Modelling Drying Processes a Reaction Engineering Approach

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http://www.cambridge.org/9781107012103

Modelling Drying Processes

This comprehensive summary of the state-of-the-art and the ideas behind the reactionengineering 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, itgoes 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 thespatial-REA, to model non-equilibrium multiphase drying. Finally, it summarises otherestablished drying models, discussing their features, limitations and comparisons withthe REA.

Application examples featured throughout help fine-tune the models and implementthem for process design, and the evaluation of existing drying processes and productquality during drying. Further uses of the principles of REA are demonstrated, includingcomputational fluid dynamics-based modelling, and further expanded to model othersimultaneous heat and mass transfer processes.

Xiao Dong Chen is currently the 1000-talent Chair Professor of Chemical Engineering atXiamen University in China, and the Head of Department of Chemical and Biochem-ical Engineering. He held previously Chair Professorships of Chemical Engineering atAuckland University, New Zealand, and Monash University, Australia, respectively from2001 to 2010. He is now a fractional Professor of Chemical Engineering and the Co-Director of the Biotechnology and Food Engineering Research Laboratory at MonashUniversity, Australia. He is an Elected Fellow of Royal Society of NZ, AustralianAcademy of Technological Sciences and Engineering, and IChemE.

Aditya Putranto holds a BE of Chemical Engineering from Bandung Institute of Technol-ogy, 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 researcharea is heat and mass transfer. He has published a dozen journal papers in peer-reviewedhard-core chemical engineering journals.

‘The Reaction Engineering Approach (REA), which captures basic drying physics, isa simple yet effective mathematical model for practical applications of diverse dryingprocesses. The intrinsic “fingerprint” of the drying phenomena can, in principle, beobtained through just one accurate drying experiment. The REA is easy to use withthe guidance of featured application examples given in this book. This book is highlyrecommended for both academics and industry practitioners involved in any aspect ofthermal drying.’

Zhanyong Li,Tianjin University of Science and Technology,

China

‘An interesting book on a novel approach to mathematical modelling of an importantprocess. Modelling Drying Processes: A Reaction Engineering Approach is the firstattempt to summarize the REA to modelling in a single comprehensive reference source.’

Sakamon Devahastin,King Mongkut’s University of Technology Thonburi,

Thailand

Modelling Drying ProcessesA Reaction Engineering Approach

XIAO DONG CHENMonash University, Australia

ADITYA PUTRANTOMonash University, Australia

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town,Singapore, Sao Paulo, Delhi, Mexico City

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9781107012103

C© Xiao Dong Chen and Aditya Putranto 2013

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission 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 dataChen, Xiao Dong.Modelling drying processes : a reaction engineering approach / Xiao Dong Chen,Monash University, Australia, Aditya Putranto, Monash University, Australia.

pages cmIncludes 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 2013664′.0284 – dc23 2013003983

ISBN 978-1-107-01210-3 Hardback

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

http://www.cambridge.org

http://www.cambridge.org/9781107012103

Contents

List of figures page ixList of tables xxviPreface xxviiHistorical background xxx

1 Introduction 1

1.1 Practical background 11.2 A ‘microstructural’ discussion of the phenomena of drying moist,

porous materials 61.3 The REA to modelling drying 15

1.3.1 The relevant classical knowledge of physical chemistry 151.3.2 General modelling approaches 171.3.3 Outline of the REA 18

1.4 Summary 29References 30

2 Reaction engineering approach I: Lumped-REA (L-REA) 34

2.1 The REA formulation 342.2 Determination of REA model parameters 362.3 Coupling the momentum, heat and mass balances 402.4 Mass or heat transfer limiting 43

2.4.1 Biot number analysis 432.4.2 Lewis number analysis 472.4.3 Combination of Biot and Lewis numbers 50

2.5 Convective drying of particulates or thin layer products modelledusing the L-REA 502.5.1 Mathematical modelling of convective drying of droplets of

whey protein concentrate (WPC) using the L-REA 512.5.2 Mathematical modelling of convective drying of a mixture of

polymer solutions using the L-REA 532.5.3 Results of modelling convective drying of droplets of WPC

using the L-REA 55

vi Contents

2.5.4 Results of modelling convective drying of a thin layer of a mixtureof polymer solutions using the L-REA 57

2.6 Convective drying of thick samples modelled using the L-REA 612.6.1 Formulation of the L-REA for convective drying of thick samples 612.6.2 Prediction of surface sample temperature 632.6.3 Modelling convective drying thick samples of mango tissues

using the L-REA 642.6.4 Results of convective drying thick samples of mango tissues

using the L-REA 662.7 The intermittent drying of food materials modelled using the L-REA 69

2.7.1 Mathematical modelling of intermittent drying of food materialsusing the L-REA 69

2.7.2 The results of modelling of intermittent drying of food materialsusing the L-REA 69

2.7.3 Analysis of surface temperature, surface relative humidity,saturated and surface vapour concentration duringintermittent drying 73

2.8 The intermittent drying of non-food materials under time-varyingtemperature and humidity modelled using the L-REA 802.8.1 Mathematical modelling using the L-REA 812.8.2 Results of intermittent drying under time-varying temperature

and humidity modelled using the L-REA 822.9 The heating of wood under linearly increased gas temperature modelled

using the L-REA 882.9.1 Mathematical modelling using the L-REA 892.9.2 Results of modelling wood heating under linearly increased gas

temperatures using the L-REA 912.10 The baking of cake modelled using the L-REA 95

2.10.1 Mathematical modelling of the baking of cake usingthe L-REA 96

2.10.2 Results of modelling of the baking of cake usingthe L-REA 97

2.11 The infrared-heat drying of a mixture of polymer solutions modelledusing the L-REA 1002.11.1 Mathematical modelling of the infrared-heat drying of a mixture

of polymer solutions using the L-REA 1012.11.2 The results of mathematical modelling of infrared-heat drying of a

mixture of polymer solutions using the L-REA 1032.12 The intermittent drying of a mixture of polymer solutions under

time-varying infrared-heat intensity modelled using the L-REA 1042.12.1 Mathematical modelling of the intermittent drying of a mixture of

polymer solutions under time-varying infrared-heat intensity usingthe L-REA 105

Contents vii

2.12.2 Results of modelling the intermittent drying of a mixture ofpolymer solutions under time-varying infrared heat intensity usingthe L-REA 106


3 Reaction engineering approach II: Spatial-REA (S-REA) 121

3.1 The S-REA formulation 1213.2 Determination of the S-REA parameters 1253.3 The S-REA for convective drying 127

3.3.1 Mathematical modelling of convective drying of mango tissuesusing the S-REA 128

3.3.2 Mathematical modelling of convective drying of potato tissuesusing the S-REA 130

3.3.3 Results of modelling of convective drying of mango tissues usingthe S-REA 133

3.3.4 Results of modelling of convective drying of potato tissues usingthe S-REA 138

3.4 The S-REA for intermittent drying 1413.4.1 The mathematical modelling of intermittent drying

using the S-REA 1413.4.2 Results of modelling intermittent drying using the S-REA 142

3.5 The S-REA to wood heating under a constant heating rate 1483.5.1 The mathematical modelling of wood heating using the S-REA 1483.5.2 The results of modelling wood heating using the S-REA 151

3.6 The S-REA for the baking of bread 1583.6.1 Mathematical modelling of the baking of bread using the S-REA 1583.6.2 The results of modelling of the baking of bread using the S-REA 160


4 Comparisons of the REA with Fickian-type drying theories, Luikov’s andWhitaker’s approaches 169

4.1 Model formulation 1694.1.1 Crank’s effective diffusion 1714.1.2 The formulation of effective diffusivity to represent

complex drying mechanisms 1724.1.3 Several diffusion-based models 173

4.2 Boundary conditions’ controversies 1774.3 A diffusion-based model with local evaporation rate 179

4.3.1 Problems in determining the local evaporation rate 1804.3.2 The equilibrium and non-equilibrium multiphase

drying models 182

viii Contents

4.4 Comparison of the diffusion-based model and the L-REA onconvective drying 185

4.5 Comparison of the diffusion-based model and the S-REA onconvective drying 188

4.6 Model formulation of Luikov’s approach 1904.7 Model formulation of Whitaker’s approach 1954.8 Comparison of the L-REA, Luikov’s and Whitaker’s approaches for

modelling heat treatment of wood under constant heating rates 2004.9 Comparison of the S-REA, Luikov’s and Whitaker’s approaches for

modelling heat treatment of wood under constant heating rates 2034.10 Summary 206References 207

Index 212

Figures

1.1 Some traditional dried products. (a) Broccoli-steam blanched and airdried (kindly provided by Ms Xin Jin, Wageningen University, TheNetherlands), (b) air-dried Chinese tea leaves (taken at XiamenUniversity laboratory), (c) spray dried aqueous herbal extract (particlesize is about 80 µm) (taken at Xiamen University laboratory), (d) timberstacked for kiln drying (kindly provided by Professor Shusheng Pang(Canterbury University, New Zealand). page 2

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. 4

1.3 ‘Air drying’ of a capillary assembly (a bundle) which consists ofidentical capillaries (diameter and wall material) – a scenario ofsymmetrical hot air drying of an infinitely large slab filled with thecapillaries (modified from Chen, 2007); the air flows along both sides ofthe symmetrical material. 7

1.4 Schematic showing a common scenario of air drying of a moist solid. 91.5 Packed particulate material. 101.6 Cellular structures in plant material. 101.7 (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 moisturecontent) for a variety of drying conditions. [Reprinted from Journal ofFood Engineering, 106, P.S. Takhar, D.E. Maier, O.H. Campanella andG. Chen, Hybrid mixture theory based moisture transport and stressdevelopment in corn kernels during drying: Validation and simulationresults, 275–282, Copyright (2012), with permission fromElsevier.] 13

1.8 Wood cellular structures employed in pore-network modelling of dryingof wood. [Reprinted from Drying Technology, 29, P. Perre, A review ofmodern computational and experimental tools relevant to the field ofdrying, 1529–1541, Copyright (2012), with permission from Taylor &Francis.] 15

1.9 Schematic illustration of the effect of temperature on final liquid watercontent (qualitatively derived from Equation 1.3.6). 20

x List of figures

1.10 (a) Drying flux versus average water content X ; (b) the CDRC(characteristic drying rate curve). [Reprinted from ChemicalEngineering Science, 9, D.A. van Meel, Adiabatic convection batchdrying with recirculation of air, 36–44, Copyright (2012), reprinted withpermission from Elsevier.] 21

1.11 Saturated water vapour concentration in air under 1 atm (Equation1.3.21). 26

1.12 Schematic diagram showing the heat of drying as a function of watercontent of a porous solid of concern (when the water content is beyondthe point where the heat of drying becomes the latent heat of pure waterevaporation, the water content may be called free water). 28

2.1 Equipment setup of convective drying of milk droplets (a) measuringdroplet shrinkage; (b) measuring droplet temperature; (c) measuringmass change. [Reprinted from Chemical Engineering Science, 66, N. Fu,M.W. Woo, S.X.Q. Lin et al., 1738–1747, Copyright (2012), withpermission from Elsevier.] (Adapted from Fu et al. (2011) ChemicalEngineering Science 66, 1738–1747). 37

2.2 The deflection of glass filament and a typical standard curve (a)measuring displacement to measure weight loss; (b) correlation betweenthe displacement and the weight. [Reprinted from Chemical EngineeringScience, 66, N. Fu, M.W. Woo, S.X.Q. Lin et al., 1738–1747, Copyright(2012), with permission from Elsevier.] 38

2.3 The relative activation energy of convective drying of 20%wt. skim milkpowder at a drying air temperature of 67.5 °C, velocity of 0.45 m s−1

and humidity of 0.0001 kg H2O kg dry air−1. [Reprinted from AIChEJournal, 51, X.D Chen and S.X.Q. Lin, Air drying of milk droplet underconstant and time-dependent conditions, 1790–1799, Copyright (2012),with permission from John Wiley & Sons, Inc.] 39

2.4 Schematic diagram showing the plug-flow spray dryer. 412.5 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 andLewis number analysis, 2239–2248, Copyright (2012), with permissionfrom Taylor & Francis.] 44

2.6 The schematic diagram showing the parameters for the definition of themodified Biot number) (Chen–Biot number). [Reprinted from DryingTechnology, 23, X.D. Chen, Air drying of food and biologicalmaterials – Modified Biot and Lewis number analysis, 2239–2248,Copyright (2012), with permission from Taylor & Francis.] 45

2.7 The relative activation energy of convective drying of WPC at differentdrying air temperatures. [Reprinted from Chemical Engineering andProcessing, 46, S.X.Q. Lin and X.D. Chen, The reaction engineeringapproach to modelling the cream and whey protein concentrate dropletdrying, 437–443, Copyright (2012), with permission from Elsevier.] 52

List of figures xi

2.8 The droplet diameter changes during convective drying of WPC.[Reprinted from Chemical Engineering and Processing, 46, S.X.Q. Linand X.D. Chen, The reaction engineering approach to modelling thecream and whey protein concentrate droplet drying, 437–443, Copyright(2012), with permission from Elsevier.] 53

2.9 Heat transfer mechanisms of the convective drying of a mixture ofpolymer solutions. [Reprinted from Chemical Engineering andProcessing: 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.] 54

2.10 Normalised activation energy and fitted curve of polyvinylalcohol/glycerol/water under convective drying at an air temperature of35 °C and relative humidity of 30%. [Reprinted from ChemicalEngineering and Processing: Process Intensification, 49, A. Putranto,X.D. Chen and P.A. Webley, Infrared and convective drying of thin layerof polyvinyl alcohol (PVA)/glycerol/water mixture – The reactionengineering approach (REA), 348–357, Copyright (2012), withpermission from Elsevier.] 55

2.11 The comparison between experimental and model prediction using theL-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 Engineeringand Processing, 46, S.X.Q. Lin and X.D. Chen, The reaction engineeringapproach to modelling the cream and whey protein concentrate dropletdrying, 437–443, Copyright (2012), with permission from Elsevier]. 56

2.12 Moisture content profile of convective drying at an air temperature of55 °C, air velocity of 2.8 m s−1 and air relative humidity of 12%.[Reprinted from Chemical Engineering and Processing: ProcessIntensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared andconvective drying of thin layer of polyvinyl alcohol(PVA)/glycerol/water mixture – The reaction engineering approach(REA), 348–357, Copyright (2010), with permission from Elsevier.] 57

2.13 Product temperature profile of convective drying at an air temperature of55 °C, air velocity of 2.8 m s−1 and air relative humidity of 12%.[Reprinted from Chemical Engineering and Processing: ProcessIntensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared andconvective drying of thin layer of polyvinyl alcohol(PVA)/glycerol/water mixture – The reaction engineering approach(REA), 348–357, Copyright (2010), with permission from Elsevier.] 58

2.14 Moisture content profile of convective drying at an air temperature of35 °C, air velocity of 1 m s−1 and air relative humidity of 30%.[Reprinted from Chemical Engineering and Processing: ProcessIntensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared andconvective drying of thin layer of polyvinyl alcohol

xii List of figures

(PVA)/glycerol/water mixture – The reaction engineering approach(REA), 348–357, Copyright (2010), with permission from Elsevier.] 59

2.15 Product temperature profile of convective drying at an air temperature of35 °C, air velocity of 1 m s−1 and air relative humidity of 30%.[Reprinted from Chemical Engineering and Processing: ProcessIntensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared andconvective drying of thin layer of polyvinyl alcohol(PVA)/glycerol/water mixture – The reaction engineering approach(REA), 348–357, Copyright (2010), with permission from Elsevier.] 59



2.18 The relative activation energy (�Ev/�Ev,b) of convective drying ofmango tissues at an air velocity of 4 m s−1, drying air temperature of55 °C, and air humidity of 0.0134 kg H2O kg dry air−1. [Reprinted fromDrying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley,Modelling of drying of food materials with thickness of severalcentimeters by the reaction engineering approach (REA), 961–973,Copyright (2012), with permission from Taylor & Francis Ltd.] 65

2.19 Moisture content profile of convective mango tissues at air temperaturesof 45, 55, and 65 °C (modelled using the L-REA which incorporates thetemperature distribution inside the sample). [Reprinted from DryingTechnology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling ofdrying of food materials with thickness of several centimeters by thereaction engineering approach (REA), 961–973, Copyright (2012), withpermission from Taylor & Francis Ltd.] 66

2.20 Temperature profile of convective mango tissues at air temperatures of45, 55, and 65 °C (modelled using the L-REA which incorporates thetemperature distribution inside the sample). [Reprinted from DryingTechnology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling ofdrying of food materials with thickness of several centimeters by thereaction engineering approach (REA), 961–973, Copyright (2012), withpermission from Taylor & Francis Ltd.] 67

List of figures xiii

2.21 Moisture content profile of convective mango tissues at air temperaturesof 45, 55, and 65 °C (modelled using the L-REA without approximationof temperature distribution inside the sample). [Reprinted from DryingTechnology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling ofdrying of food materials with thickness of several centimeters by thereaction engineering approach (REA), 961–973, Copyright (2012), withpermission from Taylor & Francis Ltd.] 67

2.22 Temperature profile of convective mango tissues at air temperatures of45, 55, and 65 °C (modelled using the L-REA without approximation oftemperature distribution inside the sample). [Reprinted from DryingTechnology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling ofdrying of food materials with thickness of several centimeters by thereaction engineering approach (REA), 961–973, Copyright (2012), withpermission from Taylor & Francis Ltd.] 68

2.23 Moisture content profile of mango tissues during intermittent drying at adrying air temperature of 45 °C and resting at 27 °C. [Reprinted fromIndustrial 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 ChemicalSociety.] 70

2.24 Temperature profile of mango tissues during intermittent drying at adrying air temperature of 45 °C and resting at 27 °C. [Reprinted fromIndustrial 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 ChemicalSociety.] 70

2.25 Moisture content profile of mango tissues during intermittent drying at adrying air temperature of 55 °C and resting at 27 °C [Reprinted fromIndustrial 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 ChemicalSociety.] 71

2.26 Temperature profile of mango tissues during intermittent drying at adrying air temperature of 55 °C and resting at 27 °C [Reprinted fromIndustrial 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 ChemicalSociety.] 71

2.27 Moisture content profile of mango tissues during intermittent drying at adrying air temperature of 65 °C and resting at 27 °C. [Reprinted from

xiv List of figures

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 ChemicalSociety.] 72

2.28 Temperature profile of mango tissues during intermittent drying at adrying air temperature of 65 °C and resting at 27 °C. [Reprinted fromIndustrial 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 ChemicalSociety.] 72

2.29 Relative activation energy profile of mango tissues during intermittentdrying 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 ofmango tissues: Implementation of the reaction engineering approach,1089–1098, Copyright (2012), with permission from the AmericanChemical Society.] 74

2.30 Surface relative humidity profile of mango tissues during intermittentdrying 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 ofmango tissues: Implementation of the reaction engineering approach,1089–1098, Copyright (2012), with permission from the AmericanChemical Society.] 75

2.31 Saturated vapour concentration and surface temperature profile ofmango tissues during intermittent drying at a drying air temperature of65 °C and resting at 27 °C. [Reprinted from Industrial EngineeringChemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A.Webley, Intermittent drying of mango tissues: Implementation of thereaction engineering approach, 1089–1098, Copyright (2012), withpermission from the American Chemical Society.] 75

2.32 Surface and saturated vapour concentration profile of mango tissuesduring intermittent drying at a drying air temperature of 65 °C andresting at 27 °C. [Reprinted from Industrial Engineering ChemistryResearch, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley,Intermittent drying of mango tissues: Implementation of the reactionengineering approach, 1089–1098, Copyright (2012), with permissionfrom the American Chemical Society.] 76

2.33 Surface vapour concentration and surface temperature profile of mangotissues during intermittent drying at a drying air temperature of 65 °Cand resting at 27 °C. [Reprinted from Industrial Engineering ChemistryResearch, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley,

List of figures xv

Intermittent drying of mango tissues: Implementation of the reactionengineering approach, 1089–1098, Copyright (2012), with permissionfrom the American Chemical Society]. 76

2.34 Moisture content profile of intermittent drying of mango tissues withheating (at a drying air temperature of 45 °C) and resting periods of4000 s each. [Reprinted from Industrial Engineering ChemistryResearch, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley,Intermittent drying of mango tissues: Implementation of the reactionengineering approach, 1089–1098, Copyright (2012), with permissionfrom the American Chemical Society.] 77

2.35 Saturated vapour concentration and surface temperature profile ofintermittent drying of mango tissues with heating (at a drying airtemperature of 45 °C) and resting periods of 4000 s each. [Reprintedfrom Industrial Engineering Chemistry Research, 50, A. Putranto,Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mangotissues: Implementation of the reaction engineering approach,1089–1098, Copyright (2012), with permission from the AmericanChemical Society.] 77

2.36 Surface vapour concentration and surface temperature profile ofintermittent drying of mango tissues with heating (at a drying airtemperature of 45 °C) and resting periods of 4000 s each. [Reprintedfrom Industrial Engineering Chemistry Research, 50, A. Putranto,Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mangotissues: Implementation of the reaction engineering approach,1089–1098, Copyright (2012), with permission from the AmericanChemical Society.] 78

2.37 Surface and saturated vapour concentration profile of intermittent dryingof mango tissues with heating (at a drying air temperature of 45 °C) andresting periods of 4000 s each. [Reprinted from Industrial EngineeringChemistry Research, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A.Webley, Intermittent drying of mango tissues: Implementation of thereaction engineering approach, 1089–1098, Copyright (2012), withpermission from the American Chemical Society.] 78

2.38 Surface vapour concentration and surface relative humidity profile ofintermittent drying of mango tissues with heating (at a drying airtemperature of 45 °C) and resting periods of 4000 s each. [Reprintedfrom Industrial Engineering Chemistry Research, 50, A. Putranto,Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mangotissues: Implementation of the reaction engineering approach,1089–1098, Copyright (2012), with permission from the AmericanChemical Society.] 79

2.39 The relative activation energy (�Ev/�Ev,b) of the convective drying ofkaolin. [Reprinted from Chemical Engineering Science, 66, A. Putranto,X.D. Chen, S. Devahastin et al., Application of the reaction engineering

xvi List of figures

approach (REA) for modelling intermittent drying under time-varyinghumidity and temperature, 2149–2156, Copyright (2012), withpermission from Elsevier.] 82

2.40 Moisture content profile of intermittent drying in Case 1 (periodicallychanged drying air temperatures between 65–43 °C). [Reprinted fromChemical 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 humidityand temperature, 2149–2156, Copyright (2012), with permission fromElsevier.] 83

2.41 Temperature profile of intermittent drying in Case 1 (periodicallychanged drying air temperatures between 65–43 °C). [Reprinted fromChemical 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 humidityand temperature, 2149–2156, Copyright (2012), with permission fromElsevier.] 83

2.42 Moisture content profile of intermittent drying in Case 2 (periodicallychanged drying air temperatures between 100–50 °C). [Reprinted fromChemical 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 humidityand temperature, 2149–2156, Copyright (2012), with permission fromElsevier.] 84

2.43 Temperature profile of intermittent drying in Case 2 (periodicallychanged drying air temperatures between 100–50 °C). [Reprinted fromChemical 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 humidityand temperature, 2149–2156, Copyright (2012), with permission fromElsevier.] 84

2.44 Moisture content profile of intermittent drying in Case 3 (periodicallychanged relative humidity between 4–12%). [Reprinted from ChemicalEngineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al.,Application of the reaction engineering approach (REA) for modellingintermittent drying under time-varying humidity and temperature,2149–2156, Copyright (2012), with permission from Elsevier.] 86

2.45 Temperature profile of intermittent drying in Case 3 (periodicallychanged relative humidity between 4–12%). [Reprinted from ChemicalEngineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al.,Application of the reaction engineering approach (REA) for modellingintermittent drying under time-varying humidity and temperature,2149–2156, Copyright (2012), with permission from Elsevier.] 87

List of figures xvii

2.46 Moisture content profile of intermittent drying in Case 4 (periodicallychanged relative humidity between 4–80%). [Reprinted from ChemicalEngineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al.,Application of the reaction engineering approach (REA) for modellingintermittent drying under time-varying humidity and temperature,2149–2156, Copyright (2012), with permission from Elsevier.] 87

2.47 Temperature profile of intermittent drying in Case 4 (periodicallychanged relative humidity between 4–80%). [Reprinted from ChemicalEngineering Science, 66, A. Putranto, X.D. Chen, S. Devahastin et al.,Application of the reaction engineering approach (REA) for modellingintermittent drying under time-varying humidity and temperature,2149–2156, Copyright (2012), with permission from Elsevier.] 88

2.48 Relative activation energy (�Ev/�Ev,b) of the dehydration of woodduring 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 ofhigh-temperature treatment of wood by using the reaction engineeringapproach (REA), 6214–6220, Copyright (2012), with permission fromElsevier.] 90

2.49 Moisture content profiles during the heat treatment of Cases 1 to 3 (referto Table 2.10). [Reprinted from Bioresource Technology, 102,A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley, Modelling ofhigh-temperature treatment of wood by using the reaction engineeringapproach (REA), 6214–6220, Copyright (2012), with permission fromElsevier.] 92

2.50 Temperature profiles during the heat treatment of Cases 1 to 3 (refer toTable 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto,X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperaturetreatment of wood by using the reaction engineering approach (REA),6214–6220, Copyright (2012), with permission from Elsevier.] 93

2.51 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 ofhigh-temperature treatment of wood by using the reaction engineeringapproach (REA), 6214–6220, Copyright (2012), with permission fromElsevier.] 94

2.52 Temperature profiles during the heat treatment of Cases 4 and 5 (refer toTable 2.10). [Reprinted from Bioresource Technology, 102, A. Putranto,X.D. Chen, Z. Xiao and P.A. Webley, Modelling of high-temperaturetreatment of wood by using the reaction engineering approach (REA),6214–6220, Copyright (2012), with permission from Elsevier.] 95

2.53 The relative activation energy (�Ev/�Ev,b) of baking of thin layer ofcake at an oven temperature of 100 °C. [Reprinted from Journal of FoodEngineering, 105, A. Putranto, X.D. Chen and W. Zhou, Modelling of

xviii List of figures

baking of cake using the reaction engineering approach (REA),306–311, Copyright (2012), with permission from Elsevier.] 97

2.54 Moisture content profiles at baking temperatures of 100, 140 and160 °C. [Reprinted from Journal of Food Engineering, 105, A. Putranto,X.D. Chen and W. Zhou, Modelling of baking of cake using the reactionengineering approach (REA), 306–311, Copyright (2012), withpermission from Elsevier.] 98

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 reactionengineering approach (REA), 306–311, Copyright (2012), withpermission from Elsevier.] 98

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 reactionengineering approach (REA), 306–311, Copyright (2012), withpermission from Elsevier.] 99

2.57 Temperature profiles at baking temperatures of 50 and 80 °C. [Reprintedfrom Journal of Food Engineering, 105, A. Putranto, X.D. Chen and W.Zhou, Modelling of baking of cake using the reaction engineeringapproach (REA), 306–311, Copyright (2012), with permission fromElsevier.] 100

2.58 Heat transfer mechanisms of convective and infrared-heat drying.[Reprinted from Chemical Engineering and Processing: ProcessIntensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared andconvective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357,Copyright (2012), with permission from Elsevier.] 101

2.59 Moisture content profile of convective and infrared drying at an airtemperature of 35 °C, air velocity of 1 m s−1, air relative humidity of18% and intensity of infrared drying of 3700 W m−2. [Reprinted fromChemical Engineering and Processing: Process Intensification, 49, A.Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying ofthin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – Thereaction engineering approach (REA), 348–357, Copyright (2012), withpermission from Elsevier.] 103

2.60 Product temperature profile of convective and infrared drying at an airtemperature of 35 °C, air velocity of 1 m s−1, air relative humidity of18% and intensity of infrared drying of 3700 W m−2. [Reprinted fromChemical Engineering and Processing: Process Intensification, 49, A.Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying ofthin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – Thereaction engineering approach (REA), 348–357, Copyright (2012), withpermission from Elsevier.] 104

List of figures xix

2.61 Sensitivity of the moisture content profile of cyclic drying, Case 1 (referto Table 2.12) towards n (on Equation 2.12.1). [Reprinted from ChemicalEngineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley,Application of the reaction engineering approach (REA) to model cyclicdrying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture,5193–5203, Copyright (2012), with permission from Elsevier.] 106

2.62 Sensitivity of the temperature profile of cyclic drying, Case 1 (refer toTable 2.12) towards n (on Equation 2.12.1). [Reprinted from ChemicalEngineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley,Application of the reaction engineering approach (REA) to model cyclicdrying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture,5193–5203, Copyright (2012), with permission from Elsevier.] 107

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), withpermission from Elsevier.] 108

2.64 Temperature profile of cyclic drying, Case 1 (refer to Table 2.12) usingthe 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), withpermission from Elsevier.] 109

2.65 Sensitivity of the moisture content profile of cyclic drying, Case 1 (referto Table 2.12) towards q (on Equation 2.12.3). [Reprinted from ChemicalEngineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley,Application of the reaction engineering approach (REA) to model cyclicdrying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture,5193–5203, Copyright (2012), with permission from Elsevier.] 110

2.66 Sensitivity of the temperature profile of cyclic drying, Case 1 (refer toTable 2.12) towards q (on Equation 2.12.3). [Reprinted from ChemicalEngineering Science, 65, A. Putranto, X.D. Chen and P.A. Webley,Application of the reaction engineering approach (REA) to model cyclicdrying of thin layers of polyvinyl alcohol (PVA)/glycerol/water mixture,5193–5203, Copyright (2012), with permission from Elsevier.] 110

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) withq = 1.8. [Reprinted from Chemical Engineering Science, 65, A.Putranto, X.D. Chen and P.A. Webley, Application of the reactionengineering approach (REA) to model cyclic drying of thin layers of

xx List of figures

polyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright(2012), with permission from Elsevier.] 111

2.68 Temperature profile of cyclic drying, Case 1 (refer to Table 2.12) usingthe 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), withpermission from Elsevier.] 111

2.69 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), withpermission from Elsevier.] 112

2.70 Temperature profile of cyclic drying, Case 2 (refer to Table 2.12) usingthe 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), withpermission from Elsevier.] 112

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) withq = 1.5. [Reprinted from Chemical Engineering Science, 65, A.Putranto, X.D. Chen and P.A. Webley, Application of the reactionengineering approach (REA) to model cyclic drying of thin layers ofpolyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright(2012), with permission from Elsevier.] 113

2.72 Temperature profile of cyclic drying, Case 2 (refer to Table 2.12) usingthe 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 engineeringapproach (REA) to model cyclic drying of thin layers of polyvinylalcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright (2012),with permission from Elsevier.] 113

2.73 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), withpermission from Elsevier.] 114

List of figures xxi

2.74 Temperature profile of cyclic drying, Case 3 (refer to Table 2.12) usingthe 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), withpermission from Elsevier.] 115

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) withq = 1.6. [Reprinted from Chemical Engineering Science, 65, A.Putranto, X.D. Chen and P.A. Webley, Application of the reactionengineering approach (REA) to model cyclic drying of thin layers ofpolyvinyl alcohol (PVA)/glycerol/water mixture, 5193–5203, Copyright(2012), with permission from Elsevier.] 115

2.76 Temperature profile of cyclic drying, Case 3 (refer to Table 2.12) usingthe 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), withpermission from Elsevier.] 116

3.1 Schematic diagram of a cube dried in a uniform convective environment. 1223.2 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 10and 200 spatial increments. [Reprinted from AIChE Journal, 59,Aditya Putranto, Xiao Dong Chen, Spatial reaction engineeringapproach as an alternative for nonequilibrium multiphase mass-transfermodel for drying of food and biological materials, 55–67, Copyright(2012), with permission from John Wiley & Sons Inc.] 131

3.3 Average moisture content profiles of mango tissues during convectivedrying at different drying air temperatures. [Reprinted from AIChEJournal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reactionengineering approach as an alternative for nonequilibrium multiphasemass-transfer model for drying of food and biological materials, 55–67,Copyright (2012), with permission from John Wiley & Sons Inc.] 134

3.4 Centre temperature profiles of mango tissues during convective drying atdifferent drying air temperatures. [Reprinted from AIChE Journal, 59,Aditya Putranto, Xiao Dong Chen, Spatial reaction engineeringapproach as an alternative for nonequilibrium multiphase mass-transfermodel for drying of food and biological materials, 55–67, Copyright(2012), with permission from John Wiley & Sons Inc.] 134

3.5 Spatial moisture content profiles of mango tissues during convectivedrying at drying air temperatures of 45 °C. [Reprinted from AIChEJournal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction

xxii List of figures

engineering approach as an alternative for nonequilibrium multiphasemass-transfer model for drying of food and biological materials, 55–67,Copyright (2012), with permission from John Wiley & Sons Inc.] 135

3.6 Spatial water vapour concentration profiles of mango tissues duringconvective drying at drying air temperatures of 45 °C. [Reprinted fromAIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reactionengineering approach as an alternative for nonequilibrium multiphasemass-transfer model for drying of food and biological materials, 55–67,Copyright (2012), with permission from John Wiley & Sons Inc.] 136

3.7 Spatial temperature profiles of mango tissues during convective dryingat drying air temperatures of 45 °C. [Reprinted from AIChE Journal, 59,Aditya Putranto, Xiao Dong Chen, Spatial reaction engineeringapproach as an alternative for nonequilibrium multiphase mass-transfermodel for drying of food and biological materials, 55–67, Copyright(2012), with permission from John Wiley & Sons Inc.] 137

3.8 Profiles of evaporation rates inside mango tissues during convectivedrying at a drying air temperature of 55 °C. [Reprinted from AIChEJournal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reactionengineering approach as an alternative for nonequilibrium multiphasemass-transfer model for drying of food and biological materials, 55–67,Copyright (2012), with permission from John Wiley & Sons Inc.] 138

3.9 Moisture content profiles in the core and cortex during convective dryingof potato tissues with a diameter of 1.4 cm. [Reprinted from AIChEJournal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reactionengineering approach as an alternative for nonequilibrium multiphasemass-transfer model for drying of food and biological materials, 55–67,Copyright (2012), with permission from John Wiley & Sons Inc.] 139

3.10 Moisture content profiles in the core and cortex during convective dryingof potato tissues with a diameter of 2.8 cm. [Reprinted from AIChEJournal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reactionengineering approach as an alternative for nonequilibrium multiphasemass-transfer model for drying of food and biological materials, 55–67,Copyright (2012), with permission from John Wiley & Sons Inc.] 140

3.11 Core temperature profiles during convective drying of potato tissueswith a diameter of 1.4 cm. [Reprinted from AIChE Journal, 59, AdityaPutranto, Xiao Dong Chen, Spatial reaction engineering approach as analternative for nonequilibrium multiphase mass-transfer model fordrying of food and biological materials, 55–67, Copyright (2012), withpermission from John Wiley & Sons Inc.] 140

3.12 Average moisture content profiles of mango tissues during intermittentdrying at different drying air temperatures. 143

3.13 Spatial moisture content profiles of mango tissues during intermittentdrying at a drying air temperature of 55 °C. 144

List of figures xxiii

3.14 Spatial water vapour concentration profiles of mango tissues duringintermittent drying at a drying air temperature of 55 °C. 144

3.15 Centre temperature profiles of mango tissues during intermittent dryingat different drying air temperatures. 145

3.16 Spatial temperature profiles of mango tissues during intermittent dryingat a drying air temperature of 55 °C. 146

3.17 Profiles of evaporation rate inside mango tissues during intermittentdrying at a drying air temperature of 55 °C. 147

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 100increments. 151

3.19 Effect of liquid diffusivity on profiles of the moisture content during heattreatment in Case 1 (refer to Table 3.5). 152

3.20 Effect of liquid diffusivity on profiles of temperature during heattreatment in Case 1 (refer to Table 3.5). 152

3.21 Profiles of average moisture content during heat treatment in Case 1(refer to Table 3.5). 153

3.22 Profiles of temperature during heat treatment in Case 1 (refer to Table3.5). 153

3.23 Profiles of average moisture content during heat treatment in Case 2(refer to Table 3.5). 154

3.24 Profiles of temperature during heat treatment in Case 2 (refer to Table3.5). 155

3.25 Profiles of spatial moisture content during heat treatment in Case 2 (referto Table 3.5). 156

3.26 Profiles of spatial water vapour concentration during heat treatment inCase 2 (refer to Table 3.5). 156

3.27 Profiles of spatial temperature during heat treatment in Case 2 (refer toTable 3.5). 157

3.28 Profiles of average moisture content during the baking of bread at abaking temperature of 150 °C. 161

3.29 Spatial profiles of moisture content during the baking of bread at abaking temperature of 150 °C and air velocity of 10 m s−1. 162

3.30 Spatial profiles of concentration of water vapour during the baking ofbread at a baking temperature of 150 °C and air velocity of 10 m s−1. 162

3.31 Profiles of top and bottom surface temperatures during the baking ofbread at a baking temperature of 150 °C and air velocity of 1 m s−1. 163

3.32 Spatial profiles of temperature during the baking of bread at a bakingtemperature of 150 °C and air velocity of 10 m s−1. 163

4.1 Experimental setup for convective drying of porcine skin. [Reprintedfrom Chemical Engineering Research and Design, 87, S. Kar, X.D.Chen, B.P. Adhikari and S.X.Q. Lin, The impact of various dryingkinetics models on the prediction of sample temperature–time and

xxiv List of figures

moisture content–time profiles during moisture removal from stratumcorneum, 739–755, Copyright (2012), with permission from Elsevier.] 174

4.2 (a) Overview of a sample/plate assembly for convective drying ofporcine 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 dryingkinetics models on the prediction of sample temperature–time andmoisture content–time profiles during moisture removal from stratumcorneum, 739–755, Copyright (2012), with permission from Elsevier.] 175

4.3 Moisture content profiles from the convective drying of mango tissuesmodelled using the L-REA and diffusion-based model (Vaquiro et al.,2009). [Reprinted from Drying Technology, 29, A. Putranto, X.D. Chenand P.A. Webley, Modelling of Drying of Food Materials with Thicknessof Several Centimeters by the Reaction Engineering Approach (REA),961–973, Copyright (2012), with permission from Taylor & Francis Ltd.] 187

4.4 Temperature profiles from convective drying of mango tissues modelledusing 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 severalcentimeters by the reaction engineering approach (REA), 961–973,Copyright (2012), with permission from Taylor & Francis Ltd.] 188

4.5 Moisture content profiles from the convective drying of mango tissuesmodelled 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 fornon-equilibrium multiphase mass-transfer model for drying of food andbiological materials, DOI 10.1002/aic.13808, Copyright (2012), withpermission from John Wiley & Sons, Inc.] 189

4.6 Temperature profiles from the convective drying of mango tissuesmodelled 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 fornon-equilibrium multiphase mass-transfer model for drying of food andbiological materials, DOI 10.1002/aic.13808, Copyright (2012), withpermission from John Wiley & Sons, Inc.] 190

4.7 Moisture content profiles from the heat treatment of wood modelledusing the L-REA and Luikov’s approach. [Reprinted from BioresourceTechnology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley,Modelling of high-temperature treatment of wood by using the reactionengineering approach (REA), 6214–6220, Copyright (2012), withpermission from Elsevier.] 200

4.8 Temperature profiles from the heat treatment of wood modelled usingthe L-REA and Luikov’s approach. [Reprinted from BioresourceTechnology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A. Webley,

List of figures xxv

Modelling of high-temperature treatment of wood by using the reactionengineering approach (REA), 6214–6220, Copyright (2012), withpermission from Elsevier.] 201

4.9 Moisture content profiles from the heat treatment of wood (refer to Table4.1) modelled using the L-REA and Whitaker’s approach. [Reprintedfrom Bioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao andP.A. Webley, Modelling of high-temperature treatment of wood by usingthe reaction engineering approach (REA), 6214–6220, Copyright(2012), with permission from Elsevier.] 202

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 fromBioresource Technology, 102, A. Putranto, X.D. Chen, Z. Xiao and P.A.Webley, Modelling of high-temperature treatment of wood by using thereaction engineering approach (REA), 6214–6220, Copyright (2012),with permission from Elsevier.] 203

4.11 Moisture content profile from the heat treatment of wood modelled usingthe S-REA and Luikov’s approach. 204

4.12 Temperature profile from the heat treatment of wood modelled using theS-REA and Luikov’s approach. 204

4.13 Moisture content profiles from the heat treatment of wood (refer to Table4.1) modelled using the S-REA and Whitaker’s approach. 205

4.14 Temperature profiles from the heat treatment of wood (refer to Table 4.1)modelled using the S-REA and Whitaker’s approach. 205

Tables

2.1 Experimental conditions of convective drying of a mixture ofpolymer solutions (Allanic et al., 2009). page 51

2.2 R2 and RMSE of modelling of a mixture of polymer solutions usingthe L-REA. 58

2.3 Experimental conditions of convective drying of mango tissues (Vaquiroet al., 2009). 61

2.4 R2 and RMSE of modelling of convective drying of mango tissues usingthe L-REA. 66

2.5 Schemes of intermittent drying of mango tissues (Vaquiro et al., 2009). 692.6 R2 and RMSE of modelling of intermittent drying of mango tissues using

the L-REA. 732.7 Settings of intermittent drying of kaolin (Kowalski and Pawlowski,

2010). 812.8 R2, RMSE, average absolute deviation and maximum absolute deviation

of profiles of moisture content predicted by and Kowalski andPawlowski’s model (2010b). 85

2.9 R2, RMSE, average absolute deviation and maximum absolute deviationof 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). 892.11 R2 of modelling using the REA. 992.12 The experimental conditions of intermittent drying of a mixture of

polymer solutions. 1053.1 Experimental conditions of convective drying of mango tissues (Vaquiro

et al., 2009). 1283.2 R2 and RMSE of convective drying of mango tissues using the S-REA. 1353.3 Scheme of intermittent drying of mango tissues (Vaquiro et al., 2009). 1413.4 R2 and RMSE of intermittent drying of mango tissues. 1423.5 Experimental settings of wood heating under a constant heating rate

(Younsi et al., 2007). 1483.6 R2 and RMSE of modelling of heat treatment of wood under a constant

heating rate using the S-REA. 1554.1 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 biologicalmaterials. Low moisture content in foods prevents the growth of bacteria responsible fortheir deterioration so foods can have extended shelf-lives. When foods became abundant,trade became possible. Today, dried products are the main materials trading round theworld but this is not limited to food products. Construction materials, textiles, electronicparts and appliances, biomass-based fuels, pharmaceutics and many other materialsimportant 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. Productquality and process parameters are interactive. Industrial drying is energy hungry; aprocess involving simultaneous heat, mass transfer and momentum transfer. Productquality is determined through compositional and structural rearrangements, as well aschemical reactions in some circumstances. For existing drying facilities, optimisation isoften needed to achieve new goals such as energy reduction, quality improvements anddevelopment of new materials. There are also opportunities in designing dryer modi-fications or even brand new dryers that are superior in performance over conventionaldevices. Modelling of drying processes is very useful for these purposes.

A number of drying models have been proposed, which are conveniently classifiedinto empirical and mechanistic models. The empirical models give advantages of beingsimple in their mathematical formulation. However, these models most often cannotexplain the physics of drying and their application is limited since they are valid onlyfor a particular set of drying conditions. On the other hand, the mechanistic models arederived based on fundamental phenomena that occur during drying. These phenomenaare crucial in material science (and materials processing) though material scientiststhemselves may not have yet come to appreciate the process engineering aspects whichimpact on the product microstructure. Some of these models can capture the physicswell. These models are, however, often mathematically complex and sometimes containtoo many parameters, which need to be determined experimentally (prior to modelpredictions).

For some decades now, a comprehensive set of macroscopic equations has been devel-oped and used to address heat and mass transfer and mechanical aspects related to drying.The application of macroscopic descriptions of drying (temperature, moisture and some-times pressure) has been perfected over the past two decades, and relevance has beenconfirmed in many drying configurations. Some of these involve irreversible thermody-namics formulations, which are lengthy and have many model coefficients. These have

xxviii Preface

become the ‘classical’ approach. However, this classical approach has serious limita-tions. 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 thermodynamicequilibrium is not attained, however, the time scales usually overlap. This is a real multi-scale configuration and challenging in terms of the great demand in computational powerand handling of mathematics. Several scales can be considered simultaneously, rangingfrom simple exchanges between macroscopic phases to comprehensive formulations inwhich time evolution of microscopic values and microscopic gradients is consideredover 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 ofdrying, see Drying Technology, 29, 1529–1541, 2011).

While exploring the detailed physics involved in drying using these multi-scale andmulti-physics approaches, it is, from an engineering viewpoint, also important to developnew ideas establishing simpler models. In general, today industrial drying applicationsrequire mathematical models that are simple and easy to use. For practical purposes, aneffective drying model should be simple, accurate, and able to capture the major physicsof drying and its application should be robust. This model should also favour shortcomputation 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, perhapsbetween the empirical and the mechanistic models, was first thought about by the firstauthor of this book, Chen, in 1996. Through much of the research on its possible applica-tions, it has been revealed that the REA is indeed simple, accurate and robust enough tomodel many cases of drying, i.e., drying in a constant or variable environment. The REAhas also been implemented in industry for prediction of spray dryer performance andshows good agreement with plant data for different scales in the dairy industry. It has alsobeen 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 effortto establish the parameters needed, compared with the more fundamental models. TheREA was first taken as a lumped model which does not need us to resolve the spatialdistribution of water content, etc.; the lumped-REA (or L-REA), but in recent times, wehave 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 isa significant amount of successful applications already illustrated. As mentioned earlier,it is a middle path between the rigorous theory that requires high-level mathematics andthe empirical models that do not represent much physics. We can see, through our ownpractices and from other colleagues in the same area who have used the REA concept, itis 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 REAapproach to drying modelling – the basic idea, rationale, mathematical descriptionand implementation procedures – for various systems. This approach has been extended,

Preface xxix

and experimented with, by several quality Ph.D. graduates, in particular, the secondauthor, Aditya Putranto. Regarding the other more established theories, this book notonly provides essential details so the readers can refer to them but also illustrates, bycomparison, the physics involved in REA concepts. The disadvantages and advantagesbetween theories are also briefly introduced. The book should benefit both academics indrying research and practicing engineers in industry. Undergraduate students in processengineering may also find it useful for quickly setting up a drying model for designpurposes. The main emphasis of this book is how to apply the REA to reality. The bookwill also elaborate on potential applications of similar thinking to more complex reactivesystems that couple with drying processes, hopefully to foster their future development.

Here, the modern ideas of microstructure development and product qualities createdby 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 atCanterbury University, Christchurch, New Zealand, (1988–1990), the main task wasto establish mechanistically the understanding of moisture influence on coal oxidationand the impact of moisture transfer in a packed coal particle bed on the developmentof spontaneous combustion. The experimental aspect was challenging both technicallyand physically. In addition to coal oxidation and its racemic measurement, I becamevery interested in the mechanisms of water evaporation and moisture transfer (liquidand vapour) in porous material. Dr Jim Stott (Reader of Chemical Engineering) was mymain supervisor and Dr John Abrahamson (Senior Lecturer), in the same department,was my cosupervisor. Jim published some of the pioneering literature on the subject ofspontaneous combustion of coal (1959) and built (largely by himself) ingenious exper-imental rigs. Dr Abrahamson was an inspirational and distinguished individual as wellwho has been credited as one of the first to have made a carbon nanotube (he called itthe ‘carbon cylinder’) (1978), a theory of ball lightening (2000) and a theory of particlecollision 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 moistcoal bed has taught me that if the coal bed were completely saturated with water vapourunder near ambient pressure (the institutional voids of the bed remain saturated withwater vapour), the maximum temperature would remain at around 80 °C. This waspredicted from a numerical spontaneous combustion model involving mass transferof moisture within the coal bed when assuming the vapour concentration in the bedis always saturated. Jim discovered this in the late 1960s, and later, in the 1970s, aPh.D. student of his proved this more comprehensively. This aspect was more or lessrepublished in 1990s by a research group in Europe (who were perhaps unaware ofthe work by Jim and his ex-students). However, if an equilibrium relationship betweenmoisture content in the coal particles and vapour concentration in the air surroundingthe particles can be adopted, a dry spot can be predicted and the maximum temperaturewill exceed the boiling temperature of water, therefore rising to an elevated temperaturedue to oxidation heat (Chen, 1992a). Of course, there are also other influences such asporosity, oxidation rate and oxygen transfer, heat transfer and, sometimes, fluid flow dueto a pressure gradient. Nevertheless, this equilibrium relationship is what we are now sofamiliar with, termed the equilibrium isotherm in drying literature. The oxidation rateof coal itself was also found, in my own experiments, to vary with the residual watercontent (Chen and Stott, 1993) and I had gone to extra lengths to try to understand this

Historical background xxxi

phenomenon. This formed the foundation of my understanding of the presence of wateraffecting chemical (and biochemical) reactions. In food drying, it would mean that theremoval of water, to some extent, could significantly reduce rate of deterioration, givinga 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 firstbook on drying principles and practice that was published in English. I had spent a lot oftime looking for information on how to model moisture transfer, coupled with chemicalreactions and heat transfer and momentum transfer. Keey’s books over the years have hadan impact on my own work related to this area (especially the latest one; Keey, 1992). Inparticular, 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 Canterburygraduate, has worked extensively on this idea, which has extended Keey’s views on thedrying of wood and some other different materials, including foods. His postdoctoralperiod (after his return from Oxford University) with Professor Keey overlapped with thefinal year of my Ph.D. (1990). Another distinguished individual whose work has affectedmy own thinking has been Professor Shusheng Pang, another Canterbury graduatesupervised by Roger Keey, who has published some key literature in wood drying relatedto the application of CDRC. CDRC captures the phenomena of drying by recognisingthe existence of a constant and falling drying rate period(s). The critical or transitionalwater content between any of the connecting rate schemes are recognised (Keey, 1992).Doctors Sandeep Chu and Peter Kho, who were student colleagues at Canterbury duringthe period of 1988–1990 and whose works were supervised by Professor Keey, also hadan impact on my later research on drying.

Some others who also influenced me positively were Professor Miles Kennedy, Dr JohnPeet and Dr Maurice Allen at Canterbury. I had read many of their works during thepeaceful evenings when I had pretty much the whole department to myself and someof the weekends during my Ph.D. study at the corner room on the top floor of Simon’sBlock. The surroundings of Canterbury University were beautiful and peaceful and gaveme great times (and spaces) to spend thinking about my work and, of course, my lovedones.

I submitted my Ph.D. thesis three years after I started in late December 1987. Istarted working at the New Zealand Dairy Research Institute (NZDRI) (which is nowthe Fonterra Research Centre based in Palmerston North of New Zealand), first asan engineer and then as a senior engineer, working on spray drying and milk pow-der agglomeration. Dr Kevin Pearce (my section manager), who was a distinguishedchemist, gave advice that I understood one has to take in order to take protein chemistryseriously when dealing with engineering problems related to dairy products. This periodof time was very constructive for my career development. After coal research, I reallywanted to move onto biotechnology and, at the time, the food industry was the nearestthing to biotechnology in which I could secure a good position. I was deeply involvedin 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

xxxii Historical background

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

Zealand.

Historical background xxxiii

make a significant contribution in the area of agglomeration (hardware improvementand macrostructural analysis) and new product development that was hampered by highstickiness, with large financial returns for the dairy industry.

My employment as an academic at the Department of Chemical and Materials Engi-neering, The University of Auckland, started in late 1993, which instantly gave megreater freedom to develop new ideas. I had great fun working at Auckland, benefittingfrom being surrounded by a number of highly positive individuals at the department andthe school. One strong influence came from my colleagues who were experts in mate-rials science. Among many other studies, in 1995–1996, I had developed an idea thatwas 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 ambitiousyoung man (me) but later was proven to be, well, ‘kinetics is just kinetics and equilib-rium 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 concentrationand the water content of the porous solid material being dried, noting that this surfacevapour concentration less the vapour concentration in the gas phase is the driving forcefor moisture transfer from the porous material to the drying environment, the modelcould be a good alternative to the CDRC model. The obvious one for surface vapourconcentration to relate to is liquid water content. At the time, I already found some issueswith the CDRC approach, as uncertainty can be great depending on the drying processesconsidered. Keey (1992) has rightly pointed out that the CDRC model was excellent forparticles or sample sizes smaller than 20 mm for constant drying conditions.

The link between the surface vapour concentration and the remaining water con-tent in the porous material as well as the material surface temperature was eventuallyconstructed using an Arrhenius-type relationship, which essentially suggested the evap-oration of pure water and ‘extraction’ of the water from the inside the porous materialwas a reaction and the condensation/adsorption was not an activation process. This isin line with a mathematical description of evaporation and a condensation mechanismformulated by Gray and Wake (1990). Professor Brian Gray (Professor of Applied Math-ematics at Sydney University at the time) is a distinguished applied mathematician (heis also a distinguished physical chemist) whom I came to know through the link betweenme and Professor Graeme Wake (another outstanding applied mathematician from NewZealand) 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 inthe systems of reactions, both exothermic and endothermic. Evaporation is viewed as anendothermic 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 andwe prepared a simple paper for Chemeca in 1997. I was also fortunate in hosting a vis-iting researcher from Xian Jiao-Tong University (China) during 1996–1997, AssociateProfessor Guozhen Xie, who was a refrigeration expert but was daring enough to pickup drying modelling as the main topic in his year of working with me. We didn’t do anyexperiments on drying but used data reported in the literature. However, in all cases, wehad to solve an energy balance to obtain (surface) temperature of the material tested for

xxxiv Historical background

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, wecould establish the activation energy in the Arrhenius equation (mentioned earlier) todemonstrate the concept.

Then, in 2000, at Auckland, I had a masters student by research, Wayne Pirini, whowas interested in drying, so we started experimenting on thin-layer drying of variousmaterials measuring both weight loss and temperature as drying proceeded. The first lotof 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 couldnot account for the conditions when evaporation occurs. This gave me an opportunityto derive a modified Biot number later on (the so-called Chen–Biot number). Then, atAuckland in the period of 2000–2004, Dr Sean Lin, my Ph.D. student at that time, did acomprehensive study on droplet drying kinetics for dairy products in particular. He hadlots of practical experience before coming to me. He designed and built an excellentcost-effective droplet drying test rig and conducted probably the most careful, accurateexperiments on dairy droplet drying. This has allowed the comprehensive establishmentof the REA model for dairy droplets that is relevant to the spray drying industry (Chenand Lin, 2005).

Following that, two Ph.D. students under my supervision who were from India, DrsSaptarshi Kar and Kamlesh Patel, had made a significant contribution to the develop-ment of the REA concept. Saptarshi applied REA to a spatial distributed case for watertransport in skin relevant to transdermal drug delivery for the first time. We deliberatelyignored the liquid diffusivity to see if it really mattered. It turned out that it really did mat-ter. Kamlesh had helped in extending the Chen–Biot number concept and helped to bringin a new concept called the ‘composite REA approach’, which describes an approachto estimating the activation energies of sugar mixtures based on the components’ ownactivation energies. They were both tremendous students with high aptitudes to pursuebasic 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) toSenior Lecturer (1995) to Associate Professor (1998) to Personal Chair Professor at theage of 36 (2001). It was the most dramatic time in my life, both in career and personallife. I had my first child, Lisa, who was born in May 2000. Sad events had taken hermother away from her in 2001. I must thank the Engineering Dean at the time, ProfessorPeter Brothers, who, in my darkest days in 2001, promised his institution’s support inallowing me to do whatever I needed to do and go wherever I needed to go withoutworrying about losing my job.

Beyond that, I enjoyed tremendous learning experiences, friendships, and supportfrom my colleagues at the Department of Chemical and Materials Engineering: Pro-fessor Geoff Duffy (who was most influential individual in my stay at Auckland),Associate Professor Kevin Free, Professor John J. J. Chen, Professor Wei Gao, ProfessorMohammed Farid, Professor Neil Broom, Professor George Fergusson and Dr NecatiOzkan. I was inspired by the genius professors such as Professor John Boys (Electri-cal Engineering), Professor Peter Hunter (Engineering Science) and Professor Debes

Historical background xxxv

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

Bhattahtrayya (Mechanical Engineering) for their innovations. I benefited tremendouslyfrom collaborating with Associate Professor Sing Kiong Nguang, who is a genius inmathematical problems in system and dynamics engineering. In that period of time atAuckland, I picked up the idea of combining process engineering and material scienceand became familiar with microscopy and material science techniques. My colleagueshave 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 ofOxford University) and Professor Diane McCarthy (Dean of Medicine at Aucklandand later President of Royal Society of New Zealand). Without their recognition ofmy ability and my contribution, my rapid promotion at Auckland would not have beenpossible.

Coming back to the main technical topic, can the REA model do the things that aCDRC model cannot? For small-size particles and constant drying conditions, CDRCseems to be very comparable with REA. With this question, and many others, I hadmoved 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 AdityaPutranto, a humble young man, joined my group at Monash to do a Ph.D. with me.

xxxvi Historical background

He demonstrated superior ability in testing and further developing REA for numerousapplications, which are presented in this book.

In 2010, I moved to Xiamen University on the southeast coast of China, from which mygrandparents graduated in history and English literature in 1930 and 1931, respectivelyas a National Expert Professor of Chemical Engineering (also known as the 1000-Elite Chair Professor). I have not stopped the excellent collaborations with Aditya andwe 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 YanJin (in 2009), and Dr Mengwai Woo (2010–2011), have also continued to contributeexperimentally, and theoretically, to the establishment of REA and its applications tothe real world. Notably, Nan generated significantly new data on the REA approachto dairy droplet drying and linked drying to crystallisation and particle solubility. Shehas extended the techniques of single droplet drying to a more powerful means inorder to understand drying-quality inter-relations. Dr Jin has comprehensively modelledthe three-dimensional transient flows in large-scale spray dryers and has incorporatedthe REA approach. Dr Woo has independently investigated the robustness of the REAapproach for modelling droplet drying in the context of computational fluid dynamicsof spray dryers.

In no way can I claim it was only me who made REA development possible, but I canclaim the original idea and model framework to be mostly mine. I sincerely thank all thepreviously mentioned individuals and others whom I have not mentioned but who havemade contributions to the development of the REA in one way or another. REA is alsothe result of a belief that engineering theory should be as simple and robust as possiblein 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 andBenjamin.

To my grandparents, my parents, my sister and brother-in-law for their neverendinglove and support.

To others whom I have loved and who have loved me selflessly.

Xiao Dong ChenXiamen City,Southeast Coast of ChinaAugust 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 otherswhom have shared, and will share the love and faithfulness of the Creator.

Aditya PutrantoMelbourneAustraliaAugust 2012

Historical background xxxvii

References

Chen, X.D., 1991. The Spontaneous Heating of Coal – Large Scale Laboratory Assessment andSupporting Theory, Ph.D. thesis, Chemical and Process Engineering Department, University ofCanterbury, New Zealand.

Chen, X.D., 1992a. On the mathematical modelling of transient process of spontaneous heatingin a moist coal stockpile. Combustion and Flame 90, 114–120.

Chen, X.D., 1992b. Whole milk powder agglomeration – Principle and practice. In Milk Powdersfor 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 modellingdrying 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-dependentconditions. AIChE Journal 51(6), 1790–1799.

Chen, X.D. and Mujumdar, A.S. (eds.), 2008. Drying Technologies in Food Processing. BlackwellPublishing Ltd: Oxford.

Chen, X.D. and Stott, J.B., 1993. The effect of moisture content on the oxidation rate of coalduring 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 inlarge 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 foodmaterials established using a simple reaction engineering model. Transactions of the Instituteof 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 wetcombustion system, Proc. Roy. Soc. A 429, 449–458.

Gray, B.F. and Wake, G.C., 1990. The ignition of hygroscopic materials by water. Combustionand 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 stepfor a wide range of human endeavours in our history. The dependence of human societyon drying is highly visible. For instance, in food production and preservation, dryingis the oldest, most popular and one of the most effective ways to make solid foodsand to preserve them as long as practically required. The textile industries need dryingprocesses. Natural fibre-based products such as those from the wood and paper industriesalso need drying as a critical step in manufacturing. In fact, anything having to do withthe particulate products, not just food particles (milk powders, vegetable soup powdersand 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 seenbefore in our history, wet chemistry is needed, which requires drying (‘dewetting’) toform 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 theoriginal water content removed either naturally, through solar drying or sometimes freezedrying and air drying (with low-humidity air in particular), or ‘unnaturally’, through theuse of specialised dryers or dehydrators powered by electricity or combustion. Theseunnatural ways include mechanical dewatering, convective air/gas drying, superheatedsteam drying, electro-osmosic processes, osmotic pressure-driven processes, refractorywindow drying, freeze drying, vacuum drying, and microwave-aided drying processes,to name a few. If one includes liquid evaporation (with liquid products also), these maybe 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, concentratednutritional value and long shelf-life because the water activity is low (Chen andMujumdar, 2008). As water content is removed, the material shrinks (leading to a smallervolume), the sugar content per unit volume of the material increases, as do nutritionalcomponents such as proteins, vitamins and so on. Today, dried fruit consumption iswidespread. 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

http://en.wikipedia.org/wiki/Dried_fruit

2 Modelling Drying Processes

(a) (b)

(c) (d)

Figure 1.1 Some traditional dried products. (a) Broccoli-steam blanched and air dried (kindlyprovided 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 (particlesize 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 inthe 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. sucrosesyrup) prior to drying to enhance sweetness and microbial stability. This means theseare not necessarily healthy products, especially for diabetics. Sugar-infused and driedpapaya and pineapples are actually candied fruit. Dried fruits are usually thought toretain most of the nutritional value of the fresh fruits. The specific nutritional content ofdried fruits reflects that of their fresh counterparts and is influenced by the processingmethod or processing technology, particularly processing temperature. In general, alldried 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 ofchronic disease.

Furthermore, dried fruits (and nuts) are not only important sources of vitamins, min-erals 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 sincethey are not essential to sustain life but play a role in health and longevity and havebeen linked to a reduction in risk of developing major chronic diseases. Convincingevidence suggests that the benefits of phytochemicals may be even greater than cur-rently understood, as they seem to affect metabolic pathways and cellular reactions.However, the precise mechanisms by which specific compounds exert their biologicaleffect remains largely hypothetical, which requires greater investigation. Certainly, asis 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 thediet and appear to be, at least partially, responsible for the potential benefits associ-ated with the consumption of diets rich in fruits and vegetables. Different dried fruitshave unique phenolic profiles (Donovan et al., 1998); for example, the most abun-dant in raisins are the flavonols (quercetin and kaempferol) and the phenolic acids(caftaric and coutaric acid) (Willamson and Carughi, 2010). Therefore, due to theirhigh polyphenol content, dried fruits are an important source of antioxidants in thediet (Wu et al., 2004; Vinson et al., 2005). Antioxidants can lower oxidative stressand so prevent oxidative damage to critical cellular components. Dried apricots andpeaches are also important sources of carotenoids. These compounds not only are pre-cursors of vitamin A but also have antioxidant activity. Dried fruits such as driedplums provide pectin, a soluble fibre that may lower blood cholesterol levels (Tinkeret al., 1991). Dried fruits such as raisins are a source of prebiotic compounds inthe diet. They contain fructooligosaccharides like inulin, naturally occurring fibrelikecarbohydrates that contribute to colon health (Camire and Dougherty, 2003). The chem-ical structures of some of these useful compounds for human health are shown inFigure 1.2.

Since drying foods may affect the chemical structure of their components, someof which could be undesirable, it is important to keep a balance between how fast orefficient the drying process is in terms of energy usage or product throughput and qualityrequirements. Sometimes, faster processing is not necessarily better. Like food material,all the solid-form products, natural or processed, have interesting structures (includingmicrostructures) and qualities.

It is worthwhile noting that removing water in its liquid form from a solid structureis often (perhaps better) called ‘dewatering’, which induces solid structural changesaround or near where the water used to be. In a more general situation, water removalmay also be called ‘dehydration’, especially when talking about materials of partially orwholly biological origin.

In this book, drying is mostly referred as those processes that use gas as a dry-ing medium so the water comes out of the material as a gas (water vapour). Later inthis book, other forms of drying, such as vacuum drying or even steam-aided dry-ing, may be employed using the reaction engineering approach (REA) when it isappropriate.

Furthermore, roasting (coffee, for instance), baking (biscuits and bread, etc.) andheating of moist material (e.g., detoxification of wood) may be considered extensions ofthe concept of drying, and their purposes are for improving material performance whilethe moisture effect simply cannot be ignored.


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Introduction 5

The powering mechanisms for drying can be solar radiation, electricity, steam,microwave and ultrasound, amongst others. In all drying operations, energy consump-tion 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 productsoften related to nanotechnology these days, nanostructure and microstructure aspectsand functionalities are an increasing concern for both the researchers and marketers. Infood and pharmaceutical products, these aspects are even more important as they affectthe metabolism and health of our organs.

Therefore, as far as drying is concerned, it provides an excellent example of processengineering interacting with product quality. In chemical engineering, the interactionsmay be considered systematically with the ideas of chemical process engineering versuschemical product engineering (Cussler and Mogoridge, 1997).

One may see engineering as a terminology which differs from technology. Engineer-ing should involve developing the predictive tools (evaluating a building design usingestablished mathematical analysis and then building it accordingly), which need goodmathematical descriptions of the physics and chemistry that go on into the relevant pro-cesses, much like in modern times, where calculations are used to investigate the validityof constructing an architecturally designed building before actually building it; i.e., civilengineering. 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 thedrying processes, one needs accurate and robust mathematical models. Better still, someof the models can actually help us explore drying mechanisms (physics, chemistry andbiochemistry), investigating the scientific aspects associated with the drying phenom-ena. Drying models are usually referred to as models that describe the mass and heattransport within the material being dried, as the exterior conditions are already nicelycovered by the conventional heat and mass transfer and momentum transfer theories. Theboundary conditions are intended to connect the drying models with exterior transporttheories.

Simple, yet effective, mathematical models of drying are welcomed by practitionersor engineers. When modelling multiphase flow in a spray dryer, for instance, one mightneed to mathematically track thousands of particles of different sizes travelling insidethe drying chamber to make the simulation more realistic. If one has to solve the spatiallydistributed variables (like water content) inside each particle, the computational effortis great; hence, the whole exercise incurs a high expense. There is also a high likelihoodof computational instability. However, if one does not need to solve for the spatialdistributions of moisture, temperature and species distribution within each particle, oneonly needs to integrate a model over time to obtain the average water content, temperatureand average concentrations of species for each particle; the computational effort is muchsmaller.

For a large piece of material being dried, a ‘lumped’ drying model that may predictdrying history accurately without the need to resolve the spatially distributed parametersis also very handy for practical purposes. Especially when many such pieces of the samematerial 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


computational input. The lumped models can be implemented using simple softwaresuch as an Excel spread sheet and sometimes even a simple, programmable scientificcalculator.

To be relevant to the specific material of concern, every drying model proposed needscareful experimentation to establish the required model parameters (constants) and themodel predictive power. In other words, these constants are mostly material specific.For a diffusion-based drying model, for example, effective mass diffusivity is the keyparameter. It may be both water content and temperature dependent, making accurateexperimental determination and data analysis difficult due to non-uniform temperaturedistribution when it comes to relatively large materials.

If there was one model that incurred only minimal experimental effort and less demandfor lab facilities, that model would be welcomed by industry. Relevant experiments haveto be accurate, possessing the required resolution, and simple, robust apparatuses andsimple 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 dryingmoist, porous materials

Drying, the process of water removal, can affect the chemical composition of where thewater 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 ortotally by a nearby species. These ‘movements’ should affect the microstructure forma-tion of the material being dried. These movements may include rearrangement of solutemolecules (in drying liquid in particular) and shrinkage (with shell formation or hardsurface 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 solidsurface 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 thebiological 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 cellwall or cell membranes and cell contents such as genetic material, enzymes, or proteinswithin, causing irreversible damage. This damage could be minimised if there wasanother structure-supporting material such as sugar (this can be done by infusing theproducts, mostly food materials, with sugar molecules in osmotic treatment) whichmay ‘replace’ the water molecules to uphold bioactivity. It is also possible that thedrying–concentration process alters the ionic conditions, which may be more suited forcells’ survival. Drying, in high-temperature instances, with still-considerable moisturecontent, can cause proteins to denature, thus affecting the solubility and heat stabilityof protein products. The colour of the food material can be altered during drying, espe-cially when heat is added. Here, Mallard reactions may be responsible where proteins

Introduction 7

Temperatureprofile

Liquid content

Symmetry

Uniformcapillaryassembly

Heat flux

T∞

Vapour flux

Mixture of watervapour and air

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 thesymmetrical 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 andcooked appearance.

Before we proceed with the REA model concept and theory, we first discuss the dryingitself in relation to microstructures to provide an important scientific background, whichmakes drying more relevant to modern material science. Microstructure is a relativelynew term compared to the classic theories of drying. There are a number of versionsof academic descriptions on how drying proceeds into initially moist materials. Here,an intuitive, microstructural view of the gas- (hot air) aided drying process for moistporous media is presented, which is simplified from Chen (2007).

First, we look into a hypothetical scenario and ideal case, where the initial temperatureof the moist material is slightly lower than the wet-bulb temperature of the dryingmedium for an ideal capillary system, as shown in Figure 1.3. The directions of heatingand water vapour transfer are opposite each other. The capillaries here have identicaldiameters at the micro level and the walls are hydrophilic, with no interexchange ofheat and mass across the capillary walls (simply, the walls are impermeable), assumingthey were initially completely filled with water and evaporation starts to happen. Theevaporation occurs uniformly here for all tubes in the same convection condition, at theirexits, provided the convection condition is the same everywhere along the side of theassembly. There will be an obvious receding front of the liquid–gas interface movinginwards as drying proceeds. The thickness of the moving evaporation front will reflectthe meniscus of the liquid–gas interface.


A mixture of complexity arises for this ideal system as different diameters of thecapillaries and permeable walls (e.g., membranes) are involved. The interexchange ofheat 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 non-uniform receding liquid–gas interfaces among the tubes, giving a distribution of theaverage liquid–water content (averaged over the lateral direction) along the horizontaldirection, broader than that which is shown in Figure 1.3. Furthermore, if the walls aremade of materials that are hygroscopic, water-molecular movement or liquid spreadingalong the wall surfaces is also possible.

Though the interexchange of moisture and heat between the tubes may attempt toeven the evaporation rates and liquid water content, a broader distribution of the liquidwater 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 offinite dimension: the occurrence of an evaporation zone at the micro- as well as at themacro-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 liquidfront 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 practicalporous media), capillaries of various dimensions are oriented in many directions andinterlinked or networked. Even locally at the micron level, the capillary diameter sizescan 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, thereceding front(s), if any, would be fuzzy, depending on the specific local microstructureand hydrophilic (or not) nature. Liquid movement may be diffusive or driven by capillaryforces and travelling in relatively easier passages. For air and vapour transfer, certaindifficult (yet wet) patches may be bypassed by a main ‘receding front’, which may beleft to dry more gradually, thus forming a relatively wet region. ‘Fingering’ phenomenamay be possible with relatively dry and relatively wet patches coexisting nearby. Here,the sorption/desorption characteristics of the materials distributed should play key rolesas well.

The capillary wall’s thickness (the apparently solid structure) and the walls’ ownporous microstructures (yet another, smaller level of pore networks or systems),and their unevenness in spatial distribution, add to the overall picture, makingthe transition more uniform. The materials that create the walls of the microstructuresare also important as they can have quite different affinities towards water molecules(these are reflected by their equilibrium isotherms or liquid water holding capacity atthe same relative humidity and temperature). Evaporation may occur mostly in the tran-sitional 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 differentaffinities towards water molecules (these are reflected by their equilibrium isotherms orliquid water holding capacities at the same relative humidity and temperature).

All these characteristics would make the liquid water content (averaged over thesemicroscopic regions) distributed over a region between the really wet core and boundary

Introduction 9

Drying airMass exchange surface

Liquid watercontent

Temperature

Moist porousmaterial

T∞

Ts

Cl.s

Cv,s

Cv.∞

xx = xsx = 0

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 ratherthan the sharp receding liquid waterfront. This is an important recognition of the dryingphenomena such as air–gas drying. In freeze drying, on the other hand, sublimationphenomena may induce a sharp solid–gas interface. In high-temperature processes, evenfor air drying, the powerful heat front may result in an apparently sharper front of watermovement than its counterpart when a much lower heat wave is encountered. However,in this case, the previously mentione intuitive analysis is still valid, though quantitativelyit may become less influential.

Figure 1.4 shows a general depiction of air drying a moist porous material. Onemay pay attention to the ‘partitioning’ between the liquid phase and the vapour at themass exchange surface (the interface between the solid domain and the air). Into theporous solid domain, a vapour phase can also coexist with air and moisture (liquid)phases.

These arguments can be readily generalised to packed particulate systems wherethe 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 bethe voids in between the packed particles. In fruits and vegetables, the cellular structureplays a very important role as the cell walls present major water transfer resistance (seeFigure 1.6).

The perception of a moving (liquid) front or the (sharp) evaporation front can leadto different approaches in drying modelling. Mass transfer from the sharp moving frontand the vapour exit surface is often modelled using a simple effective diffusion concept


Figure 1.5 Packed particulate material.

Intercellularspaces

Intact cell walls

Cell content

Figure 1.6 Cellular structures in plant material.

(with an expanding resistance layer). In general though, this is a simplification, whichis intended largely for mathematical modelling.

It is also interesting to note from the process shown in Figure 1.3 that, due to thetemperature 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 notnecessary to have the highest water vapour content at the innermost boundary whereliquid water content starts departing from the initial value. The vapour concentration canbe higher than the boundary value, or else there would be little or no drying. It is possibleto intuitively reason that there is a ‘hump’ that can exist somewhere in the transitionalregion 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 havinga high temperature and low humidity in the same air would induce an inward transportof vapour as well as one that goes outwards (thus, drying is evident). Furthermore, a partof the water evaporated in the lower part of the transitional liquid water content region istransported into the structure and condensed at the lower temperature location, as longas there is porosity (spaces for vapour to go into). This is an interesting phenomenon, asit 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 orno porosity). This mechanism has an impact on the preservation of active ingredientssuch 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 thisrelatively dry layer. Conversely, the porosity and structure inside the region between lines1 and 2 are also important, affecting vapour transport in this area. One would expectthis region to have lower porosity (thus, a lower vapour transfer coefficient – the vapourdiffusivity). In addition to this, the vapour transfer and condensation mechanism, asmentioned earlier, would make this heat conductivity effectively even higher. The lowestliquid water content near the solid–gas boundary should be determined by the nature ofthe 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 trendof the liquid water content steeper; and a ‘waterfall’-like behaviour, where the vapourwave is apparently moving inwards and a more tidal-like liquid water content versusdistance profile emerges. Furthermore, how the material swells and shrinks locally willhave an impact on the dried product’s quality.

This description has been supported by the micro-scale transient observations usingmagnetic resonance imaging (MRI) conducted mainly at low or moderate temperatures.A number of studies have directly targeted moisture transfer (Guillot et al., 1991; Hillset al., 1994; Bennett et al., 2003; Mantle et al., 2003; Reis et al., 2003; Ruiz-Cabreraet 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 capillarydiffusion of liquid water (Reis et al., 2003). The moisture transfer or transport devices,units such as capillaries, inter-cellular spaces, voids and channel networks betweenpacked particles (which themselves may also be porous presenting another, perhapsfiner, level of transfer devices or units), all naturally possess non-uniformity. The spreadof the evaporation zone, or a transitional, ‘mushy’ zone, from the still very wet coreand 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 revisingthe conventional Biot and Lewis number calculations when air drying is of interest,so conditions for model simplification can be made more realistically, and the modelconcepts may be discerned with more quantitative support (see later sections on Biot andLewis number analyses). The effective diffusivity functions published in the literaturecan be compared and discussed based on their relationships with the scientific discoveries


with MRI or other insightful tools. It is now recognised that the material microstructureand its nature (composition, pore sizes, etc.) is interactive with transfer phenomena andformation of microstructures, which has so much to do with the speed and schedule ofthe transfer processes. Drying induces shrinkage, which is the most apparent structuralchange visible. As drying occurs into the pore channels in the material, the waterremaining ‘clamps’ on the structural walls to create the ‘pulling’ strength that drags thestructure ‘in’, so to speak.

In today’s modelling exercises (also refer to Chapter 3), more and more studies arefocused on treating materials according to their structural makeup. In biological materialas an example, domains in different scales of length may be divided according to thestructural characteristics, such as structural biology (see Figure 1.6 for modelling dryingof corn kernels as an example), and solved for moisture and energy transfer with theirown specific properties (Takhar et al., 2011). This approach has been termed the ‘hybridmixture theory-based moisture transport and stress development’. The diffusivities foreach 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 werecaptured using X-ray computed tomography (CT) and the scans were performed usingan Xradia micro-XCT scanner (Xradia Inc., Concord, CA, USA) at voltage and powersettings of 40 kV and 4 W, respectively. For imaging, a total of 253 two-dimensionalslices were obtained with a voxel size of 2.7392 µm in the x, y and z directions. Someother techniques were also to ‘convert’ this information into a digital format to beused in simulations from the Comsol Multiphysics package. Here, the geometry wasrescaled to generate suitable computational meshes (Takhar et al., 2011), while the three-dimensional distributions of the computed moisture content, temperature and the stressare visual and useful to help understand the physics involved. The predictions of theaverage moisture content during drying, especially at higher temperatures such as 85 °Cor at lower temperatures such as 29 °C, were not accurate. The authors had attributedthese 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 materialpoint 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 moistsolid domain, for instance) treated as a collection of material points p = 1, 2 . . . np (Sulskyet al., 1994; Sulsky et al., 1995). Each material point carries its own properties such asposition, velocity, acceleration, strain and stress (basically the Lagrangian approach).This method allows the finite-element discretisation of rather complex material shapesto 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 plantcell level from images taken at various resolutions. For instance, an optical microscopecommands a spatial resolution of 0.5 µm with an acquisition time of 100 frames persecond; IR microscope 10 µm, a few frames per second; confocal microscope 0.2 µm, afew frames per second; Raman microscope 0.2 µm, a few frames per second; scanning

(a)

A stack of 253 slices obtained usingmicro-CT scanning at a voxel size of2.7392 microns in x, y and z directions

Exporting theAvizo geometry toComsol usingGmsh package

Rescaling to Lagrangian coordinates andmeshing using Comsol package

Digital cutting of 3D geometryacross the plane of symmetry

3D surface renderingusing Avizo package

Figure 1.7 (a) Generation of computational domains of corn geometry for the hybrid mixturetheory 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 ofFood Engineering, 106, P.S. Takhar, D.E. Maier, O.H. Campanella and G. Chen, Hybrid mixturetheory based moisture transport and stress development in corn kernels during drying: Validation

and simulation results, 275–282, Copyright (2012), with permission from Elsevier.]


(b)

0.2112

0.2061

0.201

T29 C, RH 48% T48 C, RH 25%

T85 C, RH 14%T67 C, RH 15%

0.1959

0.1908

0.1857

0.1806

0.1755

0.1704

0.1653

0.1602

0.1551

0.15

0.1449

0.1398

0.1181

0.1142

0.1102

0.1062

0.1023

0.0983

0.0943

0.0904

0.0864

0.0824

0.0785

0.0745

0.0706

0.0666

0.0626

0.094

0.0911

0.0881

0.0851

0.0822

0.0792

0.0762

0.0733

0.0703

0.0673

0.0644

0.0614

0.0584

0.0555

0.0525

0.1579

0.1529

0.148

0.1431

0.1382

0.1332

0.1283

0.1234

0.1185

0.1135

0.1086

0.1037

0.0988

0.0938

0.0889

Figure 1.7 (cont.)

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 permissionfrom Taylor & Francis.]

electron microscope (SEM) 3 nm, several seconds for one frame; transmission electronmicroscope (TEM) 1 A, <1 s; atomic force microscope (AFM) <1 nm (x-y plane) (and<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 modellingdrying process will be briefly described. A short summary of the classic modellingapproaches 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 chemistryprinciples of chemical reactions with a little originality on our part, in order to formthe basis of the REA idea. The most prominent idea in reaction engineering is theexpression of the chemical reaction rate. A chemical reaction rate of species A, involvedin the reaction of two species (A and B) yields a product commonly expressed as:

−dcA

dt= kAcn A

A cnBB , (1.3.1)

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


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

−d ln kA

dT= E A

RT 2, (1.3.2)

where EA is the activation energy of the reaction (J mol−1). This means that thevalue of ln kA change against temperature is proportional to the value of EA. Thelarger activation energy EA is, the more sensitive the reaction rate towards temperaturechange. EA can be variable against temperature when multiple reactions are occurringsimultaneously.

When the range of temperature is not large, EA may be considered a constant. In thiscase, which is more commonly adopted in real life, the rate constant is expressed as:

kA = kAoe−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 sim-ulations for argon and methanol. Vapour molecules colliding with the surface are in thecondition of almost complete capture for both fluids but, in the case of methanol, molec-ular 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 engi-neering. For decades, various experiments have been done to measure the absolute valuesof evaporation or condensation rate, but there still remains a lack of knowledge of theunderlying molecular mechanisms. There is still a lot of room available to explore thefundamental aspects of evaporation–condensation. This phenomenon is further compli-cated 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 processinvolving latent heat. One has to put in energy to ‘activate’ water molecules that couldbecome free when the material is dried. When the water molecules are not associatedwith the solid or solute molecules and they stay in the bulk liquid domain, their inter-actions between one another would be as if there was no solid. Evaporating them intoa gaseous form would be done mainly by providing energy sufficient to overcome thelatent heat of water evaporation. Of course, another condition is that the vapour can betransported into the gas medium or into a vacuum. It is interesting to note that, since thecondensation process is spontaneous and does not need to overcome a kind of activa-tion energy (an energy barrier), the activation energy for condensation is zero, and theactivation energy of the evaporation process for pure water would be equal to the heatof reaction (the latent heat of water vaporisation); this is to say, the forward reactionactivation energy is less than the reverse reaction activation energy, which is zero forcondensation. This was the basic motive and idea in formulating the drying rate as acompetition between evaporation and condensation (to be described later).

Introduction 17

1.3.2 General modelling approaches

Before REA there were three main general approaches in the literature to formulatingdrying models, summarised next (Chen and Xie, 1997):

1. The concept of a characteristic drying rate curve (CDRC model), which recognisesdifferent 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 notaffect 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 conductivitiesand 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. havebeen 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 asthe ‘lumped drying models’ in that they do not need to solve for the spatially distributedmoisture content and temperature. For (2), there have been a number of continuum-type 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 pressuregradient) driving-force mechanisms by Luikov (1986), another dual driving-force mech-anism 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 themacro- and micro-scale structures in biological materials. Three-phase (solid, liquid andvapour) conservations and their local volume-averaged behaviours are considered. Themechanisms 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 theoryis based on a known (or pre-assumed) distribution of the macro-scale and micro-scaleunit structures, which allow local volume averaging to be carried out. Pore-networkmodels have become popular in recent years, as the concept of multi-scale and multi-physics is expanding, e.g. coupling meso-scale problems and equipment scale problems(Perre, 2011). When a local thermodynamic equilibrium is not attained, the time scalesusually overlap. This is a real multi-scale configuration and challenging in terms of thegreat demand in computational power and mathematics. Several scales can be consid-ered simultaneously, ranging from simple exchanges between macroscopic phases to acomprehensive formulation, in which the time evolution of microscopic parameters andmicroscopic gradients are considered over a representative elementary volume accord-ing to a recent review by Perre (2011). These models are often mathematically highlyinvolved.


For (3), the models usually have little physics explained and it is difficult to extractany fundamental information, though some of them have attempted to show somephysical significance on somewhat weaker grounds. On the other hand, some accuratetime functions under category (3), such as the Page model, can be used to fit the datapoints to generate accurate drying rate data sets for other purposes. Some models of acomprehensive nature will be described in detailed mathematical terms in later chapterswhen the authors compare the performances of REA in various practical cases againstwell-known models.

In modern times, expanding what we have already in category (2), the comprehensivemodelling of drying in a spatially distributed manner (or, say, in a discrete manner)has frequently involved more pore-level information (pore-network models) and math-ematical techniques, which do not require volume-averaging procedures of some sort.The respective pore networks can be used to systematically study the influence of thestructure of a porous medium on drying kinetics. There are potentials of this discretemodelling for use as a virtual laboratory to improve our understanding of how struc-tures correlate with properties and how better products may be developed (Tsotas andMujumdar, 2007).

In general, several scales of problem are involved in drying, modified from thosesummarised by Tsotsas and Mujumdar 2007:

The molecular scale (water molecules interact with each other and with the other species inthe liquid or gas, and with the solid surfaces), the pore scale (the smallest entity for expressingtransport phenomena within the drying particles or single bodies), the particle scale (single dryingbody can be identified; this larger scale can include rather ‘large particles’ such as wood boardsstacked and dried in an industrial drying kiln), the particle-system scale (the equipment is designedand properly operated at this level; the interactions between the particles, the gas flow and theapparatus are considered at this scale), and finally the process-system scale (the drying systeminteracting with other engineering systems ensuring the proper operation of the entire productionplant).

1.3.3 Outline of REA

Following the basic descriptions of physical chemistry in relation to the expression ofchemical 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 inspiredby two pieces of information: a paper published by Professor Brian Gray (1990) anda 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 exothermicallyreactive (porous solid) system, which is also influenced by the presence of water, waspublished by Professor Gray, who was a Senior Professor of Mathematics at School ofChemistry in Macquarie University, Sydney, Australia. However, before that landmark,the role of water on the exothermic (solid) systems, such as spontaneous heating or spon-taneous 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 chemicalreactions (in oxidation in particular) had not been considered quantitatively. In the paperby Gray, he added a term in the mass balance and energy balance, respectively, whichaccounts for the direct participation of water in chemical reaction (exothermic), in con-junction 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-combustionsystem where the temperature of the (combustible) solid material is assumed to beuniform, as is the water content within the solid matrix:

dx

dτ= �c (1 − x) − �exe−α/u − �wxe−αw/u, (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 watercontent (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 constantsare represented by ϕ in Equation (1.3.4); this system is also, in the theory of thermalignition and combustion, known as the Semenov approach, signifying the uniformityof the variables throughout the material of concern (Bowes, 1984). This is a usefulassumption, which paves the way for a large number of mathematical analyses that haveconsiderable physical meaning relevant to practical conditions.

In Equation (1.3.4), the first term on the right-hand side represents the condensationprocess, denoted by subscript c, and the second term evaporation, denoted by e. The lastterm on the right-hand side represents the consumption of water due to the water-inducedor -involved chemical reaction (exothermic), the wet oxidation, denoted by w. When weremove this wet-oxidation term, an inert system, we have:

dx

dτ= �c (1 − x) − �exe−α/u . (1.3.5)

Equation (1.3.5) represents the water exchange (condensation less evaporation) betweenthe moist material and the environment/surrounding. (1 − x) signifies a conservation ofthe ‘total water content in the domain of interest’. The evaporation term is consideredto be first-order as far as water ‘reactant’ is concerned. The most important descriptionof the evaporation term is used in the Arrhenius dependence function (a well-knownfunction in physical chemistry), i.e. e− α

u . The α in the context of Gray’s analysis denotesthe dimensionless latent heat of water vaporisation. This may not have been completelynew, 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 effectiveformulation to fruitfully explore the wealth of the behaviours in the wet-combustionsystem. In any case, a steady state can be attained for the system described by Equation(1.3.5) as:

�c (1 − x) = �exe−α/u . (1.3.6)


T

x

1

0

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

x= �e

�ce−α/u, (1.3.7)

which essentially suggests that as the material gets hotter, the liquid water contentin the system reduces more. The activation energy of the evaporation ‘reaction’ wasassumed to be the latent heat of water vaporisation. This has good intuitive ground-ing (Chen, 1998). As the dimensionless temperature u gets infinitely large, x becomeszero (see Figure 1.9). This system, however, does not seem to capture the physicsbehind when water content in an environment becomes zero; even when tempera-ture 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 dryingsystem.

On the practical side, a drying kinetics approach, the CDRC model, had beenemployed in order to design dryers and optimise drying operations for improvingenergy efficiency. In his developing understanding of drying operations and the fun-damentals of moisture transfer, van Meel (1958) postulated that, when working withconvective batch dryers, a single characteristic drying curve could be deduced for amoist material. This model reflects the nature of the drying rate curve(s) shown inFigure 1.10(a). It is empirical but has been successful in correlating the drying kinet-ics 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 dry-ing 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 remainsunhindered.

The physics behind what is shown in Figure 1.10(a) has been described scientifically byPerre, 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 productssuch as food, the moisture removal is related to both volume reduction and porosity. Itis necessary to know whether the loss of moisture turns into volume change or into anincrease in porosity (Perre and May, 2001). Considering these factors one can interpret

Introduction 21

Critical point (1,1)

Unhindered region

Hinderedregion

Equilibriumpoint

0 φ

ξ

Cooling down

Warming up

Critical averagewater contentEquilibrium

water content

Fallingdrying fluxperiod

Dry

ing

flux

(kg

.m--

2 .g-

-1)

Constantdrying flux period

Average content (X)XcX∞0.0

(a)

(b)

Figure 1.10 (a) Drying flux versus average water content X ; (b) the CDRC (characteristic dryingrate curve). [Reprinted from Chemical Engineering Science, 9, D.A. van Meel, Adiabaticconvection 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 bethe drying flux, kg m−2 s−1) and Nc is the drying rate at the critical condition (i.e. whenthe drying regime is in transition between the unhindered rate and falling rate periods;at the critical water content, Xc).


The characteristic moisture content (dimensionless water content) is defined by thefollowing 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 equilibriumwater content (on dry basis) and Xc is the critical water content. The drying rate isnormalised to pass through the critical point and the equilibrium point, denoted bypoints (1,1) and (0,0), respectively in Figure 1.10(b).

According to Keey (1992), the characteristic curve method is attractive since it leadsto 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 behaviourof industrial drying plants. Because of the simplicity of the parameters used, this hasbeen employed in some industrial applications.

Basically, the general form of the CDRC (Keey, 1992) is expressed as follows:

ξ = f(

X), if X ≤ Xc; ξ = 1, if X > Xc. (1.3.11)

There are a number of successful applications of this approach, especially for materialsof small dimensions such as particles or thin layers (Keey, 1992). Keey (1992) has furtherspecified that a unique characteristic curve can be established at Kirpichev numbers lessthan 2 or, in effect, when the material is thinly spread and permeability to moisture ishigh (i.e. the material has a large moisture (vapour) diffusivity).

The Kirpichev number is given as:

Ki = Ncδ

ρs Xo Dv,eff, (1.3.12)

where δ is the thickness of the sample (m), ρs 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 vapourdiffusivity (m2 s−1). According to Keey (1992), in many cases the drying curve can befitted using a simple algebraic equation over a limited moisture content range of interestby:

ξ =(

X − X∞Xc − X∞

), (1.3.13)

where j is a parameter dependent on the relative difficulty of removing moisture from amaterial. 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, forthe material of concern. The author was first exposed to the idea of the CDRC duringhis Ph.D. study at Canterbury University in Christchurch, New Zealand. The exposureof the author to CDRC was significant, due to the physical presence of the well-knownadvocate of the CDRC approach (and indeed one of the greatest in drying research anddevelopment), 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 workingon the application of a variation of the CDRC approach in through-drying of wool inKeey’s laboratory (which was later published in Chemical Engineering Science, Wu andKeey, 1995). However, being young at the time, the author did not fully appreciate theusefulness of the CDRC approach, which only became apparent later in the author’sresearch career in drying.

According to the understanding gained so far, there are aspects of CDRC that need tobe noted:

(1) The critical water content(s) have to be determined experimentally, which areknown to be dependent upon drying conditions (temperature, humidity andvelocity).

(2) The quality of results of data reduction is not good, as in some cases the data pointscan be quite scattered (for instance, with some large relative errors from the meanin the falling rate period(s)).

(3) In addition, the mass flux (Nc) calculations are sometimes based on the wet-bulbtemperatures (Twb) for the gas phase.

Aspect (3) is illustrated as follows:

Nc = hm

(ρv,sat(Ts) − ρv,∞

), (1.3.14)

where the material surface temperature Ts (K) may not be exactly the same as the wet-bulb 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) andρv,∞ 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 constantexternal conditions, with moist materials of similar form and size to that of inter-est in the real industrial dryer situation. As mentioned earlier, one also needs tonote that the CDRC model must have accurate measurement, choice or predictionof the constant rate and the maximum rate, Nc. An accurate estimation of the criticalwater content as a function of the external drying conditions must be made, as wellas the ability (or, rather, lack of it) to reduce effective data for the relative dryingrates.

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 transportphenomena theory, we write the vapour flux at the boundary (solid–gas) as:

N = hm(ρv,s − ρv,∞) = hm

(RHsρv,sat (Ts) − ρv,∞

), (1.3.16)


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, whichmay be defined as:

RHs = ρv,s

ρv,sat (Ts). (1.3.17)

During a drying process, this surface relative humidity, RHs, is reducing but is anunknown quantity.

Equations (1.3.15) and (1.3.16) thus show a significant contrast between them. Equa-tion (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 firstterm in brackets on the right-hand side of the equation being the product of RHs andsaturated 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 expressingmass transfer at the boundary, i.e. Equation (1.3.16). It is an application of chemical reac-tion engineering principles to establish a workable function for RHs. In this approach,evaporation is modelled as zero-order kinetics with activation energy while condensationis described as a first-order wetting reaction with respect to drying air vapour concen-tration without an activation energy (Chen and Chen, 1997; Chen and Xie, 1997). TheREA approach employs the Arrhenius equation in the evaporation term, which has itsorigins 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 ofa simple, ordinary differential equation with respect to time. This negates the compli-cations arising from use of the partial differential equation (Chen, 2008). REA doesneed accurate experimental data to determine model parameters, accurate equilibriumisotherm and surface area measurement. It will be shown later that REA accommodatesa natural transition because of the smooth activation energy as a function of moisturecontent (Chen, 2008). When activation energy for evaporation becomes higher than thelatent 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:

msd X

dt= −Nc A = −hm A (ρv,s − ρv,∞) , (1.3.18)

where ms is the dried mass of thin layer material (kg), X is moisture content on a drybasis (kg kg−1), X is the mean water content on a dry basis (kg kg−1), t is time (s),ρv,s is the vapour concentration at interface (kg m−3), ρv,� is the vapour concentrationin the drying medium (kg m−3), hm is the mass transfer coefficient (m s−1) and A issurface area of the material (m2). The mass transfer coefficient (hm) is determined basedon established Sherwood number correlations or is established experimentally for thespecific drying conditions involved (Incropera and DeWitt, 1990). Equation (1.3.18) isbasically correct for all cases of water vapour transfer out of a porous solid. In otherwords, no assumption of uniform water content is made in this approach, even though it

Introduction 25

began with mean water content. The surface vapour concentration (ρv,s) can be correlatedin terms of saturated vapour concentration of water (ρv,sat) by the following equation(Chen and Chen, 1997; Chen and Xie, 1997):

ρv,s = exp

(−�Ev

RTs

)ρv,sat (Ts) , (1.3.19)

where �Ev represents the additional difficulty in removing moisture from materials ontop of the free water effect. It was thought it would be excellent (and indeed lucky) tobe 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, ρv,sat (kg m−3)can be estimated with the following equation (Chen, 1998):

ρv,sat (T ) = Kv exp

(− Ev

RT

), (1.3.20)

where Kv was found to be 1.61943 × 105 (kg s−1) and Ev was found to be38.99 kJ mol−1. Ev is similar to the latent heat of water vaporisation illustrating thephysics involved (Chen, 1998). This is in line with the idea that evaporation is anactivation process, whilst condensation is not. The activation energy of the pure waterevaporation reaction is equivalent to the value of the latent heat of water evaporation, assuggested earlier based on classical physical chemistry.

More widely in the range of temperature, one can use the following, which correlatesthe entire range of the data (0 °C−�200 °C) summarised by Keey (1992) (also seeFigure 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:

msd X

dt= −hm A

(exp

(−�Ev

RT

)ρv,sat (Ts) − ρv,∞

). (1.3.22)

For small objects, such as particles or thin layer materials, the material temperature Tis approximately the same as the surface temperature Ts. Basically this happens whenthe Chen–Biot number is sufficiently small (Chen and Peng, 2005; Chen, 2007; Chenand Mujumdar, 2008). In this case, uniform temperature can be assumed throughout thematerial being dried so that one only needs to couple Equation (1.3.21) with a lumpedenergy balance to ‘govern’ the drying process.

The activation energy (�Ev) is determined experimentally by rearranging Equation(1.3.22):

�Ev = −RTs ln

(−msd Xdt

1hm A + ρv,∞

ρv,sat

), (1.3.23)


0.60

0.50

0.40

0.30

0.20

0.10

0.000.0 20.0 40.0 60.0 80.0 100.0

Temperature (°C)

Sat

urat

ed v

apou

r de

nsit

y (K

g.m

−3 )

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 generatingthe REA parameters need to be conducted where the air (or gas) humidity is very low inorder to cover the widest range of �Ev versus X . The dependence of activation energyon moisture content can be normalised as:

�Ev

�Ev,∞= ς (X − X∞), (1.3.24)

where ζ is a function of moisture content difference, �Ev,� is the maximum when themoisture concentration of the sample approaches relative humidity and temperature ofthe drying air:

�Ev,∞ = −RT∞ ln(RH∞), (1.3.25)

where X� is the equilibrium moisture content corresponding to RH� and T� which canbe related to one another by the equilibrium isotherm (Keey, 1992). It worth noting againthat, so far, the experiments for gaining the relevant Equation (1.3.24) have been undervery 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 samplesize (sometimes different sizes do not matter), the relationship (Equation 1.3.24) maybe viewed as unique as many experimental results obtained under these conditions, butdifferent drying conditions for the same material, produced more or less the same trendquantitatively (Chen, 2008). This aspect will be shown later in various applicationsdescribed in the forthcoming chapters.

Introduction 27

The REA parameters for drying of a material can be obtained from one good dry-ing experiment and can then be applied to other different drying conditions (differ-ent drying air temperatures and air velocities) if the normalised activation energycollapses to the same profile in these cases. However, the REA parameters shouldbe generated from material with the same initial moisture content since the activa-tion 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 thisbook, Equation (1.3.23) holds a very pleasant outcome indeed. Of course, other formsof Equation (1.3.23) are possible and one should be worried if there is a temperaturedependence function or if a material structure parameter is involved.

When the temperature of the moist material being dried does not vary much within, auniform temperature may be considered (more quantitative assessment of this assump-tion can be found in a later part of this book that describes the modified Biot numberand 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’ (Incr-

opera and DeWitt, 1990) for the material being dried:

mCpdT

dt= h A

(T∞ − T

)+ Hdryingmsd X

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 sampleand m is the mass of the material being dried, expressed as:

m = ms

(1 + X

). (1.3.27)

The heat of drying, Hdrying (J kg−1), is more strictly expressed as a function of watercontent:

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 usuallyunknown unless separate experiments are done to obtain it, so in the present approachthis effect has been ‘absorbed’ by the water content dependence function empiricallyobtained for the activation energy (Equation 1.3.21). Nevertheless, the heat of dryingmay be related to the so-called isosteric heat, which may be derived from the equilibriumdesorption isotherms theoretically obtained for the moist material of interest. It may alsobe measured using calorimetric methods as summarised by Shen et al. (2000). The heatof drying may be plotted qualitatively as that shown in Figure 1.12.


Latent heatof vaporisation

Water content (X)

Hea

t of

dryi

ng (

kJ.k

g--1 )

0

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

dt= koe−E/RT cnw

w , (1.3.29)

where Cw is the concentration of water, E is the activation energy (J mol−1), T istemperature (K) and ko is the constant.

The order of the evaporation reaction, nw, is taken to be 1 (i.e. the first-order reac-tion). This, though not necessarily as accurate as the current REA model, may also beconsidered 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 pulpedkiwifruit leather, whey protein concentrate, lactose, skim milk powder, whole milkpowder, cream and mixtures of sugars (Chen et al., 2001; Chen and Lin, 2005; Lin andChen, 2005; 2006; 2007). Results showed that this approach models moisture contentand temperature profile along drying time very accurately. For example, modellingof drying of an aqueous lactose solution droplet showed that the average absolutedifference of the weight loss profile was about 1% of the initial weight while that of thetemperature profile was about 1.2 °C (Lin and Chen, 2006). Moreover, application ofthe REA to model the drying of cream and whey protein concentrate showed averageabsolute differences in weight profiles of 1.9% and 2.1%, respectively, while that oftemperature was about 3 °C and 1.9 °C, respectively (Lin and Chen, 2007). Modelling ofskim milk and whole milk powder by the REA was also robust and accurate (Chen andLin, 2005).

Introduction 29

The REA has also been implemented in computational fluid dynamics (CFD) basedsimulations using a spray dryer for coupling the dispersed phase (droplets dried) andthe 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 outletparticle moisture content reasonably well compared to experimental data. In addi-tion, 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 anindustrial-scale spray dryer in two and three dimensions has been conducted (Jin andChen, 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 ‘non-interacting’ solutes (maltodextrin and sucrose). The activation energy of the mixturewas determined based on mass fraction of each solute and its corresponding activationenergy. It was shown that the average relative error between experimental and calcu-lated 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 becameclear that, for large materials being dried, both temperature and water content varyin space, so simple usage of the REA concept needed to be extended. In the pastfew 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 situationsof time-varying boundary conditions, such as intermittent drying. Following an initialattempt by Kar and Chen (2010), where the L-REA formulation was employed as thesource term (evaporation and condensation) in the partial differential equation set thatgoverns the multiphase transport phenomena in drying of porous media (porcine skin,to be exact, in their studies), Putranto and Chen (2013) have established a compre-hensive set of partial differential equations (heat and mass balances) which has beenshown to be very helpful in simulating drying, referred to as the spatial-REA; theS-REA.

In the forthcoming chapters, the applications of both the L-REA and S-REA aredemonstrated in detail using worked examples. Comparisons between the REA modelapproach and the other more established models are given. Some physical insights ofthe modelling approach are discussed where appropriate.

1.4 Summary

In this chapter, some general aspects of drying related backgrounds have been describedto 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 hasbeen provided highlighting the relationship between physical chemistry and the REA.Finally, the REA has been described. The L-REA approach has been introduced, which


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

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condensation. Fluid Phase Equilibria 104, 431–439.Patel, K.C., Chen, X.D., Lin, S.X.Q. and Adhikari, B., 2009. A composite reaction engineer-

ing approach to drying of aqueous droplets containing sucrose, maltodextrin(DE6), and theirmixtures. AIChE Journal 55, 217–231.

Perre, P., 2011. A review of modern computational and experimental tools relevant to the field ofdrying. Drying Technology 29, 1529–1541.

Perre, P. and May, B., 2007. The existence of a first drying stage for potato proved by twoindependent methods. Journal of Food Engineering 78, 1134–1140.

Perre, P., Remond, R. and Turner, I.W., 2007. Comprehensive drying models based on volumeaveraging: background, application and perspective. In Modern Drying Technology, Volume 1:Computational Tools at Different Scales, E. Tsotsas and A.S. Mujumdar (eds.), Wiley-VCH,Weinheim. pp. 8–12.

Philip, J.R. and De Vries, D.A., 1957. Moisture movement in porous materials under temperaturegradients. Transactions – American Geophysical Union 38 (222–232), 594.

Putranto, A. and Chen, X.D., 2013. Spatial reaction engineering approach (S-REA) as an alter-native for non-equilibrium multiphase mass transfer model for drying of food and biologicalmaterials. AIChE Journal 59, 55–67.

Putranto, A., Chen, X.D. and Webley, P., 2010. Infrared and convective drying of thin layer ofpolyvinyl alcohol (PVA)/glycerol/water mixture – the reaction engineering approach (REA).Chemical Engineering and Processing 49, 348–357.

Putranto, A., Chen, X.D., Xiao, Z.Y., Davastin, S. and Webley, P.A., 2011. Application of the REA(reaction engineering approach) for modelling intermittent drying under time-varying humidityand temperature. Chemical Engineering Science 66, 2149–2156.

Reis, N.C., Griffiths, R.F., Mantle, M.D. and Gladden, L.F., 2003. Investigation of the evapo-ration of embedded liquid droplets from porous surfaces using magnetic resonance imaging.International Journal of Heat and Mass Transfer 46, 1279–1292.

Rostami, A., Murthy, J. and Hajaligol, M., 2003. Modelling of a smoldering cigarette. Journal ofAnalytical and Applied Pyrolysis 66, 281–301.

Ruiz-Cabrera, M.A., Foucat, L., Bonnym J.M., Renou, J.P. and Daudin, J.D., 2005a. Assessmentof water diffusivity in gelatine gels from moisture profiles I. Non-destructive measurementof 1D moisture profiles during drying from 2D nuclear resonance images. Journal of FoodEngineering 68, 209–219.

Introduction 33

Ruiz-Cabrera, M.A., Foucat, L., Bonnym J.M., Renou, J.P. and Daudin, J.D., 2005b. Assessment ofwater diffusivity in gelatine gels from moisture profiles II. Data processing adapted to materialshrinkage. Journal of Food Engineering 68, 221–231.

Shen, D., Bulow, M., Siperstein, F., Engelhard, M. and Myers, A.L., 2000. Comparison of experi-mental 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.

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Takhar, P.S., Maier, D.E., Campanella, O.H. and Chen, G., 2011. Hybrid mixture theory basedmoisture transport and stress development in corn kernels during drying: validation and simu-lation results. Journal of Food Engineering 106, 275–282.

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Tsotsas, E. and Mujumdar, A.S., 2007. Preface of Volume 1. In Modern Drying Technology Volume1: Computational Tools at Different Scales, E. Tsotsas and A.S. Mujumdar (eds.), Wiley-VCH,Weinheim. pp. xv–xviii.

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2 Reaction engineering approach ILumped-REA (L-REA)

2.1 The REA formulation

As suggested in Chapter 1, the general REA is an application of chemical reactionengineering 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 airsolvent vapour concentration without activation energy (Chen, 2008). The REA offersthe advantage of being expressed in terms of simple ordinary differential equations withrespect to time. This negates the complications arising from use of partial differentialequations (Chen and Xie, 1997). This approach has been initially employed to expressthe overall drying rate for the entire object being dried – a lumped approach. A summaryof the developments of the lumped approach of REA is given by Chen (2008) anddescribed at length in Chapter 1.

Generally, with no assumptions, the drying rate of a material can be expressed as:

msd X

dt= −hm A(ρv,s − ρv,b), (2.1.1)

where ms is the dried mass of thin layer material (kg), X is the average moisture contenton a dry basis (kg kg−1), t is time (s), ρv,s is the water vapour concentration at interface(kg m−3), ρv,b is the water vapour concentration in the drying medium (kg m−3), hm isthe 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 transfercoefficient (hm) is determined based on the established Sherwood number correlationsfor the geometry and flow condition of concern or can be established experimentallyfor the specific drying conditions involved (Lin and Chen, 2002; Kar and Chen, 2009).The surface vapour concentration (ρv,s) can then be scaled against the saturated vapourconcentration (ρv,sat) using the following equation (Chen and Chen, 1997; Chen and Xie,1997; Chen, 2008):

ρv,s = exp

(−�Ev

RTs

)ρv,sat(Ts), (2.1.2)

where �Ev represents the additional difficulty in removing moisture from the materialbeyond 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 ρv,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:

msd X

dt= −hm A

[exp

(−�Ev

RT

)ρv,sat(T ) − ρv,b

]. (2.1.4)

From Equation (2.1.4), it can be observed that the REA is expressed in the first-order ordinary differential equation with respect to time, and the model is the coreof the L-REA. It must be noted that, though the average moisture content is used, theL-REA does not assume uniform moisture content. Of course, no spatial distribution ofmoisture content can be computed using the L-REA. The L-REA may also be appliedto cases where the temperature within the material is not uniform as long as the surfacetemperature can be determined or predicted accurately.

The activation energy (�Ev: the characteristic of the material being dried; it is material-dependent) needs to be determined experimentally. Upon the attainment of the dryingdata, notably the surface temperature of the material (or the sample temperature in thecase of a thermally thin situation) and moisture loss against time, plus the informationabout the external mass transfer coefficient, one can obtain the activation energy. Thiscan be done by rearranging Equation (2.1.4) as follows:

�Ev = −RTs ln

[−msd Xdt

1hm A + ρv,b

ρv,sat (Ts)

]. (2.1.5)

The rate of moisture loss d X/dt is experimentally determined. The surface area Ashould also be recorded in drying experiments in the case of shrinkable material. Thedependence of activation energy on average moisture content on a dry basis (X ) can benormalised as:

�Ev

�Ev,b= f (X − Xb) , (2.1.6)

where f is a function of water content difference and �Ev,b is the ‘equilibrium’ activa-tion energy representing the maximum �Ev determined by the relative humidity andtemperature of the drying air:

�Ev,b = −RTb ln(RHb), (2.1.7)

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


2.2 Determination of REA model parameters

As mentioned in the previous section, the REA model parameters such as the relativeactivation energy can be generated from one set of accurate drying runs which consistsof measuring sample mass, sample surface temperature and sample surface area duringdrying. This is only true when a master curve, Equation (2.1.6), can be applied to therange of drying conditions of interest. So far in the successful applications of the REAconcept, the drying experiments are usually performed with final moisture content beingvery low. In other words, the humidity of the drying air (or gas in general) employed inthe experiments is set to be very small. This is done so the master curve can cover thewidest range of the water contents for the same material. To illustrate the procedures toobtain and determine the REA model parameters, the experiments of convective dryingof milk droplets are described next.

The convective drying of milk droplets is conducted in a glass-filament convectivedrier (Lin and Chen, 2002; 2004; Chen and Lin, 2005). The drying air conditions weredrying air temperature of 67–110 °C, drying air velocity of 0.45–1 m s−1 and dryingair humidity of 0.001 kg H2O kg dry air−1 (Chen and Lin, 2005). The details and setupof the equipment have been described previously (Chen and Lin, 2005). Generally, byusing the glass-filament method, the droplet weight is measured by deflection of theglass 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 ofthe 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 filamentwith 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. andinitial droplet diameter about 1.42 mm) in a preconditioned (humidity, temperature andvelocity) air stream. Initial solids of greater than 30%wt. have also been successfullytested (Fu, 2012). During drying, the deflection of the glass filament and the changeof droplet diameter are monitored using standard video camera fitted with four close-up lenses. The sample mass during drying is calculated by the deflection of the glassfilament during drying. The glass filament will deflect from the original position whenthere 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 tocapture the displacement history and the value of deflection is converted to that of massby 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 surfacearea during drying, the recorded images are transferred to a digital signal using a videocapture card and analysed using image analysis software. The projected droplet area ofeach figure is considered to be a perfect circle and the equivalent diameter can be thencalculated (Chen and Lin, 2005; Fu et al., 2011; Fu, 2012).

The sample temperature during drying is measured separately from the weight mea-surement by exactly same drying conditions, droplet size and compositions. A calibratedthermocouple (type K, diameter of 13 or 24 µm) is used to record the sample temperatureduring drying. The thermocouple is connected to a Picometer (Pico Technology, UK)


Suspending glass filament

Droplet

Drying chamber

Airflow from heating tower

(a)

Video camera

Mass-measuring glassfilament

(c)


Glass capillary tubes tosupport the thermocouple

Thermocouple junction

Thermocouple wires

(b)


Figure 2.1 Equipment setup of convective drying of milk droplets (a) measuring dropletshrinkage; (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 ±0.05mg while the accuracy of the temperature measurement is within 0.1 °C. The repeata-bility of the weight loss experiment is ±0.01 mg and the accuracy is ±0.05 mg whilethe accuracy of the temperature measurement is within 0.1 °C (Chen and Lin, 2005; Fuet al., 2011; Fu, 2012).


(a)

(b)

Without weight With weight

Displacement

Mass of weight standards (mg)

Dis

plac

emen

t (pi

xel)

250

200

150

100

50

00 1 2 3 4 5 6

Experimental

y = 48, 13xR2 = 0.998

Linear correlation

Figure 2.2 The deflection of glass filament and a typical standard curve (a) measuringdisplacement 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 contentchange (dX/dt) can be obtained from the weight loss curve experiment. The heat and masstransfer coefficient is determined based on established Sherwood and Nusselt numbercorrelations or established experimentally for the specific drying conditions involved. Forthe experiments of single-droplet drying in a convective dryer, the following correlationscan be used (Lin and Chen, 2002):

Sh = 1.63 + 0.54Re0.5Sc0.333, (2.2.1)

Nu = 2.04 + 0.62Re0.5 Pr0.333, (2.2.2)

where Sh, Nu, Re, Sc and Pr are the Sherwood, Nusselt, Reynolds, Schmidt and Prandtlnumbers, respectively. These differ somewhat from the classical Ranz and Marshallcorrelations due to the blowing effect (Lin and Chen, 2002).

The saturated water vapour concentration (ρv,sat) is evaluated based on Equation(2.1.3) while the ambient water vapour concentration (ρv,b) is determined based onthe humidity and temperature of the drying air employed in the laboratory. By usingthe experimental data, calculated heat and mass transfer coefficients, ρv,sat and ρv,b,


ΔE

v/Δ

Ev,

b

1.0

0.8

0.6

0.4

0.2

0.00.0 1.0 2.0 3.0 4.0 5.0

X–Xb (kg/kg)

Exp. dataThe curve fitted

Figure 2.3 The relative activation energy of convective drying of 20%wt. skim milk powder at adrying air temperature of 67.5 °C, velocity of 0.45 m s−1 and humidity of 0.0001 kg H2O kg dryair−1. [Reprinted from AIChE Journal, 51, X.D Chen and S.X.Q. Lin, Air drying of milk dropletunder 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, theequilibrium activation energy (�Ev,b) can be evaluated using Equation (2.1.7). The ρv,b

can be related to the equilibrium moisture content (Xb) through the desorption isothermsuch as GAB isotherm. The coefficients of the isotherm for milk droplets are providedin Chen and Lin (2005). The activation energy (�Ev) can be scaled (from 0 to 1) bydividing it by the equilibrium activation energy (�Ev,b) to yield the relative activationenergy (�Ev/�Ev,b) as indicated in Equation (2.1.6). The relative activation energy canbe related to the difference of moisture content during drying (X–Xb). The relativeactivation 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 dryingprogresses. When the equilibrium moisture content is achieved, the relative activationenergy is 1. This indicates that the difficulty in removing the moisture from the materialsbeing dried increases as the drying continues. It is noted that Xb for the particular testrun of concern may also be measured by prolonging the drying test under the samecondition for an extended period to achieve equilibrium.

The relationship between the relative activation energy (�Ev/�Ev,b) and the differ-ence in moisture content (X–Xb) can be expressed by a simple mathematical function(polynomial, exponential, logarithmic, etc.). Microsoft Excel R© (Microsoft Inc., 2012)can be used for fitting in order to obtain a simple algebraic function. For the convectivedrying of 20%wt. skim milk, the relative activation energy can be represented as (Chenand Lin, 2005):

�Ev

�Ev,b= 0.998 exp

[−1.405 (X − Xb)0.993] . (2.2.3)


When Equation (2.2.3) is generally applicable, the relative activation energy generatedfrom one accurate drying run can be applied to model the convective drying at otherconditions, provided the same material and similar initial moisture content is used. Therelative activation energy is the ‘fingerprint’ of the REA where the physics of the dryingis captured by it. Since only one accurate drying run is required to generate the relativeactivation energy, the REA is effective in terms of the number of experiments to generatethe REA parameters.

This makes the REA, at least in this lumped-form, differ markedly from some othermodelling approaches to drying. For instance, the diffusion-based approach is commonlyapplied in literature in which the diffusivity function needs to be estimated from severalsets of experimental data. This means several experiments are required to obtain thediffusivity. Vaquiro et al. (2009) mentioned that 10 experiments were needed to generatethe diffusivity function as function of temperature and moisture content. Some optimi-sation procedures need to be conducted subsequently to represent the dependence ofdiffusivity 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; Silvaet 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 toyield the most appropriate diffusivity function. More discussion on these will be givenin 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 dryingprocess for lactose droplets. Basically, it is drying of mono-dispersed lactose solutiondroplets with a vertical trajectory. This drying approach does not consider the droplet-droplet interactions and droplet-wall interactions for purposes of simplicity. Some inno-vative steps can be taken to implement this one-dimensional model to a real industrialsituation 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 followingto simulate spray-drying of milk droplets. This approach follows a semi-Lagrangianframework in which the model ‘follows’ or predicts the condition of the particle. Inorder to do this, the model predicts the ambient conditions corresponding to the positionof the particle within the chamber, considering the momentum, heat and energy couplingbetween the two phases. The model assumes a pluglike flow of particles and uniform airflow 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 thedrag force due to the relative motion and the buoyancy force by the air,

dvp

dt= CD

18 μb Re

24 ρp d2p

(vb − vp) + g

ρp(ρp − ρb), (2.3.1)

where vp is the droplet/particle velocity (m s−1), vb is the gas velocity (m s−1), ρp is thedroplet density (kg m−3), ρb is the gas density (kg m−3), g is the gravitational constant


Atomiser

Uniformparticle

distribution

Uniform airvelocity,

temperatureand 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 CD can be expressedby the following for 0.5 < Re < 1000:

CD = 24

Re(1 + 0.15 Re0.687). (2.3.2)

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

Re = ρb dp |vp − vb|μb

, (2.3.3)

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

dTb

d L

=(

msd Xdt

θvp

)[� HV − Cp,v(Tb − Tp)] − θ

vph Ap(Tb − Tp) − U (π Ddryer) (Tb − Tamb) − V ρb �HV

dYd L

V ρbC p,b,

(2.3.4)

where L is the length of the one-dimensional dryer (m) (hence, dL is the differentialcontrol volume), X is the water content on dry basis (kg kg−1), ms is the dry mass ofa 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


flux (kg H2O m−2 s−1), Cp,v is the specific heat of water vapour (J kg−1 K−1), Ap isthe area of droplet (m2),V is the volumetric flow rate of gas (air) (m3 s−1), Ddryer is thediameter of the spray dryer (m2), �HV is the vaporisation heat of water (J kg−1) and Uis the overall heat transfer coefficient (W m−2 K−1).

This equation was developed considering the enthalpy balance for a control volumeof air enveloping the particle. Equation (2.3.4) also considers heat loss from the dryerto the ambient. The change of bulk air humidity against the length of the chamber thentakes the following form:

dY

d L=

−msd Xdt

θvp

mb,dry, (2.3.5)

where mb,dry is the mass flow rate of the gas (air) (kg s−1) and � is the number of dropletsgoing into the control volume per unit time (s−1). Currently, for this dyer, it is basicallyhow many droplets are generated at the atomiser per unit time. This equation has beendeveloped considering the humidity balance for a control volume of air enveloping theparticles. This suggests that any change in humidity is only affected by the drying of theparticles, 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 bedefined. In the general one-dimensional framework, the single droplet drying term canbe 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 initialsolids 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 accountingfor convective heating and evaporative cooling:

dTp

dt= h Ap(Tb − Tp) + ms

d X

dt�HV , (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, respec-

tively) can be calculated using the Ranz Marshall form of the Nusselt and Sherwoodcorrelation:

hm = (2 + 0.6 Re1/2 Sc1/3) α

dp, (2.3.7)

h = (2 + 0.6 Re1/2 Pr1/3) kb

dp, (2.3.8)

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 thedroplet/particle (W m−1 K−1). These correlations can be modified by taking into accountthe vapour blowing effects in various well-known formats. All the fluid properties usedin the calculation of Equations (2.3.9) and (2.3.10) have to be evaluated at film temper-ature, 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 bedifferent from that calculated for Equation (2.3.1). The expression for the Schmidt and


Prandtl number is given next:

Sc = μb

ρb α, (2.3.9)

Pr = Cpb μb

kb. (2.3.10)

The profiles of moisture content and temperature of the droplets as well as thoseof humidity and temperature of the drying medium along the dryer can be pre-dicted by simultaneously solving the momentum, mass and balances of the droplet.The coupling of momentum, mass and heat balances of the droplets and the dry-ing medium can be used as process simulation tool for the study of plant-wide dryerperformance.

Similarly, one can apply the approach to modelling large-scale time-dependent three-dimensional spray dryers. In this case, the momentum, heat and mass balances (partialdifferential equations that govern these processes) have been already solved (by Jin andChen, 2009a; Jin and Chen, 2010; 2011). The L-REA model for single droplet dryinghas been incorporated in these simulations. Here, the advantage of using the REA is thatone 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 totrack 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 makesimulation 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 asdrying modelling is concerned. When mass transfer is the limiting process, which mayusually be the case when the temperature difference between the material to be driedand that of the drying gas is not very large, one may not need to consider the spatialdistribution of temperature and can therefore simplify the modelling process. If heattransfer is limiting, one has to worry about the temperature gradient within the mediumbeing dried (Chen, 2007).

In air drying, one expects that the spatial distribution of water content inside a materialis significant, i.e. the boundary could be relatively dry but the core could still be verywet. The question is whether the temperature within the material being dried can beapproximated to be uniform. At first sight, all these should have much to do with twoclassical numbers: the Biot number for examining the temperature uniformity and theLewis 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 unifor-mity of a material being heated or cooled. Conventionally, the Biot number is introduced


x = 0

x

x = L

L

Ts,2

Ts,1

Heat flux

T∞

Figure 2.5 The schematic diagram showing the parameters for the definition of the classical Biotnumber. [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), withpermission from Taylor & Francis.]

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

q ′′x = k

Ts,1 − Ts,2

L= h (Ts,2 − Tb) , (2.4.1)

where q ′′x 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:

Ts,1 − Ts,2

Ts,2 − Tb= hL

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 than10% of the external temperature difference, the internal temperature distribution may beneglected for simplicity in modelling. For a spherical object, the characteristic length Lmay 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 resistanceratio:

Ts,1 − Ts,2

Ts,2 − Tb= hL

k= L/k

1/h= Rcond

Rconv, (2.4.4)


x = 0x

x = L

L

Ts,2

Ts,1

Surfaceevaporation

T∞

Figure 2.6 The schematic diagram showing the parameters for the definition of the modified Biotnumber) (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 conduc-tion and convection, respectively. As such, the Biot number can also be considered tobe the ratio of the internal resistance to external resistance. A small temperature ratioand a small resistance ratio essentially suggest the same thing. More fundamentally, theresistance ratio is a better argument. The thermal conductivity of the particle would beaffected by water content and porosity (when filled with air).

Chen introduced a new formula, which accounts for evaporation from the heatexchange surface (Chen and Peng, 2005). Similar to the conventional analysis (Equation2.4.1), considering the addition of evaporative loss, the following heat balance can beobtained (here the temperature of the environment is greater than the material beingdried; see Figure 2.6):

h (Tb − Ts,2) − �HL Sv = h∗ (Tb − Ts,2) = kTs,2 − Ts,1

L. (2.4.5)

Here h* is an equivalent convection heat transfer coefficient and Sv is the surface basedevaporation or drying rate (kg m−2 s−1) (taken as a positive value for evaporation):

h∗ = h − �HL Sv

(Tb − Ts,2). (2.4.6)

This leads to a more appropriate Biot number for the evaporative system:

Bi∗ = Bi − �HL Sv

(Tb − Ts,2)

L

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 aconventional and large Bi indicates a non-uniform temperature distribution does notmean the temperature is not reasonably uniform when evaporation occurs.


Bi* is a smaller value than Bi indicating a more uniform temperature distribution. Thesignificance of this equation can be demonstrated in the following for evaporation froma water droplet (which is also relevant to drying of a coal particle with high moisturecontent 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 mmHeat 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: Sv = 1.73 × 10−3 kg s−1 m−2

Interfacial temperature: Ts = 23.4 °CDrying air temperature: Ts = 67.5 °CLatent heat of evaporation: �HV = 2445 × 103 J kg−1

The conventional analysis, based on Equation (2.4.3), yields Bi = 0.11 for the previousexample, which is slightly greater than the critical value of 0.1 mentioned earlier. Basedon Equation (2.4.7), however, one can find that the Bi required for uniform temperatureassumption is 0.21 in order to maintain the Bi* being 0.1, i.e.:

Bicri = Bi∗cri + 2445 × 103 × 1.73 × 10−3

(67.5 − 23.4)

1.43 × 10−3

2 × 0.63≈ 0.21, (2.4.8)

due to the evaporation (cooling) effect, which consumes much of the temperature drivingforce from the outside of the material being dried.

Equation (2.4.7) can be extended to account for the cases when internal mass transferresistance plays a role. Here, one can express the surface based evaporation rate usingthe overall mass transfer coefficient concept instead:

Bi∗ = Bi − �HV Um (ρv,c − ρv,b)kL (Tb − Ts,2)

, (2.4.9)

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

Um ≈ 11

hm+ δc

Deff ,v

. (2.4.10)

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


see Van der Sman, 2003). The effective vapour diffusivity may be estimated using theporosity (ε) and the tortuosity (τ ) correction (McCabe et al., 2001):

Deff ,v ≈ ε

τDv,air, (2.4.11)

where Dv,air is the vapour diffusivity in air (m2 s−1). The tortuosity (τ ) is a parameternot usually known prior, so an estimate between 2 and 20 may be used in some cases. Itis 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 andmass transfer analogies, such as the one for a flat plate (Incropera and DeWitt, 1990;2002):

h

hm≈ kair

Dv,air

(Dv,air

αair

)0.3

, (2.4.12)

which is based on the heat transfer coefficient h, depending on the geometry of the objectof 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 Sv

(Tb − Ts,2)

L

k. (2.4.13)

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

Ch−Bi = Bi − �HV

kL

(T∞ − Ts,2

ρv,c − ρv,b

)(1

hm+ δc

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 (effectiveliquid 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 characteristicthickness δc approaches zero.

2.4.2 Lewis number analysis

The Lewis number (Le) is defined as the ratio of the thermal diffusivity to the massdiffusivity (for water vapour transfer). For drying moist, porous materials, this wouldmean that if Le < 1, the heat penetration (through conduction) into the particle is slowerthan 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 ismass-transfer limited.

This becomes particularly informative and important for evaluating drying after theinitial moisture-rich condition. This process can be visualised as follows by considering


the heating of a semi-infinite porous media in a constant temperature environment assummarised by Chen (2007). It is known that the thermal penetration depth (δT) can beapproximated as:

δT ≈√

12αtT , (2.4.15)

where α is the thermal diffusivity (=k/ρCp) (m2 s−1) and tT is thermal penetrationtime (s). Correspondingly, the vapour ‘penetration’ (mass penetration denoted by thesubscript ‘M’) may be expressed as:

δM ≈√12Deff ,v tM , (2.4.16)

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

Le = tM

tT= α

Deff ,v. (2.4.17)

The physics is apparent that if Le � 1, mass transfer occurs much faster, thus heattransfer is limiting.

A conventional Lewis number analysis using the effective vapour diffusivity calculatedwith Equation (2.4.11) would, for drying a skim milk droplet, for instance, yield a valuesmaller than 0.1 (Farid, 2003), leading to the conclusion that the drying is heat-transferlimiting.

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 (withsource terms) for drying as follows:

∂Cl

∂t= ∂

∂x

(Deff ,l

∂Cl

∂x

)− Ev; (2.4.18)

∂Cv

∂t= ∂

∂x

(Deff ,v

∂Cv

∂x

)+ Ev; (2.4.19)

∂T

∂t= 1

ρC p

∂

∂x

(keff

∂T

∂x

)− �HV

ρC pEv; (2.4.20)

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 thesample specific heat (J kg−1 K−1), keff is the thermal conductivity (W m−1 K−1), Deff,l

and Deff,v are the effective liquid and vapour diffusivity, respectively (m2 s−1), and.

Ev isthe source term (kg m−3 s−1). This is for slab geometry. Similar equations can be writtenfor spherical and cylindrical or other three-dimensional objects.

The source term may be approximated as:

Ev ≈ −ρsd X

dt. (2.4.21)


This.

Ev is equivalent to the source term I in Chapter 3 where S-REA is introduced. Thisrepresents the local rate of evaporation. Assuming that Cv is the vapour concentrationthat is in equilibrium with the liquid water content inside the solid structure, which isa conservative assumption in the sense that the process of mass transfer is perhaps theslowest, the following relationship exists:

RH = φ = Cv

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

Ev ≈ −ρs

(∂ X

∂φ

∂φ

∂t+ ∂ X

∂T

∂T

∂t

)= ρs

⎡⎣∂ X

∂φ

∂(

Cv

Cv,sat(T )

)∂t

+ ∂ X

∂T

∂T

∂t

⎤⎦ .

(2.4.23)

If the temperature may be taken as an average value, especially when the equilibriumisotherm functions are insensitive to temperature in the range considered, the followingsimplified Equation (2.4.23) can be obtained:

Ev ≈ −ρs

(∂ X

∂φ

∂φ

∂t

)≈ −ρs

∂ X

∂φ

[1

Cv,sat(

(T) ∂Cv

∂t

]. (2.4.24)

Thus, the vapour conservation Equation (2.4.19) can be rewritten (corresponding toEquation 2.4.14) into:

∂Cv

∂t≈ Deff ,v

1 + ρs

[∂ X∂φ

1Cv,sat(T )

] 1

r2

∂

∂r

(r2 ∂Cv

∂r

), (2.4.25)

by taking a mean Deff ,v for simplification.The equivalent mean effective diffusivity (D

neweff ,v) is then:

Dneweff ,v ≈ Deff ,v

1 + ρs

(∂ X∂φ

1Cv,sat(T )

) . (2.4.26)

The effective Lewis number (i.e. the Chen–Lewis number) can then be written as:

Ch Le = αeff

Dneweff

=(

keff

ρC p

)/Deff ,v

1 + ρs

[∂ X∂φ

1Cv,sat(T )

] . (2.4.27)

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 orderof 100, indicating the process is mass transfer limiting.


2.4.3 Combination of Biot and Lewis numbers

Though no strict scientific proof regarding why one should combine the Bi and Lenumbers together to justify isothermal, uniform temperature or otherwise. Nevertheless,these are the two most important dimensionless parameters in literature that seem mostrelevant to drying. Sun and Meunier (1987) conducted a comprehensive numericalanalysis on non-isothermal sorption in adsorbents, which showed that the following ruleexists: the isothermal model would be valid if LeBi > 100 and the uniform temperatureprofile 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 expectedthat this rule is conservative when drying is concerned. In the desorption process thetemperature profile inside the porous material would tend to be more gradual, thusthe criteria can be relaxed. The driving force for heat transfer, i.e. the difference betweenthe 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 productsmodelled using the L-REA

The L-REA (lumped reaction engineering approach) has been used to describe theconvective drying of droplets of whey protein concentrate and thin layer of a mixtureof polymer solution (Lin and Chen, 2007; Allanic et al., 2009). For the convectivedrying of droplets of whey protein concentrate, the experimental setup is similar tothat explained in Section 2.3. The droplets are suspended in a glass-filament convectivedryer, and the mass and temperature are recorded during drying. The deflection of theglass filament is captured and converted to droplet weight. The measurement of weightalso takes into account the drag force. A video camera system is used to monitor thedroplet diameter change during drying. A calibrated thermocouple is used to recordthe sample temperature during drying. The thermocouple is connected to a picometerand the data is obtained from the data logger. The repeatability of the weight lossand temperature measurement is ±0.01 mg and 0.1 °C, respectively. Drying air withthe velocity of 0.45 m s−1 and a temperature of 70°–110 °C is used. Initial dropletdiameter 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 inthe current work are derived from the study reported by Allanic et al. (2009) and theexperimental conditions are shown in Table 2.1.

In order to better understand the modelling presented here, the details of experimentare briefly described here (Allanic et al., 2006; 2009). Materials used in this exper-iment 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 a90-mm-diameter Petri dish so the initial thickness of the sample was 1.3 mm. During


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

NumberAir velocity(m s−1)

Air temperature(°C)

Air relativehumidity (%)

1 2.8 55 122 1 35 303 1 55 12

drying, shrinkage occurred and the relationship between thickness (m) and moisturecontent 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 isthe 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 dryingair temperature with particular velocity and temperature was fed into the rectangularcasing so that it flowed gently above the sample. The drying air temperatures andvelocities used for each experiment are listed in Table 2.1. The stable humidity of dryingair was maintained and measured using a capacitive transmitter sensor. The temperatureof the sample surface was measured using an optical pyrometer and the temperature ofthe upper and lower side of the Petri dish was measured with thermocouples with anuncertainty of about 2.5 °C. The results of this previous study showed that temperaturegradient inside the Petri dish and product can be ignored (Allanic et al., 2006).

2.5.1 Mathematical modelling of convective drying of droplets of wheyprotein concentrate (WPC) using the L-REA

The L-REA explained in Section 2.1 is implemented here to model the moisture contentand temperature profiles during the convective drying of WPC. The relative activationenergy is generated from one accurate drying run. The activation energy and equilibriumactivation energy are evaluated using Equations (2.1.5) and (2.1.7), respectively. Themass balance implementing the L-REA shown in Equation (2.1.4) is used with theconvective mass transfer coefficient (hm), determined based on the work of Lin and Chen(2002):

Sh = 1.54 + 0.54 Re0.5Sc0.333, (2.5.2)

where Sh is the Sherwood number, Re is the Reynolds number and Sc is the Schmidtnumber.

The heat balance of the convective drying of WPC can be represented as:

d(mCpT )

dt≈ h A(Tb − T ) + ms

d X

dt�Hv, (2.5.3)


ΔE

/ΔE

b

1.0

0.8

0.6

0.4

0.2

0.00.0 0.5 1.0 1.5 2.0 2.5

X–Xb (kg/kg)

67.5°C87.1°C106.6°CCurve fitted

Figure 2.7 The relative activation energy of convective drying of WPC at different drying airtemperatures. [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):

Nu = 2.04 + 0.62 Re0.5Pr0.333, (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):

�Ev

�Ev,b= 1.335 − 0.3669 exp

[exp(X − Xb)0.3011

], (2.5.5)

while the droplet diameter changes during drying can be expressed as (Lin and Chen,2007):

d

d0= 0.873 + 0.127

X − Xb

X0 − Xb. (2.5.6)

The good agreement between Equation (2.5.6) and experimental diameter changes duringdrying is shown in Figure 2.8.

The profiles of moisture content and temperature during drying are generated bysolving the mass balance implementing the L-REA and the heat balance shown inEquations (2.1.5) and (2.5.3), respectively, in conjunction with the equilibrium activationenergy, relative activation energy and droplet diameter changes during drying shown inEquations (2.1.5), (2.1.7) and (2.5.6), respectively.


d/d 0

1.0

0.9

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5X–Xb (kg kg–1)

67.5°C87.1°C106.6°CCurve fitted

Figure 2.8 The droplet diameter changes during convective drying of WPC. [Reprinted fromChemical 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 polymersolutions using the L-REA

The L-REA shown in Equation (2.1.4) used here is similar to formulation of the L-REAused in the convective drying of WPC. The activation energy and equilibrium activationenergy 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 sidedue to forced convective heat transfer from the drying air, while the lower side is heatedthrough the Petri dish which is heated by drying air from below (refer to Figure 2.9).

The heat balance can be written as:

d(ρ Cp eT )

dt= htop(Tb − T ) + Ubottom(Tb − T ) + ms

A

d X

dt�Hv(T ), (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 tempera-ture (K), ms is mass of dried product (kg), A is surface area of product (m2), X ismoisture 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 repre-sents 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:

d(ρ C p eT )

dt= Utotal(Tb − T ) + ms

A

d X

dt�Hv(T ), (2.5.8)


Petri dish

0

e(t)

z

Product

Thermocoupl

Convection

Conduction

Evaporation

Diffusion

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 ofdrying. The heat balance for this period can be expressed as:

Utotal (Tb − T ) = ms

A

d X

dt�Hv(T ). (2.5.9)

For this experiment, the activation energy is determined based on previously publishedexperimental data (Allanic et al., 2009) using Equation (2.1.5). The vapour concentrationin the environment is determined from the corresponding relative humidity and drying airtemperature reported previously (shown in Table 2.1). The mass transfer coefficient wasdeduced from the established Sherwood number correlation. Based on drying kineticsdata, the relative activation energy (�Ev/�Ev,b) for convective drying calculated throughthis exercise is expressed as:

�Ev

�Ev,b= exp

[− 1.0794(X − Xb)1.28]. (2.5.10)

Only one set of drying data was necessary and this was taken from experiment at adrying air temperature of 35 °C, drying air velocity of 1 m s−1 and relative humidityof 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 excellentagreement between correlated and experimental activation energy (R2 = 0.9892). Athigh water content, moisture removal is easy, as shown by the low activation energy, andthis increases during drying as moisture content decreases indicating greater difficulty


0 1 2 3 4 5 6 7 8

ΔE

v/Δ

Ev,

b

X–Xb (kg water/kg dry solid)

–0.2

0

0.2

0.4

0.6

0.8

1.0

DataFitted curve

Figure 2.10 Normalised activation energy and fitted curve of polyvinyl alcohol/glycerol/waterunder convective drying at an air temperature of 35 °C and relative humidity of 30%. [Reprintedfrom 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


in removing moisture. Also, this correlation ensures that �Ev/�Ev,b is �1 when themoisture content approaches equilibrium (i.e. X = Xb).

In order to evaluate the moisture content and the temperature profile as a functionof 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 andrelative 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 theL-REA are shown in Figure 2.11. Generally, the predictions using the L-REA matchwell with the experimental data. For the convective drying at a drying air temperatureof 67.5 °C, the results of modelling of moisture content and temperature profiles matchthe experimental data well as shown in Figure 2.11(a). In addition, the L-REA describeswell the moisture content and temperature profiles of the convective drying of WPCat 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


Dro

plet

wei

ght (

kg)

Dro

plet

tem

pera

ture

(°C

)

4.0E-07

6.0E-07

8.0E-07

1.0E-06

1.2E-06

1.4E-06

1.6E-06

20

40

60

80

0 50 100 150 200 250 300 350

Time (s)

(a)

Model pred.Exp. data

Dro

plet

wei

ght (

kg)

Dro

plet

tem

pera

ture

(°C

)

4.0E-07

6.0E-07

8.0E-07

1.0E-06

1.2E-06

1.4E-06

1.6E-06

20

40

60

100

80

0 50 100 150 200 250 300 350Time (s)

(b)


Dro

plet

wei

ght (

kg)

Dro

plet

tem

pera

ture

(°C

)

4.0E-07

6.0E-07

8.0E-07

1.0E-06

1.2E-06

1.4E-06

1.6E-06

20

40

60

120

100

80

0 50 100 150 200 250 300Time (s)

(c)


Figure 2.11 The comparison between experimental and model prediction using the L-REA ofconvective 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, Thereaction engineering approach to modelling the cream and whey protein concentrate droplet

drying, 437–443, Copyright (2012), with permission from Elsevier].


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

X (k

g w

ater

/kg

dry

soli

d)

t(s)

0

1

2

3

4

5

6

7

8

ModelData

Figure 2.12 Moisture content profile of convective drying at air temperature of 55 °C, air velocityof 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 andconvective 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 dropletweight 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 thephysics of the convective drying of WPC. The combination between the equilibriumrelative 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 theWPC droplets during drying. Therefore, it can be said that the L-REA can describe thedrying kinetics of the particulates well.

2.5.4 Results of modelling convective drying of a thin layer of a mixture ofpolymer solutions using the L-REA

Figures 2.12 to 2.17 present results of the simulated drying profiles and temperatureprofiles using the L-REA. It can be seen that generally, all moisture content and tem-perature profiles agree well with the experimental data supported by R2 and RMSE ofmoisture content and temperature profile presented in Table 2.2. Figures 2.12 and 2.13show the moisture content and temperature profile of convective drying conducted at55 °C and relative humidity of 12% at air velocity of 2.8 m s−1. Figure 2.12 indicates that


Table 2.2 R2 and RMSE of modelling of a mixture ofpolymer solutions using the L-REA.

Number R2 X R2 T RMSE X RMSE T

1 0.999 0.958 0.071 2.2632 0.998 0.991 0.083 0.4363 0.997 0.975 0.104 1.448

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Tem

pera

ture

(K)

t(s)

295

300

305

310

315

320

325

330

ModelData

Figure 2.13 Product temperature profile of convective drying at an air temperature of 55 °C, airvelocity of 2.8 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineeringand Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infraredand convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The

reaction engineering approach (REA), 348–357, Copyright (2010), with permission fromElsevier.]

moisture content profile can be predicted very well by the L-REA. Similarly, the temper-ature profile shown by Figure 2.13 indicates very small differences between predictedand experimental data. This modelling is comparable with modelling of drying kineticsconducted by Allanic et al. (2009). Slight discrepancies with experimental temperaturedata were also shown although the model employed was based on a diffusion partialdifferential equation with fitted diffusivity (Allanic et al., 2009).

Figures 2.14 and 2.15 provide results of modelling of drying conducted at 35 °C andrelative humidity of 30% at an air velocity of 1 m s−1 using the REA. A very goodprediction of both moisture content and temperature data was observed. Compared withthe simulation using the model proposed previously, it is apparent that the L-REA gives

0 2000 4000 6000 8000 10000 12000

X (k

g w

ater

/kg

dry

soli

d)

t(s)

0

1

2

3

4

5

6

7

8

ModelData

Figure 2.14 Moisture content profile of convective drying at an air temperature of 35 °C, airvelocity of 1 m s−1 and air relative humidity of 30%. [Reprinted from Chemical Engineering and



0 2000 4000 6000 8000 10000 12000

Tem

pera

ture

(K)

t(s)

294

296

298

300

302

304

306

308

ModelData

Figure 2.15 Product temperature profile of convective drying at air temperature of 35 °C, airvelocity of 1 m s−1 and air relative humidity of 30%. [Reprinted from Chemical Engineering and



0 1000 2000 3000 4000 5000 6000 7000 8000

X (k

g w

ater

/kg

dry

soli

d)

t(s)

0

1

2

3

4

5

6

7

8

ModelData

Figure 2.16 Product temperature profile of convective drying at an air temperature of 55 °C, airvelocity of 1 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineering and



0 1000 2000 3000 4000 5000 6000 7000 8000

Tem

pera

ture

(K)

t(s)

300

305

310

315

320

325

ModelData

Figure 2.17 Product temperature profile of convective drying at an air temperature of 55 °C, airvelocity of 1 m s−1 and air relative humidity of 12%. [Reprinted from Chemical Engineering and




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



Air humidity(kg H2O kg dry air−1)

1 4 45 0.01342 4 55 0.01343 4 65 0.0134

better results because the diffusion model shows slight discrepancies in the moisturecontent profile during drying times around 4000–10 000 s (Allanic et al., 2009). TheL-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 exper-imental moisture and temperature data. Despite the simplicity of L-REA, it compareswell with the model proposed before, which shows some discrepancies of moistureprofile 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 mixtureof polyvinyl alcohol, glycerol and water. The L-REA is shown to be able to model notonly the convective drying of particulate or thin layer of food materials which has beenproven before (Chen and Lin, 2005; Lin and Chen, 2005; 2006; 2007), but also thatof thin layers of non-food materials. The accuracy of the L-REA could be due to theaccuracy of the relative activation energy in describing the change of internal behaviourduring 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, theexperimental data are derived from the work of Vaquiro et al. (2009) on convectivedrying of mango tissues. Mango tissues used for drying experiments were formed intocubes of side lengths of 2.5 cm, with initial moisture content of 9.3 kg kg−1 and aninitial temperature of 10.8 °C. Drying was conducted in a laboratory dryer describedin detail by Sanjuan et al. (2004). The drying air temperature and air velocity werecontrolled at preset values by PID control algorithms while air humidity was main-tained at a constant during drying. Details of the experimental conditions are listed inTable 2.3. The weight of the sample was measured periodically to record weight loss aswell 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 theL-REA can still be implemented. However, the temperature of concern is the surface


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

msd X

dt= −hm A

(ρv,s(Ts) − ρv,b

), (2.6.1)

where ms is the dried mass of thin layer material (kg), t is time (s), X is moisturecontent on a dry basis (kg kg−1), ρv,s is the vapour concentration at the material-airinterface (kg m−3), ρv,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 Sherwoodnumber correlations for the geometry and flow condition of concern or establishedexperimentally for the specific drying conditions involved (Lin and Chen, 2002; Karand Chen, 2009). The surface vapour concentration (ρv,s) can be scaled against saturatedvapour concentration (ρv,sat) using the following equation (Chen and Xie, 1997; Chen,2008):

ρv,s = exp

(−�Ev

RTs

)ρv,sat(Ts), (2.6.2)

where �Ev represents the additional difficulty in removing moisture from the materialbeyond the free water effect. This �Ev is moisture-content (X) dependent. Ts is thesurface temperature of the material being dried, and ρv,sat for water can be estimated atthe 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:

msd X

dt= −hm A

[exp

(−�Ev

RTx

)ρv,sat(Ts) − ρv,b

]. (2.6.4)

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

�Ev = −RTs ln

[−msd Xdt

1hm A + ρv,b

ρv,sat(Ts)

]. (2.6.5)

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


2.6.2 Prediction of surface sample temperature

For large sample slabs, prediction of sample temperature may be necessary since thetemperature may be not uniform inside the sample. The sample temperature may beapproximated using a simple parabolic equation (Chen, 2008):

T = a + bx2. (2.6.6)

If To is the centre sample temperature (K) and L is the half-thickness of the sample as acharacteristic slab length, Equation (2.6.6) is rewritten as:

T = To +(

Ts − To

L2

)x2, (2.6.7)

Tavg is determined by:

Tavg =

L∫0

T (x)dx

L. (2.6.8)

By combining Equations (2.6.7) and (2.6.8), Tavg is expressed as:

Tavg = 1

3Ts + 2

3To. (2.6.9)

For a sample heated in a convective environment, the boundary condition at samplesurface (x = L) can be written as:

h (Tb − Ts) = k

(dT

dx

)x=L

+ |Nv|�Hv. (2.6.10)

Also note that Equation (2.6.7) satisfies the boundary condition at centre (x = 0), whichcan be expressed as: (

dT

dx

)x=0

= 0. (2.6.11)

By combining Equation (2.6.7) to (2.6.10), Ts and To are expressed as:

Ts =Tavg + hL

3kTb − |Nv|�Hv

L

3k

1 + hL

3k

, (2.6.12)

To = Tavg

⎛⎜⎝3

2− 1

2 + 2hL

3k

⎞⎟⎠− 1(

2 + 2hL

3k

) (hL

3kTb − |Nv|�Hv

L

3k

). (2.6.13)

Equation (2.6.12) and (2.6.13) clearly show that Ts and To are represented as functionsof Tavg and Tb.

The temperature profile prediction described previously seems to be valid for dryingconditions suitable for the boundary conditions mentioned (i.e. there is symmetry atcentre and at the surface; heat gained by convection from drying air is balanced byconduction heat inside the sample and heat for water evaporation). The prediction isin agreement with Pang (1994), who conducted convective drying of softwood and


heartwood with a half-thickness of 2.5 cm. It was observed that the boundary conditionsindicated 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) weremeasured during the drying time and a plot of the temperature profiles against positionsduring drying time revealed parabolic profiles.

For drying of mango tissue, as mentioned before, the sample was heated uniformlyfrom all directions (Sanjuan et al., 2004). It is reasonable to assume that the temperatureprofiles would be similar in the x, y and z directions. Because of this, the approximationof the temperature profiles can be simplified into one dimension.

It is also observed that for drying of mango tissues, there is symmetry at the centreand at the surface; heat received by convection from drying air is balanced by conductionheat inside the sample and heat for water evaporation is represented by the boundaryconditions shown in Equations (2.6.10) and (2.6.11) (Sanjuan et al., 2004; Incroperaand 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 centretemperature can be represented as:

T = To +(

Ts − To

R2

)r2, (2.6.14)

Ts =Tavg + h R

3kTb − |NV |�HV

R

3k

1 + h R

3k

, (2.6.15)

To = Tavg

⎛⎜⎝3

2− 1

2 + 2h R

3k

⎞⎟⎠− 1(

2 + 2h R3k

) (h R

3kTb − |NV |�HV

R

3k

). (2.6.16)

The equivalent radius for cubes is the side length (Incropera and DeWitt, 2002; Radziem-ska 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 theside 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 (Vaquiroet al., 2009). Based on drying kinetics data, the relative activation energy (�Ev/�Ev,b)of convective drying of mango tissues is expressed as:

�Ev

�Ev,b= −9.92 × 10−4(X − Xb)3 + 9.74 × 10−3(X − Xb)2

− 0.101(X − Xb) + 1.053. (2.6.17)

A good agreement between the fitted (Equation 2.6.17) and experimental activationenergy is shown in Figure 2.18 (R2 (0.997)). This format of correlation is similar to thatproposed by Kar (2008) to describe the activation energy of drying porcine skin. The


0 1 2 3 4 5 6 7 8 9 10

ΔE

v/Δ

Ev,

b

X–Xb

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ModelData

Figure 2.18 The relative activation energy (�Ev/�Ev,b) of convective drying of mango tissues atan air velocity of 4 m s−1, drying air temperature of 55 °C and air humidity of 0.0134 kg H2O 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 reactionengineering 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 torepresent the activation energy well. It may be observed that the decrease of moisturecontent results in the increase of activation energy, which indicates greater difficulty inremoving water. This equation also yields �Ev/�Ev,b approaching �1 as the material isdried.

The heat balance for convective drying of mango tissues can be written as:

d(mC pTavg)

dt≈ h A (Tb − Ts) + ms

d X

dt�HV , (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 vaporisationof water (J kg−1). The drying rate dX/dt is negative when drying occurs. In order to yieldboth profiles of moisture content and temperature of mango tissues during drying, themass 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 activationenergy shown in Equations (2.1.7) and (2.6.17), respectively. The surface temperaturepredicted by Equation (2.6.15) is used in the mass balance implementing the L-REAand heat balance.


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

NumberVelocity(m s−1)


Air humidity(kg H2O kg dry air−1) R2 X RMSE X R2 T RMSE T

1 4 45 0.0134 0.998 0.08 0.993 0.612 4 55 0.0134 0.998 0.1 0.982 1.123 4 65 0.0134 0.996 0.14 0.984 1.41

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

X (k

g w

ater

/kg

dry

soli

d)

× 104

0

1

2

3

4

5

6

7

8

9

10

Model 65°CModel 55°CModel 45°CData 65°CData 55°CData 45°C

t(s)

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 fromTaylor & 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 predicteddata is observed for convective drying of mango tissues at drying air temperatures of45°, 55° and 65 °C. The good predictions made by using the REA are further revealed byR2 and RMSE presented in Table 2.4, which shows all modelling of these cases yield R2

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 thepredicted and experimental data. It is clear that the L-REA with the approximation of

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Cen

tre

tem

pera

ture

(K

)

× 104

280

290

300

310

320

330

340


t(s)

Figure 2.20 Temperature profile of convective mango tissues at air temperatures of 45°, 55° and65 °C (modelled using the L-REA which incorporates the temperature distribution inside thesample). [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 reactionengineering approach (REA), 961–973, Copyright (2012), with permission from Taylor &

Francis Ltd.]

0

1

2

3

4

5

6

7

8

9

10

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lid)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t(s) × 104

Model 65°C

Model 55°C

Model 45°C

Data 65°C

Data 55°C

Data 45°C

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 insidethe 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 reactionengineering approach (REA), 961–973, Copyright (2012), with permission from Taylor &

Francis Ltd.]


280

300

290

310

320

330

340

Cen

tre

tem

pera

ture

(K

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5t(s) × 104


Figure 2.22 Temperature profile of convective mango tissues at air temperatures of 45°, 55° and65 °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 reactionengineering approach (REA), 961–973, Copyright (2012), with permission from Taylor &

Francis Ltd.]

temperature distribution inside the sample is necessary, and this model describes bothmoisture content and centre sample temperature profile well during drying.

Vaquiro et al. (2009) used diffusion-based modelling to represent the data and ourREA compares well with the modelling by Vaquiro et al. (2009). Modelling by Vaquiroet al. (2009) showed a kink in the beginning of the temperature profile that was notobserved by modelling using the REA. For drying at 65 °C, both the REA and modellingproposed by Vaquiro et al. (2009) showed a slight overestimation of the temperatureprofile during drying times of 5000–20 000 s.

It can be said that the L-REA, with the prediction of sample temperature as explainedin Section 2.6.2, is accurate enough to describe continuous convective drying of mangotissues 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, themodelling itself is still simple and only requires a short computational time to predictthe 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 todescribe both the moisture content and sample temperature profile of convective dryinglarge samples of mango tissues. For this purpose, the activation energy and the saturationvapour concentration are evaluated at the surface temperature. The remaining principles


Table 2.5 Schemes of intermittent drying of mango tissues (Vaquiro et al., 2009).

Drying airtemperature (°C)

Period of firstheating (s)

Period of resting(at 27 °C ± 1.6) (s)

Period of secondheating (s)

45 16 200 10 800 36 36055 9 480 10 800 33 72065 7 800 10 800 16 200

are similar to those of the L-REA used to describe the drying kinetics of thin layers orsmall objects published previously. Results indicate that the REA models both moisturecontent and temperature of convective drying of large samples of mango tissues verywell. When compared to the experimental data published by Vaquiro et al. (2009), asimilar if not better agreement is observed against diffusion-based models. While theresults are accurate, the effectiveness of the L-REA is also revealed as the modellingitself 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 samplessubstantially. 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 iscreated 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 arelative 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 relativeactivation 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 heatbalance 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 tothe 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 tissuesusing the REA. For intermittent drying at a drying air temperature of 45 °C, the REA


00

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7t(s) × 104

X (

kg w

ater

/kg

dry

soli

d)

ModelData

Figure 2.23 Moisture content profile of mango tissues during intermittent drying at a drying airtemperature of 45 °C and resting at 27 °C. [Reprinted from Industrial Engineering ChemistryResearch, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mangotissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012),

with permission from the American Chemical Society.]

0

Cen

tre

tem

pera

ture

(K

)

280

285

290

295

300

305

310

315

320

1 2 3t(s) × 104

4 5 6 7

ModelData

Figure 2.24 Temperature profile of mango tissues during intermittent drying at a drying airtemperature of 45 °C and resting at 27 °C. [Reprinted from Industrial Engineering ChemistryResearch, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mangotissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012),



00

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6

t(s) × 104

X (

kg w

ater

/kg

dry

soli

d)

ModelData

Figure 2.25 Moisture content profile of mango tissues during intermittent drying at a drying airtemperature 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 mangotissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012),


Cen

tre

tem

pera

ture

(K

)

280

285

290

295

300

305

310

315

320

325

330

ModelData

t(s) × 1040 1 2 3 4 5 6

Figure 2.26 Temperature profile of mango tissues during intermittent drying at a drying airtemperature 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 mangotissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012),



00

1

2

3

4

5

6

7

8

9

10

0.5 1 1.5 2 2.5 3 3.5 4t(s) × 104

X (

kg w

ater

/kg

dry

soli

d)

ModelData

Figure 2.27 Moisture content profile of mango tissues during intermittent drying at a drying airtemperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering ChemistryResearch, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mangotissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012),


0 0.5 1 1.5 2 2.5 3 3.5t(s) × 104

Cen

tre

tem

pera

ture

(K

)

280

290

300

310

320

330

340

ModelData

Figure 2.28 Temperature profile of mango tissues during intermittent drying at a drying airtemperature of 65 °C and resting at 27 °C. [Reprinted from Industrial Engineering ChemistryResearch, 50, A. Putranto, Z. Xiao, X.D. Chen and P.A. Webley, Intermittent drying of mangotissues: Implementation of the reaction engineering approach, 1089–1098, Copyright (2012),



Table 2.6 R2 and RMSE of modelling of intermittent drying ofmango tissues using the L-REA.

Drying airtemperature (°C) R2 X R2 T RMSE X RMSE T

45 0.998 0.996 0.083 0.48355 0.998 0.997 0.087 0.55465 0.998 0.997 0.082 0.686

describes both moisture content and temperature profile very well. Good agreement isobserved between experimental and predicted data. Similar results are also revealed forintermittent drying at drying air temperatures between 55° and 65 °C. The predictedmoisture content and temperature match well with experimental data. The good predic-tions of moisture content and temperature profile are revealed by R2 and RMSE shown inTable 2.6.

The benchmark of modelling proposed by Vaquiro et al. (2009) employing the dif-fusion model was conducted, and it revealed the REA gives comparable or even betterresults. Modelling proposed by Vaquiro et al. (2009) showed a kink in the temperatureprofile 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 dry-ing in drying conditions of 65 °C is not revealed by the REA; as shown in the modellingby Vaquiro et al. (2009).

It can be said that the REA is accurate enough to model intermittent drying of mangotissues, particularly when it is represented in a lumped model. This is because the relativeactivation energy (�Ev/�Ev,b) implemented allows the natural transition during dryingtimes according to the drying scheme as revealed in Figure 2.29. The relative activationenergy keeps increasing during drying, indicating an increase of difficulty removingwater from materials. This increases significantly during the heating period but onlyincreases slightly during the resting period. This natural transition during drying is notobserved by empirical models and the CDRC (Baini and Langrish, 2007). It was revealedthat 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 dryingrate 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 notonly continuous drying but also intermittent drying of rather thick samples. Althoughthe results of modelling are accurate and robust, the simplicity of the modelling is stillproven and only a short computational time is required.

2.7.3 Analysis of surface temperature, surface relative humidity, saturated andsurface vapour concentration during intermittent drying

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


00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 1

ΔE

v /Δ

Ev,

b

1.5 2 2.5 3 3.5t(s) × 104

Figure 2.29 Relative activation energy profile of mango tissues during intermittent drying at adrying 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 ofmango 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, inorder to minimise energy and final moisture content. The following paragraphs discussthese parameters in the intermittent drying of mango tissues. The L-REA is appliedhere and combined with several equations to yield a profile of surface temperature, sur-face 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 decreasesduring the heating period while it increases during resting, representing an increasein surface moisture content. In addition, Figure 2.31 indicates the profiles of surfacetemperature and saturated vapour concentration during intermittent drying at a dryingair temperature of 65 °C. The profiles of saturated vapour concentration follow thesurface temperature trend. It increases in first section, decreases in second section andincreases again in third section.

However, the profile of surface vapour concentration is different from that of satu-rated vapour concentration as revealed in Figures 2.32 and 2.33 because the profile ofsurface vapour concentration is affected by both surface temperature and surface relativehumidity. It is apparent that the surface vapour concentration increases in the very earlypart of drying because of the increase in surface temperature, followed by a decrease

0 0.5 1 1.5 2 2.5 3 3.5t(s)

Sur

face

rel

ativ

e hu

mid

ity

× 104

0.1

0

0.2

0.3

0.4

0.5

0.6

0.7

Figure 2.30 Surface relative humidity profile of mango tissues during intermittent drying at adrying 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 ofmango tissues: Implementation of the reaction engineering approach, 1089–1098, Copyright

(2012), with permission from the American Chemical Society.]

Surface temperature

Saturated vapourconcentration

0.2

0.1

Sat

urat

ed w

ater

vap

our

conc

entr

atio

n (k

g.m

–3)

Sur

face

tem

pera

ture

(K

)

0

350

300

2503.50 0.5 1 1.5 2

t(s)2.5 3

× 104

Figure 2.31 Saturated vapour concentration and surface temperature profile of mango tissuesduring intermittent drying at a drying air temperature of 65 °C and resting at 27 °C. [Reprintedfrom 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 engineeringapproach, 1089–1098, Copyright (2012), with permission from the American Chemical Society.]

Wat

er v

apou

r co

ncen

trat

ion

(kg.

m–3

)

0

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

3.50 0.5 1 1.5 2t(s)

2.5 3× 104

SaturatedSurface

Figure 2.32 Surface and saturated vapour concentration profile of mango tissues duringintermittent 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 temperature

Surface vapour concentration

0.06

Sur

face

wat

er v

apou

r co

ncen

trat

ion

(kg.

m–3

)

Sur

face

tem

pera

ture

(K

)

0.01

340

290

300

310

320

330

0.02

0.03

0.04

0.05

3.50 0.5 1 1.5 2t(s)

2.5 3× 104

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

9

8

7

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lid)

6

5

4

3

2

10 1 2 3 4

t(s) × 1045 6 7

Figure 2.34 Moisture content profile of intermittent drying of mango tissues with heating (at adrying 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 temperature

Saturated vapour concentration

0.08

Sat

urat

ed w

ater

vap

our

conc

entr

atio

n (k

g.m

–3)

Sur

face

tem

pera

ture

(K

)

0.01

320

280

290

300

310

0.02

0.04

70 1 2 3 4t(s)

5 6

0.06

× 104

Figure 2.35 Saturated vapour concentration and surface temperature profile of intermittent dryingof 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 thereaction engineering approach, 1089–1098, Copyright (2012), with permission from the

American Chemical Society.]

70 1 2 3 4t(s)

5 6× 104

Surface temperature

Surface vapour concentration

Sur

face

tem

pera

ture

(K

)

300

280

3200.04

0.02

0

Sur

face

wat

er v

apou

r co

ncen

trat

ion

(kg.

m–3

)

Figure 2.36 Surface vapour concentration and surface temperature profile of intermittent dryingof 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 thereaction engineering approach, 1089–1098, Copyright (2012), with permission from the

American Chemical Society.]

0.07

Wat

er v

apou

r co

ncen

trat

ion

(kg.

m–3

)

0.01

0.02

0.04

0.03

70 1 2 3 4t(s)

5 6

0.06

0.05

× 104

SaturatedSurface

Figure 2.37 Surface and saturated vapour concentration profile of intermittent drying of mangotissues 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 reactionengineering approach, 1089–1098, Copyright (2012), with permission from the American

Chemical Society.]


70 1 2 3 4t(s)

5 6× 104

Surface vapourconcentration

Surface relative humidity

Sur

face

rel

ativ

e hu

mid

ity

0.5

0

10.04

0.02

0

Sur

face

wat

er v

apou

r co

ncen

trat

ion

(kg.

m–3

)

Figure 2.38 Surface vapour concentration and surface relative humidity profile of intermittentdrying 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 theAmerican 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 humidityincreases dramatically. This is followed by a decrease in the surface vapour concentra-tion because of the low surface temperature, leading to a decrease of saturated vapourconcentration. In the second heating period, the surface vapour concentration continuesto decrease as the surface relative humidity decreases. This analysis is in agreementwith Baini and Langrish (2007) applying a diffusion model to the intermittent drying ofbanana tissues. It was revealed that, during the resting period, the surface temperaturedecreased, while the surface moisture content increased initially as a result of an initialincrease 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 pointin extending the resting period to 18 600 s. The resting period could be shortened andfollowed by a subsequent heating period. Similarly, during the second heating period,the surface vapour concentration profile has nearly flattened after a drying time around25 000 s. The heating time could also be shortened, followed by a subsequent restingperiod to achieve higher surface vapour concentration.


From this analysis, it seems that a cycle with a higher frequency will give betterresults. Simulation of intermittent drying at a drying air temperature of 45 °C with eachheating and resting period (at 27 °C) at 4000 s and a total drying time of 64 000s (totalheating 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 Figures2.34–2.38.

It can be seen that the trends of surface-relative humidity, saturated and surface vapourconcentration are similar to those which have been discussed in previous paragraphs.Nevertheless, no flat profile of surface vapour concentration is shown during the restingperiod which means resting is not conducted for a prolonged time. It is also observedthat the final moisture content is similar to that of intermittent drying mango tissues at45 °C using scheme listed in Table 2.5, although the total heating time of this schemeis lower. The total heating time of this scheme is 32 000 s, while that at 45 °C, listed inTable 2.5, is 52 560 s. The total drying time of this scheme (64 000 s) is also similar tothat at 45 °C, listed in Table 2.5 (62 650 s). Because the heating time is shorter, whilethe total drying time and the final moisture content is not significantly altered (from asustainable processing perspective), it is beneficial to apply such a cycle. This is becauseenergy cost can be minimised while the objective of obtaining a similar target moisturecontent can be achieved.

2.8 The intermittent drying of non-food materials under time-varyingtemperature and humidity modelled using the L-REA

In this study, the experimental data of the intermittent drying are derived from thework of Kowalski and Pawlowski (2010a,b) whose experimental details are shown inTable 2.7. For better understanding of the procedures, the experimental details are brieflyreviewed 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 thecompany’s website (Surmin-Kaolin Co., 2010). Each sample was prepared by mouldingthe materials into a cylinder with a radius of 0.025 m and a height of 0.06 m, with aninitial moisture content of 0.4 kg H2O kg dry solids−1. Each sample was placed in analuminium 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-typethermocouples were inserted inside a cylinder at different positions. The temperaturemeasurement indicated that the temperature inside the sample was uniform (Kowalskiet al., 2007; Kowalski and Pawlowski, 2010a,b).

Two types of intermittent drying experiments were conducted: time-varying dryingair temperature and time-varying humidity. The first type of intermittency was enabledby supplying cool air through a special air intake. Table 2.7 shows the cases of theintermittent drying under time-varying drying air temperature and humidity. Case 1is the intermittent drying, which implemented a periodic change of the drying air


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) 65 °C (before 17 000 s) periodically changedbetween 65° and 43 °C (after 17 000 s)

2 4% (at 100 °C) 100 °C (before 7000 s) periodically changedbetween 100° and 50 °C (after 7000 s)

3 4% (before 9000 s) periodically changedbetween 4 and 12% (after 9000 s)

100

4 4% (before 9000 s) periodically changedbetween 4 and 80% (after 9000 s)

100

temperature between 65° and 43 °C (see Table 2.7), while Case 2 is similar to thefirst one but with a periodic change of temperature between 100° and 50 °C (see Table2.7). For the intermittent drying under time-varying humidity and constant drying airtemperature of 100 °C, the intermittency was created by a periodic change in vapoursupply to the drying chamber from a humidifier. Cases 3 and 4 (see Table 2.7) are theintermittent drying under time-varying humidity. Case 3 applied a periodic change ofthe relative humidity between 4° and 12%, while Case 4 implemented a change of therelative 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 implementedhere without any modification. In this study, the relative activation energy is generatedfrom a drying experiment at a constant drying air temperature of 50.8 °C (Kowalskiet al., 2007) and can be expressed as:

�Ev

�Ev,b= 0 for X − Xb > 0.2, (2.8.1)

�Ev

�Ev,b= exp

[−49.391(X − Xb)2.103]7.847

for X − Xb > 0.2. (2.8.2)

A good fit between the experimental and predicted activation energy is shown inFigure 2.39 and indicated by the R2 of 0.99. As mentioned earlier, uniform temperatureprofiles 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 thesample is essentially uniform. Indeed, Kowalski and Pawlowski (2010b) implementedmodelling which did not take into account the variations of spatial temperature insideproducts. Based on this observation, the assumption of uniform product temperatureprofile is implemented in this study. Hence, the heat balance can be represented as:

d(mC pT )

dt≈ h A (Tb − T ) + ms

d X

dt�H V , (2.8.3)


DataFitted curve

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4


ΔE

v/Δ

Ev,

b

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 fromElsevier.]

where m is the sample mass (kg), Cp is the heat capacity of the sample (J kg−1 K−1), his the heat transfer coefficient (W m−2 K−1) and �HV is the latent heat of vaporisationof 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 accordingto the corresponding drying air settings in each time period. The equilibrium activationenergy (�Ev,b) is combined with the relative activation energy (�Ev/�Ev,b) representedin Equations (2.8.1) and (2.8.2). In addition, the mass balance implementing the L-REAand heat balance shown in Equations (2.1.4) and (2.8.3), respectively, also implement thecorresponding drying air settings in each time period. The profiles of moisture contentand temperature can be yielded by solving the mass and heat balance simultaneouslyin 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 andhumidity modelled using the L-REA

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

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5 3 3.5 4t(s) × 104

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

Data

L-REA

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


280

290

300

310

320

330

340

0 0.5 1 1.5 2 2.5 3 3.5 4t(s) × 104

Tem

pera

ture

(K

)

Data

L-REA

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

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5t(s) × 104

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

Data

L-REA

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

290

300

310

320

330

340

350

360

0 0.5 1 1.5 2 2.5t(s) × 104

Tem

pera

ture

(K

)

Data

L-REA

Figure 2.43 Temperature profile of intermittent drying in Case 2 (periodically changed drying airtemperatures between 100° and 50 °C). [Reprinted from Chemical Engineering Science, 66, A.




Table 2.8 R2, RMSE, average absolute deviation and maximum absolute deviation of profiles of moisture contentpredicted by and Kowalski and Pawlowski’s model (2010b).

Casenumber

R2

REA

R2

Kowalski andPawlowski’smodel (2010b)

RMSEREA

RMSEKowalski andPawlowski’smodel (2010b)

AverageAbsoluteDeviationREA

AverageAbsoluteDeviationKowalski andPawlowski’smodel (2010b)

MaximumAbsoluteDeviationREA

MaximumAbsoluteDeviationKowalski andPawlowski’smodel (2010b)

1 0.997 0.992 0.006 0.009 0.005 0.007 0.012 0.0242 0.997 0.983 0.007 0.016 0.006 0.014 0.012 0.0313 0.991 0.994 0.011 0.009 0.009 0.006 0.023 0.0264 0.998 0.967 0.005 0.019 0.084 0.015 0.009 0.037

Table 2.9 R2, RMSE, average absolute deviation and maximum absolute deviation of profiles of temperature predictedby Kowalski and Pawlowski’s model (2010b).

Casenumber

R2

REA

R2

Kowalski andPawlowski’smodel (2010b)

RMSEREA

RMSEKowalski andPawlowski’smodel (2010b)

AverageAbsoluteDeviationREA

AverageAbsoluteDeviationKowalski andPawlowski’smodel (2010b)

MaximumAbsoluteDeviationREA

MaximumAbsoluteDeviationKowalski andPawlowski’smodel (2010b)

1 0.952 0.761 2.529 5.567 1.481 4.335 5.968 11.1582 0.953 0.842 3.399 6.254 2.554 4.892 5.501 10.0593 0.983 0.743 1.658 7.441 1.217 6.189 5.050 14.4724 0.950 0.896 4.554 6.377 0.084 3.995 4.369 6.8326

both the moisture content and temperature profiles well. Benchmarks towards the mod-elling approach implemented by Kowalski and Pawlowksi (2010b) for the intermittentdrying 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, tem-perature profiles and the experimental data than the approach implemented by Kowalskiand Pawlowksi (2010b); as shown by the results of error analysis presented in Tables2.8 and 2.9. For Case 2 (refer to Table 2.5, Figures 2.42 and 2.43), the L-REA alsoyields closer agreement between the predicted and experimental moisture content pro-files. The L-REA results in a slight deviation in the temperature profiles during dryingtimes between 15 000 and 20 000 s, while the modelling implemented by Kowalski andPawlowksi (2010b) showed a slight deviation in the temperature profiles during dryingtimes of 8000–20 000 s. The closer agreement is indeed shown by the results of erroranalysis presented in Tables 2.8 and 2.9.

It can be said that the REA can model the intermittent drying of kaolin under time-varying drying air temperature well. This could be due to the flexibility of the activationenergy in allowing a change in the drying kinetics according to the drying air settingsin each time period. The relative activation energy (�Ev/�Ev,b) shown in Equations


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5t(s) × 104

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

Data

L-REA

Figure 2.44 Moisture content profile of intermittent drying in Case 3 (periodically changedrelative humidity between 4° and 12%). [Reprinted from Chemical Engineering Science, 66, A.



(2.8.1) and (2.8.2) combined with the equilibrium activation energy (�Ev,b) indicated inEquation (2.1.7) seems to capture accurate physics during the intermittent drying undertime-varying drying air temperature. The results indicated that the application of theREA has been extended significantly, following that of the L-REA modelling the cyclicdrying of a thin layer of polymer solution mixture of under time-varying infrared-heatintensity (Putranto et al., 2010b).

In terms of the intermittent drying of kaolin under time-varying humidity, the resultsof modelling using the L-REA are shown in Figures 2.44–2.47. The results of theintermittent 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) havealso been conducted and the L-REA yields better agreement with the experimental data.The observed deviations between the experimental and predicted temperature profilesobserved from the approach of Kowalski and Pawlowski (2010b) do not appear whenthe L-REA is used.

Similarly to Case 3 (refer to Table 2.7, Figures 2.44 and 2.45), for Case 4 (refer toTable 2.7, Figures 2.46 and 2.47) the REA results are in a good agreement between boththe predicted moisture content and temperature profiles and the experimental data. Theslight deviation of the temperature profiles during drying times of 13 000 and 18 000 s arealso observed in the modelling applied by Kowalski and Pawlowski (2010b). However,

290

300

310

320

330

340

350

370

360

0 0.5 1 1.5 2 2.5t(s) × 104

Tem

pera

ture

(K

)

Data

L-REA

Figure 2.45 Temperature profile of intermittent drying in Case 3 (periodically changed relativehumidity 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) formodelling intermittent drying under time-varying humidity and temperature, 2149–2156,

Copyright (2012), with permission from Elsevier.]

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.5 1 1.5 2 2.5t(s) × 104

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

DataL-REA

Figure 2.46 Moisture content profile of intermittent drying in Case 4 (periodically changedrelative humidity between 4° and 80%). [Reprinted from Chemical Engineering Science, 66, A.




290

300

310

320

330

340

350

370

360

0 0.5 1 1.5 2 2.5t(s) × 104

Tem

pera

ture

(K

)

L-READata

Figure 2.47 Temperature profile of intermittent drying in Case 4 (periodically changed relativehumidity 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) formodelling intermittent drying under time-varying humidity and temperature, 2149–2156,


the observed overestimation at a drying time of 8000–13 000 s is not evident whenmodelling using the L-REA. Tables 2.8 and 2.9 indicate that the L-REA yields closeragreement with the experimental data than the other model.

It can be shown that the L-REA is indeed accurate enough to model intermittentdrying under time-varying humidity. The relative activation energy in conjunction withthe equilibrium activation energy seems to be able to capture the periodically changedrelative humidity well. Because of this, the L-REA is shown to be able to model inter-mittent drying, not only under time-varying temperatures but also under time-varyinghumidity, well. The simplicity of the L-REA is also proven for this challenging dryingsystem.

2.9 The heating of wood under linearly increased gastemperature modelled using the L-REA

Wood may contain several harmful chemicals as a result of chemical processing, includ-ing chromated copper arsenate (CCA), creosote and pentachloro-phenol (Younsi et al.,2006a). Several methods of disposing such harmful chemicals were reviewed in researchby Helsen and Bulck (2005). Recycling and recovery, chemical extraction, bioremedi-ation, electrodialytic remediation and thermal destruction may be attempted to remove


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

CaseFinal gastemperature (°C)

Heating rate(°C h−1)

Initial moisture content(kg H2O kg dry solids−1)

1 220 30 0.112 220 20 0.1253 220 10 0.124 160 20 0.10595 200 20 0.07

the contaminants. Reuse and recycling contaminants has several disadvantages, includ-ing contamination to people handling the process as sorting, transportation and storageare required, while chemical extraction is not an effective process since the kinetics ofextraction is relatively slow and many steps of extraction are necessary. In addition,the remediation may result in lower quality wood fibre. The thermal destruction pro-cess has advantages of possible energy recovery and significant reduction in volume.However, intensive research still needs to be carried out to evaluate and optimise theprocess (Helsen and Bulck, 2005). Thermal destruction can be carried out by high-temperature treatment processes in which wood samples are exposed to hot gas withlinearly increased temperatures beyond 200 °C. Fundamentally, it is a drying processunder 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 isshown by published experimental data from Younsi et al. (2006a; 2007). The experimen-tal details were reported in Kocaefe et al. (1990; 2007) and Younsi et al. (2006b) and arereviewed here for better understanding of the current approach. The heat treatment wasconducted in a thermogravimetric analyser (Kocaefe et al., 1990). Wood samples withdimensions of 0.035 × 0.035 × 0.2 m were heat treated by suspending the samples ona balance with an accuracy of 0.001 g. The heat treatment was conducted by exposingthe samples to hot gas whose temperature was linearly increased according to heatingrate. The humidity of the gas was controlled by injection of steam into a second furnaceplaced under the main furnace. Initial moisture content of the samples was between7 and 12%wt. (dry) and initial temperature of the samples was 20 °C (refer to Table2.10). The samples were first heated to 120 °C and held at this temperature for halfan hour, followed by heating below the preset heating rate (refer to Table 2.10) untilthe 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 byT-type thermocouples placed inside the samples, but the measurement indicated that thetemperatures 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 anymodification. As mentioned before, for modelling using the L-REA, relative activation


Experimentally determined dataFitted curve

0 0.02 0.04 0.06 0.08 0.1 0.12

ΔE

v/Δ

Ev,

b


0

0.1

0.2

0.4

0.6

0.8

1

0.3

0.5

0.7

0.9

Figure 2.48 Relative activation energy (�Ev/�Ev,b) of the dehydration of wood during heattreatment generated from the experimental data in Case 2 (refer to Table 2.10). [Reprinted fromBioresource 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 isgenerated from the experimental data for a drying run of Case 2 (refer to Table 2.10).This can be represented as:

�Ev

�Ev,b= [1 − 2.181(X − Xb)0.372

]× exp

[−3.716(X − Xb)3.135]

for X − Xb ≤ 0.05, (2.9.1)

�Ev

�Ev,b= [1 − 1.462(X − Xb)0.207

]× exp

[−3.716(X − Xb)3.137]

forX − Xb ≤ 0.05. (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 andexperimental relative activation energy is shown in Figure 2.48 and indicated by R2 of0.994 and 0.999, respectively.

During heat treatment of wood, the drying gas temperature changed linearly withtime according to heating rate (refer to Table 2.10). In order to incorporate this inthe modelling using the L-REA, the equilibrium activation energy (�Ev,b) shown byEquation (2.1.7) has been evaluated according to the gas temperature and corresponding


humidity during heat treatment. The equilibrium activation energy (�Ev,b) is combinedwith 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 duringthe heat treatment (Younsi et al., 2006b), the heat balance can be written as:

d(mC pT )

dt≈ h A (Tb − T ) + ms

d X

dt�H V , (2.9.3)

where m is sample mass (kg), Cp is the heat capacity of the sample (J kg−1 K−1), T isthe 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 ratedX/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 inEquations (2.1.4) and (2.9.3) in conjunction with the equilibrium and relative activa-tion energy shown in Equations (2.1–7), (2.9.1) and (2.9.2), results simultaneously inthe profiles of moisture content and temperature during heat treatment of wood. Theshrinkage is neglected in the modelling because Younsi et al. (2006b) indicated that theratio between the final and initial dimension is around 0.96. Similarly, the modellingimplemented by Younsi et al. (2006a,b, 2007) did not incorporate the shrinkage effect.

2.9.2 Results of modelling wood heating under linearly increased gastemperatures using the L-REA

The profiles of both moisture content and temperature during heat treatment of wood forCases 1–5 (refer to Table 2.10) are presented in Figures 2.49–2.52. Results of modellingfor Case 1 (refer to Table 2.10) are shown in Figures 2.49 and 2.50. The L-REA-basedmodel system describes both the moisture content and temperature profiles well. Thepredictions made using the L-REA match reasonably well with the experimental dataof moisture content and temperature. The slight discrepancies in the moisture contentprofile were also found in the modelling implemented by Younsi et al. (2007) usinga far more complex model. However, as depicted in Figure 2.49, the L-REA yieldssystem closer agreement of the moisture content profile with the experimental data. TheL-REA system can model the heat treatment of wood which applied linearly changedair 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 pre-dicts both the moisture content and temperature profiles well. The results of modellingmatch well with the experimental data. This is also confirmed by R2 of moisture con-tent and temperature profiles of 0.994 and 0.996. A benchmark against the modellingimplemented by Younsi et al. (2007) shows that the L-REA gives comparable or betterresults.

For the heat treatment of wood under a heating rate of 10 °C h−1 (Case 3, refer toTable 2.10), the L-REA again describes both the moisture content and temperaturereasonably well, as shown in Figures 2.49 and 2.50. Slight deviations in predictingmoisture content profiles are observed, but when benchmarking against the modelling


100

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0.1

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2 3 4 5t(s)

6 7

× 104

Case 1-experimental dataCase 1-predicted by L-REACase 2-experimental dataCase 2-predicted by L-REACase 3-experimental dataCase 3-predicted by L-REA

Figure 2.49 Moisture content profiles during the heat treatment of Cases 1 to 3 (refer to Table2.10). [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 engineeringapproach (REA), 6214–6220, Copyright (2012), with permission from Elsevier.]

applied by Younsi et al. (2007), the L-REA still yields the closer agreement towardsexperimental data. The good agreement of moisture content and temperature profile isalso indicated by R2 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 treatmentof wood using a different final gas temperature. Case 4 (refer to Table 2.10) applied theheat treatment of wood samples with a final gas temperature of 160 °C and a heating rateof 20 °C h−1. As depicted in Figure 2.51, the L-REA system was reasonably accurate inmodelling the moisture content profiles. Both the L-REA system and the modelling byYounsi et al. (2006a) predict the temperature profiles well as shown in Figure 2.52. Itsuggests that the L-REA system can model the heat treatment of wood using differentfinal gas temperatures.

In addition, Case 5 (refer to Table 2.10) implemented the heat treatment of wood withinitial 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 temperatureprofiles has been assessed. From Figure 2.51, it can be seen that the L-REA systemdescribes the profiles of moisture content well. In addition, the L-REA is accurate inmodelling the temperature profiles during the heat treatment of wood as shown in Figure2.52. Results of modelling using the L-REA match well with experimental data. The


500

450

400

350

300

2500 1 2 3

× 104t(s)

54

Case 1-experimental data

Case 1-predicted by L-REA





Tem

pera

ture

(K

)

6 7

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 anaccurate R2 of moisture content and temperature of 0.993 and 0.999, respectively. Whenbenchmarking against the modelling implemented by Younsi et al. (2006), it is observedthat the L-REA yields better results.

From Cases 1 to 5 (refer to Table 2.10), it can be said that the L-REA systemcan be implemented to describe both moisture content and temperature profiles verysuccessfully. The applicability of the L-REA for this purpose could be due to theaccuracy and flexibility of the equilibrium activation energy (�Ev,b) combined with theunique relative activation energy (�Ev/�Ev,b) in capturing the physics of drying duringheat treatment of wood. The effect of a linearly increased gas temperature accordingto the heating rate on the drying rate seems to be captured well by the combinationof (�Ev,b) and (�Ev/�Ev,b). This allows the drying kinetics to be changed flexiblyaccording to environment conditions.

Based on the study of Cases 1 to 5 (refer to Table 2.10), it is revealed that the L-REAcan 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 biomassthat employs time-varying temperature or external conditions. Several processes applythis principle, including ThermoWood Technology (Finnish ThermoWood Association,2011) developed by Finnish industries. The process is essentially heating wood following


100

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0.1

0.08

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0.04

0.02

0.14

2 3 4 5t(s)

6 7

× 104

Case 1-experimental dataCase 1-predicted by L-REACase 2-experimental dataCase 2-predicted by L-REACase 3-experimental dataCase 3-predicted by L-REA

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,


a particular schedule, with different temperatures in each heating period aimed to reducecracking and burning by protecting wood with water vapour generated from the woodsamples (Younsi et al., 2010; Finnish ThermoWood Association, 2011). Another exampleis thermal modification of wood to improve thermal insulation properties, maintainingcolour and enhancing water resistance conducted by heating of wood to temperature of160–260 °C under various gas media and schedules (Rapp, 2001). In order to predict thequality 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-scaleindustries where quick decision-making is necessary. Simulation tools can be appliedas a means to predict the quality of wood or biomass after treatment. As the L-REA isrevealed to be simple and accurate for modelling the heat treated wood, it can be usedas a simple and valuable tool in industry for predicting quality parameters of samplesunder heat treatment. This can be conducted by the L-REA by evaluating the equilibriumactivation energy and heat balance according to corresponding external conditions (i.e.humidity and temperature) in each period of heat treatment. The equations that representquality change, usually represented as function of moisture content and/or temperature,are incorporated in the REA system and solved simultaneously to yield quality profilesduring heat treatment.


0.50280

Tem

pera

ture

(K

)460

420

400

440

380

360

340

320

300

480

1 1.5 2 2.5t(s)

3 3.5

× 104

Case 4-experimental dataCase 4-predicted by L-REACase 5-experimental dataCase 5-predicted by L-REA

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,


2.10 The baking of cake modelled using the L-REA

Baking is a complex simultaneous heat and mass transfer process commonly appliedin food industries. Although the process has been practiced for long time, there is stilllimited knowledge of the physical phenomena involved (Zhang and Datta, 2006). Theheat 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 isalso affected significantly by molecular size and structure of polymers that make upstarch and protein components. The interactions between polymer chain entanglementsand branching may determine the rheological aspects that affect the deformation duringbaking (Dobraszczyk, 2004). Shrinkage could occur during baking as a result of waterloss and thermal stress, while expansion takes place because of carbon dioxide producedby 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 isvalidated by the published experimental data and details of Sakin et al. (2007). For abetter understanding of the modelling using the L-REA, the experimental details arereviewed next. The ingredients of the cake batter were 49.4%wt. of Dr Oetker’s ready


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. Theingredients 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 withinitial thickness of 3 mm. This thickness was used to minimise temperature gradientinside 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 ovenwith dimensions of 0.39 × 0.44 × 0.35 m (Teba High-01 Inox) at baking temperatures of50°, 80°, 100°, 140° and 160 °C under forced convection conditions. Fresh air entered theoven cavity and the fan on the back side was used to circulate the air at a constant speedof 0.56 m s−1 (measured by an Airflow anemometer, LCA 6000). During baking, theweight of the batter was recorded until the equilibrium moisture content was achieved.The product temperature was also measured by a thermocouple (J-type) inserted insidethe 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 modelbaking of cake without any modification. Similar to the modelling of convective dry-ing using the L-REA mentioned in the previous sections, for modelling baking a thinlayer of cake using the L-REA, the relative activation energy (�Ev/�Ev,b) needs to begenerated from one accurate baking experiment. In this study, it was generated froman experiment baking at a temperature of 100 °C, whose experimental data of moisturecontent 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 calcu-lated 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 moisturecontent on dry basis (X) by simple mathematical expression obtained by a least-squaremethod:

�Ev

�Ev,b= [1 − 1.612(X − Xb)1.151

]exp[−1.28 × 106(X − Xb)14.19

]. (2.10.1)

Figure 2.53 shows a good agreement between the experimental and fitted relative activa-tion energy, which is also confirmed by R2 of 0.998. The profile of the relative activationenergy is very reasonable since it is zero near the start and keeps increasing as bakingproceeds. When the equilibrium moisture content is achieved, the relative activationenergy 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 thinlayer of cake.

For yielding the profiles of both moisture content and temperature during baking, themass and heat balances are solved simultaneously. The mass balance using the L-REAis shown in Equation (2.1.4). The temperature distribution inside the cake during baking


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

ΔE

v/Δ

Ev,

b


0

0.2

0.1

0.4

0.3

0.6

0.5

0.8

0.7

0.9

1

DataModel

Figure 2.53 The relative activation energy (�Ev/�Ev,b) of baking of thin layer of cake at an oventemperature 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[ms(1 + X )CpT

]dt

≈ h A (Tb − T ) + msd X

dt�HV , (2.10.2)

where Cp is the heat capacity of the sample (J kg−1 K−1), T is the temperature of thesample (K), h is the heat transfer coefficient (W m−2 K−1), �HV is vaporisation heatof water (J kg−1) and Tb is the baking oven temperature (K). The drying rate dX/dt isnegative 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 bakingtemperatures are shown in Figures 2.54–2.57. Figures 2.54 and 2.55 show that the profilesof moisture content predicted using the L-REA match well with the experimental data.This is supported by the R2 values for moisture content profiles being higher than 0.982(shown in Table 2.11). Benchmarking against a diffusion-based model implemented bySakin et al. (2007) revealed that the L-REA yielded comparable or better results, althoughthe diffusion-based model employed diffusivity which was split into two forms in order toincorporate the effects of temperature and moisture content. The slight overestimation of

Moi

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00

0.1

0.2

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0.6

0.7

500 1000 1500 2000 2500 3000 3500 4000 4500 5000t(s)

L-REA 100°CData 100°CL-REA 140°CData 140°CL-REA 160°CData 160°C

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,


00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 2 3 4 5 6 7t(s) × 104

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t (kg

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)

L-REA 50°CData 50°CL-REA 80°CData 80°C

Figure 2.55 Moisture content profiles at baking temperatures of 50° and 80 °C. [Reprinted fromJournal 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), withpermission from Elsevier.]


Table 2.11 R2 of modelling using the REA.

No.Bakingtemperature (°C) R2 for X R2 for T

1. 50 0.990 0.9902. 80 0.997 0.9993. 100 0.992 0.9704. 120 0.987 0.9975. 140 0.982 0.985

L-REA 100°CData 100°CL-REA 140°CData 140°CL-REA 160°CData 160°C

0 500 1000 1500 2000

Tem

pera

ture

(K

)

2500 3000 3500 4000 4500 5000t(s)

280

300

320

340

360

380

400

420

440

460

Figure 2.56 Temperature profiles at baking temperatures of 100°, 140° and 160 °C. [Reprintedfrom 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), withpermission from Elsevier.]

the drying rate at initial baking period at oven temperatures of 100°, 140° and 160 °C wasalso observed in modelling using the diffusion-based model. At baking temperatures of50° and 80 °C both the L-REA and diffusion-based model described the moisture contentprofiles well. It can be inferred that the L-REA can model the moisture content duringthe baking of thin layer of cake very well.

Figures 2.56 and 2.57 indicate that the L-REA modelled the profiles of tempera-ture profiles well. A good agreement with the experimental data is observed and it isconfirmed by the R2 values for temperature profiles being higher than 0.970 (shownin Table 2.11). The slight discrepancies between the predicted and experimental dataduring initial period of baking at baking temperature of 100 °C might be due largely to


L-REA 50°CData 50°CL-REA 80°CData 80°C

0290

300

310

320

330

340

350

360

370

1 2 3 4 5 6 7t(s) × 104

Tem

pera

ture

(K

)

Figure 2.57 Temperature profiles at baking temperatures of 50° and 80 °C. [Reprinted fromJournal 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), withpermission from Elsevier.]

experimental error of temperature measurement, which resulted in higher temperatureprofiles than those of baking temperatures at 140° and 160 °C. Benchmarking againstthe modelling from Sakin et al. (2007) cannot be conducted as the model in this studydid not implement the heat balance. The L-REA coupled with the heat balance (shownin Equation 2.10.2) is indeed accurate in predicting the temperature profiles duringbaking.

It has been shown that the L-REA-based heat- and mass-transfer system can besuccessfully applied to model the baking of cake. The L-REA is accurate in modellingboth the profiles of moisture content and temperature during the baking of cake. It seemsthat the relative activation energy (�Ev/�Ev,b) shown in Equation (2.9.1) combinedwith the equilibrium activation energy (�Ev,b) shown in Equation (2.1.7) can capturethe physics during baking.

2.11 The infrared-heat drying of a mixture of polymer solutions modelledusing the L-REA

The experimental data for validating the accuracy of the L-REA are derived from a studyreported by Allanic et al. (2009). The experimental setup up is similar to that describedin Section 2.5 but constant infrared-heat intensity is applied here. The velocity and


0

z

Product

Petri dish

e(t)

Thermocouple

Long infrared

Convection

Conduction

Evaporation

Diffusion

Figure 2.58 Heat transfer mechanisms of convective and infrared-heat drying. [Reprinted fromChemical 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


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

2.11.1 Mathematical modelling of the infrared-heat drying of a mixture of polymersolutions using the L-REA

The L-REA shown in Equation (2.1.4) is used for the mass balance here. For infrared heat-drying, the sample was heated by an infrared emitter which increased the temperatureof 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 Petridish actually release heat to the environment by convection. The heat is released fromthe upper side and bottom side due to forced and natural convection, respectively (referto Figure 2.58). Hence, it can be written as:

d(ρC peT )

dt= αQIR − htop(T − Tb) − Ubottom(T − Tb) + ms

A

d X

dt�HV (T ),

(2.11.1)

where ρ is sample density (kg m−3), Cp is sample heat capacity (J kg−1 K−1), e is samplethickness (m), T is sample temperature (K), ms is mass of dried product (kg), A is surfacearea of product (m2), �Hv is vaporisation enthalpy of water (J kg−1), htop is the heat


transfer coefficients on top of the sample (W m−2 K−1), Ubottom represents the overallheat transfer coefficient (W m−2 K−1) from the lower side, including convection alongand conduction through the Petri dish, QIR is the intensity of radiation (W m−2) and α isthe absorptivity of the product.

Rearranging Equation (2.11.1) results in:

d(ρC peT )

dt= αQIR − Utotal(T − Tb) + ms

A

d X

dt�HV (T ). (2.11.2)

It should be highlighted that the application of the L-REA requires accurate deter-mination of activation energy as a function of moisture content. �Ev/�Ev,b becausecharacteristics of drying kinetics are used to describe the reduction of moisture contentand temperature profiles. In convective drying, the product is heated by relatively highdrying air temperatures and the maximum activation energy �Ev,b is determined usingEquation (2.1.7). However, a different condition occurs when drying is conducted usinginfrared heating because the product temperature increases to above the drying air tem-perature so the product releases heat to the air instead of gaining heat from the air. Ifone maintains constant infrared power and lets drying continue until low water contentis reached, the product temperature would reach a constant temperature determined bythe balance between infrared power input and heat loss to the surroundings. The minormodifications of the L-REA are taken here as explained next.

The heat transfer coefficient should be determined from the final part of drying insteadof using the initial constant rate period of drying because of the low evaporation rate at thefinal part of drying, so most of the heat adsorbed by the product from the infrared emitteris released to the air. In addition, at the final part of drying the product temperature isessentially constant as revealed by Allanic et al. (2009) indicating ‘thermal’ equilibriumhas been reached. The heat balance for final part of drying can be written as:

αQIR = ms

A

d X

dt�HV (T ) + Utotal(T − Tb). (2.11.3)

It is apparent that, for the final part of drying, the contribution of the first term onthe right-hand side is low because of the low evaporation rate, thus the second term isdominant. Equation (2.11.3) can be simplified:

αQIR ≈ Utotal(T − Tb), (2.11.4)

and Utotal can be determined using Equation (2.11.3) by inserting T from the recordedfinal product temperature (Allanic et al., 2009). The predetermined Utotal is then used formodelling 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 introducedbecause 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 convectivedrying run and shown in Equation (2.5.10) is used but �Ev,b has been determined fromthe final product temperature and corresponding humidity of air instead of using dryingair temperature. This can be written:

�Ev,b = −RT ln(RHb), (2.11.5)


0

1

2

3

4

5

6

7

8

0 500 1000 1500t(s)

X (

kg w

ater

/kg

dry

soli

d)

2000 2500 3000

ModelData

Figure 2.59 Moisture content profile of convective and infrared drying at an air temperature of35 °C, air velocity of 1 m s−1, air relative humidity of 18% and intensity of infrared drying of3700 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 ofpolyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA),


where T is the final product temperature (K) and RHb is the relative humidity at the finalproduct temperature and the absolute humidity. For other drying conditions, as T is notknown prior to drying experiments, T can be determined from heat balance between theinfrared power input and heat loss to the surroundings as explained previously.

In order to yield the moisture content and temperature profiles, the mass and heatbalances 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 indicatedin Equations (2.5.10) and (2.11.5), respectively. The results of modelling using the L-REA are validated towards the experimental data of Allanic et al. (2009).

2.11.2 The results of mathematical modelling of infrared-heat drying of amixture of polymer solutions using the L-REA

The results of modelling are presented in Figures 2.59 and 2.60. It could be observedthat the discrepancies between experimental and calculated data are reasonably small.Statistical analysis showed that the R2 and RMSE of the moisture content profile are0.994 and 0.181, respectively. Overestimation of the drying rate between drying timesof 600–2250 s was also predicted by the previous model (Allanic et al., 2009). TheL-REA seems to model this case better and only shows slight overestimation in drying


360

2900 500 1000 1500

t(s)

Tem

pera

ture

(K

)

2000 2500 3000

300

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ModelData

Figure 2.60 Product temperature profile of convective and infrared drying at an air temperature of35 °C, air velocity of 1 m s−1, air relative humidity of 18% and intensity of infrared drying of3700 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 ofpolyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA),


rates between drying times of 1350–2250 s. In addition, it is apparent that the REA canhandle temperature profiles quite well as shown by the R2 and RMSE of temperatureprofile, which are 0.992 and 1.712, respectively. Overestimation in the temperatureprofile of about 5 °C during drying times of 150–1200 s was indicated by the previousmodel (Allanic et al., 2009). However, this overestimation is not observed by modellingusing the L-REA.

It can be observed that �Ev/�Ev,b derived from convective drying combined with thenew quantification of �Ev,b shown in Equation (2.11.5) is appropriate for describingthe drying kinetics of infrared-heat drying. It may be applied to other infrared-heatingcases. In the case of drying, which exhibits product temperature higher than the dryingair temperature, application of the modification of �Ev,b shown by Equation (2.11.5) inconjunction with the generated �Ev/�Ev,b is shown to be appropriate.

2.12 The intermittent drying of a mixture of polymer solutions undertime-varying infrared-heat intensity modelled using the L-REA

The experimental data for validating the accuracy of the L-REA are derived from astudy reported by Allanic et al. (2006, 2009), whose experimental conditions are briefly


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

Case numberAir velocity(m s−1)


Air relativehumidity (%)

Intensity of infraredradiation (W m2)

1 (Allanic et al., 2009) 1 35 19 12.3–5.6-regulated*2 (Allanic et al., 2009) 1 35 14 13.3–5.6-regulated*3 (Allanic et al., 2006) 1 35 18 13.3–12.3-regulated*

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

2.12.1 Mathematical modelling of the intermittent drying of a mixture of polymersolutions 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 definethe ‘equilibrium’ activation energy (�Ev,b). This is necessary because each stage hasdifferent drying conditions. If each drying condition persists for a long time, a differentterminal drying state should be reached, corresponding to a unique �Ev,b. The equilib-rium activation energy (�Ev,b) is defined as representing the maximum activation energyunder 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 heatingbe 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:

�Ev,b = −RT ∗ ln(RHb), (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 activa-

tion energy (�Ev,b) and the infrared intensity. The relationship of the infrared intensitywith the equilibrium activation energy (�Ev,b) in each stage expressed as:

�Ev,b = pI q + k, (2.12.3)

where p and k are the constants obtained from linear relationship between the equilibriumactivation energy (�Ev,b) and Iq and q is the constant indicating sensitivity of theequilibrium activation energy (�Ev,b) towards infrared intensity.

For modelling the cyclic drying here, both definitions of equilibrium activationenergy (�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),


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

Data

Figure 2.61 Sensitivity of the moisture content profile of cyclic drying; Case 1 (refer to Table2.12) 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,


generated from convective drying of a mixture of polymer solution, can still be usedfor the modelling of cyclic drying here. The L-REA shown in Equation (2.1.4) servesas the mass balance while the heat balance shown in Equation (2.11.1) is used byemploying 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 describedpreviously results in the moisture content and temperature profiles.

2.12.2 Results of modelling the intermittent drying of a mixture of polymer solutionsunder 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 imple-mented. Using the first scheme (Equation 2.12.1) (T* as a function of the infraredintensity), several values of n shown in Equation (2.12.1) in the range of 1.4–1.9 wereused to describe the drying kinetics (which we found to be more favourable than eitherlower or higher values). Figures 2.61 and 2.62 show the profiles of moisture content andtemperature along the cyclic drying run with various values of n. It can be observed thatdifferent values of n did not provide a noticeable difference in both moisture contentand temperature profiles during the constant rate period of drying. This indicates anoverriding effect of the latent heat of vaporisation. However, different values of n that


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n = 1.4

n = 1.5

n = 1.6n = 1.7

n = 1.8

n = 1.9

Data

0

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. Thelower values of n yielded lower moisture content and lower temperature profiles becausethese lower values result in lower activation energy which projects a higher evaporationrate and more heat is removed from materials being dried for evaporation, shifting thetemperature profile down somewhat. From Figures 2.61 and 2.62, modelling by thevalue of n of 1.8 tends to agree well with the moisture content profile. Figure 2.63 showsthe prediction of moisture content profile whilst Figure 2.64 shows the temperature pro-file 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 R2 and RMSE of 0.996 and0.13 for moisture content, respectively while R2 and RMSE of temperature profile are0.938 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 othermodel resulted in an underestimation of evaporation rate initially and the overestimationafter 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 intensityof the infrared heating). Similarly, several values of q (Equation 2.12.3) were used to


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Data

Figure 2.63 Moisture content profile of cyclic drying; Case 1 (refer to Table 2.12) using the firstscheme (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 reactionengineering 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 showthat no deviations were observed during constant rate period of drying but differentvalues of q gave different moisture content and temperature profile after drying time of300 s. It can be examined that the value of q of 1.8 resulted in the best fit of moisturecontent profile. This can be clearly observed in Figure 2.67. A reasonable agreement oftemperature prediction is also indicated by Figure 2.68. Higher value of q resulted inhigher moisture content and temperature profile because of the higher activation energywhich decreases evaporation rate and less heat is removed from materials so this canyield a higher temperature profile. The good fit using this second scheme by applicationof q of 1.8 was indicated by R2 and RMSE of 0.995 and 0.14 for moisture content whileR2 and RMSE of temperature are 0.937 and 3.3, respectively. Moreover, modelling usingthis second scheme yielded better prediction of moisture content profile than that fromAllanic 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 thecurrent study has revealed that both the proposed schemes gave almost the same profilesof moisture content and temperature to drying times of 300 s. After that time, althoughthe values are not exactly the same, the differences between predicted moisture contentusing the first and the second schemes are below 0.15 kg kg−1 while those of temperature


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Figure 2.64 Temperature profile of cyclic drying; Case 1 (refer to Table 2.12) using the firstscheme (T* as function of infrared intensity) with n = 1.8. [Reprinted from Chemical



are below 1.5 °C. These indicate both schemes may have been successful in capturingthe physics of the process.

The results of modelling of cyclic drying of Case 2 (refer to Table 2.12) using theL-REA and the two schemes mentioned previously are represented in Figures 2.69 to2.72. Similarly to Case 1, sensitivity of n (Equation 2.12.1) and q (Equation 2.12.3) wasconducted. It was found that n = 1.5 and q = 1.5 were the most appropriate to describethe moisture content and temperature profile. A good agreement of both moisture contentand temperature profile using the first scheme and n = 1.5 can be observed from Figures2.69 and 2.70, respectively. Similarly to the previous case, the lower value of n producedlower moisture content and temperature profiles. This good fit is shown by R2 and RMSEof moisture content of 0.992 and 0.2 while R2 and RMSE of temperature are 0.946 and4.4, 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 secondscheme (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 and2.72 for moisture content and temperature profile, respectively. In addition, this good fitwas indicated by R2 and RMSE of moisture content of 0.992 and 0.2 while R2 and RMSE


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ater

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soli

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Dataq = 1.6q = 1.7q = 1.8q = 1.9q = 2

Figure 2.65 Sensitivity of the moisture content profile of cyclic drying; Case 1 (refer to Table2.12) 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,


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Dataq = 1.6q = 1.7q = 1.8q = 1.9q = 2

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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Figure 2.67 Moisture content profile of cyclic drying; Case 1 (refer to Table 2.12) using thesecond scheme (�Ev,b as function of infrared intensity) with q = 1.8. [Reprinted from Chemical



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Figure 2.68 Temperature profile of cyclic drying; Case 1 (refer to Table 2.12) using the secondscheme (�Ev,b as function of infrared intensity) with q = 1.8. [Reprinted from Chemical




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Figure 2.69 Moisture content profile of cyclic drying; Case 2 (refer to Table 2.12) using the firstscheme (T*as function of infrared intensity) with n = 1.5. [Reprinted from Chemical



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Figure 2.70 Temperature profile of cyclic drying; Case 2 (refer to Table 2.12) using the firstscheme (T* as function of infrared intensity) with n = 1.5. [Reprinted from Chemical




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Figure 2.71 Moisture content profile of cyclic drying; Case 2 (refer to Table 2.12) using thesecond scheme (�Ev,b as function of infrared intensity) with q = 1.5. [Reprinted from Chemical



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Figure 2.72 Temperature profile of cyclic drying; Case 2 (refer to Table 2.12) using the secondscheme (�Ev,b as function of infrared intensity) with q = 1.5. [Reprinted from Chemical




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Figure 2.73 Moisture content profile of cyclic drying; Case 3 (refer to Table 2.12) using the firstscheme (T*as function of infrared intensity) with n = 1.6. [Reprinted from Chemical



of temperature are 0.933 and 4.9, respectively. This scheme also yields better moisturecontent prediction than that of Allanic et al. (2009), which shows overestimation of thedrying rate after 200 s.

Results of the modelling in Case 2 using both schemes were compared and showedthe values of moisture content and temperature were almost the same for drying times of300 s. Both schemes generate the same profiles of moisture content and tempera-ture during the constant rate period of drying. In addition, after that time, negligibledifferences between predicted moisture content and temperature using both schemescan be found. The differences between moisture content and temperature were around0.03 kg kg−1 and 1 °C, respectively.

In addition, modelling of cyclic drying in Case 3 (refer to Table 2.12) was conductedusing both schemes and the results are shown in Figures 2.73–2.76. Sensitivity was alsoconducted to find the most appropriate value of n (Equation 2.12.1) and q (Equation2.12.3). For the first scheme, application of n of 1.6 matches the temperature profile verywell. Figures 2.73 and 2.74 illustrate both moisture content and temperature profile usingthe value of n = 1.6. The good fit is also shown by R2 and RMSE of moisture contentof 0.989 and 0.24, while R2 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 theresults are shown in Figures 2.75 and 2.76. The good agreement of this scheme using qof 1.6 was indicated in Figures 2.75 and 2.76. This is also supported by R2 and RMSE of


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Figure 2.74 Temperature profile of cyclic drying; Case 3 (refer to Table 2.12) using the firstscheme (T*as function of infrared intensity) with n = 1.6. [Reprinted from Chemical



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Figure 2.75 Moisture content profile of cyclic drying; Case 3 (refer to Table 2.12) using thesecond scheme (�Ev,b as function of infrared intensity) with q = 1.6. [Reprinted from ChemicalEngineering 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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Figure 2.76 Temperature profile of cyclic drying; Case 3 (refer to Table 2.12), using the secondscheme (�Ev,b as function of infrared intensity) with q = 1.6. [Reprinted from Chemical



moisture content of 0.991 and 0.23 as well as R2 and RMSE of temperature of 0.942 and4.52. The differences of moisture content and temperature profile predictions betweenthe 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 shownin 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 useof the REA framework has shown that application of the first and second schemes withthe appropriate values of n or q give almost exactly the same moisture content andtemperature profiles. All this work supports the notion that L-REA is a robust modellingframework that allows for intuitive extensions, such as the schemes proposed here, formore 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 modellingseveral drying cases are described. Within the range that material properties are invariant,the REA parameters, expressed in relative activation energy, can be applied to modelother 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 toaccurately model the average moisture content and temperature during convective drying


of food and non-food materials, convective drying of several centimetre-thick materi-als, intermittent drying, heating of wood under linearly increased gas temperatures andbaking of cake. For modelling the discussed cases, the original formulation of the L-REAcan be implemented without major modification. The estimation of temperature distri-bution inside the samples needs to be combined with the L-REA formulation to describedrying of relatively thick materials. The L-REA can be used to model intermittent dryingby evaluating the equilibrium activation energy according to the corresponding humid-ity and temperature in each period of drying. Similarly, by evaluating the equilibriumactivation energy according to the drying settings, the L-REA can accurately model theheat treatment of wood under a constant heating rate. For baking, the original formula-tion of the L-REA can be implemented without any modifications. The accuracy of theL-REA in modelling several cases of drying could be due to the accuracy of the relativeactivation energy in capturing the physics of drying. A combination of the relative acti-vation energy and equilibrium activation energy yields unique relationships of activationenergy which can change flexibly according to the external drying conditions and repre-sents the change in internal behaviour of the samples during drying. While the L-REAformulation is simple and the REA is efficient, the results are accurate. It can be usedin industrial settings for process design and maintaining product quality during drying.

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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 ChemistryResearch 50, 1089–1098.

Putranto, A., Chen, X.D., Devahastin, S., Xiao, Z. and Webley, P.A., 2011c. Application of thereaction engineering approach (REA) to model intermittent drying under time-varying humidityand temperature. Chemical Engineering Science 66, 2149–2156.

http://http://office.microsoft.com/en-au/excel/


Putranto, A., Chen, X.D., Xiao, Z. and Webley, P.A., 2011d. Modelling of high-temperaturetreatment of wood by using the reaction engineering approach (REA). Bioresource Technology102, 6214–6220.

Putranto, A., Chen, X.D. and Zhou, W., 2011e. Modelling of baking of cake using the reactionengineering approach (REA). Journal of Food Engineering 105, 306–311.

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Modelling and simulation of heat and mass transfer during drying of solids with hemisphericalshell geometry. Computers and Chemical Engineering 35, 191–199.

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http://www.quarzwerke.com

3 Reaction engineering approach IISpatial-REA (S-REA)

3.1 The S-REA formulation

In order to capture detailed information about the distributions of moisture content andtemperature throughout the material being dried, the S-REA has been developed. S-REAis a non-equilibrium multiphase drying approach in which the REA is implemented torepresent the local phase change term. It is envisaged that, for a better understanding ofthe transport phenomena, application of effective liquid diffusion alone without sourceterms in both energy and mass conservation equations may not be sufficient as it cannotrepresent the water vapour concentration during drying. This could be affected by gas inthe pore structure (Chen, 2007). Also, vapour generation and transfer may affect othervolatile transport in the same material. Traditionally, effective liquid diffusion has beenused to simulate the detailed profiles of temperature and moisture content. This will bediscussed further in Chapter 4.

Equilibrium and non-equilibrium approaches can be implemented in the multiphasedrying model mentioned previously (Zhang and Datta, 2004; Datta, 2007). By applyingthe equilibrium approach, it is assumed that vapour pressure inside the pores of thesamples equilibrates with the liquid moisture content inside the same pores and therelationship can be described by the relevant equilibrium isotherm (Zhang and Datta,2004). The equilibrium model has been applied to the baking process and a goodagreement 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 combiningthe mass conservation of water in both liquid and vapour phases, which effectivelyresulted in the elimination of the source term. The moisture content and water vapourconcentration are related by the available isotherm data (through a correlation equation)for the materials. Reasonable agreement with experimental data was shown in the casesinvestigated (Ni et al., 1999; Zhang et al., 2005; Zhang and Datta, 2006). Similarly, theequilibrium model was applied by Aversa et al. (2010) to model the convective drying offood materials. The model has been shown to represent the experimental data reasonablywell. Nevertheless, it has not been proven that use of equilibrium approach is valid inthe case of heating hygroscopic materials (Zhang and Datta, 2004).

For a more generic application of the multiphase drying model, it is suggested thatthe non-equilibrium approach is more appropriate. The good non-equilibrium multi-phase drying model is also useful in determining the appropriateness of the equilibriumapproach for the situation of concern (Zhang and Datta, 2004). In order to implement


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 appro-priately. The internal evaporation/condensation rate is implemented in the multiphasedrying model as a depletion term for the liquid phase and as a source term for the vapourphase. It has been proposed that the internal evaporation/wetting rate can be related tothe difference of equilibrium vapour pressure and the vapour pressure at a particulartime inside the pore spaces (Chong and Chen, 1999; Scarpa and Milano, 2002; Zhangand Datta, 2004). In other words, gas must be present in the pores to permit this processto occur. The REA, in its lumped format, has been proven to model the global dryingrate of various challenging drying cases accurately (Chen and Lin, 2005; Chen, Piriniand Ozilgen, 2001; Chen and Xie, 1997; Lin and Chen, 2005; 2006; 2007; Putrantoet al., 2010a,b, 2011a–e). It was expected that formulation of the L-REA may also beapplicable in representing the source term in the S-REA approach; for instance, the sameactivation energy profile can be reserved for the same material.

The S-REA consists of a mass balance of liquid water, mass balance of water vapourand 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 liquidphase (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):

∂(Cs X )

∂t= ∂

∂x

[Dw

∂(Cs X )

∂x

]+ ∂

∂y

[Dw

∂(Cs X )

∂y

]+ ∂

∂z

[Dw

∂(Cs X )

∂z

]− I ,

(3.1.1)

where X is the concentration of liquid water (kg H2O kg dry solids−1), Cs is the solid’sconcentration (kg dry solids m−3), which can change if the structure is shrinking, Dw isliquid diffusivity (m2 s−1), I is the evaporation or condensation rate (kg H2O m−3.s−1)and I is usually defined as positive when evaporation occurs locally. The liquid diffusivityrepresents the movement of liquid water inside the pore structure of the materials dueto capillary action as a result of the water concentration gradient. In practice, the liquiddiffusivity needs to be extracted from the available effective diffusivity data (Datta,2007).

Reaction engineering approach II: S-REA 123

The mass balance of water vapour is expressed as (Chen, 2007; Chong and Chen,1999; Kar and Chen, 2011; Putranto and Chen, 2013):

∂Cv

∂t= ∂

∂x

(Dv

∂Cv

∂x

)+ ∂

∂y

(Dv

∂Cv

∂y

)+ ∂

∂z

(Dv

∂Cv

∂z

)+ I , (3.1.2)

where Cv is the concentration of water vapour (kg m−3) and Dv is the effective watervapour diffusivity in pore channels (m2 s−1).

The heat balance is represented by the following equation (Chen, 2007; Chong andChen, 1999; Kar and Chen, 2011; Putranto and Chen, 2013):

ρC p∂T

∂t= ∂

∂x

(k∂T

∂x

)+ ∂

∂y

(k∂T

∂y

)+ ∂

∂z

(k∂T

∂z

)− I�Hv, (3.1.3)

where T is the sample temperature (K), �HV is the vaporisation heat of water (J kg−1), kis the sample thermal conductivity (W m−2 K−1), ρ is the sample density (kg m−3) andk and ρ may be functions of temperature and moisture content.

For cubic objects being dried as an example, the initial and boundary conditions forEquations (3.1.1) to (3.1.3) may be written as:

t = 0, X = Xo, Cv = Cvo, T = To (initial condition, uniform initial

concentrations and temperature), (3.1.4)

x = 0, y = 0, z = 0,d X

dx= 0,

d X

dy= 0,

d X

dz= 0 (symmetrical boundary), (3.1.5)

dCv

dx= 0,

dCv

dy= 0

dCv


dT

dx= 0,

dT

dy= 0,

dT


x = L ,−Cs Dw

d X

dx= hmεw

(Cv,s

ε− ρv,b

)(convective boundary

for liquid transfer), (3.1.8)

−Dv

dCv

dx= hmεv

(Cv,s

ε− ρv,b


for vapour transfer), (3.1.9)

kdT

dx= h(Tb − T ) (convective boundary for heat transfer), (3.1.10)

y = L ,−Cs Dw

d X

dy= hmεw

(Cv,s

ε− ρv,b



−Dv

dCv

dy= hmεv

(Cv,s

ε− ρv,b

)(convective boundary for vapour transfer), (3.1.12)


kdT

dy= h(Tb − T ) (convective boundary for heat transfer), (3.1.13)

z = L ,−Cs Dw

d X

dz= hmεw

(Cv,s

ε− ρv,b



−Dv

dCv

dz= hmεv

(Cv,s

ε− ρv,b

)(convective boundary for vapour,

transfer) (3.1.15)

kdT

dz= h(Tb − T ) (convective boundary for heat transfer), (3.1.16)

where εw and εv are the fractions of surface area covered by liquid water and watervapour, respectively.

The internal evaporation rate ( I ) can be described as:

I = hmin Ain(Cv,s − Cv), (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 writtenas (Kar and Chen, 2010; 2011; Putranto and Chen, 2013):

Cv,s = exp

(−�Ev

RT

)Cv,sat , (3.1.18)

where Cv,s is the internal-solid surface water vapour concentration (kg m−3), Cv,sat is theinternal 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 = hmin Ain

[exp

(−�Ev

RT

)Cv,sat − Cv

]. (3.1.19)

In Equation (3.1.19), the REA is used to describe the local evaporation rate as affectedby pore structure (porosity, shrinkage, local moisture content and local temperature).These microstructural effects can be ‘encapsulated’ in the term hm,inAin.

In particular, Ain is clearly influenced by structural or microstructural of the material ofconcern. 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 (Vrentasand Duda, 1977; Van der Sman, 2007a,b; Van der Sman et al., 2012). Free VolumeFlory–Huggins (FVFH) theory, an extension of the classical Flory–Huggins theory, canbe used to describe the thermodynamics of food materials. The chemical potential ofmoisture transfer can be assumed due to osmotic, elastic and ionic contributions.


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 derSman et al., 2012):

Dm = Q[φDs,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 describedusing Flory–Huggins theory as (Van der Sman et al., 2012):

Q = 1 − 2χφ(1 − φ). (3.1.21)

The Stokes–Einstein relation can be used to predict Ds,s given by (Van der Sman et al.,2012):

Ds,s = Ds,oηw

ηeff, (3.1.22)

where Ds,o is the polymer self-diffusivity at infinite dilution, ηw is the viscosity of waterand 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:

lnDs,w

Dw,o= −�E

RT− yw V ∗

w + ςys V •s

yw Kww(Ksw − Tg,w + T ) + ys Kws(Kss − 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), internalmass transfer coefficient (hm,in) and internal surface area per unit volume (Ain). Theprocedures used to determine these parameters are explained in this section.

The effective vapour diffusivity is deduced from (Bird et al., 2002):

Dv = Dvo

ε

τ, (3.2.1)

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 relationshipcan be represented as (Audu and Geffreys, 1975; Gimmi et al., 1993):

τ = ε−n, (3.2.3)


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 onmicrostructure 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 andChen, 2010; 2011):

Cs = 1 − ε1ρs

+ Xρw

, (3.2.4)

while ε is the porosity which is dependent on shrinkage and local moisture content. Thiscan be determined according to (Madiouli et al., 2007):

ε = 1 − V0

V(1 − ε0)

(ρs

ρwX + 1

1 + ρs

ρwX0

). (3.2.5)

Until now, there has been no method to measure ‘effective liquid diffusivity’ (Chen,2007). Many drying research papers estimated the effective liquid diffusivity based ondrying kinetics data. Several sets of drying followed by complex optimization proce-dures are used to generate the effective diffusivity function (Azzouz et al., 2002; Marianiet al., 2008; Pakowski and Adamski, 2007; Thuwapanichayanan et al., 2008; Vaquiroet al., 2009). The literature on effective diffusivity may be used as a basis to determinethe effective liquid diffusivity to be used in the S-REA. A little adjustment is requiredto the effective liquid diffusivity to generate the effective liquid diffusivity function,since the effective liquid diffusivity in these existing literatures is used to represent thewhole phenomenon in drying (liquid diffusion, vapour diffusion, Darcy flow, evapora-tion/condensation).

For example, Srikiatden and Roberts (2008) reported the effective liquid diffusivityof potato tissues as:

Deff = 1.0418 × 10−5 exp

(−25.77 × 103

8.314T

). (3.2.6)

This was uniquely generated through well-controlled isothermal drying experimentscarefully arranged so the temperature dependence is correlated against the sample tem-perature. The real liquid diffusivity is, however, expected to be smaller than the effectiveliquid diffusivity shown in Equation (3.2.6) but the temperature dependence is expectedto remain valid. Equation (3.2.6) can still be used as the basis but is altered as:

Dw = Dw0 exp

(−25.77 × 103

8.314T

). (3.2.7)

For convective drying of potato tissues, it was found that Dwo of 6.5 × 10−6 m2 s−1 givesthe best agreement with experimental data (Putranto and Chen, 2013).

The internal mass transfer coefficient (hm,in) shown in Equation (3.1.17) is associatedwith the pore surfaces (porous media) or surfaces of the particles (packed beds), andinternal to the sample being dried. Initially, moisture is present in the void spaces ofpores and within the pores. As drying proceeds, the moisture may migrate within thepores (on the pore surfaces) by liquid (surface) diffusion and from the surfaces of the


pores through evaporation (Chong and Chen, 1999; Kar and Chen, 2010; 2011). Evenat low water content, ‘surface’ diffusion of liquid could occur along the pore surfaceaccessible to air (Chen and Mujumdar, 2008). The internal mass transfer coefficientshown in Equation (3.1.17) incorporates the restriction factor as it may be affected bythe 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) (Karand Chen, 2010; 2011). The internal surface area per unit volume (Ain) can be calculatedusing the procedures described in Kar and Chen (2010; 2011). It is calculated based onthe area of a single cell and the number of cells per unit volume, dependent on solidmass of the samples and mass of single cell. Of course, Ain should also be affected bymoisture content.

By using the REA, the relative activation energy (�Ev/�Ev,b) generated from oneaccurate 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) arecalculated using Equations (2.1.5) and (2.1.7), respectively to yield the relative activationenergy shown in Equation (2.1.6). Since in the S-REA the relative activation energyis used to represent the local evaporation rate, the average moisture content (X ) inthe relative activation energy is replaced by the local moisture content (X). The spatialprofiles of moisture content, concentration of water vapour and temperature are generatedby solving a set of equations shown in Equations (3.1.1)–(3.1.3) simultaneously inconjunction 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 theexperimental data of mango tissues (Vaquiro et al., 2009) and potato tissues (Srikiatdenand Roberts, 2008). The experimental data from drying of mango tissues are derived fromthe previous study (Vaquiro et al., 2009). For better understanding of the predictions,the necessary experimental details are summarised and reviewed in this section. Thesamples of mango tissues were formed as cubes with initial side lengths of 2.5 cm whilethe 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 weightchange of the sample and the centre temperature history were recorded. The drying airtemperature and air velocity were controlled at preset values by PID control algorithmswhile air humidity was maintained constant during drying. The experimental setting forconvective drying is shown in Table 3.1.

The density, thermal conductivity, heat capacity, equilibrium moisture content andshrinkage 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 pre-vious work (Srikiatden and Roberts, 2008). Their experimental details are also reviewedhere for better understanding of the modelling approach (Roberts et al., 2002; Srikiatdenand Roberts, 2006; 2008). The cylindrical samples of Russet potatoes with diameters


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



Air humidity(kg H2O kg dry air−1)

1 4 45 0.01342 4 55 0.01343 4 65 0.0134

of 1.4 and 2.8 cm were obtained using cylindrical cutters. The samples were sealedat their top and bottom ends with epoxy to establish approximately a one-dimensional(radial direction) moisture transfer. The experiments were conducted in a laboratoryconvective 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 thebottom of the sample draws air downward and this reduces the turbulence effect nearthe 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 formeasurement of moisture content distribution, i.e. core and cortex (the core is a cylinderwith the radius of 0.35 cm derived from the inner part of the potato tissues, while thecortex is a concentric shell derived from the outer part of the potato tissues). For thesamples 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 of1.4 cm, the core is a cylinder with a radius of 0.35 cm derived from the innermost partof the potato tissues. The procedures were repeated for a number of intervals (Srikiatdenand 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 helpestablish 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 ofmango 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 waterin 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 equationsare shown in Equations (3.1.4)–(3.1.16). Since the sample dried was a cube shape drieduniformly 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 andDeWitt, 2002; Van der Sman, 2003):

∂(Cs X )

∂t= 3

∂

∂x

[Dw

∂(Cs X )

∂x

]− I , (3.3.1)


while the mass balance of water in vapour phase can be expressed as:

∂Cv

∂t= 3

∂

∂x

(Dv

∂Cv

∂x

)+ I . (3.3.2)

In addition, the heat balance can be represented as:

ρCp∂T

∂t= 3

∂

∂x

(k∂T

∂x

)− I�Hv. (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 fromone accurate drying run of convective drying mango tissues under constant environmentconditions with a drying air temperature of 55 °C (Vaquiro et al., 2009). The activationenergy during drying is evaluated using Equation (2.1.5) and divided with the equilibriumactivation energy represented in Equation (2.1.7) to yield the relative activation energy asmentioned in Equation (2.1.6). The relationship between the relative activation energyand average moisture content can be represented by a simple mathematical equationobtained by use of the least-squares method using Microsoft Excel (Microsoft Inc.,2012). The relative activation energy can be represented as:

�Ev

�Ev,b= −9.92 × 10−4(X − Xb)3 + 9.74 × 10−3(X − Xb)2

−0.101(X − Xb) + 1.053. (3.3.4)

The good agreement between the fitted and experimental relative activation energy isshown by R2 of 0.999. Although Equation (3.3.4) involves Xb as mentioned earlier, allsuccessful applications of REA so far suggest that the experiments carried out shoulddry the materials to Xbs of very small values in order to allow the correlations such asEquation (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 datacan be misleading.

The relative activation energy correlated with Equation (3.3.4) has been implementedto model the convective and intermittent drying of mango tissues using the L-REA andthe 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 shownin Equation (3.3.4) is used but the average moisture content X in Equation (3.3.4) issubstituted by the local moisture content (X) as the REA is used to represent the localevaporation rate instead of the overall drying rate of the whole sample. In addition, itis emphasised that, for the S-REA, the equilibrium relative activation energy (�Ev,b) isevaluated at corresponding humidity and temperature inside the pores of the samplesunder equilibrium condition.

The effective vapour diffusivity (Dv), tortuosity (τ ), solid concentration (Cs) andporosity (ε) are deduced using Equations (3.2.1)–(3.2.5). Similarly, the internal masstransfer coefficient (hm,in) is evaluated using the procedures described in Section 3.2.


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

Deff = 2.933 × 10−3 exp

[−38.924 × 103

8.314T

(X

X + 1

)−1.885×10−2]

. (3.3.5)

Equation (3.3.5) can be used as an approximation to determine the liquid water diffusivityof mango tissues but a little adjustment of the constant is needed in order to match theprediction with the experimental data of moisture content and temperature. The liquidwater diffusivity used in this study can be expressed as:

Dw = 2.933 × 10−3 exp

[−31.924 × 103

8.314T

(X

X + 1

)−1.885×10−2]

. (3.3.6)

In order to yield the spatial profiles of moisture content, water vapour concentrationand temperature of the convective of mango tissues, the mass and heat balances shownin Equations (3.3.1)–(3.3.3) in conjunction with the initial and boundary conditionsrepresented in Equations (3.1.4)–(3.1.16) and the relative activation energy shown byEquation (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 ordinarydifferential 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 theprofiles observed as shown in Figure 3.2.

The shrinkage (Putranto et al., 2011a) is incorporated in the modelling by a mov-ing mesh in which the number of intervals is kept constant but the intervals of eachincrement are allowed to change according to the shrinkage relationship. The movingmesh 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 evaluatedby:

X =

L(t)∫0

X (x)dx

L(t)∫0

dx

. (3.3.7)

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

3.3.2 Mathematical modelling of convective drying of potatotissues using the S-REA

In the experiments reported by Srikiatden and Roberts (2008) which are of interest, thesamples were covered at both the top and bottom end to promote the one-dimensional


Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t(s) × 104

S-REA-10 increments

Data 45°CS-REA-200 increments

0

1

2

3

4

5

6

7

8

9

10

Figure 3.2 Moisture content profiles of the convective drying of mango tissues at a drying airtemperature 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 fordrying 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 one-dimensional modelling (at radial directions) of the S-REA of the convective drying ofcylindrical potato tissues is possible and can be represented by a set of equations ofconservation 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):

∂(Cs X )

∂t= 1

r

∂

∂r

[Dwr

∂(Cs X )

∂r

]− I , (3.3.8)

where X is the concentration of liquid water (kg H2O kg dry solids−1) and Cs is thesolids concentration (kg dry solids m−3), I is the evaporation or wetting rate (kg H2Om−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):

∂Cv

∂t= 1

r

∂

∂r

(Dvr

∂Cv

∂r

)+ I , (3.3.9)

where Cv is the concentration of water vapour (kg H2O m−3).


In addition, the heat balance can be written as (Chen, 2007; Chong and Chen, 1999;Kar and Chen, 2011; Zhang and Datta, 2004):

ρC p∂T

∂t= 1

r

∂

∂r

(kr

∂T

∂r

)− I�Hv, (3.3.10)

where T is the sample temperature (K).The initial and boundary conditions of Equations (3.3.8)–(3.3.10) are:



r = 0,d X

dr= 0,

dCv

dr= 0,

dT

dr= 0 (symmetrical condition), (3.3.12)

r = R,−Cs Dw

d X

dr= hmεw

(Cv,s

ε− ρv,b

)(convective boundary for

liquid water transfer), (3.3.13)

−Dv

dCv

dr= hmεv

(Cv,s

ε− ρv,b

)(convective boundary for water

vapor transfer), (3.3.14)

kdT

dr= h(Tb − T ) (convective boundary for heat transfer). (3.3.15)

Similar to the convective drying of mango tissues, the internal solid-surface watervapour 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 tissuesis generated from one accurate drying run of the convective drying of potato tissues witha 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) basedon the experimental data of moisture content and surface temperature during drying(Srikiatden and Roberts, 2008). It is then divided with the equilibrium activation energyrepresented in Equation (2.1.7) to yield the relative activation energy as mentioned inEquation (2.1.6). The relationship between the relative activation energy and averagemoisture content can be represented by a simple mathematical equation obtained bythe least-squares method using Microsoft Excel (Microsoft Inc., 2012). The relativeactivation energy can be represented as:

�Ev

�Ev,b= exp

[−0.364(X − Xb)0.876]. (3.3.16)

Similar to modelling of convective drying of mango tissues, for modelling usingthe S-REA, the average moisture content X in Equation (3.3.16) is substituted by thelocal moisture content (X) as the REA is then able represent the local evaporation orcondensation rate here instead of the global drying rate.

The effective liquid diffusivity (Dw) is shown in Equation (3.2.7) while the effectivevapour 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 average moisture content in the core of potato tissues (Xcore) is evaluated by:

Xcore =

Rcore∫0

X (r )rdr

Rcore∫0

rdr

. (3.3.17)

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

Xcortex3 =

Rsample out∫Rcortex in

X (r )rdr

Rcortex out∫Rcortex in

rdr

. (3.3.18)

The results of modelling average moisture content in core and cortex (hence the spa-tial distribution) are validated against the experimental data of Srikiatden and Roberts(2008). Similarly to the convective drying of mango tissues, the mass and heat balancesare shown in Equations (3.3.8)–(3.3.10) in conjunction with the initial and boundaryconditions indicated in Equations (3.3.11)–(3.3.15). The application of 10 and 200increments 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 tem-peratures of 45°, 55° and 65 °C. The original formulation of the L-REA is implementedin the partial differential equation set for transport in porous media to represent the localdrying or condensation rate. It is thus coupled with the system of equations of conserva-tion to describe the spatial profiles of moisture content, water vapour concentration andtemperature. It is noted that, if locally there is no vacant pore space which is connectedwith other pores or channels, the internal mass transfer area should be considered tobe zero; hence the REA term is zero if the pores or channels are fully hydrated. Inthis study, the internal mass transfer coefficient (hm,in: see Equation 3.3.12) is chosen to0.01 m s−1 as the sensitivity analysis indicates that hm,in is likely to be higher than0.01 m s−1, but any higher than this and it does not give any noticeable difference inthe profiles of moisture content and temperature predicted. More importantly, the valueof 0.01 m s−1 is also in the order of Dv/rp (Kar and Chen, 2010; 2011); thus it is afundamental value.

The good agreement between the predicted and experimental data of the averagemoisture content and the centre temperature is shown in Figures 3.3 and 3.4. It is alsosupported by R2 of moisture content higher than 0.996 and R2 of temperature higherthan 0.985 as listed in Table 3.2. The results of the S-REA modelling match well withthe experimental data. Benchmarks against the diffusion-based model (Vaquiro et al.,

0.50

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

S-REA 45°CData 45°CS-REA 55°CData 55°CS-REA 65°CData 65°C

1 1.5 2 2.5t(s)

3 3.5 4 4.5 5

× 104

0

1

2

3

4

5

6

7

8

9

10

Figure 3.3 Average moisture content profiles of mango tissues during convective drying atdifferent drying air temperatures. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao

Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibriummultiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright

(2012), with permission from John Wiley & Sons Inc.]

0

Tem

pera

ture

(K

)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


t(s) × 104

280

290

300

310

320

330

340

Figure 3.4 Centre temperature profiles of mango tissues during convective drying at differentdrying air temperatures. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen,

Spatial reaction engineering approach as an alternative for nonequilibrium multiphasemass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), with

permission from John Wiley & Sons Inc.]


Table 3.2 R2 and RMSE of convective drying of mango tissuesusing the S-REA.

Drying airtemperature(°C) R2 for X RMSE for X R2 for T RMSE for T

45 0.998 0.103 0.998 0.28555 0.999 0.079 0.985 1.00465 0.996 0.150 0.994 0.842

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0 0.002 0.004 0.006 0.008Half thickness (m)

0.01 0.012 0.014

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

8

7

6

5

4

3

2

1

0

9

10

Figure 3.5 Spatial moisture content profiles of mango tissues during convective drying at dryingair temperatures of 45 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao DongChen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase

mass-transfer model for drying of food and biological materials, 55–67, Copyright (2012), withpermission from John Wiley & Sons Inc.]

2009) indicate that the S-REA yields comparable, or even better, agreement towards theexperimental data.

Figure 3.5 shows the spatial profiles of the moisture content during convective dryingof mango tissues at a drying air temperature of 45 °C. The moisture content at theouter part of the samples is lower than that at the inner part, which indicates the effectof moisture removal. Initially, the gradient of moisture content inside the samples isrelatively high but this decreases as the drying progresses. At the end of drying, nonoticeable gradient of moisture content is observed, which indicates the equilibriummoisture content is nearly approached. If no liquid diffusion mechanism is used, theS-REA model would not be able to project this kind of liquid water profile (Kar andChen, 2010; 2011).


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Wat

er v

apou

r co

ncen

trat

ion

(kg/

m3 )

Half thickness (m)

0

0.002

0.004

0.006

0.008

0.01

0.012

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

Figure 3.6 Spatial water vapour concentration profiles of mango tissues during convective dryingat drying air temperatures of 45 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto,

Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibriummultiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright


The S-REA can generate the spatial profiles of water vapour concentration. Thespatial profiles of water vapour concentration during convective drying of mango tissuesat a drying air temperature of 45 °C are shown in Figure 3.6. The profiles of watervapour concentration are significantly affected by the local composition and structureof the samples being dried. Along drying, the concentration of water vapour achievesa maximum at a particular position inside the samples. This could be because, at thecore of samples, the moisture content is higher than that of the outer part which makesthe porosity of the core of samples lower. The lower porosity retards the evaporationrate at the sample core. At the outer part of the samples, the water extraction rate maybe enhanced because of higher porosity but this seems to be balanced by high diffusivewater vapour transfer as a result of higher porosity and temperature at the outer part ofthe samples. The S-REA seems to capture this physics well and can model the profilesof water vapour concentration well qualitatively.

The spatial profiles of temperature are presented in Figure 3.7. The temperature ofthe outer part of the samples is higher than that of the inner part because the sam-ples 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. How-ever, the gradient of temperature inside the samples is not large which may indicatethat the temperature inside the samples is essentially uniform. This is in agreement


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Tem

pera

ture

(K

)

Half thickness (m)

305

310

315

320

325

330

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

Figure 3.7 Spatial temperature profiles of mango tissues during convective drying at drying airtemperatures of 45 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen,



with the prediction of the Chen–Biot number (Ch–Bi) (Chen and Peng, 2005; Putrantoet 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 convectivedrying at a drying air temperature of 55 °C. As drying proceeds, the evaporation rateat the inner part is smaller than that of the outer part, which could be due to highmoisture content at the inner part of the sample. This means a lower porosity therethat retards the evaporation rate. The observation is also in agreement with the intuitiveexplanation of profiles of water vapour concentration during drying by Chen (2007). Asdrying 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, theevaporation rate decreases as the moisture content inside the samples is depleted. Atthe end of drying, essentially there is not much difference in evaporation rate inside thesamples because the moisture content has nearly achieved equilibrium under the dryingconditions.

Therefore, it can be said that the S-REA approach models the convective drying ofmango tissues well and the original REA is a simple alternative approach to representthe local evaporation and condensation rates. In addition, the S-REA has been easily


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Eva

pora

tion

rat

e (k

g w

ater

/(m

3 s))

Half thickness (m)

–0.2

0

0.2

0.4

0.6

0.8

1

1.2

t = 1000st = 5000st = 10000st = 15000st = 20000st = 30000st = 35000s

Figure 3.8 Profiles of evaporation rates inside mango tissues during convective drying at a dryingair temperature of 55 °C. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen,



operated to yield the profiles of water vapour concentration, local evaporation rate andlocal 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 convectivedrying of potato tissues. The relative activation energy is generated from one accuratedrying run which is the convective drying at air temperature of 70 °C. It is representedin Equation (3.3.16). Similar to the convective drying of mango tissues, the internalmass transfer coefficient (hm,in: on Equation 3.1.17) is chosen to be 0.01 m s−1 asthe sensitivity analysis indicates that hm,in of higher than 0.01 m s−1does not give anynoticeable differences in the profiles of moisture content and temperature. Nicely, thisis 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 sensitivityanalysis and it is found that Dwo of 6.5 × 10−6 m2 s−1 gives the best agreement againstthe experimental data. It is emphasised that the temperature dependence function for theliquid diffusivity in this case was obtained in isothermal drying experiments speciallydesigned by Srikiatden and Roberts (2006), in contrast to many published studies that


0 0.5 1 1.5 2 2.5

Moi

stur

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nten

t (kg

wat

er/k

g dr

y so

lids

)

t(s) × 104

0

1

2

3

4

5

6

S-REA-coreData-coreS-REA-cortexData-cortex

Figure 3.9 Moisture content profiles in the core and cortex during convective drying of potatotissues with a diameter of 1.4 cm. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao



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 cylindricaltissue with the diameter of 1.4 cm during the convective drying. It can be shown that theresults of modelling match well with the experimental data with a correlation coefficientR2 of 0.98. Benchmarks against modelling implemented by Srikiatden and Roberts(2008) with the liquid diffusivity concept indicate that the S-REA yields comparableresults.

In addition, the profiles of moisture content for each part of samples with the diameterof 2.8 cm are shown in Figure 3.10. Again, a good agreement towards the experimentaldata is observed (R2 of 0.992). Indeed, the S-REA describes the moisture content pro-files accurately during convective drying of potato tissues with a diameter of 2.8 cm.Benchmarks towards modelling implemented by Srikiatden and Roberts (2008) indicatethat the REA yields comparable or even better results. It can be said that the S-REA canbe used to model the profiles of moisture content very well.

Figure 3.11 indicates the core temperature during convective drying of potato tissueswith the diameter of 1.4 cm. The predictions of temperature using the S-REA match

0

1

2

3

4

5

6

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0 0.5 1 1.5 2 2.5 3 3.5

t(s) × 104

S-REA-coreS-REA-cortex 1S-REA-cortex 2S-REA-cortex 3Data-coreData-cortex 1Data-cortex 2Data-cortex 3

Figure 3.10 Moisture content profiles in the core and cortex during convective drying of potatotissues with a diameter of 2.8 cm. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao



295

305

300

310

315

320

325

335

330

340

345

Cor

e te

mpe

ratu

re (

K)

0 0.5 1 1.5 2 2.5t(s) × 104

Data

S-REA

Figure 3.11 Core temperature profiles during convective drying of potato tissues with a diameterof 1.4 cm. [Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial

reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfermodel for drying of food and biological materials, 55–67, Copyright (2012), with permission

from John Wiley & Sons Inc.]


Table 3.3 Scheme of intermittent drying of mango tissues (Vaquiroet al., 2009).

Drying airtemperature(ºC)

Period of firstheating (s)

Period of resting(at 27 °C ± 1.6)(s)

Period ofsecondheating (s)

45 16 200 10 800 36 36055 9480 10 800 33 720

well with the experimental data with R2 of 0.99. Benchmarks against modelling imple-mented by Srikiatden and Roberts (2008) indicate that the S-REA yields comparableresults. Overall, S-REA seems to be a sound approach to modelling the details of spatialdistributions of temperature, liquid water and water vapour concentration in the materialbeing dried.

3.4 The S-REA for intermittent drying

The accuracy of the S-REA in modelling intermittent drying is validated by the exper-imental data of intermittent drying of mango tissues (Vaquiro et al., 2009). For betterunderstanding of the modelling approach, the experimental details are summarised andreviewed in this section. The samples of mango tissues were cubes with initial sidelengths 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. Thedrying air temperature and air velocity were controlled at preset values by PID controlalgorithms while air humidity was maintained constant during drying. The experimentalsetting for intermittent drying is shown in Table 3.3. During the resting period, the sam-ples stayed in an environment with an ambient temperature of 27 ± 1.6 °C and relativehumidity of 60% (Vaquiro et al., 2009). Determination of density, thermal conductivity,heat capacity, equilibrium moisture content and shrinkage of samples being dried hasbeen 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 weredried from three directions (x, y and z directions) so three-dimensional modelling of theS-REA for intermittent drying of mango tissues needs to be set up, which is representednext. The mass and heat balances of intermittent drying of mango tissues are similar tothose of convective drying described in Section 3.3.1.

Similarly to the convective drying of mango tissues, the mass balance of water inliquid 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).


Table 3.4 R2 and RMSE of intermittent drying of mango tissues.

R2 for X RMSE for X R2 for T RMSE for T

45 °C 0.964 0.358 0.992 0.70555 °C 0.999 0.0774 0.994 0.874

The internal-surface water vapour concentration (Cv,s) and internal evaporation rateare 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 theprocedures explained in Section 3.2.

The relative activation energy implemented for modelling of convective drying ofmango tissues shown in Equation (3.3.4) is used here to model the intermittent dryingof mango tissues. For modelling the intermittent drying of these tissues, the equilib-rium activation (�Ev,b) energy shown in Equation (2.1.7) is evaluated according to thecorresponding drying air temperature and humidity in each drying period. It is alsocombined with the relative activation energy shown in Equation (3.3.4) to yield the localdrying/condensation rate. In addition, the heat balance implements the correspondingdrying air temperature in each drying period by using the corresponding drying airtemperature 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 mangotissues, 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 tissueswhose conditions are listed in Table 3.3. As mentioned before, for modelling of theintermittent drying, the equilibrium activation energy needs to be evaluated accordingto the corresponding drying settings in each drying period. Similarly, the heat balanceimplements the corresponding drying air temperature in each drying period. SolvingEquations (3.3.9)–(3.3.11) in conjunction with the initial and boundary conditions shownin 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 con-firmed by R2 and RMSE listed in Table 3.4. The results of modelling match well with theexperimental data of moisture content. The S-REA models the average moisture contentduring 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 theREA yields better results as Vaquiro et al. (2009) showed a slight underestimation indrying rate of intermittent drying at a drying air temperature at 45 °C during drying


0 1 2 3 4 5 6 7

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

t(s) × 104

0

1

2

3

4

5

6

7

8

9

10


Figure 3.12 Average moisture content profiles of mango tissues during intermittent drying atdifferent drying air temperatures.

times between 20 000–50 000 s. Similarly, Vaquiro et al. (2009) also revealed a slightoverestimation in drying rate of intermittent drying at a drying air temperature of 65 °Cafter a drying time of 20 000 s. The slight underestimation and overestimation of thedrying rate are not shown by the modelling using the S-REA. This indicates that theS-REA can be used to describe the moisture content of intermittent drying of mangotissues well and the REA can also be applied to model the local evaporation and con-densation rate well.

The spatial profiles of moisture content during the intermittent drying at a drying airtemperature of 55 °C are indicated in Figure 3.13. The moisture content of the outerpart of the samples is lower than that in the inner part of the samples, which indicatesthat moisture migrates outwards during intermittent drying. Initially, the gradient ofmoisture content inside the samples is relatively high, but during a drying time between9480 and 20 280 s the gradient is relatively low since the samples are in the restingperiod. This period seems to allow the moisture to redistribute inside the samples andlow gradients of moisture content inside the samples are generated. Towards the endof drying, although the samples are in the heating period again, relatively uniformmoisture content is observed, which indicates that the equilibrium condition is almostapproached.

Figure 3.14 presents the spatial profiles of water vapour concentration during theintermittent drying at a drying air temperature of 55 °C. The profiles of water vapourconcentration are significantly affected by the local composition and structure of the


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

Half thickness (m)

0

1

2

3

4

5

6

7

8

9

10

t = 1000st = 3000st = 5000st = 7000st = 14000st = 20000st = 30000st = 40000st = 50000s

Figure 3.13 Spatial moisture content profiles of mango tissues during intermittent drying at adrying air temperature of 55 °C.

0.012

0.01

0.008

0.006

Wat

er v

apou

r co

ncen

trat

ion

(kg/

m3 )

0.004

0.002

00 0.002 0.004 0.006

Half thickness (m)

0.008 0.01 0.012 0.014


Figure 3.14 Spatial water vapour concentration profiles of mango tissues during intermittentdrying at a drying air temperature of 55 °C.


0 1 2 3 4 5 6 7

Cen

tre

tem

pera

ture

(K

)

t(s) × 104

280

290

300

310

320

330

340


Figure 3.15 Centre temperature profiles of mango tissues during intermittent drying at differentdrying air temperatures.

samples being dried. The water vapour concentration achieves a maximum at particularposition inside the samples. At core of the samples, the moisture content is relativelyhigh, which may result in lower porosity and retard the local evaporation rate. At outerpart of the samples, the local evaporation rate may be enhanced as a result of the higherporosity, but this seems to be balanced by higher diffusive flux of water vapour due tothe higher porosity and temperature. During first heating period, the gradient of watervapour concentration is relatively high but this decreases as drying progresses. Therelatively uniform concentration of water vapour is shown during the resting periodwhich 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 therelatively uniform moisture content at the end of drying as explained previously. Thespatial profiles of intermittent drying are similar to those of convective drying (Putrantoand 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 aparticular position inside the samples.

The profiles of temperature during the intermittent drying are indicated in Figures 3.15and 3.16. Figure 3.15 shows the profiles of the centre temperature during the intermittentdrying of mango tissues at drying air temperatures of 45°, 55° and 65 °C. A good agree-ment between the predicted and experimental data is observed which is also supportedby R2 of higher than 0.992 and RMSE lower than 0.828. The results of modelling matchwell with the experimental data. Benchmarks against modelling implemented by Vaquiro


330

325

320

315

310

Tem

pera

ture

(K

)

305

300

2950 0.002 0.004 0.006

Half thickness (m)

0.008 0.01 0.012 0.014


Figure 3.16 Spatial temperature profiles of mango tissues during intermittent drying at a dryingair 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 beginningof the first heating period. However, these are not observed by the modelling using theS-REA. This indicates that the S-REA is indeed accurate enough to model the tempera-ture profiles of intermittent drying of mango tissues.

Figure 3.16 represents the spatial profiles of temperature during the intermittent dryingat a drying air temperature of 55 °C. During the first heating period, the temperature inthe outer part of the samples is higher than that of the inner part because the samplesreceive heat by convection from the drying air used for water evaporation and thispenetrates inwards by conduction. However, the gradient of temperature inside thesamples 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 indicatedthat the temperature inside the samples is essentially uniform. During the first heatingperiod, the temperature of samples increases but the temperature decreases betweendrying times of 9480–20 280 s as a result of resting period. This is followed by a furtherincrease of temperature during the second heating period. At the end of the intermittentdrying, the temperature approaches the drying air temperature.

The S-REA has its advantages, resulting in the spatial profiles of local evaporationrate during drying as the REA is used to predict the local evaporation rate. The profiles oflocal evaporation rates during intermittent drying of Scheme 1 at a drying air temperatureof 55 °C are shown in Figure 3.17. The local evaporation rate at core of samples is lower


1.2

1

0.8

0.6

0.4

Eva

pora

tion

rat

e (k

g w

ater

/(m

3 s))

0.2

0

–0.20 0.002 0.004 0.006

Half thickness (m)0.008 0.01 0.012 0.014

t = 1000st = 3000st = 5000st = 9000st = 14000st = 16000st = 20000st = 22000st = 35000st = 40000st = 45000st = 50000st = 54200s

Figure 3.17 Profiles of evaporation rate inside mango tissues during intermittent drying at adrying air temperature of 55 °C.

than that of the outer part of samples. This could be because of lower porosity at the coreof the samples as a result of higher moisture content. During the resting period, the spatialprofiles 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 moisturecontent decreases as drying progresses, which increases the porosity. From the start ofdrying to a drying time of around 5000 s the local evaporation rate increases, whichcould be because of the increase in temperature. After this period, the local evaporationrate 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 bebecause of the relatively low temperature. During the second heating period, the localevaporation decreases. This may be because the moisture content inside the sample isdepleted. At the end of drying, essentially there is not much difference in evaporationrate inside the samples because the moisture content has nearly achieved equilibriumunder the drying conditions.

It has been demonstrated that the S-REA is very accurate in modelling the intermittentdrying of mango tissues. The S-REA can also project the concentration of water vapourand local evaporation rate during intermittent drying so that better understanding oftransport phenomena of drying processes can be gained. It is argued here that theS-REA is not only robust enough to model convective drying (Putranto and Chen, 2013)but also intermittent drying.


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

CaseFinal gastemperature (ºC)

Heating rate(ºC h−1)


1 220 10 0.112 220 20 0.125

3.5 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 constantheating rate is benchmarked by the experimental data of Younsi et al. (2007). Theexperimental 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 inKocaefe et al. (1990). Wood samples with dimensions of 0.035 × 0.035 × 0.2 m wereheat treated by suspending the samples on a balance with accuracy of 0.001 g. The heattreatment was conducted by exposing the samples to hot gas whose temperature waslinearly increased according to the heating rate. The humidity of the gas was controlledby injection of steam into the second furnace placed under the main furnace (Younsiet al., 2006a, 2007).

The samples were first heated to 120 °C and held at this temperature for half an hourfollowed by heating under the preset heating rate (refer to Table 3.5) until the finaltemperature (also refer to Table 3.5) was achieved. During the heat treatment, the weightof the samples was recorded by the balance. In addition, the temperatures were measuredby a T-type thermocouple placed inside the samples. The measurements indicated thatthe temperatures inside the samples were essentially uniform, perhaps due to the smallsize 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 interesthere, two-dimensional modelling with respect to x and y directions can be set upas the height of the sample (0.2 m) is much greater than the length (0.035 m) andwidth (0.035 m) of the sample. The mass balance of water in the liquid phase iswritten as (Chen, 2007; Chong and Chen, 1999; Putranto and Chen, 2013; Zhang andDatta, 2004):

∂(Cs X )

∂t= ∂

∂x

[Dw

∂(Cs X )

∂x

]+ ∂

∂y

[Dw

∂(Cs X )

∂y

]− I , (3.5.1)

where X is the concentration of liquid water (kg H2O kg dry solids−1) and Cs is thesolids’ concentration (kg dry solids m−3) which can change if the structure is shrinking,I is the evaporation or wetting rate (kg H2O m−3 s−1), I is >0 when evaporation occurs


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):

∂Cv

∂t= ∂

∂x

(Dv

∂Cv

∂x

)+ ∂

∂y

(Dv

∂Cv

∂y

)+ I , (3.5.2)

where Cv is the water vapour concentration (kg H2O m−3).The heat balance is represented as (Chen, 2007; Chong and Chen, 1999; Putranto and

Chen, 2013; Zhang and Datta, 2004):

ρCp∂T

∂t= ∂

∂x

(k∂T

∂x

)+ ∂

∂y

(k∂T

∂y

)− I�Hv, (3.5.3)

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 = Xo, Cv = Cvo, T = To (initial condition, uniform initial concentrations

and temperature), (3.5.4)

x = 0, y= 0,d X

dx= 0,

dCv

dx= 0,

dT

dx= 0 (symmetry boundary), (3.5.5)

x = L , y = L ,−Cs Dw

d X

dx= hmεw

(Cv,s

ε− ρv,b


for liquid water transfer),

(3.5.6)

−Dv

dCv

dx= hmεv

(Cv,s

ε− ρv,b


water vapor transfer), (3.5.7)

kdT

dx= h(Tb − T ) (convective boundary for heat transfer). (3.5.8)

Because the samples were dried uniformly from all directions and the lengths and widthswere the same, the mass balance of water in liquid phase can be simplified into (Incroperaand DeWitt, 2002; Van der Sman, 2003):

∂(Cs X )

∂t= 2

∂

∂x

(Dw

∂(Cs X )

∂x

)− I , (3.5.9)

while the mass balance of water in vapour phase can be expressed as:

∂Cv

∂t= 2

∂

∂x

(Dv

∂Cv

∂x

)+ I . (3.5.10)

In addition, the heat balance can be represented as:

ρCp∂T

∂t= 2

∂

∂x

(k∂T

∂x

)− I�HV . (3.5.11)


Similarly to the convective and intermittent drying described in Sections 3.3 and3.4, the internal evaporation rate (I), effective vapour diffusivity, tortuosity, solidsconcentration and porosity are evaluated using Equations 3.1.19, 3.2.1, 3.2.3, 3.2.4and 3.2.5, respectively.

The relative activation energy of heat treatment of wood is generated from the dryingrun in Case 2 (refer to Table 3.5) (Younsi et al., 2007). The activation energy duringdrying is evaluated using Equation (2.1.5) and divided with the equilibrium activationenergy represented in Equation (2.1.7) to yield the relative activation energy as men-tioned in Equation (2.1.6). The relationship between the relative activation energy andaverage moisture content can be represented by simple mathematical equation obtainedby the least-square method using Microsoft Excel (Microsoft Corp, 2012). The relativeactivation energy can be represented as:

�Ev

�Ev,b= [1 − 1.517(X − Xb)0.22

]exp[−3.717(X − Xb)3.135

]. (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 substitutedby 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 increasedgas temperature, the equilibrium activation energy shown in Equation (2.1.7) implementsthe corresponding gas temperature and humidity during heat treatment. In addition, thelinearly increased gas temperature is used in Equation (3.5.8).

In order to yield the spatial profiles of moisture content, water vapour concentrationand temperature in the heat treatment of wood, the mass and heat balances shown inEquations (3.5.9) to (3.5.11), in conjunction with the initial and boundary conditionsrepresented in Equations (3.5.4) to (3.5.8) and the relative activation energy shown byEquation (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 ordinarydifferential 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 profilesobserved, as shown in Figure 3.18. No shrinkage is incorporated in the modelling, asYounsi et al. (2006b) indicated that the ratio between final and initial dimension isaround 0.96. Similarly, the modelling implemented by Younsi et al. (2006a,b, 2007) didnot incorporate the shrinkage effect.

The average moisture content of wood during heat treatment is evaluated by:

X =

L∫0

X (x)dx

L∫0

dx

. (3.5.13)

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


0.14

0.12

0.1

0.08

0.06

0.04

0.02

00 0.5 1 1.5

t(s)

Ave

rage

moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

2 2.5

×104

3

S-REA 100 incrementsS-REA 10 incrementsData

Figure 3.18 Profiles of average moisture content during heat treatment in Case 2 (refer to Table3.5) 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 andthe results of modelling are presented in Figures 3.19 to 3.27. The REA is implemented tomodel the local evaporation or condensation term and coupled with a system of equationsin order to yield the spatial profiles of moisture content, water vapour concentration andtemperature. It is noted that if locally there is no vacant pore space which is connectedto other pores or channels, the internal mass transfer area is zero, hence the REA termis 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 doesnot yield any noticeable differences in the profiles of moisture content and temperatureprofiles.

As mentioned before, no method has been presented anywhere in the literature tomeasure pore liquid diffusivity, and liquid diffusivity is obtained by numerical sen-sitivity to match the prediction with the experimental data of moisture content andtemperature. Interestingly, in this study, for all cases the profiles of moisture content andtemperature are independent of the liquid diffusivity value. Further explanation aboutthis phenomenon is presented next.

For Case 1 (refer to Table 3.5), the results of modelling are presented in Figures 3.19to 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 ofmoisture content and temperature as shown in Figures 3.19 and 3.20. This may indicate


0.12

Dw = 1e-8 m2/sDw = 1e-12 m2/sDw = 1e-18 m2/sDw = 1e-24 m2/sDw = 1e-30 m2/sData

0.1

0.08

Ave

rage

moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0.06

0.04

0.02

00 1 2 3 4

t(s) × 104

5 6 7

Figure 3.19 Effect of liquid diffusivity on profiles of the moisture content during heat treatment inCase 1 (refer to Table 3.5).

500

450

400

350

300

2500 1 2 3 4

t(s)5 6

× 104

7

Ave

rage

moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

Dw = 1e-8 m2/sDw = 1e-12 m2/sDw = 1e-18 m2/sDw = 1e-24 m2/sDw = 1e-30 m2/sData

Figure 3.20 Effect of liquid diffusivity on profiles of temperature during heat treatment in Case 1(refer to Table 3.5).


100

Ave

rage

moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)0.12

0.1

0.08

0.06

0.04

0.02

0.14

2 3 4 5t(s)

6 7

× 104

ModelData

Figure 3.21 Profiles of average moisture content during heat treatment in Case 1 (refer toTable 3.5).

10250

Tem

pera

ture

(K

)

450

350

400

300

500

2 3 4 5t(s)

6 7

× 104

ModelData

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


0.500

Ave

rage

moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)0.12

0.1

0.08

0.06

0.04

0.02

0.14

1 1.5 2 2.5t(s)

3

× 104

ModelData

Figure 3.23 Profiles of average moisture content during heat treatment in Case 2 (refer toTable 3.5).

that the modelling of heat treatment in Case 1 (refer to Table 3.5) is independent ofthe liquid diffusivity so this can be neglected and the modelling will only involve thevapour diffusion and evaporation/condensation. The phenomenon could be due to theinitial moisture content being relatively low and the heating of wood implemented at arelatively high temperature.

By ignoring the liquid diffusion term on the mass balance of liquid water (referto Equation 3.5.9), a good agreement between the predicted and experimental data ofmoisture content and temperature is shown in Figures 3.23 and 3.24 as well as confirmedby R2 and RMSE indicated in Table 3.6. Benchmarks against modelling implementedby 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 1well (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 theliquid diffusivity in the range of 1 × 10−8 to 1 × 10−30 m2 s−1 have been used and thereis no noticeable effect on the profiles of moisture content and temperature. By ignoringthe 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 contentand temperature is shown in Figures 3.23 and 3.24 and confirmed by R2 of 0.995 and0.997 for moisture content and temperature, respectively and RMSE of lower than 0.005


Table 3.6 R2 and RMSE of modelling of heat treatment of wood under aconstant heating rate using the S-REA.

Case R2 for X R2 for T RMSE for X RMSE for T

1 0.988 0.992 0.004 4.7652 0.995 0.997 0.003 3.287

0.50250

Tem

pera

ture

(K

)

450

350

400

300

500

1 1.5 2 2.5t(s)

3 3.5× 104

ModelData

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 theprofiles of moisture content and temperature very well. Benchmarks against modellingimplemented by Younsi et al. (2007) reveal that the S-REA yields better agreement withthe experimental data from moisture content.

The spatial profiles of moisture content, water vapour concentration and temperatureare presented in Figures 3.25–3.27. The distribution of moisture content is not determinedby liquid diffusivity. This may suggest that liquid diffusivity can be neglected, so thedrying process is not governed by liquid diffusion as the initial moisture content isrelatively low and temperature is relatively high. Figure 3.25 indicates that the moisturecontent of the inner part of the samples is higher than that of the outer part, whichindicates the moisture migrates outwards during drying.

Similarly, as shown in Figure 3.26, the water vapour concentration of the inner part ofthe 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

0.04

0.02Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

00

Axial position (m)

0.0160.0140.0120.010.0080.0060.0040.002 0.018

t = 1000st = 3000st = 5000st = 10000st = 15000st = 20000st = 28000s

Figure 3.25 Profiles of spatial moisture content during heat treatment in Case 2 (refer toTable 3.5).

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

Wat

er v

apou

r co

ncen

trat

ion

(kg/

m3 )

00

Axial position (m)

0.0160.0140.0120.010.0080.0060.0040.002 0.018

t = 1000st = 3000st = 5000st = 10000st = 15000st = 20000st = 28 000s

Figure 3.26 Profiles of spatial water vapour concentration during heat treatment in Case 2 (referto Table 3.5).


460

440

420

400

380

360

340

320

300

Tem

pera

ture

(K

)

2800

Axial position (m)0.0160.0140.0120.010.0080.0060.0040.002 0.018

t = 1000st = 3000st = 5000st = 10000st = 15000st = 20000st = 28000s

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 thegas. The water vapour concentration increases until a heating time of 15 500 s, followedby a decrease until the end of drying. The initial increase could be because the initialmoisture content is still relatively high at the beginning of drying but as the heating pro-gresses, moisture content decreases and lower water vapour is generated. As shown inFigure 3.27, the temperature distribution of the samples is essentially uniform, in agree-ment with the observations of Younsi et al. (2007).

It can be observed that the S-REA can model the heat treatment of wood underconstant heating rate very well for all cases investigated. The liquid diffusion term canbe neglected so that the model for heat treatment of wood under a constant heating ratemay be simplified as follows:

∂(Cs X )

∂t= − I , (3.5.14)

∂Cv

∂t= ∂

∂x

(Dv

∂Cv

∂x

)+ ∂

∂y

(Dv

∂Cv

dy

)+ I , (3.5.15)

ρC p∂T

∂t= ∂

∂x

(k∂T

∂x

)+ ∂

∂y

(k∂T

∂y

)− I�HV . (3.5.16)

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


the process. It can be said that the S-REA is an effective multiphase approach tomodelling 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). Forbetter understanding of the modelling implemented, the experimental details of Banooniet al. (2008a,b) are reviewed briefly here. Baking experiments were carried out in alaboratory oven with active belt length and width of 1 m and 0.72 m, respectively. Theoven was also equipped with an electrical heater and temperature control. The dough wasmixed, divided into pieces of 250 g and kept for 15 min and then shaped and punchedto produce flat bread with an initial thickness of 0.2 cm (Banooni et al., 2008a). Thesamples of bread were put in the belt of oven in which heated air was pushed at highvelocity onto their surfaces. The air was propelled by a centrifugal fan and directed ontothe 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-100probes 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 massand heat transfer using the REA to describe the local evaporation/condensationrate.

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):

∂(Cs X )

∂t= ∂

∂x

[Dw

∂(Cs X )

∂x

]− I , (3.6.1)

where Dw is the effective liquid water diffusivity (m2 s−1), X is the concentration ofliquid water (kg H2O 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 (kgH2O 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

(Dv

∂Cv

∂x

)+ I , (3.6.2)

where Dv is the effective vapour diffusivity (m2 s−1) and Cv is the concentration of liquidwater (kg H2O kg dry solids−1).


The heat balance is represented by the following equation (Chen, 2007; Chong andChen, 1999; Putranto and Chen, 2013; Zhang and Datta, 2004):

ρC p∂T

∂t= ∂

∂x

(k∂T

∂x

)− I�HV , (3.6.3)

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 = Xo, Cv = Cvo, T = To, (3.6.4)

x = 0,d X

dx= 0,

dCv

dx= 0, −k

dT

dx= U (Tb − T ), (3.6.5)

x = L , −Cs Dw

d X

dx= hmεw

(Cv,s

ε− ρv,b

), (3.6.6)

−Dv

dCv

dx= hmεv

(Cv,s

ε− ρv,b

), (3.6.7)

kdT

dx= h(Tb − T ), (3.6.8)

where h is the top heat transfer coefficient (W m−2 K−1) and U is the overall bottomheat transfer coefficient (W m−2 K−1).

Similarly to convective and intermittent drying, as well as wood heating under constantheating rates (described in Sections 3.3, 3.4 and 3.5, respectively), the internal evapo-ration rate, effective vapour diffusivity, tortuosity, solid concentration and porosity areevaluated using Equations (3.1.19), (3.2.1), (3.2.3), (3.2.4) and (3.2.5), respectively. Inaddition, the internal mass transfer coefficient (hm,in) described in Section 3.2 is usedhere.

The relative activation energy of baking of bread is generated from one accuratebaking run; i.e. baking bread at a baking temperature of 150 °C and air velocity of10 m s−1 (Banooni et al., 2008a). The activation energy during drying is evaluatedusing Equation (2.1.5) and divided by the equilibrium activation energy represented inEquation (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 canbe represented by a simple mathematical equation obtained by the least-square methodusing Microsoft Excel (Microsoft Corp, 2012). The relative activation energy can berepresented as:

�Ev

�Ev,b= [1 − 7.424(X − Xb)4.471] exp[23.884(X − Xb)42.282]. (3.6.9)

The good agreement between the fitted and experimental relative activation energyis shown by R2 of 0.995. For modelling using the S-REA here, the relative activa-tion energy shown in Equation (3.6.9) is used but the average moisture content X inEquation (3.6.9) is substituted for the local moisture content (X) as the REA is used


to represent the local evaporation rate instead of the overall drying rate of the wholesample.

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 concentrationand temperature of the convective of mango tissues, the mass and heat balances shownin Equations (3.6.1)–(3.6.3), in conjunction with the initial and boundary conditionsrepresented in Equations (3.6.4)–(3.6.8) and the relative activation energy shown byEquation (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 ofordinary differential equations with respect to time by firstly discretising the spatialderivatives. The ordinary differential equations are then solved simultaneously by ode23sin Matlab (Mathworks Inc., 2012). The spatial derivative here is discretised into 10increments; application of 100 increments has been conducted and there is no noticeabledifference in the profiles observed. The shrinkage during the baking process can berepresented as:

V

V0= 162.69 X

2 − 207.61 X + 66.925 (for ≥ X 0.57), (3.6.11)

V

V0= 1.307 X + 1.015 (for < X 0.57), (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:

X =

L(t)∫0

X (x)dx

L(t)∫0

dx

. (3.6.13)

The profiles of average moisture content and centre temperature are then validatedagainst 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 partialdifferential equation set for transport in porous media, to represent the local evap-oration or condensation rate. It is thus coupled with the equations of conservationto describe the spatial profiles of moisture content, water vapour concentration andtemperature.

The results of modelling the baking of bread using the S-REA are shown in Figures3.28–3.32. Figure 3.28 presents the profiles of average moisture content during bakingat a baking temperature of 150 °C. The S-REA models the average moisture content well


0.65

0.6

0.55

0.5

0.45

0.4

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0t(s)

1000500 1500

Data v = 1 m/sS-REA v = 1 m/sData v = 5 m/sS-REA v = 5 m/sData v = 10 m/sS-REA v = 10 m/s

Figure 3.28 Profiles of average moisture content during the baking of bread at a bakingtemperature of 150 °C.

at various air velocities and a baking temperature of 150 °C as shown in Figure 3.28(R2 higher than 0.985).

The spatial profiles of moisture content during the baking of bread at a baking tem-perature of 150 °C and velocity of 10 m s−1 are shown in Figure 3.29. The moisturecontent at the top and bottom surfaces of the bread are lower than at the core. This mayindicate that the moisture migrates outwards during baking since the surface tempera-ture is higher than the core. The maximum moisture content is located at a particularposition inside the samples. It is not achieved at the centre of the sample, since the topand bottom surface temperatures are not similar. The top surface temperature is higherthan the bottom surface temperature which may result in a higher moisture content atthe bottom part of the samples. As baking progresses, the moisture content decreasesand should approach equilibrium at the end of baking.

Figure 3.30 shows the spatial profiles of water vapour concentration during the bakingof bread at a baking temperature of 150 °C and velocity of 10 m s−1. Initially, theconcentration 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 bottompart of the samples, the water vapour concentration is higher than that of the top part ofthe samples. It seems that the maximum concentration of water vapour is located at thebottom surface of the samples. These could be because no mass transfer occurs on thebottom surfaces since the samples are placed on top of trays, while at the top surfaces,water vapour is transferred to the baking air.


0.7

0.65

0.6

0.55

0.5

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0.45

0.4

0.350 0.005 0.01 0.015

Axial position (m)0.02 0.025 0.03 0.035

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

Figure 3.29 Spatial profiles of moisture content during the baking of bread at a bakingtemperature of 150 °C and air velocity of 10 m s−1.

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

Wat

er v

apou

r co

ncen

trat

ion

(kg/

m3 )

00 0.005 0.01 0.015

Axial position (m)

0.02 0.025 0.03 0.035


Figure 3.30 Spatial profiles of concentration of water vapour during the baking of bread at abaking temperature of 150 °C and air velocity of 10 m s−1.


Top surface-dataTop surface-modelBottom surface-dataBottom surface-model

0300

310

320

330

340

350

360

370

380

390

400

500 1000

t(s)

1500

Tem

pera

ture

(K

)

Figure 3.31 Profiles of top and bottom surface temperatures during the baking of bread at abaking temperature of 150 °C and air velocity of 1 m s−1.

0.0050300

320

340

360

380

400

420

0.01 0.015 0.02 0.025 0.03 0.035

Axial position (m)

Tem

pera

ture

(K

)


Figure 3.32 Spatial profiles of temperature during the baking of bread at a baking temperature of150 °C and air velocity of 10 m s−1.


Figure 3.31 shows the results of modelling the top and bottom surface temperaturesof 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 (R2 of 0.965 and 0.961 fortop and bottom surface temperature, respectively). The S-REA models the top surfacetemperature well and shows a slight underestimation of the bottom surface temperatureduring baking times of 400–800 s.

Figure 3.32 shows the spatial profiles of temperature during baking at a bakingtemperature of 150 °C and velocity of 10 m s−1. The minimum temperature is locatedat a particular position inside the samples but it is not at their centre. In addition, thetemperature of the top surface is higher than that at the bottom surface. This is inagreement with the results of temperature measurement which indicate that the bottomsurface temperature is higher than the top surface. Since the heat received by the samplesfrom the upper and lower side is not equal, this may result in the minimum temperaturenot being located at the centre of the samples. The profiles of temperature are also inagreement with those of moisture content explained previously in which the maximummoisture 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 bakingof bread in the future.

3.7 Summary

In this chapter, it has been shown that the S-REA is an excellent non-equilibriummultiphase modelling approach to simulate several challenging cases of drying. TheREA parameters, used in L-REA to describe the global drying rate, are implemented inS-REA for modelling local evaporation–condensation rate as affected by local variablesand structures of the same material. As mentioned before, the REA parameters can begenerated from one accurate drying run on the materials of concern and in a narrow rangeof relevant drying conditions. In S-REA, the REA is coupled with a set of equations ofconservation of heat and mass transfer to yield a spatial model.

The S-REA has been used to model the convective drying, intermittent drying, heattreatment under linearly increased gas temperatures and baking. The results of mod-elling match the experimental data well . For modelling intermittent drying and heattreatment under linearly increased gas temperatures, the equilibrium activation energy isevaluated according to the corresponding humidity and temperature in each periodof treatment. Without any modification, the S-REA can model the baking processaccurately.

The S-REA can also yield the spatial profiles of concentration of water vapour andevaporation/condensation, useful for better understanding of drying process. To the bestof our knowledge so far, the S-REA is the first model proposed and used to describe thelocal evaporation–condensation rate explicitly.


References

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Madiouli, J., Lecomte, D., Nganya, T., Chavez, S., Sghaier, J. and Sammouda, H., 2007. A methodfor determination of porosity change from shrinkage curves of deformable Materials. DryingTechnology 25, 621–628.

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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 ChemistryResearch 50, 1089–1098.

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Putranto, A., Chen, X.D., Xiao, Z. and Webley, P.A., 2011d. Modelling of high-temperaturetreatment of wood by using the reaction engineering approach (REA). Bioresource Technology102, 6214–6220.

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Srikiatden, J. and Roberts, J.S., 2006. Measuring moisture diffusivity of potato and carrot (coreand cortex) during convective hot air and isothermal drying. Journal of Food Engineering 74,143–152.

Srikiatden, J. and Roberts, J.S., 2008. Predicting moisture profiles in potato and carrot duringconvective hot air drying using isothermally measured effective diffusivity. Journal of FoodEngineering 84, 516–525

Thuwapanichayanan, R., Prachayawarakorn, S. and Soponronnarit, S., 2008. Modelling of diffu-sion with shrinkage and quality investigation of banana foam mat drying. Drying Technology26, 1326–1333.

Van der Sman, R.G.M., 2003. Simple model for estimating heat and mass transfer in regular-shapedfoods. Journal of Food Engineering 60, 383–390.

Van der Sman, R.G.M., 2007a. Moisture transport during cooking of meat: An analysis based onFlory–Rehner theory. Meat Science 76, 730–738.

Van der Sman, R.G.M., 2007b. Soft condensed matter on moisture transport in cooking of meat.AIChE Journal 53, 2986–2995.

Van der Sman, R.G.M., Jin, X. and Meinder, M.B.J., 2012. A paradigm shift of drying of foodmaterials via free volume concepts. Proceedings of the 18th International Drying Symposium(IDS 2012), Xiamen, China (11–15 September 2012).


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Younsi, R., Kocaefe, D., Poncsak, S. and Kocaefe, Y., 2006a.Transient multiphase model for thehigh-temperature thermal treatment of wood. AIChE Journal 52, 2340–2349.

Younsi, R., Kocaefe, D., Poncsak, S. and Kocaefe, Y., 2006b. Thermal modelling of the hightemperature treatment of wood based on Luikov’s approach. International Journal of EnergyResearch 30, 699–711.



Zhang, J. and Datta, A.K., 2004. Some considerations in modelling of moisture transport in heatingof hygroscopic materials. Drying Technology 22, 1983–2008.

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Zhang, J., Datta, A.K. and Mukherjee, S., 2005. Transport processes and large deformation duringbaking of bread. AIChE Journal 51, 2569–2580.

4 Comparisons of the REA withFickian-type drying theories, Luikov’sand Whitaker’s approaches

4.1 Model formulation

Fick’s law is one of the most well-known concepts of mass transfer in the literature. Itsapplications in many different physical circumstances are substantial. In his book ondiffusion, Cussler vividly introduced the story of how Fick came up with the law namedafter him (Cussler, 1984). Through an ‘impressive combination of qualitative theories,casual analogies, and quantitative experiments’ as presented by Cussler, Adolf Fick in1855, described that:

[T]he diffusion of the dissolved material . . . is left completely to the influence of the molecularforces basic to the same law . . . for the spreading of warmth in a conductor and which has alreadybeen applied with such great success to the spreading of electricity.

Basically, diffusion can be described using the same mathematics as that of Fourier’s lawof 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. Parallelto the work on heat transfer, and Fourier’s law in 1822, Fick defined a one-dimensionalflux 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 massoccurs, 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’ byFick. This quantity D is in fact the diffusion coefficient (m2 s−1). Although there are afew formats of Fick’s law that are known in literature, their essence remains similar. Theconcept has been applied to the ‘conductive’ mass transfer processes in gases, liquidsand solids.

Drying of porous material has been considered to be one of the most suitable candi-dates for applying Fick’s law of diffusion. In particular, when one is interested in knowingwhat moisture content distribution is like within a material being dried, in many casesthis cannot be accurately measured. Because of the complexity of the process and thematerial involved, a ‘pure’ application of Fick’s law is not possible. The mass diffusivityfor moisture in a porous material is taken as an ‘effective diffusivity’ to encapsulateseveral effects in addition to the pure diffusion process.


Spatial drying modelling is important since the spatial distributions of moisture con-tent 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 influ-ence 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). Degra-dation of ascorbic and carotenoids was shown to be more enhanced with the increaseof drying air temperature and product moisture content (Di Scala and Crapiste, 2008).The rehydration ability and ascorbic acid retention were dependent on the drying airtemperature (Ramallo and Mascheroni, 2012). Similarly, the increase of temperatureenhanced the loss of aroma (Timoumi et al., 2007). The survival of probiotics was alsofound 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 contenthigh gradient inside the samples (Yang et al., 2003). By predicting the distributions ofmoisture content and temperature inside the materials being dried, product quality canbe predicted. Several drying operating conditions and schemes can be made in order tomaintain 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 ten-sile stress is at maximum, while microwave drying was damaged internally as a resultof high pore pressure (Kowalski et al., 2005). Convective drying of kaolin did notresult in sample cracking but applying a combination of convection and microwavesled to fracture of the samples (Kowalski and Pawlowski, 2010a,b). Controlleddrying can give a desirable bending strength and sintered density of ceramic samples(Misra et al., 2002). Here, the distributions of moisture content and temperature mayaffect 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 mois-ture transport in literature (Mariani et al., 2008; Pakowski and Adamski, 2007; Thuwa-panichayanan et al., 2008; Vaquiro et al., 2009). The effective liquid diffusivity is usuallyused to lump the whole phenomenon during drying including liquid diffusion, vapourdiffusion, 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

(Deff

∂C

∂x

), (4.1.2)

where C can be considered to be concentration of liquid water (kg m−3), Deff is theeffective diffusivity (m2 s−1), t is time (s) and x is distance (m). Equation (4.1.2) doesnot usually consider the shrinkage velocity effect and the most important parameter Deff

needs to be determined experimentally.

Comparisons of the REA with other theories 171

4.1.1 Crank’s effective diffusion

Crank’s effective diffusion has been used by several researchers and considered as afundamental 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 Biotnumber (Bi) criteria and do not fully satisfy the required boundary conditions. Theyoften do not report the operating conditions of their experiments in detail so that clearjustification 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 effectivediffusion in Equation (4.1.1) can be written as (Crank, 1975):

X − Xe

X0 − Xe= 8

π2

∞∑n=0

1

(2n + 1)2exp

[−(2n + 1)2 π2

L2Deff ,l t

], (4.1.3)

where Deff is the effective liquid diffusivity (m2 s−1), X is the moisture content on adry basis (kg water kg dry solids−1), which can be evaluated by (X = C/ρs), X0 isthe initial moisture content (kg water kg dry solids−1), Xe is the equilibrium moisturecontent (kg water kg dry solids−1) and L is the half thickness (m) where L = 0.5b andthe materials are dried symmetrically. For long time-period of drying, only the first termof Equation (4.1.2) is significant so that Equation (4.1.2) can be simplified into (Crank,1975):

ln

(X − Xe

X0 − Xe

)≈ ln

(8

π2

)−(

π 2

4b2Deff ,l

)t. (4.1.4)

Crank’s effective diffusion should only be valid for the conditions of isothermal drying,negligible shrinkage, negligible external resistance, constant diffusivity and uniforminitial moisture content (Crank, 1975). In addition, the approach should only correlatewell with the experimental data for the time towards the end of drying (Srikatden andRoberts, 2006). However, this has been implemented largely where the assumptions maynot 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 boundaryconditions (at x = L, for slab geometry) of (Crank, 1975):

−Deff ,ld X

dx= β(Xs − Xe), (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 boundaryconditions of vapour transfer, which can be expressed as (Chen, 2007):

−Dv,effdCv

dx= hm(ρv,s − ρv,b), (4.1.6)

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


Equations (4.1.5) and (4.1.6) are similar if, the moisture content is in equilibrium withthe water vapour concentration only at the interface (Chen, 2007). However, previouspublications 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 correlateeffective diffusivity with drying air temperature, which is also fundamentally incorrectunless 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 isothermaltest rigs to conduct isothermal drying carefully, so that the effective diffusivity could becorrelated with the sample temperature.

4.1.2 The formulation of effective diffusivity to represent complexdrying mechanisms

Modelling of drying processes is relatively complex and there have been several mecha-nistic models proposed; including liquid diffusion (Lewis, 1921), capillary flow (Buck-ingham, 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 thevapour and liquid water transfer so that the flux of liquid water and water vapour isproportional to the thermal gradient and moisture potential gradient. As mentionedearlier, the coefficients of effective water liquid diffusivity, effective water vapour dif-fusivity and thermal diffusivity are implemented in linking the fluxes and gradients.Whitaker’s approach (Whitaker, 1977) implements the continuity equation of the liquidand vapour phases combined with the equation for conservation of liquid water, watervapour and energy to describe the drying process (Whitaker and Chou, 1983). The porenetwork approach has been proposed but it has not been rigorously experimentally val-idated (Nowicki et al., 1992; Prat, 1993). The applicability of the approach is limitedby geometries and distributions of pore structure and network, especially for drying offood and biomaterials (Chen, 2007). The complex models mentioned here also requirea 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 thesimplest model proposed (Chen, 2007). The effective diffusivity should be influencedby the composition of the materials. The multi-component diffusion model is oftenimplemented to couple this effect (Ferrari et al., 1989; Yoshida and Miyasita, 2002).Therefore, the effective diffusivity is essentially a lumped parameter whose variability isdependent on the drying condition, material structure and composition and, sometimes,sample size. The last aspect rules out the most fundamental nature of effective liquiddiffusivity. For moderate drying conditions, it would be better to see effective diffusivityas a liquid depletion coefficient in order to avoid confusion with the original meaning ofdiffusivity (Chen, 2007).


4.1.3 Several diffusion-based models

Various formulations of diffusion-based models have been implemented extensively bymany researchers (Adhikari et al., 2004; Cihan and Ece, 2001; Corzo et al., 2008; Karet al., 2009; Vaquiro et al., 2009). As mentioned before, Crank’s diffusion model hasbeen 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 modelsare 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 theconvective 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 berepresented as (Kar et al., 2009):

Cs∂ X

∂t= ∂

∂x

(Deff ,lCs

∂ X

∂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 effectiveliquid diffusivity (m2 s−1).

The initial and boundary conditions (Kar et al., 2009):

t = 0, X0, (4.1.8)

x = 0,d X

dx= 0, (4.1.9)

x = L , Cs Deff ,ld X

dx= −hm(ρv,s − ρv,b), (4.1.10)

where L is the sample thickness, hm is the mass transfer coefficient (m s−1), ρv,s isthe surface water vapour concentration (kg m−3) and ρv,b is the ambient water vapourconcentration (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 pdT

dt= hu A(Tb − T ) + U A(Tb − T ) − hm A(ρv,s − ρv,b)�HV , (4.1.11)

where m is the sample mass (kg), T is sample temperature (K), Cp is the specific heatof the sample (J kg−1 K−1), A is the surface area (m2), hu is the upper heat transfercoefficient (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).


BP

Computer

S3 H2

K2 K1

TC1 TC2

PV1 PV2

Powerboard

I-7

H1

I-8E-5E-4

V-5

V-6

H1-Heater 1H2-Heater 2P-Sample stand and platform assemblyB-Micro balanceTC1-Temperature controller 1TC2-Temperature controller 2PV1-Rheostat 1PV2-Rheostat 2V-6-Stop valveE-4-Drierite packed bed columnV-5-Relief valveE-5-Air filterI-8-Pressure regulatorI-7-Digital flowmeterK1-Type K thermocouple 1K2-Type K thermocouple 2

Electrical lineData signal lineAir line

Figure 4.1 Experimental setup for convective drying of porcine skin. [Reprinted from ChemicalEngineering Research and Design, 87, S. Kar, X.D. Chen, B.P. Adhikari and S.X.Q. Lin, Theimpact of various drying kinetics models on the prediction of sample temperature–time andmoisture content–time profiles during moisture removal from stratum corneum, 739–755,


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


33 mm

(a)

(b)

Drying air

Drying channel

Sample slot

Two plasticlayers

Aluminium plate(bottom surface)

Electronic balance

Support

Cardboard Top sample surface

11.30 mm

7.16 mm

Air flowdirection

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 ofvarious drying kinetics models on the prediction of sample temperature–time and moisturecontent–time profiles during moisture removal from stratum corneum, 739–755, Copyright


consists of the mass balance. The model can be written as:

∂Wp

∂t= De

(∂2Wp

∂x2+ ∂Wp

∂y2+ ∂2Wp

∂z2

), (4.1.12)

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


The initial and boundary conditions are (Garcia-Perez et al., 2009):

t = 0, Wp = Wp0, (4.1.13)

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

dx= 0,

dWp

dy= 0,

dWp

dz= 0, (4.1.14)

x = L , y = L , z = L , Wp = Wpe, (4.1.15)

where L is the sample thickness (m) and Wpe is the equilibrium moisture content (kg waterkg dry solids−1). It is noted that the external resistance and shrinkage were ignored inthe modelling (Garcia-Perez et al., 2009).

Garcia-Perez et al. (2011) implemented the diffusion-based model with and withoutexternal resistance for ultrasonic-assisted drying of a cylindrical sample. The model canbe expressed as:

∂Wp

∂t= De

(∂2Wp

∂x2+ 1

r

∂Wp

∂r+ ∂2Wp

∂r 2

), (4.1.16)

where r is the radial position (m), with the initial and boundary conditions (Garcia-Perezet al., 2011):

t = 0, Wp = Wp0, (4.1.17)

x = 0,dWp

dx= 0, (4.1.18)

r = 0,dWp

dr= 0. (4.1.19)

For the case where external resistance is present, the surface boundary condition can bewritten as (Garcia-Perez et al., 2011):

x = L, −DedWp

dx= k(ϕe − ϕair ), (4.1.20)

r = R, −DedWp

dr= k(ϕe − ϕair ), (4.1.21)

where ϕe is the activity, ϕair is the relative humidity in the drying air and k is the masstransfer 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, Wp = Wpe, (4.1.22)

r = R, Wp = Wpe (4.1.23)

It is noted that Garcia-Perez et al. (2011) did not couple the diffusion-based model withheat 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-basedmodel which incorporated the external resistance resulted in better agreement withexperimental data (Garcia-Perez et al., 2011).


More issues with boundary conditions of mass balance also occur (Chen, 2007;Zhang and Datta, 2004) and the controversies of the boundary conditions are elaboratedin more detail in Section 4.2. Zhang and Datta (2004) and Chen (2007) also highlightedthe importance of applying the multiphase drying approach and the use of the localevaporation–condensation rate. The detailed explanation and several issues with thelocal 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 multiphaseapproach of drying models should be applied. It consists of mass balance of water inliquid and vapour phases as well as heat balance. By this approach, the spatial profiles ofmoisture content, concentration of water vapour and temperature can be generated. Forthe multiphase approach, which does not use the source and depletion term, the modelfor symmetrical convective drying of a slab can be written as (Chen, 2007; Zhang andDatta, 2004) follows:

The mass balance of liquid water:

∂(Cs X )

∂t= ∂

∂x

[Dw

∂(Cs X )

∂x

]; (4.2.1)

The mass balance of water vapour:

∂Cv

∂t= ∂

∂x

(Dv

∂Cv

∂x

); (4.2.2)

The heat balance:

ρCp∂T

∂t= ∂

∂x

(k∂T

∂x

); (4.2.3)

where Cs is the solid concentration (kg solids m−3), X is the moisture content (kg waterkg 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 thesample 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):



x = 0,∂ X

∂x= 0 (symmetrical boundary), (4.2.5)

∂Cv



∂T


x = L , −Cs Dw

∂ X

∂x= 0 (no liquid water transfer), (4.2.8)

−Dv

∂Cv

∂x= hm(ρv,s − ρv,b) (convective boundary for water


k∂T

∂x= h(Tb − T ) − �HV hm(ρv,s − ρv,b) (convective boundary for

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 masstransfer coefficient (m s−1), h is the heat transfer coefficient (W m−1 K−1), �HV isthe vaporisation heat of water (J kg−1), ρv,s is the surface water vapour concentration(kg m−3) and ρv,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 amongthe liquid water and water vapour, apart from the effective diffusivity of water vapourwhich should be a function of porosity, dependent on the moisture content. In addition,the boundary conditions indicate that, at the interface, the vapour diffusive transportinside the samples is balanced by the convective water vapour. Therefore, the equilibriumrelationship between the moisture content and concentration of water vapour has to beimplemented at the boundary (Chen, 2007).

However, if Equations (4.2.1)–(4.2.3) are solved simultaneously with the initial andboundary conditions shown in Equations (4.2.4)–(4.2.9), the rate of change of averagemoisture 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 thatonly the mass balance of liquid water and heat balance are implemented. The model canthen be simplified into (Chen, 2007):

∂(Cs X )

∂t= ∂

∂x

[Dw

∂(Cs X )

∂x

], (4.2.11)

ρC p∂T

∂t= ∂

∂x

(k∂T

∂x

). (4.2.12)

The initial and boundary conditions are (Chen, 2007):

t = 0, X = X, T = To (initial condition, uniform initial


x = 0,∂ X


∂T



x = L, −Cs Dw

∂ X

∂x= hm(ρv,s − ρv,b) (convective boundary for

liquid water transfer), (4.2.16)

∂T

∂x= h(Tb − T ) − �HV hm(ρv,s − ρv,b) (convective boundary for heat

transfer with vaporization).

heat of water). (4.2.17)

Solving Equations (4.2.11) and (4.2.12) simultaneously with the initial and boundaryconditions shown in Equations (4.2.13)–(4.2.17) results in the spatial profiles of moisturecontent and temperature. However, no profiles of water vapour concentration can begenerated. Equation (4.2.15) indicates that the most water evaporation occurs at thesurface. Similarly, Equation (4.2.17) indicates that the surface receives the largest thermalimpact from heat of evaporation. This may undermine the predictions of moisture contentdistribution 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 moreappropriate to implement the multiphase drying approach with the local evaporationrate as pointed out by Chen (2007), Datta (2007) and Zhang and Datta (2004). Thelocal evaporation rate is positive when drying occurs. It needs to be coupled with massbalance in liquid and vapour phases, as well as balance. It serves as a depletion andsource term for the mass balance of liquid water and water vapour, respectively. For thesymmetrical 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; Zhangand Datta, 2004):

The mass balance of liquid water:

∂(Cs X )

∂t= ∂

∂x

[Dw

∂(Cs X )

∂x

]− I ; (4.3.1)

The mass balance of water vapour:

∂Cv

∂t= ∂

∂x

(Dv

∂Cv

∂x

)+ I ; (4.3.2)

The heat balance:

ρC p∂T

∂t= ∂

∂x

(k∂T

∂x

)− I�Hv; (4.3.3)

where Cs is the solid concentration (kg solids m–3), X is the moisture content (kg waterkg 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 istime (s), x is the axial dimension (m), ρ is the sample density (kg m–3), k is the thermal


conductivity (W m–2 K–1), Cp is the sample’s specific heat (J kg–1 K–1) and �HV is thevaporisation heat of water (J kg–1).

The initial and boundary conditions (Chen, 2007; Chong and Chen, 1999; Kar andChen, 2011; Putranto and Chen, 2013; Zhang and Datta, 2004):



x = 0,∂ X


∂Cv


∂T


x = L , u − Cs Dw

∂ X

∂x= hmεw

(Cv,s

ε− ρv,b



−Dv

∂Cv

∂x= hmεv

(Cv,s

ε− ρv,b



k∂T

∂x= h(Tb − T ) (convective boundary for heat transfer), (4.3.10)

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 tobe expressed explicitly. However, it has not been fully understood experimentally howto establish this until the REA approach is implemented (Chen, 2007).

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 localevaporation rate so that for the symmetrical convective drying of a slab, the model canbe written as (Model B):

The mass balance:

∂ M

∂t= ∂

∂x

(DM

∂ M

∂x

); (4.3.11)

The heat balance:

ρC p∂T

∂t= ∂

∂x

(k∂T

∂x

)− ρs

∂ M

∂t�Hv; (4.3.12)


where M is the moisture content (kg water kg dry solids−1), ρs is the density of drysample (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 thermalconductivity (J kg−1 K−1), Cp is the sample specific heat (J kg−1 K−1) and �Hv is thevaporisation 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), (4.3.13)

x = 0,∂ M


x = L , k∂T

∂x= h(Tb − T ) (convective boundary for liquid transfer). (4.3.15)

However, Zhang and Datta (2004) and Datta (2007) mentioned that the model witha rate of moisture content used as the local evaporation rate does not satisfy massconservation and it is more of an empirical model. Zhang and Datta (2004) analysedthat the combination of Equations (4.3.1) and (4.3.2) in Model B results in:

∂Cv

∂t− ∂(Cs X )

∂t= ∂

∂x

(Dv

∂Cv

∂x

)− ∂

∂x

[Dw

∂(Cs X )

∂x

]+ 2 I . (4.3.16)

It can be observed that the local evaporation rate should be influenced by the moisturecontent as well as water vapour concentration, not only by the rate of moisture contentas implemented in Model B. Therefore, Model B does not satisfy the conservation ofmass for water.

Moreover, Model B is found to be an empirical model as it lumps the wholephenomenon (liquid diffusion, vapour diffusion, evaporation–condensation) into diffu-sion marked by effective diffusivity (DM) shown by Equation (4.3.11). The combinationof Equations (4.3.1) and (4.3.2) of Model A yields:

∂Cv

∂t+ ∂(Cs X )

∂t= ∂

∂x

(Dv

∂Cv

∂x

)+ ∂

∂x

[Dw

∂(Cs X )

∂x

], (4.3.17)

while Model B represents the mass balance as shown in Equation (4.3.11), which meansModel B assumes:

∂ M

∂t= ∂Cv

∂t+ ∂(Cs X )

∂t, (4.3.18)

∂

∂x

(DM

∂ M

∂x

)= ∂

∂x

(Dv

∂Cv

∂x

)+ ∂

∂x

[Dw

∂(Cs 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 ofwater vapour (Cv) and moisture content (X) as M. Therefore, Model B is an empiricmodel which looks like a fundamental model of diffusion. This is in agreement withChen (2007), who mentioned that effective diffusivity should be treated as a liquiddepletion coefficient in order to avoid confusion with the original Fick’s diffusivity.


Another inaccuracy of Model B is explained by Zhang and Datta (2004). Model Bimplements boundary conditions for heat transfer indicated in Equation (4.3.15). This isin contrast with the model shown by Equations (4.2.11) and (4.2.12) with the initial andboundary conditions indicated in Equations (4.2.13)–(4.2.17). The other model (shownby Equations 4.2.11–4.2.17) is reasonable since it indicates that the most evaporationoccurs at the surface, and the surface receives the largest thermal impact due to heat ofevaporation. This model indicates that the heat of evaporation is taken from the ambientair 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 theinside so that a negative heat source is generated. Zhang and Datta (2004) suggested thatModel B results in a lower inner temperature of samples since the heat from inside is usedfor evaporation as mentioned. The other model results in more a uniform temperatureinside 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 non-equilibrium. Referring to Model A shown in Equations (4.3–1)–(4.3–3), the massbalance of liquid water and water vapour is linked by the local evaporation rate. It is nec-essary to represent the local evaporation rate explicitly in order to solve Model A, whichconsists 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 assumedto be in equilibrium with water vapour concentration at any time so that the waterisotherm can be used to relate these relationships. The equilibrium approach does notrequire 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 expressedby (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 assumingequilibrium conditions between the water vapour concentration and moisture contentinside the samples at any time, the water vapour pressure can be written as (Zhang andDatta, 2004):

pv = pv(T, W ), (4.3.21)

aw = pv(T, X )

pv,sat (T )= f (T, X ), (4.3.22)

where pv (T,W) is equilibrium water vapour pressure (Pa) and pv,sat is the saturated waterpressure 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.


For the equilibrium approach, Model A for the symmetrical convective drying of slabscan be rearranged into (Zhang and Datta, 2004):

The mass balance:

∂(Cs X )

∂t+ ∂Cv

∂t= ∂

∂x

[Dw

∂(Cs X )

∂x

]+ ∂

∂x

(Dv

∂Cv

∂x

). (4.3.23)

It can be seen that Equation (4.3.23) is obtained by adding Equation (4.3.1) and (4.3.2).The heat balance:

ρC p∂T

∂t= ∂

∂x

(k∂T

∂x

)+[∂(Cs X )

∂t− ∂

∂x

(DwCs

∂ X

∂x

)]�HV . (4.3.24)

Here, the local evaporation rate is obtained from water vapour conservation as mentionedin Equation (4.3.2). It is emphasised that local evaporation rate here is not simply basedon 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 ofmoisture content, concentration of water vapour and temperature can be generated bysolving Equations (4.3.22)–(4.3.24) simultaneously.

The equilibrium approach has been used by several researchers to describe convectivedrying and baking (Aversa et al., 2010; Ni et al., 1999; Zhang et al., 2005; Zhangand 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). Zhanget al. (2005) and Zhang and Datta (2006) implemented this approach to model the bakingof bread. The model described the moisture content profiles during baking reasonablywell. Similarly, the model resulted in a reasonable agreement with the experimental dataof surface temperature, but an underestimation in profiles of centre temperature wasobserved (Zhang et al., 2005; Zhang and Datta, 2006). Coupling between the modeland the equations of conservation of the drying air can also be used to predict the flowfield of the drying air, as well as the spatial profiles of moisture content and temperatureinside 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 canbe used to assess the validity of the equilibrium approach (Zhang and Datta, 2004). In thenon-equilibrium approach, as mentioned before, the local evaporation rate needs to beexpressed explicitly. It has been proposed that the internal evaporation rate can be relatedto the difference of equilibrium vapour pressure and the vapour pressure at particulartimes 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 ratecan be expressed as:

I = c(X − Xe)(pv,e − pv), (4.3.25)

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


c is expected to vary with the temperature and moisture content. The coefficient c shouldbe 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 bakingprocess. The local evaporation rate was expressed as (Fang and Ward, 1999; Ouseguiet al., 2010; Zhang and Datta, 2004):

I = K (ρv,eq − ρv), (4.3.26)

where ρv and ρv,eq is the water vapour concentration (kg m−3) and equilibrium watervapour concentration (kg m−3), respectively and K is the proportionality constant depen-dent on ambient conditions (heat transfer coefficient, ambient fluid, etc.). K was deter-mined by matching the model and experimental data. The increase of K was found toincrease 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 betterunderstanding 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 thenon-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 describeseveral challenging drying cases accurately, can be used to model the local evaporation–condensation rate. The internal evaporation–condensation rate can be expressed as (Karand Chen, 2010; 2011; Putranto and Chen, 2013):

I = hmin Ain(Cv,s − Cv), (4.3.27)

where hm,in is the internal mass transfer coefficient (m s−1), Ain is the internal surfacearea per unit volume (m2 m−3), and Cv,s and Cv are the internal-surface water vapourconcentration (kg m−3) and water vapour concentration (kg m−3), respectively. Theprocedures for determining the internal mass transfer coefficient (hm,in) and internalsurface area per unit volume (Ain) are presented in Kar and Chen (2010; 2011). Thevalue of hm,in should be in the order of �Dv/rp and the Ain is determined based on thearea of single cells inside the samples or particles and number of cells per unit volumeinside 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 = hmin Ain

[exp

(−�Ev

RT

)Cv,sat − Cv

], (4.3.28)

where �Ev is the activation energy (J mol−1), T is the sample temperature (K), R is idealgas 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 conservationof heat and mass transfer to yield a non-equilibrium multiphase drying model usingthe REA as a source/depletion term, the S-REA, explained in Chapter 3. The following


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

4.4 Comparison of the diffusion-based model and the L-REA onconvective drying

In this section, the results of modelling with the diffusion-based model and the L-REAon the convective drying of mango tissues are presented. The experimental data arederived from the work of Vaquiro et al. (2009). The review of experimental details hasbeen presented in Section 2.6.

Vaquiro et al. (2009) implemented the diffusion-based model which can be writtenas:

The mass balance:

∂

∂x

(De

∂ X

∂x

)+ ∂

∂y

(De

∂ X

∂y

)+ ∂

∂z

(De

∂ X

∂z

)= ∂ X

∂t, (4.4.1)

with the initial and boundary conditions:

t = 0, X = X0, (4.4.2)

x = 0, y= 0, z = 0,∂ X

∂x= ∂ X

∂y= ∂ X

∂z= 0, (4.4.3)

x = L , Deρs∂ X

∂x(L , y, z, t) = −hm

Mw

R

[ϕ(L , y, z, t)Ps(L , y, z, t)

T (L , y, z, t)− ϕ∞ Ps∞

T∞

],

(4.4.4)

y = L , Deρs∂ X

∂x(x, L , z, t) = −hm

Mw

R

[ϕ(x, L , z, t)Ps(x, L , z, t)

T (x, L , z, t)− ϕ∞ Ps∞

T∞

],

(4.4.5)

z = L , Deρs∂ X

∂x(x, y, L , t) = −hm

Mw

R

[ϕ(x, y, L , t)Ps(x, y, L , t)

T (x, y, L , t)− ϕ∞ Ps∞

T∞

],

(4.4.6)

where De is the effective diffusivity (m2 s−1), ρs is the density of solid (kg m−3), Xis the moisture content on dry basis (kg H2O 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 relativehumidity, Ps is the saturated vapour pressure (Pa), T is sample temperature (K), ϕ� isthe drying air relative humidity, Ps� is the vapour pressure in drying air (Pa), T� is thedrying air temperature and L is the thickness of sample (m) while the heat balance can


be expressed as:

∂

∂x

(k∂T

∂x

)+ ∂

∂y

(k∂T

∂y

)+ ∂

∂z

(k∂T

∂z

)

= ρds(C pds + XCpw)∂T

∂t− DeρdsCpw

(∂ X

∂x

∂T

∂x+ ∂ X

∂y

∂T

∂y+ ∂ X

∂z

∂T

∂z

),

(4.4.7)

with the initial and boundary conditions:

t = 0,T = T0, (4.4.8)

x = 0, y = 0, z = 0,∂T

∂x= ∂T

∂y= ∂T

∂z= 0, (4.4.9)

x = L , −k∂T

∂x(L , y, z, t) = h[T (L , y, z, t) − T∞] − Deρds

∂ X

∂x(L , y, z, t)Qs,

(4.4.10)

y = L , −k∂T

∂y(x, L , z, t) = h[T (x, L , z, t) − T∞] − Deρds

∂ X

∂y(x, L , z, t)Qs,

(4.4.11)

z = L , −k∂T

∂z(x, y, L , t) = h[T (x, y, L , t) − T∞] − Deρds

∂ X

∂z(x, L , z, t)Qs,

(4.4.12)

where k is the thermal conductivity of sample (W m−1 K−1), h is the heat transfercoefficient (W m−2 K−1), ρds is the density of dry solid (kg m−3), Qs is the heat ofevaporation of water (J kg−1), Cpds is the specific heat of the dry solid (J kg−1 K−1) andCpw 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 originalformulation of the L-REA, as mentioned in Equation (2.1.4), is used and combinedwith the prediction of the temperature distribution as described in Section 2.6.2 andshown in Equation (2.6.14) since the sample is relatively thick. The relative activationenergy (�Ev/�Ev,b) is generated from experiments on convective drying at a drying airtemperature of 55 °C.

The comparisons of modelling using the diffusion-based model implemented byVaquiro et al. (2009) and the L-REA are shown in Figures 4.3 and 4.4. Figure 4.3 showsthe 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 profilesof 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 dryingrate 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 dryingair temperatures of 45°, 55° and 65 °C. The results of modelling using the L-REA


0

2

1

3

4

5

10

X (

kg w

ater

/kg

dry

soli

d)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5t(s) × 104

6

7

8

9L-REA 55°C

L-REA 65°C

L-REA 45°C

Diffusion-based by Vaquiro et al. (2009) 45°CData 65°C

Data 55°C

Data 45°C

Diffusion-based by Vaquiro et al. (2009) 55°C


Figure 4.3 Moisture content profiles from the convective drying of mango tissues modelled usingthe 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 Materialswith 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 adrying air temperature of 45 °C, the diffusion-based model (Vaquiro et al., 2009) resultsin 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 temperatureprofiles at the beginning of the drying period for convective drying at drying air temper-atures of 45°, 55° and 65 °C (Vaquiro et al., 2009). On the other hand, the L-REA modelswell the centre temperature profiles of convective drying at drying air temperatures of55° and 65 °C, indicated by a R2 higher than 0.984 (Putranto et al., 2011b). Both modelsshow a slight overestimation in temperature profiles of convective drying at a drying airtemperature 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 conjunctionwith the initial and boundary conditions indicated in Equations (4.4.1)–(4.4.12) seemsto not be able to model the convective drying well, particularly the temperature profileswhich show a kink at the beginning of drying. On the other hand, the L-REA combinedwith the prediction of temperature distribution shown in Equation (2.6.14) can describeboth the moisture content and temperature profiles well. This indicates that the L-REAperforms better than the diffusion-based model in describing convective drying.


280

300

290

310

320

330

340

Cen

tre

tem

pera

ture

(K

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t(s) × 104

L-REA 55°C

L-REA 65°C

L-REA 45°C

Diffusion-based by Vaquiro et al. (2009) 45°CData 65°C

Data 55°C

Data 45°C



Figure 4.4 Temperature profiles from convective drying of mango tissues modelled using theL-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 withthickness 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 theS-REA on convective drying

In this section, the results of modelling using the diffusion-based model and the S-REAon the convective drying of mango tissues (Vaquiro et al., 2009) are compared. Asexplained in Chapter 3, the S-REA is a non-equilibrium multiphase drying model withthe 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 andboundary 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 boundaryconditions for the convective drying of mango tissues (Vaquiro et al., 2009) are presentedin 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 showsthe moisture content profiles modelled using both approaches. It can be seen that both thediffusion-based model (Vaquiro et al., 2009) and the S-REA model the moisture contentprofiles 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 contentprofiles well (R2 higher than 0.996 for both the S-REA and diffusion-based model).


0

2

1

3

4

5

10

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t(s) × 104

6

7

8

9

Diffusion-based model byVaquiro et al. (2009) 55°C

S-REA 55°C

Diffusion-based model byVaquiro et al. (2009) 65°C

S-REA 45°C

S-REA 65°CData 55°C

Diffusion-based model by Vaquiro et al. (2009) 45°C

Data 45°C

Data 65°C

Figure 4.5 Moisture content profiles from the convective drying of mango tissues modelled usingthe 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 fornon-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-basedmodel (Vaquiro et al., 2009) and the S-REA. As mentioned before, the diffusion-basedmodel shows a kink in the centre temperature profiles at the beginning of the dryingperiod for convective drying at drying air temperatures of 45°, 55° and 65 °C (Vaquiroet al., 2009). On the other hand, the S-REA models the centre temperature accurately forall cases of the convective drying of mango tissues. The results of modelling using theS-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 generat-ing 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 (Putrantoand Chen, 2013).

It can be shown here that the diffusion-based model (Vaquiro et al., 2009) doesrepresent convective drying of mango tissues well, as shown by a kink of the temperatureprofiles. On the other hand, the S-REA is accurate at modelling both the moisture contentand temperature profiles of convective drying. The S-REA formulation in conjunctionwith 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


280

300

290

310

320

330

340

Tem

pera

ture

(K

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t(s) × 104


S-REA 55°C


S-REA 45°C

S-REA 65°CData 55°C


Data 45°C

Data 65°C

Figure 4.6 Temperature profiles from the convective drying of mango tissues modelled using theS-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 fornon-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 theimportance of multiphase model with the source term as previously mentioned byZhang 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 porousbody based on irreversible thermodynamics. A porous body can be considered to consistof 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 porousbody cause the vapour and liquid water transfer so that the flux of liquid water andwater vapour is proportional to the thermal gradient and moisture potential gradient.The relationships can be explained in two- and three-way coupled system of partialdifferential equations (Luikov, 1975). The two-way coupled system assumes that thepressure inside the capillary body is constant during the process. It is postulated thatthe thermal and moisture potential gradient within a porous body cause vapour andliquid water transfer so that the flux of liquid water and water vapour is proportionalto the thermal gradient and moisture potential gradient. The heat and mass transfer


is interdependent and several coefficients are used to explain the interdependency. Forthis purpose, two sets of partial differential equations in terms of moisture content andtemperature are established (Luikov, 1975). The two-way coupled system has been usedby several researchers to model the simultaneous heat and mass transfer processes (Liuand Cheng, 1990; Mikhailov and Shishedjiev, 1975; Younsi et al., 2006a,b; 2007).

In addition, the three-way coupled system incorporates the pressure gradient inside abody as a result of the presence of water vapour, which results in moisture movement byfiltration. In this system, it is assumed that the thermal, moisture and pressure gradientwithin a capillary body lead to moisture movement within the body. Unlike the two-waysystem, another set of partial differential equations in terms of pressure is established torepresent the pressure gradient inside the body (Luikov, 1975).

For the two-way system, it is assumed that the vapour movement inside the body is dueto molecular transport and the concentration of vapour equilibrates thermodynamicallywith the concentration of liquid. The mass flow of vapour can be written as (Luikov,1975):

j1 = −ερD∇ρ10 = −am1ρ0∇u − aTm1ρ0∇t, (4.6.1)

where j1 is the flux of vapour (kg m−2 s−1), ε is the dimensionless factor characterisingresistance 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), ρ0 is the density of dry body (kg m−3), u is themoisture content of body (kg kg−1), t is temperature (K) and am1 is the vapour diffusioncoefficient which can be expressed as (Luikov, 1975):

am1 = εDρ

ρ0

(dρ10

du

)T

, (4.6.2)

where ρ10 is the relative vapour concentration, while am1T is the thermal diffusion

coefficient, which can be written as (Luikov, 1975):

aTm1 = εD

ρ

ρ0

(dρ10

du

)u

. (4.6.3)

The mass flow of liquid can be expressed in a similar way to that of vapour as (Luikov,1975):

j2 = −ερD∇ρ10 = −am2ρ0∇u − aTm2ρ0∇t, (4.6.4)

where j2 is the flux of liquid (kg m−2 s−1) and am2 and am2T are the vapour and

thermal diffusion coefficients dependent on the moisture content and temperature of thebody.

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).


Equations (4.6.1), (4.6.4) and (4.6.5) can be rearranged into (Luikov, 1975):

∂u

∂τ= K11∇2u + K12∇2t, (4.6.6)

∂t

∂τ= K21∇2u + K22∇2t, (4.6.7)

where K11 = am = am1 + am2, (4.6.8)

K12 = aTm1 + aT

m2, (4.6.9)

K21 = Lεam

c, (4.6.10)

K22 = a + Lεamδ

c, (4.6.11)

δ = aTm1 + aT

m2

am1 + am2, (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 penetra-tion of humid air through capillary system inside the body. The transfer can be describedby 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

∂τ= K11∇2u + K12∇2t + K13∇2 p, (4.6.14)

∂t

∂τ= K21∇2u + K22∇2t + K23∇2 p, (4.6.15)

∂p

∂τ= K31∇2u + K32∇2t + K33∇2 p, (4.6.16)

where K13 = amδ f , (4.6.17)

K23 = εLam

cδp, (4.6.18)

K33 = α f − εam

C fδp, (4.6.19)

K31 = −εam

C f, (4.6.20)

K31 = −εamδ

C fδ, (4.6.21)

α f = k f

ρC f, (4.6.22)

δ f = k f

amρ0, (4.6.23)

where Cf is the body capacity for the humid air with filtration.


The boundary conditions on the surface can be written as (Luikov, 1975):

amρ0(∇u)s + aTm(∇t)s + js = 0, (4.6.24)

−(k + LaTm2)(∇t)s − Lam2ρ0(∇u)s + qs = 0, (4.6.25)

where js = βρ0(us − ue), (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 theambient temperature and ts is the surface temperature.

Equation (4.6.24) implies that the moisture supplied to the surface due to thermody-namic 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 anyleft penetrates inside the body.

Luikov’s approach (Luikov, 1975) has been used by several researchers to modeldrying 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 wasimplemented to model timber drying and a reasonable agreement with the experimentaldata was shown. The model was solved numerically using the finite element method.The results of modelling confirmed that the temperature gradient inside the sample canbe 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 usingthe finite element method to model the convective drying of silicon gel. The numericalmodel matched the exact solution well. For low Fourier numbers, the results are not stablebut the stable results are achieved if high Fourier numbers, higher geometries and non-linear properties are used (Irudayaj and Wu, 1994). In a similar investigation, Irudayajand Wu (1996) showed that the model can be used to model the convective drying ofspruce samples well. The parameters of moisture conductivity and ratio of vapour tototal diffusion were found to be very sensitive to the profiles of moisture content andtemperature. Nevertheless, these parameters need to be extracted from experimental dataand determined by error-minimization (Irudayaj and Wu, 1996).

Luikov’s approach (Luikov, 1975) has been also used by (Younsi et al., 2006a) tomodel the heat treatment of wood under a constant heating rate, which is essentially adrying process under linearly increased gas temperatures. The experimental details ofYounsi et al. (2006a,b; 2007) are reviewed briefly in Section 2.9. For modelling the heattreatment of wood using Luikov’s approach (1975), it is assumed that the wood sampleis isotropic and homogeneous, shrinkage is negligible, no heat is generated inside thewood and capillary forces are much stronger than gravity. The moisture transfer can berepresented as (Younsi et al., 2006a):

ρCm∂U

∂t= ∇

[(kmδ

Cm

)∇T + km∇U

], (4.6.28)

while the heat transfer can be expressed as (Younsi et al., 2006a):

ρCq∂T

∂t= ∇

[(kq + εkmδ

Cm

)∇T + (ελkm)∇U

]. (4.6.29)


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, (4.6.31)

−kqdT

dn= hq (T − Tg) + (1 − ε)λhm(U − Ug), at � (4.6.32)

−kmdU

dn= kmδ

Cm

dT

dn+ hm(U − Ug), at � (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 masstransfer coefficient (kg H2O m−1 s−1 °M−1), Cq is the heat capacity (J kg−1 K−1), Cm

is the moisture capacity (kg H2O, kg−1 °M−1), δ is the thermal gradient coefficient(kg H2O kg−1 K−1), kq is the conductivity (W m−1 K−1), km is the moisture diffusivity(kg H2O 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 ofvapour diffusion to total moisture diffusion and � is the boundary surface of heat andmass transfer.

For a more general expression, Equations (4.6.28) and (4.6.29) can be represented as(Younsi et al., 2006a):

A11∂T

∂t+ A12

∂U

∂t= ∇(K11∇T + K12∇U ), (4.6.34)

A21∂T

∂t+ A22

∂U

∂t= ∇(K21∇T + K22∇U ). (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, (4.6.37)

K11 = kq + ελkMδ

Cm, (4.6.38)

K12 = ελkm, (4.6.39)

K21 = kmδ

Cm, (4.6.40)

K22 = km . (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 fromSimpson 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 moisturecontent and temperature during drying.


4.7 Model formulation of Whitaker’s approach

Whitaker’s approach (Whitaker, 1977) was implemented by several researchers todescribe 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 detailedmechanisms of transport in microscale and macroscale structures based on a known dis-tribution of the structures. Equations of conservation of momentum, mass and heat insolid, vapour and liquid phases and their local volume behaviours followed by volumeaveraging methods, are used to describe the mechanisms. Darcy’s law is usually used todescribe the momentum transfer in liquid and gas phases while the mass transfer consid-ers capillary action as well as evaporation–condensation. The heat transfer incorporatesthe convective transport, conduction and vaporisation–condensation heat by assuminglocal thermal equilibrium within solid, gas and liquid phases (Whitaker, 1977; Whitakerand Chou, 1983).

The detailed modelling of Whitaker’s approach can be expressed as (Whitaker, 1977):The total thermal energy equation:

〈ρ〉C pd〈T 〉

dt+ [ρβ (C p)β〈vβ〉 + 〈ργ 〉γ (C p)γ 〈vγ 〉γ ]∇T

+ hvap〈m〉 = ∇(keff ∇〈T 〉) + 〈�〉, (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), ρβ is density of the liquid phase (kg m−3), ργ 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 gasphase, 〈vβ〉 is the phase average of velocity in the liquid phase, 〈vγ 〉γ is the phaseaverage velocity in the gas phase, T is temperature (K), hvap is the vaporisation heat ofwater (J kg−1), keff is the thermal conductivity (W m−2 K−1), 〈m〉 is the evaporation rateper unit volume (kg m−3 s−1) and � is the heat generation (W m−3).

The liquid phase equation of motion:

〈vβ〉 = −εβξ Kβ

μβ

[kε∇εβ + k〈T 〉∇T − (ρβ − ργ )g], (4.7.2)

where εβ 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:

∂εβ

∂t+ ∇〈vβ〉 + 〈m〉

ρβ

= 0; (4.7.3)

The gas phase equation of motion:

〈vγ 〉 = −Kγ

μγ

{εγ [∇〈pγ − p0〉γ − ργ g]}; (4.7.4)


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 ispressure of the gas phase (N m−2).

The gas phase continuity equation:

∂

∂t(εγ 〈ργ 〉γ ) + ∇(〈ργ 〉γ 〈vγ 〉) = 〈m〉; (4.7.5)

The gas phase diffusion equation:

∂

∂t(εγ 〈ρ2〉γ ) + ∇(〈ρ2〉γ 〈vγ 〉) = ∇[〈ργ 〉γ D2

eff ∇(〈ρ2〉γ/〈ργ 〉γ ]; (4.7.6)

where 〈ρ2〉γ is the density of water vapour (kg m−3) and Deff2 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 ; (4.7.11)

〈p1〉γ = p01 exp

{−[

(2σβγ /rρβ R1〈T 〉) + �hvap

R1

(1

〈T 〉 − 1

T0

)]}; (4.7.12)

where 〈ρi 〉γ is the pressure of ith species in gas phase (N m−2), where 〈ρi 〉γ is the densityof ith species in gas phase (kg m−3), Ri is the gas constant for ith species (N m kg−1

K−1), p10 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 duringdrying. The difficulty of this approach would be in determining the transport parametersinvolved in the equations. For this purpose, a theoretical basis and simplified dryingtheory are needed to estimate the parameters (Whitaker, 1977).

Whitaker’s approach (Whitaker, 1977) has been used to model several drying processand 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 vacuumdrying of wood and the results can yield information about transport of liquid and vapourin wood. The model was also coupled with the equations of conservation in the dryingmedium so the dynamics of water vapour concentration inside the chamber could bepredicted. It was shown that the evaporation rate depends on the levels of pressure andtemperature, while the pressure inside the sample increases as drying progresses becausethe 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 Al2O3 and ceramic


spheres. It was indicated that the total drying rate and outlet steam temperature matchedwell with experimental data, but the internal temperature showed some deviations fromit. Although the model is fairly accurate, Whitaker’s approach is very computationallydemanding (Truscott and Turner, 2005).

Whitaker’s approach (1977) was also implemented by Younsi et al. (2006b; 2007) todescribe the heat treatment of wood under a constant heating rate whose experimentaldetails are presented in Section 2.6. Whitaker’s approach (1977), implemented by Younsiet al. (2006b; 2007), can be expressed as:

− ρd∂ M

∂t= ∇ J = ∂ Jx

∂x+ ∂ Jy

∂y+ ∂ Jz

∂z, (4.7.13)

where ρd 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 canbe written as (Younsi et al., 2006b; 2007):

Jv = −mv Deff

RT∇ Pv, (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 itoccurs when the moisture content is below the fibre saturation point (FSP). The boundwater flux can be written as (Stanish et al., 1986):

Jb = −Db

mv

{−[

187 + 35.1 ln

(T

298.15

)− 8.314 ln

(Pv

101.325

)]∇T

+ 8.314T

Pv

∇ Pv

}, (4.7.16)

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):

J f = −Klρl

μl∇ Pc, (4.7.17)

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


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

J f = −0.61 × 104 Klρl(Mmax − MFSP)0.61

μl(M − MFSP)∇M. (4.7.18)

where MFSP is the moisture content at the fibre saturation point and Mmax is the moisturecontent 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

∂t= ∇ (DM∇M + DT ∇T ) , (4.7.19)

where DM and DT are positive constants, which can be expressed as (Younsi et al., 2006b;2007):

DM = Psv∂�

∂ M

1.2146 × 10−4mvδ0.75

R

+ 8.314DbT

mv�

∂�

∂ M+ αβKlρl(Mmax − MF S P )β

μl(M − MF S P )(1+β), (4.7.20)

DT = 1.2146 × 10−4mvδ0.75

R

(∂ Psv

∂T� + Psv

∂ Psv

∂T

)

− Db

mv

{[187 + 35.1 ln

(T

298.15

)− 8.314 ln

(Pv

101325

)]

−8.314T

Pv

(∂ Psv

∂T� + Psv

∂�

∂T

)}, (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

(ρaua + ρvuv + ρbub + ρ f u f + ρdud

)+ ∇(Jaha + Jvhv + Jbhb + J f h f )

= ∇(k∇T ), (4.7.22)

where ρ, u, h and k indicate; the density, the specific internal energy, the specific enthalpyand the thermal conductivity, respectively. The subscripts a, v, b, f and d indicate; thedry 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

∂t= ∇ (kM∇M + keff ∇T

), (4.7.23)


where kM and keff are positive constants, which can be expressed as (Younsi et al., 2006b;2007):

kM = hv Psv∂�

∂ M

1.2146 × 10−4mvδ0.75

R+ hb8.314

DbT

mv�

∂�

∂ M

+ h fαβKlρl (Mmax − MF S P )β

μl(M − MF S P )(1+β), (4.7.24)

keff = k + hv

1.2146 × 10−4mvδ0.75

R

(∂ Psv

∂T� + Psv

∂ Psv

∂T

)

− hbDb

mv

{[187 + 35.1 ln

(T

298.15

)− 8.314 ln(

Pv

101325

)]−8.314 T

Pv

(∂ Psv∂T � + Psv

∂�∂T

)}

. (4.7.25)

For more general expression, Equations (4.7.19) and (4.7.23) can be represented as(Younsi et al., 2006b; 2007):

A11∂ M

∂t+ A12

∂T

∂t= ∇(K11∇M + K12∇T ), (4.7.26)

A21∂ M

∂t+ A22

∂T

∂t= ∇(K21∇M + K22∇T ), (4.7.27)

where Aij and Kij are coefficients, which can be represented as (Younsi et al., 2006b;2007):

A11 = ρd , (4.7.28)

A12 = A21 = 0, (4.7.29)

A22 = (ρC p)eff , (4.7.30)

K11 = DM , (4.7.31)

K12 = DT , (4.7.32)

K 21 = kM , (4.7.33)

K21 = keff . (4.7.34)

The mass and heat balances shown in Equations (4.7.26) and (4.7.27) are solved withthe 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):

Ts = T f , (4.7.37)

Cs = C f , (4.7.38)

kMd M

dn+ keff

dT

dn= k f

dT f

dn+ D�HV

dC f

dn, (4.7.39)

DMd M

dn+ DT

dT

dn= D

dC f

dn, (4.7.40)


0

0.02

0.01

0.03

0.04

0.05

0.08

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0 0.5 1 1.5 2 2.5 3

t(s) × 104

0.06

0.07

Luikov approach: Younsi et al. (2006a)

L-REA

Experimental data

Figure 4.7 Moisture content profiles from the heat treatment of wood modelled using the L-REAand 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 reactionengineering 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 waterconcentration (kg m−3), Cf is the water vapour concentration in gas, kf is the bulk gasthermal 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 treatmentof 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 inEquations (4.7.37)–(4.7.40). The linearly increased gas temperature is implemented inEquations (4.7.37), (4.7.39) and (4.7.40).

4.8 Comparison of the L-REA, Luikov’s and Whitaker’s approaches formodelling heat treatment of wood under constant heating rates

Figure 4.7 indicates the moisture content profiles of heat treatment of wood with a finaltemperature of 200 °C, heating rate of 20 °C h−1 and initial moisture content of 0.07 kgwater kg dry solids−1. Both the L-REA and Luikov’s approach (implemented by Younsiet 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 approachesslightly underestimate the drying rate during treatment times shorter than 10 000 s. TheL-REA resulted in better agreement with the experimental data during treatment times


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

CaseFinal gastemperature (ºC)

Heating rate(ºC h−1)


1 220 20 0.1252 220 10 0.12

280

320

300

340

360

380

480

Tem

pera

ture

(K

)

0 0.5 1 1.5 2 2.5 3.53

t(s) × 104

400

420

440

460


L-REA

Experimental data

Figure 4.8 Temperature profiles from the heat treatment of wood modelled using the L-REA andLuikov’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 reactionengineering 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 0.999and 0.994 for the L-REA and Luikov’s approach, respectively). It can be shown here thatthe L-REA gives comparable results with Luikov’s approach.

The results of modelling of heat treatment of wood (refer to Table 4.1) using theL-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 ofYounsi et al. (2007), the experimental details are presented in Section 2.8 and the exper-imental settings are shown in Table 4.1. Figure 4.9 shows the moisture content profilesof heat treatment of wood modelled using both approaches. For Case 1 (heating rate of20 °C h−1, refer to Table 4.1), Whitaker’s approach models the moisture content profileswell at treatment times shorter than 20 000 s, but the approach slightly overestimatesthe drying rate after the treatment time of 20 000 s (R2 of 0.991). The L-REA results in


0

0.04

0.02

0.06

0.08

0.1

0.14

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0 1 2 3 4 5 6 7

t(s) × 104

0.12

Case 1-Whitaker approachby Younsi et al. (2007)

Case 1-L-REA

Case 1-Experimental data


Case 2-L-REA


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 treatmentof wood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with


better agreement with experimental data, shown by R2 of 0.994. Both the L-REA andWhitaker’s approach match reasonably well the experimental data for modelling of heattreatment 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 experimentaldata.

The temperature profiles of heat treatment of wood (refer to Table 4.1) are indicatedin Figure 4.10. Generally, the L-REA and Whitaker’s approach model the temperatureprofiles 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 temperatureprofiles during the stage where gas temperature is held constant at 120 °C for half anhour. During this period, the L-REA resulted in a better agreement with the experimentaldata, which could be because of the flexibility of activation energy in changing accordingto ambient conditions.

In conclusion, although the L-REA is much simpler in mathematical modelling, itperforms comparably well with Luikov’s and Whitaker’s approaches (implemented byYounsi et al., 2006a; 2007) in modelling heat treatment of wood under a constant heatingrate. The L-REA is more efficient at generating the model parameters, since these canbe generated from one accurate experiment. On the other hand, the two other approachesrequire several sets of experiments and complex optimization procedures to generatethese model parameters.


250

350

300

400

450

500

Tem

pera

ture

(K

)

0 1 2 3 4 5 6 7

t(s) × 104


Case 1-L-REA



Case 2-L-REA


Figure 4.10 Temperature profiles from the heat treatment of wood (refer to Table 4.1) modelledusing 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 ofwood by using the reaction engineering approach (REA), 6214–6220, Copyright (2012), with


4.9 Comparison of the S-REA, Luikov’s and Whitaker’s approaches formodelling heat treatment of wood under constant heating rates

In this section, the results of modelling of the heat treatment of wood under various con-stant 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 2.8while 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 afinal temperature of 200 °C, heating rate of 20 °C h−1 and initial moisture contentof 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 (implementedby Younsi et al., 2006a) model the moisture content profiles reasonably well (R2 of0.99 and 0.981 for the S-REA and Luikov’s approach, respectively), but the S-REAyields a closer agreement with the experimental data as the other model results in anincrease of moisture content at the beginning of the treatment and an overestimationin the profiles before a treatment time of 10 000 s. The temperature profiles of theexperiment, modelled using both approaches are shown in Figure 4.12. Both the S-REAand Luikov’s approach (applied by Younsi et al., 2006a) model the temperature profiles


0

0.02

0.01

0.03

0.04

0.05

0.08

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0 0.5 1 1.5 2 2.5 3

t(s) × 104

0.06

0.07


S-REA

Experimental data

Figure 4.11 Moisture content profile from the heat treatment of wood modelled using the S-REAand Luikov’s approach.

280

320

300

340

360

380

480

Tem

pera

ture

(K

)

0 0.5 1 1.5 2 2.5 3.53

t(s) × 104

400

420

440

460


S-REA

Experimental data

Figure 4.12 Temperature profile from the heat treatment of wood modelled using the S-REA andLuikov’s approach.


0

0.04

0.02

0.06

0.08

0.1

0.14

Moi

stur

e co

nten

t (kg

wat

er/k

g dr

y so

lids

)

0 1 2 3 4 5 6 7

t(s) × 104

0.12


Case 1-S-REACase 1-Experimental data


Case 2-S-REACase 2-Experimental data

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.

250

350

300

400

450

500

Tem

pera

ture

(K

)

0 1 2 3 4 5 6 7

t(s) × 104


Case 1-S-REA



Case 2-S-REA


Figure 4.14 Temperature profiles from the heat treatment of wood (refer to Table 4.1) modelledusing the S-REA and Whitaker’s approach.


during the heat treatment accurately (R2 of 0.998 and 0.994 for the S-REA and Luikov’sapproach, respectively).

Figures 4.13 and 4.14 show the results of modelling of heat treatment of wood underconstant 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 approachoverestimated the drying rate after a treatment time of 20 000 s. The S-REA yields acloser agreement with the experimental data (R2 of 0.995 and 0.991 for the S-REA andWhitaker’s approach, respectively). Similarly, the S-REA performs better than the othermodel at representing heat treatment of wood in Case 2 (R2 of 0.988 and 0.971 for theS-REA and Whitaker’s approach, respectively).

The temperature profiles of the various cases of the heat treatment are shown inFigure 4.14. Both the S-REA and the Whitaker’s approach describe the temperatureprofiles well along the treatment (R2 higher than 0.992 and 0.991 for the S-REA andWhitaker’s approach, respectively). Compared to the other approach, the S-REA hasthe advantage of generating profiles of the spatial water vapour concentration so thatbetter understanding of the process can be studied. In conclusion, it can be said that theS-REA performs comparably or even better than Whitaker’s approach in modelling theheat treatment of wood under a constant heating rate.

4.10 Summary

In this chapter, the diffusion-based models, Luikov’s and Whitaker’s approaches whichhave been used extensively applied to model drying processes are analysed. The for-mulation and limitation of Crank’s diffusion-based model and several other forms ofdiffusion-based model are discussed. Crank’s diffusion-based model offers advantagesof simple mathematical modelling but it is only valid for conditions where there areno shrinkage, isothermal, constant diffusivity, negligible external resistance and uni-form initial moisture content. Nevertheless, it is often implemented in literature withoutnecessary justification. The effective diffusivity of liquid water has been interpretedin experiments with poor documtation and control of boundary conditions, inducing alarge range in orders of magnitude in diffusivity values. Various forms of diffusion-basedmodels reported in literature have also been described. The crucial issues are boundaryconditions and the implementation of a source term. For a better understanding of thedrying process transport phenomena, a multiphase diffusion-based model needs to beimplemented. For this purpose, equilibrium and non-equilibrium diffusion-based modelscan be implemented and are both described. The equilibrium one assumes the moisturecontent inside pores of the samples equilibrates with water vapour concentration, whichessentially eliminates the source term in a way. The non-equilibrium one is more generalbut it requires the explicit formulation of the source term.

Luikov’s approach assumes that the mechanisms of moisture and heat transfer aresimilar, which is due to thes gradient of moisture content and temperature. Therefore,several coefficients are used to represent the interdependency of moisture content andtemperature. Whitaker’s approach uses detailed equations of momentum, heat and mass


conservation in solid, liquid and gas phases, followed by the volume averaging methodto describe drying processes.

The accuracy of the diffusion-based model has been compared to those of the L-REA and S-REA to model convective drying. It has been shown that the L-REA per-forms comparably, or sometimes even better than the diffusion-based model. While theresults are comparable, the L-REA has the advantages of simplicity in generating theparameters and mathematical formulation. The S-REA works very well, which offersadvantages where it not only accounts for the diffusion processes (Fickian type) but alsolocal evaporation or condensation.

Similarly, the accuracy of the Luikov’s and Whitaker’s approaches are compared withthose of the L-REA and S-REA to model heat treatment of wood, which is essentiallya drying process under linearly increased gas temperatures. It has been shown that theL-REA and S-REA perform comparably or even better than Luikov’s and Whitaker’sapproaches. While the results are similar, the L-REA offers the advantages of efficiencyin generating the parameters and simplicity of mathematical modelling. The S-REA isalso 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, 11baking, vi, vii, xvii, xviii, xxiii, 3, 7, 95–100, 117,

119–121, 158–168, 183, 184, 210, 211balance, xxxiiibiological, 17Biot, v, x, xxxiv, 11, 25, 27, 30, 43–45, 47, 50, 81,

117, 119, 146, 165, 171, 208boundary, 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, 127centre sample temperature, 63Characteristic Drying Rate Curve, xxxi, 20chemistry, v, xxxi, 1, 5, 15, 18, 19, 25, 29combustion, xxx, xxxvii, 1, 18, 19, 31Comparison, 185, 200computational fluid dynamics, xxxvi, 29concentration, 15condensation, 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, 45constant rate period, 17, 54, 102, 106, 108,

114continuum, 17convection, 45convective, 40core, xxii, 8, 11, 35, 43, 128, 133, 136, 139, 140,

145, 146, 157, 161, 167, 210cortex, 133coupling, 41Crank, vii, 171, 173, 206, 208

critical water content, 21–23cycle, 74

Darcy flow, 126deflection, 36deformable, 20, 166diffusion, 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, 48discrete, 18discretization, 12distributed, xxviii, xxxiv, 5, 8, 17, 18, 43driving force, xxxiii, 8, 17, 46, 50droplet, 36

effective diffusivity, 130effective liquid diffusivity, 126empirical, xxvii, xxviii, 17, 20, 47, 73, 105, 181energy, 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, 182equilibrium model, 121evaporation, 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, 198Fick, 169, 181finite element, 12, 193, 207first order, 34fissuring, 170, 209Food, 1

Index 213

Fourier, 169, 191, 193free volume, 124, 125, 167fundamental, 170

glass-filament, 36gradient, xxx, 17, 43, 51, 96, 122, 135, 136, 143,

145–147, 170, 172, 190–194, 197, 206

heat balance, 52heat transfer coefficient, 27, 42, 44, 45, 47, 52, 53,

54, 65, 82, 91, 97, 102, 159, 173, 178, 184,186, 194

infrared-heating, 100interface, 178intermitent, 141intermittent, 69internal mass transfer coefficient, 125internal surface area per unit volume, 125isotherm, xxx, 24, 26, 39, 49, 121, 182,

183isotherma, 126

kaolin, 170

Lagrangian, 12, 40Lewis, v, x, 11, 27, 43–45, 47–50, 117, 172, 209,

210linear, 88liquid, 9local phase change term, 121Luikov, 172, 190lumped, 34Lumped-REA, 29

macro-scale, 17Magnetic resonance imaging, 11mass balance, 25, 52Mass transfer, 9, 208mass 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, 12meso-scale, xxviii, 17micro-scale, 11, 17microwave, xxviii, 1, 5, 166, 170, 209modelling, 15, 116momentum, v, xxvii, xxxi, 5, 33, 40, 41, 43, 195,

206, 211multiphase, 177multiphase drying model, 122

non-equilibrium multiphase drying approach, 121Nusselt, 38, 42, 52

Ohm, 169optimization, 40

paper, xxx, xxxiii, 1, 18, 19, 24parabolic, 63physical, v, xxxiii, 6, 15, 18, 19, 22, 25, 29, 31, 80,

95, 169physics, 18polymer, 100pore, 121porosity, 124preservation, 1, 11

quality, 170

reaction engineering approach, 15relative humidity, xi, xiiresistance, 45, 176Reynolds, 38, 41, 42, 51

Sherwood, 24, 34, 38, 42, 51, 54, 62shrinkage, 124slab, 170solid concentration, 126source term, 29, 48, 49, 121, 122, 179, 190, 206spatial reaction engineering approach, 121spatial-REA, 121Spatial-REA, 29stress, ix, 3, 12, 13, 33, 95, 170structure, xxiv, 3, 6, 8, 9, 11, 12, 18, 27, 49, 95, 121,

122, 124, 127, 136, 143, 148, 158, 172, 175surface, 16surface sample temperature, 63surface vapor concentration, 25, 34, 62, 73, 74, 79,

80symmetry, 63, 64, 149

thermal conductivity, 48thermocouple, 36thick samples, vi, 61, 64, 66, 69, 73, 119, 166, 210thickness, 63thin layer, 34timber, 193transport, 121transport phenomena, 121turtuosity, 125

uniform, 35

vapor, 9vapor diffusivity, 125vaporization heat, 52volume-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, 34whey protein concentrate, 51Whitaker, 195

Whittaker, 172wood, 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

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Modelling Drying Processes a Reaction Engineering Approach

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