modelling intermittent microwave convective drying … · modelling intermittent microwave...

MODELLING INTERMITTENT

MICROWAVE CONVECTIVE DRYING

(IMCD) OF FOOD MATERIALS

Chandan Kumar

B. Sc. In Mechanical Engineering

Submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

Chemistry, Physics and Mechanical Engineering

Science and Engineering Faculty

Queensland University of Technology

2015

To my family (Basanta Kumar, Parul Rani, Dipika Roy, Ronzan Kumar

Roy, Ety Roy, Shailendra Nath Roy, Dipti Roy, Prabir Roy Sanjib)

Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials i

Keywords

Apple

Binary diffusion

Capillary diffusion

COMSOL Multiphysics

Convection drying model

Dielectric properties

Diffusion coefficient

Drying

Effective diffusivity

Electromagnetics

Energy efficiency

Equilibrium vapour pressure

Evaporative cooing

Evaporation rate

Experimental investigation

Food drying

Food quality

Heat and mass transfer

Intermittent microwave convective drying

Lambert’s Law

Mathematical modelling

Maxwell’s equations

Microwave

Microwave power distribution

Modelling

Multicomponent transport

Multiphase porous media model

Non-equilibrium evaporation

Non-uniform heating

Vapour pressure

ii Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials

Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials iii

Abstract

Drying of foodstuffs is an important and the oldest method of food processing.

However, drying is very energy-intensive process and consumes about 20–25% of

the energy used by food processing industry. The energy efficiency of the process

and the quality of the dried product are the two most crucial concerns in food drying.

The global energy crisis and increasing demand for quality dried food further

challenge researchers to explore innovative techniques in food drying to address

these issues. Intermittent microwave convective drying (IMCD) has proved to be an

advanced technology, which improves both energy efficiency and food quality in

drying. However, the physical understanding of the heat and mass transport

mechanism of IMCD is still not understood properly. To understand and optimize

IMCD, it is critical to develop mathematical models that can provide insight into the

physics involved in the process. Although there are some experimental investigations

of IMCD, there are until now no mathematical models to describe heat and mass

transfer in IMCD process for food.

This study aims to develop a mathematical model for IMCD of food materials.

First diffusion based and then multiphase porous media based IMCD models have

been developed in the study. The final model in this thesis is the first fundamental

and the most comprehensive multiphase model for IMCD, which considers 3D

electromagnetics coupled with multiphase porous media heat and mass transport. The

3D electromagnetics considered Maxwell’s equation and a multiphase transport

model considering three different phases: solid matrix, liquid water and gas (water

vapour and air) and considered pressure-driven flow, capillary diffusion, binary

diffusion, and evaporation. Thus, the model provides an in-depth understanding of

IMCD drying enabling investigation of moisture distribution, temperature

distribution and redistribution, evaporation, and fluxes due to different mechanisms.

Water and vapour fluxes obtained from the model showed that the pressure gradient

flow of water and vapour in IMCD is about 5–20 times higher than convective

drying, which significantly reduces the drying time in IMCD. Understanding of these

factors can, in turn, lead to an improvement in the food quality, the energy

efficiency, increased ability to automation and optimization.

iv Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials

List of Publications

Journal papers

1. C. Kumar, M. A. Karim and M. U. H. Joardder (2014) Intermittent

Drying of Food Products: A Critical Review. Journal of Food

Engineering, 121, 48–57 (Impact Factor 2.27).

doi: http://dx.doi.org/10.1016/j.jfoodeng.2013.08.014

2. C.A. Perussello, C. Kumar, F. Castilhosc, and M. A. Karim (2014) Heat

and mass transfer modelling of the osmo-convective drying of Yacon roots

(Smallanthus sonchifolius). Applied Thermal Engineering, 63(1), 23–32

(Impact Factor 2.12).

doi: http://dx.doi.org/10.1016/j.applthermaleng.2013.10.020

3. C. Kumar, G. J.Millar, and M. A. Karim (2015) Effective Diffusivity and

Evaporative Cooling in Convective Drying of Food Material. Drying

Technology, 33 (2), 227–237.

doi: http://dx.doi.org/10.1080/07373937.2014.947512

4. C. Kumar, M. U. H Joardder, M. A. Karim, G. J. Millar, and Z. M. Amin

(2014) Temperature redistribution modelling during intermittent

microwave convective heating. Procedia Engineering, 90(2014), 544–

549.

doi: http://dx.doi.org/10.1016/j.proeng.2014.11.770

5. C. Kumar, M. U. H. Joardder, T.W. Farrell, G. J. Millar, M. A. Karim

(2015) Modelling of Intermittent Microwave Convective Drying (IMCD)

of Apple. Drying technology, (Under review).

6. C. Kumar, M. U. H. Joardder, T.W. Farrell, M. A. Karim and G. J. Millar

(2015) Non-equilibrium multiphase porous media model for heat and mass

transfer during food drying. Journal of Food Engineering, (Under review).

http://dx.doi.org/10.1016/j.jfoodeng.2013.08.014

http://dx.doi.org/10.1016/j.applthermaleng.2013.10.020

http://dx.doi.org/10.1080/07373937.2014.947512

http://dx.doi.org/10.1016/j.proeng.2014.11.770

Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials v

7. C. Kumar, M. U. H. Joardder, T.W. Farrell, and M. A. Karim, (2015)

Multiphase porous media model for Intermittent microwave convective

drying (IMCD) of food: Model formulation and validation. Journal of

Thermal Scienc,. (To be submitted).

8. C. Kumar, M. U. H. Joardder, T.W. Farrell, and M. A. Karim (2015) A

3D coupled electromagnetic and multiphase porous media model for

IMCD of food material. Food and Bioprocess Technology, (To be

submitted).

9. M. U. H Joardder, M.A. Karim, R.J. Brown, and C. Kumar (2014)

Determination of effective moisture diffusivity of banana using

thermogravimetric analysis. Procedia Engineering, 90, 538–543.

Doi: http://dx.doi.org/10.1016/j.proeng.2014.11.769

10. M. U. H. Joardder, M.A. Karim, and C. Kumar (2013) Effect of

temperature distribution on predicting quality of microwave-dehydrated

food. Journal of Mechanical Engineering and Sciences, 5, 562–568.

Doi: http://dx.doi.org/10.15282/jmes.5.2013.2.0053

11. M. U. H. Joardder, C. Kumar, and M. A. Karim (2015) Food Structure:

Its Formation and Relationships with Other Properties. Critical Reviews in

Food Science and Nutrition, (available online).

12. M. U. H. Joardder; C. Kumar, R. J Brown, and M. A. Karim (2015) A

micro-level investigation of the solid displacement method for porosity

determination of dried food. Journal of Food Engineering,166, 156–164

13. M. U. H Joardder, R. J. Brown, C. Kumar, and M. A. Karim (2015) Effect

of cell wall properties on porosity and shrinkage of dried apple.

International Journal of Food Properties, 18(10), 2327-2337.

14. C. Kumar, T. W. Farrell, M.A. Karum (2015), Multiphase porous media

model considering bound water for drying of agricultural product. Journal

of Thermal Science, (To be submitted).

http://dx.doi.org/10.15282/jmes.5.2013.2.0053

vi Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials

Book:

1. M. U. H. Joardder, C. Kumar, M. A. Karim, and R. J. Brown

(2015) Porosity: Establishing the Relationship between Drying Parameters

and Dried Food Quality. Springer (In press).

Peer reviewed conference paper

1. C. Kumar, G. J. Millar, T. W. Farrell, and M. A. Karim (2015)

Multiphase porous media transport in apple tissue during drying. ICEF12–

12th International Congress on Engineering and Food, 14–18 June 2015,

Québec, Canada.

2. C. Kumar, M. U. H. Joardder, T. W. Farrell, G. J. Millar, and M.A. Karim

(2014) Multiphase porous media model for heat and mass transfer during

drying of agricultural products. In 19th Australasian Fluid Mechanics

Conference, 8–11 December 2014, RMIT University, Melbourne, VIC.

3. M. U. H. Joardder, M. A. Karim, C. Kumar, and R. J. Brown (2014)

Effect of cell wall properties on porosity and shrinkage during drying of

Apple. 1st International Conference on Food Properties (iCFP2014),

Kuala Lumpur, Malaysia, January 24–26, 2014 (Best paper award).

4. C. Kumar, M. A. Karim, M. U. H. Joardder, G. J. Miller, M. A. Karim,

and Z. M. Amin (2014) Intermittent microwave convective heating:

modelling and experiments. 10th International Conference on Mechanical

Engineering, 20–21 June, 2014, BUET, Dhaka, Bangladesh.

5. M. U. H. Joardder, M. A. Karim, C. Kumar, and R. J. Brown (2013)

Fractal dimension of dried foods: A correlation between microstructure

and porosity. Food Structures, Digestion and Health International

Conference 22–24 October, 2013 - Melbourne, Australia.

6. C. Kumar, M. A. Karim, S. C. Saha, M. U. H. Joardder, R. J. Brown, and

D. Biswas (2013) Multiphysics modelling of convective drying of food

materials. Global Engineering, Science and Technology Conference, 28–

29 December 2013, Dhaka, Bangladesh.

Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials vii

7. M. U. Joardder, M. A. Karim, and C. Kumar (2013) Better Understanding

of Food Material on the Basis of Water Holding Capacity. International

Conference on Mechanical, Industrial and Material Engineering, 1–3

November, 2013, Rajshahi, Bangladesh.

8. M. U. H. Joardder, M. A. Karim, and C. Kumar (2013) Effect of moisture

and temperature distribution on dried food Microstucture and Porosity.

From Model Foods to Food Models, the DREAM Project International

Conference, 24– 26 June 2013, Nantes, France.

9. M. U. H. Joardder, M. A. Karim, and C. Kumar (2013) Determination of

Moisture Diffusivity of Banana Using Thermogravimetric Analysis. 10th

International Conference on Mechanical Engineering, 20–21 June, 2013,

BUET, Dhaka, Bangladesh.

10. C. Kumar, M. A. Karim, M. U. H. Joardder, and G. J. Miller (2012)

Modeling Heat and Mass Transfer Process during Convection Drying of

Fruit. 4th International Conference on Computational Methods, 25–27

November, 2012, Gold Coast, Australia.

Posters presentation

1. C. Kumar, M. A. Karim, and G. J. Millar (2014) Multiphase Porous

Media Model for Microwave Convective Drying of Agricultural Products.

In Research Showcase: Opportunities with ACIAR and the Rural Research

and Development Corporations, 21 February 2014.

2. J. Davies, C. Kumar, Z. M. Amin, and M. A. Karim (2013) Design and

construction of a microwave convective food dryer. In Queensland

University of Technology, Science & Engineering Faculty Showcase, 31

October 2013, Queensland University of Technology, Brisbane, QLD.

3. M. U. H. Joardder, C. Kumar, M. A. Karim and R. J. Brown (2013)

Fractal dimension of dried foods: a correlation between microstructure and

porosity. In Food Structures, Digestion and Health International

Conference, 21–24 October 2013, Bayview Eden Hotel, Melbourne, VIC.

viii Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials

4. M. U. H. Joardder, C. Kumar, M. A. Karim, and R. J. Brown (2014)

Effect of cell wall bound water on pore formation and food quality during

drying, In Research Showcase: Opportunities with ACIAR and the Rural

Research and Development Corporations, 21 February 2014, QUT,

Australia.

Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials ix

Table of Contents

Keywords .................................................................................................................................................i

Abstract ................................................................................................................................................. iii

List of Publications ................................................................................................................................iv

Table of Contents ...................................................................................................................................ix

List of Figures ...................................................................................................................................... xii

List of Tables ........................................................................................................................................ xv

List of Abbreviations ............................................................................................................................ xvi

Nomenclature ..................................................................................................................................... xvii

Statement of Original Authorship ........................................................................................................ xix

Acknowledgements ............................................................................................................................... xx

INTRODUCTION ....................................................................................................... 1 CHAPTER 1:

1.1 Background and motivation ......................................................................................................... 1

1.2 Research problems ....................................................................................................................... 2

1.3 Aims and objectives ..................................................................................................................... 3

1.4 Significance and scope ................................................................................................................. 3

1.5 Thesis outline ............................................................................................................................... 4

LITERATURE REVIEW ........................................................................................... 7 CHAPTER 2:

2.1 Food drying .................................................................................................................................. 9

2.2 Microwave assisted drying......................................................................................................... 11

2.3 Microwave Convective Drying (MCD) ..................................................................................... 13

2.4 Intermittent Microwave Convective Drying (IMCD) ................................................................ 15

2.5 Modelling of drying and challenges ........................................................................................... 17 Empirical models ............................................................................................................ 18 2.5.1

Diffusion based (single phase) models ........................................................................... 18 2.5.2

Multiphase models .......................................................................................................... 19 2.5.3

MCD and IMCD model .................................................................................................. 20 2.5.4

2.6 Summary of literature and research gaps ................................................................................... 21

SINGLE PHASE MODEL FOR CONVECTION DRYING ................................. 23 CHAPTER 3:

3.1 Abstract ...................................................................................................................................... 25

3.2 Introduction ................................................................................................................................ 25

3.3 Model development ................................................................................................................... 28 Governing equations ....................................................................................................... 29 3.3.1

Initial and boundary conditions ...................................................................................... 29 3.3.2

Input parameters ............................................................................................................. 30 3.3.3

3.4 Simulation methodology ............................................................................................................ 33

3.5 Drying experiments .................................................................................................................... 34

3.6 Results and discussion ............................................................................................................... 36

3.7 Conclusions ................................................................................................................................ 43

3.8 Acknowledgements .................................................................................................................... 44

x Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials

3.9 Funding ...................................................................................................................................... 44

SINGLE PHASE MODEL FOR IMCD USING LAMBERT’S LAW .................. 45 CHAPTER 4:

4.1 Abstract ...................................................................................................................................... 46

4.2 Introduction................................................................................................................................ 46

4.3 Mathematical modelling ............................................................................................................ 48 Governing equations ....................................................................................................... 49 4.3.1

Initial and boundary conditions ...................................................................................... 50 4.3.2

Modelling of microwave power absorption using Lamberts Law .................................. 50 4.3.3


4.4 Materials and methods ............................................................................................................... 55

4.5 Simulation procedure ................................................................................................................. 57

4.6 Results and discussion ............................................................................................................... 58 Incident power absorption by experiments ..................................................................... 58 4.6.1

Average moisture curve .................................................................................................. 59 4.6.2

Temperature .................................................................................................................... 60 4.6.3

Moisture and temperature distribution ............................................................................ 63 4.6.4

Equilibrium vapour pressure .......................................................................................... 64 4.6.5

Absorbed power distribution .......................................................................................... 65 4.6.6

4.7 Conclusions................................................................................................................................ 66

MULTIPHASE MODEL FOR CONVECTION DRYING OF FOOD ................. 68 CHAPTER 5:

5.1 Abstract ...................................................................................................................................... 69

5.2 Introduction................................................................................................................................ 69

5.3 Mathematical model .................................................................................................................. 71 Problem description and assumptions ............................................................................ 72 5.3.1

Governing equations ....................................................................................................... 73 5.3.2

Mass balance equations .................................................................................................. 73 5.3.3

Continuity equation to solve for pressure ....................................................................... 76 5.3.4

Energy equation .............................................................................................................. 76 5.3.5

Evaporation rate .............................................................................................................. 77 5.3.6

Initial conditions ............................................................................................................. 79 5.3.7

Boundary conditions ....................................................................................................... 79 5.3.8


5.4 Experiments ............................................................................................................................... 86

5.5 Numerical solution and simulation methodology ...................................................................... 86

5.6 Results and discussion ............................................................................................................... 87 Moisture content ............................................................................................................. 87 5.6.1

Distribution and evolution of water and vapour ............................................................. 88 5.6.2

Temperature curve .......................................................................................................... 89 5.6.3

Vapour pressure, equilibrium vapour pressure, and saturated pressure .......................... 90 5.6.4

Evaporation rate .............................................................................................................. 91 5.6.5

Vapour and water fluxes ................................................................................................. 92 5.6.6

5.7 Conclusions................................................................................................................................ 96

MULTIPHASE MODEL FOR IMCD USING LAMBERT’S LAW .................... 97 CHAPTER 6:

6.1 Abstract ...................................................................................................................................... 98

6.2 Introduction................................................................................................................................ 98

6.3 Mathematical model ................................................................................................................ 100 Problem description and assumptions .......................................................................... 101 6.3.1

Governing equations ..................................................................................................... 101 6.3.2

Evaporation rate ............................................................................................................ 104 6.3.3

Initial conditions ........................................................................................................... 105 6.3.4

Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials xi

Boundary conditions ..................................................................................................... 106 6.3.5

Input parameters ........................................................................................................... 106 6.3.6

Microwave power absorption ....................................................................................... 109 6.3.7

Dielectric constant ........................................................................................................ 110 6.3.8

6.4 Materials and methods ............................................................................................................. 111

6.5 Numerical solution ................................................................................................................... 112

6.6 Results and discussion ............................................................................................................. 112 Moisture content and temperature ................................................................................ 113 6.6.1

Distribution and evolution of water and vapour ........................................................... 114 6.6.2

Temperature curve ........................................................................................................ 115 6.6.3

Gas pressure .................................................................................................................. 116 6.6.4

Vapour pressure, equilibrium vapour pressure, and saturated pressure ........................ 117 6.6.5

Evaporation ................................................................................................................... 118 6.6.6

Vapour pressure distribution ......................................................................................... 119 6.6.7

Vapour and water fluxes ............................................................................................... 120 6.6.8

Limitation of Lambert’s Law ........................................................................................ 123 6.6.9

6.7 Conclusions .............................................................................................................................. 123

MULTIPHASE MODEL FOR IMCD USING MAXWELL’S EQUATIONS ... 125 CHAPTER 7:

7.1 Abstract .................................................................................................................................... 127

7.2 Introduction .............................................................................................................................. 127

7.3 Model development ................................................................................................................. 129 Geometry and problem description ............................................................................... 129 7.3.1

Maxwell’s equation for electromagnetics and heat generation ..................................... 131 7.3.2

Dielectric properties...................................................................................................... 132 7.3.3

Multiphase porous media transport model .................................................................... 132 7.3.4

Initial conditions ........................................................................................................... 136 7.3.5

Boundary conditions ..................................................................................................... 137 7.3.6

Input parameters ........................................................................................................... 137 7.3.7

Numerical solution........................................................................................................ 140 7.3.8

7.4 Materials and methods ............................................................................................................. 141

7.5 Results and discussion ............................................................................................................. 143 Experimental validation of temperature and moisture content ..................................... 143 7.5.1

Internal temperature distribution .................................................................................. 146 7.5.2

Moisture distribution .................................................................................................... 147 7.5.3

Vapour concentration distribution ................................................................................ 148 7.5.4

Pressure ......................................................................................................................... 149 7.5.5

Vapour pressure distribution ......................................................................................... 150 7.5.6

Water and vapour fluxes ............................................................................................... 150 7.5.7

7.6 Conclusions .............................................................................................................................. 154

CONCLUSION AND FUTURE RECOMMENDATION .................................... 157 CHAPTER 8:

8.1 Overall summary ...................................................................................................................... 157

8.2 Conclusions .............................................................................................................................. 158

8.3 Contribution to knowledge and significance ............................................................................ 160

8.4 Limitations ............................................................................................................................... 161

8.5 Future direction ........................................................................................................................ 162

BIBLIOGRAPHY ............................................................................................................................. 165

APPENDICES ................................................................................................................................... 177 Appendix A Implementation of the model in mathematical interface of COMSOL .............. 177 Appendix B The conversions of moisture content ................................................................... 178

xii Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials

List of Figures

Figure 1-1. Organization of the dissertation ........................................................................................... 5

Figure 2-1.Typical drying rate curve (Okos et al., 2006) ....................................................................... 9

Figure 2-2. Schematic diagram depicting the dipolar and ionic loss mechanisms and their

contributions to the dielectric properties as a function of frequency (Metaxas,

1996b). ................................................................................................................................. 11

Figure 2-3. Drying curves of whole mushrooms under different drying methods (Orsat et al.,

2007) .................................................................................................................................... 12

Figure 2-4. General classification of drying models in literature ......................................................... 17

Figure 2-5. Modelling approach of heat generation due to microwave ................................................ 20

Figure 3-1. (a) Actual geometry of the sample slice and (b) Simplified 2D axisymmetric model

domain. ................................................................................................................................. 28

Figure 3-2. Simulation strategy in COMSOL multiphysics ................................................................. 33

Figure 3-3. Moisture evolution obtained for experimental and simulation with shrinkage-

dependent, temperature-dependent and average effective diffusivities ................................ 37

Figure 3-4. Temperature evolution obtained for experimental and simulation with shrinkage-

dependent, temperature-dependent and average effective diffusivities ................................ 38

Figure 3-5. Temperature curve for shrinkage dependent diffusivity .................................................... 39

Figure 3-6. Evolution of inward (convective), outward (evaporative) and total

(convective+evaporative) heat flux ...................................................................................... 40

Figure 3-7. (a) Moisture and (b) temperature distribution in the food after 400 minutes of

drying ................................................................................................................................... 41

Figure 3-8. Moisture content for different air temperature for velocity 0.7m/s .................................... 42

Figure 3-9. Moisture content for different air velocity at 600C ............................................................ 43

Figure 4-1. 3D apple slice and 2D an axisymmetric domain showing symmetry boundary and

transfer boundary (arrow) .................................................................................................... 49

Figure 4-2. Intermittency function ........................................................................................................ 57

Figure 4-3. Flow chart showing the modelling strategy in COMSOL Multiphysics ............................ 58

Figure 4-4. Microwave power absorption for different loading volume............................................... 59

Figure 4-5. Drying curve for IMCD (experiments and model) and convective drying ........................ 60

Figure 4-6. Temperature curve obtained from the model ..................................................................... 61

Figure 4-7. Thermal images of top surface at selected times ................................................................ 62

Figure 4-8. Moisture distribution inside the sample ............................................................................. 63

Figure 4-9. Temperature distribution inside the sample ....................................................................... 64

Figure 4-10. Evolutions of equilibrium vapour pressure at the surface of the sample .......................... 65

Figure 4-11. Absorption of microwave power along the length of the sample at 75mins .................... 66

Figure 5-1. Schematic showing 3D sample, 2D axisymmetric domain and Representative

Elementary Volume (REV) with the transport mechanism of different phases ................... 72

Figure 5-2. Gas (kr,g) and water (kr,w) relative permeabilities of apple tissues as a function of

saturation. ............................................................................................................................. 83

Figure 5-3. Typical variation of capillary force as a function of liquid saturation in porous

media (Bear, 1972) ............................................................................................................... 84

Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials xiii

Figure 5-4. Mesh for the simulation. .................................................................................................... 86

Figure 5-5. Comparison between predicted and experimental values of average moisture

content during drying ........................................................................................................... 87

Figure 5-6. Spatial distribution of water saturation with times ............................................................. 88

Figure 5-7. Spatial distribution of vapour with different time .............................................................. 89

Figure 5-8. Surface and centre temperature obtained from model ........................................................ 90

Figure 5-9. Vapour pressure, equilibrium vapour pressure and saturation pressure at surface ............. 91

Figure 5-10. Spatial distribution of evaporation rate at different drying times ..................................... 92

Figure 5-11. Water flux due to capillary diffusion ............................................................................... 93

Figure 5-12. Water flux due to gas pressure ......................................................................................... 94

Figure 5-13. Vapour flux due to binary diffusion ................................................................................. 95

Figure 5-14. Vapour flux due to gas pressure ....................................................................................... 95

Figure 6-1 . Schematic showing 3D sample, 2D axisymmetric domain and Representative

Elementary Volume (REV) with the transport mechanism of different phases ................. 101

Figure 6-2. Mesh for the simulation ................................................................................................... 112

Figure 6-3. Comparison between predicted and experimental values of average moisture

content during drying ......................................................................................................... 113

Figure 6-4. Spatial distribution of water saturation with times ........................................................... 114

Figure 6-5. Spatial distribution of vapour with different times ........................................................... 115

Figure 6-6. Comparison of surface temperature between experimental and model ............................ 116

Figure 6-7. spatial distribution of total pressure across the half thickness the sample in different

times ................................................................................................................................... 117

Figure 6-8. Vapour pressure, equilibrium vapour pressure and saturation pressure at surface ........... 118

Figure 6-9. Spatial distribution of evaporation rate at different drying times ..................................... 119

Figure 6-10. Spatial distribution of evaporation rate at different drying times ................................... 120

Figure 6-11. Water flux due to capillary at different drying times ..................................................... 121

Figure 6-12. Water flux due to gas pressure at different drying times ................................................ 121

Figure 6-13. Vapour flux due to binary diffusion at different drying times ....................................... 122

Figure 6-14. Vapour flux due to gas pressure at different drying times ............................................. 122

Figure 7-1. a) The computational domain for the IMCD drying simulation, b) Food sample and

representative elementary volume (REV) showing transport mechanism involved in

the simulation ..................................................................................................................... 130

Figure 7-2. Flow chart showing the modelling strategy in COMSOL Multiphysics .......................... 141

Figure 7-3. Intermittency of microwave power considered in the simulation and experiment ........... 142

Figure 7-4. Average moisture content obtained from experiments and simulation ............................ 143

Figure 7-5. Temperature distribution obtained from experiment and simulations .............................. 145

Figure 7-6. Temperature profile along horizontal centreline of the sample at different times ............ 146

Figure 7-7. Saturation profile along horizontal centreline of the sample for different times .............. 147

Figure 7-8. Vapour density profile along horizontal centreline of the sample at different times ........ 148

Figure 7-9. Pressure profile along horizontal centreline of the sample for different times ................. 149

Figure 7-10. Vapour pressure along the horizontal centreline of the sample ...................................... 150

Figure 7-11. Water capillary flux along horizontal centreline of the sample at different times.......... 151

xiv Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials

Figure 7-12. Water flux due to gas pressure along horizontal centreline of the sample for

different times .................................................................................................................... 152

Figure 7-13. Vapour flux due to gas pressure along horizontal centreline of the sample for

different times .................................................................................................................... 153

Figure 7-14. Vapour flux due to gas pressure along horizontal centreline of the sample for

different times .................................................................................................................... 153

Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials xv

List of Tables

Table 2-1. Summery of microwave assisted convective heating and drying of food material ............... 14

Table 2-2. Summary of intermittent microwave assisted convective heating and drying of food

material ................................................................................................................................ 16

Table 3-1. Input parameters for the model ............................................................................................ 30

Table 4-1. Input properties of the model ............................................................................................... 52

Table 4-2. Power absorption ratio for microwave power (100W, 200W and 300W) for different

sample volume ...................................................................................................................... 58

Table 4-3: Centre temperature of apple surface from experiment and model ....................................... 61

Table 5-1. Input properties for the model ............................................................................................. 80

Table 6-1. Input properties for the model ........................................................................................... 107

Table 6-2. Comparison of experimental and model temperature at centre of top surface at

different times ..................................................................................................................... 114

Table 7-1. Input parameters for the model .......................................................................................... 137

xvi Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials

List of Abbreviations

IMCD Intermittent Microwave Convective Drying

CMCD continuous Microwave Convective Drying

MW Microwave

PL Power Level (microwave power level)

PR Pulse Ratio (microwave intermittency)

RH Relative humidity (%)

db Dry basis

wb Wet basis

Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials xvii

Nomenclature

wc Mass concentrations of water (kg/m3)

vc Mass concentrations of vapour (kg/m3)

ac Mass concentrations of air (kg/m3)

effpc Effective specific heat (J/kg/K)

pwc Specific heat capacity of water (J/kg/K)

pgc Specific heat capacity of gas (J/kg/K)

psc Specific heat capacity of solid (J/kg/K)

cD Capillary diffusivity (m2/s)

TD Thermal diffusivity (m2/s)

geffD , Effective binary diffusivity of vapour and air (m2/s)

vaD Binary diffusivity of vapour and air (m2/s)

H Sample thickness (m)

gh Enthalpy of gas (J)

wh Enthalpy of water (J)

fgh Latent heat of evaporation (J/kg)

mvh Mass transfer coefficient (m/s)

Th Heat transfer coefficient (W/m2/K)

Kevap Evaporation constant

wk Intrinsic permeability of water (m2)

wrk , Relative permeability of water (m2)

gk Intrinsic permeability of gas (m2)

grk , Relative permeability of gas (m2)

effk Effective thermal conductivity (W/m/K)

gthk , Thermal conductivity of gas (W/m/K)

wthk , Thermal conductivity of water (W/m/K)

sthk , Thermal conductivity of solid (W/m/K)

dbM Moisture content dry basis

wbM Moisture content wet basis

gM Molecular weight of gas (kg/mol)

vM Molecular weight of vapour (kg/mol)

wn

Water mass flux (kg/m2s)

vn

Vapour mass flux (kg/m2s)

gn

Gas mass flux (kg/m2s)

totalvn ,

Total vapour flux at the surface (kg/m

2s)

P Total gas pressure (Pa)

xviii Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials

0P Incident microwave power (W)

vp Partial pressure of vapour (Pa)

ap Partial pressure of air (Pa)

cp Capillary pressure (Pa)

eqvp , Equilibrium vapour pressure (Pa)

satvP , Saturation vapour pressure (Pa)

airvp , Vapour pressure of ambient air (Pa)

ambP Ambient pressure (Pa)

R Universal gas constant (J/mol/K)

evapR Evaporation rate of liquid water to water vapour (kg/m3s)

wS Saturation of water

gS Saturation of gas

0wS Initial water saturation

0vS Initial saturation of vapour

0gS Initial gas saturation

T Temperature of product (0C)

airT Drying air temperature (0C)

V Drying air velocity (m/s)

Z Distance from vertical axis from origin (m)

V Representative elementary volume (m3)

gV Volume of gas (m3)

wV Volume of water (m3)

sV Volume of solid (m3)

eqt Equilibration time (1/s)

Apparent porosity

0 Initial equivalent porosity,

w Viscosity of water (Pa.s)

g Viscosity of gas (Pa.s)

v Mass fraction of vapour

a Mass fraction of air

s Solid density (kg/m3)

w Density of water (kg/m3)

g Density of gas (kg/m3)

g Density of gas phase (kg/m3)

eff Effective density (kg/m3)

QUT Verified Signature

xx Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials

Acknowledgements

First I would like to thank God for giving me the opportunity, strengths and

blessing in completing this thesis. Then, I would like to thank my principle

supervisor, Dr Azharul Karim, for his supervision and effort. I also pay my complete

gratitude to him for all his great support to initiate my higher studies in QUT. Thank

you very much for your invaluable advice, encouragement and patience throughout

this research. I also appreciate the excellent support, helpful comments, guidance and

advice given by my co-supervisor, Professor Graeme Millar.

I would also like to express my love and gratitude to my beloved family

members, for their understanding and endless love. In particular, my father for his

patience and for his encouragement to completion of this thesis although being

constantly sick since I left Bangladesh. I am really proud of my younger brother,

Ronzan Kumar Roy, for taking care of parents during this time. Special thanks to my

wife, Dipika Roy, for bearing with me all during this study.

Furthermore, I would like to thank all my colleagues and friends for

motivation, advice and help rendered: special thanks to M.U.H. Joardder for his

continued support during my candidature especially with experiments and idea

development. Thanks to Dr Camila Augusto Perrusello for being party to initial

modelling works. Thanks to all undergraduate students who were involved in

different parts of the project. I would like to thanks Dr Suvash Saha, Prof. Ian

Turner, A/Prof. Troy Farrell, Dr Wijitha Senadeera, Dr Zakaria Amin, and Dr

Hussain Nyeem for their fruitful discussion at different stage of my PhD journey.

I gratefully acknowledge financial support from the Australian Postgraduate

Award (APA) scholarship and the International Postgraduate Research Scholarship

(IPRS), funded by the Australian government. I would like to acknowledge the

research facilities and software provided by QUT. Thanks to QUT IT helpdesk and

High Performance Computing (HPC) for helping with running the computation and

CARF for their experimental facilities. Last, but by no means the least, I would like

to thank the government of the People’s Republic of Bangladesh for providing me

high quality education from primary school to university for almost free.

Modelling Intermittent Microwave Convective Drying (IMCD) of Food Materials xxi

Chapter 1: Introduction 1

Introduction Chapter 1:

This chapter outlines the background and motivation of the research, research

problems and objectives. It also describes the significance and scope, followed by an

outline of the contributing chapters.

1.1 BACKGROUND AND MOTIVATION

Currently, one-third of produced foods are wasted annually due to a lack of

proper processing and preservation technique (Gustavsson et al., 2011; UN, 2007).

This loss is even greater in the developing countries, amounting to 30–40% of

seasonal fruit and vegetables (Karim & Hawlader, 2005b). On the other hand,

according to the UN food agency, everyday 18,000 children die of hunger and

malnutrition and 850 million people go to bed every night with empty stomachs (UN,

2007). The World Food Programme (WFP) identifies hunger as the number one

health risk and it kills more people every year than AIDS, malaria and tuberculosis

combined (World Food Programme, 2012). Therefore, proper food processing must

be emphasized to reduce this massive loss, promote food security and combat

hunger.

Drying is one of the easiest and oldest methods of food processing and

preservation, which prevents food from microbial spoilage. It increases shelf life,

reduces weight and volume thus minimizing packing, storage, and transportation cost

and enables storage of food under an ambient environment (Kumar et al., 2015).

However, it is probably the most energy intensive technique of the major industrial

processes (Kudra, 2004) and accounts for up to 15% of all industrial energy usage

(Chua et al., 2001a). Moreover, drying causes changes in the food qualities including

discolouration, aroma loss, textural changes, nutritional degradation, and changes in

physical appearance and shape (Quirijns, 2006). Researchers have been striving to

improve energy efficiency and product quality in food drying for many years.

Intermittent Microwave Convective Drying (IMCD) is one approach to increases

both energy efficiency and product quality (Kumar et al., 2014).

Although there are some experimental investigations of IMCD, modelling

studies of this process remain under-developed. The modelling of IMCD is essential

2 Chapter 1: Introduction

to understanding the physical mechanism of heat and mass transfer and finally to

optimize the process. Therefore, this research aims to develop a comprehensive

model for IMCD.

1.2 RESEARCH PROBLEMS

Although there have been some studies highlighting the advantages of

intermittent microwave convective drying (IMCD) of food materials, all of these are

experimental. There have been no studies that have undertaken mathematical

modelling or which have led to a physical understanding of the heat and mass

transfer phenomenon occurring within the material during IMCD drying of food

material. Proper understanding of the internal heat and mass transfer mechanism

involved is essential for optimization of the drying process. Experimental study of

IMCD showed that the IMCD is much faster compared to convective drying;

however, the mechanism behind this scenario is not well understood. This is because

the water and vapour fluxes due to the various mechanisms in IMCD and their

comparison with convection drying, has not been investigated. A comprehensive

mathematical model for IMCD representing the physics behind the process can

enhance the understanding of the transport mechanism of heat and mass, and fluxes

inside the food. However, such a model is yet to be developed for IMCD. This

research investigation will contribute to knowledge providing better insight of heat

and mass transfer process involved in IMCD.

The microwave power level and intermittency during IMCD are the two main

factors that control the heating and drying rates. The intermittency of microwave

energy during IMCD allows the temperature to drop, and thus, prevents overheating.

Therefore, the investigation of temperature rise during microwave heating period and

fall during tempering period is essential in IMCD. Currently, there is no such study

that investigated this either experimentally or by modelling. In this study, the

temperature fluctuations will be investigated during IMCD both experimentally and

mathematical modelling.

Mathematical modelling of moisture transport inside food can be developed by

two approaches: (1) a single phase model which considers only water is present in

the food material, and (2) a multiphase porous media model which considers

transport of liquid water, vapour, air insider the food materials. The multiphase


models are more comprehensive and provide better insight into the transport

mechanisms. These models are divided into two groups: equilibrium and non-

equilibrium. Most of the multiphase models of food drying consider equilibrium.

However, equilibrium condition may not be valid at the surface of food where

moisture content is lower. Therefore, the non-equilibrium approach is a more

realistic representation of the physical situation during drying (J. Zhang & Datta,

2004). In this study, non-equilibrium multiphase porous media models were

developed for food drying.

1.3 AIMS AND OBJECTIVES

The primary aim of this work is to understand the mechanisms involved in heat

and mass transfer process during IMCD by means of a comprehensive mathematical

model. Since convection-drying modelling is a prerequisite for developing an IMCD

model, a convection-drying model is initially developed and then followed by IMCD

models. These models were gradually improved by incorporating more realistic

physical phenomena that takes place during drying. In order to achieve the primary

aim, the specific objectives of this work were to:

develop single phase mathematical models for convection drying and then

for IMCD

validate the single phase models using experimental data

develop multiphase porous media models for both convection drying and

IMCD

validate the multiphase models with experimental data

investigate the heat and mass transport mechanism of different phases

during IMCD

investigate the temperature distribution and re-distribution during IMCD

1.4 SIGNIFICANCE AND SCOPE

This research has developed a novel and the first comprehensive mathematical

model of IMCD for food, which enables an understanding of heat and mass transport

during IMCD. The model considers 3D electromagnetics for microwave heating and

multiphase multicomponent transport of heat and mass. The model was used to

investigate the water fluxes due to gas pressure and capillary pressure, vapour fluxes


due to binary diffusion and gas pressure, and temperature redistribution during the

process. The temperature distribution and its fluctuations have also been investigated

using the model. Since the temperature distribution and fluctuation during IMCD is

the key to avoid overheating of the product, the model in this work can contribute

significantly to improve food product quality. The challenges of developing an

IMCD model have been identified and discussed. Both Lambert’s Law and

Maxwell’s equations were considered to model microwave heat generation during

IMCD with their respective advantages and disadvantages.

The multiphase model, which is an advanced approach to modelling the drying

process, enables investigation of temporal and spatial profiles of temperature, liquid

water, water vapour, and air inside the food material. The model was used to

illustrate all modes of transport inside the food including capillary diffusion,

convection, and evaporation, which is not possible to investigate through

experiments or using other simpler models. Moreover, the non-equilibrium

formulation of evaporation in the model has been taken into consideration to

calculate vapour phase separately. Consideration of non-equilibrium evaporation of

the model allows a direct solution to the system of equations and easy

implementation of the model in commercial software.

The insights of physical phenomena acquired from the models in this thesis

will make a great contribution to the field of drying. The mathematical models used

in this research are fundamental; therefore, they can be easily modified and adapted

to any food material without considerable effort. Furthermore, the models developed

in this research will also be useful to future researchers to develop more

comprehensive models for other food processing applications and optimization of

those processes. Successful implementation of these theoretical models in the food

industry can lead to a significant improvement in food quality, energy efficiency, and

optimization.

1.5 THESIS OUTLINE

This thesis is organized into a total of eight chapters. Chapter 1 and Chapter 2

present the introduction and literature review. Chapter 3 to Chapter 7 presents the

gradual development of comprehensive IMCD models and those chapters contribute

to individual publications.


The thesis progresses from simple diffusion based models (Chapter 3 and 4) to

comprehensive multiphase porous media models (Chapter 5 to 7). An overview of

the dissertation is also illustrated in Figure 1-1 and chapter-wise brief discussion is

given below.

Figure 1-1. Organization of the dissertation

In Chapter 1 (this chapter), background and motivation, research problems,

objectives, significant and outline of the thesis are all provided.

In Chapter 2, a review of contemporary literature is presented. This chapter

begins with the background of food drying to identify critical factors and then

reviews microwave-assisted drying. Then it provided an extensive review of

microwave convective drying, intermittent microwave convective drying and their

modelling. Finally, the key research problems and gaps are identified and presented.

In Chapter 3, a single phase convection drying model is developed considering

three different effective diffusivities, namely, moisture dependent, temperature

dependent and an average of those two. This model will provide the basis for

choosing effective diffusivity for single-phase diffusion-based model. Evaporative

cooling phenomena during the process were also observed and explained.

In Chapter 4, a single phase IMCD model was developed by adding the

convection drying model and intermittent microwave heating. This model considered


Lambert’s Law for microwave heat generation, and drawbacks and advantages of

this model were identified and analysed.

In Chapter 5, a multiphase porous media model is implemented for convection

drying. This chapter presents fundamental formulations of heat, mass, and

momentum transfer along with input parameters and variables for the model.

Transport of different phases due to pressure driven, capillary diffusion, and binary

diffusion are investigated. Moreover, the model is validated with experimental

temperature and moisture data.

In Chapter 6, a multiphase porous media IMCD model is developed

considering Lambert’s Law for microwave heat generation. The enhancements of

moisture and vapour fluxes due to incorporating microwave heat generation is

illustrated and discussed.

In Chapter 7, a three-dimensional electromagnetics heat generation using

Maxwell’s equation are coupled with a multiphase porous media model to develop

the first and most comprehensive IMCD model. From this model, non-uniform

temperature distribution data are presented as are the main advantages of using

Maxwell’s equation instead of Lamberts Law. The model is validated with

experimental temperature and moisture data. Comprehensive transport mechanisms

of different phases in IMCD were investigated.

In Chapter 8, the major conclusions, contribution to knowledge, limitations,

and recommendations of this research were presented.

Chapter 2: Literature Review 7

Literature Review Chapter 2:

This Chapter will inform our approach towards the main research topic starting

with a detail background of the subject, and a comprehensive and most relevant

literature will be presented to identify research gaps. Part of this chapter has been

published in the following review paper:

C. Kumar, M. A. Karim, and M. U. H. Joardder (2014) Intermittent

Drying of Food Products: A Critical Review. Journal of Food

Engineering, 121, 48–57 (Impact Factor 2.27).

doi: http://dx.doi.org/10.1016/j.jfoodeng.2013.08.014

The signed statement of contribution page in QUT’s format is attached on the

next page.

This chapter, firstly, will discuss the mechanism of food drying and highlights

the crucial factors involved in the food drying process (Section 2.1). The following

sections (Section 2.2 and 2.3) will discuss microwave assisted drying and microwave

convective drying (MCD). Section 2.4 will present a review of the IMCD drying

literature to identify research gaps and their significance. Finally, in Section 2.5, the

key findings of the literature review and research gaps will be presented.

http://dx.doi.org/10.1016/j.jfoodeng.2013.08.014

8 Chapter 2: Literature Review


2.1 FOOD DRYING

The main purpose of food drying is to remove moisture from food material up

to a certain limit in order to hinder the growth and reproduction of microorganisms.

The process of drying commonly involves simultaneous heat and mass transfer

(Karim & Hawlader, 2005b). Figure 2-1 shows a typical drying rate curve of food

materials. The first stage shown in the figure (A-B) is called the “initial period”

where the wet surface of the material initially reaches equilibrium with the drying

environment. In this stage, the drying rate increases and mostly free water is

removed. The second stage (B-C) is termed the “constant rate period” where the

drying rate remains constant because the amount the water migrating from the

interior to the surface is equal to moisture removal to surroundings from the surface.

Figure 2-1.Typical drying rate curve (Okos et al., 2006)

In the next step (C-E), (the third stage of drying known as the falling rate

period), the drying rate reduces due to a decrease in moisture migration from the

interior to the surface resulting in a longer drying time. By increasing the drying air

temperature in this stage, we can accelerate the drying rate. But a higher temperature

usually damages the surface of the product, resulting in case hardening (Zeki, 2009).

This case hardening limits the heat and mass transfer, and cause deterioration in food

quality (Botha et al., 2012). This prolonged drying time results in higher energy

consumption. Thus, drying becomes the most energy intensive technique of the


major industrial process (Kudra, 2004) and accounts for up to 15% of all industrial

energy usage (Chua et al., 2001a).

The quality of dried food is another important issue in food drying. The drying

causes changes in colour, aroma, texture, nutritive value, physical appearance and

shape etc. of the food product. These change impact consumer buying behaviour

(Quirijns, 2006). Consumer demand for healthy dried food with higher product

qualities challenges researchers to develop new or improved drying techniques for

food products (Nijhuis et al., 1996). Therefore, energy efficiency and food quality

are the two most crucial factors in food drying.

Several attempts have been made to improve energy efficiency and food

quality. One of them is intermittent drying, which is accomplished by the intermittent

supply of energy to the drying chamber. Energy savings and quality improvement by

applying intermittent drying are discussed in a review paper by the author of this

thesis (Kumar et al., 2014). Intermittent drying allows moisture to migrate from the

centre to the surface during the tempering period; thus, the case hardening can be

reduced. Chua et al. (2000b) have shown that applying variable temperature could

reduce colour change of potato, guava and banana by 87, 75 and 67%, respectively.

Intermittent drying is proven to reduce energy consumption and improve quality

(Chua et al., 2003; R. Jumah et al., 2007; Putranto et al., 2011; Soysal et al., 2009a).

Despite these advantages, total drying time in intermittent drying is longer than the

continuous drying (R. Y. Jumah, 1995; Kumar et al., 2014).

Freeze drying and vacuum drying are also being practiced for high quality

product but their high initial and energy consumption make those unpopular (Pan et

al., 1998). For instance, freeze drying requires a longer drying time, and the cost is

about 200–500% higher than that of hot air drying (M. Zhang et al., 2010). Whereas,

vacuum drying needs higher initial and operating cost to maintain low pressure

throughout the drying process (Gunasekaran, 1999). Combined drying is one of the

promising approaches to improve product quality and reduce drying time. In general,

microwave (MW) is often combined with convective, vacuum, osmotic and freeze

drying. MW is an attractive source of producing volumetric heat and combining it

with other drying method can significantly reduce drying time.


2.2 MICROWAVE ASSISTED DRYING

Microwave refers to electromagnetic radiation in the frequency range of

300MHz–300GHz with a wavelength 1mm–1m. It is propagation of electromagnetic

energy through space by means of time-varying electric and magnetic fields (Hao

Feng et al., 2012). Microwave penetrates the material until moisture is located and

heats up the material volumetrically thus facilitating a higher diffusion rate and

pressure gradient to drive off the moisture from inside of the material (I. W. Turner

& Jolly, 1991). There are two main mechanisms of microwave heating; dipolar re-

orientation and ionic conduction. Water molecules are dipolar in nature and try to

follow the electric field which alternates at very high frequency. For a commonly

used microwave frequency of 2.45 GHz, the electric field changes direction 2.45

billion times a second, making the dipoles move with it (A. Datta & Rakesh, 2009).

Such rotation of molecules produces friction and generates heat inside the food

material (M. Zhang et al., 2010). Ionic conduction is a second major mechanism of

microwave heating which is caused by ions, such as those present in salty food,

which migrate under the influence of the electric field. Figure 2-2 shows the main

heating mechanisms in the different frequency regions used in industry for heating

and drying.

Figure 2-2. Schematic diagram depicting the dipolar and ionic loss mechanisms and their

contributions to the dielectric properties as a function of frequency (Metaxas, 1996b).


Due to its volumetric heating capability, application of microwaves with other

drying methods can significantly increase the drying rate. The main advantages of

microwave assisted drying are:

(a) Volumetric heating: Microwave energy interacts with water molecules

within the food leading to volumetric heating and increased moisture diffusion rates

(I. W. Turner et al., 1998). This can thereby significantly reduce drying times

(Mujumdar, 2004; M. Zhang et al., 2006);

(b) Quality improvement: The quality of the dried product can be improved

by combining microwave heating with other drying methods (Dev et al., 2011);

(c) Controlled heating: The fidelity of heating can be controlled using

microwave energy as it can be applied intermittently by varying the pulse ratio and

the power level of the microwave. (Gunasekaran, 1999).

(d) Selective heating: Preferential heating of wetter areas is possible with

microwave heating and also bound water molecules can be excited by microwaves

(Gunasekaran, 1999; I. W. Turner et al., 1998). For these reasons, microwave related

drying is referred to as innovative and fourth generation drying technology.

Drying time reduction in microwave assisted drying can be illustrated in Figure

2-3. It is clear from the Figure 2-3 that microwave vacuum (MWV) and microwave

convection (MWC) drying shows remarkably lower drying time than hot air (HA)

drying (Orsat et al., 2007; Wojdyło et al., 2014; M. Zhang et al., 2010).

Figure 2-3. Drying curves of whole mushrooms under different drying methods (Orsat et al., 2007)


However, microwaves are generally combined with hot air drying, freeze

drying, vacuum drying, spouted bed drying and osmotic drying. Since freeze and

vacuum drying involve higher capital and operating cost as discussed previously,

convection drying is most widely combined with microwave to make microwave

convective drying (MCD) (Andrés et al., 2004).

2.3 MICROWAVE CONVECTIVE DRYING (MCD)

The main drawbacks of convective drying are longer drying times and

formation of a crust at the surface due to the elevated temperature. Microwaves can

mitigate these problems by increasing the diffusion rate and supplying moisture to

the surface. Thus, combining microwave with convection drying can significantly

shorten the drying time and improve product quality and energy efficiency (M.

Zhang et al., 2006).

An extensive compilation of literature regarding microwave assisted

convective drying of food is presented in Table 2-1. The table describes both

experimental and modelling studies for different food materials. From the

experimental studies, substantial reduction in the drying time (25–90%) have been

found in MCD drying when compared with convection drying (Cinquanta et al.,

2013; Izli & Isik, 2014; Prabhanjan et al., 1995). In terms of quality, MCD dried

products resulted superior quality when compared to hot air drying (Argyropoulos et

al., 2011; Cinquanta et al., 2013).

Microwave assisted drying also applied to non-food material like wood (Lehne

et al., 1999), kaolin (Kowalski et al., 2010), brick (I. W. Turner & Jolly, 1991),

agglomerated sand (Hassini et al., 2013) and found to be helpful in terms of energy

efficiency and product quality. Jindarat et al. (2011) also found that using microwave

energy in drying of a non-hygroscopic porous packed bed reduced drying time by

five times when compared to convective drying method. From the above discussion,

it can be said that MCD is a potential option to achieve better quality of dried food

and a reduction in drying time.


Table 2-1. Summery of microwave assisted convective heating and drying of food material

Material Modelling of power

distribution

Reference

Apple N/A (Marzec et al., 2010)

Apple cylinders N/A (Andrés et al., 2004)

Beetroot N/A (Figiel, 2010)

Carrot Lamberts law (Sanga et al., 2002)

Carrot cubes N/A (Prabhanjan et al., 1995)

Chinese jujube N/A (Fang et al., 2011)

Clipfish N/A (Bantle et al., 2013)

Cooked soybeans N/A (Gowen et al., 2008)

Cranberries N/A (Sunjka et al., 2004)

Garlic Lambert’s law (Abbasi Souraki & Mowla, 2008)

Garlic cloves N/A (Sharma et al., 2009)

Gel Maxwell* (Pitchai et al., 2012)

Green peeper Empirical (H. Darvishi et al., 2013b)

Green pepper Empirical (H. Darvishi et al., 2013a)

Lemon slice Empirical (Sadeghi et al., 2013)

Mashed potato Maxwell* (Chen et al., 2014)

Minced beef Lambert’s Law* (Campañone & Zaritzky, 2005)

Moringa oleifera

pods (Drumsticks)

N/A (Dev et al., 2011)

Mushrooms N/A (Argyropoulos et al., 2011)

Pineapple N/A (Botha et al., 2011)

Pistachios Empirical (Kouchakzadeh & Shafeei, 2010)

Potato Maxwell’s Equation (Malafronte et al., 2012)

Potato Lamberts law (McMinn et al., 2003)

Pumpkin slices N/A (Ilknur, 2007)

Swede, potato,

bread, and concrete

N/A (Holtz et al., 2010)

Tomato N/A (Swain et al., 2013)



distribution

Reference

Tomato slice Empirical (Workneh & Oke, 2013)

Two-percent agar

gel

Lambert’s law and Maxwell’s*

equations

(Yang & Gunasekaran, 2004)

Wheat seeds Lambert’s Law (Mohamed Hemis et al., 2012)

*-heating only (no mass transfer); N/A-Not available

Modelling of MCD is essential to understanding the physical mechanism and

optimizing the process. Empirical modelling has been conducted for green peppers

(H. Darvishi et al., 2013a), lemons (Sadeghi et al., 2013), tomato slices (Workneh &

Oke, 2013). However, the empirical model is does not help towards understanding of

heat and mass transfer and is only applicable for specific experimental ranges

(Kumar et al., 2012a; Perussello et al., 2014). To overcome the innate deficiencies of

empirical models, some diffusion based models have been developed. Both

Lambert’s Law (M. Hemis & Raghavan, 2014; McMinn et al., 2003; Sanga et al.,

2002) and Maxwell’s equation (Malafronte et al., 2012) have been taken into

consideration in these theoretical models. It is noted here that the Lambert’s Law is a

simple approximation of microwave heat generation, whereas Maxwell’s equations

are more comprehensive and accurate in predicting microwave heating

(Chandrasekaran et al., 2012; Rakesh et al., 2009). Nevertheless, none of those

models considered a multiphase porous media approach for heat and mass transfer,

which is further discussed in Section 2.5.3.

However, supplying continuous microwave energy to heat sensitive material

like food may cause uneven heating or overheating or even create hot spots. Heat and

mass transfer should be carefully balanced to avoid such overheating and to use

applied energy more efficiently (Gunasekaran, 1999). This problem could be

overcome by applying microwave energy intermittently. This also allows to limit the

temperature rise and moisture redistribution which improves product quality and

energy efficiency (Soysal et al., 2009a).

2.4 INTERMITTENT MICROWAVE CONVECTIVE DRYING (IMCD)

Intermittent application of microwave energy in convective drying is more

advantageous than continuous application and is able to overcome the problems of

overheating and uneven heating. The heating rate also can be controlled by


regulating the intermittency. The advantages of IMCD in terms of energy efficiency

and dried product quality have been reported in literature as shown in Table 2.2. For

instance, Soysal et al. (2009a) reported that IMCD of red pepper produced better

sensory attributes, appearance, colour, texture and overall liking, than MCD and

commercial drying. Soysal et al. (2009b) compared IMCD and convective drying for

oregano and found that the IMCD was 4.7–11.2 times more energy efficient

compared to convective drying and was able to provide better quality dried food.

Table 2-2. Summary of intermittent microwave assisted convective heating and drying of food

material


distribution Reference

Bananas N/A (Ahrné et al., 2007)

Carrots, mushrooms N/A (Orsat et al., 2007)

2% agar gel Lambert’s Law* (Yang & Gunasekaran, 2001)

Dill leaves Empirical (Esturk & Soysal, 2010)

2% agar gel N/A* (Gunasekaran & Yang, 2007a)

Mashed potato Maxwell’s*

(Gunasekaran & Yang, 2007b)

Oregano N/A (Soysal et al., 2009b)

Pineapple N/A (Botha et al., 2012)

Red pepper N/A (Soysal et al., 2009a)

Sage(Salvia officinalis)

Leaves N/A (Esturk, 2012; Esturk et al., 2011)

*-heating only; N/A-not available

Advantages of IMCD in terms of improving energy efficiency and product

quality, and significantly reducing drying time have been found in many other

products such as Oregano (Soysal et al., 2009b), Pineapple (Botha et al., 2012), Red

Pepper (Soysal et al., 2009a), Sage Leaves (Esturk, 2012; Esturk et al., 2011),

Bananas (Ahrné et al., 2007), and Carrots and Mushrooms (Orsat et al., 2007).

However, all the above studies reported on IMCD drying were conducted

experimentally without considering the physics behind the heat and mass transfer

involved in the process. Thus there is a lack of understating of the physical

phenomena involving heat and mass transfer in IMCD. Moreover, limiting the


temperature in certain range which is critical for preventing overheating has not been

investigated.

Physical understanding of heat and mass transfer, and interaction of

microwaves with food products is important for optimization of the drying process

(Hao Feng et al., 2012). Coupled heat and mass transfer models have to be developed

to predict the temperature and moisture distribution inside the material which will

help to improve the understanding of the underlying physics and develop better

strategies for the control of IMCD. Although there are some single phase models

which considered intermittency of microwave energy (Gunasekaran & Yang, 2007b;

Yang & Gunasekaran, 2001), these are for only heating without considering mass

transfer, therefore, cannot be applied in drying.

2.5 MODELLING OF DRYING AND CHALLENGES

Modelling is necessary for evaluating the effect of process parameters on

energy efficiency and drying time, and optimizing the drying process (Kumar et al.,

2012a; Kumar et al., 2014). Developing a physics based drying model for

agricultural products is a challenging task. This is because of the complex structural

nature of agricultural products and changes in the thermo-physical properties during

drying. Moreover, heat and mass transfer are highly coupled during drying.

Therefore, some assumptions are indispensable if mathematical models are to be

developed, but these should be carefully made to represent the physical phenomena

during the process. In this section, the modelling approaches of drying are discussed

together with their limitations. Generally, the models can be classified into three

categories: empirical models, diffusion based (single phase) models and multiphase

models as shown in Figure 2-4 and discussed in detail in the subsections of this

paragraph.

Figure 2-4. General classification of drying models in literature


Empirical models 2.5.1

The empirical or observation based models can be rapidly developed based on

experimental data. These models are generally derived from Newton’s law of cooling

and Fick’s law of diffusions (Erbay & Icier, 2010). These are simpler to apply and

often used to describe the drying curve. Despite these advantages, they do not

provide physical insight into process. In addition, these models are only valid for

specific process condition and material. In contrast to empirical models, the physics-

based models can capture the inherent drying mechanism better (Chou et al., 2000;

Chua et al., 2003; Ho et al., 2002).

Diffusion based (single phase) models 2.5.2

The diffusion-based models are very popular because of their simplicity and

good predictive capability (Javier R. Arballo et al., 2010; Kumar et al., 2012b;

Perussello et al., 2014). These models assume conductive heat transfer for energy

and diffusive transport for moisture. These models need an effective diffusivity

(phenomenological coefficient) value, which has to be experimentally determined.

The effective diffusivity calculation is crucial for drying models because it is the

main parameter that controls the process with a higher diffusion coefficient implying

increased drying rate. This effective diffusion coefficient changes during drying due

to the effects of sample temperature and moisture content (Batista et al., 2007).

Some authors considered effective diffusivity as a function of shrinkage or moisture

content (Karim & Hawlader, 2005a), whereas, others postulated it as temperature

dependent (Chandra Mohan & Talukdar, 2010). Even though there are several

modelling studies of food drying, there are limited studies that compare the impacts

of temperature dependent and moisture dependent effective diffusivities. In general,

the thermo-physical properties (thermal conductivity, specific heat, density etc.) of

food, change as drying progresses. Some diffusion models consider them as constant

which is very unrealistic. Although this model can provide a good match with

experimental results, it cannot provide an understanding of other transport

mechanisms such as pressure driven flow and evaporation. The drawbacks of these

kinds of models are discussed in detail in the work of Zhang and Datta (2004).

Lumping all the water transport as diffusion cannot be justified under all situations.

Therefore, multiphase models considering transport of liquid water, vapour, air

insider the food materials are more realistic.


Multiphase models 2.5.3

The multiphase models incorporate the transition from the individual phase at

the ‘microscopic’ level to representative average volume at the ‘macroscopic’ level,

which provides a fundamental and convincing basis of heat and mass transport (A. K.

Datta, 2007b; Whitaker, 1977). Multiphase models can be categorised into two

groups: equilibrium and non-equilibrium approach of vapour pressure. In equilibrium

formulations, the vapour pressure, vp , is assumed to be equal with equilibrium

vapour pressure, eqvp , , and vice versa. There are some multiphase models

considering equilibrium approach applied in vacuum drying of wood (Ian W. Turner

& Perré, 2004) and convection drying of wood and clay (Stanish et al., 1986)

(Chemkhi et al., 2009), microwave spouted bed drying of apple (H. Feng et al., 2001)

and large bagasse stockpiles (Farrell et al., 2012). Some multiphase porous media

models combine the liquid water and vapour equations together to eliminate the

evaporation rate term (Farrell et al., 2012; Ratanadecho et al., 2001; Suwannapum &

Rattanadecho, 2011). By doing this, the concentrations of liquid water and vapour

cannot be determined.

However, equilibrium condition may be valid at surface with lower moisture

content because equilibrium condition may not be achieved due to lower moisture

content at the surface during drying. Therefore, the non-equilibrium approach is a

more realistic representation of the physical situation during drying (J. Zhang &

Datta, 2004). Moreover, the non-equilibrium formulation for evaporation can be used

to express explicit formulation of evaporation, thus allowing calculation of each

phase separately. Furthermore, non-equilibrium multiphase models are

computationally effective and applied to wide range of food processing such as

frying (Bansal et al., 2014; H. Ni & Datta, 1999), microwave heating (Chen et al.,

2014; Rakesh et al., 2010), puffing (Rakesh & Datta, 2013) baking (J. Zhang et al.,

2005), meat cooking (Dhall & Datta, 2011) etc. However, application of these non-

equilibrium models in drying of food materials is very limited. To the authors’ best

knowledge, there is no IMCD model considering non-equilibrium multiphase porous

media.


MCD and IMCD model 2.5.4

To model MCD and IMCD, a heat source due to microwave heating is

generally added to the energy equation. This heat source can be mostly modelled

using two formulations (Lambert’s Law and Maxwell’s equations) as shown in

Figure 2-5. As mentioned before, Lambert’s Law is simple approximation of

microwave heat generation, whereas Maxwell’s equations are more comprehensive

and accurate to predict microwave heating (Chandrasekaran et al., 2012; Rakesh et

al., 2009)

Figure 2-5. Modelling approach of heat generation due to microwave

There are some modelling studies related to MCD and IMCD found in the

literature. Most of them are empirical in nature as shown in Table 2-1 and Table 2-2.

Although several authors considered Maxwell’s equation to predict the distribution

of electromagnetic field within non-food samples, these models are not applicable for

IMCD due to several reasons (Klayborworn et al., 2013; Ratanadecho et al., 2001;

Rattanadecho, 2006a, 2006b; Suwannapum & Rattanadecho, 2011). Firstly, their

research did not consider intermittency of microwave heat source and their impact on

heat and mass transfer. Secondly, the Maxwell’s equations were solved in two-

dimensional for a sample located in rectangular waveguide. However, for a

multimode cavity, the geometry cannot be reduced to two-dimension (Rakesh et al.,

2012). Thirdly, Rattanandecho (2006a) neglected mass transfer, which is unrealistic

in drying, and affects the heat transfer due to heat loss due to evaporative cooling.

Table 2-2 shows that there were some single phase diffusion based model for

microwave heating only (Chen et al., 2014; Pitchai et al., 2012), MCD (Malafronte et

al., 2012; McMinn et al., 2003), however, intermittency of microwave has not been

dealt with in these outlined models. On the other hand, from Table 2-2, it is clear that

despite there being some single phase models which considered intermittency, these

are only for heating without consideration of mass transfer (Gunasekaran & Yang,

2007b; Yang & Gunasekaran, 2001), thus they cannot be applied in drying.

Moreover, the temperature redistribution and fluctuations, which is the main


advantage of intermittency, has been overlooked in those studies. In light of the

above literature review, it is clear that there is no model that is completely dedicated

to IMCD, and therefore, the transport processes in IMCD are still poorly understood.

Not only is there no satisfactory multiphase model, there is not even a

comprehensive single-phase model for IMCD.

2.6 SUMMARY OF LITERATURE AND RESEARCH GAPS

Based on the above literature review, we can conclude the following:

Energy efficiency and food quality are the two most crucial factors in food

drying.

Microwave assisted drying technology can significantly reduce drying

time and improve product quality. However, supplying continuous

microwave energy to heat-sensitive materials, like food, may cause

overheating and uneven heating, creating cold and hot spots inside the

food and eventually degrading the quality of food.

IMCD allows redistribution of temperature and moisture, and can limit the

temperature to a certain range; consequently, reducing the problems of

overheating.

Modelling is necessary for evaluating the effect of process parameters on

drying kinetics, understanding the transport mechanism and optimizing the

drying process.

Single-phase models are simple; but, require an effective diffusion

coefficient which needs to be experimentally determined and which varies

with process conditions. In addition, effective diffusivity lumps all

moisture flux together and therefore cannot provide an understanding of

individual transport mechanisms such as pressure-driven flow and

evaporation.

Multiphase models are available in food drying; but, most of them

considered equilibrium conditions ( eqvv pp , ), which may not be valid in

drying, especially, near the surface of the product where moisture content

is low.


Non-equilibrium multiphase models are most comprehensive and

computationally effective, the non-equilibrium formulation for evaporation

can be used to express an explicit formulation of evaporation, thus

allowing calculation of each phase separately. However, there are limited

studies of non-equilibrium models in food drying.

Despite numerous experimental studies available for IMCD, there is no

complete multiphase model and not even an accurate single-phase model

for IMCD.

Physical mechanisms of heat and mass transfer in IMCD are not properly

understood.

Moisture and vapour fluxes due to different mechanisms, such as, capillary

flow, binary diffusions, pressure driven and temperature fluctuations

during IMCD have not been investigated.

From these arguments, it is clear that a comprehensive multiphase porous

media model considering non-equilibrium vapour pressure for IMCD model is

necessary which is not available in literature. Such a model will be developed to

investigate the transport mechanism for IMCD in this research.

Chapter 3: Single Phase Model for Convection Drying 23

Single Phase Model for Chapter 3:

Convection Drying

This Chapter presents a single-phase diffusion-based model for convection

drying, which was used as a basis for developing the IMCD model outlined in the

next chapter. The model was validated with experimental data; a parametric analysis

was conducted.

This Chapter has been published as:

C. Kumar, G. J. Millar, and M. A. Karim (2015) Effective Diffusivity and


Technology, 33, (2), 227–237.

The author helped to develop an osmo-convective drying model based on this

chapter’s work, which was also published as:

C. A. Perussello, C. Kumar, F. Castilhosc, and M. A. Karim (2014) Heat

and mass transfer modelling of the osmo-convective drying of Yacon roots

(Smallanthus sonchifolius). Applied Thermal Engineering, 63(1) 23–32

(Impact Factor 2.12).

24 Chapter 3: Single Phase Model for Convection Drying

The signed statements of contributions for the above mentioned paper are

given below:


3.1 ABSTRACT

This article presents mathematical equations to simulate coupled heat and mass

transfer during the convective drying of food materials using three different effective

diffusivities: shrinkage dependent, temperature dependent, and the average of those

two. Engineering simulation software COMSOL Multiphysics was utilized to

simulate the model in 2D and 3D. The simulation results were compared with

experimental data. It is found that the temperature-dependent effective diffusivity

model predicts the moisture content more accurately at the initial stage of the drying,

whereas, the shrinkage-dependent effective diffusivity model is better for the final

stage of the drying. The model with shrinkage-dependent effective diffusivity shows

evaporative cooling phenomena at the initial stage of drying. This phenomenon was

investigated and explained. Three-dimensional temperature and moisture profiles

show that even when the surface is dry, the inside of the sample may still contain a

large amount of moisture. Therefore, the drying process should be carefully dealt

with; otherwise, microbial spoilage may start from the centre of the dried food. A

parametric investigation was conducted after the validation of the model.

3.2 INTRODUCTION

Food drying is a process that involves removing moisture in order to preserve

fruits by preventing microbial spoilage. It also reduces packaging and transport cost

by reducing weight and volume. Compared to other food preservation methods,

dried food has the advantage that it can be stored at ambient conditions. However,

drying is an energy intensive process and accounts for up to 15% of all industrial

energy usage and the quality of food may degrade during the drying process (Chua et

al., 2001a; Rami Jumah & Mujumdar, 2005; Kumar et al., 2014) . The objective of

food drying is not only to remove moisture by supplying heat energy but also to

produce quality food (Mujumdar, 2004). To reduce this energy consumption and

improve product quality, a physical understanding of the drying process is essential.

Mathematical models have been proven useful to understand the physical

mechanism, optimize energy efficiency and improve product quality (Kumar et al.,

2012a). Mathematical models can be either empirical or fundamental models.

Empirical expressions are common and relatively easy to use (Kumar et al., 2014).

Many empirical models for drying have been developed and applied for different

products, for instance, banana (Wilton Pereira Silva, Silva, et al., 2014), apple (Z.


Wang et al., 2007), rice (Cihan et al., 2007), carrot (Cui et al., 2004), cocoa (Hii et

al., 2009) etc. Erbay and Icier (2010) reviewed empirical models for drying, and

found that the best-fitted model is different for different products. However, these

empirical models are only applicable in the range used to collect the experimental

parameters (Kumar et al., 2012b). In addition, typically they are not able to describe

the physics of drying. In contrast to empirical relationships, fundamental models can

satisfactorily capture the physics during drying (Chou et al., 2000; Chua et al., 2003;

Ho et al., 2002). Fundamental mathematical modelling is applicable for a wide range

of applications and optimization scenarios (Kumar et al., 2012b).

Several fundamental mathematical models have been developed for food

drying. For example, Barati and Esfahani (2011) developed a food drying model

wherein they considered the material properties to be constant. However, in reality

during the drying process physical properties such as diffusion coefficients and

dimensional changes occur as the extent of drying progresses (Joardder, Kumar, et

al., 2013). Consequently, if these latter issues are not considered, the model

predictions may be erroneous in terms of estimating temperature and moisture

content (N. Wang & Brennan, 1995). In particular, the diffusion coefficient can have

a significant effect on the drying kinetics.

Calculation of the effective diffusivity is crucial for drying models because it

is the main parameter that controls the process with a higher diffusion coefficient

implying increased drying rate. The diffusion coefficient changes during drying due

to the effects of sample temperature and moisture content (Batista et al., 2007).

Alternatively, some authors considered effective diffusivity as a function of

shrinkage or moisture content (Karim & Hawlader, 2005a); whereas, others

postulated it as temperature dependent (Chandra Mohan & Talukdar, 2010). In the

case of a temperature dependent effective diffusivity value, the diffusivity increases

as drying progresses. On the other hand, effective diffusivity decreases with time in

the case of shrinkage or moisture dependency. This latter behaviour is ascribed to the

diffusion rate decreasing as moisture gradient drops. However, Baini and Langrish

(2007) mentioned that shrinkage also tends to reduce the path length for diffusion

which results in increased diffusivity. Consequently, there are two opposite effects of

shrinkage on effective diffusivity which theoretically may cancel each other.


Silva et al.(2011) analysed the effect of considering constant and variable

effective diffusivities in banana drying. They found that the variable effective

diffusivity (moisture dependent) is more accurate than the constant effective

diffusivity in predicting the drying curve. Some authors (Karim & Hawlader, 2005a)

considered effective diffusivity as a function of moisture content whereas others

(Wilton Pereira Silva et al., 2013) considered it as a function of temperature.

However, there are limited studies comparing the influence of temperature dependent

and moisture dependent effective diffusivity. Recently, Silva et al. (2014) considered

effective diffusivity as a function of both temperature and moisture together (i.e.

D=f(T, M)), not temperature or moisture dependent diffusivities separately. Therefore,

it was not possible to compare the impact of considering temperature and moisture

dependent effective diffusivities. Moreover, they did not report the impact of variable

diffusivities on material temperature. A comparison of drying kinetics for both

temperature and moisture dependent effective diffusivities can play a vital role in

choosing the correct effective diffusivity for modelling purposes. Though there are

several modelling studies of food drying, there are limited studies that compare the

impacts of temperature dependent and moisture dependent effective diffusivities.

Understanding the exact temperature and moisture distribution in food samples

is important in food drying. Joardder et al. (2013) showed that temperature

distribution plays a critical role in determining the quality of dried food. Similarly,

moisture distribution plays a critical role in food safety and quality. Vadivambal and

Jayas (2010) showed that despite the fact that the average moisture content was

lower than what was considered a safe value, spoilage started from the higher

moisture content area. Therefore, it is crucial to know the moisture distribution in the

sample. Unfortunately, it is difficult to measure temperature and moisture

distribution inside the sample experimentally, which means that appropriate

modelling approaches are required to determine the moisture distribution. Mujumdar

and Zhonghua (2007) argued that technical innovation can be intensified by

mathematical modelling which can provide better understanding of the drying

process. Karim and Hawlader (2005a) developed a mathematical model to determine

temperature and moisture changes with time but it did not provide temperature and

moisture distribution within the sample. The moisture distribution is a key parameter

for evaporation because evaporation depends on surface moisture content.


Evaporation plays an important role during drying in terms of heat and mass

transfer, with higher evaporation resulting in enhanced drying rates. During the

initial stage of drying, the surface is almost saturated, which induces both higher

evaporation and moisture removal rates. Due to this higher evaporation rate, the

temperature drops at this stage for a short period of time (I. W. Turner & Jolly, 1991;

W. Zhang & Mujumdar, 1992). Recently Golestani et al.(2013) also observed

reduced temperature in the initial drying phase and they attributed this phenomenon

to high enthalpy of water evaporation. The temperature evolution depends upon the

heat flux. During drying, two reverse heat fluxes take place; inward convective heat

flux and outward evaporative heat flux. Again, there are limited studies that have

investigated the temperature variation during the initial stage of convection drying

based on heat flux.

In this context, the aims of this paper are threefold; to (1) develop three drying

models based on three effective diffusivities: namely, moisture dependent,

temperature dependent, and average effective diffusivities; (2) investigate the

evaporative cooling phenomena in terms of heat flux; (3) conduct a parametric study

with validated models.

3.3 MODEL DEVELOPMENT

The model developed in this research considered the cylindrical geometry of

the food product as shown in Figure 3-1.

Figure 3-1. (a) Actual geometry of the sample slice and (b) Simplified 2D axisymmetric model

domain.


Governing equations 3.3.1

It was assumed that the mass flux is only due to diffusion; therefore, the Fick’s

diffusion law is valid for mass transfer of moisture given by:

RcDt

c

eff 3-1

where, c is the moisture concentration (mol/m3), t is time, effD is the effective

diffusion coefficient (m2/s), R is the production or consumption of moisture (kg/m

3s)

which is zero here.

The heat transfer is considered by Fourier law, given by,

ep )( QTkt

Tc

3-2

where, T is the temperature of the sample (0C), is the density of sample (kg/m

3),

pc is the specific heat (J/kg/K), and k is the thermal conductivity (W/m/K), eQ is the

internal heat source or sink. The heat source term is zero for convection drying but

when electromagnetic heating such as microwave is involved then it should be non-

zero term and added to the heat transfer equation.

Initial and boundary conditions 3.3.2

Initial moisture content, 𝑀𝑑𝑏 = 4 kg/kg (db)

Initial temperature, 𝑇0 = 380𝐶

Heat transfer boundary conditions: Both convection and evaporation were

considered at the transport boundaries (Figure 3-1). Thus the heat transfer boundary

condition at the transport boundaries was defined by equation 3-3.

𝒏. (𝑘𝛻𝑇) = ℎ𝑇(𝑇𝑎𝑖𝑟 − 𝑇) − ℎ𝑚𝜌(𝑀 − 𝑀𝑒)ℎ𝑓𝑔, 3-3

where, ℎ𝑇 is heat transfer coefficient (W/m2/K), ℎ𝑚 is mass transfer coefficient

(m/s), 𝑇𝑎𝑖𝑟 is drying air temperature (0C), Me is equilibrium moisture content (kg/kg)

dry basis and ℎ𝑓𝑔 is latent heat of evaporation (J/kg).

The heat transfer boundary condition at the symmetry boundary is given by,

𝒏. (𝑘𝛻𝑇) = 0. 3-4


Mass transfer boundary conditions:

The mass transfer boundary condition at the transport boundaries are given by,

𝒏. (𝐷𝛻𝑐) = ℎm(𝑐b − 𝑐), 3-5

where, 𝑐𝑏 is bulk moisture concentration (mol/m3).

The mass transfer boundary condition at the symmetry boundary is given by

𝒏. (𝐷𝛻𝑐) = 0. 3-6

Input parameters 3.3.3

Physical properties of banana and other input parameters used in the simulation

programme are listed in Table 3-1.

Table 3-1. Input parameters for the model

Properties Value Reference

Density of Banana, 𝜌 980 (

𝑘𝑔

𝑚3)

(Karim & Hawlader, 2005a)

Initial Moisture Content (dry basis), 𝑀 4 (

𝑘𝑔

𝑘𝑔)

Measured

Latent heat of Evaporation, ℎ𝑓𝑔 2358600 (

𝐽

𝑘𝑔)

(Cengel, 2002)

Thermal Conductivity of air at 600C, 𝑘𝑎𝑖𝑟

0.0287 (𝑊

𝑚𝐾)

(Cengel, 2002)

Density of Water at 600C, 𝜌𝑤

994.59 (𝑘𝑔

𝑚3)

(Cengel, 2002)

Dynamic viscosity of air at 600C, 𝜇𝑎𝑖𝑟 1.78𝑥10−4 (𝑃𝑎. 𝑠) (Cengel, 2002)

Specific heat of air at 600C, 𝐶𝑝𝑎𝑖𝑟

1005.04 (𝐽

𝑘𝑔𝐾)

(Cengel, 2002)

Density of air at 600C, 𝜌𝑎𝑖𝑟

1.073(𝑘𝑔

𝑚3)

(Cengel, 2002)

Equilibrium moisture content, 𝑀𝑒 0.29 (

𝑘𝑔

𝑘𝑔)

(Karim & Hawlader, 2005b)

Reference diffusion coefficient, Dref 2.41𝑥10−10 (

𝑚2

𝑠)

(Karim & Hawlader, 2005a)


Variable thermos-physical properties

In food processing, thermos-physical properties play an important role in heat

and mass transfer simulation (Perussello et al., 2013). In this simulation, specific

heat and thermal conductivity were considered as function of moisture content (Mwb)

by the following equations (Bart-Plange et al., 2012).

Specific heat, 174224751.812

wbwbp MMC 3-7

Thermal conductivity, 120.06.0 wbMk 3-8

Effective Diffusivity Calculation

In this study, three simulations were performed with three different effective

diffusivities. The effective diffusivity formulations are discussed below:

Moisture- or shrinkage dependent effective diffusivity, MD

Karim & Hawlader (2005a) presented effective diffusion coefficient as a

function of moisture content for product undergoing shrinkage during drying. In this

study the following equation was used to incorporate the shrinkage dependent

diffusivity:

2

0

b

b

D

D

M

ref, 3-9

where refD is reference effective diffusivity (m2/s) which is constant and calculated

from experimental value, b0 and b are the half thickness of the material (m) at time 0

and t, respectively. Thickness ratio obtained by the following equation:

sw

swbw

M

Mbb

0

0 3-10

where, w is the density of water (kg/m3), s is the density of the sample (kg/m

3).

Temperature dependent effective diffusivity, TD

Temperature dependent diffusivity, TD , was obtained from an Arrhenius type

relationship to the temperature with the following equation (Islam et al., 2012; N.

Wang & Brennan, 1995):

TR

E

Tg

a

eDD

0 3-11


where aE is activation energy (kJ/mol), Rg is the universal gas constant (kJ/mol/K),

and D0 is integration constant (m2/s).

Average effective diffusivity, effD

The third model considered the average effective diffusivity, effD , which is

average of the temperature-and moisture dependent effective diffusivities given by

2

2

00

b

bDeD

D

ref

TR

E

eff

g

a

.

3-12

Heat and mass transfer coefficient calculation

The heat and mass transfer coefficients are calculated from well-established

correlations of Nusselt and Sherwood number for laminar and turbulent flow over

flat plates as shown in equations 3-13 to 3-16. These relationships have been used in

drying by many other researchers (Golestani et al., 2013; Karim & Hawlader, 2005a;

Montanuci et al., 2014; Perussello et al., 2014) and hence the use of these

relationships is justified.

The average heat transfer coefficient was calculated from Nusselt number (Nu)

by using equations 3-13 and 3-14 for laminar and turbulent flow, respectively (Mills,

1995).

33.05.0T PrRe664.0h

k

LNu 3-13

33.05.0T PrRe0296.0h

k

LNu

3-14

where L is characteristics length (m), Re is the Reynolds number, and Pr is the

Prandtl number.

Because Fourier’s law and Fick’s law were similar in mathematical form the

analogy was used to find the mass transfer coefficient. Nusselt and Prandtl numbers

were replaced by the Sherwood number (Sh) and the Schmidt number (Sc),

respectively, as in the following relationships:

33.05.0m Re332.0h

ScD

LSh

va

3-15


33.08.0m Re0296.0h

ScD

LSh

va

. 3-16

Here vaD is the binary diffusivity of vapour and air (m2/s).

The values of Re, Sc and Pr were calculated by

𝑅𝑒 =𝜌𝑎𝑣𝐿

𝜇𝑎,

3-17

𝑆𝑐 =𝜇𝑎

𝜌𝑎𝐷𝑣𝑎,

3-18

and 𝑃𝑟 =𝐶𝑝𝜇𝑎

𝑘𝑎,

3-19

respectively. Here, 𝜌𝑎 is density of air (𝑘𝑔

𝑚3), 𝜇𝑎 is dynamic viscosity of air(𝑃𝑎. 𝑠), 𝑣

is drying air velocity (m/s), 𝑘𝑎 is thermal conductivity of air (𝑊

𝑚𝐾).

3.4 SIMULATION METHODOLOGY

Simulation was performed by using COMSOL Multyphysics, a finite element

based engineering simulation software rpogram. The software facilitated all steps in

the modelling process, including defining geometry, meshing, specifying physics,

solving, and then visualising the results. COMSOL multiphysics can handle the

variable properties, which are a function of the independent variables. Therefore, this

software was very useful in drying simulation where material properties changed

with temperature and moisture content. The simulation methodology and

implementation strategy followed in this project is shown in

Figure 3-2. Banana was taken as a sample for this study.

Figure 3-2. Simulation strategy in COMSOL multiphysics


3.5 DRYING EXPERIMENTS

Drying tests were performed based on the American Society of Agricultural

and Biological Engineers (ASABE S448.1) Standard. The procedures for ASABE

standard are as follows:

Tests should be conducted after drying equipment has reached steady-state

conditions. Steady state is achieved when the approaching air stream

temperature variation about the set point is less than or equal to 10C.

The sample should be clean and representative in particle size. It should be

free from broken, cracked, weathered, and immature particles and other

materials that are not inherently part of the product. The sample should be

a fresh one having its natural moisture content.

The particles in the thin layer should be exposed fully to the airstream.

Air velocity approaching the product should be 0.3 m/s or more.

Nearly continuous recording of the sample mass loss during drying is

required. The corresponding recording of material temperature (surface or

internal) is optional but preferred.

The experiment should continue until the moisture ratio, MR, equals 0.05.

Me should be determined experimentally or numerically from established

equations.

A tunnel-type drying chamber was used in this experiment. The dryer was

equipped with a heater, blower fan, and two dampers. The two dampers were used to

facilitate the air recirculation and fresh air intake. Both closed-loop and open-loop

tests were possible by adjusting the dampers. A temperature controller and blower

speed controller was used to maintain constant drying air temperature and air

velocity.

The weight of the sample was measured using a load cell, which was calibrated

using standard weights. Air velocity has a considerable effect on the load cell reading

and different calibration curves were prepared for different flow velocities through

the dryer. The load cell was calibrated after installation in the dryer. The air flow rate

was calculated by measuring the air velocity at the entrance of the dying section. A

calibrated hot wire anemometer measured the air velocity. A T-type thermocouple


and humidity transmitter were used to measure the temperature and relative

humidity. All of the sensors were connected to a data logger to store the information.

For experimental investigation, ripe bananas (Musa acuminate) of

approximately the same size were used for drying. First, the bananas were peeled and

then sliced 4mm thick with a diameter of about 36 mm. Initial moisture content was

about 4.0 kg/kg (db) and the final moisture content was between 0.22 to 0.25 kg/kg

(db); that is, moisture ratio was 0.055 to 0.062. Then the slices were put on trays

made of plastic net. Plastic net was used to reduce conduction heat transfer because

this effect was neglected in the model. The plastic tray was put into the dryer after

reaching a steady state condition. Each run included approximately 600 g of material.

Following each drying test, the sample was heated at 1000C for at least 24 h to obtain

the bone-dry mass.

Uncertainty Analysis

Uncertainty analysis of the experiments was done according to Moffat (1988).

If the result R of an experiment is calculated from a set of independent variables so

that, N321 .,..........,, XXXXRR , then the overall uncertainty can be calculated

using the following expression:

2/1

1

2

i

i

.

N

i

XX

RR

3-20

and the relative uncertainty can be expressed as follows:

2/1

1

2

i

i

..1

N

i

XX

R

RR

Re

.

3-21

Uncertainty analysis of temperature:

The temperature was directly obtained from the calibrated thermocouple and

the accuracy was within the American Society of Heating, Refrigerating and Air

Conditioning Engineers’ recommended range, which is ±0.50C. Therefore, the

uncertainty of the temperature would be

5.0measured TT 3-22


Uncertainty analysis of moisture content:

The dry basis moisture content ratio of the weight of moisture, mW , to that of

bone-dry weight, dW , of the sample was calculated from the following equation:

d

d

d

m

W

WW

W

WM

3-23

Therefore,

2

d

d

d

d

d

...

W

WW

W

WW

W

MW

W

MM

3-24

and dd

d

d .

.

WWW

WW

WW

W

M

M

. 3-25

Now the relative uncertainty associated with the measurement of moisture

content of sample can be expressed as:

2/12

dd

d

2

d

m.

.

WWW

WW

WW

We

3-26

The present work considers the following value of the banana sample to be

dried in the drying chamber: gW 600 and gW 120d . Because these two values

are obtained using the same load cell, and as per manufacturer’s specification, the

percentage error of the load cell is %1.0 ; therefore, 0001.0d WW .

Substituting all the values in equation 3-26, the relative uncertainty for moisture

content, me , is obtained, and the value is found to be %06.1 .

3.6 RESULTS AND DISCUSSION

Validation of the model was done by comparing the moisture and temperature

evolutions obtained from experiment and simulation. Figure 3-3 represents a

comparison of the moisture evolutions obtained by experiments and models

considering three different effective diffusivities for drying air temperature 600C and

air velocity of 0.7m/s. Results show that simulated moisture content with temperature

dependent diffusivity closely agreed with the experimental moisture data in the initial

stage of the drying process. On the other hand, the shrinkage dependent diffusivity

model exhibited a faster drying rate in the initial stage but followed experimental


data more closely in the final stage of drying. The higher moisture loss in shrinkage

dependent effective diffusivity could be due to the reduced path lengths of moisture

This higher drying rate during the initial stage can be attributed to the higher

diffusion coefficient in that stage. Moisture dependent effective diffusivity is higher

in the initial stage as can be seen from equations 3-9 and 3-10. These two equations

show that initially the diffusivity value was greater at higher moisture content and

then decreased with moisture content. Golestani et al. (2013) also found higher

drying rate compared to the experimental results for both models obtained from two

effective diffusivities with and without shrinkage. Therefore, a more complex and

physics-based formulation is necessary to calculate effective diffusivity and predict

the moisture content more accurately. However, consideration of effective diffusivity

as an average of those two effective diffusivities provided a better match with

experimental data. A similar result was found by Golestani et al. (2013).

Figure 3-3. Moisture evolution obtained for experimental and simulation with shrinkage-dependent,

temperature-dependent and average effective diffusivities


Figure 3-4. Temperature evolution obtained for experimental and simulation with shrinkage-

dependent, temperature-dependent and average effective diffusivities

The temperature evolution of the material is shown in Figure 3-4 for drying air

temperature of 600C and velocity of 0.5m/s. The predicted temperatures agreed

reasonably well with the experimental data. However, interestingly for the shrinkage-

dependent effective diffusivity model there was a drop in temperature at the

beginning of the drying process. This was probably due to the evaporative cooling of

the product. In the initial stage of drying, the surface of the sample was saturated

with moisture and evaporation rate was higher. Thus, evaporative heat was taken

away from the material, resulting in a temperature drop. The increased evaporation

(higher drying rate) can also be seen in Figure 3-3 for the shrinkage dependent

effective diffusivity curve. For better visualization, the temperature evolution was

plotted for small time steps in Figure 3-5, wherein temperature reduction was noted

for the first few minutes of drying. A decreasing temperature in the initial stage of

drying was also obtained by Turner and Jolly (1991) and Zhang and Mujumdar

(1992) for microwave convective drying and Golestani et al. (2013) for convection

drying simulations. However, they reported these results without any interpretation


for this event. To investigate this observation further, the inward heat flux, outward

heat flux, and total heat flux were plotted in a single graph as shown in Figure 3-6.

The inward heat flux was due to convection (from air to material) and outward heat

flux was due to evaporation (from material to air). Figure 3-6 shows that for the first

15 minutes of drying the total heat flux was negative due to evaporation, which

caused a temperature drop in the product. This phenomenon is important in food

drying where an increase in temperature can cause quality degradation. If this

mechanism of cooling could be sustained longer, then the quality of dried food may

be improved. Sometimes intermittent drying can be executed to get more

evaporation when drying resumes after each tempering period. More experimentation

with continuous temperature measurement should be undertaken to further validate

this phenomenon.

Figure 3-5. Temperature curve for shrinkage dependent diffusivity


Figure 3-6. Evolution of inward (convective), outward (evaporative) and total

(convective+evaporative) heat flux

As outlined above, temperature and moisture distribution in the food at any

instance is important because spoilage can start from the higher moisture content

region. Sometimes the centre may have higher moisture content though the surface is

already dried. Consequently, investigating the temperate and moisture distribution is

critical in the case of food drying. The modelling and simulation study was helpful in

this regard, because it was difficult to measure the moisture distribution

experimentally. Figure 3-7 shows three dimensional temperature and moisture

distribution after 400 minutes of drying for drying air temperature 600C and air

velocity of 0.7m/s. It is interesting that, although the surface moisture content

ultimately became 0.2 kg/kg (db), the centre contained 0.6 kg/kg (db) moisture

(Figure 3-7a). Similar moisture profiles were obtained by Perussello et al. (2014).

Though, the drying process may appear to be visually complete, spoilage or

microbial growth could still initiate from the moist central region. Therefore, the

difficulty in removing moisture from the product centre is a major disadvantage of

convective drying.


In regard to temperature distribution, Figure 3-7b indicated that the

temperature gradient was not significant inside the material because the thickness of

the material was very small in the simulation.

Figure 3-7. (a) Moisture and (b) temperature distribution in the food after 400 minutes of drying

Parametric study

Parametric study was important in order to examine the effect of various

process parameters on drying kinetics. After the validation of the model, a parametric

analysis was conducted in COMSOL Multiphysics. Figure 3-8 illustrates the effect of

drying air temperature on drying curve at constant air velocity of 0.7m/s. It is clear

from Figure 3-8 that the increase in drying air temperature greatly increased the

drying rate. For example, it took 500, 300 and 200 minutes to reach a moisture

content value of 0.75kg/kg (db) at drying air temperature of 40, 50 and 600C,

respectively. However, the elevated drying air temperature can decrease the product

quality (e.g. nutrients). Therefore, the drying process has to be optimized and product

quality should be investigated along with drying kinetics.


Figure 3-8. Moisture content for different air temperature for velocity 0.7m/s

Figure 3-9 shows the drying curve for different air velocities at drying air

temperature 600C. It is evident that increasing the drying air velocity increased the

drying rate but the effect was not as significant as the effect of temperature. This is

because, in convective drying, drying is dominated by internal diffusion. Because the

drying rate is very high in the beginning, no constant drying rate period is evident.

The surface becomes dry quickly and the increasing velocity has not affected the

evaporation because sufficient moisture has not accumulated on the surface.

Therefore, the velocity increase has no effect on the drying rate. These finding

conforms with the drying rate curves presented by Karim and Hawlader (2005a)

showing that the drying rate is significantly different for temperature difference

whereas it is almost same for velocity changes.


Figure 3-9. Moisture content for different air velocity at 600C

3.7 CONCLUSIONS

In this study, three simulation models were developed based on three different

effective diffusivities. The models were validated with experimental results. Variable

material properties were considered in the simulation. The temperature dependent

effective diffusivity model predicted the initial stage of drying accurately, whereas

moisture dependent effective diffusivity simulations predicted the final stage well.

The evaporative cooling phenomena that occurred during the initial stage of

drying was investigated and explained. This observation may have significant

implication in regards to product quality improvement. Further research to verify this

latter phenomenon experimentally may lead to better fundamental understanding and

ultimately be applied to limit product temperature to ensure higher product quality.

Three dimensional temperature and moisture distribution were presented. The

three dimensional graphs suggested that although the surface of the product was dry,

the centre moisture content was significant. Parametric analysis showed that by

increasing drying air temperature, the drying rate can be significantly improved.

However, drying air velocity (flow rate) has negligible impact on drying rate.


3.8 ACKNOWLEDGEMENTS

The authors acknowledge the contribution of Dr Zakaria Amin and M.U.H.

Joardder for their support in checking the manuscript.

3.9 FUNDING

The author acknowledges the financial support from International Postgraduate

Research Award (IPRS) and Australian Postgraduate Award (APA) to carry out this

research.

Chapter 4: Single Phase Model for IMCD Using Lambert’s Law 45

Single Phase Model for IMCD Chapter 4:

Using Lambert’s Law

This chapter aims to develop an IMCD model considering single-phase mass

transport. The model uses Lamberts Law for microwave power absorption and the

transport model from the previous Chapter.

The chapter feature the following publication:

C. Kumar, M. U. H. Joardder, T.W. Farrell, G.J. Millar and M. A. Karim

(2015) Modelling of Intermittent Microwave Convective Drying (IMCD)

of Apple. Drying Technology, (To be submitted).

The signed statement of contribution page is inserted below:

46 Chapter 4: Single Phase Model for IMCD Using Lambert’s Law

4.1 ABSTRACT

Intermittent microwave convective drying (IMCD) is an advanced technology

which improves both energy efficiency and food quality in drying. Modelling of

IMCD is essential in order to optimize the microwave power level and intermittency

during the process. However, there is still lack of modelling studies dedicated to

IMCD. In this study, a mathematical model for IMCD is developed and validated

with experimental data. The microwave power absorption is calculated and other

inputs parameters for microwave modelling are presented in this study. The model

developed for the full drying period and the challenges in developing the model are

discussed.

4.2 INTRODUCTION

Currently, 1.3 billion tonnes of foodstuffs are lost annually due to a lack of

proper processing and preservation (Gustavsson et al., 2011). Drying is a method of

removing moisture for the purpose of preserving food from microbial spoilage.

Conventional convective drying is a very lengthy and energy intensive process

(Karim & Hawlader, 2005b). Higher drying temperatures reduce the drying times,

however under such conditions food quality and nutritional value is reduced and

higher temperatures require higher energy usage. To overcome these problems,

convective drying is combined with microwave drying. Microwaves interact with

water molecules inside the food and heats up samples volumetrically thus increases

the moisture diffusion rate which can significantly reduce the drying time and

improve energy efficiency (Kumar et al., 2014). There is a problem, however, with

the continuous application of microwaves in the drying process as this can result in

high product temperatures and uneven heating (Kumar et al., 2014). As noted above,

high drying temperatures can cause quality degradation in heat sensitive materials

such as fruits and vegetables (Joardder, Karim, et al., 2013; Joardder, Karim, et al.,

2014). This can be overcome by applying microwave power intermittently. Research

has shown that intermittent microwave convective drying increases both energy

efficiency and product quality (Kumar et al., 2014).

The main advantages of interment microwave assisted drying are: (1)

Volumetric heating: Microwave energy interacts with water molecules within the

food leading to volumetric heating and increased moisture diffusion rates heats up


volumetrically and pumps moisture to surface (I. W. Turner et al., 1998). This can

thereby significantly reduce drying times (Mujumdar, 2004; M. Zhang et al., 2006);

(2) Quality improvement: The quality of the dried product can be improved by

combining intermittent microwave heating with other drying (Dev et al., 2011); (3)

Controlled heating: The fidelity of heating can be controlled using microwave energy

as it can be applied in a pulsed manner (Gunasekaran, 1999).

Soysal et al.(2009b) investigated intermittent microwave-convective drying

(IMCD), and the results were compared with continuous microwave-convective

drying (CMCD) and traditional convective drying for oregano. They observed that

the IMCD was 4.7–11.2 times more energy efficient when compared to convective

drying. Furthermore, the drying time of the convective drying was about 4.7–17.3

times longer when compared with the IMCD drying. Ahrné et al. (2007) compared

CMCD and IMCD for banana as a heat sensitive food product. They reported drying

at variable microwave power as a more suitable drying process. They report that

IMCD produces better outcomes in that it reduced the charring of the product. Esturk

(2012) studied IMCD of sage leaves and compared the result with convective air-

drying and CMCD. Although CMCD provided the fastest drying rate, it yielded the

lowest quality (in terms of oil content). Esturk (2012) also noted that in IMCD, the

intermittency and the microwave power level significantly impacted the energy

consumption and the quality of dried product (Esturk, 2012). Therefore, the

microwave power level and pulse ratio should be carefully chosen to achieve the best

outcomes.

Mathematical modelling can help us to understand the heat and mass transfer

involved in IMCD and thereby be used to determine the optimum pulse ratio and

power levels for drying (Kumar et al., 2014). Previously mentioned work related to

IMCD has only been limited to experimental analysis. To date, relatively few studies

have presented theoretical models of the IMCD of food.

Recently, Bhattacharya et al. (2013) and Esturk (2012) have developed purely

empirical model for CMCD (not IMCD) of oyster mushroom (pleurotus ostreatus)

and sage respectively. However, these empirical models do not provide physical

insight into the process and are only applicable to a specific experimental range

(Kumar et al., 2012a; Perussello et al., 2014). Some diffusion-based single phase

models exist for CMCD (J. R. Arballo et al., 2012; Mohamed Hemis et al., 2012),


however, none of them considered intermittency of the microwave energy applied.

For this reason, they cannot be applied to IMCD, and are not capable of investigating

the temperature redistribution due to intermittency. However, there are some

simulation models considering intermittency of microwave power (Gunasekaran,

1999; Gunasekaran & Yang, 2007a, 2007b; Yang & Gunasekaran, 2004); mass

transfer was neglected in those models.

Some models of intermittent heating use constant dielectric properties (Yang &

Gunasekaran, 2001). However, dielectric properties vary with moisture content, in

particular, for fruits and vegetables because they contain large amount of moisture.

Moisture content has a significant effect on dielectric properties of fruits and

vegetables (H Feng et al., 2002). Therefore, constant dielectric properties cannot be

considered in case of drying of fruits and vegetables.

Taken together, it can be concluded that, although extensive research has been

carried out on microwave convective heating, there is very limited study dealing with

modelling the IMCD of food which considers the whole drying period, as well as

variable material properties. Furthermore, the temperature redistribution due to

intermittency of the microwave, which is crucial in IMCD for quality improvement,

is not properly investigated.

In the current study, we present a model of IMCD of food that accounts for

intermittency of the microwave power, variable thermos-physical and dielectric

properties of the material. COMSOL Multiphysics 4.4, a finite element based

engineering simulation software, is used to model coupled heat and mass transfer

model equations. The outcomes of the model are presented and discussed and they

are validated with experimental data.

4.3 MATHEMATICAL MODELLING

We consider a 2D axisymmetric geometry of a cylindrical slice apple presented

in Figure 4-1. The following assumptions are applied when developing the

mathematical model:

A homogeneous domain having a single temperature are considered;

The initial temperature and moisture distribution within the slice are

uniform;


The thermo-physical and dielectric properties vary with moisture content

of the sample;

Only single-phase water is present in the domain. This characterizes the

moisture concentration of the apple. Furthermore, moisture is transported

by diffusion towards the surface.

Figure 4-1. 3D apple slice and 2D an axisymmetric domain showing symmetry boundary and transfer

boundary (arrow)


Heat transfer:

The energy balance is characterized by a Fourier flux and such that a heat

generation term due to microwave heating, micQ (W/m3).

)()( tfQTkt

Tc micp

4-1

where, T is the temperature (0K), is the density of sample (kg/m

3), pc is the specific

heat (J/kg/K), f(t) is an intermittency function discussed in section 4.5, and k is the

thermal conductivity (W/m/K). This heat generation, micQ (W/m3), is calculated using

Lamberts Law (Abbasi Souraki & Mowla, 2008; J. R. Arballo et al., 2012; Mohamed

Hemis et al., 2012; Mihoubi & Bellagi, 2009) as discussed in section 4.3.3.

Mass transfer:

We assume that the mass flux of moisture is due to Fickian diffusion; therefore,

0

c-D

t

ceff

4-2


where, c is the moisture concentration (mol/m3), effD is the effective diffusion

coefficient (m2/s) discussed further in section 4.3.4.

Initial and boundary conditions 4.3.2

The initial conditions for heat and mass transfer are given by,

CT t

0

)0( 20 , 4-3

and 0)0( cc tw , 4-4

respectively. Here 0c is the initial moisture concentration of the apple (mol/m3).

The boundary conditions for the heat and mass transfer equations at the

transport boundaries (as shown in Figure 4-1) are given by,

fg

airveqv

mairT hRT

-pphTThTk

,)()( , 4-5

and

RT

-pphc-D airveqv

meff

, , 4-6

respectively. Here, Th is the heat transfer coefficient (W/m2/K) and airT is the drying

air temperature (0C), airvp , vapour pressure of ambient air (Pa), eqvp , is the

equilibrium vapour pressure (Pa), fgh is the latent heat of evaporation (J/kg) , R is

the universal gas constant (J/mol/K), and mh is the mass transfer coefficient (m/s).

The boundary condition for heat and mass transfer of the symmetry boundary

(as shown in Figure 4-1) are given by

0)( Tk , 4-7

and 0c-Deff . 4-8

Modelling of microwave power absorption using Lamberts Law 4.3.3

Budd & Hill (2011) compared power absorption modelled by Lamberts Law

and Maxwell’s equation and showed that for thicker material the power absorption

according to both approaches is similar.

Many researchers used Lamberts Law for microwave energy distribution in

food products during drying (Abbasi Souraki & Mowla, 2008; J. R. Arballo et al.,


2012; Mohamed Hemis et al., 2012; Khraisheh et al., 1997; Mihoubi & Bellagi,

2009; Salagnac et al., 2004; Zhou et al., 1995). Therefore, in this study, Lamberts

Law has been used to calculate microwave energy absorption inside the food

samples. It considers exponential attenuation of microwave absorption within the

product, namely,

h-zα-

mic=PP 2

0 exp . 4-9

Here, 𝑃0 the incident power at the surface (W), α is he attenuation constant

(1/m), and h is the thickness of the sample (m) and (h-z) represents the distance from

surface (m). The measurement of 0P via experiments is presented in section 0.

The attenuation constant, α is given by

2

1'

''1

'2

2

, 4-10

where is the wavelength of microwave in free space ( cm24.12 at 2450MHz

and air temperature 200C) and ε' and ε" are the dielectric constant and dielectric loss,

respectively.

The volumetric heat generation, micQ (W/m3) in equation 4-1 is then calculated

by:

V

PQ mic

mic , 4-11

where, V is the volume of apple sample (m3).


The input parameters of the model are listed in Table 4-1 and some of these

values are further discussed in this section.


Table 4-1. Input properties of the model

Parameters Value[Unit] Reference

Initial moisture content (db), 0M 6.14[kg/kg] This work

Initial temperature, iT 20[°C] This work

Molecular weight of water, wM 18[g/mol] (Çengel & Boles, 2006)

Latent heat of evaporation, fgh 2358600[J/kg] (Çengel & Boles, 2006)

Drying air temperature, airT 60°C This work

Vapour pressure of ambient air, airvp 2.7[kPa] Calculated

Diameter of the sample 40[mm] This work

Thickness of the sample 10[mm] This work

Reference diffusivity, refD 3.24e-9 [m2/s] Calculated

Heat transfer coefficient, Th 16.746 [W/(m2·K)] (Kumar et al., 2015)

Mass transfer coefficient, mh 0.067904 [m/s] (Kumar et al., 2015)

Microwave incident power absorption

The incident power at the surface, 0P , can be determined by calculating the heat

absorbed by distilled water of same volume with the sample placed in microwave

oven (Chandrasekaran et al., 2012; Mohamed Hemis et al., 2012; Lin et al.,

1995).This is one of the most difficult aspects of microwave heating (Ashim, 2001).

Arballo et al. (2012) determined 0P via the application of the formula,

t

TCmP pww

0 ,

4-12

where, wm is the mass of water (kg), pwC is he specific heat of water (J/kg/K), T is

the temperature rise of water (0C) and T is the heating time (s).

A major drawback of equation (4-12) is that it does not account for the

evaporation heat loss. The evaporation of water is not negligible at higher microwave

power. This evaporative heat loss was also taken into account in some studies

(Abbasi Souraki & Mowla, 2008). Then the absorbed power considering evaporative

heat loss can be calculated by


wfgpww mht

TCmP

0 .

4-13

Here, wm is the evaporated mass (kg) of water and fgh is the latent heat of

evaporation (J/kg).

Auxiliary equations

The moisture content (wb), wbM , can be calculated from the water

concentration by the formula:

w

wb

cMM . 4-14

Here wM is the molecular weight of water (kg/mol).

The relationship between dry basis moisture content, dbM , and wet basis

moisture content, wbM , is given by:

wb

wbdb

M

MM

1. 4-15

Equilibrium vapour pressure

The vapour pressure of the food is assumed to be always in equilibrium with

the vapour pressure given by an appropriate sorption isotherm. For apple, the

correlation of equilibrium vapour pressure with moisture and temperature is given by

(Ratti et al., 1989),

)](ln[232.0182.0exp)(0411.0949.43696.0

,, TPMeMTPP satdb

M

dbsatveqv

. 4-16

Here, dbM is the moisture content dry basis and satvP , is the saturated vapour

pressure given by (Vega-Mercado et al., 2001),

.

ln656.61001445.0

104176.00486.03915.1/2206.5800exp

37

24

,

TTx

TxTTP satv

4-17

Effective diffusivity

We note that in our single phase model the effective diffusivity, effD , accounts

for the transport of combined moisture, c. Here we adopt an expression for refD that


was developed in the previous work (Kumar et al., 2015) that is a function of both

temperature and moisture, namely,

2

2

00

b

bDeD

D

refRT

E

eff

a

. 4-18

Here refD is reference diffusivity (m2/s), aE is activation energy of diffusion of

water (J/mol), b is the half thickness of the material (m), b0 is the initial thickness

(m), and D0 is a integration constant and is usually referred to as a frequency factor

when discussing Arrhenius equation (m2/sec). The activation energy was calculated

from the slope of a ln( refD ) versus (1/T) graph resulting in the values D0 =0.09 and

aE = 50 KJ/mol.

The thickness ratio obtained by the following equation:

00 M

M

b

b

w

wbw 4-19

where w is density of water (kg/m3), is density of sample(kg/m

3), wbM is moisture

content wet basis and 0M is initial moisture content kg/kg (wb).

Thermo-physical properties of apple

Thermal conductivity and specific heat of apple can be expressed as a functions

of moisture content (Sweat, 1974), namely,

wbMk 00493.0148.0 4-20

and wbp Mc 22.34.11000 , 4-21

respectively.

Moreover, the density of apple, 𝜌, during drying changes with moisture

content, wbM . In this study, we measured the density change with moisture content

of apple by a solid displacement method (Joardder et al., 2015; Yan et al., 2008)

using a cylindrical vial and 57 µm glass beads. The relationship between 𝜌 and wbM

determined to be,

94.41501.569 wbM . 4-22


Dielectric properties of apple

The dielectric properties of the material are the most important parameters in

microwave heating and drying applications because these properties define how

materials interact with electromagnetic energy (Sosa-Morales et al., 2010). The

evaluation of dielectric properties is critical in modelling and product and process

development (Ikediala et al., 2000). Dielectric properties of a materials define how

much microwave energy will be converted to heat (Chandrasekaran et al., 2013).

Here we use the data of Martín-Esparza et al. (2006) in a quadratic regression

analysis in which the intercept of the ' and versus wbM graph was set to 0.1 in

order to avoid numerical singularity in ' and when wbM is zero. The resulting

quadratic expression are found to be:

1.0289.30638.36'2

wbwb MM 4-23

and 1.0.8150.26543.132

wbwb MM . 4-24

Heat and mass transfer coefficient

The heat and mass transfer coefficients were calculated based on the empirical

relationship discussed in previous paper (Kumar et al., 2015) and found to be Th

=16.746 W/(m2·K) and mh =0.067904 m/s, respectively.

4.4 MATERIALS AND METHODS

Sample preparation

Fresh granny smith apples used for the intermittent microwave drying

experiments were obtained from local supermarkets. The samples were stored at

5±10C to keep them as fresh as possible before they were used in the experiments.

The apples taken from the storage unit were washed and put aside for one hour to

allow their temperature to elevate to room temperature prior to each drying

experiment. The samples were cut to a thickness of 10mm and a diameter of

approximately 40mm. The initial moisture content of the apple slices was

approximately 0.868 kg/kg (wet basis).


IMCD and convection drying

The IMCD drying was achieved by heating the sample in microwave for 60s

and then drying for 120s in the convection dryer. The experiments were conducted

with a Panasonic Microwave Oven having inverter technology with cavity dimension

355mm (W) x251mm (H) x365mm (D). The inverter technology enables accurate

and continuous power supply at lower power settings. Whereas, with conventional

oven supply the lower power is achieved by turning the microwave on and off at

maximum power (Panasonic, 2013). The microwave oven is able to supply 10

accurate power levels with a maximum of 1100W at 2.45GHz frequency. The apple

slices were placed in the centre of the microwave cavity, for an even absorption of

microwave energy. The moisture loss was recorded at regular intervals at the end of

power-off times by placing the apple slices on a digital balance (specification: 0.001g

accuracy).

The convection drying was conducted to compare the results with IMCD. For

convection drying, the same samples were placed in household convection dryer and

temperature was set to 600C. The moisture loss was recorded at regular intervals of

10 mins with the digital balance (specification: 0.001g accuracy). All experiments

were done triplicate and standard deviation was calculated.

Thermal imaging

A Flir i7 thermal imaging camera was used to measure the temperature

distribution on the surface. Accurate measurement of temperature by thermal

imaging cameras depend on the emissivity values. The emissivity value for apple

was found in the range between 0.94 and 0.97 (Hellebrand et al., 2001) and set in the

camera before taking images.

Determination of incident power (P0) for Lambert’s Law

The Panasonic inverter microwave oven was used in the experiments to

determine the power absorption. The tests were conducted at three power levels,

namely; 100W, 200W and 300W with a water sample. The volume of water sample

was taken as the same volume of apple to obtain accurate power absorption. Water

was heated for 60s and thermal images were taken by the thermal imaging camera

(Flir i7) before and after heating. The water was properly agitated to measure the


average rise of temperature. The absorbed power, 0P , can be calculated by equation

4-13 for various load volume and applied microwave power.

4.5 SIMULATION PROCEDURE

COMSOL Multiphysics, advance software for modelling and simulation, was

used to implement the numerical solution of the model introduced in section 4.3. A

combination of a rectangular function and an analytic function in COMSOL

Multiphysics were used to develop an intermittency function as shown in Figure 4-2.

Then it was multiplied with the heat generation term in the energy equations to

implement intermittency of the microwave heat source.

Figure 4-2. Intermittency function

The Figure 4-3 below shows the flow chart of the simulation procedure. It

shows that the moisture dependent material properties, microwave source term, and

input properties are updated at the beginning of each iteration.


Figure 4-3. Flow chart showing the modelling strategy in COMSOL Multiphysics


Incident power absorption by experiments 4.6.1

The incident power absorbed by the sample calculated for three different power

levels for various loads was calculated by equation 4-13 and then converted to power

absorption ratio defined as the ratio of absorbed power by sample and microwave set

power. Table 4-2 shows the power absorption ratio obtained for microwave set three

different microwave set power, namely, 100W, 200W and 300W.

Table 4-2. Power absorption ratio for microwave power (100W, 200W and 300W) for different

sample volume

Sample

volume (cc)

Power absorption ratio Standard

deviation 100W 200W 300W Average

15 25 25.3 26 25.43333 0.51316

35 38 43 41 40.66667 2.516611

55 59 66 59 61.33333 4.041452

It is interesting to note that the power absorption ratios are same for a certain

volume of sample irrespective of the microwave power. Therefore, the average

power absorption ratios are plotted against the sample volume in Figure 4-4 with the

error bar showing standard deviations.


Figure 4-4. Microwave power absorption for different loading volume

The above results correlate well with those of Mudgett (1986) who also

investigated the power absorption ratio and found a similar trend.

Average moisture curve 4.6.2

The comparison of moisture content obtained from experiments and simulation

is shown in the Figure 4-5. A Pearson correlation coefficients, R2, was used to

determine the goodness of fit of the model. We see that a high correlation is obtained

between the model and experimental values with R2=0.997623. This good agreement

between experimental data and model calculations suggests the suitability of the

model to describe the drying kinetics and moisture content obtained during the

IMCD drying process.

To demonstrate the advantage of IMCD drying over convection drying, the

drying curves for both are plotted in Figure 4-5. It was found that, in 75 minutes of

drying, convection drying reduced the moisture content to 3 kg/kg db, whereas by

using IMCD reduced the moisture content to 0.4 kg/kg db.

Thus, IMCD significantly reduces the drying time. To reduce the moisture

content to 0.4 kg/kg db by convection drying took around 300min, which is 4 times

longer than the IMCD with intermittency 60s on and 120s off.


Figure 4-5. Drying curve for IMCD (experiments and model) and convective drying

Temperature 4.6.3

Figure 4-6 shows the temperature at the centre of the top surface predicted

from our model whilst Figure 4-7 shows the temperature distribution of the surface

obtained experimentally from thermal imaging. The thermal images were taken

immediately after microwave heating for 60s and after tempering for 120s in the

convection dryer. Thus, this measurement allows the investigation of temperature

rise during microwave heating and drops during tempering in the process of IMCD.


Figure 4-6. Temperature curve obtained from the model

The temperature curve from the model shows that the temperature rises after

each heating cycle. The temperature then falls (at the centre of the surface) during the

120s convection drying phase (when the microwave is turned off).

The thermal images also show a similar pattern. To illustrate this more clearly,

the centre temperatures measured on the surface and the analogous model prediction

for the selected time are presented in Table 4-3.

Table 4-3: Centre temperature of apple surface from experiment and model

Time (mins) 16 18 19 21 73 75

Microwave On Off On On Off On

Experimental temperature 70 50 70.6 45.5 114 57.3

Model temperature 70.5 57 76.5 61 114.5 95.4

We observe that there is reasonable correlation between the observed and

predicted temperatures. Certainly the periodic pattern of heating and cooling is

captured by the model. There are, however, some discrepancies observed in Table

4-3. Generally, the model seems to predict higher temperature than those observed.

We believe that his is due to the fact that the thermal images here taken after

removing the sample the microwave oven and placing them in an ambient


environment (~200C) for a short time. A further source of discrepancy could be due

to the fact that our single phase diffusion model does not account for the latent heat

losses that are associated with evaporation.

.

Figure 4-7. Thermal images of top surface at selected times

It can be seen from the above figures (Figure 4-6 and Figure 4-7) that the

temperature reaches as high as 114°C at the end of the drying. This is because the

temperature continues to rise after each cycle while it fluctuates. We note that a

similar rise in temperature with cycled microwave heating was found by Rakesh et

al. (2010) and Yang et al. (2001).


In light of these findings, it can be said that the temperature of the sample

should be controlled, particularly, at the last stage of drying. Since the higher

temperature may reduce the food quality or even burn the product, the microwave

power should be reduced or the tempering period increased to avoid burning at the

final stage of drying.

Although the above results are taken at a single point in the sample, it is

reasonable to assume that heat energy is being dissipated during tempering via

conduction (at least in part) as opposed to purely convective cooling. This

redistribution of temperature could significantly contribute improving product

quality during IMCD by selecting an optimum tempering time.

Moisture and temperature distribution 4.6.4

Moisture distribution inside the sample is shown in Figure 4-8. We observe

that the moisture content of the surface reduces to nearly zero at about 20mins of

drying, whereas at that time the moisture content at the centre is still at its maximum.

Figure 4-8. Moisture distribution inside the sample

Another observation to note from the figure is that the moisture content is

always higher in the inner part of the sample and decreases as drying progress.


These results are consistent with the idea that the surface of the sample dries first and

then the moisture from centre is removed.

Figure 4-9 shows the simulated temperature evolutions at the surface, centre

and 8mm beneath the surface. It shows that the temperature is always higher in the

interior of the apple than the surface, despite the fact that microwave power

absorption is higher at the surface, according to Lamberts Law. This is due to the

internal heating characteristics of the microwave. Although the heating is higher near

the surface, the convection and evaporative cooling reduce the temperature of the

surface. A similar pattern (higher centre temperature) has been observed in the

microwave heating (A.K. Datta, 2002; Rakesh et al., 2010). The temperature

difference between surface and centre increase as drying progress, this is because the

thermal conductivity is considered to be a function of moisture content, and it

decrease with moisture content.

Figure 4-9. Temperature distribution inside the sample

Equilibrium vapour pressure 4.6.5

Equilibrium vapour pressure, eqvP , , is an important parameter for surface

evaporation and thus moisture loss. The equilibrium vapour pressure at the surface as


determined from equation 4-16 is plotted in Figure 4-10. We observe that eqvP ,

initially increases rapidly because of increase in temperature and the higher moisture

content. However, when the material becomes drier, near the end of the drying, the

equilibrium vapour pressure decreases. This indicates that initially the moisture loss

is higher and that the drying rate starts to decrease when the equilibrium vapour

pressure starts reducing at about 15 mins. From the moisture distribution curve

(Figure 4-8), it can be seen that the surface moisture content becomes close to zero

after 15 mins of drying, therefore, after that time the vapour pressure did not rise.

However, due to diffusion the vapour pressure is still higher than the ambient vapour

pressure, airvP , , and evaporation occurs.

Figure 4-10. Evolutions of equilibrium vapour pressure at the surface of the sample

Absorbed power distribution 4.6.6

Absorbed power along the depth of the sample at the end of drying is shown in

Figure 4-11. According to Lamberts Law, the power absorbed is the maximum at the

surface and decreases exponentially inside the sample. The absorption at the top

surface is 25 W which was also calculated in Table 4-2 and it decrease to 10W at the


bottom of the surface. A similar trend of theoretical power absorption was found by

Budd et al. (2011) and they also showed that the power absorption is higher at the

surface and decays exponentially with depth in the sample. Although microwave

power absorption is higher near surface, the convection and evaporative cooling

leads to a reduction of the temperature of the surface.

Figure 4-11. Absorption of microwave power along the length of the sample at 75mins

4.7 CONCLUSIONS

In this study, a novel model of IMCD has been developed and compared with

experimental data collected from a sample of apple. The predicted moisture content

showed good agreement with experimental data. The IMCD (with microwave 60s on

and 120s off) was four times faster as compared to convection drying. The

intermittency of microwaves in IMCD allows the temperature to re-distribute and

drop, thus, IMCD helps to limit the temperature and can improve product quality.

Unlike convection drying, the temperature at the centre of the sample is higher in

IMCD. The moisture distribution from the model showed that the moisture content is

always higher in the inner part of the sample and decrease as drying progress.

Limitations of the present model have been identified and suggestions for future

improvement in IMCD modelling have been presented.

Acknowledgement:

The author acknowledges the International Postgraduate Research Award

(IPRS) and Australian Postgraduate Award (APA).

Chapter 5: Multiphase Model for Convection Drying of Food 68

Multiphase Model for Chapter 5:

Convection Drying of Food

This Chapter presents a multiphase porous media model for convection drying,

which is a more advanced model compared to the single-phase models. The transport

model in this Chapter will be used in the following Chapters (Chapters 6 and 7) to

develop IMCD models.

This Chapter features the following submitted publication:

C. Kumar, M. U. H. Joardder, T.W. Farrell, M. A. Karim, and G.J. Millar

(2015) Non-equilibrium multiphase porous media model for heat and mass

transfer during food drying. International Journal of Multiphase Flow, (To

be submitted).

The signed statement of contribution page is inserted below:


5.1 ABSTRACT

A multiphase porous media model has been developed to predict the transport

of liquid water, vapour, air and energy during the convection drying of food. The

model considered the transport of liquid water by capillary diffusion and gas

pressure, and the transport of vapour with binary diffusion and gas pressure. A non-

equilibrium formulation was used to calculate the evaporation rate, which enabled

the separate illustration of the transport of vapour and liquid water. The equations

were solved by Finite Element Method (FEM) using physics-based modelling and

simulation platform called COMSOL multiphysics. The model predictions were

validated using experimental data and good agreement was found. Spatial

distribution of liquid water and vapour saturation curves showed that the saturation

levels were lower on and near the surface compared to the centre of the food

material. The convective and diffusive fluxes of liquid water and vapour were

presented, and this data suggested that the fluxes were higher on and near the surface

of the sample.

5.2 INTRODUCTION

Modelling of food processing is very complex problem due to issues such as

the intricate physical structure of materials of interest. Many modelling efforts have

been reported in the literature for different food preparation processes such as drying

(Barati & Esfahani, 2011; Diamante et al., 2010; Karim & Hawlader, 2005a, 2005b;

Kumar et al., 2012; Kumar et al., 2015), frying (H. Ni & Datta, 1999), microwave

heating (H. Ni et al., 1999; Rakesh et al., 2010), thawing (Chamchong & Datta,

1999a), baking (J. Zhang et al., 2005), and puffing (Rakesh & Datta, 2013) etc. All

of these models can be classified into two broad categories; empirical models and

theoretical models (Kumar et al., 2014). The empirical or observation based models

can be developed rapidly and are quite effective. However, they do not provide a

physical insight into the process and exhibit limited predictive capability. In contrast,

physics based models are preferred as predictive models not only in food drying but

also in areas outside of the food industry. Among these theoretical approaches,

diffusion based models are very popular because of their simplicity and as such have

been used by many researchers (Chandra Mohan & Talukdar, 2010; Kumar et al.,

2015; Perussello et al., 2014; Perussello et al., 2012). These latter models assume

conductive heat transfer for energy, and diffusive transport for moisture. These

70 Chapter 5: Multiphase Model for Convection Drying of Food

models need diffusivity values which have to be experimentally determined.

Although these latter models can provide a good match with experimental results,

they cannot provide a detailed understanding of other transport mechanisms such as

pressure driven flow. Lumping all the water transport processes as diffusion cannot

be justified under all situations. The drawback of these kind of models has been

discussed in detail in the work of Zhang and Datta (2004).

The next group of models with improved formulation compared to diffusion

models assumes a sharp moving boundary between dry and wet regions. This type of

model has been applied in relation to deep fat frying (Farkas et al., 1996) and are

characterized as analogues to freezing and thawing models of a pure substance

(Mascarenhas et al., 1997). Recently, distributed evaporation models, in contrast to

sharp boundary models, have become popular. Datta (2007a) termed these

distributed evaporation models as mechanistic models because these models

considered heat and mass transfer equations for each phase (solid, liquid, gas plus

vapour) in porous media. Therefore, they are termed multiphase porous media

models. These latter models rigorously study the transition from the individual phase

at the ‘microscopic’ level to representative average volume at the ‘macroscopic’

level (A. K. Datta, 2007b). Furthermore, they are computationally effective and

consequently have been applied to a wide range of food processing applications such

as frying (Bansal et al., 2014; H. Ni & Datta, 1999), microwave heating (Chen et al.,

2014; Rakesh et al., 2010), puffing (Rakesh & Datta, 2013) baking (J. Zhang et al.,

2005), meat cooking (Dhall & Datta, 2011) etc. However, application of these

models to the drying processes has been very limited.

There are some multiphase models which have been applied in: (a) vacuum

drying (Ian W. Turner & Perré, 2004); (b) convection drying (Stanish et al., 1986) of

wood and clay (Chemkhi et al., 2009); (c) microwave spouted bed drying of apples

(H. Feng et al., 2001); and (d) large bagasse stockpiles (Farrell et al., 2012). A

common issue integral to those latter models is the assumption that the vapour and

water phases are in equilibrium. However, equilibrium conditions may only be valid

for lower moisture content of the sample during drying; thus, equilibrium conditions

may never be achieved at the surface since transport rates are relatively high there.

Therefore, a non-equilibrium approach appears to be a more realistic representation

of the physical situation during drying (J. Zhang & Datta, 2004). Using equilibrium


vapour pressure in a drying model is likely to overestimate the drying rate, because

equilibrium may not be achieved instantaneously at the surface due to low moisture

content. Then the non-equilibrium approach for evaporation can be used to express

evaporation, thus allowing the calculation of each phase separately. The equations

resulting from non-equilibrium models provide a better description of the physics

involved and facilitate calculation of evaporation behaviour. Recently, Zhang et al.

(2012) applied a multiphase model to a non-equilibrium formulation for evaporation.

Nevertheless, they only used this latter model for vacuum drying of corn and the

simulation results were not compared with any experimental data. This study was

aimed at development of a multiphase porous media model using non-equilibrium

evaporation rates considering all transport mechanisms such as capillary diffusion,

and pressure driven flow evaporation of all phases (water, vapour, and air). Such a

model can provide an insight into the relative contribution of each transport

mechanism. The objectives of this study were to:

Development of a multiphase porous media model using non-equilibrium

formulations for transport of liquid water, vapour, and energy during

drying of food;

Calculation of spatial and temporal profiles for liquid water, vapour and

temperature;

Comparison of the spatial and temporal profiles for capillary and

convective fluxes of liquid water, and diffusive and convective fluxes for

vapour;

Discussion of the distribution of evaporation rate, vapour pressure,

equilibrium vapour pressure and saturation vapour pressure at the surface

in relation to drying rate.

5.3 MATHEMATICAL MODEL

In this section, the model equations for multiphase porous media are developed

describing heat, mass and momentum transfer within an apple slice (food material)

during convection drying. The equations also represent the transport mechanism,

assumptions and input parameters for the model.

The mathematical formulation developed here was based on the formulations

introduced by Datta (2007a). The model developed in this research considered


transport of liquid water, vapour, and air inside food materials. The mass and energy

conservation equations included convection, diffusion, and evaporation. The non-

equilibrium formulation for evaporation has been used to describe the evaporation,

which allows implementation of the model in commercial software. Momentum

conservation was developed from Darcy’s equations. Evaporation was considered as

being distributed throughout the domain and a non-equilibrium evaporation

formulation was used for evaporation condensation-condensation phenomena.

Problem description and assumptions 5.3.1

A schematic of the problem is presented in Figure 5-1. A 2-D axisymmetic

geometry of a 3-D apple slice was considered for simulation. Heat and mass transfer

took place at all boundaries except the symmetry boundary. The apple slice was

considered as a porous medium and the pores were filled with three transportable

phases, namely liquid water, air and water vapour as shown in Figure 5-1.

Figure 5-1. Schematic showing 3D sample, 2D axisymmetric domain and Representative Elementary

Volume (REV) with the transport mechanism of different phases

All phases (solid, liquid, and gases) were continuous and local thermal

equilibrium is valid, which means that the temperatures in all three phases are equal.


Liquid water transport occurred due to convective flow resulting from gas pressure

gradient, capillary flow and evaporation. Vapour and air transport arose from gas

pressure gradients and binary diffusion.


The mathematical model consisted of conservation equations for all

transportable phases and transport mechanisms discussed above.

Mass balance equations 5.3.3

The representative elementary volume V (m3) is the sum of the volume of

three phases, namely, gas, water, and solid, namely,

swg VVVV . 5-1

where, gV is the volume of gas (m3), wV is the volume of water (m

3), and sV was

the volume of solid (m3).

The apparent porosity, , is defined as the volume fraction occupied by gas and

water. Thus,

V

VV wg

. 5-2

The water, wS , and gas, gS , Saturation are defined as the fraction of pore

volume occupied by that particular phase, namely,

V

V

VV

VS w

gw

ww

, 5-3

and w

g

gw

g

g SV

V

VV

VS

1

, 5-4

respectively.

The mass concentrations of water, wc (kg/m3), vapour, vc (kg/m

3),and air , ac

(kg/m3), are given by,

www Sc , 5-5


gv

v SRT

pc , 5-6

and ga

a SRT

pc , 5-7

respectively. Here, w is the density of water (kg/m3), R is the universal gas constant

(J/mol/K), and T is the temperature of product (K), vp is the partial pressure of

vapour (Pa), ap is the partial pressure of air (Pa).

The mass conservation equation for the liquid water is expressed by,

evapwww RnSt

,

5-8

where, wn

is water flux (kg/m2s), and evapR is the evaporation rate of liquid water to

water vapour (kg/m3s).

The total flux of the liquid water is due to the gradient of liquid pressure,

cw pPp , as given Darcy’s Law (Bear, 1972), namely,

c

w

wrw

w

w

wrw

ww

w

wrw

ww pkk

Pkk

pkk

n

,,,

. 5-9

Here, P is the total gas pressure (Pa), cp is the capillary pressure (Pa), wk is

the intrinsic permeability of water (m2), wrk , is the relative permeability of water, and

w is the viscosity of water (Pa.s). More details on these parameters are discussed in

later sections.

We note that, the first term of equation 5-9 represents the flow due to gradients

in gas pressure, which is significant only in the case of intensive heating such as

microwave heating, deep fat frying, and contact heating at high temperature. The

second term represents the flow due to capillary pressure.

The capillary pressure depends upon concentration ( wc ) and temperature (T)

for a particular material (A. K. Datta, 2007a). Therefore, equation 5-9 can be

rewritten as,


TT

pkkc

c

pkkP

kkn c

w

wrw

ww

w

c

w

wrw

w

w

wrw

ww

,,,

. 5-10

In turn, the second and third terms of equation 5-10 can be rewritten in terms of

capillary diffusivity, cD (m2/s), and thermal diffusivity, TD (m

2/s), given by,

w

c

w

wrw

wcc

pkkD

,

, 5-11

and T

pkkD c

w

wrw

wT

,

, 5-12

respectively.

The capillary diffusivity due to the temperature gradient, T (K), is known as

the Soret effect and is often less significant than the diffusivity due to concentration

gradients (A. K. Datta, 2007a), and will thus be neglected in this work.

Substituting the above into equation 5-8 the concentration of liquid water can

be written as,

evapwc

w

wrw

www RcDPkk

St

, 5-13

The conservation of water vapour can be written in terms of the mass fraction,

v , as

evapvvgg RnSt

, 5-14

where, g is the density of gas (kg/m3), v is the mass fraction of vapour and vn

is

the vapour mass flux (kg/m2s).

For a binary mixture vn

can be written as (Bird et al., 2007),

vgeffgg

v

grg

vgv DSPkk

n

,

,, 5-15

where, gk is the intrinsic permeability of gas (m2), grk , is the relative permeability of

gas (m2), and g is the viscosity of gasr (Pa.s) and geffD , is the binary diffusivity of

vapour and air (m2/s).


The gas phase is a mixture of vapour and air. After calculating the mass

fraction of vapour, v , from the above equations, the mass fraction of air, a , can be

calculated from

va 1 . 5-16

Continuity equation to solve for pressure 5.3.4

The gas phase is assumed to consist of an ideal mixture of water vapour and

air. The gas pressure, P, may be determined via a total mass balance for the gas

phase, namely,

evapgg RnS

t

g , 5-17

where, the gas flux, gn

, is given by,

Pkk

ni

grg

gg

,

. 5-18

Here g is the density of gas phase, given by,

RT

PM gg , 5-19

where, gM is the molecular weight of gas (kg/mol).

Energy equation 5.3.5

We assume that the each of the phases are in thermal equilibrium with each

other and thus the energy balance equation can be written as,

evapfgeffwwggeffpeff RhTkhnhn

t

Tc

).(

. 5-20

Here, T is the temperature (K) of each phases, gh is the enthalpy of gas (J), wh

is the enthalpy of water (J), fgh is the latent heat of evaporation (J/kg), eff is the

effective density (kg/m3),

effpc is the effective specific heat (J/kg/K), and effk is the

effective thermal conductivity (W/m/K). Equation 5-20 considers energy transport

due to conduction and convection and energy sources/sinks due to

evaporation/condensation.


The thermo-physical properties of the mixture are obtained by the volume-

weighted average of the different phases by the following equations,

swwggeff SS 1 , 5-21

pspwwpggeffp ccScSc 1 , 5-22

and sthwthwgthgeff kkSkSk ,,, 1 . 5-23

Here s is the solid density (kg/m3); pgc , pwc , and psc are the specific heat

capacity of gas, water, and solid (J/kg/K), respectively; gthk , , wthk , , and sthk , are the

thermal conductivity of gas, water, and solid, (W/m/K) respectively.

Evaporation rate 5.3.6

A non-equilibrium formulation as described in Ni et al. (1999) is considered to

calculate the evaporation rate, namely,

veqvv

evapevap ppRT

MKR , . 5-24

Here, vM is the molecular weight of vapour (kg/mol), eqvp , is the equilibrium

vapour pressure (Pa), vp is the vapour pressure (Pa), and Kevap is the evaporation

constant (s-1

) that is material and process-dependent and given by the reciprocal of

equilibration time, teq, and discussed later in this section.

The equilibrium vapour pressure can be obtained from the sorption isotherm

for different materials. Ratti et al. (1989) developed a correlation of sorption

isotherms for different materials at a particular temperature and moisture content.

The equilibrium vapour pressure, eqvp , , for apple is given by,

)](ln[232.0182.0exp)(0411.0949.43696.0

,, TPMeMTPp satdb

M

dbsatveqv

. 5-25

Here, satvP , is the saturated vapour pressure of water (Pa) and dbM is the

moisture content (dry basis), which can be related to wS via,

s

wwdb

SM

1. 5-26


The saturated vapour pressure of water, satvP , , is a function of temperature and

is given by Vega-Mercado et al. (2001) as,

.

ln656.61001445.0

104176.00486.03915.1/2206.5800exp

37

24

,

TTx

TxTTP satv

5-27

The vapour pressure, vp , is obtained from partial pressure relations given by,

Pp vv , 5-28

where, v is the mole fraction of vapour and P is the total pressure (Pa).

The mole fraction of vapour, v , can be calculated from the mass fractions and

molecular weight of vapour and air as,

vaav

avv

MM

M

, 5-29

where, aM is the molecular mass of air (kg/mol) and vM is the molecular mass of

vapour (kg/mol).

As noted above, Kevap , is given by the reciprocal of the equilibration time teq.

The value of teq depends on the ratio of the gas phase volume in the pores in which

vapour has to diffuse, and the surface area available for evaporation (Amit Halder et

al., 2010). This ratio scales as the radius of the pore in the case of simple cylindrical

pores. Ward and Fang (1999) showed that the time taken for a molecule to make the

transition from liquid water to water vapour is s1410. Using this latter condition and

assuming pure diffusion of vapour from the evaporating surface, the time to

equilibrium at one mean free path m1 away from the liquid surface was less than

s610, and that of m25 away was around s510

(Ward & Fang, 1999). The time

scale analysis presented in Halder et al. (2010) showed that all the transport time

scales are greater than the equilibration time scale for food materials with a

maximum pore size smaller than m25 (e.g., potato, meat, etc.). Experiments

showed that the pore size of the apple sample studied was approximately m50

(Joardder, Kumar, et al., 2014a). Therefore, the time equilibration time, teq, is

considered as s310and thus the value of evaporation constant to be

1310 s .


Initial conditions 5.3.7

The initial conditions for equations 5-13, 5-15, 5-17, and 5-20 are given by,

0)0( wwtw Sc , 5-30

0262.0)0( tvw , 5-31

ambt PP )0( , 5-32

and KT t 303)0( , 5-33

respectively.

Boundary conditions 5.3.8

Total vapour flux, totalvn ,

, from a hypothetical surface with only gas phase can

be written as,

RT

-pphn airvv

mvtotalv ,

,

5-34

where, totalvn ,

is the total vapour flux at the surface (kg/m

2s), airvp , vapour pressure of

ambient air (Pa) and mvh is the mass transfer coefficient (m/s).

However, in a multiphase problem, the vapour flux on the surface will be a

contribution from evaporation from liquid water, and water vapour already present at

the surface. Therefore, assuming the volume fraction is equal to the surface area

fraction, the boundary conditions for water and vapour phase can be written as,

RT

-ppShn airvv

wmvw

, 5-35

RT

-ppShn airvv

gmvv

, 5-36

respectively.

In most of the food processes, the pressure at the boundary (exposed to

environment) is equal to the ambient pressure, ambP . Hence, the boundary condition

for continuity equations 5-17 can be expressed as,

ambPP . 5-37


For the energy equation (equation 5-20), the energy can transfer by convective

heat transfer and heat can be lost due to evaporation at the surface, given by,

fg

airvv

wmvairTsurf hRT

-ppShTThq )( . 5-38

Here, Th is the heat transfer coefficient (W/m2/K) and airT is the drying air

temperature (K).


The input parameters of the model are listed in Table 5-1. The rest of the

parameters that are not listed in Table 5-1, are derived and discussed in the latter sub-

sections.

Table 5-1. Input properties for the model

Parameter Value Reference

Sample diameter, Dias 40 mm This work

Sample thickness, Ths 10 mm This work

Equivalent porosity, initial, 0.922 (Haitao Ni, 1997; Rahman,

2008).

Water saturation, initial, 0.794 (Haitao Ni, 1997; Rahman,

2008).

Initial saturation of vapour, 0.15 (Haitao Ni, 1997; Rahman,

2008).

Gas saturation, initial, 0.19 (Haitao Ni, 1997; Rahman,

2008).

Initial temperature, T0 303K

Vapour mass fraction, 0.026 Calculated

Constants

Evaporation constant, Kevap 1000 This work

Drying air temperature, Tair 333K This work

Universal gas constant, Rg 8.314 J mol-1

K-1

(Çengel & Boles, 2006)

Molecular weight of water, 𝑀𝑤 18.016 g mol-1


0

0wS

0vS

vw



Molecular weight of vapour, 𝑀𝑣 18.016 g mol-1


Molecular weight of gas (air), 𝑀𝑎 28.966 g mol-1


Latent heat of evaporation, ℎ𝑓𝑔 2.26e6 J kg-1


Ambient pressure, 𝑃𝑎𝑚𝑏 101325 Pa

Thermo-physical properties

Specific heats

Apple solid, 𝐶𝑝𝑠 3734 J kg-1

K-1

Measured

Water, 𝐶𝑝𝑤 4183 J kg-1

K-1

(Carr et al., 2013)

Vapour, 𝐶𝑝𝑣 1900 J kg-1

K-1

(Carr et al., 2013)

Air, 𝐶𝑝𝑎 1005.68 J kg-1

K-1

(Carr et al., 2013)

Thermal conductivity

Apple solid, 0.46 W m-1

K-1

(Choi & Okos, 1986)

Gas, 0.026 W m-1

K-1

(Rakesh et al., 2012)

Water, 0.644 W m-1

K-1


Density

Apple solid, 𝜌𝑠 1419 kg m-3

This study

Vapour, Ideal gas law, kg m-3

Air, Ideal gas law, kg m-3

Water, 1000, kg m-3

Permeability

Permeability is an important factor in relation to describing the water transport

due to pressure gradient in unsaturated porous media. The value of the permeability

determines the extent of pressure generation inside the material. The smaller the

permeability, the lower the moisture transport and the higher the internal pressure,

and vice versa.

sthk ,

gthk ,

wthk ,

v

v

w


The permeability of a material to a fluid, k ,is a product of intrinsic

permeability, ik , of the material and relative permeability, rik , , of the fluid to that

material (Bear, 1972), namely,

riikkk , . 5-39

Measurement of permeability values for deformable hygroscopic materials

such as food is difficult (Haitao Ni, 1997). Therefore, some reasonable

approximation has been made to calculate permeability. The intrinsic permeability

depends on the pore structure of the material and is given as a function of porosity by

Kozeny-Carman model (H. Feng et al., 2004) as,

77.039.0

110578.5

2

312

wk . 5-40

The gas intrinsic permeability kg was 21212 102.1104.7 m and

21313 104.2105.6 m at moisture level of 36.0% (db) of 60.0% (db)

respectively. In this study, we used an average of212100.4 m (H. Feng et al., 2004).

Relative permeabilities are generally expressed as functions of liquid

saturation. There are numerous studies which have developed such functions (Plumb,

1991). In this study, the relative permeablities of water, wrk , , and gas, grk , , for apple

were obtained from the measurement of Feng et al. (2004), namely,

3

, wwr Sk , 5-41

and wS

gr ek86.10

, 01.1

, 5-42

respectively.

The relative permeabilities using the above equation are plotted in Figure 5-2

for better illustration.


Figure 5-2. Gas (kr,g) and water (kr,w) relative permeabilities of apple tissues as a function of

saturation.

Viscosity of water and gas

Viscosities of water (Truscott, 2004) and gas (Gulati & Datta, 2013) as a

function of temperature are given by,

T

ww e

1540143.19

5-43

and

65.0

3

27310017.0

T

g .

5-44

Effective gas diffusivity

The effective gas diffusivity can be calculated as a function of gas saturation

and porosity according to the Bruggeman correction (Haitao Ni, 1997) given by,

3/4

, gvageff SDD . 5-45

Here, binary diffusivity, vaD , can be written as,


81.1

0

05103.2

T

T

P

PDva ,

5-46

where 𝑇0 = 256𝐾 and 𝑃0 = 1 𝑎𝑡𝑚. For simplicity, In this study effective gas

diffusivity was considered as 2.6 × 10−6 𝑚2

𝑠 (A. K. Datta, 2007b).

Capillary diffusivity of liquid water

Capillary diffusivity of liquid water is very important for both convection and

microwave drying. Capillary force is the main driving force of liquid water in

convective drying if there is no pressure gradient developed (Haitao Ni, 1997).

Although there is a large amount of effective moisture diffusivity data for apples

available in the literature, these were obtained by fitting diffusion models to

experimental drying curves, and are not equal to capillary diffusivity.

It is clear that in our formulation (Equations 5-11) the capillary diffusivity, cD ,

is proportional to w

c

S

p

, and is a function of capillary pressure. The typical

relationship between capillary pressure and water saturation is shown in Figure 5-3

(Bear, 1972).

Figure 5-3. Typical variation of capillary force as a function of liquid saturation in porous media

(Bear, 1972)


It can be seen that the capillary pressure increases significantly at lower

saturation levels and when it reaches irreducible saturation the value becomes

infinity. Therefore, that part is neglected to avoid numerical instability. From the

Figure 5-3 we can see that near 1wS , w

c

S

p

is almost infinity, therefore the cD

becomes very large. The underlying physics is that as wS approaches 1, more water

becomes free and the resistance of the solid matrix to the flow of free water is almost

zero. Therefore, cD is very large at high moisture content; as a result, the

concentration gradient is concomitantly small (Haitao Ni, 1997). As an outcome, the

capillary diffusivity can be very close to effective moisture diffusivity for very wet

material when vapour diffusion is insignificant. However, it can be quite different in

the lower moisture region. Ni (1997) used equation 5-47 for capillary diffusivity of

potato for low to high moisture content where it assumed that the capillary diffusivity

is only moisture dependent:

dbc MD 28.2exp10 8 . 5-47

In this study, a similar function was developed for apple by analysing the value

of different effective diffusivity values presented in the literature (Esturk, 2012; H.

Feng et al., 2001; H. Feng et al., 2000; Golestani et al., 2013). Considering that the

highest value corresponds to the highest saturation of water, a similar relationship

between capillary diffusivity and moisture as given by Ni (1997) is used in this

study, namely,

wbc MD 888.6exp10 8 . 5-48

Partial pressure of vapour in ambient condition

The partial pressure of vapour in an ambient condition is product of relative

humidity (RH) and saturation vapour pressure satvP , given by, satairv pRHp , . For

RH=70% and 300C, the specific humidity (moisture ratio) is 0.0188 kg/kg (dry air).

During drying, the temperature was elevated to 600C at a specific humidity of 0.0188

kg/kg (da), where the relative humidity becomes only 15%. Therefore, the partial

vapour pressure for drying air, Pap airv 29921994715.0, .


Heat and mass transfer coefficient

The heat and mass transfer coefficients were calculated based on the empirical

relationship discussed in previous paper (Kumar et al., 2015) and found to be Th

=16.746 W/(m2K) and mh =0.017904 m/s, respectively.

5.4 EXPERIMENTS

Fresh Granny Smith apples obtained from the local supermarkets were used for

the convection drying experiments. The samples were stored at 5±10C to keep them

as fresh as possible before they were used in the experiments. The apples taken from

the storage were washed and put aside for one hour to allow its temperature to

elevate to room temperature prior to each drying experiment. The samples were then

sliced 10mm thick with diameters of about 40mm. Then the samples were put into a

household convection dryer and the temperature was set to 600C. Following each

drying test, the sample was heated to 1000C for at least 24 h to get bone dry mass to

calculate initial moisture content of the apple slices which was approximately 0.868

(w.b.). The moisture losses were recorded at regular intervals of 10 mins with a

digital balance (specification: 0.001g accuracy). A Flir i7 thermal imaging camera

was used to measure the temperature distribution on the surface. The experiments

were done three times and the standard deviation was calculated.

5.5 NUMERICAL SOLUTION AND SIMULATION METHODOLOGY

The model was solved by using COMSOL Muliphysics 4.4a. COMSOL is

advance engineering simulation software used for modelling and simulating any

physical process described by partial differential equations. Non-uniform mesh with

grid refinement at the transport boundary (maximum element size 0.1mm) was

chosen as shown in Figure 5-4.

Figure 5-4. Mesh for the simulation.


To ensure that the results were grid-independent, several grid sensitivity tests

were conducted. The time stepping period was chosen as one second (1s) to solve the

equations. The simulation was performed using Windows 7 with Intel Core i7 CPU,

3.4GHz processor and 24GB of RAM and it took about 10 minutes to run the model.


In this section, profiles of moisture, temperature, pressure, fluxes and

evaporation rate are presented and discussed. Validation is also conducted by

comparing moisture content and temperature from experiments.

Moisture content 5.6.1

The evolution of average moisture content obtained from the model and

experiments are compared in Figure 5-5. It can be seen that the model provides a

satisfactory result. The drying curve presented here has a similar characteristics to

those found in the literature for apple drying (Golestani et al., 2013; Yan Bai et al.,

2002). It was found that that moisture content (dry basis) of apple slice dropped

from its initial value of 6.6 kg/kg to 2.9 kg/kg after 150 min of drying.

Figure 5-5. Comparison between predicted and experimental values of average moisture content

during drying


Distribution and evolution of water and vapour 5.6.2

The distribution of water saturation and vapour saturation along the half

thickness of the material at different times is shown in Figure 5-6 and Figure 5-7,

respectively.

Figure 5-6. Spatial distribution of water saturation with times

As expected, the graphs show that during drying the water and vapour

saturation near the surface was lower than in the centre region. The water saturation

decreases with drying time at each point within the sample. Similar moisture

distributions were found by Chemkhi et al. (2009), namely, that the surface contained

lower moisture content compared to the core region.


Figure 5-7. Spatial distribution of vapour with different time

Unlike the moisture distribution, vapour saturation was found to increase with

drying time within the sample (as shown in Figure 5-7). However, the vapour

saturation at and near the surface was lower than the centre, because the vapour

coming into the surface was immediately convected away by the drying air.

Temperature curve 5.6.3

Figure 5-8 shows the temperature evolution at the surface and centre of the

material. The surface temperature rose sharply at the beginning of the drying process

(approximately 0–15 min); this was due to the sudden exposure of the material to

higher temperature. Chemkhi et al. (2009) described this phase as the transient

period, where the material was heated until wet bulb temperature value was reached.

Then, the temperature rose at a slower rate until it reached the drying air temperature.

A similar trend of temperature at the surface during convection drying of porous

media was found by Chemki et al. (2009). They observed that, it took about 330 min

to raise the temperature to maximum (drying air temperature).

The surface temperature was always higher than the centre temperature

throughout the drying period. The difference between the surface and centre


temperature increased as drying progressed. The reason behind this was presumably

related to a decrease in the thermal conductivity with moisture content.

Figure 5-8. Surface and centre temperature obtained from model

Vapour pressure, equilibrium vapour pressure, and saturated pressure 5.6.4

Figure 5-9 represents the comparison between vapour pressure, equilibrium

vapour pressure and saturation vapour pressure at the surface. These three vapour

pressures are very important in relations to drying kinetics. The saturation vapour

pressure varied with temperature (equation 5-25) and data was available from many

sources (Çengel & Boles, 2006). This data was compared with simulated saturation

data and found to be consistent. The equilibrium vapour pressure data was calculated

from the sorption isotherm of apple (equation 5-27) and, as expected, was found to

be lower than the saturation vapour pressure. Figure 5-9 shows that the difference

between vapour pressure and equilibrium vapour pressure is higher during the initial

stage of drying resulting in higher evaporation according to equation 5-24. However,

after the initial stage of drying (approximately 18 min), these two pressures overlap

each other or sometimes vapour pressure becomes lower than equilibrium vapour

pressure. Therefore, the evaporation becomes zero or negative at the surface after

that time, which can be seen in Figure 5-10. The evaporation is explained in the next

section to confirm that the simulation captures the physical phenomena well.


Figure 5-9. Vapour pressure, equilibrium vapour pressure and saturation pressure at surface


One of the advantages of non-equilibrium approaches over other methods in

the modelling of drying is the ability to calculate evaporation. Figure 5-10 shows that

a higher evaporation rate occurred near the surface. In addition to this behaviour, the

evaporation rate was distributed over a narrow zone near the surface. It showed a

significant amount of evaporation occurred at the beginning of the drying process (0-

30 min). Halder et al. (2007b) also found similar phenomena in frying, where

production of excess amounts vapour at the beginning caused vapour to move

towards the centre.

The non-equilibrium formulation of evaporation was higher when the

difference between vapour pressure and equilibrium vapour pressure was higher.

Near the surface, this difference was higher; therefore, the evaporation was higher

near the surface. Similar higher evaporation near the surface was also found in meat

cooking by Dhall et al. (2012).


Figure 5-10. Spatial distribution of evaporation rate at different drying times

Another interesting pattern emerged from the graph in Figure 5-10, wherein

evaporation started at the inner side of the material and the rate decreased as drying

progressed. It shows that the evaporation was very high at 30 min (amounting to 1.1

kg/m3/s) and as drying progresses the peak of evaporation moves towards the centre.

The reason behind this behaviour could be that the liquid water saturation becomes

lower (drier) near the surface as drying progressed, and the difference between

vapour pressure and equilibrium vapour pressure essentially became zero as

discussed previously. Another possible explanation could be that the gradual

penetration of heat increased the kinetic energy of water molecules which moves

towards the centre. Thus the peak evaporation gradually moved towards the centre.

In addition to this observation, the decreasing evaporation rate near the surface could

be due to the possibility that there were relatively less number of water molecules

available due to lower moisture content.

Vapour and water fluxes 5.6.6

The major advantage of a multiphase porous media model is that the relative

contribution of vapour and water fluxes due to diffusion and gas pressure gradient

can be illustrated. Moisture fluxes due to capillary pressure and gas pressure

gradients are plotted in Figure 5-11 and Figure 5-12, respectively. It can be seen that

the capillary flux is higher at about 0.5–1.0mm beneath the surface and this peak is


moving towards the core. The explanation for this could be that the capillary flux (

wc cD ) is proportional to moisture gradient ( wc ). The gradient is higher initially

near that region and the peak of gradient moves towards the core with time. At and

near the surface, the water flux decreased, which could be due to the decrease in

capillary diffusivity due to lower moisture content.

Figure 5-11. Water flux due to capillary diffusion

The water flux due to gas pressure gradient (Figure 5-12) showed a similar

pattern of flux distribution, albeit with lower magnitude. The convective water flux

increased from zero (at the centre) to a peak at approximately 1mm beneath the

surface. This could be due to the higher pressure gradient near the surface resulting

higher convective flow. However, in convection drying, the gradient of pressure is

needed a closer inspection, because there may not be enough pressure development

inside the sample. Although the pressure gradient was higher at and near the surface

(0–1mm beneath the surface), the flux due to gas pressure started reducing in these

regions. This could be due to the reduction of relative permeability of water which

tends to zero at lower moisture saturation, resulting the convective term

P

kk

w

wrw

w

, near zero.


Figure 5-12. Water flux due to gas pressure

Figure 5-13 and Figure 5-14 show the spatial distribution of the diffusive and

convective fluxes of vapour, respectively. The figures show that vapour fluxes from

both sources are mainly occurring near the surface with zero in the core region. This

is due to the transport at the surface, which generated large vapour concentration and

pressure gradients near the surface which promoted higher diffusion and convective

flux, respectively. Ousegui et al. (2010) found a similar pattern of vapour flux due to

diffusion.


Figure 5-13. Vapour flux due to binary diffusion

Figure 5-14. Vapour flux due to gas pressure

Generally, the vapour and water fluxes caused by all sources showed that the

fluxes were minimal at the centre and gradually increase towards the surface.


Therefore, the surface contains higher moisture saturation, even though the surface

became dried.

5.7 CONCLUSIONS

A non-equilibrium multiphase porous media model, which was a significant

advancement relative to existing approaches, has been developed for the convection

drying of food. The model was validated by comparing experimental moisture and

temperature data and it was demonstrated that good agreement existed. The results of

this study supported the idea that the surface dried first, and then the moisture from

inside moved due to both capillary and gas pressure. The model in this paper was

used to elucidate the relative contribution of various modes of transport and phase

change, which cannot be investigated with a single phase model. For example,

parameters such as capillary diffusion, gas pressure and evaporation in overall

moisture transport were evaluated, which is not possible through experiments or by

using simpler models. The fundamental basis of the model enabled a deeper

understanding of drying kinetics and, thus, it can be an important tool in making

safety, quality and product design related predictions.

Chapter 6: Multiphase Model for IMCD Using Lambert’s Law 97

Multiphase Model for IMCD Chapter 6:

Using Lambert’s Law

This Chapter presents a multiphase porous media model for IMCD, which is

the first attempt at developing the IMCD model considering a multiphase porous

media approach. For mass transport of water, vapour, and air, the model considers

similar formulation to Chapter 5. However, the energy equation in this model

considers an additional heat generation term due to microwave energy which is

calculated by using Lambert’s Law.

This Chapter features the following publication:

C. Kumar, M. U. H. Joardder, T.W. Farrell and M. A. Karim, (2015)

Multiphase porous media model for Intermittent microwave convective

drying (IMCD) of food: Model formulation and validation. Journal of

Thermal Science, (To be submitted)

The signed “statement of contribution” for the above paper is given below:

98 Chapter 6: Multiphase Model for IMCD Using Lambert’s Law

6.1 ABSTRACT

Intermittent microwave convective (IMCD) drying is an advanced drying

technology that improves both energy efficiency and food quality during the drying

of food materials. Despite numerous experimental studies available for IMCD, there

is no complete multiphase porous media model available to describe the process. A

multiphase porous media model considering liquid water, gases and the solid matrix

inside the food during drying can provide in depth understanding of IMCD. In this

article, firstly a multiphase porous media model was developed for IMCD. Then the

model is validated against experimental data by comparing moisture content and

temperature distributions after each heating and tempering periods. The profile of

vapour pressures and evaporation during IMCD are presented and discussed. The

relative contribution of water and vapour fluxes due to gas pressure and diffusion

demonstrated that the fluxes due are relatively higher in IMCD compared to

convection drying and this makes the IMCD faster.

6.2 INTRODUCTION

Combined microwave convection drying can significantly shorten the drying

time and improve the product quality and energy efficiency (M. Zhang et al., 2006).

However, continuous application of microwave energy may overheat the product

(Gunasekaran, 1999). To overcome this problem, Intermittent Microwave

Convective Drying (IMCD) is practiced where the heating rate can be controlled by

choosing the intermittency (Gunasekaran, 1999; I. W. Turner et al., 1998). Thus,

IMCD can improve both energy efficiency and product quality during drying. Many

experimental investigations have highlighted the advantages of IMCD showing

drying time reduction and quality improvement for different food products, such as,

oregano (Soysal et al., 2009b), sage leaves (Esturk, 2012), banana (Ahrné et al.,

2007), and pineapple (Botha et al., 2012). Being a relatively new technique in food

drying, modelling studies of IMCD are very limited. In order to describe the heat and

mass transfer process during IMCD, an appropriate model has to be developed to

obtain a better strategy for applying microwaves and optimizing the process (Kumar

et al., 2014).

Researchers have attempted to develop models for continuous microwave

convective drying without considering intermittency. Those are only either empirical


models (Bhattacharya et al., 2013; Esturk, 2012), or single-phase diffusion based

models (J. R. Arballo et al., 2012; Mohamed Hemis et al., 2012). Since these models

did not account for the intermittency of microwave power, they do not provide any

understanding of the IMCD process.

There are a some empirical models available in the literature for IMCD

(Esturk & Soysal, 2010), however, the empirical models do not help towards

optimization and are only applicable for specific experimental conditions (Kumar et

al., 2012a; Perussello et al., 2014). Apart from empirical modelling, there are some

diffusion based theoretical models that consider intermittency of microwave power

(Gunasekaran, 1999; Gunasekaran & Yang, 2007a, 2007b; Yang & Gunasekaran,

2004), however, mass transfer was neglected in these models. Moreover, none of

these intermittent heating models investigated the temperature distribution and

redistribution due to intermittent microwave power, which is critical for avoiding

overheating of food. Therefore, it can be concluded that currently there is no

modelling work that can illustrate the mechanism of heat and mass transfer during

IMCD.

The theoretical models of food drying can broadly be categorized into two

groups: (1) single phase (diffusion based), (2) multiphase modes. The single-phase

models consider only diffusion inside the food product and are unable to provide an

understanding of other transport mechanism such as pressure driven flow and

evaporation. Describing all the water transport as diffusion cannot be justified under

all situations (J. Zhang & Datta, 2004). Therefore, multiphase models considering

transport of liquid water, water vapour and air insider the food materials are more

realistic. Although the final equations of multiphase models seem to be simple

conservation equations it still provides the more fundamental and convincing basis of

transport than single-phase models (A. K. Datta, 2007b; Whitaker, 1977). Multiphase

models can be categorised into two groups viz. equilibrium and non-equilibrium

approach of vapour pressure. In equilibrium formulations, the vapour pressure, vp , is

assumed to be equal with equilibrium vapour pressure, eqvp , , and vice versa (J.

Zhang & Datta, 2004). There are some multiphase models considering the

equilibrium approach applied in vacuum drying of wood (Ian W. Turner & Perré,

2004) and convection drying of wood and clay (Stanish et al., 1986) (Chemkhi et al.,


2009), microwave spouted bed drying of apple (H. Feng et al., 2001) and large

bagasse stockpiles (Farrell et al., 2012).

However, equilibrium conditions may not be achieved due to lower moisture

content at the surface during drying. Therefore, non-equilibrium multiphase models

are computationally effective and applied to a wide range of food processing such as

frying (Bansal et al., 2014; H. Ni & Datta, 1999), microwave heating (Chen et al.,

2014; Rakesh et al., 2010), puffing (Rakesh & Datta, 2013), baking (J. Zhang et al.,

2005), meat cooking (Dhall & Datta, 2011) etc. However, application of these non-

equilibrium models in drying of food materials is very limited. To the authors’ best

knowledge, not only is there no satisfactory multiphase model, there is not even a

comprehensive single-phase model for IMCD.

The objectives of this study are to: 1) develop a multiphase porous media

model for IMCD drying of food materials considering transport of liquid water,

vapour and air, 2) validate the model for IMCD of apple slice with experimental

moisture and temperature data, 3) investigate the temperature distribution and

redistribution due to intermittency of microwave, and 4) investigate the transport

mechanisms, such as, pressure driven, binary diffusion and capillary driven flow in

IMCD.

6.3 MATHEMATICAL MODEL

In this section, the equations for multiphase porous media are developed

describing heat, mass and momentum transfer for IMCD. It also presents the

transport mechanism involved in drying, assumptions and input parameters for the

model. Apple has been considered as the sample food material for this study.

The transport model presented here is based on the mathematical model section

of Chapter 5 with additional formulations for microwave heat generation using

Lambert’s Law. The model developed in this research considers transport of liquid

water, vapour and air inside food materials. The mass and energy conservation

equations include convection, diffusion and evaporation of water and vapour.

Momentum conservation is developed from Darcy’s equations. Evaporation is

considered as distributed throughout the domain and a non-equilibrium evaporation

formulation is used for evaporation condensation-condensation phenomena.


Problem description and assumptions 6.3.1

A schematic of the sample together with transport mechanisms involved in

drying is presented in Figure 6-1. A 2D axisymmetic geometry of a 3D apple slice is

considered for simulation. Heat and mass transfer takes place at all boundaries except

the symmetry boundary. The apple slice is considered as porous media and the pores

are filled with three transportable phases, namely liquid water, air and water vapour

as shown in Figure 6-1. All phases (solid, liquid, and gases) are continuous and local

thermal equilibrium is valid, which means that the temperatures in all three phases

are equal. Liquid water transport takes place due to convective flow resulting from

gas pressure gradient, capillary flow, and evaporation. Vapour and air transport

arises from gas pressure gradients and binary diffusion.

Figure 6-1 . Schematic showing 3D sample, 2D axisymmetric domain and Representative Elementary

Volume (REV) with the transport mechanism of different phases


The mathematical formulation, initial condition, boundary conditions, and

input parameters are similar to those that have been presented in Chapter 5. The only

difference is that a heat generation term, micQ (W/m3), is added in the energy equation


(equation 6-13). However, the transport equations are briefly presented here along

with the additional for the heat generation term.

Mass and momentum balance equations

The representative elementary volume V (m3) is the sum of the volume of

three phases, namely, gas, water, and solid, namely,

swg VVVV 6-1

where, gV is the volume of gas (m3), wV is the volume of water (m

3), and sV was

the volume of solid (m3).


water. Thus,

V

VV wg

. 6-2

The water, wS , and gas, gS , Saturation are defined as the fraction of pore

volume occupied by that particular phase, namely,

V

V

VV

VS w

gw

ww

, 6-3

and w

g

gw

g

g SV

V

VV

VS

1

, 6-4

respectively.


3),and air, ac

(kg/m3), are given by,

www Sc , 6-5

gv

v SRT

pc , 6-6

and ga

a SRT

pc , 6-7


respectively. Here, w is the density of water (kg/m3), R is the universal gas constant

(J/mol/K), and T is the temperature of product (K), vp is the partial pressure of

vapour (Pa), ap is the partial pressure of air (Pa).

The mass conservation equation for liquid water considers gas pressure driven

flow, capillary diffusion, and evaporation of liquid water to vapour. The detail

derivations of these were presented in Chapter 5. The final equations for mass

concentration of liquid water can be written as,

evapwc

w

wrw

www RcDPkk

St

,,

6-8

where, wk is the intrinsic permeability of water (m2), wrk , is the relative permeability

of water, and w is the viscosity of water (Pa.s), cD is the capillary diffusivity (m2/s)

and evapR is the evaporation rate of liquid water to water vapour (kg/m3s).

The mass balance equation for the vapour component of gas phase include bulk

flow, binary diffusion, and phase change (Bird et al., 2007); namely,

evapvgeffgg

v

grg

vgvgg RDSPkk

St

,

,, 6-9

where, g is the density of gas (kg/m3), v is the mass fraction of vapour, gk is the

intrinsic permeability of gas (m2), grk , is the relative permeability of gas (m

2), and

g is the viscosity of gas (Pa.s) and geffD , is the binary diffusivity of vapour and air

(m2/s).

The gas phase is a mixture of vapour and air. After calculating the mass

fraction of vapour, v , from the above equations, the mass fraction of air, a , can be

calculated from

va 1 . 6-10

Mass balance equations for gas phase

The total gas pressure, P, is calculated by solving the overcall mass balance for

the gas phase, namely,


evap

i

grg

gg RPkk

St

,

g , 6-11

where g is the density of gas phase, given by,

RT

PM gg , 6-12


Energy balance equation

Since thermal equilibrium is assumed to exist between all phases the energy

balance equation can be written as,

)().( tfQRhTkhnhnt

Tc mevapfgeffwwggeffpeff

. 6-13

Here, T is the temperature (K) of each phases, gh is the enthalpy of gas (J), wh

is the enthalpy of water (J), fgh is the latent heat of evaporation (J/kg), eff is the

effective density (kg/m3),

effpc is the effective specific heat (J/kg/K), effk is the

effective thermal conductivity (W/m/K) and f(t) is an intermittency function as

discussed in section 4.5.

The thermo-physical properties of the mixture are obtained by the volume


swwggeff SS 1 , 6-14

pspwwpggeffp ccScSc 1 , 6-15

and sthwthwgthgeff kkSkSk ,,, 1 . 6-16

Here s is the solid density (kg/m3); pgc , pwc , and psc are the specific heat

capacities of gas, water, and solid (J/kg/K), respectively; gthk , , wthk , , and sthk , are the

thermal conductivities of gas, water, and solid, (W/m/K) respectively.


A non-equilibrium formulation as described in Ni et al. (1999) is considered to

calculate the evaporation rate, namely,


veqvv

evapevap ppRT

MKR , . 6-17

Here, vM is the molecular weight of vapour (kg/mol), eqvp , is the equilibrium

vapour pressure (Pa), vp is the vapour pressure (Pa), and Kevap is evaporation

constant (1/s).

The equilibrium vapour pressure, eqvp , , is obtained the sorption isotherm of

apple given by (Ratti et al., 1989),

)](ln[232.0182.0exp)(0411.0949.43696.0

,, TPMeMTPp satdb

M

dbsatveqv

, 6-18

and the saturated vapour pressure of water, satvP , , is a function of temperature and is

given by Vega-Mercado et al. (2001) as,

.

ln656.61001445.0

104176.00486.03915.1/2206.5800exp

37

24

,

TTx

TxTTP satv

6-19

The vapour pressure, vp , is obtained from partial pressure relations given by,

Pp vv , 6-20

where, v is the mole fraction of vapour and given by,

vaav

avv

MM

M

, 6-21

where, aM is the molecular mass of air (kg/mol) and vM is the molecular mass of

vapour (kg/mol).

The moisture content (dry basis) dbM can be calculated from,

s

vwdb

ccM

1 6-22


The initial conditions for equations 6-8, 6-9, 6-11, and 6-13 are given by,

0)0( wwtw Sc , 6-23


0262.0)0( tvw , 6-24

ambt PP )0( , 6-25

and KT t 303)0( , 6-26

respectively.


The heat and mass transfer takes place at the transport boundaries as shown in

Figure 6-1. The boundary conditions for equations 6-8 and 6-9 can be written as,

RT

-ppShn airvv

wmvw

, 6-27

RT

-ppShn airvv

gmvv

, 6-28

respectively, Here where, airvp , vapour pressure of ambient air (Pa) and mvh is the

mass transfer coefficient (m/s).

The boundary condition for continuity equations 6-11 can be expressed as,

ambPP , 6-29

where ambP is the ambient pressure (Pa).

For the energy equation (equation 6-13), the boundary condition is given by,

fg

airvv

wmvairTsurf hRT

-ppShTThq )( . 6-30

Here, Th is the heat transfer coefficient (W/m2/K) and airT is the drying air

temperature (K).


The input parameters of the model are listed in Table 6-1 and some of them are

derived and discussed in the latter of this section.


Table 6-1. Input properties for the model





2008).


2008).


2008).


2008).



Constants




K-1










Ambient pressure, 𝑃𝑎𝑚𝑏 101325 Pa (Çengel & Boles, 2006)

Gas intricsic permeability, kg (H. Feng et al., 2004).

Binary diffusivity, (A. K. Datta, 2007b).

Ambient vapour pressure, Calculated

Heat transfer coefficient, 16.746 W/(m2K) Calculated

Mass transfer coefficient, 0.017904 m/s Calculated


Specific heat

0

0wS

0vS

vw

212100.4 m

vaD sm /106.2 26

airvp ,Pa2992

Th

mh




K-1

Measured


K-1

(Carr et al., 2013)


K-1

(Carr et al., 2013)

Air, 𝐶𝑝𝑎 1005.68 J kg-1

K-1

(Carr et al., 2013)



K-1

(Choi & Okos, 1986)

Gas, 0.026 W m-1

K-1


Water, 0.644 W m-1

K-1


Density


This study



Water, 1000, kg m-3

Permeability

The intrinsic permeability of water is considered as a function of porosity by


77.039.0

110578.5

2

312

wk . 6-31

The gas intrinsic premeabilityis considered to be constant and is given by,

(H. Feng et al., 2004).

212100.4 mkw

6-32

In this study, the relative permeablities were considered as a function of water

sarutaion given by (H. Feng et al., 2004),

3

, wwr Sk , 6-33

and

sthk ,

gthk ,

wthk ,

v

v

w


wS

gr ek86.10

, 01.1

, 6-34

respectively.




T

ww e

1540143.19

6-35

and

65.0

3

27310017.0

T

g . 6-36



and porosity according to Bruggeman correction (Haitao Ni, 1997) given by,

3/4


Here, vaD is binary diffusivity between air and water vapour (m2/s).


Capillary diffusivity of liquid water used in the model is a function of moisture

content as discussed in Chapter 5 and given by,

wbc MD 888.6exp10 8 . 6-38


moisture content, wbM , is given by,

wb

wbdb

M

MM

1. 6-39

Microwave power absorption 6.3.7

Lamberts Law has been widely used for developing microwave heating models

in literature (Abbasi Souraki & Mowla, 2008; J. R. Arballo et al., 2012; Mohamed

Hemis et al., 2012; Khraisheh et al., 1997; Mihoubi & Bellagi, 2009; Salagnac et al.,


2004; Zhou et al., 1995). Therefore, in this study, we also used Lambert’s Law to

calculate microwave energy absorption inside the food samples. This Law considers

exponential attenuation of microwave absorption within the product, given by,

zh

mic PP 2

0 exp . 6-40

Here, 𝑃0 the incident power at the surface (W), α is the attenuation constant, h

is the thickness of the material, and (h-z) is the distance from top surface (towards

centre). The measurement of P0 via experiments is presented in section 4.4.6.

The attenuation constant, α is given by,

2

1'

''1

'2

2

6-41

where, is the wavelength of microwave in free space ( cm24.12 at 2450MHz

and air temperature 200C) and ε' and ε" are dielectric constant and dielectric loss,

respectively.

The volumetric heat generation, micQ (W/m3) then calculated by;

V

PQ mic

mic 6-42

where, V is the volume of water sample (m3).

Dielectric constant 6.3.8

The dielectric constant, ε' and dielectric loss, ε", are the most important

parameters that control the microwave power absorption of the materials. Here we

use the data of Martín-Esparza et al. (2006) in a quadratic regression analysis in

which the intercept of the ' and versus wbM graph was set to 0.1 in order to

avoid numerical singularity in ' and when wbM is zero. The resulting quadratic

expression are found to be,

1.0289.30638.36'2

wbwb MM 6-43


and 1.0.8150.26543.132

wbwb MM . 6-44


In this section, we discuss the experimental IMCD procedures, sample

preparation, and data acquisition method.

IMCD

The IMCD was achieved by placing a sample in a microwave oven for 20s

followed by convection drying, for 80s, in the convection dryer. The experiments

were conducted with a Panasonic Microwave Oven (Model NNST663W) having

inverter technology with internal cavity dimension 352mm (W) x230mm (H)

x347mm (D). The inverter technology enables accurate and continuous power supply

at lower power settings (Panasonic, 2013). The microwave oven is able to supply 10

accurate power levels with a maximum of 1100W at 2.45GHz frequency. The apple

slices were placed in the centre of the microwave cavity, in order to achieve an even

absorption of microwave energy. The moisture loss was recorded at regular intervals

at the end of each tempering period with a digital balance (specification: 0.001g

accuracy).

Sample preparation

Fresh Granny Smith apples obtained from the local Australian supermarkets

were used for the intermittent microwave drying experiments. The samples were

stored at 5±10C to keep them as fresh as possible before they were used in the

experiments. The apples taken from the storage unit were washed and put aside for

one hour to allow their temperature to equilibrate to room temperature prior to each

drying experiment. The sample were cut into disks with a thickness of 10mm and

diameter of approximately 40mm.The initial moisture content of the apple slices was

approximately 0.868 kg/kg (wet basis.) or 6.61 kg/kg (dry basis).

Temperature measurement


distribution on the surface. Accurate measurement of temperature by thermal

imaging camera depends on the emissivity values. The emissivity value for apple

was found in the range between 0.94 and 0.97 (Hellebrand et al., 2001) and set at

0.95 in the camera before taking images.


6.5 NUMERICAL SOLUTION

Engineering simulation software COMSOL Multiphysics 4.4 was used to solve

the equations. COMSOL is an advanced software tool used for modelling and

simulating any physical process described by partial derivative equations.

Since heat and mass transport phenomena is happening at the transport

boundaries as shown in Figure 6-1, a finer mesh (maximum element size 0.1mm)

was chosen at those boundaries to capture this phenomena more accurately. Figure

6-2 shows the mesh chosen for the model.

Figure 6-2. Mesh for the simulation

To ensure that the results are grid-independent, several grid sensitivity tests

were conducted. The time stepping was set to be 1s.The simulations were performed

using a Windows 7 computer with Intel Core i7 CPU, 3.4GHz processor and 24GB

of RAM.


In this section, the results of the theoretical and experimental investigations are

given. Experimental data was also used to validate the model developed.

Specifically, moisture content and temperature obtained from experiments were

compared with that of the model. Moreover, spatial and temporal profile of moisture,

temperature, evaporation rate, pressure and fluxes are also discussed.


Moisture content and temperature 6.6.1

The average moisture content with time obtained from simulation and

experiments are presented in Figure 6-3. The model provided a quite satisfactory

match with the experimental result (with R2=0.99359). It can be seen from Figure

6-3 that moisture content (dry basis) of apple slice dropped from its initial of 6.6

kg/kg to 4.5 kg/kg after 1200s of IMCD.

Figure 6-3. Comparison between predicted and experimental values of average moisture content

during drying

The temperature at the centre of the top surface obtained from experiments and

simulation at selected times are compared in Table 6-2. The model shows good

agreement with experimental with the presented values. However, at the beginning of

drying, the predicted temperature was lower. This could be due to the limitation of

Lambert’s Law as discussed later in section 6.6.9. The table shows that the

temperature reached about and above 600C after each heating cycles (320s, 420s etc)

and then drops to about 500C after each tempering periods (400s, 500s etc).


Table 6-2. Comparison of experimental and model temperature at centre of top surface at different

times

Time (s) 320 400 420 500 520 600 620 700 720 800

Microwave power On Off On Off On Off On Off On Off

Experimental

temperature 60 50 62 51 64 51.8 64.7 47 62 45.9

Model temperature 57.6 49.5 59.6 51.0 61.6 52.5 63.5 54.0 65.4 55.5

Distribution and evolution of water and vapour 6.6.2

Predicted distributions of liquid saturation within the sample at different time

intervals are presented in Figure 6-4. It can be seen that unlike the convection drying

(in Figure 5-6), the liquid saturation is slightly higher close to the surface at the

beginning of the drying. This is due to the fact that pressure gradient is higher near

the surface making the convective flow higher. This result indicates invaluable

evidence signifying that by supplying more drying air during the initial stage of

IMCD, the excess moisture at the surface can be removed. Thus, the drying rate can

be further improved. Similar higher saturation in microwave heating was found by

(Wei et al., 1985), throughout the heating period. Since they neglected mass transfer,

the phenomena persisted throughout the process, whereas in the case of drying, it

only last first few minutes.

Figure 6-4. Spatial distribution of water saturation with times


The vapour mass fractions in different time periods are illustrated in Figure

6-5. Similar to convection drying (in Figure 5-7), the vapour mass fraction at any

instant decreases gradually with distance from the centre, and at any position the

vapour mass fraction increases steadily with time. This behaviour is similar with

vapour density obtained by Wei et al. (1985) in microwave heating. The vapour mass

fractions decrease more sharply near the surface, which will result in greater vapour

diffusive flow and this will be discussed in respective sections. It is crucial to note

that the vapour mass fraction is higher in IMCD (~0.14), compared to convection

drying (~0.06), because of higher vapour generation in IMCD.

Figure 6-5. Spatial distribution of vapour with different times

Temperature curve 6.6.3

Figure 6-6 shows the temperature evolutions at the surface and centre of the

material. It can be seen that the temperature fluctuates for both positions (on the

surface and at the centre) because it rises during the time microwave is on (e.g. 20s,

120s etc.) and drops after the tempering periods (100s, 200s etc.). It also shows that

the temperature increases after each cycle. However, this increase and the

fluctuations of temperature can be controlled by changing the tempering period

(intermittency).


Another interesting result to emerge from the figure is that the interior

temperature is higher than the surface at the beginning of the drying (approximately

about for 15 mins). This could be due to higher evaporative cooling phenomena that

have been discussed in Chapter 3 (Kumar et al., 2015). Ni (1997) also found lower

temperature at the surface compared to the centre during microwave heating

simulation using Lambert’s Law. Further, experimental evidence of higher centre

temperature was obtained by Gunasekaran & Yang (2007a).

Figure 6-6. Comparison of surface temperature between experimental and model

Gas pressure 6.6.4

The pressure distribution of the gas phase within the sample is shown in Figure

6-7. The gas pressure is found to be a maximum at the centre and gradually decreases

towards the surfaces. Due to lower gas porosity, the transport of gas is restricted;

therefore, the gas generated at the centre contributes to a rise in total gas pressure.

Although the moisture reduction increases the gas porosity, the amount of migration

from the centre is lower in that time period. In microwave heating, the similar higher

pressure in the interior was found in literature (H. Feng et al., 2001; Wei et al.,

1985).


Figure 6-7. spatial distribution of total pressure across the half thickness the sample in different times

Vapour pressure, equilibrium vapour pressure, and saturated pressure 6.6.5

Figure 6-8 represents the comparison between vapour pressure, equilibrium

vapour pressure and saturated vapour pressure at the surface. The saturated vapour

pressure varies with temperature and data is available from many sources. The

saturated vapour pressure obtained from simulation is compared with literature

(Çengel & Boles, 2006) and found to be consistent with available data. The

equilibrium vapour pressure is calculated from the sorption isotherm of apple and, as

expected, it was found to be lower than the saturation vapour pressure. Figure 6-8

that shows the difference between vapour pressure and equilibrium vapour pressure

is higher during the initial stage of drying resulting higher evaporation. Unlike the

convection drying (Figure 5-9), the vapour pressure fluctuates due to the fluctuation

in temperature resulting from the intermittent microwave heat source. Moreover, the

saturation vapour pressure in IMCD is much higher than the convection drying

because of the higher temperature due to microwave heat generation (Figure 5-9 and

Figure 6-8). However, after some time, the vapour pressure and equilibrium vapour

pressure coincide because the surface becomes dried and equilibrium vapour

pressure becomes lower.


Figure 6-8. Vapour pressure, equilibrium vapour pressure and saturation pressure at surface

Evaporation 6.6.6

Evaporation is zero in the inner part of the sample (as shown in Figure 6-9), as

the equilibrium nature of vapour pressure, due to higher moisture content, exists in

the inner part of the sample. In contrast to this, higher evaporation near the surface

indicates that a non-equilibrium condition exists on the surface (as shown in Figure

6-9). It is found that the evaporation starts about 1mm beneath the surface and the

evaporation rate increases as it moves towards the surface. This is because the

difference between equilibrium vapour pressure and vapour pressure starts about

1mm beneath the surface and that increases with the move towards the surface.

It is noted here that the magnitude of evaporation rate decreases with time in

convection drying (Figure 5-10), whereas, in IMCD, the magnitude of evaporation

increases with time (Figure 6-9). This is because the temperature in IMCD is much

higher compared to convection drying.



A remarkable result to emerge from the graph is that after 1000s of IMCD the

evaporation drops abruptly at the surface after reaching a peak at about 0.5mm

beneath the surface. The reason is that the moisture content reduces to nearly zero (as

can be seen from Figure 6-4) after 1000s which makes the equilibrium vapour

pressure equal to vapour pressure resulting drops in evaporations.

Vapour pressure distribution 6.6.7

Vapour pressure distribution within the sample at different times is shown in

Figure 6-10. It shows that vapour pressure is higher in the interior of the sample with

the maximum at the centre. Vapour pressure at the centre is expected to have a

higher value due to the higher temperature at the centre. The surface vapour can

easily be transported to surroundings and therefore, the vapour pressure at the surface

is very close to the ambient vapour pressure.



Vapour and water fluxes 6.6.8

Moisture fluxes due to capillary diffusion and gas pressure gradient are

presented in Figure 6-11 and Figure 6-12, respectively. It can be seen that the

moisture flux due to capillary diffusion is approximately double that when compared

to convection drying (Figure 5-11 and Figure 6-11). This is mainly due to the

increase in concentration gradient resulted from the faster drying rate in IMCD as

compared to convection drying.

The most striking feature from the result is that the water flux due to gas

pressure in IMCD (Figure 6-12) is approximately 20 times higher than that in

convection drying (Figure 5-12). This higher transport of the moisture to the surface

is eventually being evaporated at the surface, which is the main reason behind the

significant reduction in drying time during IMCD.


Figure 6-11. Water flux due to capillary at different drying times

Figure 6-12. Water flux due to gas pressure at different drying times


Figure 6-13. Vapour flux due to binary diffusion at different drying times

Figure 6-14. Vapour flux due to gas pressure at different drying times


Figure 6-13 and Figure 6-14 show the spatial distribution of diffusive and

convective fluxes of vapour, respectively. The figures show that the vapour fluxes

from both sources (diffusion and convection) are higher near the surface with zero in

the core region. This can be interpreted from Figure 6-5, as it shows that the gradient

of vapour is very high near the surface thus causing higher diffusive flux. Wei et al.

(1985) also found higher vapour flux near the surface of the sample during

microwave heating. Moreover, the vapour flux due to binary diffusion and pressure

gradient are about 5 times higher when compared to convention drying.

Limitation of Lambert’s Law 6.6.9

Since this study is the first attempt of using multiphase porous media to model

for IMCD, it considered Lamberts Law for microwave power absorption. However,

some limitations of using Lambert’s Law have been observed during the simulations.

Lambert’s law for power absorption due to microwave energy cannot capture the

uneven distribution, as it considers power absorption on the surface or on any other

horizontal plane is uniform, which is not the actual case. This study revealed another

limitation of Lambert’s Law, which was not mentioned before in the literature;

namely, the power absorption at the surface is always the maximum regardless of the

moisture content. It is well known that moisture or dipolar materials are mainly

responsible for microwave absorption (Joardder, Kumar, et al., 2014b) . Therefore,

when the surface becomes dry, the microwave absorption should be less. Lamberts

law fails to take this into account, giving always highest power at the surface,

irrespective of moisture content. Using Maxwell’s equations of electromagnetic field

and power absorption provide better and accurate heat generation due to microwaves.

6.7 CONCLUSIONS

A non-equilibrium multiphase porous media model has been developed for

IMCD for food. This model is first of its kind as a multiphase model has not been

implemented before for IMCD. The model was validated by comparing experimental

moisture and temperature data, which demonstrated good agreement except for some

discrepancy in temperature at the beginning. This discrepancy in temperature

prediction could be due Lambert’s Law approximation. The IMCD drying is much

faster than the convection drying, and the reason behind this was investigated by

analysing the relative contribution of various modes of transport. It showed that the


water flux due to capillary is three times higher and due to gas pressure gradient is

ten times higher when compared to convection drying. On the other hand, the vapour

fluxes due to diffusion and gas pressure is about ten times higher than for convection

drying. The fundamental basis of the model enables us to enhance the understanding

of drying kinetics and transport of heat and mass of IMCD.

Chapter 7: Multiphase Model for IMCD Using Maxwell’s Equations 125

Multiphase Model for IMCD Chapter 7:

Using Maxwell’s Equations

In the previous IMCD model described in Chapters 4 and 6, heat generation

due to microwave energy was calculated using Lambert’s Law. Since the Lambert’s

Law models showed some discrepancy in temperature prediction and failed to

capture the uneven temperature distribution of the sample, this Chapter therefore,

aims to overcome those problems. Unlike Lambert’s Law for heat generation

outlined previously, this chapter calculates heat generation due to microwaves from

the actual electromagnetic field calculated by Maxwell’s equations. Thus, this

Chapter presents the most comprehensive model for IMCD, which solves a three-

dimensional electromagnetic field using Maxwell’s equation to obtained microwave

heat generation, and multiphase porous media for heat and mass transfer.

This Chapter features the following publication:

C. Kumar, M. U. H. Joardder, M. A. Karim, and T.W. Farrell (2015) A

3D coupled electromagnetic and multiphase porous media model for

IMCD of food material. Food and Bioprocess Technology, (To be

submitted, IF 3.1)

Preliminary work of this chapter was published as:

C. Kumar, M. U. H. Joardder, M. A. Karim, G. J. Millar, and Z.M. Amin

(2014) Temperature redistribution modelling during intermittent

microwave convective heating. Procedia Engineering, 90(0), 544-549.

doi: http://dx.doi.org/10.1016/j.proeng.2014.11.770

http://dx.doi.org/10.1016/j.proeng.2014.11.770

126 Chapter 7: Multiphase Model for IMCD Using Maxwell’s Equations

The signed “statement of contribution” form for the above papers are given

below:


7.1 ABSTRACT

Intermittent Microwave Convective Drying (IMCD) improves energy

efficiency and the product quality during drying of food. However, the physical

mechanism of heat and mass transfer involved in IMCD is poorly understood due to

lack of a comprehensive mathematical model of this process. A multiphase porous

media model considering coupled electromagnetics and multiphase porous media can

potentially provide fundamental details of underlying mechanism of IMCD. An

accurate model is also necessary for optimization of the process. In this study, a

mathematical model considering electromagnetics using Maxwell’s equations

coupled with multiphase porous media in 3D is developed. The multiphase porous

media model for heat and mass transport processes includes different phases, namely,

solid, liquid water, and gas and incorporates pressure driven flow, capillary flow, and

evaporation in the material. The result shows the temperature is uneven in the

materials with hot spots and cold spots after each heating period and then

temperature re-distributed during the tempering period to level off. The water and

vapour fluxes showed asymmetric profile along the diameter of the sample due to

non-uniformity of the microwave heating pattern. The detail understanding of these

transport mechanism in IMCD, in turn, will lead to a significant improvement in food

quality, energy efficiency, increased ability to automation and optimization during

IMCD process.

7.2 INTRODUCTION

IMCD significantly reduces drying time and improves product quality

compared to convection drying, and overcome the problem of overheating in

continuous microwave convective drying (CMCD). The advantages of IMCD in

terms of energy efficiency and dried product quality have been reported in the

literature. For instance, Soysal et al.(2009a) reported that IMCD of red pepper

produced better sensory attributes, appearance, colour, texture and overall liking,

than CMCD and convection drying. Soysal et al. (2009b) compared IMCD and

convective drying for oregano and found that the IMCD were 4.7–11.2 times more

energy efficient compared to convective drying and able to provide better quality

dried food. Advantages of IMCD in terms of improving energy efficiency and

product quality, and significantly reducing drying times have been found in many

other products such as Oregano (Soysal et al., 2009b), Pineapple (Botha et al., 2012),


Red pepper (Soysal et al., 2009a), Sage (Salvia officinalis) Leaves (Esturk, 2012;

Esturk et al., 2011), Bananas (Ahrné et al., 2007), Carrots and mushrooms (Orsat et

al., 2007). However, all the above studies regarding IMCD were conducted

experimentally, without considering the physics behind the heat and mass transfer

involved in the process. Therefore, the heat and mass transport mechanism in IMCD

is not well understood.

Physical understanding of heat and mass transfer, and interaction of

microwaves with food products is crucial for optimization of the drying process (Hao

Feng et al., 2012). A coupled heat and mass transfer model has to be developed to

predict the temperature and moisture distribution inside the material that will help to

improve the understanding of the underlying physics and develop better control of

IMCD.

There are some modelling studies related to continuous microwave convective

drying (CMCD) found in the literature. Most of them are empirical in nature or

consider only single-phase transport (diffusion based) (Chen et al., 2014; Malafronte

et al., 2012; Pitchai et al., 2012). Moreover, intermittency of microwave has not been

dealt with in those models. Therefore, they are unable to provide a proper

understanding of the transport process during IMCD. On the other hand, despite

there being some single phase models which consider the intermittency of

microwave heat source (Gunasekaran & Yang, 2007b; Yang & Gunasekaran, 2001),

these are only for heating without considering mass transfer, and thus cannot be

applied in drying. Moreover, the temperature redistribution during tempering period,

which is the main advantages of IMCD, has been overlooked in those studies. Taken

together, all the above modelling approaches consider only single phase for mass

transport.

Compared to single phase models, the multiphase models are more

comprehensive and are fundamental to proper understanding of the process (A. K.

Datta, 2007b; Whitaker, 1977). Furthermore, multiphase models are computationally

effective and applied to wide range of food processing such as frying (Bansal et al.,

2014; H. Ni & Datta, 1999), microwave heating (Chen et al., 2014; Rakesh et al.,

2010), puffing (Rakesh & Datta, 2013), baking (J. Zhang et al., 2005), meat cooking

(Dhall & Datta, 2011) etc. However, application of these non-equilibrium models in


drying of food materials is very limited. In particular, for IMCD there is no

multiphase model to investigate the transport mechanism during the process.

Moreover, the above mentioned model for CMCD and intermittent microwave

heating models considered Lambert’s Law for microwave heat generation. However,

this theory considers exponential attenuation of microwave absorption within the

product which does not accurately represent food-heating situations in a multimode

cavity such as a microwave oven where the electric field varies in three directions

(Chandrasekaran et al., 2012; Rakesh et al., 2009). The limitations of Lambert’s Law

in IMCD have been discussed in Chapter 6 (section 6.6.9). Chandrasekaran et al.

(2012) reported that Maxwell’s equation provided a more accurate solution for

microwave propagation in samples. Therefore, Maxwell’s equation needs to be

implemented for the accurate calculation of microwave heat generation. In the light

of the above literature review, it is clear that a comprehensive mathematical model

considering multiphase transport of heat and mass and Maxwell’s equation for IMCD

is not available, which is essential for a comprehensive understanding of heat and

mass transfer phenomena involved.

Therefore, in this study, first a 3D coupled electromagnetics with multiphase

porous media transport model for IMCD was developed; then, the model was

validated with experimental data. Subsequently, the model is used to investigate the

different transport mechanism of vapour and water.

7.3 MODEL DEVELOPMENT

In this study, the electromagnetic field is solved in all domains including the

oven cavity, waveguide, glass tray and the sample. (as shown in Figure 7-1). The

transport of mass and energy is solved only in the food sample. Unlike the previous

IMCD models (Chapters 4 and 6), the heat generation due to microwaves in this

chapter is calculated from the actual electric field distribution obtained by Maxwell’s

equation. However, the multiphase heat and mass transfer here is also based on the

formulations developed in Chapter 5.

Geometry and problem description 7.3.1

The schematic of the computational domain along with the different transport

mechanisms involved in food is shown in Figure 7-1.


Figure 7-1. a) The computational domain for the IMCD drying simulation, b) Food sample and

representative elementary volume (REV) showing transport mechanism involved in the simulation


Figure 7-1a shows the geometric model which includes oven cavity,

waveguide, food sample, and glass tray. The microwaves are transmitted into the

cavity through the rectangular waveguide on the right side of the cavity. The food

product is placed at the centre of the turntable. The apple slice is considered as a

porous media and the pores are filled with three transportable phases, namely liquid

water, air, and water vapour as shown in Figure 7-1b. All phases (solid, liquid, and

gases) are continuous and local thermal equilibrium is valid, which means that the

temperatures in all three phases are equal. Liquid water transport takes place due to

convective flow due to gas pressure gradient, capillary flow and evaporation. Vapour

and air transport occur due to convective flow arises from gas pressure gradient and

binary diffusion.

Maxwell’s equation for electromagnetics and heat generation 7.3.2

Maxwell’s equations provide the electromagnetic field at any point in the

computational domain. In frequency domain, the Maxwell’s equation can be written

as (Chen et al., 2014; COMSOL, 2014)

0''')2(1 2

Ei

c

fE

7-1

where E

is the electric field strength (V/m), f is the microwave frequency (Hz), c is

the speed of light (m/s), ' , '' , are the dielectric constant, dielectric loss factor,

and electromagnetic permeability of the material, respectively.

The heat generation due to microwave, mQ (W/m3), given by (COMSOL,

2012),

mlrhm QQQ . 7-2

Here, rhQ is the resistive loss (W/m3) and mlQ is the magnetic loss (W/m

3). For

food products the magnetic losses are negligible, i.e. 0mlQ (Chen et al., 2014).

The resistive loss can be calculated as (Chen et al., 2014; Wentworth, 2004)

*5.0 EJQrh

7-3


where *E

is the conjugate of E

and the electric current density J

(A/m2) is given

by,

EfEJ

02 , 7-4

where, is the electrical conductivity (S/m), is the dielectric loss factor and 0 is

permittivity in free space.

Substituting the above into equation 7-2, the microwave heat generation due to

can be written as,

2

0 EfQm 7-5

which conforms with the heat generations equation derived by Metaxas (1996a).

Dielectric properties 7.3.3

The dielectric properties of a material defines how much microwave energy

will be converted to heat and thus is very important in microwave heating

(Chandrasekaran et al., 2013). Granny Smith apple was used as a sample in this study

and the dielectric properties for Granny Smith apple with moisture change are

reported by Martín et al. (2006). Here we use the data of Martín-Esparza et al. (2006)

in a quadratic regression analysis in which the intercept of the ' and versus wbM

graph was set to 0.1 in order to avoid numerical singularity in ' and when wbM

is zero. The resulting quadratic expression are found to be,

1.0289.30638.36'2

wbwb MM 7-6

and 1.0.8150.26543.132

wbwb MM . 7-7

Here, wbM is the moisture content (wet basis).

Multiphase porous media transport model 7.3.4

The multiphase heat and mass transport model in this chapter is also similar to

that in Chapter 5 with an additional heat generation term calculated from

electromagnetic field distribution as discussed above.


Mass conservation equations

The representative elementary volume, V (m3), is the sum of the volume

occupied by gas gV (m3), water wV (m

3), and solid sV (m

3), given by,

swg VVVV 7-8


water.

V

VV wg

7-9

Saturation of water, wS , and gas, gS is defined as the fraction of pore volume

occupied by that particular phase, namely,

V

V

VV

VS w

gw

ww

7-10

and w

g

gw

g

g SV

V

VV

VS

1

7-11

respectively.


3), and air, ac

(kg/m3), are given by equations 7-12, 7-13, and 7-14, respectively:

www Sc 7-12

gv

v SRT

pc 7-13

ga

a SRT

pc

7-14

where, w is the density of water (kg/m3), R is the universal gas constant (J/mol/K),

and T is the temperature of product (0K), vp is the partial pressure of vapour (Pa),

ap is the partial pressure of air (Pa).

The conservation equation for the liquid water, vapour and air in the porous

medium is given by,


evapwww RnSt

,

7-15

evapvvgg RnSt

,

7-16

and va 1 7-17

respectively.

Here, wn

is water flux (kg/m2s), vn

is the vapour mass flux (kg/m

2s), evapR is

the evaporation rate of liquid water to water vapour (kg/m3s), v is the mass fraction

of vapour, and a the mass fraction of air.

The flux values for water and vapour component are defined as:

wc

w

wrw

ww cDPkk

n

, 7-18

and vgeffgg

v

grg

vgv DSPkk

n

,

, 7-19

respectively. Here, wk is the intrinsic permeability of water (m2), wrk , is the relative

permeability of water, and w is the viscosity of water (Pa.s), cD is the capillary

diffusivity (m2/s), P total gas pressure (Pa), gk is the intrinsic permeability of gas

(m2), grk , is the relative permeability of gas (m

2), and g is the viscosity of gas (Pa.s)

and geffD , is the binary diffusivity of vapour and air (m2/s).

Continuity equation to solve for gas pressure

The gas phase consists of an ideal mixture of water vapour and air. The gas

pressure, P, may be determined via a total mass balance for the gas phase, namely,

evapgg RnS

t

g 7-20

Where, the gas flux, gn

, is given by,

Pkk

ni

grg

gg

,

7-21


and the density of gas phase, g , is given by,

RT

PM gg 7-22


Energy equation

Since the thermal equilibrium is assumed across all phases, the energy balance

equation can be written as,

)(tfQRhTkhnhnt

Tc mevapfgeffwwggeffpeff

7-23

where, T is the temperature (0C), gh is the enthalpy of gas (J), wh is the enthalpy of

water (J), fgh is the latent heat of evaporation (J/kg), eff is the effective density

(kg/m3),

effpc is the effective specific heat (J/kg/K), effk is the effective thermal

conductivity (W/m/K), f(t) is an intermittency function which was discussed in

section 4.5. Equation 5-20 considers energy transport due to conduction and

convection and energy sources/sinks due to evaporation/condensation.

The thermo physical properties of the mixture are obtained by the volume


swwggeff SS 1 7-24

pspwwpggeffp ccScSc 1 7-25

and sthwthwgthgeff kkSkSk ,,, 1 7-26

where s is the solid density (kg/m3); pgc , pwc , and psc are the specific heat

capacities of gas, water, and solid (J/kg/K), respectively; gthk , , wthk , , and sthk , are the

thermal conductivities of gas, water, and solid, (W/m/K) respectively.

Evaporation or condensation

The porous media model incorporates the change of phase between liquid

water and vapour (evaporation or condensation) through the following expression (A.

Halder et al., 2007a)


veqvv

evapevap ppRT

MKR , 7-27

where, vM is the molecular weight of vapour, eqvp , is the equilibrium vapour pressure

(Pa), vp is the vapour pressure (Pa), and Kevap is a constant, which is material and

process-dependent signifying the rate constant of evaporation (Gulati & Datta, 2013)

and is the reciprocal of equilibration time, Δt (1/s).

The equilibrium vapour pressure, eqvp , , for apple is given by (Ratti et al.,

1989),

)](ln[232.0182.0exp)(0411.0949.43696.0

,, TPMeMTPp satdb

M

dbsatveqv

7-28

where, satvP , is the saturation vapour pressure (Pa) and M is the moisture content (dry

basis), which can be calculated by water and vapour concentration:

s

vwdb

ccM

1 7-29

The saturated vapour pressure, satvP , , is calculated by (Vega-Mercado et al.,

2001),

))ln(656.61001445.0

104176.00486.03915.1/2206.5800exp(

37

24

,

TTx

TxTTP satv

7-30

The vapour pressure vp is obtained from Ideal Gas Law.


The initial conditions for equations 7-1, 7-15, 7-16, 7-20, and 7-23 are given

by,

0)0( tE

, 7-31

0)0( wwtw Sc , 7-32

0262.0)0( tvw , 7-33

ambt PP )0( , 7-34


and CT t

0

)0( 30 7-35

respectively.


The oven and waveguide walls were considered as perfect electric conductors

(PEC). Therefore, tangential component of electric field at those boundaries are

given by,

0En

. 7-36

The boundary conditions (BC) for heat and mass transfer at the transport

boundaries of food sample are given below:

BC for equation 7-15:

RT

-ppShn airvv

wmvw

7-37

BC for equation 7-16:

RT

-ppShn airvv

gmvv

7-38

BC for equation 7-20: ambPP 7-39

BC for equation7-23:

fgairvv

wmvairTsurf hRT

-ppShTThq )( .

7-40

Here where, airvp , vapour pressure of ambient air (Pa), mvh is the mass transfer

coefficient (m/s), ambP is the ambient pressure (Pa), Th is the heat transfer coefficient

(W/m2/K), and airT is the drying air temperature (K).


The input parameters used in this study are listed in the following Table 7-1.

Important parameters except listed in table are derived and discussed in different

sections below.

Table 7-1. Input parameters for the model







2008).


2008).


2008).


2008).



Constants




K-1










Ambient pressure, 𝑃𝑎𝑚𝑏 101325 Pa (Çengel & Boles, 2006)

Gas intricsic permeability, kg (H. Feng et al., 2004).

Binary diffusivity, (A. K. Datta, 2007b).

Ambient vapour pressure, Calculated

Heat transfer coefficient, 16.746 W/(m2K) Calculated

Mass transfer coefficient, 0.017904 m/s Calculated


Specific heat


K-1

Measured


K-1

(Carr et al., 2013)


K-1

(Carr et al., 2013)

0

0wS

0vS

vw

212100.4 m

vaD sm /106.2 26

airvp ,Pa2992

Th

mh



Air, 𝐶𝑝𝑎 1005.68 J kg-1

K-1

(Carr et al., 2013)



K-1

(Choi & Okos, 1986)

Gas, 0.026 W m-1

K-1


Water, 0.644 W m-1

K-1


Density


This study



Water, 1000, kg m-3

Permeability

The intrinsic permeability of water is considered as a function of porosity by


77.039.0

110578.5

2

312

wk . 7-41

The gas intrinsic for apple is considered as 212100.4 m (H. Feng et al., 2004).

In this study, the relative permeablities were considered as a function of water

sarutaion given by (H. Feng et al., 2004),

3

, wwr Sk , 7-42

and wS

gr ek86.10

, 01.1

, 7-43

respectively.




T

ww e

1540143.19

7-44

sthk ,

gthk ,

wthk ,

v

v

w


and65.0

3

27310017.0

T

g . 7-45



and porosity according to Bruggeman correction (Haitao Ni, 1997) given by,

3/4


Here, vaD is binary diffusivity between air and water vapour (m2/s).


Capillary diffusivity of liquid water assumed to be dependent on the saturation

only, as given by:

wbc MD 888.6exp101 8 . 7-47

Details of the formulation of water capillary diffusivity for apple are given in

Chapter 5.


moisture content, wbM , is given by,

wb

wbdb

M

MM

1. 7-48

Numerical solution 7.3.8

Engineering simulation software COMSOL multiphysics 4.4 was used to solve

these equations. Because of being the most complex model in this thesis, the

simulation procedure using COMSOL Multiphysics for this model is shown in

Figure 7-2.


Figure 7-2. Flow chart showing the modelling strategy in COMSOL Multiphysics

To ensure that the results are grid-independent, several grid sensitivity tests

were conducted. It was found that the current simulation results are independent of

the grid chosen in this study. The time stepping was chosen as 1s to solve the

equations. For the food sample, finer mesh with a maximum element size 3mm was

chosen. For the rest of the domains, physics controlled mesh was chosen to solve the

electromagnetics. The simulation was performed using a Windows 7 with Intel Core

i7 CPU, 3.4GHz processor and 24GB of RAM.


In this section the experimental procedure, sample preparation, and

intermittency of input microwave power is discussed. The experiments were

conducted without rotating the turntable to identify the uneven temperature

distribution of the sample.

IMCD drying

The IMCD drying was achieved by heating the sample in the microwave oven

for 20s then drying for 80s in the convection dryer. The experiments were conducted

with a Panasonic Microwave Oven (Model NNST663W) having inverter technology

with cavity dimension 352mm (W) x230mm (H) x347mm (D). The inverter


technology enables accurate and continuous power supply at lower power settings

(Panasonic, 2013). The microwave oven is able to supply 10 accurate power levels

with a maximum of 1100W at 2.45GHz frequency. The apple slices were placed at

the centre of the microwave cavity, for an even absorption of microwave energy. The

moisture loss was recorded after each tempering period by placing the apple slices on

a digital balance (specification: 0.001g accuracy).

Microwave intermittency

The microwave power is set at 100W with 20s on and 80s off as shown in

Figure 7-3.

Figure 7-3. Intermittency of microwave power considered in the simulation and experiment

Sample preparation

Fresh Granny smith apples were obtained from the local supermarkets and used

for the IMCD experiments. The samples were stored at 5±10C to keep them as fresh

as possible before they were used in experiments. The apples taken from the storage

unit were washed and put aside for one hour to allow its temperature to elevate to

room temperature prior to each drying experiment. The sample were then sliced

10mm thick and diameter of about 40mm.The initial moisture content of the apple

slices was approximately 0.868 (w.b.).

Thermal imaging


distribution on the sample surface. Accurate measurement of temperature by thermal

imaging camera depends on the emissivity values of the sample. The emissivity

value for apple was found in the range between 0.94 and 0.97 (Hellebrand et al.,

2001) and, therefore, a value of 0.95 was set in the camera before taking images.



In this section, experimental results are presented and validation of the model

is discussed, followed by description of temperature redistribution, moisture and

pressure distribution in the sample. To comprehensively understand the process, the

water and vapour fluxes due to different mechanism are also investigated using the

model.

Experimental validation of temperature and moisture content 7.5.1

Experimental validation is critical for the developed IMCD model considering

the complexity involved. Validation of the model was conducted by comparing the

average moisture content of the sample and temperature distribution and re-

distribution obtained from the thermal imaging camera. The temperature

distributions were compared for both after heating and tempering periods to clarify

the effect of intermittency of microwave heating pattern.

Average moisture content

Figure 7-4 compares the average moisture content obtained from experiments

and simulation. The model shows good agreement with the experimental data. Both

experiment and model show that the moisture content of the sample after 1000s

drops from 6.6 kg/kg dry basis to about 4.5 kg/kg dry basis.

Figure 7-4. Average moisture content obtained from experiments and simulation


Temperature redistribution

Temperature redistribution is very crucial for IMCD because this determines

the optimum microwave power level and intermittency. According to the best of

author’s knowledge, this is the first time that temperature redistribution due to

intermittent application of microwave is being investigated.

Spatial temperature distribution on the top of the surface of the sample

obtained from experiment and simulation is shown in Figure 7-5. The image presents

the temperature distribution after each heating cycle (20s, 120s etc.) followed by

temperature distribution after each tempering period (100s, 200s etc.). It can be seen

that the temperature rises after each heating period and then decreases near to the

drying air temperature (500C) during the tempering period; this occurs both in

modelling and experimentation. The model also captures the dispersion of the

concentrated hot spot (after heating) during the tempering period. Thus, the result

offers vital evidence of the temperature redistribution after each tempering period,

which helps to limit the temperature in next heating cycle. Consequently, it improves

the drying rate and prevents the material from overheating. Moreover, a desired

temperature or uniformity can be achieved by controlling the intermittency of the

microwave for IMCD.


Figure 7-5. Temperature distribution obtained from experiment and simulations

The dispersion pattern of heat energy during intermittency is also captured by

the model. However, there are some discrepancies in the location of the hot spot

predicted by the model (i.e. shifting to right). The discrepancy could be due to the

difficulties in modelling actual geometry of the microwave oven. The microwave

oven had some internal dents and a curved surface which was not considered in the

actual geometry of the model, which could be sources for discrepancies. Moreover,

positioning of the sample was done manually which also could introduce some error

in the location of the sample. However, it is very likely that if the geometry is

accurately modelled, the error would be reduced. Nonetheless, the magnitude of the

temperature rise and fall, dispersion of the hot spot during tempering is well captured

by the models which provide valuable information for understanding IMCD and

product quality.


Internal temperature distribution 7.5.2

As can be seen from the previous figure (Figure 7-5), the temperature

distribution is uneven across the diameter of the sample in IMCD, therefore, on order

to investigate the temperature distribution closely, the temperature profile along the

horizontal centreline of the sample at different times is shown in Figure 7-6.

Moreover, since the temperature along the thickness of the sample is almost the

same, the average temperature (or point temperature or temperature across thickness)

cannot provide a genuine investigation of IMCD. For this reason all the later results

from simulation are presented along the diameter of the sample.

Figure 7-6. Temperature profile along horizontal centreline of the sample at different times

In Figure 7-6 the temperature is plotted for a number of heating periods (20s,

120s etc.) and tempering periods (100s, 200s etc.). This allows investigating the

temperature distribution and redistribution inside the sample. It can be seen that the

temperature profile is asymmetric with hotter on the right hand side of the sample

demonstrating uneven heating due to the microwaves, whereas, the temperature

profiles were symmetric for convection drying (Chapter 5) and the IMCD model

with Lambert’s Law (Chapter 4 and 6). In contrast to those studies, the IMCD model

that considers Maxwell’s equation can capture the asymmetric temperature profile


which is more realistic. Similar asymmetric profile of temperature with faster heating

at the right side was found by Rakesh et al. (2010) in microwave combination

heating of food.

It is critical to note that the temperature increase from approximately 450C at

left edge of the sample to a peak around 750C at 30mm from left edge of the sample

during the heating periods (for 120s and 220s). However, after the tempering periods

(at 200s, 300s etc.), the temperature becomes uniformly distributed throughout the

sample near 40–500C (Figure 7-5 and Figure 7-6). This highlights how important the

intermittency is in IMCD. Thus, the results offer invaluable evidence for temperature

redistribution, which potentially prove the importance of this research. Finally it can

be said that potenitailly the model can be applied within the food industry to identify

hot and cold spots location and appropriate intermittency to level the temperature.

Moisture distribution 7.5.3

Microbial or chemical safety and quality attributes of dried food material are

greatly related not only to the total moisture content but moisture distribution after

drying, because spoilage can start from a high moisture content area. Therefore,

determining moisture distribution is critical for food drying.

Figure 7-7. Saturation profile along horizontal centreline of the sample for different times


The spatial liquid saturation profiles of the sample along the horizontal

centreline at different time are plotted in Figure 7-7. The figure shows asymmetric

profile saturation with lower water saturation on the right had side. The water

saturation is highest about 1mm depth from the left edge. The asymmetric behaviour

is due to the higher temperature on the right side of the sample. The higher

temperature results higher vapour pressure on the right side of the sample,

consequently, higher moisture loss. Therefore, the liquid saturation is lower on the

right side of the sample compared to the left. However, experimental verification of

this phenomenon would be valuable and assist researchers to optimize the process.

Vapour concentration distribution 7.5.4

Vapour density profile along the centreline of the sample is shown in Figure

7-8. The vapour concentration is lower on the surfaces and peaks at 30mm from the

left edge where the temperature is higher (as shown in Figure 7-6). This nature of

vapour density correlates with the temperature distribution. For this reason, the

higher density of vapour is found exactly at the point where the maximum

temperature is to be found at any particular time of drying. The lower vapour density

at the surface is because of the transport of moisture to the ambient air.

Figure 7-8. Vapour density profile along horizontal centreline of the sample at different times


Pressure 7.5.5

Pressure distribution (Figure 7-9) shows that the pressure is higher inside the

sample. The pressure also peaks at 30mm from the left edge where the temperature is

higher. However, at both surfaces (left and right) the pressure remains at ambient

pressure. The pressure profile is asymmetric and Turner (1991) found the similar

trend of pressure in CMCD of wood. Similar higher temperatures in the inner parts

and ambient temperature at the surface was found in microwave heating by Wei et al

(1985).

Figure 7-9. Pressure profile along horizontal centreline of the sample for different times

Another important observation in Figure 7-9 is that the pressure reaches its

peaks after heating period (120s, 220s) as the pressure varies with temperature and

drops back near atmospheric pressure (101325Pa) after the tempering period.

One important point is to note that the asymmetric profiles temperature, vapour

concentration, and pressure presented here cannot be captured by any simple model

that considers only a single-phase model or Lamberts Law.


Vapour pressure distribution 7.5.6

The distribution of vapour pressure within selected time period is shown in

Figure 7-10. The vapour pressures are higher near the higher temperature regions (as

can be seen from Figure 7-6 and Figure 7-10). Figure 7-10 shows that the vapour

pressure increases after each heating cycle (20s, 120s etc.), and then decreases during

the tempering periods (100s, 200s etc.). However, for all times, it is lower on the left

side of the sample because of the lower temperature in that region and remains

almost steady for about 0–10mm from left edge. It is crucial to note that the

fluctuations in vapour pressure levels are dampened during the tempering periods

therefore causing less moisture loss during those periods. Moreover, the location

moisture loss also can be explained form this graph. Since the higher vapour pressure

causes more moisture loss, the drying rate is faster on the right side of the sample.

Figure 7-10. Vapour pressure along the horizontal centreline of the sample

Water and vapour fluxes 7.5.7

Moisture fluxes due to capillary diffusion and gas pressure gradient along the

horizontal centreline of the sample are plotted in Figure 7-11 and Figure 7-12,

respectively. It shows that the flux due to capillary diffusion is positive from the

distance of 10mm from left edge and negative for less than 10mm distance from left


surface. This indicates the water flux is moving towards right edge from 10mm and

onwards, whereas, for less that 10mm from left edge of the sample the directions of

water flux towards the left edge. It is interesting to note that, the moisture flux

increase near the surface; this could be due to the higher moisture gradient near the

surface. The right hand side of the sample contains lower moisture content, and

therefore, the concentration gradient is higher and thus causing higher capillary flux.

The abrupt increase in water diffusive flux near surface was also found higher in

meat cooking by Dhall at al. (2012).

Figure 7-11. Water capillary flux along horizontal centreline of the sample at different times

The water flux due to gas pressure (Figure 7-12) shows that the after the

tempering periods (200s, 300s etc.), the flux due to pressure gradient is negligible

compared to that of after heating periods (120s, 220s etc.). It can be seen that the

moisture flux due to gas pressure is six times higher compared to convection drying

(Figure 5-12), which makes the IMCD faster. Moreover, the magnitudes of flux

increases with time in IMCD whereas in convection drying it decreases with time.

This could be due the higher pressure generation due to volumetric microwave

heating during the IMCD than in convective drying.


Figure 7-12. Water flux due to gas pressure along horizontal centreline of the sample for different

times

Figure 7-13 and Figure 7-14 shows the spatial distribution of vapour fluxes due

to binary diffusion and gas pressure gradient, respectively. The figures shows that

vapour fluxes from both sources are positive on the right edge of the sample (starting

from about 30mm) and negative for 15mm to 28mm. This can be explained from the

vapour density (Figure 7-8) and pressure (Figure 7-9) profiles, respectively. The

vapour concentration curve (Figure 7-8) shows that the vapour concentration is the

highest near 30mm from the left edge and gradually decreases with distance in both

direction. Therefore, the vapour concentration gradient is positive from 30mm

onwards indicating positive flux and flow towards the right edge of the sample and

vice versa. However, at the far left edge (0mm to 10mm), where vapour gradient is

almost zero, the vapour flux is also minimum.

Similarly, the vapour flux due to gas pressure is positive for the right side

(30mm onwards) of the sample due to positive pressure gradient and negative on the

left of 30mm, due to negative pressure gradient. In general, the fluxes of vapour from

both sources are higher during the heating periods and lower in the tempering periods

which indicating higher moisture migrations during the heating periods.


Figure 7-13. Vapour flux due to gas pressure along horizontal centreline of the sample for different

times

Figure 7-14. Vapour flux due to gas pressure along horizontal centreline of the sample for different

times


It is noted that, all the fluxes are negligible near the left edge of the sample

except for water flux due to capillary diffusion. Therefore, the transport of moisture

to the left edge is only due to capillary diffusion, whereas all other fluxes including

capillary diffusion are higher near the right edge causing more moisture loss. This

also justifies the lower water saturation on the right side of the sample. This

information provides an important implication for IMCD if a specific part of the

sample needs drying.

7.6 CONCLUSIONS

In this study, a novel IMCD model is developed which considers complex

coupling between electromagnetics using Maxwell’s equations in 3D and multiphase

porous media for transport. Therefore, this model provided more realistic and uneven

temperature distribution, showing hot spots and cold spots due to microwave heating,

which was not possible to investigate from the previous models. Moreover, this

model provided asymmetric distribution of pressure, vapour mas fractions, moisture

and vapour fluxes, in contrast to models based on Lambert’s Law which considered

symmetry. Although the coupling of different physics present unmatched difficulties,

the application of such models is required for better insight and prediction of critical

parameters such as microwave energy deposition, temperature and moisture

distribution, evaporation and fluxes due to different mechanisms. Understanding of

these factors can, in turn, lead to a significant improvement in food quality, energy

efficiency, increased ability to automation and optimization.

The major conclusions from the work can be summarized as follows:

1) Multiphase porous media models for IMCD considering Maxwell’s equation

for electromagnetics were developed and the model’s predicted values agrees with

experimental data reasonably well.

2) IMCD produces uneven temperature distribution during the heating periods

and the temperature redistributes during the tempering period creating more uniform

distribution. The model captures these phenomena which are very useful for

understanding and optimizing the process and which cannot be done by simpler

model.


3) Intermittency of microwave in IMCD can help to control the material

temperature, thus the model can be used to find the optimum value for both the

microwave power level and the intermittency for better product quality.

4) The moisture fluxes due to capillary diffusion and gas pressure gradient, and

the vapour flux due to binary diffusion and gas pressure gradient were asymmetric

and higher than those obtained through convection drying. The fluxes due to the gas

pressure gradient are also higher in magnitude during the heating period than the

tempering period which indicating higher convective flow during heating.

Chapter 8: Conclusion and Future Recommendation 157

Conclusion and Future Chapter 8:

Recommendation

8.1 OVERALL SUMMARY

IMCD is an advanced drying technology that can significantly reduce drying

times and improve product quality during drying. Mathematical modelling is

essential for understanding and optimizing it. Particularly, understanding the physical

mechanism of heat and mass transfer during IMCD is a key element for optimizing

and achieving better insight into the process. The aim of the current research was to

develop a comprehensive mathematical model that to enhance the understating of

transport of moisture and energy during IMCD, which was a significant research gap

in the arena of food drying. The outcome of the models and related innovations lead

to eight peer reviewed journal papers.

The study was initiated with a single phase convection drying model to identify

the best way of defining an effective diffusion coefficient (Chapter 3). Then the

model was used to develop first an IMCD model which calculates the microwave

power generation using Lambert’s Law (Chapter 4). The IMCD model was validated

with experimental data and then was used to investigate the temperature

redistribution during IMCD. However, these models (Chapter 3 and 4) were single

phase and considered an overall moisture flux by an effective diffusivity term

obtained from the experiments. Therefore, the subsequent three chapters (Chapter 5

to 7) endeavour to develop a multiphase porous media model, which are more

comprehensive and advanced models for transport of energy and mass inside the

porous media.

In Chapter 5, a multiphase porous media model for the convection drying of

food has been developed. Following this, two IMCD models considering Lambert’s

Law and Maxwell’s equations in Chapter 6 and 7, respectively, were developed. The

relative advantages and disadvantages of these models have been identified and

discussed. The multiphase models provide more comprehensive insight about

transport mechanisms of different phases inside food materials such as liquid water,

water vapour and air. Moreover, it enables the investigation of the evaporation rate

158 Chapter 8: Conclusion and Future Recommendation

and pressure generations inside the material. Furthermore, the fluxes of water due to

capillary diffusion and pressure driven, as well as the fluxes of vapour due to binary

diffusion and pressure driven also has been investigated in the IMCD process.

Moreover, the IMCD model considering Maxwell’s equation (Chapter 7) provides

further details of spatial temperature and moisture distribution compared to the

model that considered Lamberts Law (Chapter 6).

The overall outcome of the thesis was the better understanding of the heat and

mass transfer involved in IMCD including temperature redistribution, and moisture

and vapour fluxes due to different mechanisms achieved by mathematical modelling

and associated experimentation. The final model for IMCD which coupled non-

equilibrium multiphase porous media with 3D electromagnetic heat generation using

Maxwell’s equations has successfully fulfilled the main aim of this research. The

modelling enhances the understanding of IMCD and will help to optimize the

process for better energy efficiency and product quality. The sequential

developments of the models were presented in Chapter 3 to Chapter 7.

This is the first time a study has systematically investigated the complex drying

problem. We started with a very basic diffusion based single-phase model and

gradually improved our modelling to evolve into a comprehensive multiphase IMCD

model. An advantage of this systematic development is that the limitations and

challenges of each model, and associated assumptions are clearly identified and

demonstrated. The section below (Section 8.2) presents the major conclusion from all

the models.

8.2 CONCLUSIONS

The study investigated the heat and mass transfer process in both convection

drying and IMCD through mathematical modelling which were validated by

experimentation. The major conclusions are:

Effective diffusivity as a function of both temperature and moisture needs

to be considered for better description of drying process in single phase

drying model (Chapter 3)(Kumar et al., 2015).

The diffusion model for convection drying shows evaporative cooling

phenomena which can be used to control surface temperature and improve

product quality (Chapter 3) (Kumar et al., 2015).


The single phase model of IMCD with Lambert’s Law can predict the

overall moisture content but fails to capture the actual uneven distribution

of temperature and moisture in the sample (Chapter 4 and 6).

The single phase models lump all water fluxes with an effective diffusion

coefficient, but this assumption cannot be justified under all situations

(Chapter 3). One such situation is IMCD, where pressure driven flow and

evaporation is involved. Therefore, to investigate IMCD accurately

multiphase models need to be developed and considered (Chapter 4).

The non-equilibrium multiphase porous media model developed was able

to determine the relative contributions of various modes of transport and

phase changes such as capillary pressure, gas pressure and evaporation,

which is not possible through experiments and/or single phase models

(Chapter 5).

Multiphase porous media models for IMCD showed that the higher fluxes

of vapour and water due to gas pressure make the IMCD faster (Chapter 6

and Chapter 7). This is a significant finding as these cannot be determined

by the single-phase model.

The temperature drops during the tempering period helps limit the

maximum sample temperature, which is the main advantage of IMCD

(Chapter 6 and Chapter 7).

However, Lamberts law for microwave heat generation consideration

cannot capture the actual distribution of temperature and moisture of the

sample (Chapter 6).

The multiphase porous media IMCD model with Maxwell’s equation

provides far better insight including uneven temperature and moisture

distribution and redistribution in comparison to Lamberts law (Chapter 7).

Determination of this uneven distribution of temperature and moisture in

microwave drying establishes the necessity of intermittency in microwave

drying. Uneven temperature and moisture distribution has significant

effect on quality of the product. Once actual (uneven) distribution is

known (which is not possible with other models), necessary measures can

be taken to overcome this problem. One of the ways to deal with that is the

application of the correct microwave power level and intermittency.


The intermittency of microwaves helps to level off temperature, and

reduce the temperature differential between hot spots and cold spots, thus

it can provide better a option to control the product temperature and finally

improve the food quality (Chapter 7).

The final IMCD model was used to investigate the transport of each phase

due to various mechanisms (pressure driven, capillary driven, binary

diffusion, evaporation) and it also demonstrated the asymmetric profile

moisture, temperature and fluxes (Chapter 6 and Chapter 7).

The final multiphase IMCD models provide a very in-depth understanding of

IMCD drying mechanism enabling investigation of moisture distribution,

temperature distribution and redistribution, evaporation, and fluxes due to different

mechanisms. These understanding of IMCD are essential to understand the effect of

process parameters on food quality and drying kinetics; and eventually to optimize

the process.

8.3 CONTRIBUTION TO KNOWLEDGE AND SIGNIFICANCE

Fundamental physics-based mathematical models of IMCD and convection

drying in synergy with relevant experimentations in this thesis provide some useful

contribution to knowledge. The contributions and significance of this research are

listed below:

Compared to the numerous single phase (diffusion based) drying models,

the single-phase model developed in this research (in Chapter 3)

demonstrated better performance. The considerations of the effective

diffusion coefficient as a function of material properties and process

parameters; and material properties as a function of moisture content rather

than assuming a constant make the model more realistic.

In addition to this, the model investigated evaporative cooling phenomena

and its application in quality improvement. This has never been

investigated before.

The governing equations, input parameters and variables for implementing

a multiphase porous media model for apple is presented which will serve

the as a basis for modelling multiphase porous media modelling in food

drying (Chapter 5).


Due to the advanced general multiphase porous media approach, the model

is flexible to implement in different areas of the food process industry, for

example, frying, heating, cooking, and baking. In addition to this, this

model can also be implemented in other applications where transport in

porous media is involved, for example, coal drying, groundwater flow, soil

science, soil mechanics and some biological tissues etc.

The models can easily adapt the solid mechanics to incorporate shrinkage,

which would be more advanced and accurate for lower moisture content.

A fully coupled mathematical model considering 3D electromagnetics and

a multiphase porous media model for IMCD was used to investigate

different phenomena such as pressure driven flow, capillary flow, binary

diffusion, pressure development and evaporation. Such a model for IMCD

has not been implemented earlier (Chapter 7).

This research developed the very first IMCD model that investigates the

temperature redistribution and its effect on the transports (Chapter 4, 6 and

7).

The 3D model for couple electromagnetics and heat and mass transfer

developed in this thesis can be used in the food industry to identify hot and

cold spots, in microwave product development and in producing better

cooking instruction. Moreover, the model can play a significant role in

microwave cavity design to achieve more uniform electric field

distribution and thus improve heating.

Food researchers and scientists will experience a number of benefits from

the models, particularly simulating and understanding the transport

mechanism of heat and mass under different process conditions and

different materials (IMCD and convection).

The industry can be enormously benefited by the IMCD models which

have great potential to choose the appropriate microwave power and its

intermittency to maintain suitable temperature. Thus, it allows avoiding

overheating that improves the quality of the product.

8.4 LIMITATIONS

Though this study developed comprehensive mathematical models for food

drying application, there are still several limitation that need to be clarified.


Due to lack to time and appropriate experimental facilities, further

validations of spatial temperature and moisture distribution inside the

product were not possible in this study.

The capillary flow was assumed with a function of moisture content rather

than calculated from capillary pressure because the variation of capillary

pressure in apple tissue was not found in literature.

In this study, all the water is assumed to be free to transport i.e. the bound

water which could be bound to the cell wall have different mechanism of

transport is neglected.

The shrinkage of the sample is neglected in this study.

The uneven interior surface of the microwave cavity and actual waveguide

geometry was difficult to measure and model. A simpler geometry

(ignoring the dents and curves) was modelled.

8.5 FUTURE DIRECTION

To overcome the above-mentioned limitations and to advance the knowledge

of IMCD, the following suggestions and recommendations are proposed.

Magnetic Resonance Imaging (MRI) or other suitable method can be used

to determine the spatial moisture and temperature profile.

Two phases of liquid water; namely, bound water and free water can be

considered separately in the model. However, further justification of

considering bound water needs to be provided for doing that.

A multi-scale model which considers both porous medium (macro-scale)

with an underlying pore structure (micro-scale) would be very interesting

to see.

Incorporating shrinkage of the sample in the model could provide more

advanced and accurate results.

To study the process more comprehensively, a conjugate problems can be

solved which simulates the air flow inside the oven to determine actual

heat and mass transfer coefficients and multiphase porous media model

along with electromagnetics model of microwave heating.

Implementation of the model to determine optimum operating condition

would be beneficial for industry.

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Appendices 177

Appendices

Appendix A

Implementation of the model in mathematical interface of COMSOL

The mathematics interfaces was used to model heat and mass transfer for

solving. A coefficient form PDE was chosen to solve the equation and the coefficient

were defined to match with the formulated equations of the models.

Where u is the independent variable, ae is the mass coefficient, ad is a

damping coefficient or a mass coefficient, c is the diffusion coefficient, α is the

conservative, flux convection coefficient, β is the convection coefficient, a is the

absorption coefficient, γ is the conservative flux source term, and f is the source

term.

178 Appendices

The final equation for liquid water transport in the model is

evapwc

w

wrw

www RcDPkk

St

,. .

Therefore, to implement this equation in COMSOL the following settings were

used.

Description Value expression

Diffusion coefficient cD

Absorption coefficient wS

t

Source term evapR

Mass coefficient 0

Damping or mass coefficient wS

Conservative flux convection coefficient 0

Convection coefficient 0

Conservative flux source P

kk

w

wrw

w

,

Similar strategy were taken for vapour conservation and energy equations.

Appendix B

The conversions of moisture content

The moisture content (dry basis), dbM , cab ne written as,

s

vwdb

ccM

1

where, wc is the mass concentration of water (kg/m3), and vc is the mass

concentration of vapour (kg/m3).


moisture content, wbM , are shown by following equations,

db

dbwb

M

MM

1

wb

wbdb

M

MM

1

Appendices 179

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