranto

Historical background

During my Ph.D. study in the Chemical and Process Engineering Department at Canterbury University, Christchurch, New Zealand, (1988–1990), the main task was to establish mechanistically the understanding of moisture influence on coal oxidation and the impact of moisture transfer in a packed coal particle bed on the development of spontaneous combustion. The experimental aspect was challenging both technically and physically. In addition to coal oxidation and its racemic measurement, I became very interested in the mechanisms of water evaporation and moisture transfer (liquid and vapour) in porous material. Dr Jim Stott (Reader of Chemical Engineering) was my main supervisor and Dr John Abrahamson (Senior Lecturer), in the same department, was my cosupervisor. Jim published some of the pioneering literature on the subject of spontaneous combustion of coal (1959) and built (largely by himself) ingenious experimental rigs. Dr Abrahamson was an inspirational and distinguished individual as well who has been credited as one of the first to have made a carbon nanotube (he called it the ‘carbon cylinder’) (1978), a theory of ball lightening (2000) and a theory of particle collision frequency in a turbulent field (1972). John was Jim’s student some years back. Working with Jim on the subject of spontaneous combustion development in a moist coal bed has taught me that if the coal bed were completely saturated with water vapour under near ambient pressure (the institutional voids of the bed remain saturated with water vapour), the maximum temperature would remain at around 80 °C. This was predicted from a numerical spontaneous combustion model involving mass transfer of moisture within the coal bed when assuming the vapour concentration in the bed is always saturated. Jim discovered this in the late 1960s, and later, in the 1970s, a Ph.D. student of his proved this more comprehensively. This aspect was more or less republished in 1990s by a research group in Europe (who were perhaps unaware of the work by Jim and his ex-students). However, if an equilibrium relationship between moisture content in the coal particles and vapour concentration in the air surrounding the particles can be adopted, a dry spot can be predicted and the maximum temperature will exceed the boiling temperature of water, therefore rising to an elevated temperature due to oxidation heat (Chen, 1992a). Of course, there are also other influences such as porosity, oxidation rate and oxygen transfer, heat transfer and, sometimes, fluid flow due to a pressure gradient. Nevertheless, this equilibrium relationship is what we are now so familiar with, termed the equilibrium isotherm in drying literature. The oxidation rate of coal itself was also found, in my own experiments, to vary with the residual water content (Chen and Stott, 1993) and I had gone to extra lengths to try to understand this


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phenomenon. This formed the foundation of my understanding of the presence of water affecting chemical (and biochemical) reactions. In food drying, it would mean that the removal of water, to some extent, could significantly reduce rate of deterioration, giving a long shelf life to products (oxidative or microbial) (Chen and Mujumdar, 2008). As I became aware of moisture transfer, I became very aware of the existence of a ‘giant’ of drying in the same department, Professor Roger Keey, who wrote the first book on drying principles and practice that was published in English. I had spent a lot of time looking for information on how to model moisture transfer, coupled with chemical reactions and heat transfer and momentum transfer. Keey’s books over the years have had an impact on my own work related to this area (especially the latest one; Keey, 1992). In particular, I have picked up the essence of the characteristic drying rate curve (CDRC) approach. One of my friends in the drying area, Professor Tim Langrish, a Canterbury graduate, has worked extensively on this idea, which has extended Keey’s views on the drying of wood and some other different materials, including foods. His postdoctoral period (after his return from Oxford University) with Professor Keey overlapped with the final year of my Ph.D. (1990). Another distinguished individual whose work has affected my own thinking has been Professor Shusheng Pang, another Canterbury graduate supervised by Roger Keey, who has published some key literature in wood drying related to the application of CDRC. CDRC captures the phenomena of drying by recognising the existence of a constant and falling drying rate period(s). The critical or transitional water content between any of the connecting rate schemes are recognised (Keey, 1992). Doctors Sandeep Chu and Peter Kho, who were student colleagues at Canterbury during the period of 1988–1990 and whose works were supervised by Professor Keey, also had an impact on my later research on drying. Some others who also influenced me positively were Professor Miles Kennedy, Dr John Peet and Dr Maurice Allen at Canterbury. I had read many of their works during the peaceful evenings when I had pretty much the whole department to myself and some of the weekends during my Ph.D. study at the corner room on the top floor of Simon’s Block. The surroundings of Canterbury University were beautiful and peaceful and gave me great times (and spaces) to spend thinking about my work and, of course, my loved ones. I submitted my Ph.D. thesis three years after I started in late December 1987. I started working at the New Zealand Dairy Research Institute (NZDRI) (which is now the Fonterra Research Centre based in Palmerston North of New Zealand), first as an engineer and then as a senior engineer, working on spray drying and milk powder agglomeration. Dr Kevin Pearce (my section manager), who was a distinguished chemist, gave advice that I understood one has to take in order to take protein chemistry seriously when dealing with engineering problems related to dairy products. This period of time was very constructive for my career development. After coal research, I really wanted to move onto biotechnology and, at the time, the food industry was the nearest thing to biotechnology in which I could secure a good position. I was deeply involved in milk powder technology and have become very familiar with powder technology, dissolution properties of the powders, powder agglomeration and instantisation (Chen, 1992b), glass transition and stickiness (Lloyd et al., 1996), etc. I was lucky enough to

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Xiao Dong Chen (left) and Dr Jim Stott (right) working on the 2-m-long packed coal column investigating spontaneous heating of coal, 1989, University of Canterbury, Christchurch, New Zealand.


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make a significant contribution in the area of agglomeration (hardware improvement and macrostructural analysis) and new product development that was hampered by high stickiness, with large financial returns for the dairy industry. My employment as an academic at the Department of Chemical and Materials Engineering, The University of Auckland, started in late 1993, which instantly gave me greater freedom to develop new ideas. I had great fun working at Auckland, benefitting from being surrounded by a number of highly positive individuals at the department and the school. One strong influence came from my colleagues who were experts in materials science. Among many other studies, in 1995–1996, I had developed an idea that was initially thought to be able to ‘unify’ drying kinetics to the equilibrium relationship (Chen and Chen, 1997). The notion of ‘unified’ came from, at the time, an ambitious young man (me) but later was proven to be, well, ‘kinetics is just kinetics and equilibrium is equilibrium’, so they don’t have to be 100% linked. What had emerged, however, was that if I could find a simple relationship between the surface vapour concentration and the water content of the porous solid material being dried, noting that this surface vapour concentration less the vapour concentration in the gas phase is the driving force for moisture transfer from the porous material to the drying environment, the model could be a good alternative to the CDRC model. The obvious one for surface vapour concentration to relate to is liquid water content. At the time, I already found some issues with the CDRC approach, as uncertainty can be great depending on the drying processes considered. Keey (1992) has rightly pointed out that the CDRC model was excellent for particles or sample sizes smaller than 20 mm for constant drying conditions. The link between the surface vapour concentration and the remaining water content in the porous material as well as the material surface temperature was eventually constructed using an Arrhenius-type relationship, which essentially suggested the evaporation of pure water and ‘extraction’ of the water from the inside the porous material was a reaction and the condensation/adsorption was not an activation process. This is in line with a mathematical description of evaporation and a condensation mechanism formulated by Gray and Wake (1990). Professor Brian Gray (Professor of Applied Mathematics at Sydney University at the time) is a distinguished applied mathematician (he is also a distinguished physical chemist) whom I came to know through the link between me and Professor Graeme Wake (another outstanding applied mathematician from New Zealand) and had influenced my approaches to engineering in more than just one aspect. They were not particularly interested in drying, but they were very much interested in the systems of reactions, both exothermic and endothermic. Evaporation is viewed as an endothermic reaction mathematically speaking. My father, who was a Professor of Aerodynamics at the Chinese Academy of Sciences, visited me in New Zealand in 1996. I discussed some of my initial ideas with him and we prepared a simple paper for Chemeca in 1997. I was also fortunate in hosting a visiting researcher from Xian Jiao-Tong University (China) during 1996–1997, Associate Professor Guozhen Xie, who was a refrigeration expert but was daring enough to pick up drying modelling as the main topic in his year of working with me. We didn’t do any experiments on drying but used data reported in the literature. However, in all cases, we had to solve an energy balance to obtain (surface) temperature of the material tested for

xxxiv


the concept. Most of the examples used (Chen and Xie, 1997) were small-sized samples. Once we had the temperature-time profile for the sample of concern during drying, we could establish the activation energy in the Arrhenius equation (mentioned earlier) to demonstrate the concept. Then, in 2000, at Auckland, I had a masters student by research, Wayne Pirini, who was interested in drying, so we started experimenting on thin-layer drying of various materials measuring both weight loss and temperature as drying proceeded. The first lot of data on activation energies obtained was reported by Chen, Pirini and Mustafa in 1996. However, I was not aware that the Biot number defined in heat transfer literature could not account for the conditions when evaporation occurs. This gave me an opportunity to derive a modified Biot number later on (the so-called Chen–Biot number). Then, at Auckland in the period of 2000–2004, Dr Sean Lin, my Ph.D. student at that time, did a comprehensive study on droplet drying kinetics for dairy products in particular. He had lots of practical experience before coming to me. He designed and built an excellent cost-effective droplet drying test rig and conducted probably the most careful, accurate experiments on dairy droplet drying. This has allowed the comprehensive establishment of the REA model for dairy droplets that is relevant to the spray drying industry (Chen and Lin, 2005). Following that, two Ph.D. students under my supervision who were from India, Drs Saptarshi Kar and Kamlesh Patel, had made a significant contribution to the development of the REA concept. Saptarshi applied REA to a spatial distributed case for water transport in skin relevant to transdermal drug delivery for the first time. We deliberately ignored the liquid diffusivity to see if it really mattered. It turned out that it really did matter. Kamlesh had helped in extending the Chen–Biot number concept and helped to bring in a new concept called the ‘composite REA approach’, which describes an approach to estimating the activation energies of sugar mixtures based on the components’ own activation energies. They were both tremendous students with high aptitudes to pursue basic research. Saptarshi in particular tended towards a more theoretical rigorousness. They started their Ph.Ds at Auckland and finished at Monash University. In 12 years at Auckland, I moved from (in the English system) Lecturer (1993) to Senior Lecturer (1995) to Associate Professor (1998) to Personal Chair Professor at the age of 36 (2001). It was the most dramatic time in my life, both in career and personal life. I had my first child, Lisa, who was born in May 2000. Sad events had taken her mother away from her in 2001. I must thank the Engineering Dean at the time, Professor Peter Brothers, who, in my darkest days in 2001, promised his institution’s support in allowing me to do whatever I needed to do and go wherever I needed to go without worrying about losing my job. Beyond that, I enjoyed tremendous learning experiences, friendships, and support from my colleagues at the Department of Chemical and Materials Engineering: Professor Geoff Duffy (who was most influential individual in my stay at Auckland), Associate Professor Kevin Free, Professor John J. J. Chen, Professor Wei Gao, Professor Mohammed Farid, Professor Neil Broom, Professor George Fergusson and Dr Necati Ozkan. I was inspired by the genius professors such as Professor John Boys (Electrical Engineering), Professor Peter Hunter (Engineering Science) and Professor Debes


xxxv

Aditya Putranto (left) and Xiao Dong Chen (right), November 2012, International Drying Symposium (IDS 2012) chaired by Xiao Dong Chen, Xiamen, China.

Bhattahtrayya (Mechanical Engineering) for their innovations. I benefited tremendously from collaborating with Associate Professor Sing Kiong Nguang, who is a genius in mathematical problems in system and dynamics engineering. In that period of time at Auckland, I picked up the idea of combining process engineering and material science and became familiar with microscopy and material science techniques. My colleagues have created an incredibly creative and happy environment for me to work in. Of course, there were giants who supported me graciously over those years; Professor John Hood (Vice Chancellor of The University of Auckland and then, later, Vice Chancellor of Oxford University) and Professor Diane McCarthy (Dean of Medicine at Auckland and later President of Royal Society of New Zealand). Without their recognition of my ability and my contribution, my rapid promotion at Auckland would not have been possible. Coming back to the main technical topic, can the REA model do the things that a CDRC model cannot? For small-size particles and constant drying conditions, CDRC seems to be very comparable with REA. With this question, and many others, I had moved to the Department of Chemical Engineering at Monash University (Melbourne, Australia) to take up the Chair of Biotechnology at Monash University in 2006. In 2009, I had great fortune in that a high-calibre student from Indonesia, Dr Aditya Putranto, a humble young man, joined my group at Monash to do a Ph.D. with me.

xxxvi


He demonstrated superior ability in testing and further developing REA for numerous applications, which are presented in this book. In 2010, I moved to Xiamen University on the southeast coast of China, from which my grandparents graduated in history and English literature in 1930 and 1931, respectively as a National Expert Professor of Chemical Engineering (also known as the 1000Elite Chair Professor). I have not stopped the excellent collaborations with Aditya and we continue to expend the REA. Of course, my other great Ph.D. students, Nan Fu, Winston Wu and Sam Rogers (in the period of 2008–2011), a postdoc fellow, Dr Yan Jin (in 2009), and Dr Mengwai Woo (2010–2011), have also continued to contribute experimentally, and theoretically, to the establishment of REA and its applications to the real world. Notably, Nan generated significantly new data on the REA approach to dairy droplet drying and linked drying to crystallisation and particle solubility. She has extended the techniques of single droplet drying to a more powerful means in order to understand drying-quality inter-relations. Dr Jin has comprehensively modelled the three-dimensional transient flows in large-scale spray dryers and has incorporated the REA approach. Dr Woo has independently investigated the robustness of the REA approach for modelling droplet drying in the context of computational fluid dynamics of spray dryers. In no way can I claim it was only me who made REA development possible, but I can claim the original idea and model framework to be mostly mine. I sincerely thank all the previously mentioned individuals and others whom I have not mentioned but who have made contributions to the development of the REA in one way or another. REA is also the result of a belief that engineering theory should be as simple and robust as possible in order to enable a broad range of applications. Finally, I would like to dedicate the book as follows: To my lovely family starting from my wife Lily and the children Lisa, Nathan and Benjamin. To my grandparents, my parents, my sister and brother-in-law for their neverending love and support. To others whom I have loved and who have loved me selflessly. Xiao Dong Chen Xiamen City, Southeast Coast of China August 2012 Aditya Putranto would like dedicate this book as follows: To his parents and sister for their endless love and support. To the Creator and others whom have shared, and will share the love and faithfulness of the Creator. Aditya Putranto Melbourne Australia August 2012


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References Chen, X.D., 1991. The Spontaneous Heating of Coal – Large Scale Laboratory Assessment and Supporting Theory, Ph.D. thesis, Chemical and Process Engineering Department, University of Canterbury, New Zealand. Chen, X.D., 1992a. On the mathematical modelling of transient process of spontaneous heating in a moist coal stockpile. Combustion and Flame 90, 114–120. Chen, X.D., 1992b. Whole milk powder agglomeration – Principle and practice. In Milk Powders for the Future, X.D. Chen (ed.), Dunmore Press: Palmerston North, New Zealand. Chen, X.D. and Chen, N.X., 1997. Preliminary introduction to a unified approach to modelling drying and equilibrium isotherms of moist porous solids, Chemeca’97, Rotorua, New Zealand, Sept. 1997, Paper DR3b (on CD-ROM). Chen, X.D. and Lin, S.X.Q., 2005. Air drying of milk droplet under constant and time-dependent conditions. AIChE Journal 51(6), 1790–1799. Chen, X.D. and Mujumdar, A.S. (eds.), 2008. Drying Technologies in Food Processing. Blackwell Publishing Ltd: Oxford. Chen, X.D. and Stott, J.B., 1993. The effect of moisture content on the oxidation rate of coal during near equilibrium drying and wetting at 50 °C. Fuel 72, 787–792. Chen, X.D. and Stott, J.B., 1997. Oxidation rates of several New Zealand coals as measured in large scale one-dimensional spontaneous heating experiments. Combustion and Flame 109, 578–586. Chen, X.D. and Xie, G.Z., 1997. Fingerprints of the drying of particulate or thin layer food materials established using a simple reaction engineering model. Transactions of the Institute of Chemical Engineers Part C: Food and Bio-Product Processing 75(C), 213–222. Gray, B.F., 1990. Analysis of chemical kinetic systems over the entire parameter space 3. A wet combustion system, Proc. Roy. Soc. A 429, 449–458. Gray, B.F. and Wake, G.C., 1990. The ignition of hygroscopic materials by water. Combustion and Flame 79, 2–6. Keey, R.B., 1992. Drying of Loose and Particular Material. Hemisphere: New York. Lloyd, R.J., Chen, X.D. and Hargreaves, J., 1996. Glass transition and caking of spray dried amorphous lactose. International Journal of Food Science and Technology 31, 305–311.

1

Introduction

1.1

Practical background Drying (removing water from wet material) has been a very important processing step for a wide range of human endeavours in our history. The dependence of human society on drying is highly visible. For instance, in food production and preservation, drying is the oldest, most popular and one of the most effective ways to make solid foods and to preserve them as long as practically required. The textile industries need drying processes. Natural fibre-based products such as those from the wood and paper industries also need drying as a critical step in manufacturing. In fact, anything having to do with the particulate products, not just food particles (milk powders, vegetable soup powders and the like) but also detergents, fertilisers, and even paints: drying is critical. As modern ‘material science’ industries have started to develop at a speed never seen before in our history, wet chemistry is needed, which requires drying (‘dewetting’) to form solid products which are more usable and transportable. Some historical and typical products are shown in Figure 1.1. Food drying is conducted in many ways. The history of using sunlight to dry fruits goes back thousands of years, dating back to the fourth millennium BC in Mesopotamia ( http://en.wikipedia.org/wiki/Dried fruit). Today, dried fruits have the majority of the original water content removed either naturally, through solar drying or sometimes freeze drying and air drying (with low-humidity air in particular), or ‘unnaturally’, through the use of specialised dryers or dehydrators powered by electricity or combustion. These unnatural ways include mechanical dewatering, convective air/gas drying, superheated steam drying, electro-osmosic processes, osmotic pressure-driven processes, refractory window drying, freeze drying, vacuum drying, and microwave-aided drying processes, to name a few. If one includes liquid evaporation (with liquid products also), these may be expanded to evaporation operations, such as falling film, rising film evaporation, vacuum distillation, and the like. Dried fruits are popular products due to their enhanced sweet taste, concentrated nutritional value and long shelf-life because the water activity is low (Chen and Mujumdar, 2008). As water content is removed, the material shrinks (leading to a smaller volume), the sugar content per unit volume of the material increases, as do nutritional components such as proteins, vitamins and so on. Today, dried fruit consumption is widespread. Nearly half the dried fruits sold are raisins, followed in popularity by dates, prunes (dried plums), figs, apricots, peaches, apples and pears (Hui, 2006). Many are

2

Modelling Drying Processes

(a)

(c)

(b)

(d)

Figure 1.1 Some traditional dried products. (a) Broccoli-steam blanched and air dried (kindly provided by Ms Xin Jin, Wageningen University, The Netherlands), (b) air-dried Chinese tea leaves (taken at Xiamen University laboratory), (c) spray dried aqueous herbal extract (particle size is about 80 µm) (taken at Xiamen University laboratory), (d) timber stacked for kiln drying (kindly provided by Professor Shusheng Pang, Canterbury University, New Zealand).

referred to as ‘conventional’ or ‘traditional’ dried fruits: fruits that have been dried in the sun or in heated wind tunnel dryers. Many fruits, such as cranberries, blueberries, cherries, strawberries and mangoes, may be infused with a common sugar (e.g. sucrose syrup) prior to drying to enhance sweetness and microbial stability. This means these are not necessarily healthy products, especially for diabetics. Sugar-infused and dried papaya and pineapples are actually candied fruit. Dried fruits are usually thought to retain most of the nutritional value of the fresh fruits. The specific nutritional content of dried fruits reflects that of their fresh counterparts and is influenced by the processing method or processing technology, particularly processing temperature. In general, all dried fruits provide essential nutrients and an array of healthy protective ingredients, making them valuable tools to both improve diet quality and help reduce the risk of chronic disease. Furthermore, dried fruits (and nuts) are not only important sources of vitamins, minerals and fibre in the diet but also provide a wide array of bioactive components or

Introduction

3

phytochemicals. These plant compounds are not designated traditional nutrients since they are not essential to sustain life but play a role in health and longevity and have been linked to a reduction in risk of developing major chronic diseases. Convincing evidence suggests that the benefits of phytochemicals may be even greater than currently understood, as they seem to affect metabolic pathways and cellular reactions. However, the precise mechanisms by which specific compounds exert their biological effect remains largely hypothetical, which requires greater investigation. Certainly, as is well known, dried fruits are an excellent source of polyphenols and phenolic acids (USDA, 2007). These compounds make up the largest group of phytochemicals in the diet and appear to be, at least partially, responsible for the potential benefits associated with the consumption of diets rich in fruits and vegetables. Different dried fruits have unique phenolic profiles (Donovan et al., 1998); for example, the most abundant in raisins are the flavonols (quercetin and kaempferol) and the phenolic acids (caftaric and coutaric acid) (Willamson and Carughi, 2010). Therefore, due to their high polyphenol content, dried fruits are an important source of antioxidants in the diet (Wu et al., 2004; Vinson et al., 2005). Antioxidants can lower oxidative stress and so prevent oxidative damage to critical cellular components. Dried apricots and peaches are also important sources of carotenoids. These compounds not only are precursors of vitamin A but also have antioxidant activity. Dried fruits such as dried plums provide pectin, a soluble fibre that may lower blood cholesterol levels (Tinker et al., 1991). Dried fruits such as raisins are a source of prebiotic compounds in the diet. They contain fructooligosaccharides like inulin, naturally occurring fibrelike carbohydrates that contribute to colon health (Camire and Dougherty, 2003). The chemical structures of some of these useful compounds for human health are shown in Figure 1.2. Since drying foods may affect the chemical structure of their components, some of which could be undesirable, it is important to keep a balance between how fast or efficient the drying process is in terms of energy usage or product throughput and quality requirements. Sometimes, faster processing is not necessarily better. Like food material, all the solid-form products, natural or processed, have interesting structures (including microstructures) and qualities. It is worthwhile noting that removing water in its liquid form from a solid structure is often (perhaps better) called ‘dewatering’, which induces solid structural changes around or near where the water used to be. In a more general situation, water removal may also be called ‘dehydration’, especially when talking about materials of partially or wholly biological origin. In this book, drying is mostly referred as those processes that use gas as a drying medium so the water comes out of the material as a gas (water vapour). Later in this book, other forms of drying, such as vacuum drying or even steam-aided drying, may be employed using the reaction engineering approach (REA) when it is appropriate. Furthermore, roasting (coffee, for instance), baking (biscuits and bread, etc.) and heating of moist material (e.g., detoxification of wood) may be considered extensions of the concept of drying, and their purposes are for improving material performance while the moisture effect simply cannot be ignored.

4


(a)

2

1

3

OH

O

OH

O

O HO

OH HO

HO

OH

OH

(b)

1

CH2OH

CH2OH O

O

OH

O OH

2

CH2OH O

OH

O

OH

O

O

OH

OH

OH

CH2OH

CH2OH

O

O O

OH

O

OH

OH

3

CH3O

OH

O

O OH

CH2OH

O

HO

CH3O

O

O

O O

O O

O

OH

O

CH3

O

OH

O OH

OH OH

OH

OH

(c) H

Gly

lle

val

Glu

Gln

Cys

Cys Cys

Thr

Ser

Ser

Leu

lle

Tyr Gln Leu

A-chain

Glu HO

H

Phe

B-chain

Val

Asn

Gln

His

Leu

Asn

Cys

Tyr

Gly

Cys

Gly

Ser

Val

His

Leu

Gly

Leu

Tyr

Phe

Leu

Phe

Asn

Cys

Val

Glu

Ala

Glu Arg

Tyr HO

Thr

Lys

Pro

Thr

Figure 1.2 Chemical structures of some chemicals: (a) 1, caffeic acid; 2, gallic acid; 3, vanillic

acid; (b) 1, cellulose; 2, starch; 3, pectin; (c) human insulin.

Introduction

5

The powering mechanisms for drying can be solar radiation, electricity, steam, microwave and ultrasound, amongst others. In all drying operations, energy consumption is a critical issue in modern times; yet, in the wider range of practical interest, people are more concerned about the quality of the products. For high-value products often related to nanotechnology these days, nanostructure and microstructure aspects and functionalities are an increasing concern for both the researchers and marketers. In food and pharmaceutical products, these aspects are even more important as they affect the metabolism and health of our organs. Therefore, as far as drying is concerned, it provides an excellent example of process engineering interacting with product quality. In chemical engineering, the interactions may be considered systematically with the ideas of chemical process engineering versus chemical product engineering (Cussler and Mogoridge, 1997). One may see engineering as a terminology which differs from technology. Engineering should involve developing the predictive tools (evaluating a building design using established mathematical analysis and then building it accordingly), which need good mathematical descriptions of the physics and chemistry that go on into the relevant processes, much like in modern times, where calculations are used to investigate the validity of constructing an architecturally designed building before actually building it; i.e., civil engineering. There is no question that drying technology needs to be made predictive: this is the notion of ‘drying engineering’. In fact, for design and optimisation of the drying processes, one needs accurate and robust mathematical models. Better still, some of the models can actually help us explore drying mechanisms (physics, chemistry and biochemistry), investigating the scientific aspects associated with the drying phenomena. Drying models are usually referred to as models that describe the mass and heat transport within the material being dried, as the exterior conditions are already nicely covered by the conventional heat and mass transfer and momentum transfer theories. The boundary conditions are intended to connect the drying models with exterior transport theories. Simple, yet effective, mathematical models of drying are welcomed by practitioners or engineers. When modelling multiphase flow in a spray dryer, for instance, one might need to mathematically track thousands of particles of different sizes travelling inside the drying chamber to make the simulation more realistic. If one has to solve the spatially distributed variables (like water content) inside each particle, the computational effort is great; hence, the whole exercise incurs a high expense. There is also a high likelihood of computational instability. However, if one does not need to solve for the spatial distributions of moisture, temperature and species distribution within each particle, one only needs to integrate a model over time to obtain the average water content, temperature and average concentrations of species for each particle; the computational effort is much smaller. For a large piece of material being dried, a ‘lumped’ drying model that may predict drying history accurately without the need to resolve the spatially distributed parameters is also very handy for practical purposes. Especially when many such pieces of the same material are placed or stacked in a large chamber to dry (wood stacks for instance), the airflow patterns around these pieces are already complex and need significant

6


computational input. The lumped models can be implemented using simple software such as an Excel spread sheet and sometimes even a simple, programmable scientific calculator. To be relevant to the specific material of concern, every drying model proposed needs careful experimentation to establish the required model parameters (constants) and the model predictive power. In other words, these constants are mostly material specific. For a diffusion-based drying model, for example, effective mass diffusivity is the key parameter. It may be both water content and temperature dependent, making accurate experimental determination and data analysis difficult due to non-uniform temperature distribution when it comes to relatively large materials. If there was one model that incurred only minimal experimental effort and less demand for lab facilities, that model would be welcomed by industry. Relevant experiments have to be accurate, possessing the required resolution, and simple, robust apparatuses and simple operating procedures are desirable. In many ways, the idea of the REA, apart from its scientific merits and the authors’ own desires to make a novel model at that time (1996), is an outcome of these rationales.

1.2

A ‘microstructural’ discussion of the phenomena of drying moist, porous materials Drying, the process of water removal, can affect the chemical composition of where the water molecules stayed within the domain marked by the material surfaces before drying. The removal of water molecules leaves vacant spaces, which may be ‘filled’ partially or totally by a nearby species. These ‘movements’ should affect the microstructure formation of the material being dried. These movements may include rearrangement of solute molecules (in drying liquid in particular) and shrinkage (with shell formation or hard surface formation as well) and solid structure breakdown (crack formation, for example). Indeed, then a solution (droplet) is dried to form a particle, in a gaseous medium the solid surface formation (chemical composition rearrangement and microstructure formation) is affected significantly by water removal (Chen et al., 2011). The structural changes, once ocurring at nanostructure/microstructural levels, can affect the bioactivity of the biological species, i.e., cells or microbes. The removal of water can cause ‘permanent’ movement of the soft structures that support the physicochemical structure of the cell wall or cell membranes and cell contents such as genetic material, enzymes, or proteins within, causing irreversible damage. This damage could be minimised if there was another structure-supporting material such as sugar (this can be done by infusing the products, mostly food materials, with sugar molecules in osmotic treatment) which may ‘replace’ the water molecules to uphold bioactivity. It is also possible that the drying–concentration process alters the ionic conditions, which may be more suited for cells’ survival. Drying, in high-temperature instances, with still-considerable moisture content, can cause proteins to denature, thus affecting the solubility and heat stability of protein products. The colour of the food material can be altered during drying, especially when heat is added. Here, Mallard reactions may be responsible where proteins

Introduction

Temperature profile

7

T∞

Liquid content

Heat flux

Vapour flux Symmetry

Mixture of water vapour and air Uniform capillary assembly

Figure 1.3 ‘Air drying’ of a capillary assembly (a bundle) which consists of identical capillaries (diameter and wall material) – a scenario of symmetrical hot air drying of an infinitely large slab filled with the capillaries (modified from Chen, 2007); the air flows along both sides of the symmetrical material.

and carbohydrates are present, such as in the baking of bread. However, the ‘brown’ colour is welcomed by consumers due to perceptions of a traditional, wholesome and cooked appearance. Before we proceed with the REA model concept and theory, we first discuss the drying itself in relation to microstructures to provide an important scientific background, which makes drying more relevant to modern material science. Microstructure is a relatively new term compared to the classic theories of drying. There are a number of versions of academic descriptions on how drying proceeds into initially moist materials. Here, an intuitive, microstructural view of the gas- (hot air) aided drying process for moist porous media is presented, which is simplified from Chen (2007). First, we look into a hypothetical scenario and ideal case, where the initial temperature of the moist material is slightly lower than the wet-bulb temperature of the drying medium for an ideal capillary system, as shown in Figure 1.3. The directions of heating and water vapour transfer are opposite each other. The capillaries here have identical diameters at the micro level and the walls are hydrophilic, with no interexchange of heat and mass across the capillary walls (simply, the walls are impermeable), assuming they were initially completely filled with water and evaporation starts to happen. The evaporation occurs uniformly here for all tubes in the same convection condition, at their exits, provided the convection condition is the same everywhere along the side of the assembly. There will be an obvious receding front of the liquid–gas interface moving inwards as drying proceeds. The thickness of the moving evaporation front will reflect the meniscus of the liquid–gas interface.

8


A mixture of complexity arises for this ideal system as different diameters of the capillaries and permeable walls (e.g., membranes) are involved. The interexchange of heat is possible across the capillary wall and the evaporation rates among the tubes (under the same drying conditions applied at the exits) now differ. There will be nonuniform receding liquid–gas interfaces among the tubes, giving a distribution of the average liquid–water content (averaged over the lateral direction) along the horizontal direction, broader than that which is shown in Figure 1.3. Furthermore, if the walls are made of materials that are hygroscopic, water-molecular movement or liquid spreading along the wall surfaces is also possible. Though the interexchange of moisture and heat between the tubes may attempt to even the evaporation rates and liquid water content, a broader distribution of the liquid water content is still expected. Furthermore, due to the extended liquid–gas interfaces, evaporation will not occur only at the meniscus. Evaporation will happen in a region of finite dimension: the occurrence of an evaporation zone at the micro- as well as at the macro-level. In a non-ideal situation, such as in a normal porous media with pores (open and closed) and interlinking ‘channels’, the likelihood of the occurrence of a sharply receding liquid front is reduced. In other words, the situation shown in Figure 1.3 is not common. When the system is ‘mixed’ at the micro-level (a more realistic situation for a practical porous media), capillaries of various dimensions are oriented in many directions and interlinked or networked. Even locally at the micron level, the capillary diameter sizes can be uneven. The mass transfer is multi-directional, following the laws of physics, that is, following the directions of the driving forces. Locally and microscopically, the receding front(s), if any, would be fuzzy, depending on the specific local microstructure and hydrophilic (or not) nature. Liquid movement may be diffusive or driven by capillary forces and travelling in relatively easier passages. For air and vapour transfer, certain difficult (yet wet) patches may be bypassed by a main ‘receding front’, which may be left to dry more gradually, thus forming a relatively wet region. ‘Fingering’ phenomena may be possible with relatively dry and relatively wet patches coexisting nearby. Here, the sorption/desorption characteristics of the materials distributed should play key roles as well. The capillary wall’s thickness (the apparently solid structure) and the walls’ own porous microstructures (yet another, smaller level of pore networks or systems), and their unevenness in spatial distribution, add to the overall picture, making the transition more uniform. The materials that create the walls of the microstructures are also important as they can have quite different affinities towards water molecules (these are reflected by their equilibrium isotherms or liquid water holding capacity at the same relative humidity and temperature). Evaporation may occur mostly in the transitional region where the rate is dependent on the local driving force for vapour transfer. The ‘walls’ of the microstructures are also important as they can have quite different affinities towards water molecules (these are reflected by their equilibrium isotherms or liquid water holding capacities at the same relative humidity and temperature). All these characteristics would make the liquid water content (averaged over these microscopic regions) distributed over a region between the really wet core and boundary

Introduction

Mass exchange surface

9

Drying air

Liquid water content

T∞ Ts

Temperature Cl.s Cv,s Cv.∞ x=0

x = xs

x

Moist porous material Figure 1.4 Schematic showing a common scenario of air drying of a moist solid.

of the moist material. This leads to a spatial transition of water content distribution rather than the sharp receding liquid waterfront. This is an important recognition of the drying phenomena such as air–gas drying. In freeze drying, on the other hand, sublimation phenomena may induce a sharp solid–gas interface. In high-temperature processes, even for air drying, the powerful heat front may result in an apparently sharper front of water movement than its counterpart when a much lower heat wave is encountered. However, in this case, the previously mentione intuitive analysis is still valid, though quantitatively it may become less influential. Figure 1.4 shows a general depiction of air drying a moist porous material. One may pay attention to the ‘partitioning’ between the liquid phase and the vapour at the mass exchange surface (the interface between the solid domain and the air). Into the porous solid domain, a vapour phase can also coexist with air and moisture (liquid) phases. These arguments can be readily generalised to packed particulate systems where the individual particles can have their own macrostructure and sorption characteristics (see Figure 1.5), while the main voids (where easier vapour paths can be found) would be the voids in between the packed particles. In fruits and vegetables, the cellular structure plays a very important role as the cell walls present major water transfer resistance (see Figure 1.6). The perception of a moving (liquid) front or the (sharp) evaporation front can lead to different approaches in drying modelling. Mass transfer from the sharp moving front and the vapour exit surface is often modelled using a simple effective diffusion concept

10


Figure 1.5 Packed particulate material.

Intercellular spaces

Cell content

Intact cell walls

Figure 1.6 Cellular structures in plant material.

(with an expanding resistance layer). In general though, this is a simplification, which is intended largely for mathematical modelling. It is also interesting to note from the process shown in Figure 1.3 that, due to the temperature distribution in the moist material being dried in a normal air drying situation (where the air temperature is higher than the porous material being dried), it is not necessary to have the highest water vapour content at the innermost boundary where liquid water content starts departing from the initial value. The vapour concentration can be higher than the boundary value, or else there would be little or no drying. It is possible to intuitively reason that there is a ‘hump’ that can exist somewhere in the transitional region of the liquid water content (Chen, 2007). A condensation mechanism also may

Introduction

11

exist in the region marked in the same spot as an area of uncertainty. The process of having a high temperature and low humidity in the same air would induce an inward transport of vapour as well as one that goes outwards (thus, drying is evident). Furthermore, a part of the water evaporated in the lower part of the transitional liquid water content region is transported into the structure and condensed at the lower temperature location, as long as there is porosity (spaces for vapour to go into). This is an interesting phenomenon, as it is clearly a more effective heat transfer mechanism than just heat conduction. Hence, this phenomenon helps increase the temperature of the core wet region more rapidly, which by itself has higher heat conductivity due to the higher water content (lower or no porosity). This mechanism has an impact on the preservation of active ingredients such as probiotic bacteria encapsulated inside a wet porous matrix subjected to drying. The structure and the porosity have a large impact on the thermal conductivity of this relatively dry layer. Conversely, the porosity and structure inside the region between lines 1 and 2 are also important, affecting vapour transport in this area. One would expect this region to have lower porosity (thus, a lower vapour transfer coefficient – the vapour diffusivity). In addition to this, the vapour transfer and condensation mechanism, as mentioned earlier, would make this heat conductivity effectively even higher. The lowest liquid water content near the solid–gas boundary should be determined by the nature of the material and the drying air conditions through the equilibrium water content concept (equilibrium isotherms) (Chen, 2007). As mentioned previously, to a great extent increasing temperature may make the trend of the liquid water content steeper; and a ‘waterfall’-like behaviour, where the vapour wave is apparently moving inwards and a more tidal-like liquid water content versus distance profile emerges. Furthermore, how the material swells and shrinks locally will have an impact on the dried product’s quality. This description has been supported by the micro-scale transient observations using magnetic resonance imaging (MRI) conducted mainly at low or moderate temperatures. A number of studies have directly targeted moisture transfer (Guillot et al., 1991; Hills et al., 1994; Bennett et al., 2003; Mantle et al., 2003; Reis et al., 2003; Ruiz-Cabrera et al., 2005a,b). There is no sharp front of evaporation observed in the latest MRI studies. The spread of the lowering liquid water content as drying proceeds relies on capillary diffusion of liquid water (Reis et al., 2003). The moisture transfer or transport devices, units such as capillaries, inter-cellular spaces, voids and channel networks between packed particles (which themselves may also be porous presenting another, perhaps finer, level of transfer devices or units), all naturally possess non-uniformity. The spread of the evaporation zone, or a transitional, ‘mushy’ zone, from the still very wet core and the already dried surface region is, therefore, expected. The pre-treatment (soaking) using surface-active reagent solutions may help accelerate the water transfer process. This understanding has been particularly helpful in supporting the ideas of revising the conventional Biot and Lewis number calculations when air drying is of interest, so conditions for model simplification can be made more realistically, and the model concepts may be discerned with more quantitative support (see later sections on Biot and Lewis number analyses). The effective diffusivity functions published in the literature can be compared and discussed based on their relationships with the scientific discoveries

12


with MRI or other insightful tools. It is now recognised that the material microstructure and its nature (composition, pore sizes, etc.) is interactive with transfer phenomena and formation of microstructures, which has so much to do with the speed and schedule of the transfer processes. Drying induces shrinkage, which is the most apparent structural change visible. As drying occurs into the pore channels in the material, the water remaining ‘clamps’ on the structural walls to create the ‘pulling’ strength that drags the structure ‘in’, so to speak. In today’s modelling exercises (also refer to Chapter 3), more and more studies are focused on treating materials according to their structural makeup. In biological material as an example, domains in different scales of length may be divided according to the structural characteristics, such as structural biology (see Figure 1.6 for modelling drying of corn kernels as an example), and solved for moisture and energy transfer with their own specific properties (Takhar et al., 2011). This approach has been termed the ‘hybrid mixture theory-based moisture transport and stress development’. The diffusivities for each domain were established separately as functions of temperature (Arrhenius’ law) and water content (power law) (Chen et al., 2009). The geometry of corn kernels shown in Figure 1.7 is a complex three-dimensional one. The computational domains representing different parts of the biological structure were captured using X-ray computed tomography (CT) and the scans were performed using an Xradia micro-XCT scanner (Xradia Inc., Concord, CA, USA) at voltage and power settings of 40 kV and 4 W, respectively. For imaging, a total of 253 two-dimensional slices were obtained with a voxel size of 2.7392 µm in the x, y and z directions. Some other techniques were also to ‘convert’ this information into a digital format to be used in simulations from the Comsol Multiphysics package. Here, the geometry was rescaled to generate suitable computational meshes (Takhar et al., 2011), while the threedimensional distributions of the computed moisture content, temperature and the stress are visual and useful to help understand the physics involved. The predictions of the average moisture content during drying, especially at higher temperatures such as 85 °C or at lower temperatures such as 29 °C, were not accurate. The authors had attributed these to the extrapolations of diffusivities evaluated at a narrower temperature range, i.e., 35°–65 °C, to both the higher and the lower temperatures explored in the simulations. Figure 1.8 shows compression of an early wood spruce, which shows how the material point method (MPM) can be used to describe the movement of the ‘shrinking material’ (Frank and Perre, 2010). MPM has been defined as the domain of interest (initial moist solid domain, for instance) treated as a collection of material points p = 1, 2 . . . np (Sulsky et al., 1994; Sulsky et al., 1995). Each material point carries its own properties such as position, velocity, acceleration, strain and stress (basically the Lagrangian approach). This method allows the finite-element discretisation of rather complex material shapes to be made, based on two- or three-dimensional images, in a robust and efficient way. As a result, it is now possible to handle such complex material structures at the plant cell level from images taken at various resolutions. For instance, an optical microscope commands a spatial resolution of 0.5 µm with an acquisition time of 100 frames per second; IR microscope 10 µm, a few frames per second; confocal microscope 0.2 µm, a few frames per second; Raman microscope 0.2 µm, a few frames per second; scanning

A stack of 253 slices obtained using micro-CT scanning at a voxel size of 2.7392 microns in x, y and z directions

3D surface rendering using Avizo package

Exporting the Avizo geometry to Comsol using Gmsh package

Rescaling to Lagrangian coordinates and meshing using Comsol package

Digital cutting of 3D geometry across the plane of symmetry (a)

Figure 1.7 (a) Generation of computational domains of corn geometry for the hybrid mixture theory of corn kernels (adapted from Takhar et al. (2011)). (b) The simulated results (isosurface plots of corn moisture content) for a variety of drying conditions. [Reprinted from Journal of Food Engineering, 106, P.S. Takhar, D.E. Maier, O.H. Campanella and G. Chen, Hybrid mixture theory based moisture transport and stress development in corn kernels during drying: Validation and simulation results, 275–282, Copyright (2012), with permission from Elsevier.]

14


T29 C, RH 48%

T67 C, RH 15%

0.2112

0.1579

0.2061

0.1529

0.201

0.148

0.1959

0.1431

0.1908

0.1382

0.1857

0.1332

0.1806

0.1283

0.1755

0.1234

0.1704

0.1185

0.1653

0.1135

0.1602

0.1086

0.1551

0.1037

0.15

0.0988

0.1449

0.0938

0.1398

T48 C, RH 25%

0.0889

0.1181

0.094

0.1142

0.0911

0.1102

0.0881

0.1062

0.0851

0.1023

0.0822

0.0983

0.0792

0.0943

0.0762

0.0904

0.0733

0.0864

0.0703

0.0824

0.0673

0.0785

0.0644

0.0745

0.0614

0.0706

0.0584

0.0666

0.0555

T85 C, RH 14%

0.0626 (b) Figure 1.7 (cont.)

0.0525

Introduction

15

Figure 1.8 Wood cellular structures employed in pore-network modelling of drying of wood.

[Reprinted from Drying Technology, 29, P. Perre, A review of modern computational and experimental tools relevant to the field of drying, 1529–1541, Copyright (2012), with permission from Taylor & Francis.]

electron microscope (SEM) 3 nm, several seconds for one frame; transmission electron ˚ <1 s; atomic force microscope (AFM) <1 nm (x-y plane) (and microscope (TEM) 1 A, ˚ <1 A at z axis), several minutes (Perre, 2011).

1.3

The REA to modelling drying

1.3.1

The relevant classical knowledge of physical chemistry In the forthcoming sections, the ideas and the development behind the REA for modelling drying process will be briefly described. A short summary of the classic modelling approaches is given first, and the newer REA approach will be subsequently introduced. To begin the discussion on the REA concepts, we need to outline physical chemistry principles of chemical reactions with a little originality on our part, in order to form the basis of the REA idea. The most prominent idea in reaction engineering is the expression of the chemical reaction rate. A chemical reaction rate of species A, involved in the reaction of two species (A and B) yields a product commonly expressed as: dc A = k A cnAA cnBB , (1.3.1) dt where nA and nB are the orders of reactions associated with species A and B, respectively, kA is a rate constant, and CA and CB are the concentrations of species A and B, respectively. This rate constant increases with temperature T, approximately increasing by two to −

16


four times with a temperature increment of 10 K. The relationship between a reaction rate constant, k, and temperature, T, has been generically described using the famous Arrhenius equation (Fogler, 1992): −

d ln k A EA , = dT RT 2

(1.3.2)

where EA is the activation energy of the reaction (J mol−1 ). This means that the value of ln kA change against temperature is proportional to the value of EA . The larger activation energy EA is, the more sensitive the reaction rate towards temperature change. EA can be variable against temperature when multiple reactions are occurring simultaneously. When the range of temperature is not large, EA may be considered a constant. In this case, which is more commonly adopted in real life, the rate constant is expressed as: k A = k Ao e−E A / RT ,

(1.3.3)

where kAo is a constant. Molecular mechanisms of evaporation and condensation at free liquid surfaces under the vapour–liquid equilibrium are investigated with molecular dynamics computer simulations for argon and methanol. Vapour molecules colliding with the surface are in the condition of almost complete capture for both fluids but, in the case of methanol, molecular exchanges strongly affect the evaporation–condensation rate (Matsumoto et al., 1995) (presumably water evaporation–condensation behaves similarly). Evaporation– condensation is one of the fundamental processes in many fields of science and engineering. For decades, various experiments have been done to measure the absolute values of evaporation or condensation rate, but there still remains a lack of knowledge of the underlying molecular mechanisms. There is still a lot of room available to explore the fundamental aspects of evaporation–condensation. This phenomenon is further complicated by the presence of species other than water during the drying process. One could, however, understand intuitively that removal of water (in vapour) is an energetic process involving latent heat. One has to put in energy to ‘activate’ water molecules that could become free when the material is dried. When the water molecules are not associated with the solid or solute molecules and they stay in the bulk liquid domain, their interactions between one another would be as if there was no solid. Evaporating them into a gaseous form would be done mainly by providing energy sufficient to overcome the latent heat of water evaporation. Of course, another condition is that the vapour can be transported into the gas medium or into a vacuum. It is interesting to note that, since the condensation process is spontaneous and does not need to overcome a kind of activation energy (an energy barrier), the activation energy for condensation is zero, and the activation energy of the evaporation process for pure water would be equal to the heat of reaction (the latent heat of water vaporisation); this is to say, the forward reaction activation energy is less than the reverse reaction activation energy, which is zero for condensation. This was the basic motive and idea in formulating the drying rate as a competition between evaporation and condensation (to be described later).

Introduction

1.3.2

17

General modelling approaches Before REA there were three main general approaches in the literature to formulating drying models, summarised next (Chen and Xie, 1997): 1. The concept of a characteristic drying rate curve (CDRC model), which recognises different drying stages; e.g. the constant rate period (which may also be called the ‘unhindered drying’ period, where the internal transport of the moisture does not affect the surface evaporation) and falling rate (‘hindered drying’) period(s). 2. The distributed-parameter models, mostly those based on volume averaging concepts, employing coupled heat and mass diffusion equations involving heat conductivities and mass diffusivities, etc. 3. The empirical models obtained entirely by simple or multivariable regression methods (often, a series of known time-dependence functions such as the Page model etc. have been used to simply correlate the weight loss over time). There are many models in categories (1) and (3). Both (1) and (3) may be regarded as the ‘lumped drying models’ in that they do not need to solve for the spatially distributed moisture content and temperature. For (2), there have been a number of continuumtype mechanisms proposed and the corresponding mathematical models established. These include effective liquid diffusion, capillary flow, evaporation–condensation, dual (temperature, water content gradient) and triple (temperature, water content and pressure gradient) driving-force mechanisms by Luikov (1986), another dual driving-force mechanism by Philip and De Vries (1957) and De Vries (1958), and, finally, the dual-phase (liquid and vapour) transfer mechanism of Krischer as summarised by Fortes and Okos (1980). Whitaker (1977; 1999) has proposed detailed transport equations to account for the macro- and micro-scale structures in biological materials. Three-phase (solid, liquid and vapour) conservations and their local volume-averaged behaviours are considered. The mechanisms for moisture transfer are largely the same as those proposed by Luikov (1975; 1986) and Philip and De Vries (1957), except that the small-scale phenomena (local pores, pore channels, shells, voids, etc.) have been taken into account. This theory is based on a known (or pre-assumed) distribution of the macro-scale and micro-scale unit structures, which allow local volume averaging to be carried out. Pore-network models have become popular in recent years, as the concept of multi-scale and multiphysics is expanding, e.g. coupling meso-scale problems and equipment scale problems (Perre, 2011). When a local thermodynamic equilibrium is not attained, the time scales usually overlap. This is a real multi-scale configuration and challenging in terms of the great demand in computational power and mathematics. Several scales can be considered simultaneously, ranging from simple exchanges between macroscopic phases to a comprehensive formulation, in which the time evolution of microscopic parameters and microscopic gradients are considered over a representative elementary volume according to a recent review by Perre (2011). These models are often mathematically highly involved.

18


For (3), the models usually have little physics explained and it is difficult to extract any fundamental information, though some of them have attempted to show some physical significance on somewhat weaker grounds. On the other hand, some accurate time functions under category (3), such as the Page model, can be used to fit the data points to generate accurate drying rate data sets for other purposes. Some models of a comprehensive nature will be described in detailed mathematical terms in later chapters when the authors compare the performances of REA in various practical cases against well-known models. In modern times, expanding what we have already in category (2), the comprehensive modelling of drying in a spatially distributed manner (or, say, in a discrete manner) has frequently involved more pore-level information (pore-network models) and mathematical techniques, which do not require volume-averaging procedures of some sort. The respective pore networks can be used to systematically study the influence of the structure of a porous medium on drying kinetics. There are potentials of this discrete modelling for use as a virtual laboratory to improve our understanding of how structures correlate with properties and how better products may be developed (Tsotas and Mujumdar, 2007). In general, several scales of problem are involved in drying, modified from those summarised by Tsotsas and Mujumdar 2007: The molecular scale (water molecules interact with each other and with the other species in the liquid or gas, and with the solid surfaces), the pore scale (the smallest entity for expressing transport phenomena within the drying particles or single bodies), the particle scale (single drying body can be identified; this larger scale can include rather ‘large particles’ such as wood boards stacked and dried in an industrial drying kiln), the particle-system scale (the equipment is designed and properly operated at this level; the interactions between the particles, the gas flow and the apparatus are considered at this scale), and finally the process-system scale (the drying system interacting with other engineering systems ensuring the proper operation of the entire production plant).

1.3.3

Outline of REA Following the basic descriptions of physical chemistry in relation to the expression of chemical reaction rate as described earlier, the REA was proposed by the author (XDC) in 1996, and in the subsequent year, a couple of papers were published (Chen and Chen, 1997; Chen and Xie, 1997; Chen and Pirini, 2004). The idea was also partially inspired by two pieces of information: a paper published by Professor Brian Gray (1990) and a series of works on the long-established CDRC model (van Meel, 1958; Keey, 1978; 1992). In 1990, a mathematical model for a wet-combustion system, the exothermically reactive (porous solid) system, which is also influenced by the presence of water, was published by Professor Gray, who was a Senior Professor of Mathematics at School of Chemistry in Macquarie University, Sydney, Australia. However, before that landmark, the role of water on the exothermic (solid) systems, such as spontaneous heating or spontaneous combustion, had been proposed in several models from more of an engineering

Introduction

19

perspective (Chen, 1991). However, the role of water as a direct participant in chemical reactions (in oxidation in particular) had not been considered quantitatively. In the paper by Gray, he added a term in the mass balance and energy balance, respectively, which accounts for the direct participation of water in chemical reaction (exothermic), in conjunction with the water effect through evaporation (liquid to vapour) and condensation (vapour to liquid). In the same paper, Gray wrote the following equations to describe a wet-combustion system where the temperature of the (combustible) solid material is assumed to be uniform, as is the water content within the solid matrix: dx = c (1 − x) − e xe−α/u − w xe−αw /u , dτ

(1.3.4)

where x, u and τ are the dimensionless liquid water concentration, temperature of the (combustible) material, and time, respectively. Term x is defined as the current water content (g) divided by the initial water content (g) that is available in liquid form (in fact, it is the total water content in the system boundary), i.e. mw /mw,o . The model constants are represented by ϕ in Equation (1.3.4); this system is also, in the theory of thermal ignition and combustion, known as the Semenov approach, signifying the uniformity of the variables throughout the material of concern (Bowes, 1984). This is a useful assumption, which paves the way for a large number of mathematical analyses that have considerable physical meaning relevant to practical conditions. In Equation (1.3.4), the first term on the right-hand side represents the condensation process, denoted by subscript c, and the second term evaporation, denoted by e. The last term on the right-hand side represents the consumption of water due to the water-induced or -involved chemical reaction (exothermic), the wet oxidation, denoted by w. When we remove this wet-oxidation term, an inert system, we have: dx = c (1 − x) − e xe−α/u . dτ

(1.3.5)

Equation (1.3.5) represents the water exchange (condensation less evaporation) between the moist material and the environment/surrounding. (1 − x) signifies a conservation of the ‘total water content in the domain of interest’. The evaporation term is considered to be first-order as far as water ‘reactant’ is concerned. The most important description of the evaporation term is used in the Arrhenius dependence function (a well-known α function in physical chemistry), i.e. e− u . The α in the context of Gray’s analysis denotes the dimensionless latent heat of water vaporisation. This may not have been completely new, even at that time, but certainly was a great approach to describe the physical picture, as we tried to understand moisture movement in and out of a porous solid matrix. Gray (1990) and Gray and Wake (1990) took full advantage of this simple and effective formulation to fruitfully explore the wealth of the behaviours in the wet-combustion system. In any case, a steady state can be attained for the system described by Equation (1.3.5) as: c (1 − x) = e xe−α/u .

(1.3.6)

20


x 1

0

T

Figure 1.9 Schematic illustration of the effect of temperature on final liquid water content

(qualitatively derived from Equation 1.3.6).

Rearrange to give: 1−x e −α/u e , = x c

(1.3.7)

which essentially suggests that as the material gets hotter, the liquid water content in the system reduces more. The activation energy of the evaporation ‘reaction’ was assumed to be the latent heat of water vaporisation. This has good intuitive grounding (Chen, 1998). As the dimensionless temperature u gets infinitely large, x becomes zero (see Figure 1.9). This system, however, does not seem to capture the physics behind when water content in an environment becomes zero; even when temperature is moderate, the liquid water content inside the material can also become zero. In other words, this system might have neglected another dimension of the drying system. On the practical side, a drying kinetics approach, the CDRC model, had been employed in order to design dryers and optimise drying operations for improving energy efficiency. In his developing understanding of drying operations and the fundamentals of moisture transfer, van Meel (1958) postulated that, when working with convective batch dryers, a single characteristic drying curve could be deduced for a moist material. This model reflects the nature of the drying rate curve(s) shown in Figure 1.10(a). It is empirical but has been successful in correlating the drying kinetics of small particles. The model assumes that, for any given sample water content, a unique relative drying rate exists. This rate is relative to the initial unhindered drying rate and is independent of the external drying conditions (refer to Figure 1.10(b)). These conditions include the temperature, humidity and pressure of the drying gas. The model also implies that a region exists where, for a period of time, the rate remains unhindered. The physics behind what is shown in Figure 1.10(a) has been described scientifically by Perre, Remond and Turner (2007). For non-deformable materials (negligible shrinkage) such as building materials and natural mineral products (including fragmented rocks), the relationship between porosity and moisture content is obvious. As drying proceeds, the moisture content is simply replaced by drying gas. For highly deformable products such as food, the moisture removal is related to both volume reduction and porosity. It is necessary to know whether the loss of moisture turns into volume change or into an increase in porosity (Perre and May, 2001). Considering these factors one can interpret

21

Introduction

Drying flux (kg.m--2.g--1)

Constant drying flux period Cooling down Falling drying flux period

Warming up

Critical average water content

Equilibrium water content 0.0

Xc (a)

X∞ ξ

Average content (X)

Critical point (1,1) Hindered region Unhindered region

0 Equilibrium point

φ (b)

Figure 1.10 (a) Drying flux versus average water content X ; (b) the CDRC (characteristic drying

rate curve). [Reprinted from Chemical Engineering Science, 9, D.A. van Meel, Adiabatic convection batch drying with recirculation of air, 36–44, Copyright (2012), reprinted with permission from Elsevier.]

the results on constant drying rate, which should be the constant drying flux, properly. The mass exchange surface area is an important parameter involved here. Here, the relative drying rate is defined as: ξ=

N , Nc

(1.3.8)

where ξ is the relative drying rate, N is the instantaneous drying rate (best defined to be the drying flux, kg m−2 s−1 ) and Nc is the drying rate at the critical condition (i.e. when the drying regime is in transition between the unhindered rate and falling rate periods; at the critical water content, Xc ).

22


The characteristic moisture content (dimensionless water content) is defined by the following equation: =

X − X∞ , Xc − X∞

(1.3.9)

where X is the average water content (on dry basis) at any time t, X is the equilibrium water content (on dry basis) and Xc is the critical water content. The drying rate is normalised to pass through the critical point and the equilibrium point, denoted by points (1,1) and (0,0), respectively in Figure 1.10(b). According to Keey (1992), the characteristic curve method is attractive since it leads to a simple lumped-parameter expression for the drying rate, in the following form: N = ξ Nc .

(1.3.10)

This expression has been used extensively as the basis for understanding the behaviour of industrial drying plants. Because of the simplicity of the parameters used, this has been employed in some industrial applications. Basically, the general form of the CDRC (Keey, 1992) is expressed as follows: ξ = f X , if X ≤ X c ; ξ = 1, if X > X c . (1.3.11) There are a number of successful applications of this approach, especially for materials of small dimensions such as particles or thin layers (Keey, 1992). Keey (1992) has further specified that a unique characteristic curve can be established at Kirpichev numbers less than 2 or, in effect, when the material is thinly spread and permeability to moisture is high (i.e. the material has a large moisture (vapour) diffusivity). The Kirpichev number is given as: Ki =

Nc δ , ρs X o Dv,eff

(1.3.12)

where δ is the thickness of the sample (m), ρ s is density of the dry solid (kg m−3 ), Xo is the initial water content (on dry basis) (kg kg−1 ) and Dv,eff is the effective vapour diffusivity (m2 s−1 ). According to Keey (1992), in many cases the drying curve can be fitted using a simple algebraic equation over a limited moisture content range of interest by: X − X∞ , (1.3.13) ξ= Xc − X∞ where j is a parameter dependent on the relative difficulty of removing moisture from a material. For example, j was found to be about 0.5 for cellulosic fibres (Langrish, 2008). Equation (1.3.10) is a nice, simple and user-friendly expression, if accurate enough, for the material of concern. The author was first exposed to the idea of the CDRC during his Ph.D. study at Canterbury University in Christchurch, New Zealand. The exposure of the author to CDRC was significant, due to the physical presence of the well-known advocate of the CDRC approach (and indeed one of the greatest in drying research and development), Professor Roger Keey at Canterbury University, at the time. During the

Introduction

23

same period of time, a visiting scholar from China, Professor Yuan Wu, was working on the application of a variation of the CDRC approach in through-drying of wool in Keey’s laboratory (which was later published in Chemical Engineering Science, Wu and Keey, 1995). However, being young at the time, the author did not fully appreciate the usefulness of the CDRC approach, which only became apparent later in the author’s research career in drying. According to the understanding gained so far, there are aspects of CDRC that need to be noted: (1) The critical water content(s) have to be determined experimentally, which are known to be dependent upon drying conditions (temperature, humidity and velocity). (2) The quality of results of data reduction is not good, as in some cases the data points can be quite scattered (for instance, with some large relative errors from the mean in the falling rate period(s)). (3) In addition, the mass flux (Nc ) calculations are sometimes based on the wet-bulb temperatures (Twb ) for the gas phase. Aspect (3) is illustrated as follows: Nc = h m ρv,sat (Ts ) − ρv,∞ ,

(1.3.14)

where the material surface temperature Ts (K) may not be exactly the same as the wetbulb temperature Twb (K) and hm is the mass transfer coefficient (m s− ). Here, ρv,sat (Ts ) is the saturated water vapour concentration at the surface temperature Ts (kg m−3 ) and ρv,∞ is the water vapour concentration in the environment or in the drying gas medium (kg m−3 ). A CDRC model is usually obtained from laboratory experiments under constant external conditions, with moist materials of similar form and size to that of interest in the real industrial dryer situation. As mentioned earlier, one also needs to note that the CDRC model must have accurate measurement, choice or prediction of the constant rate and the maximum rate, Nc . An accurate estimation of the critical water content as a function of the external drying conditions must be made, as well as the ability (or, rather, lack of it) to reduce effective data for the relative drying rates. For CDRC, it is clear that the actual rate of drying (or drying flux) N is written as: N = f ()Nc f () ρv,sat (Ts ) − ρv,∞ .

(1.3.15)

To introduce the REA concept in mathematical terms, based on the conventional transport phenomena theory, we write the vapour flux at the boundary (solid–gas) as: N = h m (ρv,s − ρv,∞ ) = h m R Hs ρv,sat (Ts ) − ρv,∞ ,

(1.3.16)

24


where ρv,s is the vapour concentration at the interface of solid and gas (kg m−3 ) and RHs is the relative humidity of the air (or gas in general) at the solid–gas boundary, which may be defined as: ρv,s . (1.3.17) R Hs = ρv,sat (Ts ) During a drying process, this surface relative humidity, RHs , is reducing but is an unknown quantity. Equations (1.3.15) and (1.3.16) thus show a significant contrast between them. Equation (1.3.15) defines the instantaneous drying flux as a fraction of the maximum possible; Equation (1.3.16) follows the widely accepted mass transfer expression with the first term in brackets on the right-hand side of the equation being the product of RHs and saturated vapour concentration at the interface. The difficulty in using Equation (1.3.16) is how one can express RHs as a function of some known quantities. The concept of REA has incorporated the more conventional approaches to expressing mass transfer at the boundary, i.e. Equation (1.3.16). It is an application of chemical reaction engineering principles to establish a workable function for RHs . In this approach, evaporation is modelled as zero-order kinetics with activation energy while condensation is described as a first-order wetting reaction with respect to drying air vapour concentration without an activation energy (Chen and Chen, 1997; Chen and Xie, 1997). The REA approach employs the Arrhenius equation in the evaporation term, which has its origins in the paper by Gray (1990) but it differs markedly overall from what he proposed (Chen, 2008). The REA approach offers the advantage of being expressed in terms of a simple, ordinary differential equation with respect to time. This negates the complications arising from use of the partial differential equation (Chen, 2008). REA does need accurate experimental data to determine model parameters, accurate equilibrium isotherm and surface area measurement. It will be shown later that REA accommodates a natural transition because of the smooth activation energy as a function of moisture content (Chen, 2008). When activation energy for evaporation becomes higher than the latent heat of water evaporation, free water should have already been removed (Chen, 2008). As a lumped approach (REA was initially proposed as a lumped parameter model), the drying rate (flux multiplied by surface area) of materials can be expressed as: dX = −Nc A = −h m A (ρv,s − ρv,∞ ) , (1.3.18) dt where ms is the dried mass of thin layer material (kg), X is moisture content on a dry basis (kg kg−1 ), X is the mean water content on a dry basis (kg kg−1 ), t is time (s), ρ v,s is the vapour concentration at interface (kg m−3 ), ρ v, is the vapour concentration in the drying medium (kg m−3 ), hm is the mass transfer coefficient (m s−1 ) and A is surface area of the material (m2 ). The mass transfer coefficient (hm ) is determined based on established Sherwood number correlations or is established experimentally for the specific drying conditions involved (Incropera and DeWitt, 1990). Equation (1.3.18) is basically correct for all cases of water vapour transfer out of a porous solid. In other words, no assumption of uniform water content is made in this approach, even though it ms

25

Introduction

began with mean water content. The surface vapour concentration (ρ v,s ) can be correlated in terms of saturated vapour concentration of water (ρ v,sat ) by the following equation (Chen and Chen, 1997; Chen and Xie, 1997): E v ρv,sat (Ts ) , (1.3.19) ρv,s = exp − RTs where Ev represents the additional difficulty in removing moisture from materials on top of the free water effect. It was thought it would be excellent (and indeed lucky) to be able to relate Ev to average water content of the material. In other words, this Ev is ideally moisture content (X) dependent. T is temperature of the material being dried (K). For a small temperature range, say from 0 °C to just over 100 °C, ρ v,sat (kg m−3 ) can be estimated with the following equation (Chen, 1998): Ev , (1.3.20) ρv,sat (T ) = K v exp − RT where Kv was found to be 1.61943 × 105 (kg s−1 ) and Ev was found to be 38.99 kJ mol−1 . Ev is similar to the latent heat of water vaporisation illustrating the physics involved (Chen, 1998). This is in line with the idea that evaporation is an activation process, whilst condensation is not. The activation energy of the pure water evaporation reaction is equivalent to the value of the latent heat of water evaporation, as suggested earlier based on classical physical chemistry. More widely in the range of temperature, one can use the following, which correlates the entire range of the data (0 °C−200 °C) summarised by Keey (1992) (also see Figure 1.11): ρv,sat = 4.844 × 10−9 (T − 273)4 − 1.4807 × 10−7 (T − 273)3 + 2.6572 × 10−5 (T − 273)2 − 4.8613 × 10−5 (T − 273) + 8.342 × 10−3 , (1.3.21) where T is temperature (K) based on the given data (Putranto et al., 2010). The mass balance is then expressed as: dX −E v ms = −h m A exp ρv,sat (Ts ) − ρv,∞ . dt RT

(1.3.22)

For small objects, such as particles or thin layer materials, the material temperature T is approximately the same as the surface temperature Ts . Basically this happens when the Chen–Biot number is sufficiently small (Chen and Peng, 2005; Chen, 2007; Chen and Mujumdar, 2008). In this case, uniform temperature can be assumed throughout the material being dried so that one only needs to couple Equation (1.3.21) with a lumped energy balance to ‘govern’ the drying process. The activation energy (Ev ) is determined experimentally by rearranging Equation (1.3.22): −m s ddtX h m1 A + ρv,∞ , (1.3.23) E v = −RTs ln ρv,sat


0.60

Saturated vapour density (Kg.m−3)

26

0.50

0.40

0.30

0.20

0.10

0.00 0.0

20.0

40.0 60.0 Temperature (°C)

80.0

100.0

Figure 1.11 Saturated water vapour concentration in air under 1 atm (Equation 1.3.21).

where d X /dt is determined from experimental data on weight loss. It has been found, based on practical experiences of using REA, that drying experiments for generating the REA parameters need to be conducted where the air (or gas) humidity is very low in order to cover the widest range of E v versus X . The dependence of activation energy on moisture content can be normalised as: E v = ς(X − X ∞ ), E v,∞

(1.3.24)

where ζ is a function of moisture content difference, Ev , is the maximum when the moisture concentration of the sample approaches relative humidity and temperature of the drying air: E v,∞ = −RT∞ ln(R H∞ ),

(1.3.25)

where X is the equilibrium moisture content corresponding to RH and T which can be related to one another by the equilibrium isotherm (Keey, 1992). It worth noting again that, so far, the experiments for gaining the relevant Equation (1.3.24) have been under very dry air conditions, so the final water content attained is usually low. For the same material, the same initial water content and the same initial sample size (sometimes different sizes do not matter), the relationship (Equation 1.3.24) may be viewed as unique as many experimental results obtained under these conditions, but different drying conditions for the same material, produced more or less the same trend quantitatively (Chen, 2008). This aspect will be shown later in various applications described in the forthcoming chapters.

Introduction

27

The REA parameters for drying of a material can be obtained from one good drying experiment and can then be applied to other different drying conditions (different drying air temperatures and air velocities) if the normalised activation energy collapses to the same profile in these cases. However, the REA parameters should be generated from material with the same initial moisture content since the activation energy has been found to be dependent on initial moisture content, too (Chen, 2008). In many scenarios tested involving different materials to be described later in this book, Equation (1.3.23) holds a very pleasant outcome indeed. Of course, other forms of Equation (1.3.23) are possible and one should be worried if there is a temperature dependence function or if a material structure parameter is involved. When the temperature of the moist material being dried does not vary much within, a uniform temperature may be considered (more quantitative assessment of this assumption can be found in a later part of this book that describes the modified Biot number and modified Lewis number). This leads to: Ts ≈ T , where T represents the mean temperature of the material (K). This has allowed us to present REA energy balance in a ‘lumped capacitance’ (Incropera and DeWitt, 1990) for the material being dried: mC p

dT dX = h A T∞ − T + Hdrying m s , dt dt

(1.3.26)

where h is the heat transfer coefficient (W m−2 K−1 ), T is the drying air temperature (K), Cp is the specific heat of the sample (J kg−1 K−1 ), ms is the dried mass of the sample and m is the mass of the material being dried, expressed as: m = ms 1 + X .

(1.3.27)

The heat of drying, Hdr ying (J kg−1 ), is more strictly expressed as a function of water content: Hdrying = f (X ) ,

(1.3.28)

which increases as water content X reduces and reaches a level as X approaches zero, which can be much higher than the latent heat of pure water evaporation (Chen, 1992; Chen and Stott, 1992; Chen, 1994; Chong and Chen, 1999). This value is usually unknown unless separate experiments are done to obtain it, so in the present approach this effect has been ‘absorbed’ by the water content dependence function empirically obtained for the activation energy (Equation 1.3.21). Nevertheless, the heat of drying may be related to the so-called isosteric heat, which may be derived from the equilibrium desorption isotherms theoretically obtained for the moist material of interest. It may also be measured using calorimetric methods as summarised by Shen et al. (2000). The heat of drying may be plotted qualitatively as that shown in Figure 1.12.


Heat of drying (kJ.kg--1)

28

Latent heat of vaporisation 0

Water content (X)

Figure 1.12 Schematic diagram showing the heat of drying as a function of water content of

a porous solid of concern (when the water content is beyond the point where the heat of drying becomes the latent heat of pure water evaporation, the water content may be called free water).

One more interesting point to note is that in some of the literature (Rostami et al., 2003), the rate of evaporation is expressed as: −

dcw = ko e−E / RT cwn w , dt

(1.3.29)

where Cw is the concentration of water, E is the activation energy (J mol−1 ), T is temperature (K) and ko is the constant. The order of the evaporation reaction, nw , is taken to be 1 (i.e. the first-order reaction). This, though not necessarily as accurate as the current REA model, may also be considered to be in the category of the REA. The simulation that uses this formulation, however, does not consider the phenomena of condensation. REA has been used to model drying of a range of food materials such as pulped kiwifruit leather, whey protein concentrate, lactose, skim milk powder, whole milk powder, cream and mixtures of sugars (Chen et al., 2001; Chen and Lin, 2005; Lin and Chen, 2005; 2006; 2007). Results showed that this approach models moisture content and temperature profile along drying time very accurately. For example, modelling of drying of an aqueous lactose solution droplet showed that the average absolute difference of the weight loss profile was about 1% of the initial weight while that of the temperature profile was about 1.2 °C (Lin and Chen, 2006). Moreover, application of the REA to model the drying of cream and whey protein concentrate showed average absolute differences in weight profiles of 1.9% and 2.1%, respectively, while that of temperature was about 3 °C and 1.9 °C, respectively (Lin and Chen, 2007). Modelling of skim milk and whole milk powder by the REA was also robust and accurate (Chen and Lin, 2005).

Introduction

29

The REA has also been implemented in computational fluid dynamics (CFD) based simulations using a spray dryer for coupling the dispersed phase (droplets dried) and the continuous phase (drying air) (Woo et al., 2008; Jin and Chen, 2009a; 2009b; 2010). CFD simulations using the REA can predict outlet air temperature and outlet particle moisture content reasonably well compared to experimental data. In addition, the REA was also implemented to predict the evaporation zone, drying rate, trajectory of particles, and deposition of particles in a spray dryer (Woo et al., 2008). Application of CFD in conjunction with the REA to describe the performance of an industrial-scale spray dryer in two and three dimensions has been conducted (Jin and Chen, 2009a; 2009b). Patel et al. (2009) have extended the ‘single solid component’ approach of the REA to a composite REA model – drying kinetics for mixtures of ‘noninteracting’ solutes (maltodextrin and sucrose). The activation energy of the mixture was determined based on mass fraction of each solute and its corresponding activation energy. It was shown that the average relative error between experimental and calculated data was below 1.5% for droplet weight and below 3% for droplet temperature (Patel et al., 2009). After years of exercising the REA concepts, summarised by Chen (2008), it became clear that, for large materials being dried, both temperature and water content vary in space, so simple usage of the REA concept needed to be extended. In the past few years, the approach has been divided into two categories: lumped-REA (L-REA, described previously) and spatial-REA (S-REA, to be described in full in Chapter 3). Putranto et al. (2011) have applied the L-REA to successfully describe the situations of time-varying boundary conditions, such as intermittent drying. Following an initial attempt by Kar and Chen (2010), where the L-REA formulation was employed as the source term (evaporation and condensation) in the partial differential equation set that governs the multiphase transport phenomena in drying of porous media (porcine skin, to be exact, in their studies), Putranto and Chen (2013) have established a comprehensive set of partial differential equations (heat and mass balances) which has been shown to be very helpful in simulating drying, referred to as the spatial-REA; the S-REA. In the forthcoming chapters, the applications of both the L-REA and S-REA are demonstrated in detail using worked examples. Comparisons between the REA model approach and the other more established models are given. Some physical insights of the modelling approach are discussed where appropriate.

1.4

Summary In this chapter, some general aspects of drying related backgrounds have been described to set the scene. A brief review of the approaches to drying modelling has been given. A brief outline of the physical chemistry aspects of reaction engineering/kinetics has been provided highlighting the relationship between physical chemistry and the REA. Finally, the REA has been described. The L-REA approach has been introduced, which

30


is the basis for the later development of the more advanced S-REA, both to be described later in this book.

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Introduction

33

Ruiz-Cabrera, M.A., Foucat, L., Bonnym J.M., Renou, J.P. and Daudin, J.D., 2005b. Assessment of water diffusivity in gelatine gels from moisture profiles II. Data processing adapted to material shrinkage. Journal of Food Engineering 68, 221–231. Shen, D., Bulow, M., Siperstein, F., Engelhard, M. and Myers, A.L., 2000. Comparison of experimental techniques for measuring isosteric heat of adsorption. Adsorption 6, 275–286. Sulsky, D., Chen, Z. and Schreyer, H.L., 1994. A particle method for history-dependent materials. Computer Methods in Applied Mechanics and Engineering 118, 179–196. Sulsky, D., Zhou, S.J. and Schreyer, H.L., 1995. Application of a particle-in-cell method to solid mechanics. Computer Physics Communications, 87, 236–252. Takhar, P.S., Maier, D.E., Campanella, O.H. and Chen, G., 2011. Hybrid mixture theory based moisture transport and stress development in corn kernels during drying: validation and simulation results. Journal of Food Engineering 106, 275–282. Tinker, L.F., Schneeman, G.O., Davis, P.A., Gallaher, D.D. and Waggoner, C.R., 1991. Consumption of prunes as a source of dietary fiber in men with mild hypercholesterolemia. American Journal of Clinical Nutrition 53, 1259–1265. Tsotsas, E. and Mujumdar, A.S., 2007. Preface of Volume 1. In Modern Drying Technology Volume 1: Computational Tools at Different Scales, E. Tsotsas and A.S. Mujumdar (eds.), Wiley-VCH, Weinheim. pp. xv–xviii. USDA, 2007. USDA database for the flavonoid content of selected foods (Release 2.1 2007). van Meel, D.A., 1958. Adiabatic convection batch drying with recirculation of air. Chemical Engineering Science 9, 36–44. Vinson, J.A., Zubic, L., Bose, P., Samman, N. and Proch, J., 2005. Dried fruits: excellent in vivo and in vitro antioxidants. Journal of American College of Nutrition 24, 44–50. Wang, Z., Zhou, Y., Li, S. and Liu, J. (eds.), 2007. Physical Chemistry, Volume 2, 4th ed. Higher Education Press, Beijing, pp. 196–223 (in Chinese). Whitaker, S., 1977. Simultaneous heat, mass and momentum transfer in porous media: A theory of drying. Advances in Heat Transfer 13, 119–203. Whitaker, S., 1999. The Method of Volume Averaging. Kluwer Academic Publishers, Dordrecht. Williamson, G. and Carughi, A., 2010. Polyphenol content and health benefits of raisins. Nutrition Research 30, 511–519. Woo, M.W., Daud, W.R.W., Mujumdar, A.S., Wu, Z.H., Talib, M.Z.M. and Tasirin, S.M., 2008. CFD evaluation of droplet drying models in a spray dryer fitted with a rotary atomizer. Drying Technology 26, 1180–1198. Wu, X., Beecher, G.R., Holden, J.M., Gebhardt, S.E. and Prior, S.L., 2004. Lipophilic and hydrophilic antioxidant capacities of common foods in the United States. Journal of Agriculture and Food Chemistry 52, 4026–4037. Wu, Y. and Keey, R.B., 1995. Design of staged through-circulation drying with air reversals. Chemical Engineering Science 50, 99–104.

2

Reaction engineering approach I Lumped-REA (L-REA)

2.1

The REA formulation As suggested in Chapter 1, the general REA is an application of chemical reaction engineering principles to model drying kinetics, first reported in 1997 (Chen, 2008). In this approach, evaporation is modelled as zero-order kinetics with activation energy, while condensation is treated as a first-order wetting reaction with respect to drying air solvent vapour concentration without activation energy (Chen, 2008). The REA offers the advantage of being expressed in terms of simple ordinary differential equations with respect to time. This negates the complications arising from use of partial differential equations (Chen and Xie, 1997). This approach has been initially employed to express the overall drying rate for the entire object being dried – a lumped approach. A summary of the developments of the lumped approach of REA is given by Chen (2008) and described at length in Chapter 1. Generally, with no assumptions, the drying rate of a material can be expressed as: ms

dX = −h m A(ρv,s − ρv,b ), dt

(2.1.1)

where ms is the dried mass of thin layer material (kg), X is the average moisture content on a dry basis (kg kg−1 ), t is time (s), ρ v,s is the water vapour concentration at interface (kg m−3 ), ρ v,b is the water vapour concentration in the drying medium (kg m−3 ), hm is the mass transfer coefficient (m s−1 ) and A is surface area of the material (m2 ). Equation (2.1.1) is a basic mass transfer equation. The convective mass transfer coefficient (hm ) is determined based on the established Sherwood number correlations for the geometry and flow condition of concern or can be established experimentally for the specific drying conditions involved (Lin and Chen, 2002; Kar and Chen, 2009). The surface vapour concentration (ρ v,s ) can then be scaled against the saturated vapour concentration (ρ v,sat ) using the following equation (Chen and Chen, 1997; Chen and Xie, 1997; Chen, 2008): −E v ρv,s = exp (2.1.2) ρv,sat (Ts ), RTs where Ev represents the additional difficulty in removing moisture from the material beyond the free water effect. This Ev is the average moisture content (X) dependent. Ts is the surface temperature of the material being dried (K) and ρ v,sat for water vapour

Reaction engineering approach I: L-REA

35

can still be estimated by: ρv,sat = 4.844 × 10−9 (Ts − 273)4 − 1.4807 × 10−7 (Ts − 273)3 + 2.6572 × 10−5 (Ts − 273)2 − 4.8613 × 10−5 (Ts − 273) + 8.342 × 10−3 , (2.1.3) based on the data summarised by Keey (1992). When material is ‘thermally’ thin, the surface temperature is considered to be the same as the sample temperature (Chen and Peng, 2005; Patel and Chen, 2008), i.e. Ts T. The mass balance (Equation 2.1.1) is then neatly expressed as: dX −E v = −h m A exp ρv,sat (T ) − ρv,b . ms (2.1.4) dt RT From Equation (2.1.4), it can be observed that the REA is expressed in the firstorder ordinary differential equation with respect to time, and the model is the core of the L-REA. It must be noted that, though the average moisture content is used, the L-REA does not assume uniform moisture content. Of course, no spatial distribution of moisture content can be computed using the L-REA. The L-REA may also be applied to cases where the temperature within the material is not uniform as long as the surface temperature can be determined or predicted accurately. The activation energy (Ev : the characteristic of the material being dried; it is materialdependent) needs to be determined experimentally. Upon the attainment of the drying data, notably the surface temperature of the material (or the sample temperature in the case of a thermally thin situation) and moisture loss against time, plus the information about the external mass transfer coefficient, one can obtain the activation energy. This can be done by rearranging Equation (2.1.4) as follows:

−m s ddtX h m1 A + ρv,b E v = −RTs ln . (2.1.5) ρv,sat (Ts ) The rate of moisture loss d X/dt is experimentally determined. The surface area A should also be recorded in drying experiments in the case of shrinkable material. The dependence of activation energy on average moisture content on a dry basis (X ) can be normalised as: E v = f (X − X b ) , E v,b

(2.1.6)

where f is a function of water content difference and Ev,b is the ‘equilibrium’ activation energy representing the maximum Ev determined by the relative humidity and temperature of the drying air: E v,b = −RTb ln(R Hb ),

(2.1.7)

where RHb is the relative humidity of drying air, Xb is the equilibrium moisture content under the condition of the drying air (kg kg−1 ) and Tb is the drying air temperature (K).

36


2.2

Determination of REA model parameters As mentioned in the previous section, the REA model parameters such as the relative activation energy can be generated from one set of accurate drying runs which consists of measuring sample mass, sample surface temperature and sample surface area during drying. This is only true when a master curve, Equation (2.1.6), can be applied to the range of drying conditions of interest. So far in the successful applications of the REA concept, the drying experiments are usually performed with final moisture content being very low. In other words, the humidity of the drying air (or gas in general) employed in the experiments is set to be very small. This is done so the master curve can cover the widest range of the water contents for the same material. To illustrate the procedures to obtain and determine the REA model parameters, the experiments of convective drying of milk droplets are described next. The convective drying of milk droplets is conducted in a glass-filament convective drier (Lin and Chen, 2002; 2004; Chen and Lin, 2005). The drying air conditions were drying air temperature of 67–110 °C, drying air velocity of 0.45–1 m s−1 and drying air humidity of 0.001 kg H2 O kg dry air−1 (Chen and Lin, 2005). The details and setup of the equipment have been described previously (Chen and Lin, 2005). Generally, by using the glass-filament method, the droplet weight is measured by deflection of the glass filament as long as the values are corrected for the corresponding drag forces. The single droplet is attached to the tip of a special fine glass filament. The details of the experiments are as follows (refer to Figures 2.1 and 2.2). The cross section of the drying tunnel is 29.97 × 29.97 mm. The glass filament with a thin tip section of 30–100 µm (whose end has a glass knob of 100–220 µm) is used to suspend the milk droplet (initial solids concentration of 20–30%wt. and initial droplet diameter about 1.42 mm) in a preconditioned (humidity, temperature and velocity) air stream. Initial solids of greater than 30%wt. have also been successfully tested (Fu, 2012). During drying, the deflection of the glass filament and the change of droplet diameter are monitored using standard video camera fitted with four closeup lenses. The sample mass during drying is calculated by the deflection of the glass filament during drying. The glass filament will deflect from the original position when there is a mass hanging up on the tip (refer to Figure 2.1). As the drying progresses, the deflection will reduce because of moisture evaporation. The camcorder is used to capture the displacement history and the value of deflection is converted to that of mass by calibration which shows the linear relationship between the deflection and mass. The example of the linear relationship is shown in Figure 2.2. For calculating surface area during drying, the recorded images are transferred to a digital signal using a video capture card and analysed using image analysis software. The projected droplet area of each figure is considered to be a perfect circle and the equivalent diameter can be then calculated (Chen and Lin, 2005; Fu et al., 2011; Fu, 2012). The sample temperature during drying is measured separately from the weight measurement by exactly same drying conditions, droplet size and compositions. A calibrated thermocouple (type K, diameter of 13 or 24 µm) is used to record the sample temperature during drying. The thermocouple is connected to a Picometer (Pico Technology, UK)


(a)

37

Suspending glass filament Video camera Droplet Drying chamber

Airflow from heating tower (b)

Glass capillary tubes to support the thermocouple Thermocouple junction

Thermocouple wires

Airflow from heating tower (c) Mass-measuring glass filament

Airflow from heating tower Figure 2.1 Equipment setup of convective drying of milk droplets (a) measuring droplet shrinkage; (b) measuring droplet temperature; (c) measuring mass change. [Reprinted from Chemical Engineering Science, 66, N. Fu, M.W. Woo, S.X.Q. Lin et al., 1738–1747, Copyright (2012), with permission from Elsevier.] (Adapted from Fu et al. (2011) Chemical Engineering Science, 66, 1738–1747.)

and the data are obtained from the data logger (Picolog R5, 17, Pico Technology, UK). The repeatability of the weight loss experiment is ±0.01 mg and the accuracy is ±0.05 mg while the accuracy of the temperature measurement is within 0.1 °C. The repeatability of the weight loss experiment is ±0.01 mg and the accuracy is ±0.05 mg while the accuracy of the temperature measurement is within 0.1 °C (Chen and Lin, 2005; Fu et al., 2011; Fu, 2012).


(a)

Without weight

With weight

Displacement

(b)

250 Displacement (pixel)

38

200 y = 48, 13x R2 = 0.998

150 100

Experimental Linear correlation

50 0

0

1

2 3 4 5 Mass of weight standards (mg)

6

Figure 2.2 The deflection of glass filament and a typical standard curve (a) measuring displacement to measure weight loss; (b) correlation between the displacement and the weight. [Reprinted from Chemical Engineering Science, 66, N. Fu, M.W. Woo, S.X.Q. Lin et al., 1738–1747, Copyright (2012), with permission from Elsevier.]

From the experiments mentioned previously, the experimental data of sample mass, surface area, volume and temperature can be obtained. The rate of moisture content change (dX/dt) can be obtained from the weight loss curve experiment. The heat and mass transfer coefficient is determined based on established Sherwood and Nusselt number correlations or established experimentally for the specific drying conditions involved. For the experiments of single-droplet drying in a convective dryer, the following correlations can be used (Lin and Chen, 2002): Sh = 1.63 + 0.54Re0.5 Sc0.333 , N u = 2.04 + 0.62Re

0.5

Pr

0.333

,

(2.2.1) (2.2.2)

where Sh, Nu, Re, Sc and Pr are the Sherwood, Nusselt, Reynolds, Schmidt and Prandtl numbers, respectively. These differ somewhat from the classical Ranz and Marshall correlations due to the blowing effect (Lin and Chen, 2002). The saturated water vapour concentration (ρ v,sat ) is evaluated based on Equation (2.1.3) while the ambient water vapour concentration (ρ v,b ) is determined based on the humidity and temperature of the drying air employed in the laboratory. By using the experimental data, calculated heat and mass transfer coefficients, ρ v,sat and ρ v,b ,


39

1.0 Exp. data The curve fitted

ΔEv/ΔEv,b

0.8 0.6 0.4 0.2 0.0 0.0

1.0

2.0 3.0 X–Xb (kg/kg)

4.0

5.0

Figure 2.3 The relative activation energy of convective drying of 20%wt. skim milk powder at a drying air temperature of 67.5 °C, velocity of 0.45 m s−1 and humidity of 0.0001 kg H2 O kg dry air−1 . [Reprinted from AIChE Journal, 51, X.D Chen and S.X.Q. Lin, Air drying of milk droplet under constant and time-dependent conditions, 1790–1799, Copyright (2012), with permission from John Wiley & Sons, Inc.]

the activation energy (Ev ) can be calculated using Equation (2.1.5). In addition, the equilibrium activation energy (Ev,b ) can be evaluated using Equation (2.1.7). The ρ v,b can be related to the equilibrium moisture content (Xb ) through the desorption isotherm such as GAB isotherm. The coefficients of the isotherm for milk droplets are provided in Chen and Lin (2005). The activation energy (Ev ) can be scaled (from 0 to 1) by dividing it by the equilibrium activation energy (Ev,b ) to yield the relative activation energy (Ev /Ev,b ) as indicated in Equation (2.1.6). The relative activation energy can be related to the difference of moisture content during drying (X–Xb ). The relative activation energy (Ev /Ev,b ) of 20%wt. of skim milk powder is shown in Figure 2.3. It shows that, initially, the relative activation energy is zero and this increases as drying progresses. When the equilibrium moisture content is achieved, the relative activation energy is 1. This indicates that the difficulty in removing the moisture from the materials being dried increases as the drying continues. It is noted that Xb for the particular test run of concern may also be measured by prolonging the drying test under the same condition for an extended period to achieve equilibrium. The relationship between the relative activation energy (Ev /Ev,b ) and the difference in moisture content (X–Xb ) can be expressed by a simple mathematical function R (polynomial, exponential, logarithmic, etc.). Microsoft Excel (Microsoft Inc., 2012) can be used for fitting in order to obtain a simple algebraic function. For the convective drying of 20%wt. skim milk, the relative activation energy can be represented as (Chen and Lin, 2005):

E v = 0.998 exp −1.405 (X − X b )0.993 . E v,b

(2.2.3)

40


When Equation (2.2.3) is generally applicable, the relative activation energy generated from one accurate drying run can be applied to model the convective drying at other conditions, provided the same material and similar initial moisture content is used. The relative activation energy is the ‘fingerprint’ of the REA where the physics of the drying is captured by it. Since only one accurate drying run is required to generate the relative activation energy, the REA is effective in terms of the number of experiments to generate the REA parameters. This makes the REA, at least in this lumped-form, differ markedly from some other modelling approaches to drying. For instance, the diffusion-based approach is commonly applied in literature in which the diffusivity function needs to be estimated from several sets of experimental data. This means several experiments are required to obtain the diffusivity. Vaquiro et al. (2009) mentioned that 10 experiments were needed to generate the diffusivity function as function of temperature and moisture content. Some optimisation procedures need to be conducted subsequently to represent the dependence of diffusivity on moisture content and/or temperature (Azzouz et al., 2002; Mariani et al., 2008; Pakowski and Adamski, 2007; Ramos et al., 2010; Ruiz-Lopez et al., 2011; Silva et al., 2010; Vaquiro et al., 2009). Pakowski and Adamski (2007) and Ruiz-Lopez et al., 2011 show that various mathematical expressions of diffusivity need to be attempted to yield the most appropriate diffusivity function. More discussion on these will be given in Chapter 4.

2.3

Coupling the momentum, heat and mass balances As an example, the REA has been implemented to model a ‘plug-flow’ spray drying process for lactose droplets. Basically, it is drying of mono-dispersed lactose solution droplets with a vertical trajectory. This drying approach does not consider the dropletdroplet interactions and droplet-wall interactions for purposes of simplicity. Some innovative steps can be taken to implement this one-dimensional model to a real industrial situation in order to obtain practical and meaningful predictions (Patel et al., 2010). According to the methodology of Patel et al. (2010), one can devise the following to simulate spray-drying of milk droplets. This approach follows a semi-Lagrangian framework in which the model ‘follows’ or predicts the condition of the particle. In order to do this, the model predicts the ambient conditions corresponding to the position of the particle within the chamber, considering the momentum, heat and energy coupling between the two phases. The model assumes a pluglike flow of particles and uniform air flow across the diameter of the dryer. This is illustrated in Figure 2.4. The change in particle axial trajectory is modelled by taking the balance between the drag force due to the relative motion and the buoyancy force by the air, g 18 μb Re dv p = CD (vb − v p ) + (ρ p − ρb ), dt 24 ρ p d 2p ρp

(2.3.1)

where vp is the droplet/particle velocity (m s−1 ), vb is the gas velocity (m s−1 ), ρ p is the droplet density (kg m−3 ), ρ b is the gas density (kg m−3 ), g is the gravitational constant

41


Atomiser

Uniform particle distribution Uniform air velocity, temperature and humidity

Figure 2.4 Schematic diagram showing the plug-flow spray dryer.

(m2 s−1 ) and dp is the droplet diameter (m). The drag coefficient C D can be expressed by the following for 0.5 < Re < 1000: CD =

24 (1 + 0.15 Re0.687 ). Re

(2.3.2)

The Reynolds number is calculated based on relative velocity between the air and the particle taking the diameter of the particle as the characteristic length, Re =

ρb d p |v p − vb | , μb

(2.3.3)

where μb is the viscosity of the gas (Pa.s). It can be seen from Equation (2.3.1) that the momentum of the particle is coupled with the drying gas (air). Only one-way momentum coupling is considered and the effect of the particle motion on the air is assumed negligible. Corresponding to the position of the particle, the change of the air temperature against the length of the chamber is given next: dTb dL

=

m s ddtX

θ vp

[ HV − C p,v (Tb − T p )] −

θ vp

h A p (Tb − T p ) − U (π Ddryer ) (Tb − Tamb ) − V˙ ρb HV V˙ ρb C p,b

dY dL

,

(2.3.4)

where L is the length of the one-dimensional dryer (m) (hence, dL is the differential control volume), X is the water content on dry basis (kg kg−1 ), ms is the dry mass of a single droplet (kg), Tb is the gas (air) temperature (K), Tp is the particle temperature (K), Cp,b is the specific heat of the drying medium (J kg−1 K−1 ), Nv is the evaporation

42


flux (kg H2 O m−2 s−1 ), Cp,v is the specific heat of water vapour (J kg−1 K−1 ), Ap is the area of droplet (m2 ),V˙ is the volumetric flow rate of gas (air) (m3 s−1 ), Ddryer is the diameter of the spray dryer (m2 ), HV is the vaporisation heat of water (J kg−1 ) and U is the overall heat transfer coefficient (W m−2 K−1 ). This equation was developed considering the enthalpy balance for a control volume of air enveloping the particle. Equation (2.3.4) also considers heat loss from the dryer to the ambient. The change of bulk air humidity against the length of the chamber then takes the following form: −m s ddtX vθp dY , = m˙ b,dr y dL

(2.3.5)

where m˙ b,dr y is the mass flow rate of the gas (air) (kg s−1 ) and is the number of droplets going into the control volume per unit time (s−1 ). Currently, for this dyer, it is basically how many droplets are generated at the atomiser per unit time. This equation has been developed considering the humidity balance for a control volume of air enveloping the particles. This suggests that any change in humidity is only affected by the drying of the particles, releasing vapour into the bulk air stream. In Equations (2.3.4) and (2.3.5), the rate of drying of a single droplet needs to be defined. In the general one-dimensional framework, the single droplet drying term can be determined using the REA model already mentioned previously in Equation (2.1.4). The parameters need to be relevant to the material being dried; for skim milk with initial solids content of 20 %wt., the relative activation energy function is thus Equation (2.2.3). Heat transfer across the surface of the droplet takes the following form accounting for convective heating and evaporative cooling: dT p dX = h A p (Tb − T p ) + m s HV , dt dt

(2.3.6)

noting that dX/dt is negative when drying takes place. In Equation (2.1.4) and (2.3.6), the heat and mass transfer coefficients (hm , h, respectively) can be calculated using the Ranz Marshall form of the Nusselt and Sherwood correlation: α h m = 2 + 0.6 Re1/2 Sc1/3 , (2.3.7) dp kb , (2.3.8) h = 2 + 0.6 Re1/2 Pr 1/3 dp where α is the film thermal diffusivity of the vapour-air system around the droplet/particle (m2 s−1 ) and kb is the thermal conductivity of the vapour-air system around the droplet/particle (W m−1 K−1 ). These correlations can be modified by taking into account the vapour blowing effects in various well-known formats. All the fluid properties used in the calculation of Equations (2.3.9) and (2.3.10) have to be evaluated at film temperature, which can be approximated as the average of the bulk and particle temperatures. Therefore, the Reynolds number calculated for Equations (2.3.9) and (2.3.10) will be different from that calculated for Equation (2.3.1). The expression for the Schmidt and


43

Prandtl number is given next: μb , ρb α C pb μb Pr = . kb Sc =

(2.3.9) (2.3.10)

The profiles of moisture content and temperature of the droplets as well as those of humidity and temperature of the drying medium along the dryer can be predicted by simultaneously solving the momentum, mass and balances of the droplet. The coupling of momentum, mass and heat balances of the droplets and the drying medium can be used as process simulation tool for the study of plant-wide dryer performance. Similarly, one can apply the approach to modelling large-scale time-dependent threedimensional spray dryers. In this case, the momentum, heat and mass balances (partial differential equations that govern these processes) have been already solved (by Jin and Chen, 2009a; Jin and Chen, 2010; 2011). The L-REA model for single droplet drying has been incorporated in these simulations. Here, the advantage of using the REA is that one does not need to solve the spatial distribution of the moisture within the particles. For example, one does not need to use a diffusion equation for each particle. Trying to track the moisture changes using a spatially distributed model for the particle inside, for literally thousands if not hundreds of thousands of particles in a spray dryer to make simulation realistic, would take far more computational time.

2.4

Mass or heat transfer limiting The question of mass or heat transfer limitation is not a straightforward concept as far as drying modelling is concerned. When mass transfer is the limiting process, which may usually be the case when the temperature difference between the material to be dried and that of the drying gas is not very large, one may not need to consider the spatial distribution of temperature and can therefore simplify the modelling process. If heat transfer is limiting, one has to worry about the temperature gradient within the medium being dried (Chen, 2007). In air drying, one expects that the spatial distribution of water content inside a material is significant, i.e. the boundary could be relatively dry but the core could still be very wet. The question is whether the temperature within the material being dried can be approximated to be uniform. At first sight, all these should have much to do with two classical numbers: the Biot number for examining the temperature uniformity and the Lewis number for examining heat or mass transfer limiting.

2.4.1

Biot number analysis The Biot number criterion is well known and used to investigate the temperature uniformity of a material being heated or cooled. Conventionally, the Biot number is introduced

44


Heat flux Ts,1 Ts,2

L T∞ x x=0

x=L

Figure 2.5 The schematic diagram showing the parameters for the definition of the classical Biot

number. [Reprinted from Drying Technology, 23, X.D. Chen, Air drying of food and biological materials – Modified Biot and Lewis number analysis, 2239–2248, Copyright (2012), with permission from Taylor & Francis.]

through steady-state heat conduction in a slab with one side cooled by convection (see Figure 2.5). The conductive heat flux through the wall is set to be equal to the heat flux due to convection (Incropera and DeWitt, 1990; 2002): qx = k

Ts,1 − Ts,2 = h (Ts,2 − Tb ) , L

(2.4.1)

where qx is the heat flux (W m−2 ), k is the thermal conductivity of the solid material (slab) (W m−1 K−1 ), L is the thickness of the material (m), h is the heat transfer coefficient (due to convection) (W m−2 K−1 ), Ts represents the surface temperatures, the subscripts ‘1’ and ‘2’ represent the high and the low temperature of the slab, respectively (K), Tb is the temperature of the gas (air, for instance) (K) and k is the thermal conductivity (W m−1 K−1 ). The ratio of the temperature differences can then be expressed as: hL Ts,1 − Ts,2 = . Ts,2 − Tb k

(2.4.2)

The classical Biot number is thus defined as the temperature ratio by: Bi =

hL . k

(2.4.3)

When this ratio is less than 0.1, i.e. the internal temperature difference is smaller than 10% of the external temperature difference, the internal temperature distribution may be neglected for simplicity in modelling. For a spherical object, the characteristic length L may be set to be the radius of the sphere. One can rewrite Equation (2.4.2), in the form of the temperature ratio and the resistance ratio: Ts,1 − Ts,2 L/k Rcond hL = = = , Ts,2 − Tb k 1/ h Rconv

(2.4.4)


45

Surface evaporation T∞

Ts,2 Ts,1

L

x=0

x=L

x

Figure 2.6 The schematic diagram showing the parameters for the definition of the modified Biot number) (Chen–Biot number). [Reprinted from Drying Technology, 23, X.D. Chen, Air drying of food and biological materials – Modified Biot and Lewis number analysis, 2239–2248, Copyright (2012), with permission from Taylor & Francis.]

where R represents the thermal resistances, and ‘cond’ and ‘conv’ represent the conduction and convection, respectively. As such, the Biot number can also be considered to be the ratio of the internal resistance to external resistance. A small temperature ratio and a small resistance ratio essentially suggest the same thing. More fundamentally, the resistance ratio is a better argument. The thermal conductivity of the particle would be affected by water content and porosity (when filled with air). Chen introduced a new formula, which accounts for evaporation from the heat exchange surface (Chen and Peng, 2005). Similar to the conventional analysis (Equation 2.4.1), considering the addition of evaporative loss, the following heat balance can be obtained (here the temperature of the environment is greater than the material being dried; see Figure 2.6): Ts,2 − Ts,1 . (2.4.5) h (Tb − Ts,2 ) − HL S˙v = h ∗ (Tb − Ts,2 ) = k L Here h* is an equivalent convection heat transfer coefficient and S˙ v is the surface based evaporation or drying rate (kg m−2 s−1 ) (taken as a positive value for evaporation): h∗ = h −

HL S˙v . (Tb − Ts,2 )

(2.4.6)

This leads to a more appropriate Biot number for the evaporative system: Bi ∗ = Bi −

HL S˙v L . (Tb − Ts,2 ) k

(2.4.7)

When evaporation occurs from the heat exchanger surface, the modified Biot number (Bi* ) represents the temperature uniformity more precisely than Bi. Just because a conventional and large Bi indicates a non-uniform temperature distribution does not mean the temperature is not reasonably uniform when evaporation occurs.

46


Bi* is a smaller value than Bi indicating a more uniform temperature distribution. The significance of this equation can be demonstrated in the following for evaporation from a water droplet (which is also relevant to drying of a coal particle with high moisture content at the beginning of the drying process). The following calculations are based on the laboratory data obtained by Lin and Chen (2002) and Lin (2004) on evaporating of a single water droplet (spherical condition). The laboratory conditions and the techniques employed are given: Droplet diameter: 2L = 1.43 mm Heat transfer coefficient measured: h = 99.1 W m−2 K−1 Thermal conductivity of water: k = 0.63 W m−1 K−1 Evaporation rate: S˙v = 1.73 × 10−3 kg s−1 m−2 Interfacial temperature: Ts = 23.4 °C Drying air temperature: Ts = 67.5 °C Latent heat of evaporation: HV = 2445 × 103 J kg−1 The conventional analysis, based on Equation (2.4.3), yields Bi = 0.11 for the previous example, which is slightly greater than the critical value of 0.1 mentioned earlier. Based on Equation (2.4.7), however, one can find that the Bi required for uniform temperature assumption is 0.21 in order to maintain the Bi* being 0.1, i.e.: ∗ Bi cri = Bi cri +

2445 × 103 × 1.73 × 10−3 1.43 × 10−3 ≈ 0.21, (67.5 − 23.4) 2 × 0.63

(2.4.8)

due to the evaporation (cooling) effect, which consumes much of the temperature driving force from the outside of the material being dried. Equation (2.4.7) can be extended to account for the cases when internal mass transfer resistance plays a role. Here, one can express the surface based evaporation rate using the overall mass transfer coefficient concept instead: Bi ∗ = Bi −

HV Um (ρv,c − ρv,b ) , k (Tb − Ts,2 ) L

(2.4.9)

where L may be used as the characteristic length generically of slab, cylinder or a sphere etc. (m), ρv,b is the vapour concentration in the drying medium and ρv,c is the vapour concentration (based in bulk) at the location marked by the characteristic dimension (δ c ) inside the material, which may be taken as the saturated vapour concentration at the material (mean) temperature (a high bound estimate). The overall mass transfer coefficient (Um ) may be expressed approximately as the following by Chen (2007): Um ≈

1 hm

1 . + Dδeffc ,v

(2.4.10)

The characteristic dimension (δc ) signifies the effect of the vapour concentration profile and the solid matrix resistance to vapour transfer. For a symmetric material being dried, this dimension should be smaller than the corresponding characteristic dimension for heat conduction (for example, < the 0.25 of the half-thickness for a slab being heated:


47

see Van der Sman, 2003). The effective vapour diffusivity may be estimated using the porosity (ε) and the tortuosity (τ ) correction (McCabe et al., 2001): ε (2.4.11) Deff ,v ≈ Dv,air , τ where Dv,air is the vapour diffusivity in air (m2 s−1 ). The tortuosity (τ ) is a parameter not usually known prior, so an estimate between 2 and 20 may be used in some cases. It is estimated as an empirical function of ε (McCabe et al., 2001). The mass transfer coefficient (hm ) could also be estimated using one of the heat and mass transfer analogies, such as the one for a flat plate (Incropera and DeWitt, 1990; 2002): Dv,air 0.3 h kair ≈ , (2.4.12) hm Dv,air αair which is based on the heat transfer coefficient h, depending on the geometry of the object of concern. One can see that the new number, the Chen–Biot (Ch–Bi) number, can be defined as (Chen, 2007) follows for the surface evaporation case shown in Figure 2.6: Ch−Bi = Bi −

HV S˙v L . (Tb − Ts,2 ) k

(2.4.13)

It is difficult to evaluate the Chen–Biot number for a case where drying occurs inside the material. As mentioned earlier, the following formula has been proposed (Chen, 2005a; Chen, 2007) for cases where evaporation also takes place within:

Ch−Bi = Bi − k L

HV

T∞ − Ts,2 ρv,c − ρv,b

δc 1 + hm Deff ,v

.

(2.4.14)

Alternatively, the term for mass transfer resistance due to vapour diffusion in Equation (2.4.14) is ‘replaced’ by that using the effective liquid water diffusivity, Deff,l (effective liquid water diffusivity), as it is most commonly measured or cited in drying literature. Equation (2.4.14) reduces to the surface evaporation case when the characteristic thickness δ c approaches zero.

2.4.2

Lewis number analysis The Lewis number (Le) is defined as the ratio of the thermal diffusivity to the mass diffusivity (for water vapour transfer). For drying moist, porous materials, this would mean that if Le < 1, the heat penetration (through conduction) into the particle is slower than the ‘penetration’ of the water vapour front. If Le 1, it means that the heat transfer is limiting the drying process. When Le 1, the heat input and moisture removal are highly coupled. When Le 1, the process is mass-transfer limited. This becomes particularly informative and important for evaluating drying after the initial moisture-rich condition. This process can be visualised as follows by considering

48


the heating of a semi-infinite porous media in a constant temperature environment as summarised by Chen (2007). It is known that the thermal penetration depth (δ T ) can be approximated as: δT ≈ 12αtT , (2.4.15) where α is the thermal diffusivity (=k/ρCp ) (m2 s−1 ) and tT is thermal penetration time (s). Correspondingly, the vapour ‘penetration’ (mass penetration denoted by the subscript ‘M’) may be expressed as: δM ≈

12Deff ,v t M ,

(2.4.16)

where tM is the mass ‘penetration’ time (s). For reaching the same distance into the material, i.e. δT = δ M , the ratio of time required for mass to ‘penetrate’ to that for heat to penetrate is essentially the Lewis number: Le =

α tM = . tT Deff ,v

(2.4.17)

The physics is apparent that if Le 1, mass transfer occurs much faster, thus heat transfer is limiting. A conventional Lewis number analysis using the effective vapour diffusivity calculated with Equation (2.4.11) would, for drying a skim milk droplet, for instance, yield a value smaller than 0.1 (Farid, 2003), leading to the conclusion that the drying is heat-transfer limiting. In order to derive the new Lewis number that considers the phase change process, Chen (2005a) has employed a fuller account of the energy and mass balances (with source terms) for drying as follows: ∂ ∂Cl ∂Cl = Deff ,l − E˙ v ; (2.4.18) ∂t ∂x ∂x ∂ ∂C v ∂Cv = Deff ,v + E˙ v ; (2.4.19) ∂t ∂x ∂x 1 ∂ HV ˙ ∂T ∂T (2.4.20) Ev ; = keff − ∂t ρC p ∂ x ∂x ρC p where Cl and Cv are the concentrations of liquid water and water vapour, respectively (kg m−3 ), T is the sample temperature (K), ρ is the sample density (kg m−3 ), Cp is the sample specific heat (J kg−1 K−1 ), keff is the thermal conductivity (W m−1 K−1 ), D. eff,l and Deff,v are the effective liquid and vapour diffusivity, respectively (m2 s−1 ), and E v is the source term (kg m−3 s−1 ). This is for slab geometry. Similar equations can be written for spherical and cylindrical or other three-dimensional objects. The source term may be approximated as: dX . E˙ v ≈ −ρs dt

(2.4.21)


49

.

This E v is equivalent to the source term I˙ in Chapter 3 where S-REA is introduced. This represents the local rate of evaporation. Assuming that Cv is the vapour concentration that is in equilibrium with the liquid water content inside the solid structure, which is a conservative assumption in the sense that the process of mass transfer is perhaps the slowest, the following relationship exists: RH = φ =

Cv = f (X, T ) or X = F(φ, T ), Cv,sat (T )

(2.4.22)

where RH is the relative humidity and Cv,sat is the saturated water vapour concentration (kg m−3 ). Equation (2.4.22) is the equilibrium isotherm function. Therefore, one may write the following: ⎡ ⎤ Cv ∂ ∂ X ∂φ ∂ X ∂T ∂ X ∂T ⎦ ∂X Cv,sat (T ) E˙ v ≈ −ρs + = ρs ⎣ + . ∂φ ∂t ∂ T ∂t ∂φ ∂t ∂ T ∂t (2.4.23) If the temperature may be taken as an average value, especially when the equilibrium isotherm functions are insensitive to temperature in the range considered, the following simplified Equation (2.4.23) can be obtained:

∂ X ∂φ 1 ∂C ∂ X v ≈ −ρs . (2.4.24) E˙ v ≈ −ρs ∂φ ∂t ∂φ Cv,sat( T ∂t Thus, the vapour conservation Equation (2.4.19) can be rewritten (corresponding to Equation 2.4.14) into: 1 ∂ D ∂C v ∂Cv 2 eff ,v ≈ r2 , (2.4.25) ∂t r ∂r ∂r ∂X 1 1 + ρs ∂φ C T v,sat ( ) by taking a mean D eff ,v for simplification. new The equivalent mean effective diffusivity (D eff ,v ) is then: D eff ,v

new

D eff ,v ≈

1 + ρs

∂X 1 ∂φ C v,sat (T )

.

(2.4.26)

The effective Lewis number (i.e. the Chen–Lewis number) can then be written as: α eff k eff D . eff ,v Ch Le = new = (2.4.27) ρC p D eff ∂X 1 1 + ρs ∂φ C T v,sat ( ) Equation (2.4.27) may be used to estimate a more likely high bound of the Lewis number (here the Chen–Lewis number). For the same skim milk drying case as mentioned earlier, Equation (2.4.27) can yield an estimate of the Chen–Lewis number (Ch–Le) of the order of 100, indicating the process is mass transfer limiting.

50


2.4.3

Combination of Biot and Lewis numbers Though no strict scientific proof regarding why one should combine the Bi and Le numbers together to justify isothermal, uniform temperature or otherwise. Nevertheless, these are the two most important dimensionless parameters in literature that seem most relevant to drying. Sun and Meunier (1987) conducted a comprehensive numerical analysis on non-isothermal sorption in adsorbents, which showed that the following rule exists: the isothermal model would be valid if LeBi > 100 and the uniform temperature profile model would be a good model if Le > 10. Note that here the Le and Bi are all based on the conventional definitions. It is expected that this rule is conservative when drying is concerned. In the desorption process the temperature profile inside the porous material would tend to be more gradual, thus the criteria can be relaxed. The driving force for heat transfer, i.e. the difference between the drying air (or the drying medium in general) and the porous material, i.e. (Tb –Ts ), would also affect the temperature uniformity when drying proceeds, the ‘waterfall’ effect.

2.5

Convective drying of particulates or thin layer products modelled using the L-REA The L-REA (lumped reaction engineering approach) has been used to describe the convective drying of droplets of whey protein concentrate and thin layer of a mixture of polymer solution (Lin and Chen, 2007; Allanic et al., 2009). For the convective drying of droplets of whey protein concentrate, the experimental setup is similar to that explained in Section 2.3. The droplets are suspended in a glass-filament convective dryer, and the mass and temperature are recorded during drying. The deflection of the glass filament is captured and converted to droplet weight. The measurement of weight also takes into account the drag force. A video camera system is used to monitor the droplet diameter change during drying. A calibrated thermocouple is used to record the sample temperature during drying. The thermocouple is connected to a picometer and the data is obtained from the data logger. The repeatability of the weight loss and temperature measurement is ±0.01 mg and 0.1 °C, respectively. Drying air with the velocity of 0.45 m s−1 and a temperature of 70°–110 °C is used. Initial droplet diameter and solids concentration are 1.45 mm and 30% wt., respectively (Lin and Chen, 2007). For convective drying a mixture of polymer solutions, the experimental data used in the current work are derived from the study reported by Allanic et al. (2009) and the experimental conditions are shown in Table 2.1. In order to better understand the modelling presented here, the details of experiment are briefly described here (Allanic et al., 2006; 2009). Materials used in this experiment were a mixture of equal proportions of partially hydrolysed polyvinyl alcohol (80%wt.) and glycerol with 88%wt. of water, then 8 ml of the mixture was poured into a 90-mm-diameter Petri dish so the initial thickness of the sample was 1.3 mm. During


51

Table 2.1 Experimental conditions of convective drying of a mixture of polymer solutions (Allanic et al., 2009).

Number

Air velocity (m s−1 )

Air temperature (°C)

Air relative humidity (%)

1 2 3

2.8 1 1

55 35 55

12 30 12

drying, shrinkage occurred and the relationship between thickness (m) and moisture content on a dry basis could be correlated in the linear form: e = ed (1 + λX ),

(2.5.1)

where e is the thickness of product (m), ed is the thickness of the dried product (m), X is the moisture content on a dry basis (kg kg−1 ), and λ is the linear shrinkage coefficient (=1.3). The weight measurement was accurate to about 0.2 g. During drying, regulated drying air temperature with particular velocity and temperature was fed into the rectangular casing so that it flowed gently above the sample. The drying air temperatures and velocities used for each experiment are listed in Table 2.1. The stable humidity of drying air was maintained and measured using a capacitive transmitter sensor. The temperature of the sample surface was measured using an optical pyrometer and the temperature of the upper and lower side of the Petri dish was measured with thermocouples with an uncertainty of about 2.5 °C. The results of this previous study showed that temperature gradient inside the Petri dish and product can be ignored (Allanic et al., 2006).

2.5.1

Mathematical modelling of convective drying of droplets of whey protein concentrate (WPC) using the L-REA The L-REA explained in Section 2.1 is implemented here to model the moisture content and temperature profiles during the convective drying of WPC. The relative activation energy is generated from one accurate drying run. The activation energy and equilibrium activation energy are evaluated using Equations (2.1.5) and (2.1.7), respectively. The mass balance implementing the L-REA shown in Equation (2.1.4) is used with the convective mass transfer coefficient (hm ), determined based on the work of Lin and Chen (2002): Sh = 1.54 + 0.54 Re0.5 Sc0.333 ,

(2.5.2)

where Sh is the Sherwood number, Re is the Reynolds number and Sc is the Schmidt number. The heat balance of the convective drying of WPC can be represented as: d(mC p T ) dX ≈ h A(Tb − T ) + m s Hv , dt dt

(2.5.3)


1.0 67.5°C 87.1°C 106.6°C Curve fitted

0.8

ΔE/ΔEb

52

0.6 0.4 0.2 0.0 0.0

0.5

1.0 1.5 X–Xb (kg/kg)

2.0

2.5

Figure 2.7 The relative activation energy of convective drying of WPC at different drying air temperatures. [Reprinted from Chemical Engineering and Processing, 46, S.X.Q. Lin and X.D. Chen, The reaction engineering approach to modelling the cream and whey protein concentrate droplet drying, 437–443, Copyright (2012), with permission from Elsevier.]

where m is the mass of droplets during drying (kg), Cp is the specific heat of the samples (J kg−1 K−1 ), T is the sample temperature (K), Tb is the drying medium temperature (K), Hv is the vaporisation heat of water (J kg−1 ) and h is the heat transfer coefficient (W m−2 K−1 ) which can be evaluated by (Lin and Chen, 2002): N u = 2.04 + 0.62 Re0.5 Pr0.333 ,

(2.5.4)

where Nu is the Nusselt number and Pr is the Prandtl number. The relative activation energy of the WPC is generated from one accurate drying run. It is shown in Figure 2.7 and can be expressed as (Lin and Chen, 2007):

E v = 1.335 − 0.3669 exp exp(X − X b )0.3011 , E v,b

(2.5.5)

while the droplet diameter changes during drying can be expressed as (Lin and Chen, 2007): d X − Xb = 0.873 + 0.127 . d0 X0 − Xb

(2.5.6)

The good agreement between Equation (2.5.6) and experimental diameter changes during drying is shown in Figure 2.8. The profiles of moisture content and temperature during drying are generated by solving the mass balance implementing the L-REA and the heat balance shown in Equations (2.1.5) and (2.5.3), respectively, in conjunction with the equilibrium activation energy, relative activation energy and droplet diameter changes during drying shown in Equations (2.1.5), (2.1.7) and (2.5.6), respectively.


53

1.0

d/d0

1.0

0.9

0.9 0.0

67.5°C 87.1°C 106.6°C Curve fitted 0.5

1.0 1.5 X–Xb (kg kg–1)

2.0

2.5

Figure 2.8 The droplet diameter changes during convective drying of WPC. [Reprinted from Chemical Engineering and Processing, 46, S.X.Q. Lin and X.D. Chen, The reaction engineering approach to modelling the cream and whey protein concentrate droplet drying, 437–443, Copyright (2012), with permission from Elsevier.]

2.5.2

Mathematical modelling of convective drying of a mixture of polymer solutions using the L-REA The L-REA shown in Equation (2.1.4) used here is similar to formulation of the L-REA used in the convective drying of WPC. The activation energy and equilibrium activation energy is also calculated using Equations (2.1.5) and (2.1.7), respectively. During convective drying, it can be seen that the sample is heated from the upper side due to forced convective heat transfer from the drying air, while the lower side is heated through the Petri dish which is heated by drying air from below (refer to Figure 2.9). The heat balance can be written as: ms d X d(ρ C p eT ) = h top (Tb − T ) + Ubottom (Tb − T ) + Hv (T ), dt A dt

(2.5.7)

where ρ is sample density (kg m−3 ), Cp is sample heat capacity (J kg−1 K−1 ), e is sample thickness (m), T is sample temperature (K), Tb is drying air temperature (K), ms is mass of dried product (kg), A is surface area of product (m2 ), X is moisture content of product (kg kg−1 ), Hv is enthalpy of vaporisation (J kg−1 ), htop is heat transfer coefficients at the upper surface of the dish and Ubottom represents the overall heat transfer coefficient from the lower side including convection (natural) along and conduction through the Petri dish. Rearranging Equation (2.5.7) results in: ms d X d(ρ C p eT ) = Utotal (Tb − T ) + Hv (T ), dt A dt

(2.5.8)

54


0

Product

Petri dish

e(t)

z Thermocoupl

Evaporation

Convection

Diffusion

Conduction Figure 2.9 Heat transfer mechanisms of the convective drying of a mixture of polymer solutions. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2012), with permission from Elsevier.]

where Utotal represents the sum of the convective heat transfer coefficient (W m−2 K−1 ) at the top and that at bottom. This value can be deduced from the constant rate period of drying. The heat balance for this period can be expressed as: Utotal (Tb − T ) =

ms d X Hv (T ). A dt

(2.5.9)

For this experiment, the activation energy is determined based on previously published experimental data (Allanic et al., 2009) using Equation (2.1.5). The vapour concentration in the environment is determined from the corresponding relative humidity and drying air temperature reported previously (shown in Table 2.1). The mass transfer coefficient was deduced from the established Sherwood number correlation. Based on drying kinetics data, the relative activation energy (Ev /Ev ,b ) for convective drying calculated through this exercise is expressed as:

E v = exp − 1.0794(X − X b )1.28 . E v,b

(2.5.10)

Only one set of drying data was necessary and this was taken from experiment at a drying air temperature of 35 °C, drying air velocity of 1 m s−1 and relative humidity of 30% (Allanic et al., 2009). This is of a similar format to that proposed previously (Chen and Xie, 1997; Chen and Lin, 2005). As Figure 2.10 shows, there is excellent agreement between correlated and experimental activation energy (R2 = 0.9892). At high water content, moisture removal is easy, as shown by the low activation energy, and this increases during drying as moisture content decreases indicating greater difficulty

55


1.0 Data Fitted curve 0.8

ΔEv/ΔEv,b

0.6

0.4

0.2

0

–0.2

0

1

2

3 4 5 X–Xb (kg water/kg dry solid)

6

7

8

Figure 2.10 Normalised activation energy and fitted curve of polyvinyl alcohol/glycerol/water under convective drying at an air temperature of 35 °C and relative humidity of 30%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2012), with permission from Elsevier.]

in removing moisture. Also, this correlation ensures that Ev /Ev ,b is 1 when the moisture content approaches equilibrium (i.e. X = Xb ). In order to evaluate the moisture content and the temperature profile as a function of drying time, the mass balance and heat balance expressed in Equations (2.1.4) and (2.5.8), respectively were solved simultaneously in conjunction with the equilibrium and relative activation energy shown in Equations (2.1.7) and (2.5.10), respectively.

2.5.3

Results of modelling convective drying of droplets of WPC using the L-REA The results of modelling of the convective drying of the droplets of WPC using the L-REA are shown in Figure 2.11. Generally, the predictions using the L-REA match well with the experimental data. For the convective drying at a drying air temperature of 67.5 °C, the results of modelling of moisture content and temperature profiles match the experimental data well as shown in Figure 2.11(a). In addition, the L-REA describes well the moisture content and temperature profiles of the convective drying of WPC at drying air temperatures of 87.1° and 106.6 °C, as depicted in Figures 2.11(b) and (c), respectively. For all experiments, the average absolute differences between the


1.6E-06

1.2E-06

60

1.0E-06 8.0E-07

40

6.0E-07 4.0E-07

(b)

0

50

100

150 200 Time (s)

250

300

1.6E-06

20 350

100

Droplet weight (kg)

1.4E-06 Model pred. Exp. data

1.2E-06 1.0E-06

80

60

8.0E-07 40 6.0E-07 4.0E-07

(c)

0

50

100

150 200 Time (s)

250

300

1.6E-06

20 350 120

1.4E-06

100 Model pred. Exp. data

1.2E-06

80

1.0E-06 60 8.0E-07 40

6.0E-07 4.0E-07

Droplet temperature (°C)

Droplet weight (kg)

1.4E-06


80 Model pred. Exp. data

0

50

100

150 Time (s)

200

250


(a)

Droplet weight (kg)

56

20 300

Figure 2.11 The comparison between experimental and model prediction using the L-REA of

convective drying of WPC at drying air temperatures of (a) 67.5 °C (b) 87.1 °C (c) 106.6 °C. [Reprinted from Chemical Engineering and Processing, 46, S.X.Q. Lin and X.D. Chen, The reaction engineering approach to modelling the cream and whey protein concentrate droplet drying, 437–443, Copyright (2012), with permission from Elsevier].


57

8 Model Data

7

X (kg water/kg dry solid)

6 5 4 3 2 1 0

0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000 t(s)

Figure 2.12 Moisture content profile of convective drying at air temperature of 55 °C, air velocity

of 2.8 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]

experiments and predictions are about 2.1% of initial droplet weight for the droplet weight prediction and about 1.9 °C for the temperature prediction (Lin and Chen, 2007). It has been shown that the L-REA can model the convective drying of WPC accurately. This could be due to the accuracy of the relative activation energy in capturing the physics of the convective drying of WPC. The combination between the equilibrium relative energy and relative activation energy shown by Equations (2.1.7) and (2.5.10), respectively seems to be sufficient to describe the change of internal behaviour of the WPC droplets during drying. Therefore, it can be said that the L-REA can describe the drying kinetics of the particulates well.

2.5.4

Results of modelling convective drying of a thin layer of a mixture of polymer solutions using the L-REA Figures 2.12 to 2.17 present results of the simulated drying profiles and temperature profiles using the L-REA. It can be seen that generally, all moisture content and temperature profiles agree well with the experimental data supported by R2 and RMSE of moisture content and temperature profile presented in Table 2.2. Figures 2.12 and 2.13 show the moisture content and temperature profile of convective drying conducted at 55 °C and relative humidity of 12% at air velocity of 2.8 m s−1 . Figure 2.12 indicates that


Table 2.2 R 2 and RMSE of modelling of a mixture of polymer solutions using the L-REA. Number

R2 X

R2 T

RMSE X

RMSE T

1 2 3

0.999 0.998 0.997

0.958 0.991 0.975

0.071 0.083 0.104

2.263 0.436 1.448

330 325 320 Temperature (K)

58

315 310 305 300 295

Model Data 0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000 t(s)

Figure 2.13 Product temperature profile of convective drying at an air temperature of 55 °C, air velocity of 2.8 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]

moisture content profile can be predicted very well by the L-REA. Similarly, the temperature profile shown by Figure 2.13 indicates very small differences between predicted and experimental data. This modelling is comparable with modelling of drying kinetics conducted by Allanic et al. (2009). Slight discrepancies with experimental temperature data were also shown although the model employed was based on a diffusion partial differential equation with fitted diffusivity (Allanic et al., 2009). Figures 2.14 and 2.15 provide results of modelling of drying conducted at 35 °C and relative humidity of 30% at an air velocity of 1 m s−1 using the REA. A very good prediction of both moisture content and temperature data was observed. Compared with the simulation using the model proposed previously, it is apparent that the L-REA gives

8 Model Data

7


6 5 4 3 2 1 0

0

2000

4000

6000 t(s)

8000

10000

12000

Figure 2.14 Moisture content profile of convective drying at an air temperature of 35 °C, air velocity of 1 m s−1 and air relative humidity of 30%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]

308 306

Temperature (K)

304 302 300 298 296 294

Model Data 0

2000

4000

6000 t(s)

8000

10000

12000

Figure 2.15 Product temperature profile of convective drying at air temperature of 35 °C, air velocity of 1 m s−1 and air relative humidity of 30%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]

8 Model Data

7


6 5 4 3 2 1 0

0

1000

2000

3000

4000 t(s)

5000

6000

7000

8000

Figure 2.16 Product temperature profile of convective drying at an air temperature of 55 °C, air velocity of 1 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]

325

Temperature (K)

320

315

310

305 Model Data 300

0

1000

2000

3000

4000 t(s)

5000

6000

7000

8000

Figure 2.17 Product temperature profile of convective drying at an air temperature of 55 °C, air velocity of 1 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]


61

Table 2.3 Experimental conditions of convective drying of mango tissues (Vaquiro et al., 2009).

Number



Air humidity (kg H2 O kg dry air−1 )

1 2 3

4 4 4

45 55 65

0.0134 0.0134 0.0134

better results because the diffusion model shows slight discrepancies in the moisture content profile during drying times around 4000–10 000 s (Allanic et al., 2009). The L-REA can be used to describe this well, as shown in Figure 2.14. Similarly, Figures 2.16 and 2.17 show a good agreement between estimated and experimental moisture and temperature data. Despite the simplicity of L-REA, it compares well with the model proposed before, which shows some discrepancies of moisture profile during drying times around 3000–6000 s (Allanic et al., 2009). Overall the L-REA can be used successfully to model the thin layer drying of a mixture of polyvinyl alcohol, glycerol and water. The L-REA is shown to be able to model not only the convective drying of particulate or thin layer of food materials which has been proven before (Chen and Lin, 2005; Lin and Chen, 2005; 2006; 2007), but also that of thin layers of non-food materials. The accuracy of the L-REA could be due to the accuracy of the relative activation energy in describing the change of internal behaviour during drying.

2.6

Convective drying of thick samples modelled using the L-REA For studying simulations of convective drying of thick samples using the L-REA, the experimental data are derived from the work of Vaquiro et al. (2009) on convective drying of mango tissues. Mango tissues used for drying experiments were formed into cubes of side lengths of 2.5 cm, with initial moisture content of 9.3 kg kg−1 and an initial temperature of 10.8 °C. Drying was conducted in a laboratory dryer described in detail by Sanjuan et al. (2004). The drying air temperature and air velocity were controlled at preset values by PID control algorithms while air humidity was maintained at a constant during drying. Details of the experimental conditions are listed in Table 2.3. The weight of the sample was measured periodically to record weight loss as well as centre temperatures every 2 min.

2.6.1

Formulation of the L-REA for convective drying of thick samples In order to model the convective drying of thick samples, the original formulation of the L-REA can still be implemented. However, the temperature of concern is the surface

62


temperature (Ts ) (Putranto et al., 2011a,b). Therefore, the drying rate of the material can be expressed as: ms

dX = −h m A ρv,s (Ts ) − ρv,b , dt

(2.6.1)

where ms is the dried mass of thin layer material (kg), t is time (s), X is moisture content on a dry basis (kg kg−1 ), ρ v,s is the vapour concentration at the material-air interface (kg m−3 ), ρ v,b is the vapour concentration in the drying medium (kg m−3 ), hm is the mass transfer coefficient (m s−1 ) and A is the surface area of the material (m2 ). The mass transfer coefficient (hm ) is determined based on the established Sherwood number correlations for the geometry and flow condition of concern or established experimentally for the specific drying conditions involved (Lin and Chen, 2002; Kar and Chen, 2009). The surface vapour concentration (ρ v,s ) can be scaled against saturated vapour concentration (ρ v,sat ) using the following equation (Chen and Xie, 1997; Chen, 2008): ρv,s = exp

−E v RTs

ρv,sat (Ts ),

(2.6.2)

where Ev represents the additional difficulty in removing moisture from the material beyond the free water effect. This Ev is moisture-content (X) dependent. Ts is the surface temperature of the material being dried, and ρ v,sat for water can be estimated at the surface material being dried by the following equation: ρv,sat = 4.844 × 10−9 (Ts − 273)4 − 1.4807 × 10−7 (Ts − 273)3 + 2.6572 ×10−5 (Ts − 273)2 − 4.8613 × 10−5 (Ts − 273) + 8.342 × 10−3 ,

(2.6.3)

based on the data summarised by Keey (1992). The mass balance (Equation 2.6.1) is then expressed as: dX −E v = −h m A exp ρv,sat (Ts ) − ρv,b . ms dt RTx

(2.6.4)

The activation energy (Ev ) is determined experimentally by placing the parameters required for Equation (2.6.4) in its rearranged form:

E v = −RTs ln

−m s ddtX

1 hm A

+ ρv,b

ρv,sat (Ts )

.

(2.6.5)

The equilibrium activation energy (Ev,b ) is still evaluated by Equation (2.1.7). It can be shown that the general formulation of the L-REA shown in Equation (2.1.4) can still be implemented but the temperature of concern is the surface temperature (Ts ).


2.6.2

63

Prediction of surface sample temperature For large sample slabs, prediction of sample temperature may be necessary since the temperature may be not uniform inside the sample. The sample temperature may be approximated using a simple parabolic equation (Chen, 2008): T = a + bx 2 .

(2.6.6)

If To is the centre sample temperature (K) and L is the half-thickness of the sample as a characteristic slab length, Equation (2.6.6) is rewritten as: Ts − To (2.6.7) T = To + x 2, L2 Tavg is determined by: L

T (x)d x

0

. (2.6.8) L By combining Equations (2.6.7) and (2.6.8), Tavg is expressed as: 1 2 Tavg = Ts + To . (2.6.9) 3 3 For a sample heated in a convective environment, the boundary condition at sample surface (x = L) can be written as: dT + |Nv |Hv . (2.6.10) h (Tb − Ts ) = k d x x=L Tavg =

Also note that Equation (2.6.7) satisfies the boundary condition at centre (x = 0), which can be expressed as: dT = 0. (2.6.11) d x x=0 By combining Equation (2.6.7) to (2.6.10), Ts and To are expressed as: L hL Tavg + Tb − |Nv |Hv 3k 3k , (2.6.12) Ts = hL 1+ 3k ⎞ ⎛ 1 hL L 1 ⎟ ⎜3 Tb − |Nv |Hv . (2.6.13) To = Tavg ⎝ − − 2h L ⎠ 2h L 2 3k 3k 2+ 2+ 3k 3k Equation (2.6.12) and (2.6.13) clearly show that Ts and To are represented as functions of Tavg and Tb . The temperature profile prediction described previously seems to be valid for drying conditions suitable for the boundary conditions mentioned (i.e. there is symmetry at centre and at the surface; heat gained by convection from drying air is balanced by conduction heat inside the sample and heat for water evaporation). The prediction is in agreement with Pang (1994), who conducted convective drying of softwood and

64


heartwood with a half-thickness of 2.5 cm. It was observed that the boundary conditions indicated in Equations (2.6.10) and (2.6.11) fulfil the drying conditions of Pang (1994). The temperatures in several positions (x = 0, 7, 13, 19 and 25 cm from centre) were measured during the drying time and a plot of the temperature profiles against positions during drying time revealed parabolic profiles. For drying of mango tissue, as mentioned before, the sample was heated uniformly from all directions (Sanjuan et al., 2004). It is reasonable to assume that the temperature profiles would be similar in the x, y and z directions. Because of this, the approximation of the temperature profiles can be simplified into one dimension. It is also observed that for drying of mango tissues, there is symmetry at the centre and at the surface; heat received by convection from drying air is balanced by conduction heat inside the sample and heat for water evaporation is represented by the boundary conditions shown in Equations (2.6.10) and (2.6.11) (Sanjuan et al., 2004; Incropera and DeWitt, 2002). Therefore, similarly to Equations (2.6.7), (2.6.12) and (2.6.13), showing the temperature distribution inside mango and apple tissues, surface and centre temperature can be represented as: Ts − To (2.6.14) T = To + r 2, R2 R hR Tavg + Tb − |N V |HV 3k 3k , Ts = (2.6.15) hR 1+ 3k ⎞ ⎛ hR R 1 1 ⎟ ⎜3 − |N |H − T . (2.6.16) To = Tavg ⎝ − ⎠ b V V 2h R 2 3k 3k 2 + 2h3kR 2+ 3k The equivalent radius for cubes is the side length (Incropera and DeWitt, 2002; Radziemska and Lewandowski, 2008). Because of the symmetry principle used for Equations (2.6.14) to (2.6.16), the equivalent radius (r) used for cubes in this study are half the side length.

2.6.3

Modelling convective drying thick samples of mango tissues using the L-REA For drying thick samples of mango tissues convectively, the relative activation energy (Ev /Ev,b ) is generated from continuous convective drying runs at 55 °C (Vaquiro et al., 2009). Based on drying kinetics data, the relative activation energy (Ev /Ev,b ) of convective drying of mango tissues is expressed as: E v = −9.92 × 10−4 (X − X b )3 + 9.74 × 10−3 (X − X b )2 E v,b − 0.101(X − X b ) + 1.053.

(2.6.17)

A good agreement between the fitted (Equation 2.6.17) and experimental activation energy is shown in Figure 2.18 (R 2 (0.997)). This format of correlation is similar to that proposed by Kar (2008) to describe the activation energy of drying porcine skin. The

65


1 Data Model

0.9 0.8

ΔEv/ΔEv,b

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

1

2

3

4

5 X–Xb

6

7

8

9

10

Figure 2.18 The relative activation energy (Ev /Ev,b ) of convective drying of mango tissues at an air velocity of 4 m s−1 , drying air temperature of 55 °C and air humidity of 0.0134 kg H2 O kg dry air−1 . [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.]

format of the equation could be varied but for this study the Equation (2.6.17) seems to represent the activation energy well. It may be observed that the decrease of moisture content results in the increase of activation energy, which indicates greater difficulty in removing water. This equation also yields Ev /Ev,b approaching 1 as the material is dried. The heat balance for convective drying of mango tissues can be written as: dX d(mC p Tavg ) ≈ h A (Tb − Ts ) + m s HV , dt dt

(2.6.18)

where m is the sample mass (kg), Cp is the heat capacity of the sample (J kg−1 K−1 ), h is the heat transfer coefficient (W m−2 K−1 ) and HV is the latent heat of vaporisation of water (J kg−1 ). The drying rate dX/dt is negative when drying occurs. In order to yield both profiles of moisture content and temperature of mango tissues during drying, the mass implementing the L-REA and heat balance shown in Equations (2.6.4) and (2.6.18) are solved simultaneously in conjunction with the equilibrium and relative activation energy shown in Equations (2.1.7) and (2.6.17), respectively. The surface temperature predicted by Equation (2.6.15) is used in the mass balance implementing the L-REA and heat balance.

66


Table 2.4 R 2 and RMSE of modelling of convective drying of mango tissues using the L-REA.

Number

Velocity (m s−1 )



R2 X

RMSE X

R2 T

RMSE T

1 2 3

4 4 4

45 55 65

0.0134 0.0134 0.0134

0.998 0.998 0.996

0.08 0.1 0.14

0.993 0.982 0.984

0.61 1.12 1.41

10 Data 45°C Data 55°C Data 65°C Model 45°C Model 55°C Model 65°C

9


8 7 6 5 4 3 2 1 0

0

0.5

1

1.5

2

2.5 t(s)

3

3.5

4

4.5

5 × 104

Figure 2.19 Moisture content profile of convective mango tissues at air temperatures of 45°, 55° and 65 °C (modelled using the L-REA which incorporates the temperature distribution inside the sample). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.]

2.6.4

Results of convective drying thick samples of mango tissues using the L-REA From Figures 2.19 and 2.20, a good agreement between the experimental and predicted data is observed for convective drying of mango tissues at drying air temperatures of 45°, 55° and 65 °C. The good predictions made by using the REA are further revealed by R 2 and RMSE presented in Table 2.4, which shows all modelling of these cases yield R 2 of moisture content and temperature profiles higher than 0.996 and 0.982, respectively, as well as RMSE of moisture content and temperature profiles lower than 0.14 and 1.41, respectively. On the other hand, Figures 2.21 and 2.22 show discrepancies between the predicted and experimental data. It is clear that the L-REA with the approximation of

340

Centre temperature (K)

330

320


300

290

280

0

0.5

1

1.5

2

2.5 t(s)

3

3.5

4

4.5

5 × 104

Figure 2.20 Temperature profile of convective mango tissues at air temperatures of 45°, 55° and

65 °C (modelled using the L-REA which incorporates the temperature distribution inside the sample). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.] 10 Data 45°C

Moisture content (kg water/kg dry solid)

9

Data 55°C 8

Data 65°C Model 45°C

7

Model 55°C

6

Model 65°C

5 4 3 2 1 0

0

0.5

1

1.5

2

2.5 t(s)

3

3.5

4

4.5

5 × 104

Figure 2.21 Moisture content profile of convective mango tissues at air temperatures of 45°, 55° and 65 °C (modelled using the L-REA without approximation of temperature distribution inside the sample). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.]


340

330 Centre temperature (K)

68

320


300

290

280

0

0.5

1

1.5

2

2.5 t(s)

3

3.5

4

4.5

5 × 104

Figure 2.22 Temperature profile of convective mango tissues at air temperatures of 45°, 55° and 65 °C (modelled using the L-REA without approximation of temperature distribution inside the sample). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.]

temperature distribution inside the sample is necessary, and this model describes both moisture content and centre sample temperature profile well during drying. Vaquiro et al. (2009) used diffusion-based modelling to represent the data and our REA compares well with the modelling by Vaquiro et al. (2009). Modelling by Vaquiro et al. (2009) showed a kink in the beginning of the temperature profile that was not observed by modelling using the REA. For drying at 65 °C, both the REA and modelling proposed by Vaquiro et al. (2009) showed a slight overestimation of the temperature profile during drying times of 5000–20 000 s. It can be said that the L-REA, with the prediction of sample temperature as explained in Section 2.6.2, is accurate enough to describe continuous convective drying of mango tissues well. It also compares favourably with the model proposed by Vaquiro et al. (2009) in spite of the simplicity of the L-REA. While the results are accurate, the modelling itself is still simple and only requires a short computational time to predict the drying kinetics accurately. This shows that the L-REA is effective for modelling ‘thick’ samples of mango tissues. A new and innovative application of the L-REA has been implemented in this study to describe both the moisture content and sample temperature profile of convective drying large samples of mango tissues. For this purpose, the activation energy and the saturation vapour concentration are evaluated at the surface temperature. The remaining principles


69

Table 2.5 Schemes of intermittent drying of mango tissues (Vaquiro et al., 2009). Drying air temperature (°C)

Period of first heating (s)

Period of resting (at 27 °C ± 1.6) (s)

Period of second heating (s)

45 55 65

16 200 9 480 7 800

10 800 10 800 10 800

36 360 33 720 16 200

are similar to those of the L-REA used to describe the drying kinetics of thin layers or small objects published previously. Results indicate that the REA models both moisture content and temperature of convective drying of large samples of mango tissues very well. When compared to the experimental data published by Vaquiro et al. (2009), a similar if not better agreement is observed against diffusion-based models. While the results are accurate, the effectiveness of the L-REA is also revealed as the modelling itself is still simple and only requires a short amount of computational time. Therefore, this work has extended the application of the L-REA to handle drying of thick samples substantially. The L-REA can model not only the drying of thinlayer or small objects, but also drying of thick samples.

2.7

The intermittent drying of food materials modelled using the L-REA In this study, the experimental data are derived from the work of Vaquiro et al. (2009) whose experimental details are briefly reviewed in Section 2.6. The intermittency is created by the heating and resting period listed in Table 2.5. During the resting period, the samples stay in an environment with an ambient temperature of 27 ± 1.6 °C and a relative humidity of 60%.

2.7.1

Mathematical modelling of intermittent drying of food materials using the L-REA Since the sample is relatively thick, the surface temperature is incorporated in modelling. The L-REA shown in Equation (2.6.4) is used for modelling. Equations (2.6.15) and (2.6.16) are used to predict the surface and centre temperatures, respectively. The relative activation energy shown in Equation (2.6.17) and the heat balance shown in Equation (2.6.18) are also applied to the modelling here. For modelling intermittent drying, the heat balance is employed according to the drying air temperature in each section. In addition, the equilibrium activation energy shown in Equation (2.1.7) is evaluated according to the corresponding drying air temperature and humidity in each drying period.

2.7.2

The results of modelling of intermittent drying of food materials using the L-REA Figures 2.23 to 2.28 show the results of modelling of intermittent drying of mango tissues using the REA. For intermittent drying at a drying air temperature of 45 °C, the REA


10 Model Data

9


8 7 6 5 4 3 2 1 0

0

1

2

3

4

5

6

t(s)

7 × 104

Figure 2.23 Moisture content profile of mango tissues during intermittent drying at a drying air

temperature of 45 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

320 315 Centre temperature (K)

70

310 305 300 295 290 Model Data

285 280

0

1

2

3

4 t(s)

5

6

7 × 104

Figure 2.24 Temperature profile of mango tissues during intermittent drying at a drying air temperature of 45 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]


71

10 Model Data

9


8 7 6 5 4 3 2 1 0

0

1

2

3 t(s)

4

5

6 × 104


temperature of 55 °C and resting at 27 °C [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.] 330 325


320 315 310 305 300 295 290 Model Data

285 280

0

1

2

3 t(s)

4

5

6 × 104

Figure 2.26 Temperature profile of mango tissues during intermittent drying at a drying air temperature of 55 °C and resting at 27 °C [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]


10 Model Data

9


8 7 6 5 4 3 2 1 0

0

0.5

1

1.5

2 t(s)

2.5

3

3.5

4 × 104


temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

340

330 Centre temperature (K)

72

320

310

300

290

280 0

Model Data 0.5

1

1.5

2 t(s)

2.5

3

3.5 × 104

Figure 2.28 Temperature profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]


73

Table 2.6 R 2 and RMSE of modelling of intermittent drying of mango tissues using the L-REA. Drying air temperature (°C)

R2 X

R2 T

RMSE X

RMSE T

45 55 65

0.998 0.998 0.998

0.996 0.997 0.997

0.083 0.087 0.082

0.483 0.554 0.686

describes both moisture content and temperature profile very well. Good agreement is observed between experimental and predicted data. Similar results are also revealed for intermittent drying at drying air temperatures between 55° and 65 °C. The predicted moisture content and temperature match well with experimental data. The good predictions of moisture content and temperature profile are revealed by R 2 and RMSE shown in Table 2.6. The benchmark of modelling proposed by Vaquiro et al. (2009) employing the diffusion model was conducted, and it revealed the REA gives comparable or even better results. Modelling proposed by Vaquiro et al. (2009) showed a kink in the temperature profile at the beginning of drying; which was not observed by modelling using the REA. In addition, the underestimation of the moisture content profile at the last period of drying in drying conditions of 65 °C is not revealed by the REA; as shown in the modelling by Vaquiro et al. (2009). It can be said that the REA is accurate enough to model intermittent drying of mango tissues, particularly when it is represented in a lumped model. This is because the relative activation energy (Ev /Ev,b ) implemented allows the natural transition during drying times according to the drying scheme as revealed in Figure 2.29. The relative activation energy keeps increasing during drying, indicating an increase of difficulty removing water from materials. This increases significantly during the heating period but only increases slightly during the resting period. This natural transition during drying is not observed by empirical models and the CDRC (Baini and Langrish, 2007). It was revealed that empirical approaches could not model the intermittent drying of banana tissues well. The CDRC might not be able to handle this type of material since the intermittent drying rate could not be represented simply as a linear and exponential decreasing drying rate (Baini and Langrish, 2007). Therefore, this has extended the application of the REA significantly to model not only continuous drying but also intermittent drying of rather thick samples. Although the results of modelling are accurate and robust, the simplicity of the modelling is still proven and only a short computational time is required.

2.7.3

Analysis of surface temperature, surface relative humidity, saturated and surface vapour concentration during intermittent drying Analysis of surface temperature, surface relative humidity, as well as saturated and surface water vapour concentration during intermittent drying will assist the determination


1 0.9 0.8 0.7 ΔEv /ΔEv,b

74

0.6 0.5 0.4 0.3

0.2 0.1 0

0.5

1

1.5

2 t(s)

2.5

3

3.5 × 104

Figure 2.29 Relative activation energy profile of mango tissues during intermittent drying at a

drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

of the appropriate cycle conditions; that is the length of the drying and resting periods, in order to minimise energy and final moisture content. The following paragraphs discuss these parameters in the intermittent drying of mango tissues. The L-REA is applied here and combined with several equations to yield a profile of surface temperature, surface relative humidity, saturated vapour concentration and surface vapour concentration. The saturated vapour concentration and surface temperature are evaluated by Equations (2.1.3) and (2.6.15) while the surface vapour concentration is calculated by Equation (2.1.2). The profile of surface relative humidity is shown in Figure 2.30. Humidity decreases during the heating period while it increases during resting, representing an increase in surface moisture content. In addition, Figure 2.31 indicates the profiles of surface temperature and saturated vapour concentration during intermittent drying at a drying air temperature of 65 °C. The profiles of saturated vapour concentration follow the surface temperature trend. It increases in first section, decreases in second section and increases again in third section. However, the profile of surface vapour concentration is different from that of saturated vapour concentration as revealed in Figures 2.32 and 2.33 because the profile of surface vapour concentration is affected by both surface temperature and surface relative humidity. It is apparent that the surface vapour concentration increases in the very early part of drying because of the increase in surface temperature, followed by a decrease

0.7

Surface relative humidity

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

2

2.5

3

t(s)

3.5 × 104

Figure 2.30 Surface relative humidity profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

350

0.2

0.1

0

Saturated vapour concentration

0

0.5

1

1.5

2 t(s)

2.5

3

300

Surface temperature (K)

Saturated water vapour concentration (kg.m–3)

Surface temperature

250 3.5 × 104

Figure 2.31 Saturated vapour concentration and surface temperature profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

0.16

Water vapour concentration (kg.m–3)

0.14 0.12 0.1 Saturated Surface

0.08 0.06 0.04 0.02 0

0

0.5

1

1.5

2

2.5

3

3.5 × 104

t(s)

0.06

340

0.05

330 Surface temperature

0.04

320

0.03

310 Surface vapour concentration 300

0.02

0.01


Surface water vapour concentration (kg.m–3)

Figure 2.32 Surface and saturated vapour concentration profile of mango tissues during intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

0

0.5

1

1.5

2 t(s)

2.5

3

290 3.5 × 104

Figure 2.33 Surface vapour concentration and surface temperature profile of mango tissues during

intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

10

Moisture content (kg water/kg dry solid)

9 8 7 6 5 4 3 2 1

0

1

2

3

5

4

6

t(s)

7 × 104

Figure 2.34 Moisture content profile of intermittent drying of mango tissues with heating (at a drying air temperature of 45 °C) and resting periods of 4000 s each. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

0.08

320

0.06

310

0.04

300

0.02

290


Saturated water vapour concentration (kg.m–3)

Surface temperature

Saturated vapour concentration 0.01

0

1

2

3

4 t(s)

5

6

7

280

× 104

Figure 2.35 Saturated vapour concentration and surface temperature profile of intermittent drying of mango tissues with heating (at a drying air temperature of 45 °C) and resting periods of 4000 s each. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

0.04

320

0.02

300



Surface temperature

Surface vapour concentration 0

0

1

2

3

4

5

7 × 104

6

t(s)

280

Figure 2.36 Surface vapour concentration and surface temperature profile of intermittent drying of mango tissues with heating (at a drying air temperature of 45 °C) and resting periods of 4000 s each. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

0.07

Water vapour concentration (kg.m–3)

Saturated Surface 0.06

0.05

0.04

0.03

0.02

0.01

0

1

2

3

4 t(s)

5

6

7 × 104

Figure 2.37 Surface and saturated vapour concentration profile of intermittent drying of mango

tissues with heating (at a drying air temperature of 45 °C) and resting periods of 4000 s each. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

79


1 Surface vapour concentration

0.02

0.5

Surface relative humidity


0.04

Surface relative humidity 0

0

1

2

3

4 t(s)

5

6

7 × 104

0

Figure 2.38 Surface vapour concentration and surface relative humidity profile of intermittent drying of mango tissues with heating (at a drying air temperature of 45 °C) and resting periods of 4000 s each. [Reprinted from Industrial Engineering Chemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

due to decrease in surface relative humidity. During the initial part of the resting period, the surface vapour concentration increases significantly as the surface relative humidity increases dramatically. This is followed by a decrease in the surface vapour concentration because of the low surface temperature, leading to a decrease of saturated vapour concentration. In the second heating period, the surface vapour concentration continues to decrease as the surface relative humidity decreases. This analysis is in agreement with Baini and Langrish (2007) applying a diffusion model to the intermittent drying of banana tissues. It was revealed that, during the resting period, the surface temperature decreased, while the surface moisture content increased initially as a result of an initial increase in surface relative humidity. It can be seen that, during the resting period of drying times higher than 12 000 s, the surface vapour concentration reaches a plateau. It means there is actually no point in extending the resting period to 18 600 s. The resting period could be shortened and followed by a subsequent heating period. Similarly, during the second heating period, the surface vapour concentration profile has nearly flattened after a drying time around 25 000 s. The heating time could also be shortened, followed by a subsequent resting period to achieve higher surface vapour concentration.

80


From this analysis, it seems that a cycle with a higher frequency will give better results. Simulation of intermittent drying at a drying air temperature of 45 °C with each heating and resting period (at 27 °C) at 4000 s and a total drying time of 64 000s (total heating time at 45 °C of 32 000 s) was conducted to illustrate profiles of this scheme. Results of the simulation, including the profiles of moisture content, surface temperature, surface-relative humidity, and saturated surface temperature, are presented in Figures 2.34–2.38. It can be seen that the trends of surface-relative humidity, saturated and surface vapour concentration are similar to those which have been discussed in previous paragraphs. Nevertheless, no flat profile of surface vapour concentration is shown during the resting period which means resting is not conducted for a prolonged time. It is also observed that the final moisture content is similar to that of intermittent drying mango tissues at 45 °C using scheme listed in Table 2.5, although the total heating time of this scheme is lower. The total heating time of this scheme is 32 000 s, while that at 45 °C, listed in Table 2.5, is 52 560 s. The total drying time of this scheme (64 000 s) is also similar to that at 45 °C, listed in Table 2.5 (62 650 s). Because the heating time is shorter, while the total drying time and the final moisture content is not significantly altered (from a sustainable processing perspective), it is beneficial to apply such a cycle. This is because energy cost can be minimised while the objective of obtaining a similar target moisture content can be achieved.

2.8

The intermittent drying of non-food materials under time-varying temperature and humidity modelled using the L-REA In this study, the experimental data of the intermittent drying are derived from the work of Kowalski and Pawlowski (2010a,b) whose experimental details are shown in Table 2.7. For better understanding of the procedures, the experimental details are briefly reviewed here. The materials used for drying is KOC kaolin clay supplied by Surmin-Kaolin, SA Co., Poland. The detailed physical and chemical properties of the samples are provided on the company’s website (Surmin-Kaolin Co., 2010). Each sample was prepared by moulding the materials into a cylinder with a radius of 0.025 m and a height of 0.06 m, with an initial moisture content of 0.4 kg H2 O kg dry solids−1 . Each sample was placed in an aluminium container suspended on an electronic balance with an accuracy of ±0.01 g. For measurement of the sample temperature, a parallel experiment was conducted. T-type thermocouples were inserted inside a cylinder at different positions. The temperature measurement indicated that the temperature inside the sample was uniform (Kowalski et al., 2007; Kowalski and Pawlowski, 2010a,b). Two types of intermittent drying experiments were conducted: time-varying drying air temperature and time-varying humidity. The first type of intermittency was enabled by supplying cool air through a special air intake. Table 2.7 shows the cases of the intermittent drying under time-varying drying air temperature and humidity. Case 1 is the intermittent drying, which implemented a periodic change of the drying air


81

Table 2.7 Settings of intermittent drying of kaolin (Kowalski and Pawlowski, 2010). Case number

Relative humidity

Drying air temperature (°C)

1

7.2% (at 65 °C)

2

4% (at 100 °C)

3

4% (before 9000 s) periodically changed between 4 and 12% (after 9000 s) 4% (before 9000 s) periodically changed between 4 and 80% (after 9000 s)

65 °C (before 17 000 s) periodically changed between 65° and 43 °C (after 17 000 s) 100 °C (before 7000 s) periodically changed between 100° and 50 °C (after 7000 s) 100

4

100

temperature between 65° and 43 °C (see Table 2.7), while Case 2 is similar to the first one but with a periodic change of temperature between 100° and 50 °C (see Table 2.7). For the intermittent drying under time-varying humidity and constant drying air temperature of 100 °C, the intermittency was created by a periodic change in vapour supply to the drying chamber from a humidifier. Cases 3 and 4 (see Table 2.7) are the intermittent drying under time-varying humidity. Case 3 applied a periodic change of the relative humidity between 4° and 12%, while Case 4 implemented a change of the relative humidity between 4–80% (Kowalski and Pawlowski, 2010a,b).

2.8.1

Mathematical modelling using the L-REA The original formulation of the L-REA described in Section 2.1 is still implemented here without any modification. In this study, the relative activation energy is generated from a drying experiment at a constant drying air temperature of 50.8 °C (Kowalski et al., 2007) and can be expressed as: E v = 0 for X − X b > 0.2, E v,b

7.847 E v = exp −49.391(X − X b )2.103 E v,b

for X − X b > 0.2.

(2.8.1) (2.8.2)

A good fit between the experimental and predicted activation energy is shown in Figure 2.39 and indicated by the R 2 of 0.99. As mentioned earlier, uniform temperature profiles inside the product were observed in the work of Kowalski and Pawlowski (2010a). It has been noted that the Chen–Biot number (Ch–Bi) (Chen and Peng, 2005) for intermittent drying of kaolin is 0.03, which indicates the temperature inside the sample is essentially uniform. Indeed, Kowalski and Pawlowski (2010b) implemented modelling which did not take into account the variations of spatial temperature inside products. Based on this observation, the assumption of uniform product temperature profile is implemented in this study. Hence, the heat balance can be represented as: d(mC p T ) dX ≈ h A (Tb − T ) + m s H V , dt dt

(2.8.3)

82


1 Data Fitted curve

0.9 0.8

ΔEv /ΔEv,b

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.05

0.1

0.15 0.2 0.25 0.3 X–Xb (kg water/kg dry solid)

0.35

0.4

Figure 2.39 The relative activation energy (Ev /Ev,b ) of the convective drying of kaolin. [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.]

where m is the sample mass (kg), Cp is the heat capacity of the sample (J kg−1 K−1 ), h is the heat transfer coefficient (W m−2 K−1 ) and HV is the latent heat of vaporisation of water (J kg−1 ). In order to incorporate the effects of time-varying drying air temperature or humidity, the equilibrium activation energy (Ev,b ) shown in Equation (2.1.7) is defined according to the corresponding drying air settings in each time period. The equilibrium activation energy (Ev,b ) is combined with the relative activation energy (Ev /Ev,b ) represented in Equations (2.8.1) and (2.8.2). In addition, the mass balance implementing the L-REA and heat balance shown in Equations (2.1.4) and (2.8.3), respectively, also implement the corresponding drying air settings in each time period. The profiles of moisture content and temperature can be yielded by solving the mass and heat balance simultaneously in conjunction with the equilibrium and relative activation energy shown in Equations (2.1.7), (2.8.1) and (2.8.2).

2.8.2

Results of intermittent drying under time-varying temperature and humidity modelled using the L-REA Figures 2.40 to 2.43 show the results of modelling of the intermittent drying under timevarying drying air temperature using the L-REA. It can be seen that the L-REA describes

Moisture content (kg water/kg dry solids)

0.45 L-REA Data

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.5

1

1.5

2 t(s)

2.5

3

3.5

4 × 104

Figure 2.40 Moisture content profile of intermittent drying in Case 1 (periodically changed drying

air temperatures between 65° and 43 °C). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.]

340

Temperature (K)

330

320

310

300

290

280

L-REA Data 0

0.5

1

1.5

2 t(s)

2.5

3

3.5

4 × 10

4

Figure 2.41 Temperature profile of intermittent drying in Case 1 (periodically changed drying air

temperatures between 65° and 43 °C). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.]


0.45 L-REA Data

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.5

1

1.5

2

t(s)

2.5 × 104

Figure 2.42 Moisture content profile of intermittent drying in Case 2 (periodically changed drying

air temperatures between 100° and 50 °C). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.]

360 350

Temperature (K)

340 330 320 310 300 290

L-REA Data 0

0.5

1

1.5 t(s)

2

2.5 × 104

Figure 2.43 Temperature profile of intermittent drying in Case 2 (periodically changed drying air

temperatures between 100° and 50 °C). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.]


85

Table 2.8 R 2 , RMSE, average absolute deviation and maximum absolute deviation of profiles of moisture content predicted by and Kowalski and Pawlowski’s model (2010b). Average Absolute R2 RMSE Average Deviation Kowalski and Kowalski and Absolute Kowalski and Case R2 Pawlowski’s RMSE Pawlowski’s Deviation Pawlowski’s number REA model (2010b) REA model (2010b) REA model (2010b)

Maximum Absolute Maximum Deviation Absolute Kowalski and Deviation Pawlowski’s REA model (2010b)

1 2 3 4

0.012 0.012 0.023 0.009

0.997 0.997 0.991 0.998

0.992 0.983 0.994 0.967

0.006 0.007 0.011 0.005

0.009 0.016 0.009 0.019

0.005 0.006 0.009 0.084

0.007 0.014 0.006 0.015

0.024 0.031 0.026 0.037

Table 2.9 R 2 , RMSE, average absolute deviation and maximum absolute deviation of profiles of temperature predicted by Kowalski and Pawlowski’s model (2010b). Average Absolute R2 RMSE Average Deviation Kowalski and Kowalski and Absolute Kowalski and Pawlowski’s RMSE Pawlowski’s Deviation Pawlowski’s Case R2 model (2010b) number REA model (2010b) REA model (2010b) REA

Maximum Absolute Maximum Deviation Absolute Kowalski and Deviation Pawlowski’s REA model (2010b)

1 2 3 4

5.968 5.501 5.050 4.369

0.952 0.953 0.983 0.950

0.761 0.842 0.743 0.896

2.529 3.399 1.658 4.554

5.567 6.254 7.441 6.377

1.481 2.554 1.217 0.084

4.335 4.892 6.189 3.995

11.158 10.059 14.472 6.8326

both the moisture content and temperature profiles well. Benchmarks towards the modelling approach implemented by Kowalski and Pawlowksi (2010b) for the intermittent drying have been conducted. For Case 1 (refer to Table 2.7, Figures 2.40 and 2.41), L-REA can give even better agreement between the predicted moisture content, temperature profiles and the experimental data than the approach implemented by Kowalski and Pawlowksi (2010b); as shown by the results of error analysis presented in Tables 2.8 and 2.9. For Case 2 (refer to Table 2.5, Figures 2.42 and 2.43), the L-REA also yields closer agreement between the predicted and experimental moisture content profiles. The L-REA results in a slight deviation in the temperature profiles during drying times between 15 000 and 20 000 s, while the modelling implemented by Kowalski and Pawlowksi (2010b) showed a slight deviation in the temperature profiles during drying times of 8000–20 000 s. The closer agreement is indeed shown by the results of error analysis presented in Tables 2.8 and 2.9. It can be said that the REA can model the intermittent drying of kaolin under timevarying drying air temperature well. This could be due to the flexibility of the activation energy in allowing a change in the drying kinetics according to the drying air settings in each time period. The relative activation energy (Ev /Ev,b ) shown in Equations


0.45 Moisture content (kg water/kg dry solids)

86

L-REA Data

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.5

1

1.5 t(s)

2

2.5 × 104

Figure 2.44 Moisture content profile of intermittent drying in Case 3 (periodically changed relative humidity between 4° and 12%). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.]

(2.8.1) and (2.8.2) combined with the equilibrium activation energy (Ev,b ) indicated in Equation (2.1.7) seems to capture accurate physics during the intermittent drying under time-varying drying air temperature. The results indicated that the application of the REA has been extended significantly, following that of the L-REA modelling the cyclic drying of a thin layer of polymer solution mixture of under time-varying infrared-heat intensity (Putranto et al., 2010b). In terms of the intermittent drying of kaolin under time-varying humidity, the results of modelling using the L-REA are shown in Figures 2.44–2.47. The results of the intermittent drying for Case 3 (refer to Table 2.7) are indicated in Figures 2.44 and 2.45. The L-REA models both the moisture content and temperature profiles reasonably well. Benchmarks towards modelling implemented by Kowalski and Pawlowski (2010b) have also been conducted and the L-REA yields better agreement with the experimental data. The observed deviations between the experimental and predicted temperature profiles observed from the approach of Kowalski and Pawlowski (2010b) do not appear when the L-REA is used. Similarly to Case 3 (refer to Table 2.7, Figures 2.44 and 2.45), for Case 4 (refer to Table 2.7, Figures 2.46 and 2.47) the REA results are in a good agreement between both the predicted moisture content and temperature profiles and the experimental data. The slight deviation of the temperature profiles during drying times of 13 000 and 18 000 s are also observed in the modelling applied by Kowalski and Pawlowski (2010b). However,

370 360

Temperature (K)

350 340 330 320 310 L-REA Data

300 290

0

0.5

1

1.5

2

t(s)

2.5 × 104

Figure 2.45 Temperature profile of intermittent drying in Case 3 (periodically changed relative humidity between 4° and 12%). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.]


0.4 L-REA Data

0.35 0.3 0.25 0.2 0.15 0.1 0.05

0

0.5

1

1.5 t(s)

2

2.5 × 104

Figure 2.46 Moisture content profile of intermittent drying in Case 4 (periodically changed relative humidity between 4° and 80%). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.]

88


370 360

Temperature (K)

350 340 330 320 310 L-REA Data

300 290

0

0.5

1

1.5 t(s)

2

2.5 × 104

Figure 2.47 Temperature profile of intermittent drying in Case 4 (periodically changed relative humidity between 4° and 80%). [Reprinted from Chemical Engineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al., Application of the reaction engineering approach (REA) for modelling intermittent drying under time-varying humidity and temperature, 2149–2156, Copyright (2012), with permission from Elsevier.]

the observed overestimation at a drying time of 8000–13 000 s is not evident when modelling using the L-REA. Tables 2.8 and 2.9 indicate that the L-REA yields closer agreement with the experimental data than the other model. It can be shown that the L-REA is indeed accurate enough to model intermittent drying under time-varying humidity. The relative activation energy in conjunction with the equilibrium activation energy seems to be able to capture the periodically changed relative humidity well. Because of this, the L-REA is shown to be able to model intermittent drying, not only under time-varying temperatures but also under time-varying humidity, well. The simplicity of the L-REA is also proven for this challenging drying system.

2.9

The heating of wood under linearly increased gas temperature modelled using the L-REA Wood may contain several harmful chemicals as a result of chemical processing, including chromated copper arsenate (CCA), creosote and pentachloro-phenol (Younsi et al., 2006a). Several methods of disposing such harmful chemicals were reviewed in research by Helsen and Bulck (2005). Recycling and recovery, chemical extraction, bioremediation, electrodialytic remediation and thermal destruction may be attempted to remove


89

Table 2.10 Settings of heat treatment of wood samples (Younsi et al., 2006a; 2007).

Case

Final gas temperature (°C)

Heating rate (°C h−1 )

Initial moisture content (kg H2 O kg dry solids−1 )

1 2 3 4 5

220 220 220 160 200

30 20 10 20 20

0.11 0.125 0.12 0.1059 0.07

the contaminants. Reuse and recycling contaminants has several disadvantages, including contamination to people handling the process as sorting, transportation and storage are required, while chemical extraction is not an effective process since the kinetics of extraction is relatively slow and many steps of extraction are necessary. In addition, the remediation may result in lower quality wood fibre. The thermal destruction process has advantages of possible energy recovery and significant reduction in volume. However, intensive research still needs to be carried out to evaluate and optimise the process (Helsen and Bulck, 2005). Thermal destruction can be carried out by hightemperature treatment processes in which wood samples are exposed to hot gas with linearly increased temperatures beyond 200 °C. Fundamentally, it is a drying process under linearly increased gas temperature according to heating rate (Younsi et al., 2006a). The accuracy and robustness of the REA for high-temperature treatment of wood is shown by published experimental data from Younsi et al. (2006a; 2007). The experimental details were reported in Kocaefe et al. (1990; 2007) and Younsi et al. (2006b) and are reviewed here for better understanding of the current approach. The heat treatment was conducted in a thermogravimetric analyser (Kocaefe et al., 1990). Wood samples with dimensions of 0.035 × 0.035 × 0.2 m were heat treated by suspending the samples on a balance with an accuracy of 0.001 g. The heat treatment was conducted by exposing the samples to hot gas whose temperature was linearly increased according to heating rate. The humidity of the gas was controlled by injection of steam into a second furnace placed under the main furnace. Initial moisture content of the samples was between 7 and 12%wt. (dry) and initial temperature of the samples was 20 °C (refer to Table 2.10). The samples were first heated to 120 °C and held at this temperature for half an hour, followed by heating below the preset heating rate (refer to Table 2.10) until the final temperature (also refer to Table 2.10) was achieved. During heat treatment, the weights of the samples were recorded. In addition, temperatures were measured by T-type thermocouples placed inside the samples, but the measurement indicated that the temperatures inside the samples were essentially uniform because of their small size (Younsi et al., 2006b).

2.9.1

Mathematical modelling using the L-REA The original formulation of the L-REA described in Section 2.1 is used here without any modification. As mentioned before, for modelling using the L-REA, relative activation


1 Experimentally determined data Fitted curve

0.9 0.8 0.7 ΔEv /ΔEv,b

90

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.02

0.04 0.06 0.08 X–Xb (kg water/kg dry solid)

0.1

0.12

Figure 2.48 Relative activation energy (Ev /Ev,b ) of the dehydration of wood during heat treatment generated from the experimental data in Case 2 (refer to Table 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.]

energy (Ev /Ev,b ) needs to be generated. In this study, the relative activation energy is generated from the experimental data for a drying run of Case 2 (refer to Table 2.10). This can be represented as:

E v = 1 − 2.181(X − X b )0.372 E v,b

× exp −3.716(X − X b )3.135

for X − X b ≤ 0.05,

E v = 1 − 1.462(X − X b )0.207 E v,b

× exp −3.716(X − X b )3.137 forX − X b ≤ 0.05.

(2.9.1)

(2.9.2)

The format of the relative activation energy can be varied but, in this case, Equations (2.9.1) and (2.9.2) seem to be sufficient. The good agreement between the fitted and experimental relative activation energy is shown in Figure 2.48 and indicated by R 2 of 0.994 and 0.999, respectively. During heat treatment of wood, the drying gas temperature changed linearly with time according to heating rate (refer to Table 2.10). In order to incorporate this in the modelling using the L-REA, the equilibrium activation energy (Ev,b ) shown by Equation (2.1.7) has been evaluated according to the gas temperature and corresponding


91

humidity during heat treatment. The equilibrium activation energy (Ev,b ) is combined with the relative activation energy (Ev /Ev,b ) shown in Equations (2.9.1) and (2.9.2). Since the temperature distributions inside the samples were essentially uniform during the heat treatment (Younsi et al., 2006b), the heat balance can be written as: d(mC p T ) dX ≈ h A (Tb − T ) + m s H V , dt dt

(2.9.3)

where m is sample mass (kg), Cp is the heat capacity of the sample (J kg−1 K−1 ), T is the temperature of the sample (K), h is the heat transfer coefficient (W m−2 K−1 ), HV is vaporisation heat of water (J kg−1 ) and Tb is the gas temperature (K). The drying rate dX/dt is negative when drying occurs. In this case, the linearly increased gas temperature is applied in Equation (2.9.3). Solving the mass balance, and implementing the L-REA and heat balance shown in Equations (2.1.4) and (2.9.3) in conjunction with the equilibrium and relative activation energy shown in Equations (2.1–7), (2.9.1) and (2.9.2), results simultaneously in the profiles of moisture content and temperature during heat treatment of wood. The shrinkage is neglected in the modelling because Younsi et al. (2006b) indicated that the ratio between the final and initial dimension is around 0.96. Similarly, the modelling implemented by Younsi et al. (2006a,b, 2007) did not incorporate the shrinkage effect.

2.9.2

Results of modelling wood heating under linearly increased gas temperatures using the L-REA The profiles of both moisture content and temperature during heat treatment of wood for Cases 1–5 (refer to Table 2.10) are presented in Figures 2.49–2.52. Results of modelling for Case 1 (refer to Table 2.10) are shown in Figures 2.49 and 2.50. The L-REA-based model system describes both the moisture content and temperature profiles well. The predictions made using the L-REA match reasonably well with the experimental data of moisture content and temperature. The slight discrepancies in the moisture content profile were also found in the modelling implemented by Younsi et al. (2007) using a far more complex model. However, as depicted in Figure 2.49, the L-REA yields system closer agreement of the moisture content profile with the experimental data. The L-REA system can model the heat treatment of wood which applied linearly changed air temperature with a heating rate of 30 °C h−1 . For Case 2 (refer to Table 2.10), Figures 2.49 and 2.50 indicate that the L-REA predicts both the moisture content and temperature profiles well. The results of modelling match well with the experimental data. This is also confirmed by R 2 of moisture content and temperature profiles of 0.994 and 0.996. A benchmark against the modelling implemented by Younsi et al. (2007) shows that the L-REA gives comparable or better results. For the heat treatment of wood under a heating rate of 10 °C h−1 (Case 3, refer to Table 2.10), the L-REA again describes both the moisture content and temperature reasonably well, as shown in Figures 2.49 and 2.50. Slight deviations in predicting moisture content profiles are observed, but when benchmarking against the modelling


0.14 Case 1-experimental data Case 1-predicted by L-REA Case 2-experimental data Case 2-predicted by L-REA Case 3-experimental data Case 3-predicted by L-REA


92

0.1

0.08

0.06

0.04

0.02

0

0

1

2

3

4 t(s)

5

6

7 4

× 10

Figure 2.49 Moisture content profiles during the heat treatment of Cases 1 to 3 (refer to Table 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.]

applied by Younsi et al. (2007), the L-REA still yields the closer agreement towards experimental data. The good agreement of moisture content and temperature profile is also indicated by R 2 of moisture content and temperature profiles of 0.987 and 0.993. It is also important to assess the accuracy of the L-REA to model the heat treatment of wood using a different final gas temperature. Case 4 (refer to Table 2.10) applied the heat treatment of wood samples with a final gas temperature of 160 °C and a heating rate of 20 °C h−1 . As depicted in Figure 2.51, the L-REA system was reasonably accurate in modelling the moisture content profiles. Both the L-REA system and the modelling by Younsi et al. (2006a) predict the temperature profiles well as shown in Figure 2.52. It suggests that the L-REA system can model the heat treatment of wood using different final gas temperatures. In addition, Case 5 (refer to Table 2.10) implemented the heat treatment of wood with initial moisture content of 7%wt. (dry basis), slightly lower than that of other cases. The accuracy of the L-REA in predicting both the moisture content and temperature profiles has been assessed. From Figure 2.51, it can be seen that the L-REA system describes the profiles of moisture content well. In addition, the L-REA is accurate in modelling the temperature profiles during the heat treatment of wood as shown in Figure 2.52. Results of modelling using the L-REA match well with experimental data. The

93


500

Temperature (K)

450

400

350

Case 1-experimental data Case 1-predicted by L-REA Case 2-experimental data Case 2-predicted by L-REA Case 3-experimental data Case 3-predicted by L-REA

300

250

0

1

2

4

3 t(s)

5

6

7 × 10

4

Figure 2.50 Temperature profiles during the heat treatment of Cases 1 to 3 (refer to Table 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.]

agreement towards the profiles of moisture content and temperature is indicated by an accurate R 2 of moisture content and temperature of 0.993 and 0.999, respectively. When benchmarking against the modelling implemented by Younsi et al. (2006), it is observed that the L-REA yields better results. From Cases 1 to 5 (refer to Table 2.10), it can be said that the L-REA system can be implemented to describe both moisture content and temperature profiles very successfully. The applicability of the L-REA for this purpose could be due to the accuracy and flexibility of the equilibrium activation energy (Ev,b ) combined with the unique relative activation energy (Ev /Ev,b ) in capturing the physics of drying during heat treatment of wood. The effect of a linearly increased gas temperature according to the heating rate on the drying rate seems to be captured well by the combination of (Ev,b ) and (Ev /Ev,b ). This allows the drying kinetics to be changed flexibly according to environment conditions. Based on the study of Cases 1 to 5 (refer to Table 2.10), it is revealed that the L-REA can be implemented in modelling the heat treatment of wood with various heating rates. Therefore, the L-REA may also be applied to the similar thermal processing of biomass that employs time-varying temperature or external conditions. Several processes apply this principle, including ThermoWood Technology (Finnish ThermoWood Association, 2011) developed by Finnish industries. The process is essentially heating wood following


0.14 Case 1-experimental data Case 1-predicted by L-REA Case 2-experimental data Case 2-predicted by L-REA Case 3-experimental data Case 3-predicted by L-REA


94

0.1

0.08

0.06

0.04

0.02

0

0

1

2

3

4 t(s)

5

6

7 4

× 10

Figure 2.51 Moisture content profiles during the heat treatment of Cases 1–3 (refer to Table 2.10).

[Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.]

a particular schedule, with different temperatures in each heating period aimed to reduce cracking and burning by protecting wood with water vapour generated from the wood samples (Younsi et al., 2010; Finnish ThermoWood Association, 2011). Another example is thermal modification of wood to improve thermal insulation properties, maintaining colour and enhancing water resistance conducted by heating of wood to temperature of 160–260 °C under various gas media and schedules (Rapp, 2001). In order to predict the quality changes resulting from the process, extensive experiments can be carried out. However, these may require a lot of resources and may not be suitable for large-scale industries where quick decision-making is necessary. Simulation tools can be applied as a means to predict the quality of wood or biomass after treatment. As the L-REA is revealed to be simple and accurate for modelling the heat treated wood, it can be used as a simple and valuable tool in industry for predicting quality parameters of samples under heat treatment. This can be conducted by the L-REA by evaluating the equilibrium activation energy and heat balance according to corresponding external conditions (i.e. humidity and temperature) in each period of heat treatment. The equations that represent quality change, usually represented as function of moisture content and/or temperature, are incorporated in the REA system and solved simultaneously to yield quality profiles during heat treatment.

95


480 460 440

Temperature (K)

420 400 380 360 340 Case 4-experimental data Case 4-predicted by L-REA Case 5-experimental data Case 5-predicted by L-REA

320 300 280

0

0.5

1

1.5

2 t(s)

2.5

3

3.5 × 104

Figure 2.52 Temperature profiles during the heat treatment of Cases 4 and 5 (refer to Table 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.]

2.10

The baking of cake modelled using the L-REA Baking is a complex simultaneous heat and mass transfer process commonly applied in food industries. Although the process has been practiced for long time, there is still limited knowledge of the physical phenomena involved (Zhang and Datta, 2006). The heat and mass transfer process is relatively complicated since it is related to physical, chemical and structural changes during baking (Lostie et al., 2002a). The process is also affected significantly by molecular size and structure of polymers that make up starch and protein components. The interactions between polymer chain entanglements and branching may determine the rheological aspects that affect the deformation during baking (Dobraszczyk, 2004). Shrinkage could occur during baking as a result of water loss and thermal stress, while expansion takes place because of carbon dioxide produced by leavening agents and water vapour inside the porous medium (Lostie et al., 2002a,b; Mayor and Sereno, 2004). The applicability of the L-REA to modelling the baking of a thin layer of cake is validated by the published experimental data and details of Sakin et al. (2007). For a better understanding of the modelling using the L-REA, the experimental details are reviewed next. The ingredients of the cake batter were 49.4%wt. of Dr Oetker’s ready

96


dry cake mix (consists of wheat flour, sugar, corn starch and baking powder), 24.7%wt. pasteurised whole liquid egg, 16.2%wt. vegetable margarine and 9.7%wt. water. The ingredients were mixed thoroughly using a three-stage mixing method and a hand mixer. The mixture was spread on a baking tray with diameter of 220 mm to create a batter with initial thickness of 3 mm. This thickness was used to minimise temperature gradient inside the samples. The initial moisture content of the samples was 53%wt. (dry basis) (Sakin et al., 2007). The baking experiments were conducted in an electrical baking oven with dimensions of 0.39 × 0.44 × 0.35 m (Teba High-01 Inox) at baking temperatures of 50°, 80°, 100°, 140° and 160 °C under forced convection conditions. Fresh air entered the oven cavity and the fan on the back side was used to circulate the air at a constant speed of 0.56 m s−1 (measured by an Airflow anemometer, LCA 6000). During baking, the weight of the batter was recorded until the equilibrium moisture content was achieved. The product temperature was also measured by a thermocouple (J-type) inserted inside the samples. In addition, the thickness was measured by a digital calliper (Sakin et al., 2007).

2.10.1

Mathematical modelling of the baking of cake using the L-REA The original formulation of the L-REA described in Section 2.1 is used here to model baking of cake without any modification. Similar to the modelling of convective drying using the L-REA mentioned in the previous sections, for modelling baking a thin layer of cake using the L-REA, the relative activation energy (Ev /Ev,b ) needs to be generated from one accurate baking experiment. In this study, it was generated from an experiment baking at a temperature of 100 °C, whose experimental data of moisture content and temperature (Sakin et al., 2007) were used to evaluate the activation energy (Ev ) shown in Equation (2.1.5). The relative activation energy (Ev /Ev,b ) is calculated by dividing the activation energy with the equilibrium activation energy (Ev,b ) indicated in Equation (2.1.7). The relative activation energy is related with the moisture content on dry basis (X) by simple mathematical expression obtained by a least-square method:

E v = 1 − 1.612(X − X b )1.151 exp −1.28 × 106 (X − X b )14.19 . E v,b

(2.10.1)

Figure 2.53 shows a good agreement between the experimental and fitted relative activation energy, which is also confirmed by R 2 of 0.998. The profile of the relative activation energy is very reasonable since it is zero near the start and keeps increasing as baking proceeds. When the equilibrium moisture content is achieved, the relative activation energy is 1. The format of Equation (2.10.1) can be varied but in this case, Equation (2.10.1) seems to be sufficient to represent the relative activation energy of baking a thin layer of cake. For yielding the profiles of both moisture content and temperature during baking, the mass and heat balances are solved simultaneously. The mass balance using the L-REA is shown in Equation (2.1.4). The temperature distribution inside the cake during baking

97


1 Data Model

0.9 0.8

ΔEv /ΔEv,b

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.05

0.1

0.15 0.2 0.25 0.3 0.35 X–Xb (kg water/kg dry solid)

0.4

0.45

0.5

Figure 2.53 The relative activation energy (Ev /Ev,b ) of baking of thin layer of cake at an oven

temperature of 100 °C. [Reprinted from Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of baking of cake using the reaction engineering approach (REA), 306–311, Copyright (2012), with permission from Elsevier.]

can be neglected as the thickness of the cake was around 3 mm (Sakin et al., 2007). Therefore, the heat balance can be expressed as:

d m s (1 + X )C p T dX (2.10.2) ≈ h A (Tb − T ) + m s HV , dt dt where Cp is the heat capacity of the sample (J kg−1 K−1 ), T is the temperature of the sample (K), h is the heat transfer coefficient (W m−2 K−1 ), HV is vaporisation heat of water (J kg−1 ) and Tb is the baking oven temperature (K). The drying rate dX/dt is negative when baking occurs.

2.10.2

Results of modelling of the baking of cake using the L-REA Profiles of moisture content and temperature during baking cake at different baking temperatures are shown in Figures 2.54–2.57. Figures 2.54 and 2.55 show that the profiles of moisture content predicted using the L-REA match well with the experimental data. This is supported by the R 2 values for moisture content profiles being higher than 0.982 (shown in Table 2.11). Benchmarking against a diffusion-based model implemented by Sakin et al. (2007) revealed that the L-REA yielded comparable or better results, although the diffusion-based model employed diffusivity which was split into two forms in order to incorporate the effects of temperature and moisture content. The slight overestimation of


0.7 L-REA 100°C Data 100°C L-REA 140°C Data 140°C L-REA 160°C Data 160°C

0.6 0.5 0.4 0.3 0.2 0.1 0

0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000 t(s)

Figure 2.54 Moisture content profiles at baking temperatures of 100°, 140° and 160 °C. [Reprinted from Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of baking of cake using the reaction engineering approach (REA), 306–311, Copyright (2012), with permission from Elsevier.]

0.5 L-REA 50°C Data 50°C L-REA 80°C Data 80°C


0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

1

2

3

4 t(s)

5

6

7 × 10

4

Figure 2.55 Moisture content profiles at baking temperatures of 50° and 80 °C. [Reprinted from Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of baking of cake using the reaction engineering approach (REA), 306–311, Copyright (2012), with permission from Elsevier.]


99

Table 2.11 R 2 of modelling using the REA.

No.

Baking temperature (°C)

R 2 for X

R 2 for T

1. 2. 3. 4. 5.

50 80 100 120 140

0.990 0.997 0.992 0.987 0.982

0.990 0.999 0.970 0.997 0.985

460 440

Temperature (K)

420 400 380 360 340

L-REA 100°C Data 100°C L-REA 140°C Data 140°C L-REA 160°C Data 160°C

320 300 280

0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000 t(s)

Figure 2.56 Temperature profiles at baking temperatures of 100°, 140° and 160 °C. [Reprinted

from Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of baking of cake using the reaction engineering approach (REA), 306–311, Copyright (2012), with permission from Elsevier.]

the drying rate at initial baking period at oven temperatures of 100°, 140° and 160 °C was also observed in modelling using the diffusion-based model. At baking temperatures of 50° and 80 °C both the L-REA and diffusion-based model described the moisture content profiles well. It can be inferred that the L-REA can model the moisture content during the baking of thin layer of cake very well. Figures 2.56 and 2.57 indicate that the L-REA modelled the profiles of temperature profiles well. A good agreement with the experimental data is observed and it is confirmed by the R 2 values for temperature profiles being higher than 0.970 (shown in Table 2.11). The slight discrepancies between the predicted and experimental data during initial period of baking at baking temperature of 100 °C might be due largely to

100


370 360

Temperature (K)

350 340 330 320 310

L-REA 50°C Data 50°C L-REA 80°C Data 80°C

300 290

0

1

2

3

4 t(s)

5

6

7 4

× 10

Figure 2.57 Temperature profiles at baking temperatures of 50° and 80 °C. [Reprinted from Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of baking of cake using the reaction engineering approach (REA), 306–311, Copyright (2012), with permission from Elsevier.]

experimental error of temperature measurement, which resulted in higher temperature profiles than those of baking temperatures at 140° and 160 °C. Benchmarking against the modelling from Sakin et al. (2007) cannot be conducted as the model in this study did not implement the heat balance. The L-REA coupled with the heat balance (shown in Equation 2.10.2) is indeed accurate in predicting the temperature profiles during baking. It has been shown that the L-REA-based heat- and mass-transfer system can be successfully applied to model the baking of cake. The L-REA is accurate in modelling both the profiles of moisture content and temperature during the baking of cake. It seems that the relative activation energy (Ev /Ev,b ) shown in Equation (2.9.1) combined with the equilibrium activation energy (Ev,b ) shown in Equation (2.1.7) can capture the physics during baking.

2.11

The infrared-heat drying of a mixture of polymer solutions modelled using the L-REA The experimental data for validating the accuracy of the L-REA are derived from a study reported by Allanic et al. (2009). The experimental setup up is similar to that described in Section 2.5 but constant infrared-heat intensity is applied here. The velocity and


101

0 Product e(t) Petri dish

z

Thermocouple

Conduction

Long infrared

Evaporation

Convection

Diffusion

Figure 2.58 Heat transfer mechanisms of convective and infrared-heat drying. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2012), with permission from Elsevier.]

temperature of the drying medium were set to 1 m s−1 and 35 °C, respectively was fed into the canal. Constant infrared-heating with an intensity of 3.7 kW m−2 was maintained throughout the experimental run (Allanic et al., 2009).

2.11.1

Mathematical modelling of the infrared-heat drying of a mixture of polymer solutions using the L-REA The L-REA shown in Equation (2.1.4) is used for the mass balance here. For infrared heatdrying, the sample was heated by an infrared emitter which increased the temperature of the sample and the Petri dish (refer to Figure 2.58). Because of the relatively low temperature of the drying air, the sample and the Petri dish actually release heat to the environment by convection. The heat is released from the upper side and bottom side due to forced and natural convection, respectively (refer to Figure 2.58). Hence, it can be written as: d(ρC p eT ) ms d X = α Q IR − h top (T − Tb ) − Ubottom (T − Tb ) + HV (T ), dt A dt (2.11.1) where ρ is sample density (kg m−3 ), Cp is sample heat capacity (J kg−1 K−1 ), e is sample thickness (m), T is sample temperature (K), ms is mass of dried product (kg), A is surface area of product (m2 ), Hv is vaporisation enthalpy of water (J kg−1 ), htop is the heat

102


transfer coefficients on top of the sample (W m−2 K−1 ), Ubottom represents the overall heat transfer coefficient (W m−2 K−1 ) from the lower side, including convection along and conduction through the Petri dish, QIR is the intensity of radiation (W m−2 ) and α is the absorptivity of the product. Rearranging Equation (2.11.1) results in: ms d X d(ρC p eT ) = α Q IR − Utotal (T − Tb ) + HV (T ). dt A dt

(2.11.2)

It should be highlighted that the application of the L-REA requires accurate determination of activation energy as a function of moisture content. Ev /Ev ,b because characteristics of drying kinetics are used to describe the reduction of moisture content and temperature profiles. In convective drying, the product is heated by relatively high drying air temperatures and the maximum activation energy Ev ,b is determined using Equation (2.1.7). However, a different condition occurs when drying is conducted using infrared heating because the product temperature increases to above the drying air temperature so the product releases heat to the air instead of gaining heat from the air. If one maintains constant infrared power and lets drying continue until low water content is reached, the product temperature would reach a constant temperature determined by the balance between infrared power input and heat loss to the surroundings. The minor modifications of the L-REA are taken here as explained next. The heat transfer coefficient should be determined from the final part of drying instead of using the initial constant rate period of drying because of the low evaporation rate at the final part of drying, so most of the heat adsorbed by the product from the infrared emitter is released to the air. In addition, at the final part of drying the product temperature is essentially constant as revealed by Allanic et al. (2009) indicating ‘thermal’ equilibrium has been reached. The heat balance for final part of drying can be written as: α Q IR =

ms d X HV (T ) + Utotal (T − Tb ). A dt

(2.11.3)

It is apparent that, for the final part of drying, the contribution of the first term on the right-hand side is low because of the low evaporation rate, thus the second term is dominant. Equation (2.11.3) can be simplified: α Q IR ≈ Utotal (T − Tb ),

(2.11.4)

and Utotal can be determined using Equation (2.11.3) by inserting T from the recorded final product temperature (Allanic et al., 2009). The predetermined Utotal is then used for modelling of moisture content and temperature profile. It is emphasised that Equation (2.11.4) only holds at this point. In addition, a new definition of maximum activation energy (Ev ,b ) is introduced because the product is not heated only by air so the definition of Ev ,b shown by Equation (2.1.7) is not appropriate. The relative activation energy generated from the convective drying run and shown in Equation (2.5.10) is used but Ev ,b has been determined from the final product temperature and corresponding humidity of air instead of using drying air temperature. This can be written: E v,b = −RT ln(R Hb ),

(2.11.5)

103


8 Model Data

7


6 5 4 3 2 1 0

0

500

1000

1500 t(s)

2000

2500

3000

Figure 2.59 Moisture content profile of convective and infrared drying at an air temperature of

35 °C, air velocity of 1 m s−1 , air relative humidity of 18% and intensity of infrared drying of 3700 W m−2 . [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2012), with permission from Elsevier.]

where T is the final product temperature (K) and RHb is the relative humidity at the final product temperature and the absolute humidity. For other drying conditions, as T is not known prior to drying experiments, T can be determined from heat balance between the infrared power input and heat loss to the surroundings as explained previously. In order to yield the moisture content and temperature profiles, the mass and heat balances shown in Equations (2.1.4) and (2.11.1), respectively are solved simultaneously. The equations are combined with the relative and equilibrium activation energy indicated in Equations (2.5.10) and (2.11.5), respectively. The results of modelling using the LREA are validated towards the experimental data of Allanic et al. (2009).

2.11.2

The results of mathematical modelling of infrared-heat drying of a mixture of polymer solutions using the L-REA The results of modelling are presented in Figures 2.59 and 2.60. It could be observed that the discrepancies between experimental and calculated data are reasonably small. Statistical analysis showed that the R 2 and RMSE of the moisture content profile are 0.994 and 0.181, respectively. Overestimation of the drying rate between drying times of 600–2250 s was also predicted by the previous model (Allanic et al., 2009). The L-REA seems to model this case better and only shows slight overestimation in drying

104


360 350

Temperature (K)

340 330 320 310 300 290

Model Data 0

500

1000

1500 t(s)

2000

2500

3000

Figure 2.60 Product temperature profile of convective and infrared drying at an air temperature of

35 °C, air velocity of 1 m s−1 , air relative humidity of 18% and intensity of infrared drying of 3700 W m−2 . [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright (2012), with permission from Elsevier.]

rates between drying times of 1350–2250 s. In addition, it is apparent that the REA can handle temperature profiles quite well as shown by the R 2 and RMSE of temperature profile, which are 0.992 and 1.712, respectively. Overestimation in the temperature profile of about 5 °C during drying times of 150–1200 s was indicated by the previous model (Allanic et al., 2009). However, this overestimation is not observed by modelling using the L-REA. It can be observed that Ev /Ev ,b derived from convective drying combined with the new quantification of Ev ,b shown in Equation (2.11.5) is appropriate for describing the drying kinetics of infrared-heat drying. It may be applied to other infrared-heating cases. In the case of drying, which exhibits product temperature higher than the drying air temperature, application of the modification of Ev ,b shown by Equation (2.11.5) in conjunction with the generated Ev /Ev ,b is shown to be appropriate.

2.12

The intermittent drying of a mixture of polymer solutions under time-varying infrared-heat intensity modelled using the L-REA The experimental data for validating the accuracy of the L-REA are derived from a study reported by Allanic et al. (2006, 2009), whose experimental conditions are briefly

105


Table 2.12 The experimental conditions of intermittent drying of a mixture of polymer solutions.

Case number



Air relative humidity (%)

Intensity of infrared radiation (W m2 )

1 (Allanic et al., 2009) 2 (Allanic et al., 2009) 3 (Allanic et al., 2006)

1 1 1

35 35 35

19 14 18

12.3–5.6-regulated* 13.3–5.6-regulated* 13.3–12.3-regulated*

reviewed in Section 2.11. The conditions of experiments are listed in Table 2.12, which indicates that different intensities of the infrared heater power were applied in each stage to induce a different condition for drying in the corresponding stage.

2.12.1

Mathematical modelling of the intermittent drying of a mixture of polymer solutions under time-varying infrared-heat intensity using the L-REA For modelling of this kind of cyclic drying, a new idea has to be introduced to define the ‘equilibrium’ activation energy (Ev,b ). This is necessary because each stage has different drying conditions. If each drying condition persists for a long time, a different terminal drying state should be reached, corresponding to a unique Ev,b . The equilibrium activation energy (Ev,b ) is defined as representing the maximum activation energy under conditions in each stage of drying. For this purpose, two new definitions of equilibrium activation energy are introduced. The first employs a relationship between the infrared intensity in each stage and T* : T ∗ = m I n + c,

(2.12.1)

where T* is the final product temperature in each stage should the infrared heating be prolonged to equilibrium (K), I is the infrared intensity employed in each stage (kW m−2 ), m and c are the empirical constants obtained from the linear relationship, T* and In , and n is a constant indicating sensitivity of T* towards the infrared intensity. Using this expression, Ev,b in each stage is determined from T* and relative humidity of air at corresponding T* and corresponding humidity. This can be written as: E v,b = −RT ∗ ln(R Hb ),

(2.12.2)

where RHb is the relative humidity of air at T* and the corresponding humidity. Alternatively, a second scheme could be used to relate directly the equilibrium activation energy (Ev,b ) and the infrared intensity. The relationship of the infrared intensity with the equilibrium activation energy (Ev,b ) in each stage expressed as: E v,b = p I q + k,

(2.12.3)

where p and k are the constants obtained from linear relationship between the equilibrium activation energy (Ev,b ) and Iq and q is the constant indicating sensitivity of the equilibrium activation energy (Ev,b ) towards infrared intensity. For modelling the cyclic drying here, both definitions of equilibrium activation energy (Ev,b ) need to be combined with the relative activation energy (Ev /Ev,b ) shown in Equation (2.5.10). The relative activation energy shown in Equation (2.5.10),

106


8 Data n = 1.4 n = 1.5 n = 1.6 n = 1.7 n = 1.8 n = 1.9

7


6 5 4 3 2 1 0

0

200

400

600 t(s)

800

1000

1200

Figure 2.61 Sensitivity of the moisture content profile of cyclic drying; Case 1 (refer to Table 2.12) towards n (on Equation 2.12.1). [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]

generated from convective drying of a mixture of polymer solution, can still be used for the modelling of cyclic drying here. The L-REA shown in Equation (2.1.4) serves as the mass balance while the heat balance shown in Equation (2.11.1) is used by employing the different heating intensity in each drying period. Solving Equations (2.1.4), (2.5.10) and (2.11.1) and the appropriate equilibrium energy function described previously results in the moisture content and temperature profiles.

2.12.2

Results of modelling the intermittent drying of a mixture of polymer solutions under time-varying infrared heat intensity using the L-REA For Case 1 of the cyclic drying process (refer to Table 2.12), both schemes were implemented. Using the first scheme (Equation 2.12.1) (T* as a function of the infrared intensity), several values of n shown in Equation (2.12.1) in the range of 1.4–1.9 were used to describe the drying kinetics (which we found to be more favourable than either lower or higher values). Figures 2.61 and 2.62 show the profiles of moisture content and temperature along the cyclic drying run with various values of n. It can be observed that different values of n did not provide a noticeable difference in both moisture content and temperature profiles during the constant rate period of drying. This indicates an overriding effect of the latent heat of vaporisation. However, different values of n that


107

380 370 360

Temperature (K)

350 340 330 Data n = 1.4 n = 1.5 n = 1.6 n = 1.7 n = 1.8 n = 1.9

320 310 300 290

0

200

400

600 t(s)

800

1000

1200

Figure 2.62 Sensitivity of the temperature profile of cyclic drying; Case 1 (refer to Table 2.12) towards n (on Equation 2.12.1). [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]

gave deviations in both profiles were observed after 300 s of the drying process. The lower values of n yielded lower moisture content and lower temperature profiles because these lower values result in lower activation energy which projects a higher evaporation rate and more heat is removed from materials being dried for evaporation, shifting the temperature profile down somewhat. From Figures 2.61 and 2.62, modelling by the value of n of 1.8 tends to agree well with the moisture content profile. Figure 2.63 shows the prediction of moisture content profile whilst Figure 2.64 shows the temperature profile using the L-REA and the first scheme (Equation 2.12.1) in conjunction with n = 1.8. In addition, the good fit of this model approach is shown by R 2 and RMSE of 0.996 and 0.13 for moisture content, respectively while R 2 and RMSE of temperature profile are 0.938 and 3.3, respectively. Compared with results of modelling of Allanic et al. (2009) results of the current model seem to describe moisture content profile better (the other model resulted in an underestimation of evaporation rate initially and the overestimation after a drying time of 600 s). Figures 2.65 to 2.68 indicate the results of modelling of Case 1 (refer to Table 2.12) by the second scheme shown in Equation 2.12.3 (Ev,b as function of the intensity of the infrared heating). Similarly, several values of q (Equation 2.12.3) were used to


8 Model Data

7 6 X (kg water/kg dry solid)

108

5 4 3 2 1 0

0

200

400

600 t(s)

800

1000

1200

Figure 2.63 Moisture content profile of cyclic drying; Case 1 (refer to Table 2.12) using the first

scheme (T* as function of infrared intensity) with n = 1.8. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]

describe both moisture content and temperature profiles. Figures 2.65 and 2.66 show that no deviations were observed during constant rate period of drying but different values of q gave different moisture content and temperature profile after drying time of 300 s. It can be examined that the value of q of 1.8 resulted in the best fit of moisture content profile. This can be clearly observed in Figure 2.67. A reasonable agreement of temperature prediction is also indicated by Figure 2.68. Higher value of q resulted in higher moisture content and temperature profile because of the higher activation energy which decreases evaporation rate and less heat is removed from materials so this can yield a higher temperature profile. The good fit using this second scheme by application of q of 1.8 was indicated by R 2 and RMSE of 0.995 and 0.14 for moisture content while R2 and RMSE of temperature are 0.937 and 3.3, respectively. Moreover, modelling using this second scheme yielded better prediction of moisture content profile than that from Allanic et al. (2009) similar to that described previously for the first scheme. Comparison of modelling using the first and the second scheme for Case 1 in the current study has revealed that both the proposed schemes gave almost the same profiles of moisture content and temperature to drying times of 300 s. After that time, although the values are not exactly the same, the differences between predicted moisture content using the first and the second schemes are below 0.15 kg kg−1 while those of temperature


109

380 370 360

Temperature (K)

350 340 330 320 310 Model Data

300 290

0

200

400

600 t(s)

800

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Figure 2.64 Temperature profile of cyclic drying; Case 1 (refer to Table 2.12) using the first


are below 1.5 °C. These indicate both schemes may have been successful in capturing the physics of the process. The results of modelling of cyclic drying of Case 2 (refer to Table 2.12) using the L-REA and the two schemes mentioned previously are represented in Figures 2.69 to 2.72. Similarly to Case 1, sensitivity of n (Equation 2.12.1) and q (Equation 2.12.3) was conducted. It was found that n = 1.5 and q = 1.5 were the most appropriate to describe the moisture content and temperature profile. A good agreement of both moisture content and temperature profile using the first scheme and n = 1.5 can be observed from Figures 2.69 and 2.70, respectively. Similarly to the previous case, the lower value of n produced lower moisture content and temperature profiles. This good fit is shown by R 2 and RMSE of moisture content of 0.992 and 0.2 while R 2 and RMSE of temperature are 0.946 and 4.4, respectively. When compared to the modelling published by Allanic et al. (2009), this model again seems to represent moisture content better. Results of the modelling in Case 2 (refer to Table 2.12) by implementing the second scheme (Equation 2.12.3) (Ev,b as function of intensity of infrared intensity) and q = 1.5 are shown in Figures 2.71 and 2.72. The good agreement is shown in Figures 2.71 and 2.72 for moisture content and temperature profile, respectively. In addition, this good fit was indicated by R 2 and RMSE of moisture content of 0.992 and 0.2 while R 2 and RMSE


8 Data q = 1.6 q = 1.7 q = 1.8 q = 1.9 q=2

7


6 5 4 3 2 1 0

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Figure 2.65 Sensitivity of the moisture content profile of cyclic drying; Case 1 (refer to Table 2.12) towards q (on Equation 2.12.3). [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]

380 370 360 350 Temperature (K)

110

340 330 Data q = 1.6 q = 1.7 q = 1.8 q = 1.9 q=2

320 310 300 290

0

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600 t(s)

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Figure 2.66 Sensitivity of the temperature profile of cyclic drying; Case 1 (refer to Table 2.12) towards q (on Equation 2.12.3). [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]

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8 Model

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600 t(s)

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Figure 2.67 Moisture content profile of cyclic drying; Case 1 (refer to Table 2.12) using the second scheme (Ev ,b as function of infrared intensity) with q = 1.8. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]

380 370

Temperature (K)

360 350 340 330 320 310 Model 300 290

Data 0

200

400

600 t(s)

800

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Figure 2.68 Temperature profile of cyclic drying; Case 1 (refer to Table 2.12) using the second

scheme (Ev,b as function of infrared intensity) with q = 1.8. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]


8 Model

7

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6 5 4 3 2 1 0

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scheme (T* as function of infrared intensity) with n = 1.5. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] 380 370 360

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113


8 Model

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Figure 2.71 Moisture content profile of cyclic drying; Case 2 (refer to Table 2.12) using the second scheme (Ev ,b as function of infrared intensity) with q = 1.5. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]

380 370 360

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Data 0

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300

400

500

600

700

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t(s) Figure 2.72 Temperature profile of cyclic drying; Case 2 (refer to Table 2.12) using the second

scheme (Ev ,b as function of infrared intensity) with q = 1.5. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]


8 Model

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of temperature are 0.933 and 4.9, respectively. This scheme also yields better moisture content prediction than that of Allanic et al. (2009), which shows overestimation of the drying rate after 200 s. Results of the modelling in Case 2 using both schemes were compared and showed the values of moisture content and temperature were almost the same for drying times of 300 s. Both schemes generate the same profiles of moisture content and temperature during the constant rate period of drying. In addition, after that time, negligible differences between predicted moisture content and temperature using both schemes can be found. The differences between moisture content and temperature were around 0.03 kg kg−1 and 1 °C, respectively. In addition, modelling of cyclic drying in Case 3 (refer to Table 2.12) was conducted using both schemes and the results are shown in Figures 2.73–2.76. Sensitivity was also conducted to find the most appropriate value of n (Equation 2.12.1) and q (Equation 2.12.3). For the first scheme, application of n of 1.6 matches the temperature profile very well. Figures 2.73 and 2.74 illustrate both moisture content and temperature profile using the value of n = 1.6. The good fit is also shown by R 2 and RMSE of moisture content of 0.989 and 0.24, while R 2 and RMSE of temperature are 0.964 and 3.55, respectively. The second scheme was also implemented to describe cyclic drying of Case 3 and the results are shown in Figures 2.75 and 2.76. The good agreement of this scheme using q of 1.6 was indicated in Figures 2.75 and 2.76. This is also supported by R 2 and RMSE of

115


380 370 360

Temperature (K)

350 340 330 320 310 Model 300 290

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scheme (T* as function of infrared intensity) with n = 1.6. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.] 8 Model

7

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6 5 4 3 2 1 0

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300 t(s)

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Figure 2.75 Moisture content profile of cyclic drying; Case 3 (refer to Table 2.12) using the second scheme (Ev,b as function of infrared intensity) with q = 1.6. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]

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380 370 360

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350 340 330 320 310 Model 300 290

Data 0

100

200

300 t(s)

400

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600

Figure 2.76 Temperature profile of cyclic drying; Case 3 (refer to Table 2.12), using the second

scheme (Ev,b as function of infrared intensity) with q = 1.6. [Reprinted from Chemical Engineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley, Application of the reaction engineering approach (REA) to model cyclic drying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012), with permission from Elsevier.]

moisture content of 0.991 and 0.23 as well as R 2 and RMSE of temperature of 0.942 and 4.52. The differences of moisture content and temperature profile predictions between the two schemes are around 0.03 kg kg−1 and 1 °C. Therefore, the two new formulations of Ev,b combined with Ev /Ev,b shown in Equation (2.5.10) can predict the moisture content and temperature profiles well. Comparison in modelling of the different cases of cyclic drying conditions under use of the REA framework has shown that application of the first and second schemes with the appropriate values of n or q give almost exactly the same moisture content and temperature profiles. All this work supports the notion that L-REA is a robust modelling framework that allows for intuitive extensions, such as the schemes proposed here, for more complex situations. The modelling itself remains fairly simple and effective.

2.13

Summary In this chapter, the applications of the L-REA as an accurate approach for modelling several drying cases are described. Within the range that material properties are invariant, the REA parameters, expressed in relative activation energy, can be applied to model other drying runs of the same materials, provided there is similar initial moisture content. Besides milk-droplet drying, the L-REA has been innovatively applied and shown to accurately model the average moisture content and temperature during convective drying


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of food and non-food materials, convective drying of several centimetre-thick materials, intermittent drying, heating of wood under linearly increased gas temperatures and baking of cake. For modelling the discussed cases, the original formulation of the L-REA can be implemented without major modification. The estimation of temperature distribution inside the samples needs to be combined with the L-REA formulation to describe drying of relatively thick materials. The L-REA can be used to model intermittent drying by evaluating the equilibrium activation energy according to the corresponding humidity and temperature in each period of drying. Similarly, by evaluating the equilibrium activation energy according to the drying settings, the L-REA can accurately model the heat treatment of wood under a constant heating rate. For baking, the original formulation of the L-REA can be implemented without any modifications. The accuracy of the L-REA in modelling several cases of drying could be due to the accuracy of the relative activation energy in capturing the physics of drying. A combination of the relative activation energy and equilibrium activation energy yields unique relationships of activation energy which can change flexibly according to the external drying conditions and represents the change in internal behaviour of the samples during drying. While the L-REA formulation is simple and the REA is efficient, the results are accurate. It can be used in industrial settings for process design and maintaining product quality during drying.

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Lin, S.X.Q., 2004. Drying of Single Milk Droplets. Ph.D. thesis, Department of Chemical and Materials Engineering, The University of Auckland, New Zealand. Lin, S.X.Q. and Chen, X.D., 2002. Improving the glass-filament method for accurate measurement of drying kinetics of liquid droplets. Trans IChemE Part A: Chemical Engineering Research and Design 80, 401–440. Lin, S.X.Q. and Chen, X.D., 2005. Prediction of air drying of milk droplets under relatively high humidity using the reaction engineering approach, Drying Technology 23, 1396–1406. Lin, S.X.Q. and Chen, X.D., 2006. A model for drying of an aqueous lactose droplet using the reaction engineering approach. Drying Technology 24, 1329–1334. Lin, S.X.Q. and Chen, X.D., 2007. The reaction engineering approach to modelling the cream and whey protein concentrate droplet drying. Chemical Engineering and Processing 46, 437– 443. Lostie, M., Peczalski, R., Andrieu, J. and Laurent, M., 2002a. Study of sponge cake batter baking process. I: Experimental data. Journal of Food Engineering 51, 131–137. Lostie, M., Peczalski, R., Andrieu, J. and Laurent, M., 2002b. Study of sponge cake batter baking process. II. Modelling and parameter estimation. Journal of Food Engineering 55, 349– 357. Mariani, V.C., de Lima, A.G.B. and Coelho, L.S., 2008. Apparent thermal diffusivity estimation of the banana during drying using inverse method. Journal of Food Engineering 85, 569–579. Mayor, L. and Sereno, A. M., 2004. Modelling shrinkage during convective drying of food materials: A review. Journal of Food Engineering 61, 373–386. McCabe, W.L., Smith, J.C. and Harriott, P., 2001. Unit Operations of Chemical Engineering, 6th ed. McGraw Hill, Boston. Microsoft Corp., 2012. http://office.microsoft.com/en-au/excel/ (accessed 21 November, 2012). Pakowski, Z. and Adamski, A., 2007. The comparison of two models of convective drying of shrinking materials using apple tissue as an example. Drying Technology 25, 1139–1147. Pang, S., 1994. High-Temperature Drying of Pinus Radiata Boards in a Batch Kiln. Ph.D. thesis, University of Canterbury, New Zealand. Patel, K.C. and Chen, X.D., 2008. Surface-center temperature differences within milk droplets during convective drying and drying-based Biot number analysis. AIChE Journal 54, 3273– 3290. Patel, K., Chen, X.D., Jeantet, R. and Schuck, P., 2010. One-dimensional simulation of co-current, dairy spray drying systems – pros and cons. Dairy Science and Technology 90, 181–210. Putranto, A., Chen, X.D. and Webley, P.A., 2010a. Infrared and Convective Drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA). Chemical Engineering and Processing: Process Intensification 49, 348–357. Putranto, A., Chen, X.D. and Webley, P.A., 2010b. Application of the reaction engineering approach (REA) to model cyclic drying of polyvinyl alcohol (PVA)/glycerol/water mixture. Chemical Engineering Science 65, 5193–5203. Putranto, A., Chen, X.D. and Webley, P.A., 2011a. Modelling of drying of thick samples of mango and apple tissues using the reaction engineering approach (REA). Drying Technology 29, 961–973. Putranto, A, Xiao, Z., Chen, X.D. and Webley, P.A., 2011b. Intermittent drying of mango tissues: implementation of the reaction engineering approach (REA). Industrial Engineering Chemistry Research 50, 1089–1098. Putranto, A., Chen, X.D., Devahastin, S., Xiao, Z. and Webley, P.A., 2011c. Application of the reaction engineering approach (REA) to model intermittent drying under time-varying humidity and temperature. Chemical Engineering Science 66, 2149–2156.

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Putranto, A., Chen, X.D., Xiao, Z. and Webley, P.A., 2011d. Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA). Bioresource Technology 102, 6214–6220. Putranto, A., Chen, X.D. and Zhou, W., 2011e. Modelling of baking of cake using the reaction engineering approach (REA). Journal of Food Engineering 105, 306–311. Putranto, A., Xiao, Z., Chen, X.D. and Webley, P.A., 2011f. Intermittent drying of mango tissues: Implementation of the reaction engineering approach. Industrial Engineering Chemistry Research 50, 1089–1098. Radziemska, E. and Lewandowski, W.M., 2008. Experimental verification of natural convective heat transfer phenomenon from isothermal cuboids. Experimental Thermal and Fluid Science 32, 1034–1038. Ramos, I.N., Miranda, J.M.R., Brandao, T.R.S. and Silva, C.L.M. 2010. Estimation of water diffusivity parameters on grape dynamic drying. Journal of Food Engineering 97, 519–525. Rapp, A.O., 2001. Review of heat treatment of wood. Proceedings of Antibes Seminar, France. Ruiz-Lopez, I.I.R., Ruiz-Espinosa, H., Luna-Guevara, M.L. and Garc´ıa-Alvarado, M.A., 2011. Modelling and simulation of heat and mass transfer during drying of solids with hemispherical shell geometry. Computers and Chemical Engineering 35, 191–199. Sakin, M., Kaymak-Ertekin, F. and Ilicali, C., 2007. Modelling the moisture transfer during baking of white cake. Journal of Food Engineering 80, 822–831. Sanjuan, N., Lozano, M., Garcia-Pascual, P. and Mulet, A., 2004. Dehydration kinetics of red pepper (Capsicum annuum L var Jaranda). Journal of the Science of Food and Agriculture 83, 697–701. Silva, W.P., Precker, J.W., Silva, C.M.P.D.S. and Gomes, J.P., 2010. Determination of effective diffusivity and convective mass transfer coefficient for cylindrical solids via analytical solution and inverse method: Application to the drying of rough rice. Journal of Food Engineering 98, 302–308. Sun, L.M. and Meunier, F., 1987. A detailed model for non-isothermal sorption in porous adsorbents. Chemical Engineering Science 42, 1585–1593. Surmin-Kaolin, 2010. Company website. (Available at http://www.quarzwerke.com/surmin,/ accessed 21 November, 2012.) Van der Sman, R.G.M., 2003 Simple model for estimating heat and mass transfer in regular-shaped foods. Journal of Food Engineering 60, 383–390. Vaquiro, H.A., Clemente, G., Garcia Perez, J.V., Mulet, A. and Bon, J., 2009. Enthalpy driven optimization of intermittent drying of Mangifera indica L. Chemical Engineering Research and Design 87, 885–898. Younsi, R., Kocaefe, D., Poncsak, S. and Kocaefe, Y., 2006a. Transient multiphase model for the high-temperature thermal treatment of wood. AIChE Journal 52, 2340–2349. Younsi, R., Kocaefe, D., Poncsak, S. and Kocaefe, Y., 2006b. Thermal modelling of the high temperature treatment of wood based on Luikov’s approach. International Journal of Energy Research 30, 699–711. Younsi, R., Kocaefe, D., Poncsak, S. and Kocaefe, Y., 2007. Computational modelling of heat and mass transfer during the high-temperature heat treatment of wood. Applied Thermal Engineering 27, 1424–1431. Younsi, R., Kocaefe, D., Poncsak, S. and Kocaefe, Y., 2010. Computational and experimental analysis of high temperature thermal treatment of wood based on ThermoWood technology. International Communications in Heat and Mass Transfer 37, 21–28.

3

Reaction engineering approach II Spatial-REA (S-REA)

3.1

The S-REA formulation In order to capture detailed information about the distributions of moisture content and temperature throughout the material being dried, the S-REA has been developed. S-REA is a non-equilibrium multiphase drying approach in which the REA is implemented to represent the local phase change term. It is envisaged that, for a better understanding of the transport phenomena, application of effective liquid diffusion alone without source terms in both energy and mass conservation equations may not be sufficient as it cannot represent the water vapour concentration during drying. This could be affected by gas in the pore structure (Chen, 2007). Also, vapour generation and transfer may affect other volatile transport in the same material. Traditionally, effective liquid diffusion has been used to simulate the detailed profiles of temperature and moisture content. This will be discussed further in Chapter 4. Equilibrium and non-equilibrium approaches can be implemented in the multiphase drying model mentioned previously (Zhang and Datta, 2004; Datta, 2007). By applying the equilibrium approach, it is assumed that vapour pressure inside the pores of the samples equilibrates with the liquid moisture content inside the same pores and the relationship can be described by the relevant equilibrium isotherm (Zhang and Datta, 2004). The equilibrium model has been applied to the baking process and a good agreement with experimental data has been shown (Ni et al., 1999; Zhang et al., 2005; Zhang and Datta, 2006). The equilibrium model has been implemented by combining the mass conservation of water in both liquid and vapour phases, which effectively resulted in the elimination of the source term. The moisture content and water vapour concentration are related by the available isotherm data (through a correlation equation) for the materials. Reasonable agreement with experimental data was shown in the cases investigated (Ni et al., 1999; Zhang et al., 2005; Zhang and Datta, 2006). Similarly, the equilibrium model was applied by Aversa et al. (2010) to model the convective drying of food materials. The model has been shown to represent the experimental data reasonably well. Nevertheless, it has not been proven that use of equilibrium approach is valid in the case of heating hygroscopic materials (Zhang and Datta, 2004). For a more generic application of the multiphase drying model, it is suggested that the non-equilibrium approach is more appropriate. The good non-equilibrium multiphase drying model is also useful in determining the appropriateness of the equilibrium approach for the situation of concern (Zhang and Datta, 2004). In order to implement

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z

y

x Figure 3.1 Schematic diagram of a cube dried in a uniform convective environment.

the model, it is necessary to represent the internal evaporation rate explicitly and appropriately. The internal evaporation/condensation rate is implemented in the multiphase drying model as a depletion term for the liquid phase and as a source term for the vapour phase. It has been proposed that the internal evaporation/wetting rate can be related to the difference of equilibrium vapour pressure and the vapour pressure at a particular time inside the pore spaces (Chong and Chen, 1999; Scarpa and Milano, 2002; Zhang and Datta, 2004). In other words, gas must be present in the pores to permit this process to occur. The REA, in its lumped format, has been proven to model the global drying rate of various challenging drying cases accurately (Chen and Lin, 2005; Chen, Pirini and Ozilgen, 2001; Chen and Xie, 1997; Lin and Chen, 2005; 2006; 2007; Putranto et al., 2010a,b, 2011a–e). It was expected that formulation of the L-REA may also be applicable in representing the source term in the S-REA approach; for instance, the same activation energy profile can be reserved for the same material. The S-REA consists of a mass balance of liquid water, mass balance of water vapour and heat balance. For uniform convective drying of cubic object in a heated environment, three-dimensional modelling can be established. The mass balance of water in the liquid phase (liquid water) is written as (refer to Figure 3.1 and Chong and Chen, 1999; Chen, 2007; Kar and Chen, 2011; Putranto and Chen, 2013; Zhang and Datta, 2004): ∂(C s X ) ∂(Cs X ) ∂(C s X ) ∂ ∂ ∂ ∂(C s X ) = Dw + Dw + Dw − I˙, ∂t ∂x ∂x ∂y ∂y ∂z ∂z (3.1.1) where X is the concentration of liquid water (kg H2 O kg dry solids−1 ), Cs is the solid’s concentration (kg dry solids m−3 ), which can change if the structure is shrinking, Dw is liquid diffusivity (m2 s−1 ), I˙ is the evaporation or condensation rate (kg H2 O m−3 .s−1 ) and I˙ is usually defined as positive when evaporation occurs locally. The liquid diffusivity represents the movement of liquid water inside the pore structure of the materials due to capillary action as a result of the water concentration gradient. In practice, the liquid diffusivity needs to be extracted from the available effective diffusivity data (Datta, 2007).

Reaction engineering approach II: S-REA

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The mass balance of water vapour is expressed as (Chen, 2007; Chong and Chen, 1999; Kar and Chen, 2011; Putranto and Chen, 2013): ∂Cv ∂Cv ∂Cv ∂ ∂ ∂ ∂Cv = Dv + Dv + Dv + I˙, (3.1.2) ∂t ∂x ∂x ∂y ∂y ∂z ∂z where Cv is the concentration of water vapour (kg m−3 ) and Dv is the effective water vapour diffusivity in pore channels (m2 s−1 ). The heat balance is represented by the following equation (Chen, 2007; Chong and Chen, 1999; Kar and Chen, 2011; Putranto and Chen, 2013): ∂T ∂ ∂T ∂ ∂T ∂ ∂T ρC p (3.1.3) = k + k + k − I˙Hv , ∂t ∂x ∂x ∂y ∂y ∂z ∂z where T is the sample temperature (K), HV is the vaporisation heat of water (J kg−1 ), k is the sample thermal conductivity (W m−2 K−1 ), ρ is the sample density (kg m−3 ) and k and ρ may be functions of temperature and moisture content. For cubic objects being dried as an example, the initial and boundary conditions for Equations (3.1.1) to (3.1.3) may be written as: t = 0, X = X o , C v = Cvo , T = To (initial condition, uniform initial concentrations and temperature), (3.1.4) dX dX dX = 0, = 0, = 0 (symmetrical boundary), dx dy dz dCv dC v dC v = 0, =0 = 0 (symmetrical boundary), dx dy dz dT dT dT = 0, = 0, = 0 (symmetrical boundary), dx dy dz Cv,s dX x = L , −Cs Dw = h m εw − ρv,b (convective boundary dx ε for liquid transfer),

x = 0, y = 0, z = 0,

dCv −Dv = h m εv dx

Cv,s − ρv,b ε

dCv = h m εv dy

Cv,s − ρv,b ε

(3.1.7)

(3.1.8)

(convective boundary

dT = h(Tb − T ) (convective boundary for heat transfer), dx Cv,s dX y = L , −Cs Dw = h m εw − ρv,b (convective boundary dy ε for liquid transfer), −Dv

(3.1.6)

for vapour transfer),

k

(3.1.5)

(3.1.9) (3.1.10)

(3.1.11)

(convective boundary for vapour transfer), (3.1.12)

124


dT = h(Tb − T ) (convective boundary for heat transfer), (3.1.13) dy Cv,s dX = h m εw − ρv,b (convective boundary z = L , −Cs Dw dz ε for liquid transfer), (3.1.14) k

dCv = h m εv −Dv dz

Cv,s − ρv,b ε

(convective boundary for vapour, transfer)

k

dT = h(Tb − T ) (convective boundary for heat transfer), dz

(3.1.15) (3.1.16)

where εw and εv are the fractions of surface area covered by liquid water and water vapour, respectively. The internal evaporation rate ( I˙) can be described as: I˙ = h m in Ain (Cv,s − C v ),

(3.1.17)

where Ain is the internal surface area per unit volume available for phase change (m2 m−3 ) and hm,in is the internal surface mass transfer coefficient (m s−1 ). By implementing the REA, internal-surface water vapour concentration can be written as (Kar and Chen, 2010; 2011; Putranto and Chen, 2013): −E v (3.1.18) C v,sat , Cv,s = exp RT where Cv,s is the internal-solid surface water vapour concentration (kg m−3 ), Cv,sat is the internal saturated water vapour concentration (kg m−3 ) and Ev is the activation energy (J mol−1 ) similar to the one described in Equation (2.1.4). Therefore, the internal evaporation rate can be expressed as (Kar and Chen, 2010; 2011; Putranto and Chen, 2013): ˙I = h m in Ain exp −E v Cv,sat − C v . (3.1.19) RT In Equation (3.1.19), the REA is used to describe the local evaporation rate as affected by pore structure (porosity, shrinkage, local moisture content and local temperature). These microstructural effects can be ‘encapsulated’ in the term hm,in Ain . In particular, Ain is clearly influenced by structural or microstructural of the material of concern. It is interesting to note that when Ain is zero (i.e. for a ‘non-porous’ or ‘voidless’ material), I˙ becomes zero. Then, the liquid transfer (may be a kind of diffusion form) is predominant. In this case, for soft materials such as polymeric and biological entities, the free volume concept may be used to predict the effective liquid diffusivity (Vrentas and Duda, 1977; Van der Sman, 2007a,b; Van der Sman et al., 2012). Free Volume Flory–Huggins (FVFH) theory, an extension of the classical Flory–Huggins theory, can be used to describe the thermodynamics of food materials. The chemical potential of moisture transfer can be assumed due to osmotic, elastic and ionic contributions.


125

The Flory–Huggins theory can also be implemented to predict the mutual diffusivity (Dm ). By using Darken’s relation, the mutual diffusivity can be expressed as (Van der Sman et al., 2012):

Dm = Q φ Ds,s + (1 − φ)Ds,w , (3.1.20) where φ is the volume fraction of polymer, χ is the Flory–Huggins interaction parameter, Ds,s and Ds,w are the self-diffusivity of polymer and water, respectively, and Q is described using Flory–Huggins theory as (Van der Sman et al., 2012): Q = 1 − 2χ φ(1 − φ).

(3.1.21)

The Stokes–Einstein relation can be used to predict Ds,s given by (Van der Sman et al., 2012): ηw , (3.1.22) Ds,s = Ds,o ηeff where Ds,o is the polymer self-diffusivity at infinite dilution, ηw is the viscosity of water and ηeff is the viscosity of the polymer solution. Ds,w can be predicted by Vrentas and Duda’s free volume theory (Vrentas and Duda, 1977), which can be written as: ln

E y w Vw∗ + ς ys Vs• Ds,w =− − , Dw,o RT yw K ww (K sw − Tg,w + T ) + ys K ws (K ss − Tg,s − T ) (3.1.23)

where E is the activation energy, Kij is free volume parameter(s), Vi * is parameters related to the volume of the molecule and ζ is the shape factor. The mutual diffusivity (Dm ) may then be used as effective liquid diffusivity (Van der Sman, 2012).

3.2

Determination of the S-REA parameters In S-REA, there are several parameters involved; i.e. effective vapour diffusivity (Dv ), tortuosity (τ ), porosity (ε), solid concentration (Cs ), capillary diffusivity (Dw ), internal mass transfer coefficient (hm,in ) and internal surface area per unit volume (Ain ). The procedures used to determine these parameters are explained in this section. The effective vapour diffusivity is deduced from (Bird et al., 2002): ε (3.2.1) Dv = Dvo , τ while Dvo is the water vapour diffusivity (m2 s−1 ), which is dependent on temperature. For food and biological materials, Dv can be expressed as (Slattery and Bird, 1958): Dvo = 2.09 × 10−5 + 2.137 × 10−7 (T − 273.15).

(3.2.2)

The tortuosity (τ ) of the samples is generally related to the porosity. The relationship can be represented as (Audu and Geffreys, 1975; Gimmi et al., 1993): τ = ε−n ,

(3.2.3)

126


where n is the value between 0 and 0.5 (Audu and Geffreys, 1975; Gimmi et al., 1993). A more refined approach would be to somehow incorporate the information on microstructure in mathematical terms in order to evaluate τ and ε. Cs is the solid concentration (kg m−3 ) which can be expressed by (Kar, 2008; Kar and Chen, 2010; 2011): Cs =

1−ε , + ρXw

1 ρs

(3.2.4)

while ε is the porosity which is dependent on shrinkage and local moisture content. This can be determined according to (Madiouli et al., 2007): ρs X +1 V0 ρw ε = 1 − (1 − ε0 ) . (3.2.5) V 1 + ρρws X 0 Until now, there has been no method to measure ‘effective liquid diffusivity’ (Chen, 2007). Many drying research papers estimated the effective liquid diffusivity based on drying kinetics data. Several sets of drying followed by complex optimization procedures are used to generate the effective diffusivity function (Azzouz et al., 2002; Mariani et al., 2008; Pakowski and Adamski, 2007; Thuwapanichayanan et al., 2008; Vaquiro et al., 2009). The literature on effective diffusivity may be used as a basis to determine the effective liquid diffusivity to be used in the S-REA. A little adjustment is required to the effective liquid diffusivity to generate the effective liquid diffusivity function, since the effective liquid diffusivity in these existing literatures is used to represent the whole phenomenon in drying (liquid diffusion, vapour diffusion, Darcy flow, evaporation/condensation). For example, Srikiatden and Roberts (2008) reported the effective liquid diffusivity of potato tissues as: 25.77 × 103 . (3.2.6) Deff = 1.0418 × 10−5 exp − 8.314T This was uniquely generated through well-controlled isothermal drying experiments carefully arranged so the temperature dependence is correlated against the sample temperature. The real liquid diffusivity is, however, expected to be smaller than the effective liquid diffusivity shown in Equation (3.2.6) but the temperature dependence is expected to remain valid. Equation (3.2.6) can still be used as the basis but is altered as: 25.77 × 103 Dw = Dw0 exp − . (3.2.7) 8.314T For convective drying of potato tissues, it was found that Dwo of 6.5 × 10−6 m2 s−1 gives the best agreement with experimental data (Putranto and Chen, 2013). The internal mass transfer coefficient (hm,in ) shown in Equation (3.1.17) is associated with the pore surfaces (porous media) or surfaces of the particles (packed beds), and internal to the sample being dried. Initially, moisture is present in the void spaces of pores and within the pores. As drying proceeds, the moisture may migrate within the pores (on the pore surfaces) by liquid (surface) diffusion and from the surfaces of the


127

pores through evaporation (Chong and Chen, 1999; Kar and Chen, 2010; 2011). Even at low water content, ‘surface’ diffusion of liquid could occur along the pore surface accessible to air (Chen and Mujumdar, 2008). The internal mass transfer coefficient shown in Equation (3.1.17) incorporates the restriction factor as it may be affected by the pore structure and pore network inside the samples. This makes the value of hm,in grow from a small value to the value of Dv /rp (when the constriction factor = 1) (Kar and Chen, 2010; 2011). The internal surface area per unit volume (Ain ) can be calculated using the procedures described in Kar and Chen (2010; 2011). It is calculated based on the area of a single cell and the number of cells per unit volume, dependent on solid mass of the samples and mass of single cell. Of course, Ain should also be affected by moisture content. By using the REA, the relative activation energy (Ev /Ev,b ) generated from one accurate drying run is used to describe the local evaporation rate shown in Equation (3.1.19). The activation energy (Ev ) and equilibrium activation energy (Ev,b ) are calculated using Equations (2.1.5) and (2.1.7), respectively to yield the relative activation energy shown in Equation (2.1.6). Since in the S-REA the relative activation energy is used to represent the local evaporation rate, the average moisture content (X ) in the relative activation energy is replaced by the local moisture content (X). The spatial profiles of moisture content, concentration of water vapour and temperature are generated by solving a set of equations shown in Equations (3.1.1)–(3.1.3) simultaneously in conjunction with the initial and boundary conditions indicated in Equations (3.1.4)– (3.1.16).

3.3

The S-REA for convective drying The validity of the S-REA in modelling convective drying is benchmarked against the experimental data of mango tissues (Vaquiro et al., 2009) and potato tissues (Srikiatden and Roberts, 2008). The experimental data from drying of mango tissues are derived from the previous study (Vaquiro et al., 2009). For better understanding of the predictions, the necessary experimental details are summarised and reviewed in this section. The samples of mango tissues were formed as cubes with initial side lengths of 2.5 cm while the initial moisture content and temperature were 9.3 kg kg−1 and 10.8 ºC, respectively. The laboratory drier was described in Sanjuan et al. (2004). During drying the weight change of the sample and the centre temperature history were recorded. The drying air temperature and air velocity were controlled at preset values by PID control algorithms while air humidity was maintained constant during drying. The experimental setting for convective drying is shown in Table 3.1. The density, thermal conductivity, heat capacity, equilibrium moisture content and shrinkage of the samples are presented in previous publication (Putranto et al., 2011a). For convective drying of potato tissues, the experimental data were taken from the previous work (Srikiatden and Roberts, 2008). Their experimental details are also reviewed here for better understanding of the modelling approach (Roberts et al., 2002; Srikiatden and Roberts, 2006; 2008). The cylindrical samples of Russet potatoes with diameters

128


Table 3.1 Experimental conditions of convective drying of mango tissues (Vaquiro et al., 2009).

Number




1 2 3

4 4 4

45 55 65

0.0134 0.0134 0.0134

of 1.4 and 2.8 cm were obtained using cylindrical cutters. The samples were sealed at their top and bottom ends with epoxy to establish approximately a one-dimensional (radial direction) moisture transfer. The experiments were conducted in a laboratory convective dryer with a drying air temperature of 70 °C and axial velocity of 1.5 m s−1 . The experimental setup can be found in Srikiatden and Roberts (2006). The fan at the bottom of the sample draws air downward and this reduces the turbulence effect near the sample as the air moves downwards (Roberts et al., 2002). The samples with the diameter of 1.4 cm were cut into two concentric parts for measurement of moisture content distribution, i.e. core and cortex (the core is a cylinder with the radius of 0.35 cm derived from the inner part of the potato tissues, while the cortex is a concentric shell derived from the outer part of the potato tissues). For the samples with the diameter of 2.8 cm, the samples were cut into four concentric parts; i.e. core, cortex 1, cortex 2 and cortex 3. Similarly to the samples with the diameter of 1.4 cm, the core is a cylinder with a radius of 0.35 cm derived from the innermost part of the potato tissues. The procedures were repeated for a number of intervals (Srikiatden and Roberts, 2006).

3.3.1

Mathematical modelling of convective drying of mango tissues using the S-REA Based on the experiments reported by Vaquiro et al. (2009) which have been used to help establish the REA, the samples were dried from three directions (x, y and z directions) so three-dimensional modelling of the S-REA for convective and intermittent drying of mango tissues needs to be set up, which is presented next. The mass balance of water in the liquid phase (liquid water), the mass balance of water in the vapour phase (water vapour) and the heat balance are shown in Equations (3.1.1), (3.1.2) and (3.1.3), respectively, while the initial and boundary conditions for equations are shown in Equations (3.1.4)–(3.1.16). Since the sample dried was a cube shape dried uniformly from all directions (x, y and z directions) (Sanjuan et al., 2004; Vaquiro et al., 2009), the mass balance of water in liquid phase can be simplified into (Incropera and DeWitt, 2002; Van der Sman, 2003): ∂(Cs X ) ∂(C s X ) ∂ =3 Dw − I˙, (3.3.1) ∂t ∂x ∂x


while the mass balance of water in vapour phase can be expressed as: ∂ ∂Cv ∂C v =3 Dv + I˙. ∂t ∂x ∂x In addition, the heat balance can be represented as: ∂T ∂T ∂ ρC p =3 k − I˙Hv . ∂t ∂x ∂x

129

(3.3.2)

(3.3.3)

The internal-surface water vapour concentration (Cv,s ) and internal evaporation rate ( I˙) are evaluated using Equations (3.1.18) and (3.1.19). The relative activation energy of convective drying of mango tissues is generated from one accurate drying run of convective drying mango tissues under constant environment conditions with a drying air temperature of 55 °C (Vaquiro et al., 2009). The activation energy during drying is evaluated using Equation (2.1.5) and divided with the equilibrium activation energy represented in Equation (2.1.7) to yield the relative activation energy as mentioned in Equation (2.1.6). The relationship between the relative activation energy and average moisture content can be represented by a simple mathematical equation obtained by use of the least-squares method using Microsoft Excel (Microsoft Inc., 2012). The relative activation energy can be represented as: E v = −9.92 × 10−4 (X − X b )3 + 9.74 × 10−3 (X − X b )2 E v,b −0.101(X − X b ) + 1.053.

(3.3.4)

The good agreement between the fitted and experimental relative activation energy is shown by R 2 of 0.999. Although Equation (3.3.4) involves Xb as mentioned earlier, all successful applications of REA so far suggest that the experiments carried out should dry the materials to Xb s of very small values in order to allow the correlations such as Equation (3.3–15) to cover the widest range of water content of practical interest. If Xb is close to initial water content, the activation energy calculated from the laboratory data can be misleading. The relative activation energy correlated with Equation (3.3.4) has been implemented to model the convective and intermittent drying of mango tissues using the L-REA and the results of modelling already matched well with experimental data (Putranto et al., 2011a,b). For modelling using the S-REA here, the relative activation energy shown in Equation (3.3.4) is used but the average moisture content X in Equation (3.3.4) is substituted by the local moisture content (X) as the REA is used to represent the local evaporation rate instead of the overall drying rate of the whole sample. In addition, it is emphasised that, for the S-REA, the equilibrium relative activation energy (Ev,b ) is evaluated at corresponding humidity and temperature inside the pores of the samples under equilibrium condition. The effective vapour diffusivity (Dv ), tortuosity (τ ), solid concentration (Cs ) and porosity (ε) are deduced using Equations (3.2.1)–(3.2.5). Similarly, the internal mass transfer coefficient (hm,in ) is evaluated using the procedures described in Section 3.2.

130


The effective diffusivity of mango tissues presented by Vaquiro et al. (2009) is expressed as:

−1.885×10−2 X 38.924 × 103 −3 . (3.3.5) Deff = 2.933 × 10 exp − 8.314T X +1 Equation (3.3.5) can be used as an approximation to determine the liquid water diffusivity of mango tissues but a little adjustment of the constant is needed in order to match the prediction with the experimental data of moisture content and temperature. The liquid water diffusivity used in this study can be expressed as:

−1.885×10−2 X 31.924 × 103 −3 . (3.3.6) Dw = 2.933 × 10 exp − 8.314T X +1 In order to yield the spatial profiles of moisture content, water vapour concentration and temperature of the convective of mango tissues, the mass and heat balances shown in Equations (3.3.1)–(3.3.3) in conjunction with the initial and boundary conditions represented in Equations (3.1.4)–(3.1.16) and the relative activation energy shown by Equation (3.3.4) are solved by the method of lines (Chapra, 2006; Constatinides, 1999). By this method, the partial differential equations are transformed into a set of ordinary differential equations with respect to time by firstly discretising the spatial derivatives. The ordinary differential equations are then solved simultaneously by ode23s in Matlab (Mathworks Inc., 2012). The spatial derivative here is discretised into 10 increments; application of 200 increments has been conducted and there is no real difference in the profiles observed as shown in Figure 3.2. The shrinkage (Putranto et al., 2011a) is incorporated in the modelling by a moving mesh in which the number of intervals is kept constant but the intervals of each increment are allowed to change according to the shrinkage relationship. The moving mesh was found to give better agreement with experimental data than a fixed coordinate (immobilising boundary) (Thuwapanichayanan et al., 2008). The average moisture content of mango tissues during convective drying is evaluated by: L(t)

X=

X (x)d x

0 L(t)

.

(3.3.7)

dx

0

The profiles of average moisture content and centre temperature are then validated against the experimental data of Vaquiro et al. (2009).

3.3.2

Mathematical modelling of convective drying of potato tissues using the S-REA In the experiments reported by Srikiatden and Roberts (2008) which are of interest, the samples were covered at both the top and bottom end to promote the one-dimensional

131


10 S-REA-10 increments S-REA-200 increments Data 45°C


9 8 7 6 5 4 3 2 1 0

0

0.5

1

1.5

2

2.5

3

3.5

t(s)

4

4.5

5 × 104

Figure 3.2 Moisture content profiles of the convective drying of mango tissues at a drying air

temperature of 45 °C solved by the method of lines with 10 and 200 spatial increments. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]

drying condition with respect to radial direction (Srikiatden and Roberts, 2008) so onedimensional modelling (at radial directions) of the S-REA of the convective drying of cylindrical potato tissues is possible and can be represented by a set of equations of conservation next. The mass balance of liquid water can be represented as (Chen, 2007; Chong and Chen, 1999; Kar and Chen, 2011; Zhang and Datta, 2004): 1 ∂ ∂(Cs X ) ∂(Cs X ) = Dw r − I˙, (3.3.8) ∂t r ∂r ∂r where X is the concentration of liquid water (kg H2 O kg dry solids−1 ) and Cs is the solids concentration (kg dry solids m−3 ), I˙ is the evaporation or wetting rate (kg H2 O m−3 s−1 ) and I˙ is >0 when evaporation occurs locally. The mass balance of water vapour can be expressed as (Chen, 2007; Chong and Chen, 1999; Kar and Chen, 2011; Zhang and Datta, 2004): 1 ∂ ∂Cv ∂Cv = Dv r + I˙, (3.3.9) ∂t r ∂r ∂r where Cv is the concentration of water vapour (kg H2 O m−3 ).

132


In addition, the heat balance can be written as (Chen, 2007; Chong and Chen, 1999; Kar and Chen, 2011; Zhang and Datta, 2004): ∂T 1 ∂ ∂T = kr − I˙Hv , ρC p (3.3.10) ∂t r ∂r ∂r where T is the sample temperature (K). The initial and boundary conditions of Equations (3.3.8)–(3.3.10) are: t = 0, X = X o , Cv = Cvo , T = To (initial condition, uniform initial concentrations and temperature),

(3.3.11)

dCv dT dX = 0, = 0, = 0 (symmetrical condition), (3.3.12) dr dr dr dX Cv,s r = R, −Cs Dw = h m εw − ρv,b (convective boundary for dr ε liquid water transfer), (3.3.13) r = 0,

dCv = h m εv −Dv dr

Cv,s − ρv,b ε

(convective boundary for water vapor transfer),

(3.3.14)

dT (3.3.15) = h(Tb − T ) (convective boundary for heat transfer). dr Similar to the convective drying of mango tissues, the internal solid-surface water vapour concentration and the local evaporation rate (I˙ ) are evaluated using Equations (3.1.17) and (3.1.18). The relative activation energy of convective drying of potato tissues is generated from one accurate drying run of the convective drying of potato tissues with a diameter of 1.4 cm at a drying air temperature of 70 °C (Srikiatden and Roberts, 2008). The activation energy during drying was evaluated using Equation (2.1.5) based on the experimental data of moisture content and surface temperature during drying (Srikiatden and Roberts, 2008). It is then divided with the equilibrium activation energy represented in Equation (2.1.7) to yield the relative activation energy as mentioned in Equation (2.1.6). The relationship between the relative activation energy and average moisture content can be represented by a simple mathematical equation obtained by the least-squares method using Microsoft Excel (Microsoft Inc., 2012). The relative activation energy can be represented as:

Ev = exp −0.364(X − X b )0.876 . (3.3.16) E v,b k

Similar to modelling of convective drying of mango tissues, for modelling using the S-REA, the average moisture content X in Equation (3.3.16) is substituted by the local moisture content (X) as the REA is then able represent the local evaporation or condensation rate here instead of the global drying rate. The effective liquid diffusivity (Dw ) is shown in Equation (3.2.7) while the effective vapour diffusivity (Dv ), tortuosity (τ ), solid concentration (Cs ) and porosity (ε) are

133


deduced using Equations (3.2.1)–(3.2.5). Similarly, the internal mass transfer coefficient (hm,in ) is evaluated using the procedures explained in Section 3.2. The average moisture content in the core of potato tissues (Xcore ) is evaluated by: Rcore

X core =

X (r )r dr

0 Rcore

.

(3.3.17)

rdr

0

The average moisture content in cortex (Xcortex ) is evaluated by: Rsample

X cortex3 =

Rcortex

out

X (r )r dr .

in

Rcortex

out

Rcortex

in

(3.3.18)

r dr

The results of modelling average moisture content in core and cortex (hence the spatial distribution) are validated against the experimental data of Srikiatden and Roberts (2008). Similarly to the convective drying of mango tissues, the mass and heat balances are shown in Equations (3.3.8)–(3.3.10) in conjunction with the initial and boundary conditions indicated in Equations (3.3.11)–(3.3.15). The application of 10 and 200 increments did not result in noticeable differences in the profiles.

3.3.3

Results of modelling of convective drying of mango tissues using the S-REA The S-REA is used to model the convective drying of mango tissues at drying air temperatures of 45°, 55° and 65 °C. The original formulation of the L-REA is implemented in the partial differential equation set for transport in porous media to represent the local drying or condensation rate. It is thus coupled with the system of equations of conservation to describe the spatial profiles of moisture content, water vapour concentration and temperature. It is noted that, if locally there is no vacant pore space which is connected with other pores or channels, the internal mass transfer area should be considered to be zero; hence the REA term is zero if the pores or channels are fully hydrated. In this study, the internal mass transfer coefficient (hm,in : see Equation 3.3.12) is chosen to 0.01 m s−1 as the sensitivity analysis indicates that hm,in is likely to be higher than 0.01 m s−1 , but any higher than this and it does not give any noticeable difference in the profiles of moisture content and temperature predicted. More importantly, the value of 0.01 m s−1 is also in the order of Dv /rp (Kar and Chen, 2010; 2011); thus it is a fundamental value. The good agreement between the predicted and experimental data of the average moisture content and the centre temperature is shown in Figures 3.3 and 3.4. It is also supported by R 2 of moisture content higher than 0.996 and R 2 of temperature higher than 0.985 as listed in Table 3.2. The results of the S-REA modelling match well with the experimental data. Benchmarks against the diffusion-based model (Vaquiro et al.,

10 S-REA 45°C Data 45°C S-REA 55°C Data 55°C S-REA 65°C Data 65°C


9 8 7 6 5 4 3 2 1 0

0

0.5

1

1.5

2

2.5 t(s)

3

3.5

4

4.5

5 4

× 10

Figure 3.3 Average moisture content profiles of mango tissues during convective drying at different drying air temperatures. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]

340

Temperature (K)

330

320

310

S-REA 45°C Data 45°C S-REA 55°C Data 55°C S-REA 65°C Data 65°C

300

290

280

0

0.5

1

1.5

2

2.5 t(s)

3

3.5

4

4.5

5 4

× 10

Figure 3.4 Centre temperature profiles of mango tissues during convective drying at different drying air temperatures. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]


135

Table 3.2 R2 and RMSE of convective drying of mango tissues using the S-REA. Drying air temperature (°C)

R2 for X

RMSE for X

R2 for T

RMSE for T

45 55 65

0.998 0.999 0.996

0.103 0.079 0.150

0.998 0.985 0.994

0.285 1.004 0.842

10 Moisture content (kg water/kg dry solids)

9 8 7 6 5 t = 1000s t = 3000s t = 5000s t = 10000s t = 20000s t = 30000s t = 35000s

4 3 2 1 0 0

0.002

0.004

0.006 0.008 Half thickness (m)

0.01

0.012

0.014

Figure 3.5 Spatial moisture content profiles of mango tissues during convective drying at drying

air temperatures of 45 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]

2009) indicate that the S-REA yields comparable, or even better, agreement towards the experimental data. Figure 3.5 shows the spatial profiles of the moisture content during convective drying of mango tissues at a drying air temperature of 45 °C. The moisture content at the outer part of the samples is lower than that at the inner part, which indicates the effect of moisture removal. Initially, the gradient of moisture content inside the samples is relatively high but this decreases as the drying progresses. At the end of drying, no noticeable gradient of moisture content is observed, which indicates the equilibrium moisture content is nearly approached. If no liquid diffusion mechanism is used, the S-REA model would not be able to project this kind of liquid water profile (Kar and Chen, 2010; 2011).


0.012

Water vapour concentration (kg/m3)

136

t = 1000s t = 3000s t = 5000s t = 10000s t = 20000s t = 30000s t = 35000s

0.01

0.008

0.006

0.004

0.002

0

0

0.002

0.004


0.01

0.012

0.014

Figure 3.6 Spatial water vapour concentration profiles of mango tissues during convective drying

at drying air temperatures of 45 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]

The S-REA can generate the spatial profiles of water vapour concentration. The spatial profiles of water vapour concentration during convective drying of mango tissues at a drying air temperature of 45 °C are shown in Figure 3.6. The profiles of water vapour concentration are significantly affected by the local composition and structure of the samples being dried. Along drying, the concentration of water vapour achieves a maximum at a particular position inside the samples. This could be because, at the core of samples, the moisture content is higher than that of the outer part which makes the porosity of the core of samples lower. The lower porosity retards the evaporation rate at the sample core. At the outer part of the samples, the water extraction rate may be enhanced because of higher porosity but this seems to be balanced by high diffusive water vapour transfer as a result of higher porosity and temperature at the outer part of the samples. The S-REA seems to capture this physics well and can model the profiles of water vapour concentration well qualitatively. The spatial profiles of temperature are presented in Figure 3.7. The temperature of the outer part of the samples is higher than that of the inner part because the samples receive heat by convection from the drying air and this is used for vaporisation; if any is left over as such, this would penetrate further inwards by conduction. However, the gradient of temperature inside the samples is not large which may indicate that the temperature inside the samples is essentially uniform. This is in agreement

137


330 t = 1000s t = 3000s t = 5000s t = 10000s t = 20000s t = 30000s t = 35000s

Temperature (K)

325

320

315

310

305

0

0.002

0.004


0.01

0.012

0.014

Figure 3.7 Spatial temperature profiles of mango tissues during convective drying at drying air temperatures of 45 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]

with the prediction of the Chen–Biot number (Ch–Bi) (Chen and Peng, 2005; Putranto et al., 2011a) which remains low (less than 0.3) during drying reported previously (Putranto et al., 2011a). Figure 3.8 indicates the local evaporation rate inside mango tissues during convective drying at a drying air temperature of 55 °C. As drying proceeds, the evaporation rate at the inner part is smaller than that of the outer part, which could be due to high moisture content at the inner part of the sample. This means a lower porosity there that retards the evaporation rate. The observation is also in agreement with the intuitive explanation of profiles of water vapour concentration during drying by Chen (2007). As drying progresses, the evaporation rate increases as the temperature increases. However, the increase is observed up to a drying time of around 15 000 s. After this period, the evaporation rate decreases as the moisture content inside the samples is depleted. At the end of drying, essentially there is not much difference in evaporation rate inside the samples because the moisture content has nearly achieved equilibrium under the drying conditions. Therefore, it can be said that the S-REA approach models the convective drying of mango tissues well and the original REA is a simple alternative approach to represent the local evaporation and condensation rates. In addition, the S-REA has been easily

138


1.2 t = 1000s t = 5000s t = 10 000s t = 15 000s t = 20 000s t = 30 000s t = 35 000s

Evaporation rate (kg water/(m3s))

1 0.8 0.6 0.4 0.2 0 –0.2

0

0.002

0.004


0.01

0.012

0.014

Figure 3.8 Profiles of evaporation rates inside mango tissues during convective drying at a drying air temperature of 55 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]

operated to yield the profiles of water vapour concentration, local evaporation rate and local heat evaporation rate inside the mango tissues during drying.

3.3.4

Results of modelling of convective drying of potato tissues using the S-REA As mentioned earlier, the S-REA has also been implemented to model the convective drying of potato tissues. The relative activation energy is generated from one accurate drying run which is the convective drying at air temperature of 70 °C. It is represented in Equation (3.3.16). Similar to the convective drying of mango tissues, the internal mass transfer coefficient (hm,in : on Equation 3.1.17) is chosen to be 0.01 m s−1 as the sensitivity analysis indicates that hm,in of higher than 0.01 m s−1 does not give any noticeable differences in the profiles of moisture content and temperature. Nicely, this is also in the order of Dv /rp as suggested by Kar and Chen (2010; 2011), hence hm,in is a fundamental value. However, Dwo (in Equation 3.2.7) is determined by sensitivity analysis and it is found that Dwo of 6.5 × 10−6 m2 s−1 gives the best agreement against the experimental data. It is emphasised that the temperature dependence function for the liquid diffusivity in this case was obtained in isothermal drying experiments specially designed by Srikiatden and Roberts (2006), in contrast to many published studies that


139


6 S-REA-core Data-core S-REA-cortex Data-cortex

5

4

3

2

1

0

0

0.5

1

1.5 t(s)

2

2.5 × 104

Figure 3.9 Moisture content profiles in the core and cortex during convective drying of potato tissues with a diameter of 1.4 cm. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]

suggest the diffusivity in the material is related to drying air temperature instead (Chen, 2007). Results of modelling convective drying of potato tissues are shown in Figures 3.9– 3.11. Figure 3.9 shows the profiles of moisture content of each part of potato cylindrical tissue with the diameter of 1.4 cm during the convective drying. It can be shown that the results of modelling match well with the experimental data with a correlation coefficient R2 of 0.98. Benchmarks against modelling implemented by Srikiatden and Roberts (2008) with the liquid diffusivity concept indicate that the S-REA yields comparable results. In addition, the profiles of moisture content for each part of samples with the diameter of 2.8 cm are shown in Figure 3.10. Again, a good agreement towards the experimental data is observed (R2 of 0.992). Indeed, the S-REA describes the moisture content profiles accurately during convective drying of potato tissues with a diameter of 2.8 cm. Benchmarks towards modelling implemented by Srikiatden and Roberts (2008) indicate that the REA yields comparable or even better results. It can be said that the S-REA can be used to model the profiles of moisture content very well. Figure 3.11 indicates the core temperature during convective drying of potato tissues with the diameter of 1.4 cm. The predictions of temperature using the S-REA match


6

5

4

3 S-REA-core S-REA-cortex 1 S-REA-cortex 2 S-REA-cortex 3 Data-core Data-cortex 1 Data-cortex 2 Data-cortex 3

2

1

0

0

0.5

1

1.5

2

2.5

3

t(s)

3.5 × 104

Figure 3.10 Moisture content profiles in the core and cortex during convective drying of potato

tissues with a diameter of 2.8 cm. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]

345 340

Core temperature (K)

335 330 325 320 315 310 305 S-REA Data

300 295

0

0.5

1

1.5 t(s)

2

2.5 × 104

Figure 3.11 Core temperature profiles during convective drying of potato tissues with a diameter

of 1.4 cm. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]


141

Table 3.3 Scheme of intermittent drying of mango tissues (Vaquiro et al., 2009). Drying air temperature (ºC)

Period of first heating (s)

Period of resting (at 27 °C ± 1.6) (s)

Period of second heating (s)

45 55

16 200 9480

10 800 10 800

36 360 33 720

well with the experimental data with R2 of 0.99. Benchmarks against modelling implemented by Srikiatden and Roberts (2008) indicate that the S-REA yields comparable results. Overall, S-REA seems to be a sound approach to modelling the details of spatial distributions of temperature, liquid water and water vapour concentration in the material being dried.

3.4

The S-REA for intermittent drying The accuracy of the S-REA in modelling intermittent drying is validated by the experimental data of intermittent drying of mango tissues (Vaquiro et al., 2009). For better understanding of the modelling approach, the experimental details are summarised and reviewed in this section. The samples of mango tissues were cubes with initial side lengths of 2.5 cm, while the initial moisture content and temperature were 9.3 kg kg−1 and 10.8 ºC, respectively. The laboratory drier was described in Sanjuan et al. (2004). During drying, the weight of the sample and the centre temperature were recorded. The drying air temperature and air velocity were controlled at preset values by PID control algorithms while air humidity was maintained constant during drying. The experimental setting for intermittent drying is shown in Table 3.3. During the resting period, the samples stayed in an environment with an ambient temperature of 27 ± 1.6 °C and relative humidity of 60% (Vaquiro et al., 2009). Determination of density, thermal conductivity, heat capacity, equilibrium moisture content and shrinkage of samples being dried has been described previously (Putranto et al., 2011a).

3.4.1

The mathematical modelling of intermittent drying using the S-REA In experiments reported by Vaquiro et al. (2009), the subject of interest, the samples were dried from three directions (x, y and z directions) so three-dimensional modelling of the S-REA for intermittent drying of mango tissues needs to be set up, which is represented next. The mass and heat balances of intermittent drying of mango tissues are similar to those of convective drying described in Section 3.3.1. Similarly to the convective drying of mango tissues, the mass balance of water in liquid phase liquid water, the mass balance of water in the vapour phase (water vapour) and the heat balance are shown in Equations (3.3.1), (3.3.2) and (3.3.3), respectively, while the initial and boundary conditions for equations are shown in Equations (3.1.4)– (3.1.16).

142


Table 3.4 R 2 and RMSE of intermittent drying of mango tissues.

45 °C 55 °C

R 2 for X

RMSE for X

R 2 for T

RMSE for T

0.964 0.999

0.358 0.0774

0.992 0.994

0.705 0.874

The internal-surface water vapour concentration (Cv,s ) and internal evaporation rate are evaluated using Equations (3.1.18) and (3.1.19), respectively. The liquid diffusivity (Dw ) is shown in Equation (3.2.7) while the effective vapour diffusivity (Dv ), tortuosity (τ ), solid concentration (Cs ) and porosity (ε) are deduced using Equations (3.2.1)– (3.2.5). Similarly, the internal mass transfer coefficient (hm,in ) is evaluated using the procedures explained in Section 3.2. The relative activation energy implemented for modelling of convective drying of mango tissues shown in Equation (3.3.4) is used here to model the intermittent drying of mango tissues. For modelling the intermittent drying of these tissues, the equilibrium activation (Ev,b ) energy shown in Equation (2.1.7) is evaluated according to the corresponding drying air temperature and humidity in each drying period. It is also combined with the relative activation energy shown in Equation (3.3.4) to yield the local drying/condensation rate. In addition, the heat balance implements the corresponding drying air temperature in each drying period by using the corresponding drying air temperature in the boundary conditions indicated in Equations (3.1.10), (3.1.13) and (3.1.16). The solution procedures are similar to the one for convective drying of mango tissues, described in Section 3.3.1.

3.4.2

Results of modelling intermittent drying using the S-REA The S-REA is implemented here to model the intermittent drying of mango tissues whose conditions are listed in Table 3.3. As mentioned before, for modelling of the intermittent drying, the equilibrium activation energy needs to be evaluated according to the corresponding drying settings in each drying period. Similarly, the heat balance implements the corresponding drying air temperature in each drying period. Solving Equations (3.3.9)–(3.3.11) in conjunction with the initial and boundary conditions shown in Equations (3.3.4) to (3.3.8) simultaneously yields the profiles of moisture content, concentration of water vapour and temperature during intermittent drying. Figures 3.12– 3.17 show the results of modelling of the intermittent drying. The profiles of moisture content during intermittent drying are shown in Figure 3.12. A good agreement between the predicted and experimental data is observed and confirmed by R 2 and RMSE listed in Table 3.4. The results of modelling match well with the experimental data of moisture content. The S-REA models the average moisture content during the intermittent drying at drying air temperature of 45°, 55° and 65 °C very well. Benchmarks against modelling implemented by Vaquiro et al. (2009) revealed that the REA yields better results as Vaquiro et al. (2009) showed a slight underestimation in drying rate of intermittent drying at a drying air temperature at 45 °C during drying

143


10 S-REA 45°C Data 45°C S-REA 55°C Data 55°C S-REA 65°C Data 65°C


9 8 7 6 5 4 3 2 1 0

0

1

2

3

4 t(s)

5

6

7 × 104

Figure 3.12 Average moisture content profiles of mango tissues during intermittent drying at

different drying air temperatures.

times between 20 000–50 000 s. Similarly, Vaquiro et al. (2009) also revealed a slight overestimation in drying rate of intermittent drying at a drying air temperature of 65 °C after a drying time of 20 000 s. The slight underestimation and overestimation of the drying rate are not shown by the modelling using the S-REA. This indicates that the S-REA can be used to describe the moisture content of intermittent drying of mango tissues well and the REA can also be applied to model the local evaporation and condensation rate well. The spatial profiles of moisture content during the intermittent drying at a drying air temperature of 55 °C are indicated in Figure 3.13. The moisture content of the outer part of the samples is lower than that in the inner part of the samples, which indicates that moisture migrates outwards during intermittent drying. Initially, the gradient of moisture content inside the samples is relatively high, but during a drying time between 9480 and 20 280 s the gradient is relatively low since the samples are in the resting period. This period seems to allow the moisture to redistribute inside the samples and low gradients of moisture content inside the samples are generated. Towards the end of drying, although the samples are in the heating period again, relatively uniform moisture content is observed, which indicates that the equilibrium condition is almost approached. Figure 3.14 presents the spatial profiles of water vapour concentration during the intermittent drying at a drying air temperature of 55 °C. The profiles of water vapour concentration are significantly affected by the local composition and structure of the


10 Moisture content (kg water/kg dry solids)

9 8 7 6 5

t = 1000s t = 3000s t = 5000s t = 7000s t = 14 000s t = 20 000s t = 30 000s t = 40 000s t = 50 000s

4 3 2 1 0

0

0.002

0.004


0.01

0.012

0.014

Figure 3.13 Spatial moisture content profiles of mango tissues during intermittent drying at a

drying air temperature of 55 °C.

0.012

Water vapour concentration (kg/m3)

144

t = 1000s t = 3000s t = 5000s t = 7000s t = 14 000s t = 20 000s t = 30 000s t = 40 000s t = 50 000s

0.01

0.008

0.006

0.004

0.002

0

0

0.002

0.004


0.01

0.012

0.014

Figure 3.14 Spatial water vapour concentration profiles of mango tissues during intermittent

drying at a drying air temperature of 55 °C.

145


340


330

320

310

300

S-REA 45°C Data 45°C S-REA 55°C Data 55°C S-REA 65°C Data 65°C

290

280

0

1

2

3

4 t(s)

5

6

7 4

× 10

Figure 3.15 Centre temperature profiles of mango tissues during intermittent drying at different

drying air temperatures.

samples being dried. The water vapour concentration achieves a maximum at particular position inside the samples. At core of the samples, the moisture content is relatively high, which may result in lower porosity and retard the local evaporation rate. At outer part of the samples, the local evaporation rate may be enhanced as a result of the higher porosity, but this seems to be balanced by higher diffusive flux of water vapour due to the higher porosity and temperature. During first heating period, the gradient of water vapour concentration is relatively high but this decreases as drying progresses. The relatively uniform concentration of water vapour is shown during the resting period which could be due to a relatively low temperature. During the second heating period, relatively uniform water vapour concentration is observed. This is in agreement with the relatively uniform moisture content at the end of drying as explained previously. The spatial profiles of intermittent drying are similar to those of convective drying (Putranto and Chen, 2013). In addition, this is in agreement with a qualitative prediction by Chen (2007) which explained that the maximum water vapour concentration is achieved at a particular position inside the samples. The profiles of temperature during the intermittent drying are indicated in Figures 3.15 and 3.16. Figure 3.15 shows the profiles of the centre temperature during the intermittent drying of mango tissues at drying air temperatures of 45°, 55° and 65 °C. A good agreement between the predicted and experimental data is observed which is also supported by R2 of higher than 0.992 and RMSE lower than 0.828. The results of modelling match well with the experimental data. Benchmarks against modelling implemented by Vaquiro



146

315 t = 1000s t = 3000s t = 5000s t = 7000s t = 14 000s t = 20 000s t = 30 000s t = 40 000s t = 50 000s

310 305 300 295

0

0.002

0.004


0.01

0.012

0.014

Figure 3.16 Spatial temperature profiles of mango tissues during intermittent drying at a drying

air temperature of 55 °C.

et al. (2009) show that the REA yields better results since the modelling by Vaquiro et al. (2009) indicated kinks and an underestimation of temperature profiles in the beginning of the first heating period. However, these are not observed by the modelling using the S-REA. This indicates that the S-REA is indeed accurate enough to model the temperature profiles of intermittent drying of mango tissues. Figure 3.16 represents the spatial profiles of temperature during the intermittent drying at a drying air temperature of 55 °C. During the first heating period, the temperature in the outer part of the samples is higher than that of the inner part because the samples receive heat by convection from the drying air used for water evaporation and this penetrates inwards by conduction. However, the gradient of temperature inside the samples is not high, in agreement with the prediction of Chen–Biot number (Ch–Bi) (Chen and Peng, 2005) reported previously (Putranto et al., 2011b), which indicated that the temperature inside the samples is essentially uniform. During the first heating period, the temperature of samples increases but the temperature decreases between drying times of 9480–20 280 s as a result of resting period. This is followed by a further increase of temperature during the second heating period. At the end of the intermittent drying, the temperature approaches the drying air temperature. The S-REA has its advantages, resulting in the spatial profiles of local evaporation rate during drying as the REA is used to predict the local evaporation rate. The profiles of local evaporation rates during intermittent drying of Scheme 1 at a drying air temperature of 55 °C are shown in Figure 3.17. The local evaporation rate at core of samples is lower


147

1.2

Evaporation rate (kg water/(m3s))

1 0.8

t = 1000s t = 3000s t = 5000s t = 9000s t = 14 000s t = 16 000s t = 20 000s t = 22 000s t = 35 000s t = 40 000s t = 45 000s t = 50 000s t = 54 200s

0.6 0.4 0.2 0 –0.2

0

0.002

0.004


0.01

0.012

0.014

Figure 3.17 Profiles of evaporation rate inside mango tissues during intermittent drying at a

drying air temperature of 55 °C.

than that of the outer part of samples. This could be because of lower porosity at the core of the samples as a result of higher moisture content. During the resting period, the spatial profiles of local evaporation rate are more uniform than those during first heating period, which may be due to a lower temperature inside the samples. Towards the end of drying, the gradient of local evaporation rate decreases. This could be because the moisture content decreases as drying progresses, which increases the porosity. From the start of drying to a drying time of around 5000 s the local evaporation rate increases, which could be because of the increase in temperature. After this period, the local evaporation rate tends to decrease, which could be because of the decrease in moisture content. During the resting period, the local evaporation rate changes only slightly, which may be because of the relatively low temperature. During the second heating period, the local evaporation decreases. This may be because the moisture content inside the sample is depleted. At the end of drying, essentially there is not much difference in evaporation rate inside the samples because the moisture content has nearly achieved equilibrium under the drying conditions. It has been demonstrated that the S-REA is very accurate in modelling the intermittent drying of mango tissues. The S-REA can also project the concentration of water vapour and local evaporation rate during intermittent drying so that better understanding of transport phenomena of drying processes can be gained. It is argued here that the S-REA is not only robust enough to model convective drying (Putranto and Chen, 2013) but also intermittent drying.

148


Table 3.5 Experimental settings of wood heating under a constant heating rate (Younsi et al., 2007).

3.5

Case

Final gas temperature (ºC)

Heating rate (ºC h−1 )


1 2

220 220

10 20

0.11 0.125

The S-REA to wood heating under a constant heating rate The accuracy and robustness of the S-REA for heat treatment of wood under a constant heating rate is benchmarked by the experimental data of Younsi et al. (2007). The experimental details were reported in Kocaefe et al. (1990; 2007) and Younsi et al. (2006b) and these are reviewed here for better understanding of the current approach. A thermogravimetric analyser was used for the heat treatment of wood as shown in Kocaefe et al. (1990). Wood samples with dimensions of 0.035 × 0.035 × 0.2 m were heat treated by suspending the samples on a balance with accuracy of 0.001 g. The heat treatment was conducted by exposing the samples to hot gas whose temperature was linearly increased according to the heating rate. The humidity of the gas was controlled by injection of steam into the second furnace placed under the main furnace (Younsi et al., 2006a, 2007). The samples were first heated to 120 °C and held at this temperature for half an hour followed by heating under the preset heating rate (refer to Table 3.5) until the final temperature (also refer to Table 3.5) was achieved. During the heat treatment, the weight of the samples was recorded by the balance. In addition, the temperatures were measured by a T-type thermocouple placed inside the samples. The measurements indicated that the temperatures inside the samples were essentially uniform, perhaps due to the small size of the samples (Younsi et al., 2006b).

3.5.1

The mathematical modelling of wood heating using the S-REA In the experiments reported by Younsi et al. (2006a; 2007), the subject of interest here, two-dimensional modelling with respect to x and y directions can be set up as the height of the sample (0.2 m) is much greater than the length (0.035 m) and width (0.035 m) of the sample. The mass balance of water in the liquid phase is written as (Chen, 2007; Chong and Chen, 1999; Putranto and Chen, 2013; Zhang and Datta, 2004): ∂(C s X ) ∂(Cs X ) ∂(Cs X ) ∂ ∂ = Dw + Dw − I˙, (3.5.1) ∂t ∂x ∂x ∂y ∂y where X is the concentration of liquid water (kg H2 O kg dry solids−1 ) and Cs is the solids’ concentration (kg dry solids m−3 ) which can change if the structure is shrinking, I˙ is the evaporation or wetting rate (kg H2 O m−3 s−1 ), I is >0 when evaporation occurs


149

locally, while the mass balance of water in the vapour phase is expressed as (Chen, 2007; Chong and Chen, 1999; Putranto and Chen, 2013; Zhang and Datta, 2004): ∂C v ∂C v ∂Cv ∂ ∂ = Dv + Dv + I˙, (3.5.2) ∂t ∂x ∂x ∂y ∂y where Cv is the water vapour concentration (kg H2 O m−3 ). The heat balance is represented as (Chen, 2007; Chong and Chen, 1999; Putranto and Chen, 2013; Zhang and Datta, 2004): ∂ ∂T ∂ ∂T ∂T (3.5.3) = k + k − I˙Hv , ρC p ∂t ∂x ∂x ∂y ∂y where T is the sample temperature (K), k is thermal conductivity of sample (W m−1 K−1 ), ρ is the sample density (kg m−3 ) and Hv is the vaporisation heat of water (J kg−1 ). The initial and boundary conditions for Equations (3.5.1) to (3.5.3) are: t = 0, X = X o , C v = Cvo , T = To (initial condition, uniform initial concentrations and temperature),

(3.5.4)

dX dC v dT = 0, = 0, = 0 (symmetry boundary), (3.5.5) dx dx dx dX Cv,s = h m εw − ρv,b (convective boundary x = L , y = L , −Cs Dw dx ε for liquid water transfer), x = 0, y= 0,

dCv = h m εv −Dv dx

Cv,s − ρv,b ε

(3.5.6)

(convective boundary for water vapor transfer),

(3.5.7)

dT = h(Tb − T ) (convective boundary for heat transfer). (3.5.8) dx Because the samples were dried uniformly from all directions and the lengths and widths were the same, the mass balance of water in liquid phase can be simplified into (Incropera and DeWitt, 2002; Van der Sman, 2003): ∂(Cs X ) ∂ ∂(Cs X ) =2 Dw − I˙, (3.5.9) ∂t ∂x ∂x k

while the mass balance of water in vapour phase can be expressed as: ∂ ∂Cv ∂C v =2 Dv + I˙. ∂t ∂x ∂x In addition, the heat balance can be represented as: ∂T ∂ ∂T ρC p =2 k − I˙HV . ∂t ∂x ∂x

(3.5.10)

(3.5.11)

150


Similarly to the convective and intermittent drying described in Sections 3.3 and 3.4, the internal evaporation rate (I˙), effective vapour diffusivity, tortuosity, solids concentration and porosity are evaluated using Equations 3.1.19, 3.2.1, 3.2.3, 3.2.4 and 3.2.5, respectively. The relative activation energy of heat treatment of wood is generated from the drying run in Case 2 (refer to Table 3.5) (Younsi et al., 2007). The activation energy during drying is evaluated using Equation (2.1.5) and divided with the equilibrium activation energy represented in Equation (2.1.7) to yield the relative activation energy as mentioned in Equation (2.1.6). The relationship between the relative activation energy and average moisture content can be represented by simple mathematical equation obtained by the least-square method using Microsoft Excel (Microsoft Corp, 2012). The relative activation energy can be represented as:

E v = 1 − 1.517(X − X b )0.22 exp −3.717(X − X b )3.135 . E v,b

(3.5.12)

For modelling using the S-REA here, the relative activation energy shown in Equation (3.5.12) is used, but the average moisture content X in Equation (3.5.12) is substituted by the local moisture content (X) as the REA represents the local evaporation rate here, instead of the global drying rate. In order to incorporate the effect of linearly increased gas temperature, the equilibrium activation energy shown in Equation (2.1.7) implements the corresponding gas temperature and humidity during heat treatment. In addition, the linearly increased gas temperature is used in Equation (3.5.8). In order to yield the spatial profiles of moisture content, water vapour concentration and temperature in the heat treatment of wood, the mass and heat balances shown in Equations (3.5.9) to (3.5.11), in conjunction with the initial and boundary conditions represented in Equations (3.5.4) to (3.5.8) and the relative activation energy shown by Equation (3.5.12), are solved by method of lines (Chapra, 2006; Constantinides, 1999). By this method, the partial differential equations are transformed into a set of ordinary differential equations with respect to time by firstly discretising the spatial derivatives. The ordinary differential equations are then solved simultaneously by ode23s in Matlab (Mathworks Inc., 2012). The spatial derivative here is discretised into 10 increments; application of 100 increments has been conducted and there is no difference in the profiles observed, as shown in Figure 3.18. No shrinkage is incorporated in the modelling, as Younsi et al. (2006b) indicated that the ratio between final and initial dimension is around 0.96. Similarly, the modelling implemented by Younsi et al. (2006a,b, 2007) did not incorporate the shrinkage effect. The average moisture content of wood during heat treatment is evaluated by: L X=

X (x)d x

0

L

.

(3.5.13)

dx

0

The profiles of average moisture content and temperature are then validated towards the experimental data of Younsi et al. (2007).

151


Average moisture content (kg water/kg dry solids)

0.14 S-REA 100 increments S-REA 10 increments Data

0.12 0.1 0.08 0.06 0.04 0.02 0

0

0.5

1

1.5 t(s)

2

2.5

3 ×104

Figure 3.18 Profiles of average moisture content during heat treatment in Case 2 (refer to Table

3.5) solved by the method of lines using 10 and 100 increments.

3.5.2

The results of modelling wood heating using the S-REA The S-REA is used to model the heat treatment of wood under a constant heating rate and the results of modelling are presented in Figures 3.19 to 3.27. The REA is implemented to model the local evaporation or condensation term and coupled with a system of equations in order to yield the spatial profiles of moisture content, water vapour concentration and temperature. It is noted that if locally there is no vacant pore space which is connected to other pores or channels, the internal mass transfer area is zero, hence the REA term is zero. In this study, the internal mass transfer coefficient (hm,in ) shown in Equation (3.1.17) is chosen to be 0.001 m s−1 . Application of hm,in higher than 0.001 m s−1 does not yield any noticeable differences in the profiles of moisture content and temperature profiles. As mentioned before, no method has been presented anywhere in the literature to measure pore liquid diffusivity, and liquid diffusivity is obtained by numerical sensitivity to match the prediction with the experimental data of moisture content and temperature. Interestingly, in this study, for all cases the profiles of moisture content and temperature are independent of the liquid diffusivity value. Further explanation about this phenomenon is presented next. For Case 1 (refer to Table 3.5), the results of modelling are presented in Figures 3.19 to 3.22. The varied values of the liquid diffusivity in the range of 1 × 10−8 to 1 × 10−30 m2 s−1 have been used and there are no noticeable differences in the profiles of moisture content and temperature as shown in Figures 3.19 and 3.20. This may indicate



0.12 Dw = 1e-8 m2/s Dw = 1e-12 m2/s Dw = 1e-18 m2/s Dw = 1e-24 m2/s Dw = 1e-30 m2/s Data

0.1

0.08

0.06

0.04

0.02

0

0

1

2

3

4

5

6

t(s)

7 × 10

4

Figure 3.19 Effect of liquid diffusivity on profiles of the moisture content during heat treatment in

Case 1 (refer to Table 3.5).

500 Average moisture content (kg water/kg dry solids)

152

450

400

350

Dw = 1e-8 m2/s Dw = 1e-12 m2/s Dw = 1e-18 m2/s Dw = 1e-24 m2/s Dw = 1e-30 m2/s Data

300

250

0

1

2

3

4 t(s)

5

7

6

4

× 10

Figure 3.20 Effect of liquid diffusivity on profiles of temperature during heat treatment in Case 1

(refer to Table 3.5).

153



0.14 Model Data

0.12

0.1 0.08 0.06 0.04 0.02 0

0

1

2

3

4

5

6

t(s)

7 × 104

Figure 3.21 Profiles of average moisture content during heat treatment in Case 1 (refer to

Table 3.5).

500

Temperature (K)

450

400

350

300 Model Data 250 0

1

2

3

4 t(s)

5

6

7 × 104

Figure 3.22 Profiles of temperature during heat treatment in Case 1 (refer to Table 3.5).


0.14 Average moisture content (kg water/kg dry solids)

154

Model Data

0.12 0.1 0.08 0.06 0.04 0.02 0

0

0.5

1

1.5 t(s)

2

2.5

3 × 10

4

Figure 3.23 Profiles of average moisture content during heat treatment in Case 2 (refer to

Table 3.5).

that the modelling of heat treatment in Case 1 (refer to Table 3.5) is independent of the liquid diffusivity so this can be neglected and the modelling will only involve the vapour diffusion and evaporation/condensation. The phenomenon could be due to the initial moisture content being relatively low and the heating of wood implemented at a relatively high temperature. By ignoring the liquid diffusion term on the mass balance of liquid water (refer to Equation 3.5.9), a good agreement between the predicted and experimental data of moisture content and temperature is shown in Figures 3.23 and 3.24 as well as confirmed by R 2 and RMSE indicated in Table 3.6. Benchmarks against modelling implemented by Younsi et al. (2007) indicate that the S-REA yields comparable or even better results. Therefore, it can be said that the S-REA models the heat treatment of wood in Case 1 well (refer to Table 3.5). Figures 3.23 to 3.27 show the results of modelling of heat treatment of wood in Case 2 (refer to Table 3.5). Similarly to Case 1 (refer to Table 3.5), the varied values of the liquid diffusivity in the range of 1 × 10−8 to 1 × 10−30 m2 s−1 have been used and there is no noticeable effect on the profiles of moisture content and temperature. By ignoring the liquid diffusion term in the mass balance of liquid water (refer to Equation 3.5.9), a good agreement between the predicted and experimental data of moisture content and temperature is shown in Figures 3.23 and 3.24 and confirmed by R 2 of 0.995 and 0.997 for moisture content and temperature, respectively and RMSE of lower than 0.005

155


Table 3.6 R 2 and RMSE of modelling of heat treatment of wood under a constant heating rate using the S-REA. Case

R 2 for X

R 2 for T

RMSE for X

RMSE for T

1 2

0.988 0.995

0.992 0.997

0.004 0.003

4.765 3.287

500

Temperature (K)

450

400

350

300

Model Data 250

0

0.5

1

1.5

2 t(s)

2.5

3

3.5 × 104

Figure 3.24 Profiles of temperature during heat treatment in Case 2 (refer to Table 3.5).

and 3.287 for moisture content and temperature, respectively. The S-REA describes the profiles of moisture content and temperature very well. Benchmarks against modelling implemented by Younsi et al. (2007) reveal that the S-REA yields better agreement with the experimental data from moisture content. The spatial profiles of moisture content, water vapour concentration and temperature are presented in Figures 3.25–3.27. The distribution of moisture content is not determined by liquid diffusivity. This may suggest that liquid diffusivity can be neglected, so the drying process is not governed by liquid diffusion as the initial moisture content is relatively low and temperature is relatively high. Figure 3.25 indicates that the moisture content of the inner part of the samples is higher than that of the outer part, which indicates the moisture migrates outwards during drying. Similarly, as shown in Figure 3.26, the water vapour concentration of the inner part of the samples is higher than that of the outer part. This could be because of the relatively



0.14 0.12 0.1 0.08 0.06 t = 1000s t = 3000s t = 5000s t = 10 000s t = 15 000s t = 20 000s t = 28 000s

0.04 0.02 0

0

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Axial position (m)

Figure 3.25 Profiles of spatial moisture content during heat treatment in Case 2 (refer to

Table 3.5).

0.09 0.08 Water vapour concentration (kg/m3)

156

0.07 0.06 0.05 0.04 t = 1000s t = 3000s t = 5000s t = 10 000s t = 15 000s t = 20 000s t = 28 000s

0.03 0.02 0.01 0

0

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Axial position (m)

Figure 3.26 Profiles of spatial water vapour concentration during heat treatment in Case 2 (refer

to Table 3.5).

157


460 440

Temperature (K)

420 400 380 360 t = 1000s t = 3000s t = 5000s t = 10 000s t = 15 000s t = 20 000s t = 28 000s

340 320 300 280 0

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Axial position (m)

Figure 3.27 Profiles of spatial temperature during heat treatment in Case 2 (refer to Table 3.5).

high initial porosity of the samples, which allows evaporation at the core of the samples. The water vapour seems to migrate outwards and at the surface it is removed by the gas. The water vapour concentration increases until a heating time of 15 500 s, followed by a decrease until the end of drying. The initial increase could be because the initial moisture content is still relatively high at the beginning of drying but as the heating progresses, moisture content decreases and lower water vapour is generated. As shown in Figure 3.27, the temperature distribution of the samples is essentially uniform, in agreement with the observations of Younsi et al. (2007). It can be observed that the S-REA can model the heat treatment of wood under constant heating rate very well for all cases investigated. The liquid diffusion term can be neglected so that the model for heat treatment of wood under a constant heating rate may be simplified as follows: ∂(C s X ) = − I˙, ∂t ∂C v ∂C v ∂Cv ∂ ∂ = Dv + Dv + I˙, ∂t ∂x ∂x ∂y dy ∂T ∂ ∂T ∂ ∂T = k + k − I˙HV . ρC p ∂t ∂x ∂x ∂y ∂y

(3.5.14) (3.5.15) (3.5.16)

The S-REA has also the advantages of yielding profiles of water vapour concentration during the process. This enables better understanding of the transport phenomena during

158


the process. It can be said that the S-REA is an effective multiphase approach to modelling the heat treatment of wood.

3.6

The S-REA for the baking of bread The S-REA is validated against the experimental data of Banooni et al. (2008a,b). For better understanding of the modelling implemented, the experimental details of Banooni et al. (2008a,b) are reviewed briefly here. Baking experiments were carried out in a laboratory oven with active belt length and width of 1 m and 0.72 m, respectively. The oven was also equipped with an electrical heater and temperature control. The dough was mixed, divided into pieces of 250 g and kept for 15 min and then shaped and punched to produce flat bread with an initial thickness of 0.2 cm (Banooni et al., 2008a). The samples of bread were put in the belt of oven in which heated air was pushed at high velocity onto their surfaces. The air was propelled by a centrifugal fan and directed onto the samples through ‘fingers’ with jet holes of 1.2 cm; its velocity was 1–10 m s−1 . During the baking, the online system monitored the weight of the samples and Pt-100 probes were used to measure the top and bottom surface temperatures of the samples (Banooni et al., 2008a,b).

3.6.1

Mathematical modelling of the baking of bread using the S-REA Here, the S-REA is set up based on the experiments reported by Banooni et al. (2008a,b). It consists of a set of equations describing the conservation of mass and heat transfer using the REA to describe the local evaporation/condensation rate. The mass balance of water in the liquid phase (liquid water) is written as (Chen, 2007; Chong and Chen, 1999; Putranto and Chen, 2013; Zhang and Datta, 2004): ∂(C s X ) ∂ ∂(Cs X ) = Dw − I˙, ∂t ∂x ∂x

(3.6.1)

where Dw is the effective liquid water diffusivity (m2 s−1 ), X is the concentration of liquid water (kg H2 O kg dry solids−1 ), Cs is the solid concentration (kg dry solids m−3 ), which can change if the structure changes, I˙ is the evaporation or condensation rate (kg H2 O m−3 s−1 ) and I˙ is >0 when evaporation occurs locally. The mass balance of water in the vapour phase (water vapour) is expressed as (Chen, 2007; Chong and Chen, 1999; Putranto and Chen, 2013; Zhang and Datta, 2004): ∂Cv ∂ = ∂t ∂x

∂Cv Dv ∂x

+ I˙,

(3.6.2)

where Dv is the effective vapour diffusivity (m2 s−1 ) and Cv is the concentration of liquid water (kg H2 O kg dry solids−1 ).


159

The heat balance is represented by the following equation (Chen, 2007; Chong and Chen, 1999; Putranto and Chen, 2013; Zhang and Datta, 2004): ∂T ∂T ∂ ρC p (3.6.3) = k − I˙HV , ∂t ∂x ∂x where T is the sample temperature (K), k is thermal conductivity of sample (W m−2 K−1 ), ρ is the sample density (kg m−3 ) and Hv is the vaporisation heat of water (J kg−1 ). The initial and boundary conditions for Equations (3.6.1–3.6.3) are: t = 0, X = X o , C v = Cvo , T = To , dX dCv dT x = 0, = 0, = 0, −k = U (Tb − T ), dx dx dx dX Cv,s x = L , −C s Dw = h m εw − ρv,b , dx ε Cv,s dCv = h m εv − ρv,b , −Dv dx ε dT k = h(Tb − T ), dx

(3.6.4) (3.6.5) (3.6.6) (3.6.7) (3.6.8)

where h is the top heat transfer coefficient (W m−2 K−1 ) and U is the overall bottom heat transfer coefficient (W m−2 K−1 ). Similarly to convective and intermittent drying, as well as wood heating under constant heating rates (described in Sections 3.3, 3.4 and 3.5, respectively), the internal evaporation rate, effective vapour diffusivity, tortuosity, solid concentration and porosity are evaluated using Equations (3.1.19), (3.2.1), (3.2.3), (3.2.4) and (3.2.5), respectively. In addition, the internal mass transfer coefficient (hm,in ) described in Section 3.2 is used here. The relative activation energy of baking of bread is generated from one accurate baking run; i.e. baking bread at a baking temperature of 150 °C and air velocity of 10 m s−1 (Banooni et al., 2008a). The activation energy during drying is evaluated using Equation (2.1.5) and divided by the equilibrium activation energy represented in Equation (2.1.7) to yield the relative activation energy as mentioned in Equation (2.1.6). The relationship between relative activation energy and average moisture content can be represented by a simple mathematical equation obtained by the least-square method using Microsoft Excel (Microsoft Corp, 2012). The relative activation energy can be represented as: E v = [1 − 7.424(X − X b )4.471 ] exp[23.884(X − X b )42.282 ]. E v,b

(3.6.9)

The good agreement between the fitted and experimental relative activation energy is shown by R 2 of 0.995. For modelling using the S-REA here, the relative activation energy shown in Equation (3.6.9) is used but the average moisture content X in Equation (3.6.9) is substituted for the local moisture content (X) as the REA is used

160


to represent the local evaporation rate instead of the overall drying rate of the whole sample. The effective liquid diffusivity (Dw ) of the baking of bread presented is expressed as (Ni et al., 1999): Dw = 1 × 10−6 exp(−2.8 + 2X )ε.

(3.6.10)

In order to yield the spatial profiles of moisture content, water vapour concentration and temperature of the convective of mango tissues, the mass and heat balances shown in Equations (3.6.1)–(3.6.3), in conjunction with the initial and boundary conditions represented in Equations (3.6.4)–(3.6.8) and the relative activation energy shown by Equation (3.6.9), are solved by the method of lines (Chapra, 2006; Constantinides, 1999). In this method, the partial differential equations are transformed into a set of ordinary differential equations with respect to time by firstly discretising the spatial derivatives. The ordinary differential equations are then solved simultaneously by ode23s in Matlab (Mathworks Inc., 2012). The spatial derivative here is discretised into 10 increments; application of 100 increments has been conducted and there is no noticeable difference in the profiles observed. The shrinkage during the baking process can be represented as: V 2 = 162.69 X − 207.61 X + 66.925 (for ≥ X 0.57), V0 V = 1.307 X + 1.015 (for < X 0.57), V0

(3.6.11) (3.6.12)

where V is the volume of sample (m3 ) and V0 is the initial volume of sample (m3 ). The average moisture content of bread baking is evaluated by: L(t)

X=

X (x)d x

0 L(t)

.

(3.6.13)

dx

0

The profiles of average moisture content and centre temperature are then validated against the experimental data of Banooni et al. (2008a).

3.6.2

The results of modelling of the baking of bread using the S-REA The S-REA is used to model the baking of bread at baking temperatures of 150° and 200 °C. The original formulation of the L-REA is implemented in the partial differential equation set for transport in porous media, to represent the local evaporation or condensation rate. It is thus coupled with the equations of conservation to describe the spatial profiles of moisture content, water vapour concentration and temperature. The results of modelling the baking of bread using the S-REA are shown in Figures 3.28–3.32. Figure 3.28 presents the profiles of average moisture content during baking at a baking temperature of 150 °C. The S-REA models the average moisture content well

161



0.65 Data v = 1 m/s S-REA v = 1 m/s Data v = 5 m/s S-REA v = 5 m/s Data v = 10 m/s S-REA v = 10 m/s

0.6

0.55

0.5

0.45

0.4

0

500

1000

1500

t(s) Figure 3.28 Profiles of average moisture content during the baking of bread at a baking

temperature of 150 °C.

at various air velocities and a baking temperature of 150 °C as shown in Figure 3.28 (R 2 higher than 0.985). The spatial profiles of moisture content during the baking of bread at a baking temperature of 150 °C and velocity of 10 m s−1 are shown in Figure 3.29. The moisture content at the top and bottom surfaces of the bread are lower than at the core. This may indicate that the moisture migrates outwards during baking since the surface temperature is higher than the core. The maximum moisture content is located at a particular position inside the samples. It is not achieved at the centre of the sample, since the top and bottom surface temperatures are not similar. The top surface temperature is higher than the bottom surface temperature which may result in a higher moisture content at the bottom part of the samples. As baking progresses, the moisture content decreases and should approach equilibrium at the end of baking. Figure 3.30 shows the spatial profiles of water vapour concentration during the baking of bread at a baking temperature of 150 °C and velocity of 10 m s−1 . Initially, the concentration of water vapour is relatively high and this decreases as baking progresses, which could be because of the depletion of moisture during the baking. At the bottom part of the samples, the water vapour concentration is higher than that of the top part of the samples. It seems that the maximum concentration of water vapour is located at the bottom surface of the samples. These could be because no mass transfer occurs on the bottom surfaces since the samples are placed on top of trays, while at the top surfaces, water vapour is transferred to the baking air.


t = 20s t = 50s t = 100s t = 200s t = 400s t = 600s t = 800s t = 1000s t = 1200s t = 1500s


0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35

0

0.005

0.01

0.015 0.02 Axial position (m)

0.025

0.03

0.035

Figure 3.29 Spatial profiles of moisture content during the baking of bread at a baking

temperature of 150 °C and air velocity of 10 m s−1 .

0.45 t = 20s t = 50s t = 100s t = 200s t = 400s t = 600s t = 800s t = 1000s t = 1200s t = 1500s

0.4 Water vapour concentration (kg/m3)

162

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.005

0.01


0.025

0.03

0.035

Figure 3.30 Spatial profiles of concentration of water vapour during the baking of bread at a

baking temperature of 150 °C and air velocity of 10 m s−1 .

163


400 390 380

Temperature (K)

370 360 350 340 330 Top surface-data Top surface-model Bottom surface-data Bottom surface-model

320 310 300 0

500

1000

1500

t(s) Figure 3.31 Profiles of top and bottom surface temperatures during the baking of bread at a

baking temperature of 150 °C and air velocity of 1 m s−1 .

420

400

Temperature (K)

380 t = 20s t = 50s t = 100s t = 200s t = 400s t = 600s t = 800s t = 1000s t = 1200s t = 1500s

360

340

320

300 0

0.005

0.01


0.025

0.03

0.035

Figure 3.32 Spatial profiles of temperature during the baking of bread at a baking temperature of

150 °C and air velocity of 10 m s−1 .

164


Figure 3.31 shows the results of modelling the top and bottom surface temperatures of bread during baking at a baking temperature of 150 °C and air velocity of 1 m s−1 . A good agreement with the experimental data is shown (R 2 of 0.965 and 0.961 for top and bottom surface temperature, respectively). The S-REA models the top surface temperature well and shows a slight underestimation of the bottom surface temperature during baking times of 400–800 s. Figure 3.32 shows the spatial profiles of temperature during baking at a baking temperature of 150 °C and velocity of 10 m s−1 . The minimum temperature is located at a particular position inside the samples but it is not at their centre. In addition, the temperature of the top surface is higher than that at the bottom surface. This is in agreement with the results of temperature measurement which indicate that the bottom surface temperature is higher than the top surface. Since the heat received by the samples from the upper and lower side is not equal, this may result in the minimum temperature not being located at the centre of the samples. The profiles of temperature are also in agreement with those of moisture content explained previously in which the maximum moisture content is not located the centre of the samples. The study indicates that the S-REA is excellent for modelling the baking of bread. This provides a good basis for the S-REA modelling changes in quality during baking of bread in the future.

3.7

Summary In this chapter, it has been shown that the S-REA is an excellent non-equilibrium multiphase modelling approach to simulate several challenging cases of drying. The REA parameters, used in L-REA to describe the global drying rate, are implemented in S-REA for modelling local evaporation–condensation rate as affected by local variables and structures of the same material. As mentioned before, the REA parameters can be generated from one accurate drying run on the materials of concern and in a narrow range of relevant drying conditions. In S-REA, the REA is coupled with a set of equations of conservation of heat and mass transfer to yield a spatial model. The S-REA has been used to model the convective drying, intermittent drying, heat treatment under linearly increased gas temperatures and baking. The results of modelling match the experimental data well . For modelling intermittent drying and heat treatment under linearly increased gas temperatures, the equilibrium activation energy is evaluated according to the corresponding humidity and temperature in each period of treatment. Without any modification, the S-REA can model the baking process accurately. The S-REA can also yield the spatial profiles of concentration of water vapour and evaporation/condensation, useful for better understanding of drying process. To the best of our knowledge so far, the S-REA is the first model proposed and used to describe the local evaporation–condensation rate explicitly.


165

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4

Comparisons of the REA with Fickian-type drying theories, Luikov’s and Whitaker’s approaches

4.1

Model formulation Fick’s law is one of the most well-known concepts of mass transfer in the literature. Its applications in many different physical circumstances are substantial. In his book on diffusion, Cussler vividly introduced the story of how Fick came up with the law named after him (Cussler, 1984). Through an ‘impressive combination of qualitative theories, casual analogies, and quantitative experiments’ as presented by Cussler, Adolf Fick in 1855, described that: [T]he diffusion of the dissolved material . . . is left completely to the influence of the molecular forces basic to the same law . . . for the spreading of warmth in a conductor and which has already been applied with such great success to the spreading of electricity.

Basically, diffusion can be described using the same mathematics as that of Fourier’s law of heat conduction or that of Ohm’s law of electrical conduction. In reality, of course, diffusion is a mass transfer process that involves a dynamic molecular process. Parallel to the work on heat transfer, and Fourier’s law in 1822, Fick defined a one-dimensional flux of mass J: J = −AD

∂c1 , ∂z

(4.1.1)

where J takes the units of kg m−2 s−1 , A is the area across which diffusion of mass occurs, c1 is the concentration of the species of concern (kg m−3 ) and z is distance (m). The quantity D was called, ‘the constant depending on the nature of the substances’ by Fick. This quantity D is in fact the diffusion coefficient (m2 s−1 ). Although there are a few formats of Fick’s law that are known in literature, their essence remains similar. The concept has been applied to the ‘conductive’ mass transfer processes in gases, liquids and solids. Drying of porous material has been considered to be one of the most suitable candidates for applying Fick’s law of diffusion. In particular, when one is interested in knowing what moisture content distribution is like within a material being dried, in many cases this cannot be accurately measured. Because of the complexity of the process and the material involved, a ‘pure’ application of Fick’s law is not possible. The mass diffusivity for moisture in a porous material is taken as an ‘effective diffusivity’ to encapsulate several effects in addition to the pure diffusion process.

170


Spatial drying modelling is important since the spatial distributions of moisture content and/or temperature affect the product quality of materials being dried (Huang et al., 2009; Li et al., 1999; Mrad et al., 2012). For food materials, drying was shown to influence the loss of ascorbic acid, volatiles, aroma and carotenoids (Di Scala and Crapiste, 2008; Mrad et al., 2012; Ramallo and Mascheroni, 2012; Timoumi et al., 2007). Degradation of ascorbic and carotenoids was shown to be more enhanced with the increase of drying air temperature and product moisture content (Di Scala and Crapiste, 2008). The rehydration ability and ascorbic acid retention were dependent on the drying air temperature (Ramallo and Mascheroni, 2012). Similarly, the increase of temperature enhanced the loss of aroma (Timoumi et al., 2007). The survival of probiotics was also found to be very dependent on the temperature and moisture content (Huang et al., 2009). Drying was shown to induce fissuring of rice as a result of a moisture content high gradient inside the samples (Yang et al., 2003). By predicting the distributions of moisture content and temperature inside the materials being dried, product quality can be predicted. Several drying operating conditions and schemes can be made in order to maintain the product quality during drying (Chen, 2007). For non-food materials, drying was shown to induce cracking of kaolin samples. Convective drying resulted in cracking at the top of cylindrical samples where the tensile stress is at maximum, while microwave drying was damaged internally as a result of high pore pressure (Kowalski et al., 2005). Convective drying of kaolin did not result in sample cracking but applying a combination of convection and microwaves led to fracture of the samples (Kowalski and Pawlowski, 2010a,b). Controlled drying can give a desirable bending strength and sintered density of ceramic samples (Misra et al., 2002). Here, the distributions of moisture content and temperature may affect the local stress formation and the physicochemical states of the materials (Chen, 2007). The effective diffusion has been considered to be a fundamental mechanism of moisture transport in literature (Mariani et al., 2008; Pakowski and Adamski, 2007; Thuwapanichayanan et al., 2008; Vaquiro et al., 2009). The effective liquid diffusivity is usually used to lump the whole phenomenon during drying including liquid diffusion, vapour diffusion, Darcy’s flow, capillary flow and evaporation/condensation (Mariani et al., 2008; Pakowski and Adamski, 2007; Thuwapanichayanan et al., 2008; Vaquiro et al., 2009). For one-dimensional convective drying of a slab, the simplest (and indeed typical) mathematical model of moisture diffusion is expressed as: ∂ ∂C = ∂t ∂x

∂C Deff ∂x

,

(4.1.2)

where C can be considered to be concentration of liquid water (kg m−3 ), Deff is the effective diffusivity (m2 s−1 ), t is time (s) and x is distance (m). Equation (4.1.2) does not usually consider the shrinkage velocity effect and the most important parameter Deff needs to be determined experimentally.

Comparisons of the REA with other theories

4.1.1

171

Crank’s effective diffusion Crank’s effective diffusion has been used by several researchers and considered as a fundamental drying model (Arslan and Ozcan, 2011; Cihan and Ece, 2001; Corzo et al., 2008; Crank, 1975; Hassini et al., 2007). However, these studies usually neglect the Biot number (Bi) criteria and do not fully satisfy the required boundary conditions. They often do not report the operating conditions of their experiments in detail so that clear justification for the requiring validity of Crank’s effective diffusion approach (Crank, 1975) cannot be made (Chen, 2007). For slab geometry, the solution of Crank’s effective diffusion in Equation (4.1.1) can be written as (Crank, 1975): ∞ 2 X − Xe 1 8 2π = 2 exp −(2n + 1) 2 Deff ,l t , (4.1.3) X0 − Xe π n=0 (2n + 1)2 L where Deff is the effective liquid diffusivity (m2 s−1 ), X is the moisture content on a dry basis (kg water kg dry solids−1 ), which can be evaluated by (X = C/ρ s ), X0 is the initial moisture content (kg water kg dry solids−1 ), Xe is the equilibrium moisture content (kg water kg dry solids−1 ) and L is the half thickness (m) where L = 0.5b and the materials are dried symmetrically. For long time-period of drying, only the first term of Equation (4.1.2) is significant so that Equation (4.1.2) can be simplified into (Crank, 1975): 2 8 π X − Xe ≈ ln − (4.1.4) D eff ,l t. ln X0 − Xe π2 4b2 Crank’s effective diffusion should only be valid for the conditions of isothermal drying, negligible shrinkage, negligible external resistance, constant diffusivity and uniform initial moisture content (Crank, 1975). In addition, the approach should only correlate well with the experimental data for the time towards the end of drying (Srikatden and Roberts, 2006). However, this has been implemented largely where the assumptions may not be fulfilled and justifications are not made (Arslan and Ozcan, 2011; Cihan and Ece, 2001; Corzo et al., 2008; Hassini et al., 2007). For instance, Crank’s effective diffusion theory is based on the surface boundary conditions (at x = L, for slab geometry) of (Crank, 1975): −Deff ,l

dX = β(X s − X e ), dx

(4.1.5)

where Xs is the surface moisture (liquid) content (kg water kg dry solids−1 ) and β is a kind of mass transfer coefficient. Equation (4.1.5) is in contrast to the boundary conditions of vapour transfer, which can be expressed as (Chen, 2007): −Dv,eff

dCv = h m (ρv,s − ρv,b ), dx

(4.1.6)

where Dv,eff is the effective vapour diffusivity (m2 s−1 ), hm is the mass transfer coefficient in the conventional sense (m2 s−1 ), Cv is the concentration of water vapour (kg m−3 ), ρ v,s is the surface water vapour concentration (kg m−3 ) and ρ v,b is the water vapour concentration in the drying air.

172


Equations (4.1.5) and (4.1.6) are similar if, the moisture content is in equilibrium with the water vapour concentration only at the interface (Chen, 2007). However, previous publications assume this without the necessary justifications (Arslan and Ozcan, 2011; Castell-Palou, 2011; Cihan and Ece, 2001; Corzo et al., 2008; Hassini et al., 2007). Several researchers (Corzo et al., 2008; Doymaz, 2004; Kaya et al., 2007) also correlate effective diffusivity with drying air temperature, which is also fundamentally incorrect unless the material being dried has a temperature similar to that of the air (Chen, 2007). This is in contrast with Srikiatden and Roberts (2006), who inventively set up isothermal test rigs to conduct isothermal drying carefully, so that the effective diffusivity could be correlated with the sample temperature.

4.1.2

The formulation of effective diffusivity to represent complex drying mechanisms Modelling of drying processes is relatively complex and there have been several mechanistic models proposed; including liquid diffusion (Lewis, 1921), capillary flow (Buckingham, 1907), evaporation condensation (Henry, 1939), the Luikov approach (Luikov, 1975) and the Whitaker approach (Whitaker, 1977). Luikov’s approach (Luikov, 1975) assumes the thermal and moisture potential gradient within a porous body cause the vapour and liquid water transfer so that the flux of liquid water and water vapour is proportional to the thermal gradient and moisture potential gradient. As mentioned earlier, the coefficients of effective water liquid diffusivity, effective water vapour diffusivity and thermal diffusivity are implemented in linking the fluxes and gradients. Whitaker’s approach (Whitaker, 1977) implements the continuity equation of the liquid and vapour phases combined with the equation for conservation of liquid water, water vapour and energy to describe the drying process (Whitaker and Chou, 1983). The pore network approach has been proposed but it has not been rigorously experimentally validated (Nowicki et al., 1992; Prat, 1993). The applicability of the approach is limited by geometries and distributions of pore structure and network, especially for drying of food and biomaterials (Chen, 2007). The complex models mentioned here also require a lot of constants, which need to be established from several sets of drying experiments (Chen, 2007). Among these, there is little question that the effective liquid diffusion model is the simplest model proposed (Chen, 2007). The effective diffusivity should be influenced by the composition of the materials. The multi-component diffusion model is often implemented to couple this effect (Ferrari et al., 1989; Yoshida and Miyasita, 2002). Therefore, the effective diffusivity is essentially a lumped parameter whose variability is dependent on the drying condition, material structure and composition and, sometimes, sample size. The last aspect rules out the most fundamental nature of effective liquid diffusivity. For moderate drying conditions, it would be better to see effective diffusivity as a liquid depletion coefficient in order to avoid confusion with the original meaning of diffusivity (Chen, 2007).


4.1.3

173

Several diffusion-based models Various formulations of diffusion-based models have been implemented extensively by many researchers (Adhikari et al., 2004; Cihan and Ece, 2001; Corzo et al., 2008; Kar et al., 2009; Vaquiro et al., 2009). As mentioned before, Crank’s diffusion model has been applied to model convective drying (Arslan and Ozcan, 2011; Cihan and Ece, 2001; Corzo et al., 2008; Crank, 1975; Hassini et al., 2007). The diffusion-based models are usually implemented as mass balance coupled with heat balance (Adhikari et al., 2004; Guine, 2008; Kar et al., 2009; Pakowski and Adamski, 2007). As an example, Kar et al. (2009) used the diffusion-based model coupled with heat balance to model the convective drying of porcine skin (see Figures 4.1 and 4.2 for experimental setup). The skin was very thin (approximately 200 µm). The diffusion-based models can be represented as (Kar et al., 2009): ∂ ∂X = Cs ∂t ∂x

∂X Deff ,l Cs ∂x

,

(4.1.7)

where Cs is the solids concentration (kg m−3 ), X is the moisture content on a dry basis (kg water kg dry basis−1 ), x is the axial position (m), t is time (s) and Deff,l is the effective liquid diffusivity (m2 s−1 ). The initial and boundary conditions (Kar et al., 2009): t = 0, X 0 , x = 0, x = L , Cs Deff ,l

dX = 0, dx

dX = −h m (ρv,s − ρv,b ), dx

(4.1.8) (4.1.9) (4.1.10)

where L is the sample thickness, hm is the mass transfer coefficient (m s−1 ), ρ v,s is the surface water vapour concentration (kg m−3 ) and ρ v,b is the ambient water vapour concentration (kg m−3 ). Since the temperature inside the sample is essentially uniform (Chen and Peng, 2005; Kar, 2008; Kar et al., 2009), the heat balance can be written as (Kar et al., 2009): mC p

dT = h u A(Tb − T ) + U A(Tb − T ) − h m A(ρv,s − ρv,b )HV , (4.1.11) dt

where m is the sample mass (kg), T is sample temperature (K), Cp is the specific heat of the sample (J kg−1 K−1 ), A is the surface area (m2 ), hu is the upper heat transfer coefficient (W m−2 K−1 ), U is the overall bottom heat transfer coefficient (W m−2 K−1 ), Tb is the ambient temperature (K) and HV is the vaporisation heat of water (J kg−1 ). It is noted that shrinkage was also incorporated in the modelling (Kar, 2008; Kar et al., 2009).

174


K2

K1 H2

S3

H1

P PV1

B

PV2

TC2

TC1 Power board

Computer

V-6

E-5

E-4

I-8

I-7

V-5 H1-Heater 1 H2-Heater 2 P-Sample stand and platform assembly B-Micro balance TC1-Temperature controller 1 TC2-Temperature controller 2 PV1-Rheostat 1 PV2-Rheostat 2 V-6-Stop valve E-4-Drierite packed bed column V-5-Relief valve E-5-Air filter I-8-Pressure regulator I-7-Digital flowmeter K1-Type K thermocouple 1 K2-Type K thermocouple 2

Electrical line Data signal line Air line

Figure 4.1 Experimental setup for convective drying of porcine skin. [Reprinted from Chemical

Engineering Research and Design, 87, S. Kar, X.D. Chen, B.P. Adhikari and S.X.Q. Lin, The impact of various drying kinetics models on the prediction of sample temperature–time and moisture content–time profiles during moisture removal from stratum corneum, 739–755, Copyright (2012), with permission from Elsevier.]

However, as other examples, several researchers (Batista et al., 2007; Garcia-Perez et al., 2009; Loulou et al., 2006; Viollaz and Rovedo, 2002) implemented the diffusionbased model without coupling with heat balance. Garcia-Perez et al. (2009) implemented the diffusion-based model for ultrasonic-assisted drying of cube samples, which only


(a)

175

Air flow direction 33 mm 11.30 mm

7.16 mm

(b)

Drying channel

Cardboard

Top sample surface

Sample slot Two plastic layers

Drying air

Support

Aluminium plate (bottom surface)

Electronic balance Figure 4.2 (a) Overview of a sample/plate assembly for convective drying of porcine skin. (b)

Detailed of layering structure of sample support. [Reprinted from Chemical Engineering Research and Design, 87, S. Kar, X.D. Chen, B.P. Adhikari and S.X.Q. Lin, The impact of various drying kinetics models on the prediction of sample temperature–time and moisture content–time profiles during moisture removal from stratum corneum, 739–755, Copyright (2012), with permission from Elsevier.]

consists of the mass balance. The model can be written as: 2 ∂Wp ∂ Wp ∂ Wp ∂2Wp , = De + + ∂t ∂x2 ∂ y2 ∂z 2

(4.1.12)

where Wp is the moisture content on a dry basis (kg water kg dry solids−1 ), De is the effective diffusivity (m2 s−1 ), and x, y and z are the axial positions (m).

176


The initial and boundary conditions are (Garcia-Perez et al., 2009): t = 0, W p = W p0 , x = 0, y = 0, z = 0,

dW p dW p dW p = 0, = 0, = 0, dx dy dz

x = L , y = L , z = L , W p = W pe ,

(4.1.13) (4.1.14) (4.1.15)

where L is the sample thickness (m) and Wpe is the equilibrium moisture content (kg water kg dry solids−1 ). It is noted that the external resistance and shrinkage were ignored in the modelling (Garcia-Perez et al., 2009). Garcia-Perez et al. (2011) implemented the diffusion-based model with and without external resistance for ultrasonic-assisted drying of a cylindrical sample. The model can be expressed as: 2 1 ∂Wp ∂2Wp ∂ Wp ∂Wp + = De + , (4.1.16) ∂t ∂x2 r ∂r ∂r 2 where r is the radial position (m), with the initial and boundary conditions (Garcia-Perez et al., 2011): t = 0, W p = W p0 ,

(4.1.17)

x = 0,

dW p = 0, dx

(4.1.18)

r = 0,

dW p = 0. dr

(4.1.19)

For the case where external resistance is present, the surface boundary condition can be written as (Garcia-Perez et al., 2011): x = L, −De

dW p = k(ϕe − ϕair ), dx

(4.1.20)

r = R, −De

dW p = k(ϕe − ϕair ), dr

(4.1.21)

where ϕ e is the activity, ϕ air is the relative humidity in the drying air and k is the mass transfer coefficient (kg m−2 s−1 ). By ignoring external resistance, the surface boundary condition can be expressed as (Garcia-Perez et al., 2011): x = L, W p = W pe ,

(4.1.22)

r = R, W p = W pe

(4.1.23)

It is noted that Garcia-Perez et al. (2011) did not couple the diffusion-based model with heat balance. It is noted that the shrinkage was also not incorporated in the modelling (Garcia-Perez et al., 2011). The results of modelling indicated that the diffusion-based model which incorporated the external resistance resulted in better agreement with experimental data (Garcia-Perez et al., 2011).


177

More issues with boundary conditions of mass balance also occur (Chen, 2007; Zhang and Datta, 2004) and the controversies of the boundary conditions are elaborated in more detail in Section 4.2. Zhang and Datta (2004) and Chen (2007) also highlighted the importance of applying the multiphase drying approach and the use of the local evaporation–condensation rate. The detailed explanation and several issues with the local evaporation–condensation rate are discussed in Section 4.3.

4.2

Boundary conditions’ controversies For a better understanding of transport phenomena during a drying process, a multiphase approach of drying models should be applied. It consists of mass balance of water in liquid and vapour phases as well as heat balance. By this approach, the spatial profiles of moisture content, concentration of water vapour and temperature can be generated. For the multiphase approach, which does not use the source and depletion term, the model for symmetrical convective drying of a slab can be written as (Chen, 2007; Zhang and Datta, 2004) follows: The mass balance of liquid water: ∂(Cs X ) ∂ ∂(Cs X ) = Dw ; (4.2.1) ∂t ∂x ∂x The mass balance of water vapour: ∂ ∂Cv = ∂t ∂x

The heat balance: ρC p

∂C v ∂x

Dv

∂ ∂T = ∂t ∂x

k

∂T ∂x

;

(4.2.2)

;

(4.2.3)

where Cs is the solid concentration (kg solids m−3 ), X is the moisture content (kg water kg dry solids−1 ), Cv is the water vapour concentration (kg m−3 ), T is temperature (K), Dw is the capillary diffusivity (m2 s−1 ), Dv is the effective vapour diffusivity (m2 s−1 ), t is time (s), x is the axial dimension (m), ρ is the sample density (kg m−3 ), Cp is the sample specific heat (J kg−1 K−1 ) and k is thermal conductivity (W m−2 K−1 ). The initial conditions of Equations (4.2.1)–(4.2.3) are (Chen, 2007; Zhang and Datta, 2004): t = 0, X = Xo , Cv = Cv o, T = To (initial condition, uniform initial concentrations and temperature), (4.2.4) x = 0,

∂X = 0 (symmetrical boundary), ∂x

∂C v = 0 (symmetrical boundary), ∂x

(4.2.5)

(4.2.6)

178


∂T = 0 (symmetrical boundary), ∂x ∂X = 0 (no liquid water transfer), x = L , −Cs Dw ∂x −Dv

k

∂C v = h m (ρv,s − ρv,b ) (convective boundary for water ∂x vapor transfer),

(4.2.7) (4.2.8)

(4.2.9)

∂T = h(Tb − T ) − HV h m (ρv,s − ρ v,b ) (convective boundary for ∂x heat transfer with vaporization heat of water), (4.2.10)

where L is the sample at half thickness, Tb is drying air temperature (K), hm is the mass transfer coefficient (m s−1 ), h is the heat transfer coefficient (W m−1 K−1 ), HV is the vaporisation heat of water (J kg−1 ), ρ v,s is the surface water vapour concentration (kg m−3 ) and ρ v,b is the water vapour concentration at the drying medium (kg m−3 ). From Equations (4.2.1)–(4.2.3), it can be observed that there is no interaction among the liquid water and water vapour, apart from the effective diffusivity of water vapour which should be a function of porosity, dependent on the moisture content. In addition, the boundary conditions indicate that, at the interface, the vapour diffusive transport inside the samples is balanced by the convective water vapour. Therefore, the equilibrium relationship between the moisture content and concentration of water vapour has to be implemented at the boundary (Chen, 2007). However, if Equations (4.2.1)–(4.2.3) are solved simultaneously with the initial and boundary conditions shown in Equations (4.2.4)–(4.2.9), the rate of change of average moisture content would be zero, which means no drying occurs. This is not reasonable. In order to make this model work well, Equation (4.2.2) needs to be removed so that only the mass balance of liquid water and heat balance are implemented. The model can then be simplified into (Chen, 2007): ∂ ∂(Cs X ) ∂(Cs X ) = Dw , (4.2.11) ∂t ∂x ∂x ∂ ∂T ∂T ρC p = k . (4.2.12) ∂t ∂x ∂x The initial and boundary conditions are (Chen, 2007): t = 0, X = X, T = To (initial condition, uniform initial concentrations and temperature), x = 0,


∂T = 0 (symmetrical boundary), ∂x

(4.2.13) (4.2.14) (4.2.15)


x = L, −Cs Dw

∂X = h m (ρv,s − ρv,b ) (convective boundary for ∂x liquid water transfer),

179

(4.2.16)

∂T = h(Tb − T ) − HV h m (ρv,s − ρ v,b ) (convective boundary for heat ∂x transfer with vaporization). heat of water).

(4.2.17)

Solving Equations (4.2.11) and (4.2.12) simultaneously with the initial and boundary conditions shown in Equations (4.2.13)–(4.2.17) results in the spatial profiles of moisture content and temperature. However, no profiles of water vapour concentration can be generated. Equation (4.2.15) indicates that the most water evaporation occurs at the surface. Similarly, Equation (4.2.17) indicates that the surface receives the largest thermal impact from heat of evaporation. This may undermine the predictions of moisture content distribution and temperature inside the samples (Chen, 2007).

4.3

A diffusion-based model with local evaporation rate Due to the controversies of boundary conditions explained previously, it may be more appropriate to implement the multiphase drying approach with the local evaporation rate as pointed out by Chen (2007), Datta (2007) and Zhang and Datta (2004). The local evaporation rate is positive when drying occurs. It needs to be coupled with mass balance in liquid and vapour phases, as well as balance. It serves as a depletion and source term for the mass balance of liquid water and water vapour, respectively. For the symmetrical convective drying of a slab, the model can be written as (Model A) (Chen, 2007; Chong and Chen, 1999; Kar and Chen, 2011; Putranto and Chen, 2013; Zhang and Datta, 2004): The mass balance of liquid water: ∂ ∂(C s X ) ∂(C s X ) = Dw − I˙; (4.3.1) ∂t ∂x ∂x The mass balance of water vapour: ∂C v ∂ = ∂t ∂x The heat balance: ρC p

∂ ∂T = ∂t ∂x

Dv

∂Cv ∂x

+ I˙;

∂T k − I˙Hv ; ∂x

(4.3.2)

(4.3.3)

where Cs is the solid concentration (kg solids m–3 ), X is the moisture content (kg water kg dry solids–1 ), Cv is the concentration of water vapour (kg m–3 ), T is temperature (K), Dw is the capillary diffusivity (m2 s–1 ), Dv is the effective vapour diffusivity (m2 s–1 ), t is time (s), x is the axial dimension (m), ρ is the sample density (kg m–3 ), k is the thermal

180


conductivity (W m–2 K–1 ), Cp is the sample’s specific heat (J kg–1 K–1 ) and HV is the vaporisation heat of water (J kg–1 ). The initial and boundary conditions (Chen, 2007; Chong and Chen, 1999; Kar and Chen, 2011; Putranto and Chen, 2013; Zhang and Datta, 2004): t = 0, X = X o , Cv = Cvo , T = To (initial condition, uniform initial concentrations and temperature), x = 0,


∂C v = 0 (symmetrical boundary), ∂x

(4.3.4) (4.3.5) (4.3.6)

∂T = 0 (symmetrical boundary), (4.3.7) ∂x Cv,s ∂X (convective boundary x = L , u − C s Dw = h m εw − ρv,b ∂x ε for liquid transfer), (4.3.8) ∂C v Cv,s −Dv = h m εv − ρv,b (convective boundary for ∂x ε vapor transfer), (4.3.9) ∂T = h(Tb − T ) (convective boundary for heat transfer), (4.3.10) ∂x where εw and εv are the fraction of surface area covered by liquid water and water vapour, respectively. It can be observed that Equations (4.3.1)–(4.3.3) require the local evaporation rate to be expressed explicitly. However, it has not been fully understood experimentally how to establish this until the REA approach is implemented (Chen, 2007). k

4.3.1

Problems in determining the local evaporation rate Several researchers (Lu et al., 1998; Sablani et al., 1998; Sahin and Dincer, 2002; Srikiaden and Roberts, 2006) use the moisture content rate during drying as the local evaporation rate so that for the symmetrical convective drying of a slab, the model can be written as (Model B): The mass balance: ∂M ∂ ∂M = DM ; (4.3.11) ∂t ∂x ∂x The heat balance: ρC p

∂T ∂ = ∂t ∂x

k

∂T ∂x

− ρs

∂M Hv ; ∂t

(4.3.12)


181

where M is the moisture content (kg water kg dry solids−1 ), ρ s is the density of dry sample (kg m−3 ), DM is the effective diffusivity (m2 s−1 ), T is temperature (K), t is time (s), x is the axial dimension (m), ρ is the sample density (kg m−3 ), k is the thermal conductivity (J kg−1 K−1 ), Cp is the sample specific heat (J kg−1 K−1 ) and Hv is the vaporisation heat of water (J kg−1 ). The initial and boundary conditions: t =o , M = Mo , T = To (initial conditions, uniform initial moisture content and temperature), x = 0,

∂M = 0 (symmetrical boundary), ∂x

(4.3.13) (4.3.14)

∂T = h(Tb − T ) (convective boundary for liquid transfer). (4.3.15) ∂x However, Zhang and Datta (2004) and Datta (2007) mentioned that the model with a rate of moisture content used as the local evaporation rate does not satisfy mass conservation and it is more of an empirical model. Zhang and Datta (2004) analysed that the combination of Equations (4.3.1) and (4.3.2) in Model B results in: ∂Cv ∂(Cs X ) ∂Cv ∂(Cs X ) ∂ ∂ − = Dv − Dw + 2 I˙. (4.3.16) ∂t ∂t ∂x ∂x ∂x ∂x x = L, k

It can be observed that the local evaporation rate should be influenced by the moisture content as well as water vapour concentration, not only by the rate of moisture content as implemented in Model B. Therefore, Model B does not satisfy the conservation of mass for water. Moreover, Model B is found to be an empirical model as it lumps the whole phenomenon (liquid diffusion, vapour diffusion, evaporation–condensation) into diffusion marked by effective diffusivity (DM ) shown by Equation (4.3.11). The combination of Equations (4.3.1) and (4.3.2) of Model A yields: ∂Cv ∂Cv ∂(C s X ) ∂(Cs X ) ∂ ∂ + = Dv + Dw , (4.3.17) ∂t ∂t ∂x ∂x ∂x ∂x while Model B represents the mass balance as shown in Equation (4.3.11), which means Model B assumes: ∂C v ∂(C s X ) ∂M = + , (4.3.18) ∂t ∂t ∂t ∂ ∂x

DM

∂M ∂x

=

∂ ∂x

Dv

∂C v ∂x

+

∂ ∂(C s X ) Dw . ∂x ∂x

(4.3.19)

Model B uses an effective diffusivity (DM ) to substitute the effective vapour diffusivity (Dv ) and capillary diffusivity (Dw ). Similarly, Model B represents the concentration of water vapour (Cv ) and moisture content (X) as M. Therefore, Model B is an empiric model which looks like a fundamental model of diffusion. This is in agreement with Chen (2007), who mentioned that effective diffusivity should be treated as a liquid depletion coefficient in order to avoid confusion with the original Fick’s diffusivity.

182


Another inaccuracy of Model B is explained by Zhang and Datta (2004). Model B implements boundary conditions for heat transfer indicated in Equation (4.3.15). This is in contrast with the model shown by Equations (4.2.11) and (4.2.12) with the initial and boundary conditions indicated in Equations (4.2.13)–(4.2.17). The other model (shown by Equations 4.2.11–4.2.17) is reasonable since it indicates that the most evaporation occurs at the surface, and the surface receives the largest thermal impact due to heat of evaporation. This model indicates that the heat of evaporation is taken from the ambient air and used for evaporation of water, while the heat left penetrates inside by conduction. On the other hand, Model B indicates that the heat of evaporation is taken from the inside so that a negative heat source is generated. Zhang and Datta (2004) suggested that Model B results in a lower inner temperature of samples since the heat from inside is used for evaporation as mentioned. The other model results in more a uniform temperature inside the sample, as the heat for evaporation is taken from the ambient air.

4.3.2

The equilibrium and non-equilibrium multiphase drying models Two approaches can be used in multiphase drying model; equilibrium and nonequilibrium. Referring to Model A shown in Equations (4.3–1)–(4.3–3), the mass balance of liquid water and water vapour is linked by the local evaporation rate. It is necessary to represent the local evaporation rate explicitly in order to solve Model A, which consists of three partial differential equations with three dependent variables (i.e. X, Cv and T ). In the equilibrium approach, the moisture content inside the samples is assumed to be in equilibrium with water vapour concentration at any time so that the water isotherm can be used to relate these relationships. The equilibrium approach does not require the expression of local evaporation rate (Datta, 2007; Zhang and Datta, 2004). By assuming water vapour is an ideal gas, water vapour concentration can be expressed by (Zhang and Datta, 2004): Cv =

pv M , RT

(4.3.20)

where pv is water vapour pressure (Pa), M is the molecular weight of water vapour (kg kmol−1 ), R is 8314 J kmol−1 K−1 and T is the temperature (K). By assuming equilibrium conditions between the water vapour concentration and moisture content inside the samples at any time, the water vapour pressure can be written as (Zhang and Datta, 2004):

aw =

pv = pv (T, W ),

(4.3.21)

pv (T, X ) = f (T, X ), pv,sat (T )

(4.3.22)

where pv (T,W) is equilibrium water vapour pressure (Pa) and pv,sat is the saturated water pressure at particular temperature (Pa). The moisture sorption isotherm, such as GAB, Henderson, Oswin and BET (Brunauer et al., 1938; Oswin, 1946; Thompson et al., 1968; van den Berg, 1984), can be used to describe these relationships.


183

For the equilibrium approach, Model A for the symmetrical convective drying of slabs can be rearranged into (Zhang and Datta, 2004): The mass balance: ∂(Cs X ) ∂Cv ∂(C s X ) ∂Cv ∂ ∂ + = Dw + Dv . (4.3.23) ∂t ∂t ∂x ∂x ∂x ∂x It can be seen that Equation (4.3.23) is obtained by adding Equation (4.3.1) and (4.3.2). The heat balance: ∂T ∂X ∂ ∂T ∂(Cs X ) ∂ = k + − Dw C s HV . (4.3.24) ρC p ∂t ∂x ∂x ∂t ∂x ∂x Here, the local evaporation rate is obtained from water vapour conservation as mentioned in Equation (4.3.2). It is emphasised that local evaporation rate here is not simply based on the rate of moisture content during drying, as used by Model B (Zhang and Datta, 2004). One more equation, i.e. the moisture sorption isotherm indicated in Equation (4.3.22), is required to create the equilibrium multiphase drying approach. The spatial profiles of moisture content, concentration of water vapour and temperature can be generated by solving Equations (4.3.22)–(4.3.24) simultaneously. The equilibrium approach has been used by several researchers to describe convective drying and baking (Aversa et al., 2010; Ni et al., 1999; Zhang et al., 2005; Zhang and Datta, 2006). Generally, results have been in agreement with the experimental data (Aversa et al., 2010; Ni et al., 1999; Zhang et al., 2005; Zhang and Datta, 2006). Zhang et al. (2005) and Zhang and Datta (2006) implemented this approach to model the baking of bread. The model described the moisture content profiles during baking reasonably well. Similarly, the model resulted in a reasonable agreement with the experimental data of surface temperature, but an underestimation in profiles of centre temperature was observed (Zhang et al., 2005; Zhang and Datta, 2006). Coupling between the model and the equations of conservation of the drying air can also be used to predict the flow field of the drying air, as well as the spatial profiles of moisture content and temperature inside the product (Aversa et al., 2010). Although the equilibrium approach can model the drying process reasonably well, the use of the non-equilibrium approach is recommended as it is more generic and can be used to assess the validity of the equilibrium approach (Zhang and Datta, 2004). In the non-equilibrium approach, as mentioned before, the local evaporation rate needs to be expressed explicitly. It has been proposed that the internal evaporation rate can be related to the difference of equilibrium vapour pressure and the vapour pressure at particular times inside the pore spaces (Bixler, 1985; Chong and Chen, 1999; Scarpa and Milano, 2002; Zhang and Datta, 2004). Bixler (1985) proposed that the local evaporation rate can be expressed as: I˙ = c(X − X e )( pv,e − pv ),

(4.3.25)

where Xe is the equilibrium moisture content (kg water kg dry solids−1 ), pv,e is the equilibrium water vapour pressure (Pa) and c is the coefficient. However, the actual application of Equation (4.3.25) has not been tested so far (Bixler, 1985). The coefficient

184


c is expected to vary with the temperature and moisture content. The coefficient c should be arbitrarily chosen in order to match the model with the experimental data. Similarly, Ousegui et al. (2010) implemented the non-equilibrium model to describe the baking process. The local evaporation rate was expressed as (Fang and Ward, 1999; Ousegui et al., 2010; Zhang and Datta, 2004): I˙ = K (ρv,eq − ρv ),

(4.3.26)

where ρ v and ρ v,eq is the water vapour concentration (kg m−3 ) and equilibrium water vapour concentration (kg m−3 ), respectively and K is the proportionality constant dependent on ambient conditions (heat transfer coefficient, ambient fluid, etc.). K was determined by matching the model and experimental data. The increase of K was found to increase the local evaporation rate and, thus, result in the greater overall drying rate (Ousegui et al., 2010). As mentioned before, the multiphase drying model should be employed for a better understanding of transport phenomena of drying. However, as shown in Section 4.3.1, the use of moisture loss as local evaporation–condensation rate is not appropriate; the application of equilibrium multiphase drying model is restricted and use of the non-equilibrium multiphase drying model is suggested (Chen, 2007; Zhang and Datta, 2004). The REA in its lumped format, which has been shown in Chapter 2 to describe several challenging drying cases accurately, can be used to model the local evaporation– condensation rate. The internal evaporation–condensation rate can be expressed as (Kar and Chen, 2010; 2011; Putranto and Chen, 2013): I˙ = h m in Ain (Cv,s − C v ),

(4.3.27)

where hm,in is the internal mass transfer coefficient (m s−1 ), Ain is the internal surface area per unit volume (m2 m−3 ), and Cv,s and Cv are the internal-surface water vapour concentration (kg m−3 ) and water vapour concentration (kg m−3 ), respectively. The procedures for determining the internal mass transfer coefficient (hm,in ) and internal surface area per unit volume (Ain ) are presented in Kar and Chen (2010; 2011). The value of hm,in should be in the order of Dv /rp and the Ain is determined based on the area of single cells inside the samples or particles and number of cells per unit volume inside the samples (Kar and Chen, 2010; 2011; Putranto and Chen, 2013). Using the REA, Equation (4.3.27) can be rearranged into (Kar and Chen, 2010; 2011; Putranto and Chen, 2013): ˙I = h m in Ain exp −E v Cv,sat − C v , (4.3.28) RT where Ev is the activation energy (J mol−1 ), T is the sample temperature (K), R is ideal gas constant (8.314 J mol−1 K−1 ) and Cv,sat is the saturated water vapour concentration (kg m−3 ). Equation (4.3.28) can then be combined with a system of equations for conservation of heat and mass transfer to yield a non-equilibrium multiphase drying model using the REA as a source/depletion term, the S-REA, explained in Chapter 3. The following


185

section provides comparisons of the results of modelling using the diffusion-based model and the L-REA, as well as the S-REA.

4.4

Comparison of the diffusion-based model and the L-REA on convective drying In this section, the results of modelling with the diffusion-based model and the L-REA on the convective drying of mango tissues are presented. The experimental data are derived from the work of Vaquiro et al. (2009). The review of experimental details has been presented in Section 2.6. Vaquiro et al. (2009) implemented the diffusion-based model which can be written as: The mass balance: ∂X ∂X ∂X ∂ ∂ ∂X ∂ De + De + De = , (4.4.1) ∂x ∂x ∂y ∂y ∂z ∂z ∂t with the initial and boundary conditions: t = 0, X = X 0 , x = 0, y= 0, z = 0,

x = L , De ρs

∂X Mw (L , y, z, t) = −h m ∂x R

∂X Mw y = L , De ρs (x, L , z, t) = −h m ∂x R

∂X Mw z = L , De ρs (x, y, L , t) = −h m ∂x R

∂X ∂X ∂X = = = 0, ∂x ∂y ∂z

(4.4.2)

(4.4.3)

ϕ(L , y, z, t)Ps (L , y, z, t) ϕ∞ P s∞ , − T (L , y, z, t) T∞ (4.4.4)

ϕ(x, L , z, t)Ps (x, L , z, t) ϕ∞ P s∞ − , T (x, L , z, t) T∞ (4.4.5)

ϕ(x, y, L , t)Ps (x, y, L , t) ϕ∞ P s∞ , − T (x, y, L , t) T∞ (4.4.6)

where De is the effective diffusivity (m2 s−1 ), ρ s is the density of solid (kg m−3 ), X is the moisture content on dry basis (kg H2 O m−3 ), hm is the mass transfer coefficient (m s−1 ), Mw is the molecular weight of water (kg kmol−1 ), ϕ is the surface relative humidity, Ps is the saturated vapour pressure (Pa), T is sample temperature (K), ϕ is the drying air relative humidity, Ps is the vapour pressure in drying air (Pa), T is the drying air temperature and L is the thickness of sample (m) while the heat balance can

186


be expressed as: ∂T ∂ ∂T ∂ ∂T ∂ k + k + k ∂x ∂x ∂y ∂y ∂z ∂z ∂T ∂ X ∂T ∂ X ∂T ∂ X ∂T − De ρds C pw + + , = ρds (C pds + XC pw ) ∂t ∂x ∂x ∂y ∂y ∂z ∂z (4.4.7) with the initial and boundary conditions: t = 0,T = T0 , x = 0, y = 0, z = 0,

x = L , −k

y = L , −k

z = L , −k

∂T ∂T ∂T = = = 0, ∂x ∂y ∂z

(4.4.8)

(4.4.9)

∂X ∂T (L , y, z, t) = h[T (L , y, z, t) − T∞ ] − De ρds (L , y, z, t)Q s , ∂x ∂x (4.4.10) ∂T ∂X (x, L , z, t) = h[T (x, L , z, t) − T∞ ] − De ρds (x, L , z, t)Q s , ∂y ∂y (4.4.11) ∂X ∂T (x, y, L , t) = h[T (x, y, L , t) − T∞ ] − De ρds (x, L , z, t)Q s , ∂z ∂z (4.4.12)

where k is the thermal conductivity of sample (W m−1 K−1 ), h is the heat transfer coefficient (W m−2 K−1 ), ρ ds is the density of dry solid (kg m−3 ), Qs is the heat of evaporation of water (J kg−1 ), Cpds is the specific heat of the dry solid (J kg−1 K−1 ) and Cpw is the specific heat of water (J kg−1 K−1 ). The L-REA, as explained in Section 2.1, is implemented here. Basically, the original formulation of the L-REA, as mentioned in Equation (2.1.4), is used and combined with the prediction of the temperature distribution as described in Section 2.6.2 and shown in Equation (2.6.14) since the sample is relatively thick. The relative activation energy (Ev /Ev,b ) is generated from experiments on convective drying at a drying air temperature of 55 °C. The comparisons of modelling using the diffusion-based model implemented by Vaquiro et al. (2009) and the L-REA are shown in Figures 4.3 and 4.4. Figure 4.3 shows the profiles of moisture content modelled using the diffusion-based model (Vaquiro et al., 2009) and the L-REA. The diffusion-based model describes the moisture content profiles of the convective drying at drying air temperatures of 55° and 65 °C well, indicated by R2 higher than 0.996 (Vaquiro et al., 2009). However, a slight overestimation of the drying rate is shown for the convective drying at a drying air temperature of 45 °C (Vaquiro et al., 2009). The L-REA models the moisture content well for convective drying at drying air temperatures of 45°, 55° and 65 °C. The results of modelling using the L-REA

187


10 Data 45°C L-REA 45°C Data 55°C L-REA 55°C Data 65°C L-REA 65°C Diffusion-based by Vaquiro et al. (2009) 45°C Diffusion-based by Vaquiro et al. (2009) 55°C Diffusion-based by Vaquiro et al. (2009) 65°C

9


8 7 6 5 4 3 2 1 0

0

0.5

1

1.5

2

2.5 t(s)

3

3.5

4

4.5

5 × 104

Figure 4.3 Moisture content profiles from the convective drying of mango tissues modelled using

the L-REA and diffusion-based model (Vaquiro et al., 2009). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of Drying of Food Materials with Thickness of Several Centimeters by the Reaction Engineering Approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.]

match well the experimental data indicated by an R2 higher than 0.996 (Putranto et al., 2011a). Figure 4.4 shows the centre temperature profiles modelled using both models. At a drying air temperature of 45 °C, the diffusion-based model (Vaquiro et al., 2009) results in a kink at the beginning of the drying period not shown by the L-REA. Similarly, the diffusion-based (Vaquiro et al., 2009) model shows a kink in the centre temperature profiles at the beginning of the drying period for convective drying at drying air temperatures of 45°, 55° and 65 °C (Vaquiro et al., 2009). On the other hand, the L-REA models well the centre temperature profiles of convective drying at drying air temperatures of 55° and 65 °C, indicated by a R2 higher than 0.984 (Putranto et al., 2011b). Both models show a slight overestimation in temperature profiles of convective drying at a drying air temperature of 65 °C between drying times of 10 000 and 30 000 s. Based on the case study mentioned, the diffusion-based model shown in conjunction with the initial and boundary conditions indicated in Equations (4.4.1)–(4.4.12) seems to not be able to model the convective drying well, particularly the temperature profiles which show a kink at the beginning of drying. On the other hand, the L-REA combined with the prediction of temperature distribution shown in Equation (2.6.14) can describe both the moisture content and temperature profiles well. This indicates that the L-REA performs better than the diffusion-based model in describing convective drying.

188


340


330

320

310 Data 45°C L-REA 45°C Data 55°C L-REA 55°C Data 65°C L-REA 65°C Diffusion-based by Vaquiro et al. (2009) 45°C Diffusion-based by Vaquiro et al. (2009) 55°C Diffusion-based by Vaquiro et al. (2009) 65°C

300

290

280

0

0.5

1

1.5

2

2.5 t(s)

3

3.5

4

4.5

5 × 104

Figure 4.4 Temperature profiles from convective drying of mango tissues modelled using the L-REA and diffusion-based model (Vaquiro et al., 2009). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.]

4.5

Comparison of the diffusion-based model and the S-REA on convective drying In this section, the results of modelling using the diffusion-based model and the S-REA on the convective drying of mango tissues (Vaquiro et al., 2009) are compared. As explained in Chapter 3, the S-REA is a non-equilibrium multiphase drying model with the REA as the local evaporation rate. The experimental details of Vaquiro et al. (2009) are presented in Section 2.6. The diffusion-based model, together with the initial and boundary conditions, is presented in Section 4.4 and indicated in Equations (4.4.1) to (4.4.12). The details of S-REA modelling in conjunction with the initial and boundary conditions for the convective drying of mango tissues (Vaquiro et al., 2009) are presented in Section 3.3.1. Figures 4.5–4.6 show the comparisons of the results of modelling convective drying (Vaquiro et al., 2009) using the diffusion-based model and the S-REA. Figure 4.5 shows the moisture content profiles modelled using both approaches. It can be seen that both the diffusion-based model (Vaquiro et al., 2009) and the S-REA model the moisture content profiles well during drying (Putranto and Chen, 2013). Both the diffusion-based model (Vaquiro et al., 2009) and the S-REA are accurate enough to model the moisture content profiles well (R2 higher than 0.996 for both the S-REA and diffusion-based model).

189


10 S-REA 45°C Diffusion-based model by Vaquiro et al. (2009) 45°C Data 45°C S-REA 55°C Diffusion-based model by Vaquiro et al. (2009) 55°C Data 55°C S-REA 65°C Diffusion-based model by Vaquiro et al. (2009) 65°C Data 65°C


9 8 7 6 5 4 3 2 1 0

0

0.5

1

1.5

2

2.5 t(s)

3

3.5

4

4.5

5 × 104

Figure 4.5 Moisture content profiles from the convective drying of mango tissues modelled using the S-REA and diffusion-based model (Vaquiro et al., 2009). [Reprinted from AIChE Journal, A. Putranto and X.D. Chen, Spatial reaction engineering approach as an alternative for non-equilibrium multiphase mass-transfer model for drying of food and biological materials, DOI 10.1002/aic.13808, Copyright (2012), with permission from John Wiley & Sons, Inc.]

Figure 4.6 shows the centre temperature profiles modelled using the diffusion-based model (Vaquiro et al., 2009) and the S-REA. As mentioned before, the diffusion-based model shows a kink in the centre temperature profiles at the beginning of the drying period for convective drying at drying air temperatures of 45°, 55° and 65 °C (Vaquiro et al., 2009). On the other hand, the S-REA models the centre temperature accurately for all cases of the convective drying of mango tissues. The results of modelling using the S-REA match well with the experimental data, indicated by R2 higher than 0.985 (Putranto and Chen, 2013). In addition, the S-REA model yields advantages of generating the profiles of concentration of water vapour (as shown in Figure 3.6 in Chapter 3), as well as local evaporation rate during drying (as shown in Figure 3.8 of Chapter 3) which is useful for better understanding of transport phenomena during drying (Putranto and Chen, 2013). It can be shown here that the diffusion-based model (Vaquiro et al., 2009) does represent convective drying of mango tissues well, as shown by a kink of the temperature profiles. On the other hand, the S-REA is accurate at modelling both the moisture content and temperature profiles of convective drying. The S-REA formulation in conjunction with the initial and boundary conditions can represent the convective drying very well (Putranto and Chen, 2013). The accuracy of the S-REA, which is a non-equilibrium

190


340

330

Temperature (K)

320

310

Data 45°C S-REA 45°C Data 55°C S-REA 55°C Data 65°C S-REA 65°C Diffusion-based model by Vaquiro et al. (2009) 45°C Diffusion-based model by Vaquiro et al. (2009) 55°C Diffusion-based model by Vaquiro et al. (2009) 65°C

300

290

280

0

0.5

1

1.5

2

2.5 t(s)

3

3.5

4

4.5

5 × 104

Figure 4.6 Temperature profiles from the convective drying of mango tissues modelled using the

S-REA and diffusion-based model (Vaquiro et al., 2009). [Reprinted from AIChE Journal, A. Putranto and X.D. Chen, Spatial reaction engineering approach as an alternative for non-equilibrium multiphase mass-transfer model for drying of food and biological materials, DOI 10.1002/aic.13808, Copyright (2012), with permission from John Wiley & Sons, Inc.]

multiphase model, with the use of REA for local evaporation rate, may prove the importance of multiphase model with the source term as previously mentioned by Zhang and Datta (2004) and Chen (2007).

4.6

Model formulation of Luikov’s approach Luikov (1975) developed a theory of simultaneous heat and mass transfer in a porous body based on irreversible thermodynamics. A porous body can be considered to consist of four components: a dry solid, water vapour, liquid water and air within the pore. It was postulated that the thermal and moisture potential gradient within a porous body cause the vapour and liquid water transfer so that the flux of liquid water and water vapour is proportional to the thermal gradient and moisture potential gradient. The relationships can be explained in two- and three-way coupled system of partial differential equations (Luikov, 1975). The two-way coupled system assumes that the pressure inside the capillary body is constant during the process. It is postulated that the thermal and moisture potential gradient within a porous body cause vapour and liquid water transfer so that the flux of liquid water and water vapour is proportional to the thermal gradient and moisture potential gradient. The heat and mass transfer

191


is interdependent and several coefficients are used to explain the interdependency. For this purpose, two sets of partial differential equations in terms of moisture content and temperature are established (Luikov, 1975). The two-way coupled system has been used by several researchers to model the simultaneous heat and mass transfer processes (Liu and Cheng, 1990; Mikhailov and Shishedjiev, 1975; Younsi et al., 2006a,b; 2007). In addition, the three-way coupled system incorporates the pressure gradient inside a body as a result of the presence of water vapour, which results in moisture movement by filtration. In this system, it is assumed that the thermal, moisture and pressure gradient within a capillary body lead to moisture movement within the body. Unlike the two-way system, another set of partial differential equations in terms of pressure is established to represent the pressure gradient inside the body (Luikov, 1975). For the two-way system, it is assumed that the vapour movement inside the body is due to molecular transport and the concentration of vapour equilibrates thermodynamically with the concentration of liquid. The mass flow of vapour can be written as (Luikov, 1975): T ρ0 ∇t, j1 = −ερ D∇ρ10 = −am1 ρ0 ∇u − am1

(4.6.1)

where j1 is the flux of vapour (kg m−2 s−1 ), ε is the dimensionless factor characterising resistance to vapour diffusion in the moisture body, D is the diffusivity of vapour in air (m2 s−1 ), ρ is the density (kg m−3 ), ρ 0 is the density of dry body (kg m−3 ), u is the moisture content of body (kg kg−1 ), t is temperature (K) and am1 is the vapour diffusion coefficient which can be expressed as (Luikov, 1975): ρ dρ10 am1 = ε D , (4.6.2) ρ0 du T where ρ 10 is the relative vapour concentration, while am1 T is the thermal diffusion coefficient, which can be written as (Luikov, 1975): ρ dρ10 T am1 = εD . (4.6.3) ρ0 du u The mass flow of liquid can be expressed in a similar way to that of vapour as (Luikov, 1975): T ρ0 ∇t, j2 = −ερ D∇ρ10 = −am2 ρ0 ∇u − am2

(4.6.4)

where j2 is the flux of liquid (kg m−2 s−1 ) and am2 and am2 T are the vapour and thermal diffusion coefficients dependent on the moisture content and temperature of the body. Fourier’s law can be used to explain the heat flux inside the body as (Luikov, 1975): q = −k∇t,

(4.6.5)

where q is the heat flux (W m−2 ) and k is the thermal conductivity (W m−2 K−1 ).

192


Equations (4.6.1), (4.6.4) and (4.6.5) can be rearranged into (Luikov, 1975): ∂u = K 11 ∇ 2 u + K 12 ∇ 2 t, ∂τ ∂t = K 21 ∇ 2 u + K 22 ∇ 2 t, ∂τ where K 11 = am = am1 + am2 , K 12 =

T am1

+

T am2 ,

Lεam , c Lεam δ K 22 = a + , c T a T + am2 , δ = m1 am1 + am2 K 21 =

(4.6.6) (4.6.7) (4.6.8) (4.6.9) (4.6.10) (4.6.11) (4.6.12)

where L is the specific heat of phase transition (J kg−1 K−1 ), α is the thermal diffusivity (m2 s−1 ) and c is the specific heat of sample (J kg−1 K−1 ). For a three-way coupling system, the pressure gradient may occur due to the penetration of humid air through capillary system inside the body. The transfer can be described by Darcy’s law as (Luikov, 1975): j f = −k f ∇ p,

(4.6.13)

where kf is the total filtration coefficient and p is pressure. By incorporating Equation (4.6.13), the mass and heat transfer inside the capillary body can be expressed as (Luikov, 1975): ∂u = K 11 ∇ 2 u + K 12 ∇ 2 t + K 13 ∇ 2 p, ∂τ ∂t = K 21 ∇ 2 u + K 22 ∇ 2 t + K 23 ∇ 2 p, ∂τ ∂p = K 31 ∇ 2 u + K 32 ∇ 2 t + K 33 ∇ 2 p, ∂τ where K 13 = am δ f , am K 23 = εL δ p , c εam K 33 = α f − δp, Cf am , K 31 = −ε Cf am δ K 31 = −ε δ, Cf kf , αf = ρC f kf , δf = am ρ 0 where Cf is the body capacity for the humid air with filtration.

(4.6.14) (4.6.15) (4.6.16) (4.6.17) (4.6.18) (4.6.19) (4.6.20) (4.6.21) (4.6.22) (4.6.23)


193

The boundary conditions on the surface can be written as (Luikov, 1975): am ρ0 (∇u)s + amT (∇t)s + js = 0, −(k +

T )(∇t)s Lam2

where

− Lam2 ρ0 (∇u)s + qs = 0,

(4.6.24) (4.6.25)

js = βρ0 (u s − u e ),

(4.6.26)

qs = h(ta − ts ),

(4.6.27)

where js is the mass flux at surface (kg m−2 s−1 ), qs is heat flux at surface (W m−2 K−1 ), us is the surface moisture content, ue is the equilibrium moisture content, ta is the ambient temperature and ts is the surface temperature. Equation (4.6.24) implies that the moisture supplied to the surface due to thermodynamic forces is equal to the one left from the surface to the ambience, while Equation (4.6.25) indicates that heat received from the ambience is used for evaporation and any left penetrates inside the body. Luikov’s approach (Luikov, 1975) has been used by several researchers to model drying of porous materials (Irudayaj and Wu, 1994; 1996; Kulasiri and Samarasinghe, 1996; Thomas et al., 1980). The two-way coupled system of Luikov’s approach was implemented to model timber drying and a reasonable agreement with the experimental data was shown. The model was solved numerically using the finite element method. The results of modelling confirmed that the temperature gradient inside the sample can be neglected (Kulasiri and Samarasinghe, 1996; Thomas et al., 1980). Irudayaj and Wu (1994) used the three-way coupled version of Luikov’s approach and solved it by using the finite element method to model the convective drying of silicon gel. The numerical model matched the exact solution well. For low Fourier numbers, the results are not stable but the stable results are achieved if high Fourier numbers, higher geometries and nonlinear properties are used (Irudayaj and Wu, 1994). In a similar investigation, Irudayaj and Wu (1996) showed that the model can be used to model the convective drying of spruce samples well. The parameters of moisture conductivity and ratio of vapour to total diffusion were found to be very sensitive to the profiles of moisture content and temperature. Nevertheless, these parameters need to be extracted from experimental data and determined by error-minimization (Irudayaj and Wu, 1996). Luikov’s approach (Luikov, 1975) has been also used by (Younsi et al., 2006a) to model the heat treatment of wood under a constant heating rate, which is essentially a drying process under linearly increased gas temperatures. The experimental details of Younsi et al. (2006a,b; 2007) are reviewed briefly in Section 2.9. For modelling the heat treatment of wood using Luikov’s approach (1975), it is assumed that the wood sample is isotropic and homogeneous, shrinkage is negligible, no heat is generated inside the wood and capillary forces are much stronger than gravity. The moisture transfer can be represented as (Younsi et al., 2006a): ∂U km δ (4.6.28) =∇ ∇T + km ∇U , ρCm ∂t Cm while the heat transfer can be expressed as (Younsi et al., 2006a): ∂T εkm δ ρCq =∇ kq + ∇T + (ελkm )∇U . ∂t Cm

(4.6.29)

194


The initial and boundary conditions of Equations (4.6.28) and (4.6.29) are (Younsi et al., 2006a): t = 0, U = U0 ,

(4.6.30)

t = 0, T = T0 , dT −kq = h q (T − Tg ) + (1 − ε)λh m (U − Ug ), at dn dU km δ dT −km = + h m (U − Ug ), at dn Cm dn

(4.6.31) (4.6.32) (4.6.33)

where t is time (s), n is the spatial direction (x,y,z), U is the moisture potential (°M−1 ), T is temperature (K), hq is the heat transfer coefficient (W m−2 K−1 ), hm is the mass transfer coefficient (kg H2 O m−1 s−1 °M−1 ), Cq is the heat capacity (J kg−1 K−1 ), Cm is the moisture capacity (kg H2 O, kg−1 °M−1 ), δ is the thermal gradient coefficient (kg H2 O kg−1 K−1 ), kq is the conductivity (W m−1 K−1 ), km is the moisture diffusivity (kg H2 O m−1 s−1 °M−1 ), λ is the latent heat (J kg−1 ), ρ is the dry body density (kg m−3 ), Ug is the gas moisture potential (°M−1 ), Tg is the gas temperature (K), ε is the ratio of vapour diffusion to total moisture diffusion and is the boundary surface of heat and mass transfer. For a more general expression, Equations (4.6.28) and (4.6.29) can be represented as (Younsi et al., 2006a): ∂T ∂U + A12 = ∇(K 11 ∇T + K 12 ∇U ), ∂t ∂t ∂T ∂U + A22 = ∇(K 21 ∇T + K 22 ∇U ). A21 ∂t ∂t

A11

(4.6.34) (4.6.35)

Aij and Kij are coefficients which can be represented as (Younsi et al., 2006a): A11 = ρCq ,

(4.6.36)

A12 = A21 = 0, ελk M δ K 11 = kq + , Cm K 12 = ελkm , km δ K 21 = , Cm K 22 = km .

(4.6.37) (4.6.38) (4.6.39) (4.6.40) (4.6.41)

The heat and mass transfer coefficient can be estimated from established correlations (Incropera and DeWitt, 2002) while the properties of wood can be predicted from Simpson and Tenwold (1999). The heat and mass balances shown in Equations (4.6.34) and (4.6.35) in conjunction with the initial and boundary conditions shown in Equations (4.6.36)–(4.6.41) are solved simultaneously in order to give the profiles of moisture content and temperature during drying.


4.7

195

Model formulation of Whitaker’s approach Whitaker’s approach (Whitaker, 1977) was implemented by several researchers to describe drying of a porous body (Baggio et al., 1997; Pavon-Melendez et al., 2002; Whitaker and Chou, 1983; Younsi et al., 2006a,b; 2007). This approach proposed detailed mechanisms of transport in microscale and macroscale structures based on a known distribution of the structures. Equations of conservation of momentum, mass and heat in solid, vapour and liquid phases and their local volume behaviours followed by volume averaging methods, are used to describe the mechanisms. Darcy’s law is usually used to describe the momentum transfer in liquid and gas phases while the mass transfer considers capillary action as well as evaporation–condensation. The heat transfer incorporates the convective transport, conduction and vaporisation–condensation heat by assuming local thermal equilibrium within solid, gas and liquid phases (Whitaker, 1977; Whitaker and Chou, 1983). The detailed modelling of Whitaker’s approach can be expressed as (Whitaker, 1977): The total thermal energy equation:

ρC p

d T + [ρβ (C p )β vβ + ργ γ (C p )γ vγ γ ]∇T dt ˙ = ∇(keff ∇ T ) + , + h vap m

(4.7.1)

where t is time (s), ρ is the spatial average of density (kg m−3 ), Cp is specific heat (J kg−1 K−1 ), ρ β is density of the liquid phase (kg m−3 ), ρ γ is density in the gas phase (kg m−3 ), (Cp )β is specific heat in the liquid phase, (Cp )γ is specific heat in the gas phase, vβ is the phase average of velocity in the liquid phase, vγ γ is the phase average velocity in the gas phase, T is temperature (K), hvap is the vaporisation heat of ˙ is the evaporation rate water (J kg−1 ), keff is the thermal conductivity (W m−2 K−1 ), m per unit volume (kg m−3 s−1 ) and is the heat generation (W m−3 ). The liquid phase equation of motion:

vβ =

−εβ ξ K β [kε ∇εβ + k T ∇T − (ρβ − ργ )g], μβ

(4.7.2)

where ε β is the volume fraction of liquid phase, Kβ is the liquid phase permeability (m2 s−1 ), μβ is the viscosity of liquid phase (N s m−2 ), kε is −∂ pc /∂εβ , k T is −∂ pc /∂ T , g is the gravitational constant (m2 s−1 ), ξ is a function topology of liquid phase and pc is the capillary pressure (N m−2 ). The liquid phase continuity equation: ˙

m ∂εβ = 0; + ∇ vβ + ∂t ρβ

(4.7.3)

The gas phase equation of motion:

vγ =

−K γ {εγ [∇ pγ − p0 γ − ργ g]}; μγ

(4.7.4)

196


where εγ is the volume fraction of the gas phase, Kγ is gas phase permeability (m2 s−1 ), μγ is viscosity of gas phase (N s m−2 ), p0 is the reference pressure (N m−2 ) and pγ is pressure of the gas phase (N m−2 ). The gas phase continuity equation: ∂ ˙ (εγ ργ γ ) + ∇( ργ γ vγ ) = m; ∂t The gas phase diffusion equation: ∂ 2 ∇( ρ2 γ/ ργ γ ]; (εγ ρ2 γ ) + ∇( ρ2 γ vγ ) = ∇[ ργ γ Deff ∂t

(4.7.5)

(4.7.6)

where ρ2 γ is the density of water vapour (kg m−3 ) and Deff 2 is the effective diffusivity of water vapour in gas (m2 s−1 ). The volume constraint: εσ + εβ + εγ = 1;

(4.7.7)

where ετ is the volume fraction of solid phase. The thermodynamics relations:

p1 γ = ρ1 γ R1 T ;

(4.7.8)

p2 γ = ρ2 γ R2 T ;

(4.7.9)

ργ γ = ρ1 γ + ρ2 γ ;

(4.7.10)

pγ γ = p1 γ + p2 γ ; h vap 1 1 γ 0

p1 = p1 exp − (2σβγ /rρβ R1 T ) + − ; R1

T T0

(4.7.11) (4.7.12)

where ρi γ is the pressure of ith species in gas phase (N m−2 ), where ρi γ is the density of ith species in gas phase (kg m−3 ), Ri is the gas constant for ith species (N m kg−1 K−1 ), p1 0 is the reference pressure of ith species in the gas phase (N m−2 ) and T0 is the reference temperature. As a result, Whitaker’s approach provides the 12 equations shown in Equations (4.7.1)– (4.7.12) which need to be solved simultaneously in order to yield the profiles during drying. The difficulty of this approach would be in determining the transport parameters involved in the equations. For this purpose, a theoretical basis and simplified drying theory are needed to estimate the parameters (Whitaker, 1977). Whitaker’s approach (Whitaker, 1977) has been used to model several drying process and the results have good agreement with the experimental data (Hager et al., 2000; Torres et al., 2011). Torres et al. (2011) implemented the approach to model the vacuum drying of wood and the results can yield information about transport of liquid and vapour in wood. The model was also coupled with the equations of conservation in the drying medium so the dynamics of water vapour concentration inside the chamber could be predicted. It was shown that the evaporation rate depends on the levels of pressure and temperature, while the pressure inside the sample increases as drying progresses because the liquid water is expulsed during drying (Torres et al., 2011). In addition, Hager et al. (2000) use Whitaker’s approach to model steam drying of porous Al2 O3 and ceramic


197

spheres. It was indicated that the total drying rate and outlet steam temperature matched well with experimental data, but the internal temperature showed some deviations from it. Although the model is fairly accurate, Whitaker’s approach is very computationally demanding (Truscott and Turner, 2005). Whitaker’s approach (1977) was also implemented by Younsi et al. (2006b; 2007) to describe the heat treatment of wood under a constant heating rate whose experimental details are presented in Section 2.6. Whitaker’s approach (1977), implemented by Younsi et al. (2006b; 2007), can be expressed as: − ρd

∂M ∂ Jx ∂ Jy ∂ Jz = ∇J = + + , ∂t ∂x ∂y ∂z

(4.7.13)

where ρ d is the dry wood density (kg m−3 ) and J is the total mass flux vector (kg m−2 s−1 ), which can be written as: J = Jv + Jb + J f ,

(4.7.14)

where Jv , Jb and Jf are the water vapour flux (kg m−2 s−1 ), bound water flux (kg m−2 s−1 ) and liquid water flux (kg m−2 s−1 ), respectively. The vapour flow is postulated due to the vapour pressure gradient, since the total pressure gradient inside the sample can be neglected so that the water vapour flux can be written as (Younsi et al., 2006b; 2007): Jv =

−m v Deff ∇ Pv , RT

(4.7.15)

where mv is the molar mass of vapour (kg mol−1 ), R is the ideal gas constant (J mol1 ), Pv is the partial vapour pressure of water (Pa) and Deff is effective vapour diffusivity (m2 s−1 ). The bound water diffusion is governed by the chemical potential difference and it occurs when the moisture content is below the fibre saturation point (FSP). The bound water flux can be written as (Stanish et al., 1986): −Db Pv T Jb = − 8.314 ln ∇T − 187 + 35.1 ln mv 298.15 101.325 T (4.7.16) + 8.314 ∇ Pv , Pv where Db is the diffusion coefficient of bound water (m−2 s−1 ) and T is temperature (K). When moisture content is higher than FSP, the free water transfer occurs due to capillary flow. The liquid water flux can be represented as (Younsi et al., 2006b; 2007): Jf =

−K l ρl ∇ Pc , μl

(4.7.17)

where Pc is the capillary pressure (Pa), Kl is the permeability, ρ l is the density of liquid water (kg m−3 ) and μl is the viscosity of liquid water (kg m−1 s−1 ).

198


Equation (4.7.17) can be rearranged into (Younsi et al., 2006b; 2007): Jf =

−0.61 × 104 K l ρl (Mmax − MFSP )0.61 ∇ M. μl (M − MFSP )

(4.7.18)

where MFSP is the moisture content at the fibre saturation point and Mmax is the moisture content if the entire void is occupied by water. By substituting Equations (4.7.15), (4.7.16) and (4.7.18) into Equations (4.7.13) and (4.7.14), the conservation of mass can be written as (Younsi et al., 2006b; 2007): ρd

∂M = ∇ (D M ∇ M + DT ∇T ) , ∂t

(4.7.19)

where DM and DT are positive constants, which can be expressed as (Younsi et al., 2006b; 2007): ∂ 1.2146 × 10−4 m v δ 0.75 ∂M R Db T ∂ αβ K l ρl (Mmax − M F S P )β + 8.314 , + mv ∂ M μl (M − M F S P )(1+β)

D M = Psv

DT =

∂ Psv 1.2146 × 10−4 m v δ 0.75 ∂ Psv + Psv R ∂T ∂T Db Pv T − − 8.314 ln 187 + 35.1 ln mv 298.15 101325 ∂ T ∂ Psv + Psv , −8.314 Pv ∂T ∂T

(4.7.20)

(4.7.21)

where Psv is the saturation vapour pressure of water (Pa) and ψ is the coefficient, α = 104 , β = 0.61. The equation of conservation of energy can be written as (Younsi et al., 2006b; 2007): −ρd ρa u a + ρv u v + ρb u b + ρ f u f + ρd u d + ∇(Ja h a + Jv h v + Jb h b + J f h f ) = ∇(k∇T ),

(4.7.22)

where ρ, u, h and k indicate; the density, the specific internal energy, the specific enthalpy and the thermal conductivity, respectively. The subscripts a, v, b, f and d indicate; the dry air, the water vapour, the bound water, the free water and the dry wood, respectively. The energy balance can be represented as (Younsi et al., 2006b; 2007): ρC p

∂T = ∇ k M ∇ M + keff ∇T , ∂t

(4.7.23)


199

where kM and keff are positive constants, which can be expressed as (Younsi et al., 2006b; 2007): ∂ 1.2146 × 10−4 m v δ 0.75 Db T ∂ + h b 8.314 ∂M R mv ∂ M αβ K l ρl (Mmax − M F S P )β +hf , μl (M − M F S P )(1+β) 1.2146 × 10−4 m v δ 0.75 ∂ Psv ∂ Psv = k + hv + Psv R ∂T ∂T P ! T v Db − 8.314 187 + 35.1 298.15 ∂ln ln 101325 − hb . Psv ∂ T m v −8.314 Pv ∂ T + Psv ∂ T

k M = h v Psv

keff

(4.7.24)

(4.7.25)

For more general expression, Equations (4.7.19) and (4.7.23) can be represented as (Younsi et al., 2006b; 2007): ∂M ∂T (4.7.26) + A12 = ∇(K 11 ∇ M + K 12 ∇T ), ∂t ∂t ∂M ∂T A21 (4.7.27) + A22 = ∇(K 21 ∇ M + K 22 ∇T ), ∂t ∂t where Aij and Kij are coefficients, which can be represented as (Younsi et al., 2006b; 2007): A11

A11 = ρd ,

(4.7.28)

A12 = A21 = 0,

(4.7.29)

A22 = (ρC p )eff ,

(4.7.30)

K 11 = D M ,

(4.7.31)

K 12 = DT ,

(4.7.32)

K 21 = k M ,

(4.7.33)

K 21 = keff .

(4.7.34)

The mass and heat balances shown in Equations (4.7.26) and (4.7.27) are solved with the initial and boundary conditions, which can be written as: t = 0, M = M0 ,

(4.7.35)

t = 0, T = T0 .

(4.7.36)

At the surface, the boundary conditions can be written as (Younsi et al., 2006b; 2007):

kM

Ts = T f ,

(4.7.37)

Cs = C f ,

(4.7.38)

dM dT dT f dC f + keff = kf + DHV , dn dn dn dn dM dT dC f + DT =D , DM dn dn dn

(4.7.39) (4.7.40)

200



0.08 Experimental data L-REA Luikov approach: Younsi et al. (2006a)

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

0.5

1

1.5 t(s)

2

2.5

3 × 104

Figure 4.7 Moisture content profiles from the heat treatment of wood modelled using the L-REA

and Luikov’s approach. [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.]

where Ts is the temperature (K), Tf is the gas temperature (K), Cs is the surface water concentration (kg m−3 ), Cf is the water vapour concentration in gas, kf is the bulk gas thermal conductivity (W m−1 K−1 ), D is the diffusivity of water in bulk gas (m2 s−1 ) and HV is the vaporisation heat (J kg−1 ). The spatial profiles of moisture content and temperature during the heat treatment of wood can be generated by solving the heat and mass balances shown in Equations (4.7.26) and (4.7.27), in conjunction with the initial and boundary conditions shown in Equations (4.7.37)–(4.7.40). The linearly increased gas temperature is implemented in Equations (4.7.37), (4.7.39) and (4.7.40).

4.8

Comparison of the L-REA, Luikov’s and Whitaker’s approaches for modelling heat treatment of wood under constant heating rates Figure 4.7 indicates the moisture content profiles of heat treatment of wood with a final temperature of 200 °C, heating rate of 20 °C h−1 and initial moisture content of 0.07 kg water kg dry solids−1 . Both the L-REA and Luikov’s approach (implemented by Younsi et al., 2006a) match reasonably well with the experimental data of moisture content (R2 of 0.993 and 0.987 for the L-REA and Luikov’s approach, respectively). Both approaches slightly underestimate the drying rate during treatment times shorter than 10 000 s. The L-REA resulted in better agreement with the experimental data during treatment times

201


Table 4.1 Experimental settings of the heat treatment of wood (Younsi et al., 2007).

Case

Final gas temperature (ºC)

Heating rate (ºC h−1 )


1 2

220 220

20 10

0.125 0.12

480 Experimental data L-REA Luikov approach: Younsi et al. (2006a)

460 440

Temperature (K)

420 400 380 360 340 320 300 280

0

0.5

1

1.5

2 t(s)

2.5

3

3.5 × 104

Figure 4.8 Temperature profiles from the heat treatment of wood modelled using the L-REA and

Luikov’s approach. [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.]

above 10 000 s. The temperature profiles of the experiment are shown in Figure 4.8. Both the L-REA and Luikov’s approach match well with experimental data (R2 of 0.999 and 0.994 for the L-REA and Luikov’s approach, respectively). It can be shown here that the L-REA gives comparable results with Luikov’s approach. The results of modelling of heat treatment of wood (refer to Table 4.1) using the L-REA and Whitaker’s approach (Whitaker, 1977, implemented by Younsi et al. (2007)) are shown in Figures 4.9 and 4.10. The experimental data are derived from the work of Younsi et al. (2007), the experimental details are presented in Section 2.8 and the experimental settings are shown in Table 4.1. Figure 4.9 shows the moisture content profiles of heat treatment of wood modelled using both approaches. For Case 1 (heating rate of 20 °C h−1 , refer to Table 4.1), Whitaker’s approach models the moisture content profiles well at treatment times shorter than 20 000 s, but the approach slightly overestimates the drying rate after the treatment time of 20 000 s (R2 of 0.991). The L-REA results in


0.14


202

Case 1-Experimental data Case 1-L-REA

0.12

Case 1-Whitaker approach by Younsi et al. (2007) Case 2-Experimental data Case 2-L-REA

0.1 0.08

Case 2-Whitaker approach by Younsi et al. (2007)

0.06 0.04 0.02 0

0

1

2

3

4 t(s)

5

6

7 × 104

Figure 4.9 Moisture content profiles from the heat treatment of wood (refer to Table 4.1) modelled using the L-REA and Whitaker’s approach. [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.]

better agreement with experimental data, shown by R2 of 0.994. Both the L-REA and Whitaker’s approach match reasonably well the experimental data for modelling of heat treatment in Case 2 (R2 of 0.987 and 0.971 for the L-REA and Whitaker’s approach, respectively). Nevertheless, the L-REA yields a better agreement with experimental data. The temperature profiles of heat treatment of wood (refer to Table 4.1) are indicated in Figure 4.10. Generally, the L-REA and Whitaker’s approach model the temperature profiles well (R2 higher than 0.993 and 0.991 for the L-REA and Whitaker’s approach, respectively). However, Whitaker’s approach slightly underestimates the temperature profiles during the stage where gas temperature is held constant at 120 °C for half an hour. During this period, the L-REA resulted in a better agreement with the experimental data, which could be because of the flexibility of activation energy in changing according to ambient conditions. In conclusion, although the L-REA is much simpler in mathematical modelling, it performs comparably well with Luikov’s and Whitaker’s approaches (implemented by Younsi et al., 2006a; 2007) in modelling heat treatment of wood under a constant heating rate. The L-REA is more efficient at generating the model parameters, since these can be generated from one accurate experiment. On the other hand, the two other approaches require several sets of experiments and complex optimization procedures to generate these model parameters.

203


500

Temperature (K)

450

400 Case 1-Experimental data Case 1-L-REA Case 1-Whitaker approach by Younsi et al. (2007) Case 2-Experimental data Case 2-L-REA Case 2-Whitaker approach by Younsi et al. (2007)

350

300

250

0

1

2

3

4 t(s)

5

6

7 × 104

Figure 4.10 Temperature profiles from the heat treatment of wood (refer to Table 4.1) modelled

using the L-REA and Whitaker’s approach. [Reprinted from Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperature treatment of wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.]

4.9

Comparison of the S-REA, Luikov’s and Whitaker’s approaches for modelling heat treatment of wood under constant heating rates In this section, the results of modelling of the heat treatment of wood under various constant heating rates using the S-REA, Luikov’s (Luikov, 1975) and Whitaker’s approaches (Whitaker, 1977) are compared. The experimental details are described in Section 2.8 while the modelling using Luikov’s (Luikov, 1975) and Whitaker’s approaches (Whitaker, 1977) are presented in Section 4.6 and 4.7, respectively. Figure 4.11 shows the moisture content profiles of heat treatment of wood with a final temperature of 200 °C, heating rate of 20 °C h−1 and initial moisture content of 0.07 kg water kg dry solids−1 , modelled using the S-REA and Luikov’s approach (applied by Younsi et al., 2006a). Both the S-REA and Luikov’s approach (implemented by Younsi et al., 2006a) model the moisture content profiles reasonably well (R2 of 0.99 and 0.981 for the S-REA and Luikov’s approach, respectively), but the S-REA yields a closer agreement with the experimental data as the other model results in an increase of moisture content at the beginning of the treatment and an overestimation in the profiles before a treatment time of 10 000 s. The temperature profiles of the experiment, modelled using both approaches are shown in Figure 4.12. Both the S-REA and Luikov’s approach (applied by Younsi et al., 2006a) model the temperature profiles



0.08 Experimental data S-REA Luikov approach: Younsi et al. (2006a)

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

0.5

1

1.5 t(s)

2

2.5

3 × 104

Figure 4.11 Moisture content profile from the heat treatment of wood modelled using the S-REA

and Luikov’s approach. 480 Experimental data S-REA Luikov approach: Younsi et al. (2006a)


204

400 380 360 340 320 300 280

0

0.5

1

1.5

2 t(s)

2.5

3

3.5 × 104

Figure 4.12 Temperature profile from the heat treatment of wood modelled using the S-REA and

Luikov’s approach.

205



0.14 Case 1-Experimental data Case 1-S-REA Case 1-Whitaker approach by Younsi et al. (2007) Case 2-Experimental data Case 2-S-REA Case 2-Whitaker approach by Younsi et al. (2007)

0.12 0.1 0.08 0.06 0.04 0.02 0

0

1

2

3

4

5

6

t(s)

7 × 104

Figure 4.13 Moisture content profiles from the heat treatment of wood (refer to Table 4.1)

modelled using the S-REA and Whitaker’s approach.

500

Temperature (K)

450

400 Case 1-Experimental data Case 1-S-REA Case 1-Whitaker approach by Younsi et al. (2007) Case 2-Experimental data Case 2-S-REA Case 2-Whitaker approach by Younsi et al. (2007)

350

300

250

0

1

2

3

4

5

t(s)

6

7 × 104

Figure 4.14 Temperature profiles from the heat treatment of wood (refer to Table 4.1) modelled

using the S-REA and Whitaker’s approach.

206


during the heat treatment accurately (R2 of 0.998 and 0.994 for the S-REA and Luikov’s approach, respectively). Figures 4.13 and 4.14 show the results of modelling of heat treatment of wood under constant heating rates (refer to Table 4.1) using the S-REA and Whitaker’s approach (used by Younsi et al. (2007)). For Case 1 (refer to Table 4.1), Whitaker’s approach overestimated the drying rate after a treatment time of 20 000 s. The S-REA yields a closer agreement with the experimental data (R2 of 0.995 and 0.991 for the S-REA and Whitaker’s approach, respectively). Similarly, the S-REA performs better than the other model at representing heat treatment of wood in Case 2 (R2 of 0.988 and 0.971 for the S-REA and Whitaker’s approach, respectively). The temperature profiles of the various cases of the heat treatment are shown in Figure 4.14. Both the S-REA and the Whitaker’s approach describe the temperature profiles well along the treatment (R2 higher than 0.992 and 0.991 for the S-REA and Whitaker’s approach, respectively). Compared to the other approach, the S-REA has the advantage of generating profiles of the spatial water vapour concentration so that better understanding of the process can be studied. In conclusion, it can be said that the S-REA performs comparably or even better than Whitaker’s approach in modelling the heat treatment of wood under a constant heating rate.

4.10

Summary In this chapter, the diffusion-based models, Luikov’s and Whitaker’s approaches which have been used extensively applied to model drying processes are analysed. The formulation and limitation of Crank’s diffusion-based model and several other forms of diffusion-based model are discussed. Crank’s diffusion-based model offers advantages of simple mathematical modelling but it is only valid for conditions where there are no shrinkage, isothermal, constant diffusivity, negligible external resistance and uniform initial moisture content. Nevertheless, it is often implemented in literature without necessary justification. The effective diffusivity of liquid water has been interpreted in experiments with poor documtation and control of boundary conditions, inducing a large range in orders of magnitude in diffusivity values. Various forms of diffusion-based models reported in literature have also been described. The crucial issues are boundary conditions and the implementation of a source term. For a better understanding of the drying process transport phenomena, a multiphase diffusion-based model needs to be implemented. For this purpose, equilibrium and non-equilibrium diffusion-based models can be implemented and are both described. The equilibrium one assumes the moisture content inside pores of the samples equilibrates with water vapour concentration, which essentially eliminates the source term in a way. The non-equilibrium one is more general but it requires the explicit formulation of the source term. Luikov’s approach assumes that the mechanisms of moisture and heat transfer are similar, which is due to thes gradient of moisture content and temperature. Therefore, several coefficients are used to represent the interdependency of moisture content and temperature. Whitaker’s approach uses detailed equations of momentum, heat and mass


207

conservation in solid, liquid and gas phases, followed by the volume averaging method to describe drying processes. The accuracy of the diffusion-based model has been compared to those of the LREA and S-REA to model convective drying. It has been shown that the L-REA performs comparably, or sometimes even better than the diffusion-based model. While the results are comparable, the L-REA has the advantages of simplicity in generating the parameters and mathematical formulation. The S-REA works very well, which offers advantages where it not only accounts for the diffusion processes (Fickian type) but also local evaporation or condensation. Similarly, the accuracy of the Luikov’s and Whitaker’s approaches are compared with those of the L-REA and S-REA to model heat treatment of wood, which is essentially a drying process under linearly increased gas temperatures. It has been shown that the L-REA and S-REA perform comparably or even better than Luikov’s and Whitaker’s approaches. While the results are similar, the L-REA offers the advantages of efficiency in generating the parameters and simplicity of mathematical modelling. The S-REA is also more efficient in generating the parameters than Luikov’s and Whitaker’s approaches.

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Index

activation energy, xxxiv, 16, 24, 27, 34, 35, 39, 40, 51–54, 57, 64, 69, 73, 82, 85, 88, 90, 91, 93, 96, 100, 102, 105, 117, 127, 129, 132, 142, 150, 159 average water content, x, 5, 21, 22, 25 bacteria, xxvii, 11 baking, vi, vii, xvii, xviii, xxiii, 3, 7, 95–100, 117, 119–121, 158–168, 183, 184, 210, 211 balance, xxxiii biological, 17 Biot, v, x, xxxiv, 11, 25, 27, 30, 43–45, 47, 50, 81, 117, 119, 146, 165, 171, 208 boundary, 5, 8, 10, 19, 23, 24, 29, 43, 63, 64, 123, 124, 127, 128, 130, 132, 133, 141, 142, 149, 150, 159, 160, 171, 173, 176–179, 181, 182, 185–189, 193, 194, 199, 200, 206 capillary, ix, 7, 8, 11, 17, 122, 125, 170, 172, 177, 179, 181, 190–193, 195, 197, 208, 209 cell, 127 centre sample temperature, 63 Characteristic Drying Rate Curve, xxxi, 20 chemistry, v, xxxi, 1, 5, 15, 18, 19, 25, 29 combustion, xxx, xxxvii, 1, 18, 19, 31 Comparison, 185, 200 computational fluid dynamics, xxxvi, 29 concentration, 15 condensation, xxxiii, 10, 16, 17, 19, 24, 25, 28, 29, 32, 34, 122, 126, 132, 133, 137, 142, 143, 151, 154, 158, 160, 164, 170, 172, 177, 181, 184, 195, 207 conduction, 45 constant rate period, 17, 54, 102, 106, 108, 114 continuum, 17 convection, 45 convective, 40 core, xxii, 8, 11, 35, 43, 128, 133, 136, 139, 140, 145, 146, 157, 161, 167, 210 cortex, 133 coupling, 41 Crank, vii, 171, 173, 206, 208

critical water content, 21–23 cycle, 74 Darcy flow, 126 deflection, 36 deformable, 20, 166 diffusion, vii, viii, xxiv, 6, 9, 11, 17, 40, 43, 47, 58, 61, 68, 69, 73, 79, 97, 117, 121, 124, 126, 133, 135, 154, 155, 157, 166, 167, 169–174, 176, 181, 185–191, 193–195, 197, 206–210 diffusivity, 48 discrete, 18 discretization, 12 distributed, xxviii, xxxiv, 5, 8, 17, 18, 43 driving force, xxxiii, 8, 17, 46, 50 droplet, 36 effective diffusivity, 130 effective liquid diffusivity, 126 empirical, xxvii, xxviii, 17, 20, 47, 73, 105, 181 energy, x, xi, xii, xiv, xv, xvii, xxvii, xxxiii, 3, 5, 12, 16, 19, 20, 24–29, 34–36, 39, 40, 42, 48, 51–55, 57, 61, 62, 64, 65, 68, 69, 73, 74, 80–82, 85, 88–91, 93, 94, 96, 97, 100, 102, 103, 105, 107, 108, 116, 117, 121, 122, 124, 125, 127, 129, 130, 132, 138, 142, 150, 159, 160, 164, 172, 184, 186, 195, 198, 202, 207, 208 equilibrium, 35, 182 equilibrium model, 121 evaporation, vii, x, xxii, xxiii, xxx, xxxiii, xxxiv, 1, 7–9, 11, 16, 17, 19, 20, 24, 25, 27, 28, 29, 32, 34, 36, 41, 45–47, 49, 63, 64, 102, 107, 108, 122, 124, 126, 127, 129, 131, 132, 136, 137, 138, 142, 143, 145–148, 150, 151, 154, 157–160, 164, 170, 172, 177, 179–184, 186, 188–190, 193, 195, 196, 207, 208, 210 fiber, 1, 2, 30, 33, 89, 197, 198 Fick, 169, 181 finite element, 12, 193, 207 first order, 34 fissuring, 170, 209 Food, 1

Index

Fourier, 169, 191, 193 free volume, 124, 125, 167 fundamental, 170 glass-filament, 36 gradient, xxx, 17, 43, 51, 96, 122, 135, 136, 143, 145–147, 170, 172, 190–194, 197, 206 heat balance, 52 heat transfer coefficient, 27, 42, 44, 45, 47, 52, 53, 54, 65, 82, 91, 97, 102, 159, 173, 178, 184, 186, 194 infrared-heating, 100 interface, 178 intermitent, 141 intermittent, 69 internal mass transfer coefficient, 125 internal surface area per unit volume, 125 isotherm, xxx, 24, 26, 39, 49, 121, 182, 183 isotherma, 126 kaolin, 170 Lagrangian, 12, 40 Lewis, v, x, 11, 27, 43–45, 47–50, 117, 172, 209, 210 linear, 88 liquid, 9 local phase change term, 121 Luikov, 172, 190 lumped, 34 Lumped-REA, 29 macro-scale, 17 Magnetic resonance imaging, 11 mass balance, 25, 52 Mass transfer, 9, 208 mass transfer coefficient, 23, 24, 34, 35, 38, 46, 47, 51, 54, 62, 124–126, 129, 133, 138, 142, 151, 159, 171, 173, 176, 178, 184, 185, 194 material, ix, xxvii, xxx, xxxiii, xxxv, 1, 3, 5–7, 9–12, 16, 19, 20, 22–27, 33–36, 40, 42–48, 50, 62, 65, 116, 121, 122, 124, 139, 141, 164, 169, 172 Material Point Method, 12 meso-scale, xxviii, 17 micro-scale, 11, 17 microwave, xxviii, 1, 5, 166, 170, 209 modelling, 15, 116 momentum, v, xxvii, xxxi, 5, 33, 40, 41, 43, 195, 206, 211 multiphase, 177 multiphase drying model, 122

213

non-equilibrium multiphase drying approach, 121 Nusselt, 38, 42, 52 Ohm, 169 optimization, 40 paper, xxx, xxxiii, 1, 18, 19, 24 parabolic, 63 physical, v, xxxiii, 6, 15, 18, 19, 22, 25, 29, 31, 80, 95, 169 physics, 18 polymer, 100 pore, 121 porosity, 124 preservation, 1, 11 quality, 170 reaction engineering approach, 15 relative humidity, xi, xii resistance, 45, 176 Reynolds, 38, 41, 42, 51 Sherwood, 24, 34, 38, 42, 51, 54, 62 shrinkage, 124 slab, 170 solid concentration, 126 source term, 29, 48, 49, 121, 122, 179, 190, 206 spatial reaction engineering approach, 121 spatial-REA, 121 Spatial-REA, 29 stress, ix, 3, 12, 13, 33, 95, 170 structure, xxiv, 3, 6, 8, 9, 11, 12, 18, 27, 49, 95, 121, 122, 124, 127, 136, 143, 148, 158, 172, 175 surface, 16 surface sample temperature, 63 surface vapor concentration, 25, 34, 62, 73, 74, 79, 80 symmetry, 63, 64, 149 thermal conductivity, 48 thermocouple, 36 thick samples, vi, 61, 64, 66, 69, 73, 119, 166, 210 thickness, 63 thin layer, 34 timber, 193 transport, 121 transport phenomena, 121 turtuosity, 125 uniform, 35 vapor, 9 vapor diffusivity, 125 vaporization heat, 52 volume-averaged, 17

214

Index

water vapor concentration, x, xxii, xxiii, 23, 26, 34, 38, 49, 73, 121, 124, 129, 130, 132, 133, 136, 138, 141–144, 149–151, 154–157, 160, 171–173, 178, 184, 206 wetting, 34 whey protein concentrate, 51 Whitaker, 195

Whittaker, 172 wood, vi, vii, viii, ix, xvii, xxiv, xxv, xxvi, xxxi, 1, 3, 5, 12, 15, 18, 88–95, 116, 118, 120, 148, 150, 151, 154, 155, 157–159, 166–168, 193, 194, 196–198, 200–207, 210, 211 zero order, 24

Modelling Drying Processes a Reaction Engineering Approach

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