SIMULATION OF A BATCH DRYER BY THE FINITE DIFFERENCE ... · I wish to thank Suzan Tireki, Halil...

SIMULATION OF A BATCH DRYER BY THE FINITE DIFFERENCE METHOD

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

UMUT TURAN

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE IN

FOOD ENGINEERING

AUGUST 2005

Approval of the Graduate School of Natural and Applied Sciences

Prof. Dr. Canan Özgen Director

I certify that this thesis satisfies all the requirements as a thesis for the degree of Master of Science.

Prof. Dr. Levent Bayındırlı

Head of Department This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science. Assoc. Prof. Dr. Hami Alpas Prof. Dr. Ali Esin Co-Supervisor Supervisor Examining Committee Members Prof. Dr. Mehmet Mutlu (Hacettepe Unv., FDE)

Prof. Dr. Ali Esin (METU, FDE)

Assoc. Prof. Dr. Hami Alpas (METU, FDE)

Assoc. Prof. Dr. Gülüm Şumnu (METU, FDE)

Inst. Dr. Deniz Çekmecelioglu (METU, FDE)

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I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last name: Umut Turan

Signature :

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ABSTRACT

SIMULATION OF A BATCH DRYER BY THE FINITE DIFFERENCE METHOD

Turan, Umut

M. Sc., Department of Food Engineering

Supervisor: Prof. Dr. Ali Esin

Co-Supervisor: Assoc. Prof. Dr. Hami Alpas

August 2005, 166 pages

The objectives of this study are to investigate the dynamic behavior of an apple slab

subjected to drying at constant external conditions and under changing in the drying

temperatures and to determine the effects of temperature and time combinations at

different steps during drying on the process dynamics parameters, time constant and

process gain of the system. For this purpose, a semi-batch dryer system was

simulated by using integral method of analysis.

Initially, the dynamic behavior of the drying temperature was investigated by using

first order system dynamic model. Process dynamic parameters, time constant and

process gain of the system, for change in drying temperature were determined.

Secondly, investigation of the drying kinetics of the apple slab was carried out under

constant external conditions in a semi-batch dryer. A mathematical model for

diffusion mechanism assumed in one dimensional transient analysis of moisture

distribution was solved by using explicit finite difference method of analysis.

v

Thirdly, investigation of the drying kinetics of the apple slab was carried out under

change in drying temperature at different time steps during drying. Inverse response

system model was used for the representation of the dynamic behavior of drying.

Process dynamic parameters, time constant and process gain of the system were

determined.

Model predicted results for apple slab drying under constant external condition and

under step change in the drying temperature were compared with the experimental

data.

Keywords: Drying, Response to step input, Apple slab, Finite difference method of

analysis

vi

ÖZ

SÜREYE BAĞLI BİR KURUTMA SİSTEMİNDE SONLU FARKLAR

YÖNTEMİYLE BENZETİŞİM YAPILMASI

Turan, Umut

Y. Lisans, Gıda Mühendisliği

Tez Yöneticisi: Prof. Dr. Ali Esin

Yardımcı Tez Yöneticisi: Doç. Dr. Hami Alpas

Ağustos 2005, 166 sayfa

Bu çalışmanın amacı elma dilimi dinamiğinin sabit dış ortam kurutma koşullarında

ve kurutma sıcaklığı değişikliği yapılarak yürütülen kurutma işlemi sırasında

incelenmesi, kurutma işlemi sırasında değişik zaman aralıklarında uygulanan sıcaklık

zaman birleşimlerinin, işlem dinamik parametreleri olan zaman sabitine ve oransal

banda etkisinin araştırılmasıdır. Bu amaç doğrultusunda süreye bağlı bir kurutma

sisteminin entegral metot analizi yöntemiyle benzetişimi yapılmıştır.

Başlangıçta kurutma sıcaklığı dinamiği birinci derece işlem dinamiği ile

incelenmiştir. Kurutma sıcaklığı değişikliği için işlem dinamik parametreleri olan

zaman sabiti ve oransal bant belirlenmiştir.

İkinci olarak elma dilimi kurutma işlem kinetiği sabit dış ortam koşullarında süreye

bağlı bir kurutma sisteminde araştırılmıştır. Süreye bağlı olarak tek boyutlu nem

dağılımı mekanizması sonlu farklar yöntemi analizi ile çözülmüştür.

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Üçüncü olarak elma dilimi kurutma işlem kinetiği değişen kurutma sıcaklıklarında

araştırılmıştır. Kurutma işlemi dinamiği ters etkilenme sistem modeli ile

açıklanmıştır. İşlem dinamik parametreleri, zaman sabiti ve oransal bant

hesaplanmıştır.

Sabit dış ortam koşullarında ve kurutma sıcaklığında değişiklik yapılarak yürütülen

elma dilimi kurutma işlemi sırasında elde edilen model sonuçları, deneysel sonuçlar

ile karşılaştırılmıştır.

Anahtar kelimeler: Kurutma, Basamak uyarıya tepki, Elma dilimi, Sonlu farklar

yöntemi analizi

viii

To My Family

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ACKNOWLEGMENTS

I would like to express my deepest gratitude to my supervisor Prof. Dr. Ali Esin for

his guidance, helpful comments, suggestions and encouragement throughout this

study, and in writing of this report.

I extend my sincere appreciation to my co-supervisor Assoc. Prof. Dr. Hami Alpas

for his suggestions, helpful comments and guidance throughout this study.

I would like to thank Inst. Dr. Deniz Çekmecelioglu for his helpful suggestions and

solutions on theoretical part of this study.

I wish to thank Suzan Tireki, Halil İbrahim Çetin, Salih Obut, Aybeniz Yiğit, Adnan

Özkıranartlı, Ahmet Sümer for their great help and support.

Special thank are due to Sevil Yılmaz for her endless support and love.

Finally, I would like to thank to my father and mother, Ali Osman and Hatice Turan,

for their love, help and encouragement in all my life. I would also like to thank my

sister, Dilek Turan for her love and encouragement in all my life.

x

TABLE OF CONTENTS

PLAGIARISM……………………………………………………………………… iii

ABSTRACT………………………………………………………………………… iv

ÖZ…………………………………………………………………………………… vi

ACKNOWLEDGMENTS…………………………………………………………... ix

TABLE OF CONTENTS………………………………………….………………… x

LIST OF TABLES…………………………………………………….…………… xiv

LIST OF FIGURES………………………………………………………………... xix

CHAPTER

1. INTRODUCTION……………….……………………………………………..…1

1.1. Drying Theory…………….……………………………………...………… 1

1.1.1. Moisture Diffusions in Foods………………………………………... 3

1.1.2. Effect of Drying Parameters on Product Quality………………….…. 4

1.2. Analytical Method……………………………………………………...….. 6

1.2.1. Moisture Content…………………………………………………...... 6

1.2.2. Drying Rate…………………………………………………………... 7

1.2.3. Effective Diffusivity…………………………………………………. 7

1.3. Numerical Methods……………………………………………………..….. 8

1.3.1. Finite Difference Method of Analysis……………………………….. 8

1.4. Dynamic Behavior and Modeling of the Process………………………… 11

1.5. Evaporation of Water…………………………………………………….. 20

1.6. Explicit Finite Difference Method in Drying of a Slab Sample………….. 23

1.6.1. Stability Criteria…………………………………………………….. 26

1.7. Objectives of the Study…………………………………………………… 26

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2. MATERIALS AND METHODS……………………………………………….. 28

2.1. Experimental Setup……………………………………………………….. 28

2.2. Measurement……………………………………………………………… 30

2.2.1. Weight………………………………………………………………. 30

2.2.2. Temperature………………………………………………………… 30

2.2.3. Velocity……………………………………………………………... 30

2.2.4. Air Humidity………………………………………………………... 31

2.3. Samples…………………………………………………………………… 31

2.4. Determination of the Dynamic Behavior of the Sample Point…………… 32

2.5. Preliminary Work…………………………………………………………. 35

2.5.1. Evaporation of Distilled Water……………………………………... 35

2.6. Apple Slab Drying………………………………………………………... 37

2.6.1. Property of the Apple Slab…………………………………………. 37

2.6.2. Drying Under Constant External Conditions……………………….. 38

2.6.2.1. Application of the Explicit Finite Difference Method

in Drying……………………………………………………………….. 39

2.6.2.2. Solution by the Explicit Finite Difference Method…………… 41

2.6.2.3. Calculation of the Time Dependent Correction Factors……… 42

2.6.3. Dynamic Behavior of the Apple Slab Drying Under A Step Change in

the Drying Temperature………………………………………………. 43

3. RESULTS AND DISCUSSION…………………………………………...…... 44

3.1. Overall Heat transfer Coefficient Value of the Tunnel Dryer……..……....44

3.2. Dynamic Behavior of the Sample Point Temperature…………….……….45

3.3. Dynamic Behavior of the Water Evaporation……………………………..50

3.4. Apple Slab Drying Under Constant External Conditions…………………52

3.5. Application of the Explicit Finite Difference Method of Analysis………..55

3.5.1. Time Dependent Correction Factors……..………..……….……......58

3.6. Apple Slab Drying Under Step Change in the Drying Temperature……....61

3.6.1. Comparisons of the experimental and predicted drying

rates….……………………………………………………………...….71

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4. CONCLUSION AND RECOMMENDATION……………………………….... 76

REFERENCES……………………………….....………………………….……..... 78

APPENDICES……………………………….....…………………………..……..... 82

A. TUNNEL DRYING MODELING……...………………………….……..…......82

A.1. Psychometric Data for Air……...………………………….………................. 82

A.2. Dynamic Behavior Analysis of the Sample Point Temperature…………..….. 84

B. DISTILLED WATER EVAPORATION……………………………………..... 90

B.1. Evaporation Data…………………………………………………………...… 90

B.2. Dynamic Behavior Analysis of the Distilled Water Evaporation………..…… 91

C. APPLE SLAB DRYING UNDER CONSTANT EXTERNAL

CONTIONTS……...………………………...……………….…………..…........... 93

C.1. Drying Data...………………………...……………….……..……..…........... 93

D. EXPLICIT FINITE DIFFERENCE METHOD OF ANALYSIS FOR APPLE

SLAB DRYING UNDER CONSTANT EXTERNAL CONDIONTS..............102

D.1. Parameters Used in the Explicit Finite Difference Method..……..….............102

D.2. Predicted Average Moisture Content Values, pX ..……..….......……...........103

D.3. Error and Time Dependent Correction Factor Analysis……………………. 111

D.4. Final Values of the Model Predicted Average Moisture Content, f-pX ……. 112

D.5. Final Values of the Model Predicted Drying Rate, f-fR …………………… 116

E. APPLE SLAB DRYING UNDER CHANGE IN THE SAMPLE POINT

(DRYING) TEMPERATURE………………………………….……….......... 120

E.1. Drying Data …………………………………….…………….……….......... 120

E.2. Nonlinear Regression Analysis for the Last 10 Drying Rate Data of the Apple

Slab Obtained Under New Steady-state External Conditions After Change in

Sample Point Temperature……………………...…………….………............. 158

E.3. Process Dynamic Parameters of the Slab Drying Under Change in the Sample

Point Temperature………….……………………...………….………............. 159

F. COMPUTER CODES IN MATLAB…………..…...………….………........... 162

F.1. Flow Chart for the Determination of the Average Moisture Content in Finite

Difference Method by Matlab Computer Code………………………………. 162

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F.2. Computer Code Developed for the Solution of the One Dimensional Transient

Analysis of Moisture Distribution in Apple Slab Drying……………...…....... 163

F.3. Computer Code Developed for the Nonlinear Regression Analysis for the

Inverse Response Dynamic Behavior of the Apple Slab

Drying……………..…...…............................................................................... 165

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LIST OF TABLES TABLE

3.1. Dynamic parameters for the sample point temperature found by nonlinear

regression analysis ………..………………………………………………... 45

3.2. Model constants found by integral method of analysis for dynamic behavior of

the sample point temperature…………………………………………………… 46

3.3. Effective diffusion coefficients and activation energies in apple slab drying

under constant external conditions……………………………………………... 55

A.1.1. Physical properties of the air in the tunnel dryer at the arithmetic averages of

the temperatures in the air inlet and at the sample point………………………... 82

A.1.2. Physical properties of the air at the average sample point temperatures…... .83

A.2.1. Data for sample point temperature change from 55oC to 96oC after step

change in the inlet air temperature from 66oC to 135oC. ……..…………..….…84


change in the inlet air temperature from 86.5oC to 135oC. ……………………. 85


change in the inlet air temperature from 100.5oC to 135oC. …………………... 86


change in the inlet air temperature from 135oC to 66oC……….……………..… 87


change in the inlet air temperature from 135oC to 86.5oC……………………... 88


change in the inlet air temperature from 135oC to 100.5oC…………………….. 89

B.1.1. Data for the water evaporation under change in sample point temperature

from 97oC to 56oC after step change in the inlet air temperature from 135oC to

66oC ( av = 0.04m/s) …………………………………...…………...………..….90

B.2.1. Psychometric data for air and parameters used in the dynamic behavior of the

distilled water evaporation……………………………………………….……... 91

xv

B.2.2. Change in the evaporation rate data found according to Equation 1.40 under

change in sample point temperature from 97oC to 56oC after step change in the

inlet air temperature from 135oC to 66oC at 0.04m/s air velocity …….…..…… 92

B.2.3. Value of the constants found by nonlinear regression analysis for the dynamic

behavior of distilled water evaporation under change in sample point temperature


66oC at 0.04m/s air velocity…………………………………………………...... 92

C.1.1. Drying data for apple slab at 55oC (%RH = 4.2, av = 0.04m/s) ………..….. 93

C.1.2. Drying data for apple slab at 65oC (%RH = 4.5, av = 0.04m/s) …………… 96

C.1.3. Drying data for apple slab at 75oC (%RH = 3.3, av = 0.04m/s) ….………... 98

C.1.4. Drying data for apple slab at 96oC (%RH = 1.5, av = 0.04m/s) …………...100

C.1.5. Equilibrium moisture contents values of the apple slab, eX ………….…. 101

D.1.1. Space and time intervals used for apple slab drying at 55oC and 65oC….. 102

D.1.2. Space and time intervals used for apple slab drying at 75oC and 96oC….. 102

D.2.1. Predicted data found for apple slab drying at 55oC ……..……..…………. 103

D.2.2. Predicted data found for apple slab drying at 65oC……..……..……..…… 106

D.2.3. Predicted data found for apple slab drying at 75oC ……..……..……..…... 108

D.2.4. Predicted data found for apple slab drying at 96oC ……..……..……..…....110

D.3.1. Root Mean Square Error (RMSE) and Sum Square Error (SSE) values

between predicted and experimental average moisture content data with respect

to node numbers (m) for apple slab drying at 55oC and 65oC………………… 111

D.3.2. Root Mean Square Error (RMSE) and Sum Square Error (SSE) values

between predicted and experimental average moisture content data with respect

to node numbers (m) for apple slab drying at 75oC and 96oC………………… 111

D.3.3. Model constants found by nonlinear regression analysis according to

Equation 2.19 (for node-41) ……..……..……..….……..……..……..………. 111

D.4.1. Predicted data found for apple slab drying at 55oC (for node-41) ……...... 112

D.4.2. Predicted data found for apple slab drying at 65oC (for node-41) ….…..... 113

D.4.3. Predicted data found for apple slab drying at 75oC (for node-41) ………... 114

D.4.4. Predicted data found for apple slab drying at 96oC (for node-41) …….….. 115

D.5.1. Predicted data found for apple slab drying at 55oC (for node-41) ………... 116

xvi

D.5.2. Predicted data found for apple slab drying at 65oC (for node-41) ….....….. 117

D.5.3. Predicted data found for apple slab drying at 75oC (for node-41) ……….. 118

D.5.4. Predicted data found for apple slab drying at 96oC (for node-41) ……….. 119

E.1.1. Drying rate of apple slab under change in sample point temperature from

55oC to 96oC after step change in the inlet air temperature from 66oC to 135oC

at 1800th s. ( av = 0.04m/s) …..……..…..……..…..……..…..……..…..……… 120



at 10200th s ( av = 0.04m/s) ……..……..……..……..……..……..……..…….. 121



at 16200th s ( av = 0.04m/s) ……..……..……..……..……..……..……..…….. 123



at 19200th s ( av = 0.04m/s) ……..……..……..……..……..……..……..…….. 125


65oC to 96oC after step change in the inlet air temperature from 86.5oC to 135oC

at 3600th s ( av = 0.04m/s) ……..……..………....……..……..……..………… 127



at 7800th s ( av = 0.04m/s) ……..……..……..…..……..……..……..………… 128



at 14400th s ( av = 0.04m/s) ……..……..…..……..……..……..……..……….. 129



at 18600th s ( av = 0.04m/s) ……..……..…..……..……..……..……..……….. 130



at 3600th s ( av = 0.04m/s) ……..…..……..……..……..……..……..………… 132

xvii



at 6600th s ( av = 0.04m/s) ……..……..……..……..……..……..……..………. 133



at 11400th s ( av = 0.04m/s) ……..……..……..……..……..……..……..…….. 134



at 19200th s ( av = 0.04m/s) ……..……..……..……..……..……..……………. 135


96oC to 55oC after step change in the inlet air temperature from 135oC to 66oC to

at 3000th s ( av = 0.04m/s) ……..……..……..……..……..……..……..……… 137



at 7200th s ( av = 0.04m/s) ……..……..……..……..……..……..……..……… 138



at 12600th s ( av = 0.04m/s) ……..………..………..………..………..……….. 140



at 18600th s ( av = 0.04m/s) ……..………..………..………..………..……….. 142


96oC to 65oC after step change in the inlet air temperature from 135oC to 86.5oC

to at 3000th s ( av = 0.04m/s) ……..………..………..………..………..……… 144



to at 6600th s ( av = 0.04m/s) ……..………..………..………..………..……… 145



to at 12600th s ( av = 0.04m/s) ……..………..………..………..………..…….. 147

xviii



to at 18600th s ( av = 0.04m/s) ……..………..………..………..………..….…. 149



to at 2400th s ( av = 0.04m/s) ……..………..………..………..………..……… 151



to at 6000th s ( av = 0.04m/s) ……..………..………..………..………..……… 152



to at 12000th s ( av = 0.04m/s) ……..………..………..………..………..…….. 154



to at 18600th s ( av = 0.04m/s) ……..………..………..………..………..…….. 156

E.2.1. Model constants found by nonlinear regression analysis for the change in

sample point temperature from 96oC to 55oC ……..………..………..……….. 158


sample point temperature from 96oC to 65oC……..………..………..……….. 158


sample point temperature from 96oC to 75oC……..………..………..……….. 158

E.3.1. Dynamic parameters for apple slab drying under change in sample point

temperature from 55oC to 96oC ……..………..………..………..……………. 159


temperature from 65oC to 96oC ……..………..………..………..………..…… 159







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xx

LIST OF FIGURES

FIGURE

1.1. Error term representations in finite difference analysis with node intervals…. 11

1.2. Response of the process output to the step change in the process input in the

first order system……………….………………….……………….…………… 15

1.3. Two first order systems connected in series……………….…………………. 15

1.4. Two first order systems connected in parallel………………………..………. 17

1.5. Response of the second order systems having an overshot

and inverse response……………………………………………………………. 19

1.6. Heat transfer mechanism in water evaporation from a metal cup………….… 21

1.7. Application of the finite difference method in slab geometry…………..…… 23

2.1. Laboratory scale tunnel dryer…………..………………..…….…………..…. 29

2.2. Front view (A) and side view (B) of the apple slab…..……..….…..……..….. 37

3.1. Overall heat transfer coefficient of the tunnel dryer, dU with respect to

arithmetic average of the temperatures at the inlet and sample points…………. 44

3.2. Change in the sample point temperature from 55oC to 96oC after step change in

the inlet temperature from 66oC to 135oC…..……..……..……..……..……..… 47


the inlet temperature from 86.5oC to 135oC…..……..……..……..……..……… 47


the inlet temperature from 100.5oC to 135oC…..………..………..………..…… 48


the inlet temperature from 135oC to 66oC…..……..…………..………..……… 48


the inlet temperature from 135oC to 86.5oC……………………………………. 49


the inlet temperature from 135oC to 100.5oC…………………………………… 49

xxi

3.8. Variation of water evaporation rate with time under a change in sample point

temperature from 97oC to 56oC after a step change in the inlet air temperature

from 135oC to 66oC…………………………………………………………….. 50

3.9. Nonlinear regression analysis for the response part of the time versus

evaporation rate curve under change in the sample point (evaporation)

temperature from 96oC to 57oC………………………………………………… 52

3.10. Moisture content values in apple slab drying under constant external conditions

at 55oC, 65oC, 75oC, 96oC………………………………………………………. 53

3.11. Dimensionless moisture content data in apple slab drying under constant

external conditions at 55oC, 65oC, 75oC, 96oC…………………………………. 54

3.12. Comparison of the experimental and predicted average moisture contents for

apple slab drying at 55oC…………………………………….…………………. 56


apple slab drying at 65oC……………………………………………………….. 56





3.16. Nonlinear regression analysis of the ( )tC vs. time plot for apple slab drying at

55oC…………………………………………………………………………….. 59


65oC…………………………………………………………………………….. 59


75oC…………………………………………………………………………….. 60


96oC…………………………………………………………………………….. 60

3.20. Drying rate of apple slab under change in the sample point (drying) temperature

from 55oC to 96oC………………………………………………………………. 61


from 65oC to 96oC………………………………………………………………. 62

xxii


from 75oC to 96oC………………………………………………………………. 62


from 96oC to 55oC………………………………………………………………. 63


from 96oC to 65oC………………………………………………………………. 63


from 96oC to 75oC………………………………………………………………. 64

3.26. Nonlinear regression analysis for the determination of the response parts of the

drying rate versus time curve in drying under change in the sample point (drying)

temperature from 96oC to 55oC at 3000th s……………………………….……. 65



temperature from 96oC to 55oC at 7200th s………………………….…………. 65



temperature from 96oC to 55oC at 12600th s…………………………….……... 66

3.29. Response part of the apple slab drying rate versus time curve under change in

the sample point (drying) temperature from 55oC to 96oC …………..………... 68


the sample point (drying) temperature from 65oC to 96oC ……………………. 69


the sample point (drying) temperature from 75oC to 96oC.……………………. 69


the sample point (drying) temperature from 96oC to55oC ……………………... 70


the sample point (drying) temperature from 96oC to 65oC....…………………... 70


the sample point (drying) temperature from 96oC to 75oC ……………….…..... 71

3.35. Predicted and experimental drying rate data under change in the sample point

(drying) temperature from 55oC to 96oC at 16200th s ………………………..... 72

xxiii


(drying) temperature from 65oC to 96oC at 14400th s ……………..……...…..... 73


(drying) temperature from 75oC to 96oC at 11400th s ..…………………..…...... 73


(drying) temperature from 96oC to 55oC at 7200th s …...………………………. 74


(drying) temperature from 96oC to 65oC at 6600th s .…...……………………… 74


(drying) temperature from 96oC to 75oC at 6000th s …...………………..…….. 75

1

CHAPTER 1

INTRODUCTION

1.1 Drying Theory Drying is a commonly used food preservation method by removing of the volatile

substances, organic liquids and solvents from a solid or liquid solution by applying

energy to yield solid product. In a large number of industrial applications to foods

water is the volatile substance (Geankoplis, 1993; Keey, 1972).

Drying is one of the oldest food preservation methods. It reduces water content of the

food materials that inhibits the microbial growth and the enzymatic activities causing

chemical changes in foods. By this way it prolongs the shelf life of the product. Also,

from the economical point of view, by reducing size and weight of the food

materials, drying minimizes storage and transportation capacity and cost, which

makes handling easier (Geankoplis, 1993; Guiné and Castro, 2003).

All drying processes are aimed to produce desired quality products at minimum cost

at the minimum process time. Hygienic ways of preserving foods in a cost effective

way has a high importance to food suppliers throughout the world. In order to

improve the quality of the dried products, reduce the product losses, lower the usage

of energy, maintain high efficiency for heat and mass transfer in the operation and

control the operation easily, process simulation models has to be done for drying

operations.

In the drying industry, examination of temperature and moisture distribution inside

the dryer during operation can be a good way to analyze performance of the dryer

and product quality. Besides this, desired drying temperature and time combinations

at different steps in the operation can be an applicable way to minimize time and cost

2

of drying operation to reach desired quality of foods at the end. To achieve these

purposes, understanding of the whole operation should be done in better and in the

most cost effective way.

Drying systems are categorized by mode of operation as batch systems, where the

material is placed in the dryer and drying is carried out until desired moisture level in

the product is reached, and continuous systems, where the wet material is moving in

the system and gets progressively dried to the final level of desired moisture and

exits at the discharge end continuously (Geankoplis, 1993). The third kind of drying

system is the semi-batch method where the material is placed in the dryer and the hot

air circulates over it. For all drying operations, final moisture content of the product

depends on the characteristics of foods and the process parameters.

The source of the energy for drying operation is supplied either individually or in

combination by conductive, convective, radiative and dielectric means (Keey, 1972).

In the conduction, energy transfer from a hot surface is carried through molecules in

direct contact. In the convective type drying, moisture transfer from the material is

executed by simultaneous heat and mass transfer between drying medium (air) and

material (Geankoplis, 1993). External conditions including temperature, humidity

and velocity of the drying air are the main parameters controlling drying operation

and affect the product quality (Saravacos and Maroulis, 2001). In the radiative type,

electromagnetic radiation sources are used and infrared drying is an example for this

type of drying where radiative energy absorbed at the surface is conducted inside of

material causing evaporation of the water to move from inside to out. In dielectric

method energy is generated within the material by application of high-frequency

electrical energy in an electrical field oscillating like microwave rapidly causing

temperature within the material to increase (Geankoplis, 1993; Keey, 1972). Besides

these techniques, freeze drying is also a commonly used drying method in which

water as ice is sublimed without changing into liquid phase (Heldman and Lund,

1992).

3

1.1.1 Moisture Diffusions in Foods In the conventional method of drying, heat transfer from the outer surface of material

to the interior occurs causing transfer of water as liquid or vapor within the solid and

as vapor from the air exposed surface of the material. Moisture migration in drying is

governed by several mechanisms involving internal liquid flow depending on the

solid structure, and external conditions involving temperature, humidity and velocity

of the drying medium. Internal mechanism affecting moisture migration within the

solid body involve capillary flow mechanism explaining moisture flow through the

cell cavities by surface-tension forces for the granular and porous material (Keey,

1972), diffusion mechanism explaining moisture migration within the solid body by

concentration differences in the homogenous nonporous material (Akit, 1976).

Besides these, shrinkage and vaporization-condensation sequence are the other

factors affecting moisture migration during drying (Say, 1968).

Generally, the hot air circulation over the food material is used as energy source

needed for moisture to evaporate out of the material in drying during which three

different periods are observed. First one is the adaptation period where the surface

temperature of the material is heated up or cooled down to wet bulb temperature of

drying air. This period can be considered as short settling down period prior to

constant rate period of drying. When surface temperature reaches wet bulb

temperature, constant rate of evaporation on the material surface is observed. Film of

unbound water on surface of wet material acts as independent of solid and evaporates

from the surface at the wet bulb temperature of the drying air. This period is known

as constant rate period. During evaporation of unbound water from the solid surface,

driving forces for the surface evaporation are wet-bulb depression between saturated

surface and moisture concentration differences on the surface and drying medium. In

this period, temperature and flow rate of the air has effective role for the rate of

evaporation from the material surface. Amount of latent heat of vaporization for

removal of water vapor from the material surface is controlled by the rate of heat

transfer. In the constant rate period, drying rate of the material remains unchanged

because the amount of water evaporated is available on the solid surface. When there

4

is insufficient amount of water present in the material to keep a layer of water film

on the surface, surface starts to dry, drying rate starts to decrease and falling rate

period starts. At this period drying rate is controlled by internal moisture transfer

mechanisms because the rate of evaporation from the surface is much higher

compared to the rate of moisture migration within the material. Since the amount of

evaporation from the surface decreases during the falling rate period, the heat

supplied by air increases the surface and internal temperature of the solid. Eventually

equilibrium moisture content of the material is reached depending on the external

conditions of the drying medium (Geankoplis, 1993; Karel et all., 1964).

1.1.2 Effect of Drying Parameters on Product Quality During drying of food materials, some quality changes related with the physical,

chemical, enzymatic and microbial characteristics of the food products occur

depending on the process parameters used in drying. Changes in color and structure

of the material like shrinkage and surface crack formation can be considered as

physical changes occurring during drying. Oxidation reactions like lipid oxidation

resulting in loss of volatiles, flavors and vitamins, browning reactions like Maillard

reactions causing discoloration and caramelization are the common expected

chemical reactions during drying of foods (Erenturk et all., 2005).

To get desired quality product in terms of traditional and consumer acceptance at the

end of the drying, relationship between the process parameters and quality

characteristics of the foods should be studied well. Further, design and operational

parameters of the dryers should be optimized by considering cost of operation beside

the product quality factors. For this purpose, time-varying temperature at different

time periods during drying can be used to prevent quality loss as observed for

ascorbic acid and non-enzymatic browning degradation in drying of potatoes (Ho et

al., 2002) and banana pieces (Chua et al., 2001). Also it can be used to reduce the

drying time and operating cost reported for grain drying by Devahastin and Mujurdar

(1999). Since, quality changes in foods during drying depend not only on the

5

operation temperature but also the moisture content (McMinn and Macgee, 1997),

time-varying temperature applications should be managed at the certain moisture

content of foods during drying.

Sometimes microorganisms may grow before and after drying in the food materials

by causing quality deterioration and food poisoning if drying temperature is not

suitable to inactivate microbial growth. For this reason, high temperature treatment in

the initial period of the drying can be helpful for thermal death of the

microorganisms which may cause food poisoning and diseases (Güler, 2002). Beside

this, thin dry layer on the food surface can be formed to eliminate loss of the volatile

substance in the initial period of the drying (Heldman and Lund, 1992), showing

mechanical resistance to deformation and shrinkage effect as reported for apple

drying by Lewicki and Jakubczyk (2003). Sometimes, low temperature treatment can

be applied to prevent shrinkage (Van Arsdel, 1963) but increase in the drying time

increases the shrinkage effect as reported for drying of apple disc by Ferna´ndez et

al., (2004). Moreover, high temperature treatment in the initial period of the drying

may also help to minimize ascorbic acid degradation in the initial drying period

because degradation of ascorbic acid occurs much at high moisture content (Mishkin

et al., 1982). To prevent color loss during drying, relationships between browning

reactions, enzyme activities, drying temperature, moisture content and drying times

should be examined well before applying temperature variation during drying.

According to Sacilik and Elicin (2005), browning of apple slices increases with

drying temperature. Moreover, in low temperature application, increase in the drying

time increases the browning of apple discs (Ferna´ndez et al., 2004).

6

1.2 Analytical Method 1.2.1 Moisture Content Moisture content of solids, x can be explained on wet or dry basis. Unless stated

otherwise any reported value is considered as wet-basis either as fraction or percent.

W

Ww=x (1.1)

and

dsw W-WW = (1.2)

where, W is the mass of the sample (w), wW is the mass of water in the sample (w)

and dsW is the mass of bone dry solids (w bds).

However, in most calculations moisture content on dry-basis is preferred as,

ds

w

W

W

1=

−=Χ

x

x

(1.3)

Hence, when W is the initial mass (w), iW , ix represents the initial moisture content

(w/w bds) and when eWW = it becomes the equilibrium moisture content, where,

eW is the equilibrium weight of the sample (w) at specific constant external

conditions.

7

1.2.2 Drying Rate

When the change in moisture content with time is expressed as dt

dΧ− it becomes the

drying rate. Experimental determination of the drying rate of a sample is given by

Equation 1.4.

d

dt

R∆

∆Χ= (1.4)

where, dR is the drying rate of the sample (w/w bds.t), ∆Χ is the change in the

moisture content of the sample (w/w bds) with time during drying and dt∆ is the

time interval (t).

1.2.3 Effective Diffusivity When moisture migration in the falling rate drying period is controlled by the

diffusional moisture transport mechanism, effective diffusion coefficient, effD (m2/s)

is used to describe drying behavior of the system.

For the drying of an infinite slab, the value of the effective diffusion coefficient is

determined by solution of Fick’s second law of diffusion (GEANKOPLIS, 1993).

2

2

effz

XD

t

X

∂

∂=

∂

∂ (1.5)

8

Solution of which is,

( )( )

⋅

⋅⋅+⋅⋅−⋅

⋅+⋅=

−

−∑

=2

22

eff

1

022

ei

e

L4

t1n2Dexp

1n2

8

XX

XX π

πn

(1.6)

where, L is the half thickness of the slab (m) and t is the time (s).

If drying takes long time, slab thickness is small and hence dimensionless Fourier

Number, FoN is greater than 0.01, the series solution is simplified to its first term as,

⋅

⋅⋅−⋅=

−

−2

2eff

2ei

e

L4

tDexp

8

XX

XX π

π (1.7)

By taking logarithm of the both sides of the Equation1.7, linear form of the equation

is obtained as,

⋅

⋅⋅−

=

−

−2

2eff

2ei

e

L4

Dt

8ln

XX

XXln

π

π (1.8)

Effective diffusion coefficient value is calculated from the slope of the straight line

on the

−

−

ei

e

XX

XXln against time plot (GEANKOPLIS, 1993).

1.3 Numerical Methods 1.3.1 Finite Difference Method of Analysis Finite difference method of analysis depends on the representation of differential

equations characterizing the system by algebraic equations at a selected number of

points in the divided volume element of the system (Smith, 1978). The average value

9

of the property in interest (moisture) is expressed at the midpoints of each volume

element called nodes. Application of the governing differential balance equations

depending on the property is carried out between each node of volume element.

Development of the finite difference formulation for the system is carried out by

making approximation of the differential quantities in the balance equations in terms

of small differences by using Taylor Series expansion written in the form given in

Equation 1.9 (Buchanan and Turner, 1992).

( ) ( ) ( ) ( )...

dx

xfdx

2

1

dx

xdfxxfxxf

2

22 +⋅∆⋅+⋅∆+=∆+ (1.9)

Expression of the first derivative in the form of difference is done by truncating the

series 2x∆ after the second term.

( ) ( ) ( )

x

xfxxf

dx

xdf

∆

−∆+≅ (1.10)

Expression of the second derivative in difference form is done by truncating after the

2x∆ term.

( ) ( ) ( ) ( )22

2

x

xxfxf2x-xf

dx

xfd

∆

∆++⋅−∆≅ (1.11)

Approximation of the partial derivatives by finite difference can be done by using the

explicit finite difference (forward difference), the implicit finite difference

(backward difference) and the central difference method. Approximation of the time

derivative is done by explicit finite difference (forward difference). Representation of

the space derivatives is done by using one of the three approximations mentioned

above (Çengel, Y. A., 1998). Thus, by using the explicit finite difference (forward

difference) approximation for the time derivative and by using central difference

10

approximation in explicit and implicit form for the space derivative of a nonlinear,

second order partial differential equation of parabolic type given below

2

2

x

u

t

u

∂

∂=

∂

∂ (1.12)

can be represented as,

Explicit finite difference form : 2

i

1j

i

j

i

1j

i

j

1i

j

z

uu2u

t

uu

∆

+⋅−=

∆

− +−

+

(1.13)

Implicit finite difference form : 2

1i

1j

1i

j

1i

1j

i

j

1i

j

z

uu2u

t

uu

∆

+⋅−=

∆

−+

+

++

−

+

(1.14)

where, i and j represent the time step and node point, t∆ and z∆ represents the time

and space interval respectively.

Since property in interest is time and position dependent, discretization in time and

space is carried out by choosing appropriate values of z∆ and t∆ for the desired

accuracy in the solution of the algebraic equations. In these approximations, some

truncation or discritization errors appear in the result due to truncation of the terms in

the Taylor Series Expansion.

Solution of the system of finite difference formulations at each node point within the

sample is carried out with time by using a computer code. In the solution, number of

algebraic equations is solved in a matrix by starting from the initial conditions of the

system. Unknown node properties in the next time (i+1) is determined by the known

node properties at the previous time (i) (Incropera and DeWitt, 2002). During

computation, due to limitation in the significant digits of the computer, round-off

error may come out (Özişik, 1994). To minimize the round-off error term in the

11

result, number of calculation should be minimized by decreasing node number by

increasing the node interval which causes truncation error to increase. Total error

term in the result is sum of round-off and truncation error terms and is given in

Figure 1.1.

Figure 1.1 Error term representations in finite difference analysis with node

intervals.

1.4 Dynamic Behavior and Modeling of the Process Many industrial processes and operations are dynamic in nature showing time

depending behaviors during operations. The understanding of the dynamic behavior

of these by using a relevant mathematical model is important for design, optimization

and control of the systems. Integral method of analysis is one of the applicable

methods used in the mathematical modeling. Rate of input, output, accumulation in

the control volume and rate of generation terms are expressed in terms of the

conservation of the property on an integral basis (Brodkey and Hershey, 1988).

12

According to the integral method of analysis, conservation of energy including flow

terms for input and output and the accumulation term for the control volume with

kinetic and potential energy changes in the system neglected is,

∫∫∫∫∫ ⋅

⋅⋅⋅⋅+⋅⋅⋅

∂

∂=

→→

A

p

V

p dAnvTCdVTCtdt

dEρρ (1.15)

where, T is the temperature (oC), ρ is the density (kg/m3), pC is the heat capacity

(W/m2.K) and v is the velocity of the air (m/s).

Modeling of any operation or process starts with characterizing the system by using

appropriate differential equations. For this purpose, one of the system approaches is

chosen; (1) Lumped parameter systems where the variable of the system changes

only with time, (2) Distributed parameter systems where the variable of system

changes not only with time but also with position (Seborg et al., 1989). Then

dependent and independent variables of the system are determined. Selection of the

appropriate system approach in the modeling depends on the behavior of the

dependent parameters in the system to changes in the independent parameters. This

relation between dependent and independent variables is expressed as the transfer

function, G(s) of the system, where process output responses, Y(s) are explained by

the process input changes, X(s) (step, pulse, ramp and sinusoidally) in Laplace

domain (Bequette, 1998).

X(s)G(s)Y(s) ⋅= (1.16)

Transfer function of the system is nearly always expressed as a polynomial of s

which can be first, second or higher orders. In many industrial processes, dynamic

behavior of the systems are mostly of first or second order process characteristics

where some may also have input dynamics (Coughanowr and Koppel, 1965).

13

∆X

aSlope =

Step

Ramp

0 0t < ∆X(t) ∆X 0t ≥

( )s

XsX

∆=∆

0 0t < ∆X(t) at 0t ≥

( )2s

asX =∆

time

time

Rectangular

Pulse

0 0t <

∆X(t) h Rtt0 <≤

ta ⋅ Rtt ≥

0 0t <

∆X(t)

Asin(wt) 0t ≥

h

Rt time

time

Sinusodial

A

14

The first order process dynamics when a step change in the input variable occurs, is

represented in the Laplace domain as (Seborg et al., 1989),

( )1s

st-expK

s

XY(s) d

+

⋅⋅

∆=

τ (1.17)

or in time domain as,

( )( )

−⋅∆⋅=∆

τdt-t-

exp1XKty (1.18)

where, dt is the dead time, τ is the time constant and K is the steady-state gain of

the system.

Dead time, dt is defined as the time required for the process output to respond to

changing in the input variable (Marlin, 2000). Steady-state gain of the process, K is

defined as how change in the process input affects the change in the process output

when steady-state is attained and is determined as ratio between them (Bequette,

1998) as,

X

YK

∆

∆= (1.19)

where, X∆ is the change in process input and Y∆ is the change in process output at

the new steady-state condition.

A process with a high gain reacts more to the changes in the process inputs, vice

versa. After dead time, the process output starts to change and it attains 63.3% of its

final value within a time equal to the time constant of the process, τ . This indicates

how fast response of the process output after change in the process input occurs

15

(Murrill, 1991). The time required for the process output to reach its new steady state

value is defined as response time. Response of the process output to a step change in

the process input for a first order system is shown in Figure 1.2 where subscripts 1

and 2 represents the initial and final values of the process input and output properties,

respectively.

Figure 1.2 Response of the process output to the step change in the process input in

the first order system.

In some cases two first order systems are connected in series as given in Figure 1.3.

This dynamic behavior of the system shows a second order process dynamics.

Figure 1.3 Two first order systems connected in series.

K1

( 1τ s+1)

K2

( 2τ s+1)

Y(s)

X(s)

Y2

Y1

X1

X2

output

input

time

16

Transfer function of this system is represented in the Laplace domain as (Shinners,

1992),

( )1s2s

KG(s)

22 ++=

ξττ (1.20)

where, τ is the overall time constant, ξ is the damping coefficient and K is the

overall gain represented respectively as,

21 τττ ⋅= (1.21)

τ

ττξ

⋅

+=

221 (1.22)

21 KKK ⋅= (1.23)

where 1τ is the time constant of the first process, 2τ is the time constant of the

second process, 1K is the steady-state gain of the first process and 2K is the steady-

state gain of the second process.

Response of the second order systems to a step change in the input depends on the

value of the damping coefficient ξ of the system (Marlin, 2000).

If 1>ξ , second order process response is overdamped and is represented in time

domain as,

( )

−⋅

−−

−⋅

−+⋅∆⋅=∆

212

2

112

1 texp

texp1XKty

τττ

τ

τττ

τ (1.24)

17

If 1=ξ , second order process response is critically-damped and is represented in

time domain as,

( )

−⋅

+−⋅∆⋅=∆

ττ

texp

t11XKty (1.25)

If 10 << ξ , second order process response is underdamped and is represented in

time domain as,

( )

+⋅

⋅−⋅

∆⋅∆⋅=∆ α

τ

ξ

τ

ξ

ξt

-1sin

texp

-1

XK-XKty

2

2 (1.26)

where,

= −

ξ

ξα

21 -1

tan

In some cases second order systems can arise when two first order systems are

connected in parallel between a process input and an output as given in Figure 1.4.

Figure 1.4 Two first order systems connected in parallel.

K1

( 1τ s+1)

K2

( 2τ s+1)

+

+ Y(s)

X(s)

18

Transfer function of this second order system is represented in the Laplace domain as

(Marlin, 2000),

( )( )( )1s1s

1sKG(s)

21

3

++

+⋅=

ττ

τ (1.27)

where, 1τ is the time constant of the first process, 2τ is the time constant of the

second process, 3τ is the overall time constant, 1K is the steady-state gain of the first

process, 2K is the steady-state gain of the second process and K is the overall gain.

21 KKK += (1.28)

K

KK 12213

τττ

⋅+⋅= (1.29)

Response of the this second order system to a step change in the input is given in

time domain as,

( )

−⋅

−

−+

−⋅

−

−+⋅∆⋅=∆

212

23

121

13 texp

texp1XKty

τττ

ττ

τττ

ττ (1.30)

Depending on the 3τ value, overshoot or inverse response may occur for a step input.

If 03 <τ , step response of the system shows inverse response behavior where

“response of the process output initially moves in an opposite direction to its final

steady state value” (Marlin, 2000).

19

Relationship between process parameters of two first order systems connected in

parallel by showing response in opposing direction is given by Equation 1.31

(Seborg et al., 1989).

1

2

1

2

K

K

τ

τ>− (1.31)

Initially the second process, which reacts faster than the first according to Equation

1.31, dominates the response of the overall system. But then, first process, having a

higher steady-state gain value, dominates the response of the overall system in the

opposite direction.

If 312 τττ << , step response of the system shows overshoot (Seborg et al., 1989).

Response of the second order systems having an overshot and inverse response is

given in Figure 1.4.

Figure 1.5 Response of the second order systems having an overshot and inverse response.

130 ττ <<

31 ττ <

Y1

Y2

03 <τ time

20

1.5 Evaporation of Water Evaporation is the removal of a volatile liquid from a solution by applying energy to

the system (Geankoplis, 1993). In most of the time, water refers to evaporating

liquid. When energy is applied by the flow of the hot air, rate of evaporation depends

on the temperature and concentration in the evaporating substance and in the air,

flow rate of the air and the atmospheric pressure in the system. Its value is

determined experimentally by measuring the change in the weight of the solution

with time and is given as,

t

WR

∆

∆= (1.32)

where, R is the rate of evaporation (w/t), W∆ is the change in the weight of

solution (w) over a time interval, t∆ (t).

A mathematical model for water evaporation in a metal cup at constant external

condition by hot air circulation starts by expressing heat and mass transfer

mechanism in the system shown in Figure 1.6.

To express heat transfer mechanism during evaporation, the general energy balance

for the system is carried assuming that; 1) During evaporation of the water at

constant external condition, temperature of the water is constant at the wet-bulb

temperature of the air, Twb, 2) Convective heat transfer occurs between water surface

at Twb (oC) and hot air at T (oC), 3) Convective heat transfer occurs through the

bottom of the metal cup at Tc (oC) and hot air at T (oC), 4) Conductive heat transfer

occurs between bottom of the metal cup at Tc (oC) and water at Twb (oC), 5) Heat

transfer from the side surfaces of the metal cup is negligible due to small side surface

areas compared to top and bottom heat transfer surface areas.

21

Figure 1.6 Heat transfer mechanism in water evaporation from a metal cup. According to the conditions shown above the overall heat transfer coefficient of the

system, eU (W/m2.K) under constant external conditions is expressed as,

+∆⋅

+

+⋅=

w

air

c

cairaire

h

h

k

xh1

11hU (1.33)

where, ck is the thermal conductivity of the metal cup (W/m.K),

wh is the

convective heat transfer coefficient of the water (W/m2.K), airh is the convective

heat transfer coefficient of the hot air (W/m2.K) and cx∆ is the thickness of the metal

cup (m).

Amount of energy given to the system by hot air during evaporation is given in

Equation 1.34.

( )wbwe T-TAUQ ⋅⋅= (1.34)

where, Q is the heat flux into the system (W) and wA is the water evaporation

surface area (m2).

Air Twb

T

T

Tc

hair

hw

22

During evaporation of the water at constant external conditions, heat added by hot air

is totally used for vaporization. Rate of the water evaporation is expressed in

Equation 1.35.

( )

wbgl

wbwee

h

T-TAUR

∆

⋅⋅= (1.35)

where,

wbglh∆ is the latent heat of vaporization of water (kJ/kg). Its temperature

dependency is expressed in Equation 1.36 (Esin, 1993).

wbwbgl T2.439-2503h ⋅=∆ for 0<T<130 (1.36)

New form of the equation for rate of evaporation becomes,

( )

wb

wbwee

T2.439-2503

T-TAUR

⋅

⋅⋅= (1.37)

In the dynamic behavior analysis of the evaporation rate under a step change in the

air temperature, wet-bulb temperature of the system is expressed as a function of dry-

bulb temperature of the air at the same air humidity (0.006 g/g) as,

12.31 + T0.2252Twb ⋅= (1.38)

Final form of the equation for rate of evaporation becomes,

⋅

⋅⋅⋅=

T0.55-2473

12.31-T0.77AUR wee (1.39)

23

Dynamic behavior analysis of the evaporation rate under a step change in the air

temperature is expressed in Equation 1.39 as,

∆⋅

∆⋅⋅⋅=∆

T0.55-2473

12.31-T0.77AUR wee (1.40)

where, eR∆ is the change in evaporation rate with time (w/t) and T∆ is the step

change in the air temperature (oC).

1.6 Explicit Finite Difference Method in Drying of a Slab Sample In the application of the explicit finite difference method for moisture distribution

analysis in a slab sample for one dimensional diffusional moisture transport

mechanism during drying, sample geometry is divided into a number of volume

elements. The value of the moisture contents are expressed at each midpoints (node

points) of the volume elements represented by dashed lines shown in Figure 1.7.

Figure 1.7 Application of the finite difference method in slab geometry.

0 1 2 m-1 m

z = 0 z = 2L

∆z

2L

24

For this purpose, appropriate differential equations expressing the moisture migration

in the system during drying operation are determined at the boundaries (air exposed

surface and within the sample).

The differential equation governing the drying process within one dimensional slab

sample is given by Equation 1.5, for which the convective boundary condition at the

air exposed surfaces of the slab is,

( )eaireff X-Xkdz

dXD ⋅=⋅ (1.41)

where, airk is the moisture transfer coefficient of the air (m/s).

The average moisture transfer coefficient for evaporation of the liquid into the gas

phase during laminar flow of air when flow is parallel to the surface

(GEANKOPLIS, 1993) is,

( ) ( ) 3/1

Sc

2/1

L-Re

aw

airSh NN664.0

D

kN ⋅⋅=

⋅=

l (1.42)

where, all physical properties of air were calculated at the arithmetic average of the

wet-bulb ( s-wbT ) and dry-bulb ( sT ) temperature of the air at the sample point part of

the dryer ( s-aveT ). LReN is Reynolds Number, ScN is Schmidt number, ShN is

Sherwood Number, awD is the water vapor diffusivity in the air (m2/s) and l is the

slab surface dimension parallel to flow direction (m) calculated as,

4

14.3d 2

a ⋅=l (1.43)

where, ad is the average diameter of the apple slab.

25

In the explicit finite difference method, moisture distribution in the apple slab is

represented with time and position as i

jX by replacing differential equations with the

algebraic ones. Node point is represented by j ( m,...,3,2,1,0j = ) in the slab

geometry given in Figure 1.7 where 0 and m represent the initial and final node at the

boundaries respectively. Calculation of the value of m is given as,

1z

2L m +∆

= (1.44)

where z∆ is the node interval (m).

Time step used to find moisture contents at each node is represented by i

( n,...,3,2,1,0i = ) where n represents the number of total time steps used in the

method. Calculation method for the value of n is given in Equation 1.45.

∆=

t

tn d (1.45)

where dt and t∆ is the total drying and the time interval used in the method in (s)

respectively.

Representations of the time and space derivatives for Equation 1.5 and Equation 1.41

with explicit finite differences are given as,

∆

+⋅−⋅=

∆

− +−

+

2

i

1j

i

j

i

1j

eff

i

j

1i

j

z

XX2XD

t

XX (1.46)

( )e

i

jair

i

j

1i

j

eff XXkz

XXD +⋅=

∆

−⋅

+

(1.47)

26

Solution of the algebraic equation starts with an assumption of the appropriate node

interval, z∆ initially, then value of the time interval, t∆ is determined by stability

criteria.

1.6.1 Stability Criteria In the explicit finite difference method of analysis, the value of the time interval, t∆

is upper limited by stability criteria because “the coefficient associated with the node

of interest at the previous time had to be greater than or equal to zero” (Incropera and

DeWitt, 2002). Determination of the t∆ value is carried by applying stability criteria

for the algebraic equation within the interior node. The governing equation for

stability criteria is given as,

eff

2

D2

zt

⋅

∆≤∆ (1.48)

1.7 Objectives of the Study The objective of this study is to simulate a semi-batch drying system to examine the

dynamic behavior of an apple slab during drying. For this purpose, drying

temperature is considered as independent variable and drying rate and process

dynamics parameters are considered as dependent variables of the system. Dynamic

behavior of the distilled water evaporation was studied with evaporation temperature

as parameter in the preliminary work. Then the dynamic behavior of an apple slab

subjected to drying under constant external conditions with the drying temperature as

parameter was studied in a semi-batch dryer. To predict the moisture distribution

within the apple slab and the drying rate under constant external conditions, explicit

finite difference method of analysis was applied for diffusional mechanism in one

dimensional transient analysis of moisture migration during drying. To model the

27

dynamic behavior of the drying operation and to determine the process dynamic

parameters of the system under change in the drying temperature, nonlinear

regression analysis was used. The predicted values were compared with the

experimental ones.

28

CHAPTER 2

MATERIALS AND METHODS

2.1 Experimental Setup Evaporation of distilled water and drying of an apple slab were carried in a

laboratory scale tunnel dryer (1.5×28×0.28 m3, Armfield Limited, D.27412,

England) shown in Figure 2.1. It consists of a rate adjustable fan and an adjustable

electrical heater with setting switches. For temperature and sample weight

measurements, a digital temperature indicator and a digital balance were connected

to the tunnel with a sample holder in the sample placement part of the dryer.

During drying and evaporation under constant external conditions, flow rate and

temperature of the inlet air was adjusted by settling the knobs of the fan and the

electrical heater in the air inlet part of the dryer shown in Figure 2.1. Hot air flowing

through the tunnel reached the sample holder located 1.5m away from the air inlet.

Thus, during the air flow along the drying tunnel, heat loss from the dryer to

surrounding occurred. Hence, at the sample point, air flowed parallel to the sample

surface at its local steady-state temperature value.

Before starting the experiments under constant external conditions, temperature in

the sample point to attain the steady-state value took some time. Then the sample

was hanged to the sample holder and change in the weight, inlet air and sample point

temperatures were recorded every 10 minutes. Since the quantity of the air supplied

was large compared to the sample, humidity of air was assumed constant during each

experiment.

29

Sample Point

o o o o o

Air Heater

Air Fan

Air Outlet

Digital Balance

Digital Temperature Indicator

Hot Air Inlet

Heater Settling Knob

Sample

Air Rate Settling Knob

1.5m

0.28m

Ambient Air

Sampling Part

Figure 2.1 Laboratory scale tunnel dryer.

30

2.2 Measurement 2.2.1 Weight Sample weight (distilled water and apple slab) was determined by a digital balance

(3000 ± 0.01gr, Avery Berkel) connected to the sample holder in the sample

placement part of the dryer. It continuously displayed the weight of the sample

during evaporation and drying.

2.2.2 Temperature During evaporation and drying experiments, ambient air temperature and hot air

temperature in the air inlet and sample point in the dryer were measured and

monitored by means of thermocouples with digital display (Nel Electronic

Equipments, NR900, Turkey) with 0.1oC accuracy. Sample point temperature was

assumed to be drying and evaporation temperature depending on which experiment

was carried out. In the experiments, the temperature values were selected in the range

of 55oC - 97oC considering most of the food drying operations.

2.2.3 Velocity

The velocity of the air, av , in the drying tunnel was measured at the sample point by

using a vane anemometer supplied with the dryer (measurement range of 0–30 m/s).

During evaporation and drying operations at constant external conditions steady-state

value of the air velocity was adjusted as 0.04m/s.

31

2.2.4 Air Humidity Humidity of the air circulating through the drying tunnel was determined by

measuring the dry-bulb and wet-bulb temperature of the ambient air by using

Psychrometer (Cole-Parmer Instrument Company, Model No. 3312-40, USA).

Ambient air humidity was almost constant at 0.006 g/g throughout the experimental

studies.

2.3 Samples 1. Distilled water (500g) placed in a stainless steel cup of 0.28×0.19×0.01 m3 and

weight of 500 g.

2. Apple slab of thickness of 0.02 m in width, 0.075 m and 0.08 m in diameter of

both slab surface areas.

The dependent and the independent variables of the system were set before starting

the mathematical modeling of the system. Temperature of the air was taken as the

independent variable. Evaporation and drying rates, process dynamics parameters

including time constant and process gain of the systems were considered to be the

temperature dependent variables in the system. The physical properties of the hot air

were assigned as the temperature dependent. In the analysis, the arithmetic average

values of the temperatures at the inlet and sample points were used. The physical

properties of the distilled water and apple slab were assumed to be constant during

each experiment.

32

2.4 Determination of the Dynamic Behavior of the Sample Point Initially, it was required to determine the dynamic behavior of the sample point air

temperature, sT , for a step change in the inlet air temperature, iT . For this purpose,

Equation 1.15 was used with the arithmetic average temperature in this zone.

Thus, the final form of the energy balance for the system from Equation 1.15

become,

( )

−

+⋅⋅−⋅⋅=⋅⋅⋅ amb

siddsia-pa

sa-pad T

2

TTA U-TTCm

dt

dTCV ρ (2.1)

where Ti is the air temperature in the inlet part of the dryer (oC), ambT is the ambient

air temperature (oC), Ac is the cross sectional area for air flow (0.0784m2), Ad is the

wall surface area of the drying tunnel (1.68m2), Va is the volume of the drying tunnel

(0.1176m3) and am is the mass flow rate of the hot air (kg/s).

During air flow through the drying tunnel, heat loss from the dryer to surrounding

occurred. Heat loss from the drying tunnel to surrounding was explained by the

overall heat transfer coefficient of the dryer, Ud which was assumed to be

temperature dependent.

The overall heat transfer coefficient values were calculated when the tunnel dryer

was operated under four different constant external conditions for the inlet air

temperatures at 66oC, 86.5oC, 100.5oC, 135oC and corresponding sample point

temperatures at 55oC, 65oC, 75oC, 96oC respectively with a constant air velocity of

0.04m/s. During each experiment, psychometric data for air in the sample point in

the dryer and surrounding air was obtained at different time steps. In the calculation

of the overall heat transfer coefficient, the ambient temperature, ambT was taken as

33

20oC. Overall heat transfer coefficient in the drying chamber was obtained from

Equation 2.1.

( )

+⋅

−⋅⋅=

ambsi

d

sia-pa

d

T-2

TTA

TTCmU (2.2)

To determine the dynamic behavior of the sample point temperature, step change in

the inlet air temperature of the dryer were applied as from 66oC to 135oC, 86.5oC to

135oC, 100.5oC to 135oC and 135oC to 66oC, 135oC to 86.5oC, 135oC to 100.5oC to

change the corresponding sample point temperature from 55oC to 96oC, 65oC to

96oC, 75oC to 96oC and 96oC to 55oC, 96oC to 65oC, 96oC to 75oC respectively while

keeping the velocity of the air in the dryer constant at 0.04m/s. Temperature change

at the sample point was recorded by means of digital thermocouple in every 10

seconds and obtained data were plotted with time.

Modeling of the dynamic behavior of the sample point temperature was carried out

in two different ways; mathematically by using integral method of analysis and

experimentally by nonlinear regression analysis of the experimental data using first

order system dynamic equation for the dynamic behavior of the sample point

temperature respectively.

During mathematical modeling of the dynamic behavior of the sample point

temperature by using integral method analysis, changes in the inlet and sample point

temperatures, iT∆ and sT∆ were represented as,

i-if-ii T-TT =∆ (2.3)

i-sf-ss T-TT =∆ (2.4)

34

where Ti-i and Ti-f were the initial and final temperature values of the air in the air

inlet part of the dryer (oC), Ts-i and Ts-f were the initial and final temperature values

of the air in the sample point (oC).

To determine the dynamic behavior of the sample point temperature under a step

change in the inlet temperature of the dryer, Equation 2.1 was used. By using

Laplace and inverse Laplace transform respectively, the dynamics equation of the

sample point temperature of the dryer was determined as,

( ) ( ) )exp(-t/1KTtT mmis τ−⋅⋅∆=∆ (2.5)

where τm (s) and Km (oC/oC) were the time constant and gain of the system in terms

of the model constants 1τ and 1k .

2

k1 1

1m

+

=τ

τ (2.6)

2

k1

2

k1

K1

1

m

+

−= (2.7)

The values of the 1τ and 1k were calculated from,

a

ad1

m

V ρτ

⋅= (2.8)

a-pa

dd1

C m

AUk

⋅

⋅= (2.9)

35

where, average value of the overall heat transfer coefficient was used.

To model dynamic behavior of the sample point temperature by using nonlinear

regression analysis for the experimental data, Equation 1.18 was used for the step

response part of the time versus sample point temperature graphs by neglecting dead

time, dt .

2.5 Preliminary Work 2.5.1 Evaporation of Distilled Water Dynamic behavior of the distilled water evaporation under a step change in the inlet

air temperature from 135oC to 66oC resulting with a change in the sample point

(evaporation) temperature from 97oC to 56oC was studied in the tunnel dryer prior to

examine the dynamic behavior of the apple slab during drying.

Before starting the evaporation experiments, temperature in the sample point to attain

the steady-state value took some time. Then distilled water in the stainless steel cup

was connected to the digital balance in the sample part of the dryer and evaporation

started at constant external conditions of air at 97oC with 1.013 %RH and at a

velocity 0.04m/s. During evaporation, weight of the distilled water cup, w-dW was

recorded with 10 minute intervals. The weight data obtained for the experiment were

converted to evaporation rate, eR using Equation 1.32. Then step change in the inlet

air temperature of the dryer was applied from 135oC to 66oC to change

corresponding sample point temperature from 97oC to 56oC.

Determination of the dynamic behavior of the distilled water evaporation was carried

out by two different ways i) applying mathematical model analysis for the system

and ii) making nonlinear regression analysis of the experimental data for the

response part of the evaporation rate versus time graph respectively.

36

In the mathematical model analysis of the heat transfer mechanism during the

evaporation, flow of air in the drying tunnel was laminar and 7.0N Pr > according to

operational, physical and the design parameters of the dryer. The convective heat

transfer coefficient of the hot air, airh was calculated at the arithmetic average of the

inlet and sample point temperatures of the air in the drying tunnel, aveT by assuming

laminar flow of fluid inside horizontal tube in forced convection according to

Equation 2.10 (GEANKOPLIS, 1993).

3/1

hPrL-Re

c-a

hairNu

L

DNN86.1

k

DhN

⋅⋅⋅=

⋅= (2.10)

where, c-ak was the thermal conductivity of air (W/m.K), L was the length of the

drying tunnel, hD was the hydraulic diameter (m) calculated by considering a square

tube (Munson at all., 1998) as,

aa4

a4D

2

h =⋅

⋅= (2.11)

where, a was the side length of the cross sectional area of the drying tunnel (0.28 m).

Thermal conductivity of the stainless steel cup, ck was taken as 50.2 W/m.K. Since

the value of the Biot number was less than 0.1 and the ratio of the heat transfer

coefficient of the air to water was small, it was assumed that, heat transfers was

controlled by the air side. Thus, the overall heat transfer coefficient for the system

according to Equation 1.33 was reduced to

aire h2U ⋅= (2.12)

37

where, for the value of the heat transfer coefficient of the air, airh , the arithmetic

averages of its initial and final values calculated in the initial and final steady-state

external conditions was used.

Dynamic behavior of the evaporation rate under a change in the sample point

temperature was determined in time domain according to Equation 1.40 and change

in the evaporation rate with respect to change in the sample point temperature was

calculated. Nonlinear regression analysis of the experimental data for the response

part of the evaporation rate versus time graph to the change in the sample point

temperature was carried out according to Equation 1.30 by neglecting dead time, dt .

Dynamic parameters of the system (time constant and gain) were determined.

2.6 Apple Slab Drying 2.6.1 Property of the Apple Slab Apples used in the experiment were purchased from the local supermarket and stored

in a refrigerator. They were cut in to a slab of thickness 0.02m and 0.075m and

0.08m in diameter of both slab surfaces prior to drying (Figure 2.2).

Figure 2.2 Front view (A) and side view (B) of the apple slab.

2L ≅ 20mm D1 ≅ 75mm D2 ≅ 80mm

A B

z

38

2.6.2 Drying Under Constant External Conditions Before starting a drying experiment, it was waited for temperature at the sample

point to attain the steady-state value. Then apple slab sample was connected to the

digital balance in the sample placement part of the dryer. Drying of apple slab was

carried out under constant external conditions at 55oC, 65oC, 75oC and 96oC at 4.2,

4.5, 3.3, 1.5 %RH using constant air velocity of 0.04m/s. During drying, weight of

the apple slab was recorded with 10 minute intervals.

During the experiments for each condition and sample the initial and equilibrium

moistures and the dry solid contents were determined using the standard gravimetric

method (Saravacos and Maroulis, 2001). In parallel to drying initial moisture content

of apple slab, iX and its dry solid content was determined by placing 11g slab

sample from the apple in a laboratory oven (Nüve, KD 400) according to Equation

1.3. The recorded sample weight data, aW was converted to moisture content value

on wet ( w-aW ) and dry basis ( aX ) by using Equations 1.1 and 1.3. Percent moisture

content, a%X and drying rate values, dR were calculated according to Equations 1.1

and 1.4 respectively. Average of these samples was taken. To determine the

equilibrium moisture content values of the apple slab, eX for drying under constant

external conditions, apple samples in slabs (0.002m in thick and 0.05m in diameter)

were dried in a laboratory scale batch tray dryer at 55oC, 65oC, 75oC and 96oC

respectively. For this purpose, three apple slab samples were spread over the tray

surface for each set of experiment and weigh data were recorded for every 10 min

during drying. Dry-bulb and wet-bulb temperatures of the surrounding air were

measured by using psychrometer to calculate percent relative humidity inside of the

dryer, %RH . To determine their dry solid contents, standard gravimetric method

was used and after about 24 hours constant weights of the samples were attained.

Then the equilibrium moisture content value of each apple slab was calculated using

Equation 1.3 and the average of the three equilibrium moisture contents was

considered to be the equilibrium moisture content value of the apple slab dried at that

temperature. To determine dry solid content of the apple slabs for drying under

39

sample point (drying) temperature change, apple slabs dried in the tunnel dryer were

put in to a laboratory oven at 100oC at the end of the experiments. Then same

procedure was followed for determination of the dry solid content of the samples.

During apple slab drying, moisture migration in the falling rate period was assumed

to be controlled by diffusion mechanism. The effective diffusion coefficient values,

Deff used to describe the moisture migration in the falling rate period were

determined from the respective drying rate curves under constant external conditions

according to Equation 1.8.

2.6.2.1 Application of the Explicit Finite Difference Method in Drying During drying, air flowed parallel to side view of the apple slab surfaces exposed to

hot air. Rate of moisture evaporation was assumed to occur in the same amount from

both surfaces of the apple slab exposed to hot air due to the symmetry neglecting the

slight difference in the side surface cross sectional areas. Since the lateral surface

area of the apple slab, shown in Figure 2.2, was kept with its peel with wax, major

barrier to water loss in fruits (Krajayklang, 2001), water evaporation through that

surface was assumed to be negligible compared to the side surfaces. For that reason

one dimensional moisture migration in nonporous solid body was assumed in the

apple slab drying.

Basic assumptions made in the analysis are given below.

(1) Apple was a homogenous and nonporous material.

(2) Initial moisture content was uniform within the sample.

(3) Temperature dependency of the moisture diffusion coefficient was negligible so

can be taken constant during drying.

(4) External conditions in the drying medium involving dry bulb temperature and

humidity of the air were at their steady-state values.

40

(5) Moisture content of the air at the air exposed surface of the apple slab was taken

as equilibrium moisture content of apple slab at that temperature.

(6) Heat generation within the apple slab was negligible.

(7) Shrinkage during drying was negligible.

In the explicit finite difference method of analysis, boundary conditions according to

Equation 1.5 and Equation 1.41 were expressed in explicit form as,

( ))N(1

XNXX

Sh

eSh

i

1ji

j+

⋅+=

+ for 0i > and 0j = (2.13)

( ))N(1

XNXX

Sh

eSh

i

1-ji

j+

⋅+= for 0i > and mj = (2.14)

where eff

airSh

D

zkN

∆⋅=

( ) i

1j

i

1-j

i

j

1i

j XFoXFoXFo21X +

+⋅+⋅+⋅⋅−= for 0i > and mj0 << (2.15)

where 2

eff

z

tDFo

∆

∆⋅=

Initial condition at 0t = was expressed in explicit form as,

i

i

j XX = for 0i = for mj0 ≤≤ (2.16)

Moisture transfer coefficient of the air, airk , was calculated according to Equation

1.42 where physical properties of air were taken at the arithmetic average of the

sample point and wet-bulb temperature of the air. The value of Schmidt Number,

ScN and moisture transfer coefficient of the air, airk was given in Table A.1.2. In the

41

calculation moisture diffusion coefficient in air, waD at 20oC was taken as 41025.0 −⋅

m2/s and corrected for the temperature using the 3/2T dependence (GEANKOPLIS,

1993).

In the finite difference method of analysis, node interval, z∆ were taken as 0.005m,

0.002m, 0.001m and 0.0005m. The corresponding node numbers were calculated as

5, 11, 21, 41 according to Equation 1.44. Time interval value, t∆ used in the solution

was determined according to stability criteria by using Equation 1.48.

2.6.2.2 Solution by the Explicit Finite Difference Method Solution of the explicit finite difference relations, used for the moisture distribution

in the slab shaped sample during drying, is carried out by developing computer code

in Matlab. The inputs of the program are the product thickness, moisture transfer

coefficients of the air on the slab surface, initial moisture content of the sample,

equilibrium moisture content of the sample, effective diffusion coefficient of the

sample, drying time, node interval, z∆ , time step calculated by stability criteria, t∆ ,

Fourier Number ( FoN ) and Schmidt Number ( ShN ). Outcomes of the program are

predicted average moisture content values, pX .

In the computer code, determination of the predicted average moisture content values

were carried as,

∑=

⋅+

=m

0j

i

j

i

p X1m

1X for n,...,3,2,1,0i = (2.17)

42

2.6.2.3 Calculation of the Time Dependent Correction Factors To fit the predicted average moisture content values to the experimental data, time

dependent correction factors are determined according to Equation 2.18.

( ) ( )( )

=

tX

tXtC

p

(2.18)

where ( )tX is the experimental moisture content (g/g) and ( )tX p is the predicted

average moisture content (g/g) calculated by explicit finite difference method.

The predicted model equation for the nonlinear regression analysis of the time versus

( )tC plot is given in Equation 2.19.

( ) t))exp(-c-(1batC ⋅⋅+= (2.19)

After multiplication of the predicted average moisture content values with the

Equation 2.19, final values of the predicted average moisture contents, f-pX (g/g)

were determined. Then final values of the predicted drying rates, f-fR (g/g.s) are

determined according to Equation 1.4.

2.6.3 Dynamic Behavior of the Apple Slab Drying Under A Step Change in the

Drying Temperature

To investigate the effect of the drying temperature on the dynamic behavior of the

apple slab, change the sample point (drying) temperature was applied at four

different times while drying of apple slab was carried out under constant external

conditions. For this purpose, initially, drying air temperature at the sample point and

air velocity were adjusted to constant values of 55oC, 65oC, 75oC, 96oC and 0.04m/s

43

respectively. Drying of the apple slab was started when sample point temperature

was attained at steady-state value. During drying of the sample weight was recorded

at every 10 min. After the adaptation period of drying was passed, a step change in

the inlet air temperature of the dryer was applied as 66oC to 135oC, 86.5oC to 135oC,

100.5oC to 135oC and 135oC to 66oC, 135oC to 86.5oC, 135oC to 100.5oC to change

the corresponding sample point (drying) temperature values from 55oC to 96oC, 65oC

to 96oC, 75oC to 96oC and 96oC to 55oC, 96oC to 65oC, 96oC to 75oC respectively.

In the low temperature to high temperature changes in the sample point (drying)

temperature (from 55oC to 96oC, 65oC to 96oC and 75oC to 96oC), experimental data

points for the response parts of the drying rate curve to the step change in the sample

point temperature were determined starting from change in drying rate values up to

the new drying rate values. In the high temperature to low temperature changes in the

sample point (drying) temperature (from 96oC to 55oC, 96oC to 65oC and 96oC to

75oC), experimental data points for the response parts of the drying rate curve to the

step change in the sample point temperature were determined starting from change in

drying rate values up to drying rate data determined at the intersection point of the

experimental data. For this purpose regression analysis for the last 10 new conditions

data values of the drying rate was applied.

Determination of the dynamic parameters of the apple slab drying (time constants

and gain of the system) was carried by developing computer code in the Matlab

computer program by using Equation 1.30 for the response parts of the drying rate

curve to the step change in the sample point temperature in the time versus drying

rate plots.

44

CHAPTER 3

RESULTS AND DISCUSSION 3.1 Overall Heat transfer Coefficient Value of the Tunnel Dryer Psychometric data and physical properties of air used for determination of the overall

heat transfer coefficient of the tunnel dryer and dU are given in Table A.1.1. The

variation of the calculated values of the overall heat transfer coefficient with respect

to arithmetic average of the temperatures at the inlet and sample points are shown in

Figure 3.1.

0.00

0.20

0.40

0.60

0.80

1.00

0 20 40 60 80 100 120 140

temperature (oC)

Ud (

W/m

2 .K)

Figure 3.1 Overall heat transfer coefficient of the tunnel dryer, dU with respect to

arithmetic average of the temperatures at the inlet and sample points.

45

As can be observed from Figure 3.1, increase in the air temperature gradually

increased the overall heat transfer coefficient in the tunnel dryer, dU towards a finite

value due to the physical properties and thus the heat loss from the dryer.

3.2 Dynamic Behavior of the Sample Point Temperature Experimentally obtained air temperature data at the sample point in the dryer after a

step change in the inlet air temperatures are tabulated in Table A.2.1-2.6 and given in

Figure 3.2-3.7. Dynamic parameters of the system, mτ , mK were determined by

nonlinear regression according to Equation 2.6-2.7 and constants, 1τ and 1k were

calculated according to Equation 2.6-2.7 by using calculated values of mτ and mK .

Calculated values of 1τ and 1k were compared with the ones ( i1−τ and i1k − ) calculated

by integral method of analysis using Equation 2.8-2.9. Results are given in Table 3.1-

3.2 with the average initial and final steady-state values of the temperatures at the air

inlet part and sample point of the dryer.

Table 3.1 Dynamic parameters for the sample point temperature found by nonlinear

regression analysis.

i-iT

(oC) f-iT

(oC) i-sT

(oC) f-sT

(oC) 1k

(m2.s/kg) 1τ

(s) mK

(oC/oC) mτ

(s) R2

135 100.5 96 75 0.56 32.96 0.56 25.71 0.95 135 86.5 96 65 0.52 33.81 0.59 26.88 0.96 135 66 96 55 0.56 31.97 0.56 24.94 0.99

100.5 135 75 96 0.50 36.55 0.60 29.24 0.99 86.5 135 65 96 0.45 30.60 0.63 24.94 0.99 66 135 55 96 0.52 42.64 0.59 33.90 0.99

46

Table 3.2 Model constants found by integral method of analysis for dynamic

behavior of the sample point temperature.

iT

(oC) sT

(oC) i1k −

(m2.s/kg) i1−τ

(s) 135 96 0.29 39.47

100.5 75 0.35 39.47 86.5 65 0.38 39.47 66 55 0.41 39.47

Since physical and operational parameters in the system were assumed to be

constant in the calculation, it was observed from the results that, magnitude of the

changes in the inlet air temperature did not significantly affect the values of the

model parameters, mτ and mK . Besides these, since there was no significant

difference between the values of the model constants, 1τ and 1k calculated by using

the values of mτ and mK and by using integral method of analysis, lumped analysis

assumption made for the dynamic behavior of the sample point temperature is

justified.

Expression of the dynamic behavior of the temperature in the tunnel dryer by using

integral method of analysis with lumped parameter system assumption for the time-

varying temperature treatment in the drying of apple slab is a new study in the

scientific literature about drying. For that reason, there is no available information in

the literature for the comparison with the experimental results.

47

0 50 100 150 2000

20

40

60

80

100

120

time (s)

tem

per

atu

re (

oC

)

experimentalpredicted by Eq. 1.18

Figure 3.2 Change in the sample point temperature from 55oC to 96oC after step

change in the inlet temperature from 66oC to 135oC.

time (s)

0 20 40 60 80 100 120 140

tem

per

atu

re (

o C)

0

20

40

60

80

100

120



change in the inlet temperature from 86.5oC to 135oC.

48

0 50 100 150 2000

20

40

60

80

100

120

tem

per

atu

re (

oC

)

time (s)



change in the inlet temperature from 100.5oC to 135oC.

time (s)

0 50 100 150 200 250 300

tem

per

atu

re (

o C)

0

20

40

60

80

100

120



change in the inlet temperature from 135oC to 66oC.

49

time (s)

0 50 100 150 200 250 300

tem

per

atu

re (

o C)

0

20

40

60

80

100

120



change in the inlet temperature from 135oC to 86.5oC.

time (s)

0 50 100 150 200 250

tem

per

atu

re (

o C)

0

20

40

60

80

100

120



change in the inlet temperature from 135oC to 100.5oC.

50

3.3 Dynamic Behavior of the Water Evaporation Experimental data for distilled water evaporation, at the studied air inlet and sample

point temperature values are given in Table B.1.1. Time versus evaporation rate

graph is shown in Figure 3.8.

Psychometric data and physical properties of the air calculated at the arithmetic

average of the wet-bulb and dry-bulb temperatures of the air at the sample point part

of the dryer and the dimensionless numbers (NPr, NRe, NNu, Biot) in forced convective

evaporation over the water surface used for the determination of the convective heat

transfer coefficient of the air, airh during evaporation are tabulated in Table B.2.1.

time (s)

0 2000 4000 6000 8000 10000

evap

orat

ion

rat

e (g

/s)

0.00

0.01

0.02

0.03

0.04

Adaptationperiod

Constantrate

period

Response period to the change in the

evaporation temperature

Figure 3.8 Variation of water evaporation rate with time under a change in sample

point temperature from 97oC to 56oC after a step change in the inlet air temperature

from 135oC to 66oC.

51

Change in evaporation rate, eR∆ calculated by the mathematical analysis using

Equation 1.40 is given in Table B.2.2.

It was observed from Figure 3.8 that, in the initial part of the evaporation under

constant external conditions, adaptation period was observed from the time versus

evaporation rate plot. Then rate of evaporation increased while temperature of water

increased up to wet-bulb temperature of the air. When water temperature reached

wet-bulb temperature evaporation occurred at constant rate. After a step change in

the inlet air temperature was applied from 135oC to 66oC for the corresponding

sample point (evaporation) temperature change from 97oC to 56oC, the observed

dynamic behavior is given in Figure 3.8. The maximum observed at about the 4000th

second can be due to unbalanced heat and mass transfer where through the air

temperature changed from 97oC to 56oC, while the vapor pressure on the water

surface was still behaving as if it was at the previous constant external conditions.

After temperature of the water started to decrease to adapt to the new constant

external conditions at 56oC, vapor pressure on the water surface also decreased

resulting in decrease in the driving force for the evaporation. When temperature of

the water attained wet-bulb temperature of the new constant external conditions at

56oC, the new constant rate of evaporation was observed.

In the nonlinear regression analysis for the response parts of the time versus

evaporation rate graph to the change in sample point temperature, Equation 1.30 was

used. Process dynamic parameters, time constant and gain of the system are given in

Table B.2.3. Representation of the nonlinear regression analysis for the response part

of the time versus evaporation rate plot was given in Figure 3.9.

52

0 1200 2400 3600 4800 6000 7200-20

-15

-10

-5

0

5x 10

-3

time (s)

evap

orat

ion

rat

e (g

/s)

Experimental

Model

Figure 3.9 Nonlinear regression analysis for the response part of the time versus

evaporation rate curve under change in the sample point (evaporation) temperature

from 96oC to 57oC.

3.4 Apple Slab Drying Under Constant External Conditions The experimental results for the apple slab drying under constant external condition

in the tunnel dryer are given in Table C.1.1-1.4. Moisture content data are shown in

Figure 3.10. Equilibrium moisture content data for apple slab drying at 55oC, 65oC,

75oC and 96oC are given in Table C.1.5.

53

time (s)

0 10000 20000 30000 40000 50000 60000

moi

stu

re c

onte

nt

(g/g

bd

s)

0

2

4

6

8

10

at 55oC

at 65oC

at 75oC

at 96oC

Figure 3.10 Moisture content values in apple slab drying under constant external

conditions at 55oC, 65oC, 75oC, 96oC.

As can be observed from Figure 3.10, moisture content inside the apple slab

decreased rapidly in the early period of drying. This is explained by high moisture

gradient causing fast moisture migration within the apple slab according to Hussain

M.M., and Dincer I. (2003). Also drying rate increased as the air temperature was

increased. These findings are in accordance with the theory as due to increase in the

heat transfer between the air and the apple slab the moisture migration from inside to

surface is enhanced. Further, qualitatively and quantitatively it can be stated that the

constant rate period could not be observed where nearly the entire drying was in the

falling rate period.

The same data plotted according to Equation 1.8 is show in Figure 3.11 for which the

values of the values of the effective diffusion coefficient, effD , Fourier Number

( FoN ) and correlation coefficient, R2 are given in Table 3.3.

54

time (s)

0 10000 20000 30000 40000 50000 60000

ln((

Xa-

Xe)

/(X

i-X

e))

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

at 96oC

at 75oC

at 65oC

at 55oCLinear

Figure 3.11 Dimensionless moisture content data in apple slab drying under constant

external conditions at 55oC, 65oC, 75oC, 96oC.

As already stated above, rate of moisture diffusion during drying of the apple slab

was to a high degree affected by the drying temperature only and hence the

assumption of negligible effects of shrinkage and moisture concentration were valid.

Using an Arrhenius type relation the dependency of effD and T was obtained as,

⋅⋅=

TR

EexpDD a

0eff (3.1)

where, the magnitude of both the diffusivity and the activation energy given in

Table 3.3 were in the order reported by other researchers (Saravacos and Maroulis,

2001) but effective diffusivity values were found to be higher compared to the values

ranging from -1010 2.27 ⋅ to -1010 4.97 ⋅ m2/s in the drying of apple slices with

thicknesses of 0.005 and 0.009 m at 40–60oC of the temperature interval (Sacilik and

Elicin, 2005).

55

Table 3.3 Effective diffusion coefficients and activation energies in apple slab

drying under constant external conditions.

sT (oC) 96 75 65 55 -10

eff 10D ⋅ (m2/s) 21.6832 13.1892 9.9886 7.4997

FoN 5189 8482 9955 13116

R2 0.99 0.99 0.99 0.99 -7

0 10D ⋅ (m/s2) 169.20

aE (kJ/mol) 27

R2 0.99 3.5 Application of the Explicit Finite Difference Method of Analysis Physical properties and mass transfer coefficient of the air at the average sample

point temperatures are given in Table A.1.2. Node interval values, node numbers,

time step values, number of iterations and dimensionless numbers ( FoN , ShN ) found

in the transient analysis of the moisture distribution by explicit finite difference

method are given in Table D.1.1-1.2 for apple slab drying under constant external

conditions at 55oC, 65oC, 75oC, 96oC.

Flow chart of the computer code developed in the Matlab and computer program

used for the solution of the explicit finite difference method are given in APPENDIX

F.1-F-2. Predicted average moisture content values, pX are given in Table D.2.1-2.4.

Comparisons of the predicted average moisture content values with the experimental

data with respect to node numbers are shown in Figure 3.12-3.15.

56

0.00

2.00

4.00

6.00

8.00

10.00

0 10000 20000 30000 40000 50000 60000

time (s)

moi

stu

re c

onte

nt

(g/g

bd

s)

node 11 node 21 node 41 experimental

55oC

Figure 3.12 Comparison of the experimental and predicted average moisture

contents for apple slab drying at 55oC.

0.00

2.00

4.00

6.00

8.00

10.00

0 10000 20000 30000 40000 50000

time (s)

moi

stu

re c

onte

nt

(g/g

bd

s)


65oC



57

0.00

2.00

4.00

6.00

8.00

10.00

0 10000 20000 30000 40000

time (s)

moi

stu

re c

onte

nt

(g/g

bd

s)


75oC



0.00

2.00

4.00

6.00

8.00

10.00

0 5000 10000 15000 20000 25000

time (s)

moi

stu

re c

onte

nt

(g/g

bd

s)


96oC



58

It can be observed from the figures that, the optimum value of the node interval was

0.0005m (node 41) to get minimum error from the result. Root Mean Square Error

(RMSE) and Sum Square Error (SSE) values between predicted and experimental

average moisture content data are given in Table D.3.1-3.2. However, the results of

the transient analysis of the moisture distribution in apple slab depending solely on

the assumption of the diffusional mechanism in nonporous material indicates that, a

satisfactory simulation between predicted average moisture contents and

experimental data was not obtained. This might be due to the complex mechanism

governing internal moisture diffusion within the apple slab during drying and the

assumption of nonporous structure might not be sufficient to describe the entire

phase of drying. Besides these, since in the falling rate period mass transfer interface

moves from the surface to inside of the apple slab, calculated value of the moisture

transfer coefficient, airk , according to Equation 1.42 can not be sufficient to describe

the moisture transfer on the surface during drying. Due to non-availability in the

scientific literature about one dimensional moisture distribution analysis inside the

apple slab during drying, comparison of predicted results with the scientific literature

can not be made. There is one study available in the scientific literature about two

dimensional moisture transfer analysis in the cylindrically shaped apple by explicit

finite difference method (Hussain M.M. and Dincer I., 2003).

3.5.1 Time Dependent Correction Factors Nonlinear regression analysis carried out by using Equation 2.19 with the time versus

time dependent correction factor, ( )tC graphs are plotted in Figure 3.16-3.19. Values

of the model constants with the correlation coefficient value, R2 are tabulated in

Table D.3.1.

Final values of the predicted average moisture contents, f-pX are given in Table

D.4.1-4.4. Final values of the predicted drying rates, f-fR were given in Table D.5.1-

5.4.

59

time (s)

0 10000 20000 30000 40000 50000 60000

Xa

/ Xp

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26

1.28

C(t)predicted

Figure 3.16 Nonlinear regression analysis of the time vs. ( )tC plot for apple slab

drying at 55oC.

time (s)

0 10000 20000 30000 40000 50000

Xa

/ Xp

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26

1.28

C(t)predicted


drying at 65oC.

60

time (s)

0 10000 20000 30000 40000

Xa

/ Xp

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26

1.28

C(t)predicted


drying at 75oC.

time (s)

0 5000 10000 15000 20000 25000

Xa

/ Xp

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26

1.28

C(t)predicted


drying at 96oC.

61

3.6 Apple Slab Drying Under Step Change in the Drying Temperature Changes in drying rates with respect to a change at the sample point (drying)

temperatures are given in Figure 3.20-3.25. The sources of the data are tabulated in

Tables E.1.1-1.24.

0.000000

0.000100

0.000200

0.000300

0.000400

0 10000 20000 30000 40000 50000 60000

time (s)

dryi

ng r

ate

(g/g

bds

.s)

at 96C at 55C change in 1800th s

change in 10200th s change in 16200th s change in 19200th s

55oC to 96oC

Figure 3.20 Drying rate of apple slab under change in the sample point (drying)

temperature from 55oC to 96oC.

62

0.000000

0.000100

0.000200

0.000300

0.000400

0 10000 20000 30000 40000 50000

time (s)

dryi

ng r

ate

(g/g

bds

.s)



65oC to 96oC



0.000000

0.000100

0.000200

0.000300

0.000400

0 10000 20000 30000 40000 50000

time (s)

dry

ing

rate

(g/

g b

ds.

s)



75oC to 96oC



63

0.000000

0.000100

0.000200

0.000300

0.000400

0 10000 20000 30000 40000 50000 60000

time (s)

dry

ing

rate

(g/

g b

ds.

s)



96oC to 55oC



0.000000

0.000100

0.000200

0.000300

0.000400

0 10000 20000 30000 40000 50000

time (s)

dry

ing

tim

e (g

/g.s

)



96oC to 65oC



64

0.000000

0.000100

0.000200

0.000300

0.000400

0 10000 20000 30000 40000

time (s)

dry

ing

rate

(g/

g b

ds.

s)



96oC to 75oC



To determine the response part of the time versus drying rate curves after the step

change in sample point (drying) temperature from 96oC to 55oC, 96oC to 65oC and

96oC to 75oC Equation 2.19 was used as the governing model for the nonlinear

regression analysis for the last 10 drying rate data.

Some of the nonlinear regression analyses according to Equation 2.19 are shown in

Figure 3.26-3.28 where black points indicate the starting time for the response of the

drying rate curve to the change in the sample point (drying) temperature. Model

equation constants are given in Table E.2.1-2.3.

65

time (s)

0 5000 10000 15000 20000 25000 30000

dry

ing

rate

(g/

g b

ds.

s)

0.000000

0.000100

0.000200

0.000300

0.000400

0.000500

experimentalpredicted

Figure 3.26 Nonlinear regression analysis for the determination of the response parts

of the time versus drying rate curve in apple slab drying under change in the sample

point (drying) temperature from 96oC to 55oC at 12600th s.

time (s)

0 5000 10000 15000 20000 25000 30000

dry

ing

rate

(g/

g d

bs.

s)

0.000000

0.000100

0.000200

0.000300

0.000400

0.000500





66

time (s)

0 5000 10000 15000 20000 25000 30000

dry

ing

rate

(g/

g b

ds.

s)

0.000000

0.000100

0.000200

0.000300

0.000400

0.000500





According to Figure 3.20-3.25, when a step change in the inlet air temperature from

low value to high value was applied during drying of apple slab, initially decrease in

the drying rate was observed as it has been reported for step-wise change in the

temperature during potato drying by other researchers (Chua et all, 2001). Also

Devahastin and Mujurdar (1999) reported that higher drying rate values of grains

under step-down temperature treatment was observed compared to above those for

constant air drying temperature. In other words, dynamic behavior of the response

part of the time versus drying rate plots under change in the sample point (drying)

temperature showed inverse response dynamics. This can be due to the reversal of

the phenomena encountered in the water evaporation. That is; when a step change in

the inlet air temperature from low temperature to high temperature was applied

during drying, this time heat transfer is increased from apple slab surface to inner by

increasing partial pressure of the vapor from product surface to inner gradually while

the partial pressure of the vapor within the apple slab was still behaving as if it was at

the previous constant external conditions. That phenomenon caused decrease in the

67

partial pressure differences controlling the moisture migration from inside of the

apple slab to the surface resulting in a decrease in the drying rate just after the

sample point temperature was increased. Then temperature within the apple slab

started to increase to adapt to the new constant external conditions and partial

pressure of the vapor within the apple slab started to increase causing increase in the

driving force for moisture migration to the surface thus, increase in the drying rate.

When conditions within the apple slab adapted for the new constant external

conditions after a lag time for heat flux to conduct inside of the material (Ho et all.,

2002), drying rate started to decrease with the decrease in the moisture content.

When from high temperature to low temperature change in the sample point (drying)

temperature was applied in drying, reverse phenomenon in heat transfer from inside

to product surface occurred as reported by Chua et all., (2001). Surface temperature

started to decrease causing decrease in the partial pressure of the vapor from surface

to inner while the partial pressure of the vapor within the apple slab was still

behaving as if it was at the previous constant external conditions. That phenomenon

caused increase in the partial pressure differences between inside of the apple slab

and the surface resulting in increase in the drying rate just after the sample point

temperature was decreased. When temperature within the apple slab started to adapt

to the new constant external conditions, partial pressure of the vapor within the apple

slab started to decrease causing decrease in the drying rate with decrease in the

moisture content.

Determination of the process dynamic parameters of the apple slab drying under

change in drying temperature is the first study conducted in that scientific area.

Equation 1.30 representing the dynamic behavior of the inverse response system was

used in the nonlinear regression analysis for the determination of the dynamic

parameters of the system (time constant values 1τ , 2τ and 3τ and total process gain

value, K ). Values of the 1K and 2K were calculated according to Equation 1.28-

1.29. Computer code developed for the nonlinear regression analysis by using

Equation 1.30 is given in APPENDIX F.3. Process dynamic parameters of the system

found in the nonlinear regression analysis are given in Table E.3.1-3.6. Due to non

68

availability in the scientific literature about process dynamics of the apples in drying

under change in air temperature, predicted results can not be compared with the data

obtained from the scientific literature. According to results it was concluded that;

change in the drying rate directly proportional with the magnitude of the change in

the drying temperature. Total gain of the system increases with decrease in the

magnitude of the step change in the inlet air temperature from high temperature to

low temperature and decreases with the time period for the application of the step

change during drying due to decrease in the moisture content. Total gain of the

system is almost constant for step change in the inlet air temperature from low value

to high value however, inverse response characteristics increases with the decrease in

the moisture content.

Results of the nonlinear regression analysis for the response parts of the apple slab

drying rate versus time curves with respect to change in sample point (drying)

temperature are given in Figure 3.29-3.34.

0 600 1200 1800 2400-2

-1

0

1

2x 10

-4

time (s)

dry

ing

rate

(g/

g b

ds.

s)

Experimentalat 1800th sExperimentalat 10200th sExperimentalat 16200th sExperimentalat 19200th s

Figure 3.29 Response part of the apple slab drying rate versus time curve under

change in the sample point (drying) temperature from 55oC to 96oC.

69

0 600 1200 1800 2400 3000 3600 4200-2

-1

0

1

2x 10

-4

time (s)

dry

ing

rate

(g/

g b

ds.

s)Experimentalat 3600th sExperimentalat 8400th sExperimentalat 14400th sExperimental18600th s



0 1200 2400 3600 4800-2

-1

0

1

2x 10

-4

time (s)

dry

ing

rate

(g/

g b

ds.

s)




70

0 2400 4800 7200-3

-2

-1

0

1

2

3x 10

-4

time (s)

dry

ing

rate

(g/

g b

ds.

s)



change in the sample point (drying) temperature from 96oC to55oC.

0 2400 4800 7200-3

-2

-1

0

1

2

3x 10

-4

time (s)

dry

ing

rate

(g/

g b

ds.

s)




71

0 2400 4800 7200-3

-1-1

0

1

2

3x 10

-4

time (s)

dry

ing

rate

(g/

g b

ds.

s)




3.6.1 Comparisons of the experimental and predicted drying rates Drying rate data predicted for apple slab drying under constant external conditions

and change in the sample point (drying) temperature were combined for the

comparison with the experimental drying rate data obtained drying under change in

the sample point (drying) temperature. For this purpose initially, value of the drying

rates under initial constant external conditions in the time versus drying rate curve

had the value of the drying rate data, f-fR predicted by the time dependent correction

factor analysis. Then, response part of the time versus drying rate curve was formed

by adding response part of the drying rate data, calculated by using dynamic

parameters of the system in Equation 1.30 with response time, to the predicted initial

drying rate data obtained under constant external conditions. Finally, final values of

the drying rate data, f-fR predicted by the time dependent correction factor analysis

were added to response part of the drying rate curve to determine drying rate data

72

under new constant external conditions. For this purpose, the last predicted drying

rate data in the response part of the time versus drying rate curve was assumed to be

first drying rate data predicted by the time dependent correction factor analysis. It

was assumed that under constant external conditions, drying rate depended on only

moisture content in the apple slab.

Some of the comparisons for the predicted and experimental drying rate data are

given in Figure 3.35-3.40 respectively. It was observed from the figures that, high

agreement is found between the predicted and experimental values of the apple slab

drying rates under time-varying temperature treatment during drying.

0.000000

0.000100

0.000200

0.000300

0.000400

0.000500

0 10000 20000 30000

time (s)

dry

ing

rate

(g/

g b

ds.

s)

experimental predicted

Figure 3.35 Predicted and experimental drying rate data under change in the sample


73

0.000000

0.000100

0.000200

0.000300

0.000400

0.000500

0 10000 20000 30000

time (s)

dryi

ng r

ate

(g/g

bds

.s)




0,000000

0,000100

0,000200

0,000300

0,000400

0,000500

0 10000 20000 30000

time (s)

dryi

ng r

ate

(g/g

bds

.s)




74

0,000000

0,000100

0,000200

0,000300

0,000400

0,000500

0 10000 20000 30000

time (s)

dry

ing

rate

(g/

g b

ds.

s)




0.000000

0.000100

0.000200

0.000300

0.000400

0.000500

0 10000 20000 30000

time (s)

dryi

ng r

ate

(g/g

bds

.s)




75

0.000000

0.000100

0.000200

0.000300

0.000400

0.000500

0 10000 20000 30000

time (s)

dryi

ng r

ate

(g/g

bds

.s)




76

CHAPTER 4

CONCLUSION AND RECOMMENDATIONS

In this study dynamic behavior of the temperature in the tunnel dryer was determined

under change in the inlet air temperature. For this purpose mathematical analysis of

the system was carried by using integral method of analysis with lumped parameter

system assumption. In the light of the results the following conclusions could be

drawn;

The results indicate that, lumped parameter assumption is applicable for the

investigation of the dynamic behavior of the temperature in the tunnel dryer

according to which it can be represented with a first order dynamics. Thus, tuning of

the controllers for the control of air temperature in such tunnel dryers can be done

well.

In the numerical analysis of the moisture content distribution in apple slab drying,

comparison of the predicted results with the experimental ones indicate that,

diffusional moisture transfer mechanism in an apple slab during drying is not

sufficient to explain the whole drying operation itself due to complexity of the drying

process. Because during apple slab drying, predomination of the moisture transfer

mechanisms may change and combination of them may govern the whole operation

at different stages of drying. For that reason, numerical analysis for apple slab drying

should be carried for the combined capillary and diffusional mechanisms in the

future studies.

Change in the drying rate of apple slab under constant external conditions with a step

change in the temperature showed second order system dynamics with inverse

response or overshoot due to unbalanced heat and mass transfer phenomena.

Dynamic parameters of the system indicate that change in the drying rate directly

77

proportional with the magnitude of the change in the drying temperature. Inverse

response characteristics of the system depend on the moisture content of the system

for change in the drying temperature from low value to high value. Total gain of the

system increases with decrease in the magnitude of the change in the drying

temperature from high value to low value and decreases with the time period for the

application of the step change during drying due to decrease in the moisture content.

With these findings relationships between the moisture content and process time

under time-varying temperature treatment during drying can be understood well.

For the future studies, it can be recommended to examine the relationship between

the moisture content and the some quality changes related with the physical,

chemical, enzymatic and microbial characteristics of the food products under time-

varying temperature treatment during drying can be done for drying different foods.

Also new investigations for the dynamic behavior of the drying of apple slab with

different dimensions and geometry under different constant external conditions can

be carried. Besides these, economical analysis and optimizations in the operational

parameters by considering product quality and operation cost can be done for the

time-varying temperature treatment during drying in the further studies.

78

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80

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81

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82

APPENDIX A

TUNNEL DRYER MODELING

A.1 Psychometric Data for Air Table A.1.1 Physical properties of the air in the tunnel dryer at the arithmetic

averages of the temperatures in the air inlet and at the sample point.

iT (oC) 67 84 101 135

sT (oC) 55 65 75 96

aveT (oC) 61 75 88 116

ambT (oC) 20 19 21 20

aρ (kg/m3) 1.06 1.02 0.98 0.91

aµ (kg/m.s) 0.000020 0.000020 0.000021 0.000022

a-pC (J/kg.K) 1007.4 1008.6 1009.8 1012.8

c-ak (W/m.K) 0.03 0.03 0.03 0.03

ReN 569.89 530.14 497.16 436.61

PrN 0.69 0.69 0.69 0.68

NuN 7.80 7.60 7.44 7.11

airh (W/m2.K) 0.80 0.81 0.82 0.83

dU (J/kg.K) 0.55 0.63 0.67 0.66

83

Table A.1.2 Physical properties of the air at the average sample point temperatures.

sT (oC) 55 65 75 96

s-wbT (oC) 24.5 27.2 29.6 34

s-aveT (oC) 39.8 46 52.3 65

aρ (kg/m3) 1.13 1.11 1.09 1.05

aµ (kg/m.s) 0.000019 0.000019 0.000019 0.000020

a-pC (J/kg.K) 1006.06 1006.42 1006.81 1007.72

ReN 152.94 147.69 142.66 133.32

ScN 0.56 0.56 0.55 0.54

ShN 6.77 6.63 6.50 6.24

airk (m/s) 0.0030 0.0031 0.0031 0.0033

84

Table A.2 Dynamic Behavior Analysis of the Sample Point Temperature Table A.2.1 Data for sample point temperature change from 55oC to 96oC after step

change in the inlet air temperature from 66oC to 135oC.

time (s)

sT

(oC)

time (s)

sT

(oC)

time (s)

sT

(oC)

0 55.1 310 94.3 620 95.3

10 59.7 320 94.4 630 95.5

20 72.8 330 94.8 640 96.1

30 80.8 340 94.6 650 96

40 85.9 350 95.3 660 96

50 87.4 360 95.1 670 95.9

60 88.6 370 95 680 96

70 89.3 380 95.2

80 89.7 390 95.1

90 89.2 400 94.6

100 90.4 410 95.3

110 91.1 420 95.4

120 91.3 430 96.1

130 92 440 95.8

140 92.5 450 95.6

150 92.7 460 95.7

160 92.3 470 95.6

170 92.6 480 95.7

180 93 490 95.9

190 93.2 500 95.8

200 93.6 510 95.7

210 93.5 520 95.9

220 93.4 530 96

230 93.6 540 95.3

240 93.4 550 95.5

250 94 560 96.1

260 93.8 570 96

270 94.1 580 95.9

280 94.2 590 96

290 94.4 600 95.3

300 94.2 610 95.5

85

Table A.2.2 Data for sample point temperature change from 65oC to 96oC after step

change in the inlet air temperature from 86.5oC to 135oC.

time (s)

sT

(oC)

time (s)

sT

(oC) 0 64.9 440 95.9 10 70 450 96 20 79.4 460 95.7 30 86.1 470 95.4 40 89.4 480 95.9 50 90.5 490 95.7 60 91.5 500 95.9 70 91.6 510 95.9 80 92.5 520 96 90 93 530 95.7 100 93.1 110 93.8 120 93.8 130 93.6 140 93 150 93.8 160 94.1 170 94 180 94.1 190 94.2 200 94.5 210 94.6 220 95.3 230 95.6 240 94.9 250 94.7 260 94.7 270 95.4 280 95.7 290 95.3 300 95.2 310 96 320 96.1 330 96.2 340 96.1 350 96 360 95.4 400 96.7 410 95.4 420 95.9 430 95.7

86


change in the inlet air temperature from 100.5oC to 135oC.

time (s)

sT

(oC) 0 75.2 10 77.6 20 84.8 30 88.7 40 90 50 90.9 60 92.1 70 91.9 80 93.7 90 93.6 100 94.4 110 94.7 120 95.1 130 94.7 140 94.7 150 94 160 94.5 170 94.8 180 94.5 190 95.1 200 95.1 210 95.8 220 95.4 230 96 240 95.3 250 95.6 260 96.2 270 96 280 96.3 290 95.8 300 95.8 310 95.6 320 95.9 330 96.3 340 95.8 350 95.5 360 95.6 370 96.3 380 95.8 390 95.7 400 95.8 410 95.6

87


change in the inlet air temperature from 135oC to 66oC.

time (s)

sT

(oC)

time (s)

sT

(oC) 0 96.1 420 57.4 10 88.9 430 57.2 20 76.1 440 57.2 30 68.5 450 57.1 40 65.4 460 56.9 50 63.8 470 56.6 60 62.7 480 57 70 61.8 490 56.9 80 61.2 500 56.6 90 61 510 56.2 100 60.5 520 56.3 110 60.4 530 56.4 120 60.3 540 56.4 130 60.1 550 56.1 140 59.8 560 56 150 59.9 570 56.2 160 59.4 580 56 170 59.2 590 56.1 180 59 600 56 190 58.9 610 55.9 200 58.8 620 55.7 210 58.3 630 55.9 220 58.6 640 55.7 230 58.4 650 55.6 240 58.4 660 55.8 250 58.4 670 55.6 260 58.3 680 55.7 270 58.6 690 55.5 280 58.5 700 55.5 290 58.2 710 55.6 300 58.2 720 55.7 310 57.9 730 55.5 320 58.2 740 55.6 330 58 750 55.6 340 57.9 760 55.5 350 58 770 55.5 360 58 780 55.6 370 57.4 790 55.7 380 57.6 800 55.7 390 57.5 810 55.5 400 57.2 820 55.6 410 57.5

88


change in the inlet air temperature from 135oC to 86.5oC.

time (s)

sT

(oC)

time (s)

sT

(oC) 0 96.7 420 66.2 10 90.4 430 66.1 20 81.2 440 66.3 30 76.2 450 66.2 40 73.9 460 66.1 50 72.9 470 66.1 60 72 480 65.9 70 71.6 490 66.1 80 70.7 500 65.8 90 70.6 510 65.7 100 70.2 520 65.8 110 69.9 530 65.7 120 69.4 540 65.6 130 69.1 550 65.7 140 69.2 560 65.8 150 69 570 65.7 160 68.7 170 68.4 180 68.6 190 68 200 68.3 210 68 220 67.5 230 67.7 240 67.8 250 67.6 260 67.8 270 67.7 280 67.7 290 67.4 300 67 310 66.9 320 66.8 330 67.3 340 66.9 350 66.7 360 66.8 370 66.9 380 67 390 66.3 400 67 410 66.4

89


change in the inlet air temperature from 135oC to 100.5oC.

time (s)

sT

(oC)

time (s)

sT

(oC) 0 96.2 420 75.4 10 91.3 430 76.2 20 87 440 75.3 30 83.3 450 76.1 40 80.2 460 75.5 50 79.6 470 75.9 60 79.1 480 75.7 70 78.5 490 75.8 80 79.1 500 75.6 90 78.1 510 75.5 100 78.1 520 76.2 110 78.4 530 75.6 120 77.9 540 75.6 130 78.2 550 75.6 140 78.4 560 75.7 150 78.2 570 75.8 160 77.6 580 75.6 170 77.4 590 75.5 180 77.2 600 75.6 190 77.1 610 75.6 200 76.7 620 76.1 210 76.5 630 75.5 220 77 640 75.9 230 76.9 240 77 250 76.6 260 76.6 270 76.6 280 76.5 290 75.8 300 76.1 310 76 320 76.1 330 76.2 340 76.3 350 76.4 360 75.6 370 76.4 380 75.9 390 76.1 400 76.4 410 75.4

90

APPENDIX B

DISTILLED WATER EVAPORATION

B.1 Evaporation Data Table B.1.1 Data for the water evaporation under change in sample point

temperature from 97oC to 56oC after step change in the inlet air temperature from

135oC to 66oC ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) w-dW

(g) eR

(g/s)

0 133.0 94.7 999.72 0.027930

600 132.6 95.6 982.96 0.030780

1200 133.0 96.8 964.49 0.032530

1800 136.3 96.6 944.97 0.033580

2400 133.3 97.8 924.82 0.034220

3000 134.5 97.5 904.29 0.034280

3600 134.3 97.2 883.72 0.034200

4200 134.5 97.5 863.20 0.034570

4800 70.5 62.9 842.46 0.028030

5400 69.7 59.7 825.64 0.022280

6000 67.5 58.6 812.27 0.019100

6600 67.5 58.9 800.81 0.017350

7200 68.2 57.9 790.40 0.016420

7800 68.7 58.7 780.55 0.015920

8400 68.9 59.0 771.00 0.015570

9000 68.5 58.1 761.66 0.015300

9600 68.3 58.5 752.48 0.015220

10200 68.1 58.4 743.35 0.015183

10800 68.4 58.8 734.24

91

B.2 Dynamic Behavior Analysis of the Distilled Water Evaporation Table B.2.1 Psychometric data for air and parameters used in the dynamic behavior

of the distilled water evaporation.

ambT (oC) 21

amb-wbT (oC) 13.5

sT (oC) 56 97

s-wbT (oC) 24.5 34

s-aveT (oC) 39.8 65

H (g/g) 0.0054 0.0059

%RH 6.2 1.0

PrN 0.70 0.70

ReN 412 359

NuN 11.94 11.12

airh (W/m2.K) 1.80 1.80

ck (W/m.K) 50.2 50.2

cx∆ (m) 0.001 0.001

Bi -5103.59 ⋅ -5103.59 ⋅

amb-wbT : Wet-bulb temperature of the ambient air.

cx∆ : Thickness of the metal cup.

92

Table B.2.2 Change in the evaporation rate data found according to Equation 1.40

under change in sample point temperature from 97oC to 56oC after step change in the

inlet air temperature from 135oC to 66oC at 0.04m/s air velocity.

Ti-i

(oC) Ti-f (oC)

Ts-i

(oC) Ts-f (oC)

eR∆

(g/s) 135 66 97 56 ⋅− 37.3 10-3

Table B.2.3 Value of the constants found by nonlinear regression analysis for the

dynamic behavior of distilled water evaporation under change in sample point

temperature from 97oC to 56oC after step change in the inlet air temperature from

135oC to 66oC at 0.04m/s air velocity.

sT∆ (oC) -41

1K (g/s. oC) ⋅1.20- 10-4

2K (g/s. oC) ⋅17.1 10-4

K (g/s. oC) ⋅00.5 10-4

1τ (min) 8.11

2τ (min) 14.68

3τ (min) -7.65

Absolute dif. ⋅3.07- 10-9

93

APPENDIX C

APPLE SLAB DRYING UNDER CONSTANT EXTERNAL CONTIONTS

C.1 Drying Data Table C.1.1 Drying data for apple slab at 55oC (%RH = 4.2, av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) ei

ea

XX

XX

−

−

0 63.3 55.1 67.92 59.52 87.63 7.09 0.000107 0.000

600 64.8 54.9 67.38 58.98 87.53 7.02 0.000161 -0.009

1200 64.1 55.2 66.57 58.17 87.38 6.93 0.000165 -0.023

1800 64.2 54.8 65.74 57.34 87.22 6.83 0.000165 -0.038

2400 64.6 55.1 64.91 56.51 87.06 6.73 0.000161 -0.052

3000 63.3 55.0 64.10 55.70 86.90 6.63 0.000157 -0.067

3600 64.1 54.9 63.31 54.91 86.73 6.54 0.000155 -0.081

4200 63.9 55.2 62.53 54.13 86.57 6.44 0.000145 -0.096

4800 63.6 54.9 61.80 53.40 86.41 6.36 0.000141 -0.109

5400 64.0 55.2 61.09 52.69 86.25 6.27 0.000139 -0.123

6000 63.8 55.2 60.39 51.99 86.09 6.19 0.000133 -0.136

6600 63.3 55.0 59.72 51.32 85.93 6.11 0.000133 -0.149

7200 63.6 55.3 59.05 50.65 85.77 6.03 0.000125 -0.163

7800 63.7 55.0 58.42 50.02 85.62 5.95 0.000117 -0.175

8400 64.0 55.2 57.83 49.43 85.47 5.88 0.000115 -0.187

9000 63.8 55.1 57.25 48.85 85.33 5.82 0.000113 -0.199

9600 63.2 54.9 56.68 48.28 85.18 5.75 0.000103 -0.211

10200 64.1 55.2 56.16 47.76 85.04 5.69 0.000107 -0.222

10800 63.2 55.2 55.62 47.22 84.90 5.62 0.000103 -0.233

11400 63.5 54.9 55.10 46.70 84.75 5.56 0.000095 -0.245

12000 63.9 55.2 54.62 46.22 84.62 5.50 0.000093 -0.255

12600 63.1 54.9 54.15 45.75 84.49 5.45 0.000091 -0.265

13200 63.9 55.0 53.69 45.29 84.35 5.39 0.000093 -0.276

13800 64.0 55.3 53.22 44.82 84.22 5.34 0.000093 -0.286

14400 64.0 55.0 52.75 44.35 84.08 5.28 0.000083 -0.297

94

Table C.1.1 (continued)

15000 63.9 55.2 52.33 43.93 83.95 5.23 0.000083 -0.306

15600 63.0 54.9 51.91 43.51 83.82 5.18 0.000087 -0.316

16200 64.1 55.3 51.47 43.07 83.68 5.13 0.000083 -0.326

16800 64.1 55.1 51.05 42.65 83.55 5.08 0.000081 -0.336

17400 63.1 55.0 50.64 42.24 83.41 5.03 0.000079 -0.346

18000 63.1 55.0 50.24 41.84 83.28 4.98 0.000081 -0.356

18600 63.2 54.7 49.83 41.43 83.14 4.93 0.000075 -0.365

19200 63.9 54.8 49.45 41.05 83.01 4.89 0.000079 -0.375

19800 63.9 55.2 49.05 40.65 82.87 4.84 0.000073 -0.385

20400 63.5 55.0 48.68 40.28 82.74 4.80 0.000073 -0.394

21000 63.4 55.4 48.31 39.91 82.61 4.75 0.000073 -0.403

21600 64.1 55.3 47.94 39.54 82.48 4.71 0.000073 -0.413

22200 64.2 55.0 47.57 39.17 82.34 4.66 0.000071 -0.422

22800 63.8 55.4 47.21 38.81 82.21 4.62 0.000075 -0.432

23400 63.3 55.3 46.83 38.43 82.06 4.58 0.000069 -0.441

24000 63.0 54.5 46.48 38.08 81.93 4.53 0.000069 -0.451

24600 63.2 55.0 46.13 37.73 81.79 4.49 0.000069 -0.460

25200 63.0 55.1 45.78 37.38 81.65 4.45 0.000071 -0.469

25800 63.1 55.2 45.42 37.02 81.51 4.41 0.000065 -0.479

26400 62.5 55.4 45.09 36.69 81.37 4.37 0.000073 -0.488

27000 63.2 55.1 44.72 36.32 81.22 4.32 0.000065 -0.499

27600 64.3 55.1 44.39 35.99 81.08 4.28 0.000067 -0.508

28200 63.8 55.4 44.05 35.65 80.93 4.24 0.000069 -0.517

28800 62.8 55.0 43.70 35.30 80.78 4.20 0.000065 -0.527

29400 63.4 55.2 43.37 34.97 80.63 4.16 0.000069 -0.537

30000 64.0 55.3 43.02 34.62 80.47 4.12 0.000067 -0.547

30600 62.8 55.2 42.68 34.28 80.32 4.08 0.000062 -0.557

31200 63.9 55.3 42.37 33.97 80.17 4.04 0.000063 -0.566

31800 63.0 55.0 42.05 33.65 80.02 4.01 0.000062 -0.576

32400 62.8 54.8 41.74 33.34 79.88 3.97 0.000065 -0.585

33000 63.1 55.1 41.41 33.01 79.72 3.93 0.000063 -0.595

33600 63.0 55.1 41.09 32.69 79.56 3.89 0.000063 -0.605

34200 63.8 55.4 40.77 32.37 79.40 3.85 0.000063 -0.615

34800 63.4 55.1 40.45 32.05 79.23 3.82 0.000062 -0.625

36000 62.8 54.8 39.82 31.42 78.91 3.74 0.000065 -0.645

36600 64.2 55.2 39.49 31.09 78.73 3.70 0.000063 -0.656

37200 63.8 55.5 39.17 30.77 78.56 3.66 0.000062 -0.667

37800 64.1 55.1 38.86 30.46 78.38 3.63 0.000063 -0.677

95


38400 63.2 55.2 38.54 30.14 78.20 3.59 0.000062 -0.688

39000 62.7 54.8 38.23 29.83 78.03 3.55 0.000062 -0.698

39600 63.4 54.9 37.92 29.52 77.85 3.51 0.000060 -0.709

40200 64.5 55.0 37.62 29.22 77.67 3.48 0.000058 -0.719

40800 62.9 55.0 37.33 28.93 77.50 3.44 0.000062 -0.729

41400 63.0 55.0 37.02 28.62 77.31 3.41 0.000058 -0.740

42000 63.4 55.3 36.73 28.33 77.13 3.37 0.000060 -0.750

42600 62.7 55.1 36.43 28.03 76.94 3.34 0.000060 -0.761

43200 63.5 55.2 36.13 27.73 76.75 3.30 0.000058 -0.772

43800 64.1 55.0 35.84 27.44 76.56 3.27 0.000058 -0.783

44400 63.8 54.8 35.55 27.15 76.37 3.23 0.000060 -0.794

45000 63.2 54.6 35.25 26.85 76.17 3.20 0.000056 -0.805

45600 63.0 54.8 34.97 26.57 75.98 3.16 0.000058 -0.816

46200 64.2 54.9 34.68 26.28 75.78 3.13 0.000054 -0.827

46800 62.5 55.0 34.41 26.01 75.59 3.10 0.000058 -0.837

47400 62.8 54.7 34.12 25.72 75.38 3.06 0.000056 -0.849

48000 62.9 54.8 33.84 25.44 75.18 3.03 0.000056 -0.860

48600 63.1 54.8 33.56 25.16 74.97 3.00 0.000058 -0.871

49200 64.0 55.2 33.27 24.87 74.75 2.96 0.000056 -0.883

49800 63.5 55.2 32.99 24.59 74.54 2.93 0.000058 -0.894

50400 63.7 55.1 32.70 24.30 74.31 2.89 0.000054 -0.906

51000 63.2 55.3 32.43 24.03 74.10 2.86 0.000056 -0.918

51600 63.5 55.0 32.15 23.75 73.87 2.83 0.000058 -0.930

52200 62.9 55.2 31.86 23.46 73.63 2.79 0.000056 -0.942

52800 64.2 54.6 31.58 23.18 73.40 2.76 0.000058 -0.954

53400 64.1 54.8 31.29 22.89 73.15 2.73 0.000058 -0.967

54000 63.8 54.7 31.00 22.60 72.90 2.69 -0.980

96

Table C.1.2 Drying data for apple slab at 65oC (%RH = 4.5, av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) ei

ea

XX

XX

−

−

0 74.9 65.2 66.37 57.63 86.83 6.59 0.000149 0.000

600 75.0 65.4 65.59 56.85 86.67 6.50 0.000189 -0.014

1200 74.7 64.6 64.60 55.86 86.47 6.39 0.000195 -0.031

1800 76.3 65.2 63.58 54.84 86.25 6.27 0.000196 -0.050

2400 74.2 64.8 62.55 53.81 86.03 6.16 0.000189 -0.069

3000 75.0 65.0 61.56 52.82 85.80 6.04 0.000181 -0.088

3600 75.4 65.2 60.61 51.87 85.58 5.93 0.000172 -0.106

4200 74.1 65.2 59.71 50.97 85.36 5.83 0.000170 -0.123

4800 74.3 64.5 58.82 50.08 85.14 5.73 0.000162 -0.141

5400 73.9 65.0 57.97 49.23 84.92 5.63 0.000158 -0.158

6000 74.4 64.8 57.14 48.40 84.70 5.54 0.000149 -0.175

6600 75.0 64.6 56.36 47.62 84.49 5.45 0.000145 -0.192

7200 75.9 65.2 55.60 46.86 84.28 5.36 0.000141 -0.208

7800 75.0 65.0 54.86 46.12 84.07 5.28 0.000139 -0.224

8400 73.4 64.6 54.13 45.39 83.85 5.19 0.000133 -0.240

9000 74.2 65.0 53.43 44.69 83.64 5.11 0.000133 -0.256

9600 73.4 64.9 52.73 43.99 83.42 5.03 0.000124 -0.272

10200 73.8 64.3 52.08 43.34 83.22 4.96 0.000122 -0.287

10800 74.0 64.9 51.44 42.70 83.01 4.89 0.000120 -0.302

11400 74.2 65.2 50.81 42.07 82.80 4.81 0.000118 -0.317

12000 74.0 65.2 50.19 41.45 82.59 4.74 0.000113 -0.331

12600 75.2 65.4 49.60 40.86 82.38 4.68 0.000114 -0.346

13200 74.1 64.4 49.00 40.26 82.16 4.61 0.000113 -0.361

13800 74.0 64.5 48.41 39.67 81.95 4.54 0.000109 -0.376

14400 74.0 64.8 47.84 39.10 81.73 4.47 0.000105 -0.390

15000 74.9 65.2 47.29 38.55 81.52 4.41 0.000101 -0.405

15600 75.1 65.1 46.76 38.02 81.31 4.35 0.000103 -0.418

16200 75.3 65.0 46.22 37.48 81.09 4.29 0.000101 -0.433

16800 74.9 65.5 45.69 36.95 80.87 4.23 0.000101 -0.447

17400 74.9 65.3 45.16 36.42 80.65 4.17 0.000095 -0.462

18000 74.7 65.1 44.66 35.92 80.43 4.11 0.000097 -0.476

18600 75.1 64.8 44.15 35.41 80.20 4.05 0.000093 -0.490

19200 74.9 64.9 43.66 34.92 79.98 4.00 0.000095 -0.504

19800 75.1 65.5 43.16 34.42 79.75 3.94 0.000092 -0.519

20400 73.4 64.3 42.68 33.94 79.52 3.88 0.000088 -0.533

21000 73.0 64.6 42.22 33.48 79.30 3.83 0.000090 -0.547

97


21600 74.5 64.6 41.75 33.01 79.07 3.78 0.000084 -0.561

22200 74.0 64.9 41.31 32.57 78.84 3.73 0.000084 -0.574

22800 75.1 65.0 40.87 32.13 78.62 3.68 0.000084 -0.588

23400 74.8 64.8 40.43 31.69 78.38 3.63 0.000082 -0.602

24000 75.0 65.0 40.00 31.26 78.15 3.58 0.000082 -0.616

24600 74.9 65.0 39.57 30.83 77.91 3.53 0.000082 -0.630

25200 73.9 64.9 39.14 30.40 77.67 3.48 0.000084 -0.644

25800 75.3 65.6 38.70 29.96 77.42 3.43 0.000080 -0.659

26400 74.5 65.1 38.28 29.54 77.17 3.38 0.000078 -0.673

27000 73.7 64.2 37.87 29.13 76.92 3.33 0.000080 -0.687

27600 74.7 65.2 37.45 28.71 76.66 3.28 0.000074 -0.702

28200 74.4 65.2 37.06 28.32 76.42 3.24 0.000076 -0.716

28800 75.5 65.2 36.66 27.92 76.16 3.19 0.000076 -0.730

29400 75.8 65.5 36.26 27.52 75.90 3.15 0.000074 -0.744

30000 74.4 65.1 35.87 27.13 75.63 3.10 0.000076 -0.759

30600 75.2 64.5 35.47 26.73 75.36 3.06 0.000072 -0.774

31200 75.0 65.7 35.09 26.35 75.09 3.01 0.000071 -0.788

31800 74.7 65.1 34.72 25.98 74.83 2.97 0.000071 -0.803

32400 75.3 65.1 34.35 25.61 74.56 2.93 0.000072 -0.817

33000 73.7 65.1 33.97 25.23 74.27 2.89 0.000071 -0.832

33600 74.3 64.9 33.60 24.86 73.99 2.84 0.000069 -0.847

34200 74.7 65.2 33.24 24.50 73.71 2.80 0.000071 -0.862

34800 74.0 65.0 32.87 24.13 73.41 2.76 0.000071 -0.877

35400 74.1 64.7 32.50 23.76 73.11 2.72 0.000065 -0.893

36000 74.0 65.0 32.16 23.42 72.82 2.68 0.000065 -0.908

36600 73.5 64.5 31.82 23.08 72.53 2.64 0.000065 -0.922

37200 73.2 64.4 31.48 22.74 72.24 2.60 0.000067 -0.937

37800 75.6 65.1 31.13 22.39 71.92 2.56 0.000063 -0.953

38400 75.4 65.2 30.80 22.06 71.62 2.52 0.000063 -0.968

39000 75.0 65.6 30.47 21.73 71.32 2.49 0.000065 -0.983

39600 75.4 64.9 30.13 21.39 70.99 2.45 0.000065 -0.999

40200 75.0 64.8 29.79 21.05 70.66 2.41 -1.016

98


time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) ei

ea

XX

XX

−

−

0 87.2 74.6 69.08 59.74 86.48 6.40 0.000132 0.000 600 87.5 74.8 68.34 59.00 86.34 6.32 0.000189 -0.013 1200 87.9 75.3 67.28 57.94 86.12 6.21 0.000214 -0.031 1800 88.8 75.2 66.08 56.74 85.87 6.08 0.000221 -0.052 2400 87.9 74.8 64.84 55.50 85.60 5.94 0.000223 -0.074 3000 87.4 75.4 63.59 54.25 85.32 5.81 0.000211 -0.097 3600 87.5 75.6 62.41 53.07 85.04 5.68 0.000205 -0.119 4200 87.7 75 61.26 51.92 84.76 5.56 0.000198 -0.141 4800 88.4 75.7 60.15 50.81 84.48 5.44 0.000193 -0.162 5400 88.5 74.6 59.07 49.73 84.19 5.33 0.000184 -0.184 6000 88 75.3 58.04 48.70 83.91 5.22 0.000179 -0.205 6600 88.3 75.4 57.04 47.70 83.63 5.11 0.000177 -0.226 7200 89.1 75.5 56.05 46.71 83.34 5.00 0.000162 -0.247 7800 87.6 75.3 55.14 45.80 83.07 4.91 0.000162 -0.267 8400 87.8 75 54.23 44.89 82.78 4.81 0.000152 -0.287 9000 88.4 74.8 53.38 44.04 82.51 4.72 0.000157 -0.306 9600 88.2 74.8 52.50 43.16 82.22 4.62 0.000145 -0.326 10200 89.3 75.2 51.69 42.35 81.94 4.54 0.000139 -0.345 10800 88.4 75 50.91 41.57 81.66 4.45 0.000137 -0.364 11400 87.5 75.1 50.14 40.80 81.38 4.37 0.000130 -0.383 12000 88 75.3 49.41 40.07 81.10 4.29 0.000129 -0.401 12600 86.6 75 48.69 39.35 80.83 4.22 0.000129 -0.419 13200 87.6 74.8 47.97 38.63 80.54 4.14 0.000121 -0.438 13800 87.1 75.1 47.29 37.95 80.26 4.07 0.000123 -0.456 14400 88.8 75.4 46.60 37.26 79.97 3.99 0.000120 -0.474 15000 88 75.7 45.93 36.59 79.67 3.92 0.000118 -0.492 15600 87.3 75.2 45.27 35.93 79.38 3.85 0.000121 -0.511 16200 87.7 75.3 44.59 35.25 79.06 3.78 0.000118 -0.530 16800 88.6 75.5 43.93 34.59 78.75 3.71 0.000112 -0.549 17400 88.7 75.4 43.30 33.96 78.44 3.64 0.000116 -0.567 18000 86.7 74.9 42.65 33.31 78.11 3.57 0.000114 -0.587 18600 88.7 75.1 42.01 32.67 77.78 3.50 0.000109 -0.606 19200 85.7 75 41.40 32.06 77.45 3.43 0.000111 -0.625 19800 88 74.6 40.78 31.44 77.11 3.37 0.000104 -0.645 20400 87.8 75.1 40.20 30.86 76.78 3.31 0.000104 -0.664 21000 87.3 75 39.62 30.28 76.44 3.24 0.000105 -0.683 21600 86.6 74.6 39.03 29.69 76.08 3.18 0.000098 -0.702 22200 86.9 74.4 38.48 29.14 75.74 3.12 0.000098 -0.721 22800 87.7 74.7 37.93 28.59 75.39 3.06 0.000095 -0.740 23400 87.9 74.4 37.40 28.06 75.04 3.01 0.000095 -0.759 24000 87.7 74.7 36.87 27.53 74.68 2.95 0.000100 -0.779 24600 88 75.2 36.31 26.97 74.29 2.89 0.000093 -0.799 25200 87.3 75.2 35.79 26.45 73.91 2.83 0.000089 -0.819 25800 88.7 75.2 35.29 25.95 73.54 2.78 0.000095 -0.838

99


26400 88.8 75.3 34.76 25.42 73.14 2.72 0.000086 -0.859 27000 87.8 75.4 34.28 24.94 72.76 2.67 0.000089 -0.878 27600 88.3 74.9 33.78 24.44 72.36 2.62 0.000089 -0.898 28200 89 75.5 33.28 23.94 71.95 2.56 0.000080 -0.919 28800 88.6 75.2 32.83 23.49 71.56 2.52 0.000080 -0.938 29400 89.1 75.6 32.38 23.04 71.17 2.47 0.000080 -0.958 30000 87.6 75 31.93 22.59 70.76 2.42 0.000082 -0.978 30600 88.2 74.9 31.47 22.13 70.33 2.37 0.000084 -0.999 31200 88.6 74.9 31.00 21.66 69.88 2.32 0.000080 -1.020 31800 88.1 75.4 30.55 21.21 69.44 2.27 0.000089 -1.041 32400 87.5 74.7 30.05 20.71 68.93 2.22 0.000077 -1.065 33000 86.8 74.9 29.62 20.28 68.48 2.17 0.000075 -1.087 33600 87.6 75.3 29.20 19.86 68.03 2.13 -1.108

100


time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) ei

ea

XX

XX

−

−

0 132.3 94.0 63.38 54.79 86.45 6.38 0.000266 0.000

600 131.0 94.1 62.01 53.42 86.15 6.22 0.000326 -0.025

1200 131.4 94.6 60.33 51.74 85.77 6.03 0.000338 -0.057

1800 131.0 96.3 58.59 50.00 85.35 5.82 0.000324 -0.092

2400 132.1 95.7 56.92 48.33 84.92 5.63 0.000307 -0.125

3000 131.5 96.1 55.34 46.75 84.49 5.45 0.000295 -0.159

3600 134.0 96.3 53.82 45.23 84.05 5.27 0.000305 -0.192

4200 130.1 95.0 52.25 43.66 83.57 5.09 0.000264 -0.227

4800 133.5 96.3 50.89 42.30 83.13 4.93 0.000262 -0.259

5400 133.0 96.6 49.54 40.95 82.67 4.77 0.000252 -0.291

6000 131.6 95.9 48.24 39.65 82.20 4.62 0.000247 -0.324

6600 132.7 96.3 46.97 38.38 81.72 4.47 0.000233 -0.356

7200 132.0 95.7 45.77 37.18 81.24 4.33 0.000217 -0.388

7800 131.0 96.0 44.65 36.06 80.77 4.20 0.000214 -0.418

8400 132.8 96.4 43.55 34.96 80.28 4.07 0.000204 -0.449

9000 132.3 96.3 42.50 33.91 79.80 3.95 0.000194 -0.480

9600 133.0 96.8 41.50 32.91 79.31 3.83 0.000198 -0.510

10200 131.8 95.7 40.48 31.89 78.79 3.71 0.000198 -0.541

10800 131.0 96.0 39.46 30.87 78.24 3.60 0.000186 -0.574

11400 132.4 96.2 38.50 29.91 77.70 3.48 0.000186 -0.606

12000 131.6 95.5 37.54 28.95 77.13 3.37 0.000171 -0.638

12600 132.0 96.2 36.66 28.07 76.58 3.27 0.000173 -0.669

13200 131.0 95.6 35.77 27.18 76.00 3.17 0.000169 -0.701

13800 131.0 95.4 34.90 26.31 75.40 3.06 0.000149 -0.734

14400 131.8 96.0 34.13 25.54 74.84 2.98 0.000161 -0.764

15000 131.1 96.2 33.30 24.71 74.22 2.88 0.000149 -0.797

15600 132.0 96.0 32.53 23.94 73.61 2.79 0.000153 -0.828

16200 132.4 96.4 31.74 23.15 72.95 2.70 0.000142 -0.862

16800 132.5 96.7 31.01 22.42 72.31 2.61 0.000144 -0.894

17400 132.8 96.4 30.27 21.68 71.64 2.53 0.000142 -0.927

18000 132.0 96.3 29.54 20.95 70.93 2.44 0.000130 -0.962

18600 132.5 96.5 28.87 20.28 70.26 2.36 0.000142 -0.994

19200 132.1 96.0 28.14 19.55 69.49 2.28 0.000118 -1.031

19800 132.5 96.4 27.53 18.94 68.81 2.21 -1.063

101

Table C.1.5 Equilibrium moisture contents values of the apple slab, eX .

ambT

(oC) 19.5 21 18.5 20

amb-wbT

(oC) 13 12.5 11 13

ovT

(oC) 55 65 75 96

%RH 6.93 3.85 2.29 1.12

iW

(g) 11 11 11 11 11 11 11 11 11 11 11 11

fW

(g) 1.53 1.50 1.51 1.49 1.53 1.50 1.51 1.49 1.53 1.50 1.51 1.44

s-dW

(g) 1.45 1.43 1.44 1.44 1.45 1.43 1.44 1.44 1.45 1.43 1.44 1.44

iX

(g/g bds) 6.59 6.69 6.64 6.64 6.59 6.69 6.64 6.64 6.59 6.69 6.64 6.639

eX

(g/g bds) 0.06 0.05 0.05 0.03 0.06 0.05 0.05 0.03 0.06 0.05 0.05 0.000

ave-eX

(g/g bds) 0.05 0.03 0.02 0.01

ovT : Temperature of the air in the laboratory scale batch tray dryer.

ave-eX : Average value of the equilibrium moisture content of apple sample.

102

APPENDIX D

EXPLICIT FINITE DIFFERENCE METHOD OF ANALYSIS FOR APPLE SLAB DRYING UNDER CONSTANT EXTERNAL

CONDITIONS D.1 Parameters Used in the Explicit Finite Difference Method Table D.1.1 Space and time intervals used for apple slab drying at 55oC and 65oC.

Ts (oC) 55oC 65oC

∆z (m) 0.002 0.001 0.0005 0.002 0.001 0.0005 ∆t (s) 2400 600 150 1800 300 120

ShN 8560.2 4280.1 2140 6207.1 3103.5 1551.8

FoN 0.42055 0.42055 0.42055 0.49949 0.29966 0.47945

m 11 21 41 11 21 41 n 22 90 360 22 134 335

Table D.1.2 Space and time intervals used for apple slab drying at 75oC and 96oC.

Ts (oC) 75oC 96oC

∆z (m) 0.002 0.001 0.0005 0.002 0.001 0.0005 ∆t (s) 1200 300 60 600 200 30

ShN 4700.8 2350.4 1175.2 3043.8 1521.9 760.96

FoN 0.39568 0.39568 0.31654 0.32525 0.43366 0.26020

m 11 21 41 11 21 41 n 22 90 360 22 134 335

103

Table D.2 Predicted Average Moisture Content Values, pX

Table D.2.1 Predicted data found for apple slab drying at 55oC.

node-11 node-21 node-41

time (s)

pX

(g/g bds)

time (s)

pX

(g/g bds)

time (s)

pX

(g/g bds) 2400 5.81 600 6.42 600 6.43

4800 5.27 1200 6.13 1200 6.22

7200 4.96 1800 5.97 1800 6.06

9600 4.68 2400 5.83 2400 5.92

12000 4.45 3000 5.71 3000 5.80

14400 4.24 3600 5.60 3600 5.69

16800 4.05 4200 5.50 4200 5.59

19200 3.88 4800 5.40 4800 5.50

21600 3.71 5400 5.32 5400 5.41

24000 3.55 6000 5.24 6000 5.33

26400 3.41 6600 5.16 6600 5.25

28800 3.27 7200 5.08 7200 5.18

31200 3.13 7800 5.01 7800 5.11

33600 3.01 8400 4.95 8400 5.04

36000 2.88 9000 4.88 9000 4.97

38400 2.77 9600 4.82 9600 4.91

40800 2.65 10200 4.76 10200 4.85

43200 2.55 10800 4.70 10800 4.79

45600 2.44 11400 4.64 11400 4.73

48000 2.35 12000 4.58 12000 4.67

50400 2.25 12600 4.53 12600 4.62

52800 2.16 13200 4.48 13200 4.56

13800 4.42 13800 4.51

14400 4.37 14400 4.46

15000 4.32 15000 4.41

15600 4.27 15600 4.36

16200 4.22 16200 4.31

16800 4.18 16800 4.26

17400 4.13 17400 4.21

18000 4.09 18000 4.17

18600 4.04 18600 4.12

19200 4.00 19200 4.08

19800 3.95 19800 4.03

20400 3.91 20400 3.99

104

Table D.2.1 (continued)

21000 3.87 21000 3.95

21600 3.83 21600 3.90

22200 3.79 22200 3.86

22800 3.75 22800 3.82

23400 3.71 23400 3.78

24000 3.67 24000 3.74

24600 3.63 24600 3.70

25200 3.59 25200 3.66

25800 3.55 25800 3.63

26400 3.52 26400 3.59

27000 3.48 27000 3.55

27600 3.44 27600 3.51

28200 3.41 28200 3.48

28800 3.37 28800 3.44

29400 3.34 29400 3.41

30000 3.30 30000 3.37

30600 3.27 30600 3.34

31200 3.24 31200 3.30

31800 3.20 31800 3.27

32400 3.17 32400 3.23

33000 3.14 33000 3.20

33600 3.10 33600 3.17

34200 3.07 34200 3.14

34800 3.04 34800 3.10

35400 3.01 35400 3.07

36000 2.98 36000 3.04

36600 2.95 36600 3.01

37200 2.92 37200 2.98

37800 2.89 37800 2.95

38400 2.86 38400 2.92

39000 2.83 39000 2.89

39600 2.80 39600 2.86

40200 2.77 40200 2.83

40800 2.74 40800 2.80

41400 2.72 41400 2.77

42000 2.69 42000 2.74

42600 2.66 42600 2.72

43200 2.63 43200 2.69

43800 2.61 43800 2.66

44400 2.58 44400 2.63

105


45000 2.56 45000 2.61

45600 2.53 45600 2.58

46200 2.50 46200 2.56

46800 2.48 46800 2.53

47400 2.45 47400 2.50

48000 2.43 48000 2.48

48600 2.40 48600 2.45

49200 2.38 49200 2.43

49800 2.35 49800 2.40

50400 2.33 50400 2.38

51000 2.31 51000 2.36

51600 2.28 51600 2.33

52200 2.26 52200 2.31

52800 2.24 52800 2.28

53400 2.22 53400 2.26

54000 2.19 54000 2.24

106



time (s)

pX

(g/g bds)

time (s)

pX

(g/g bds)

time (s)

pX

(g/g bds) 1800 5.40 600 5.78 600 5.90

3600 4.86 1200 5.54 1200 5.66

5400 4.57 1800 5.36 1800 5.48

7200 4.30 2400 5.21 2400 5.33

9000 4.08 3000 5.08 3000 5.20

10800 3.87 3600 4.96 3600 5.08

12600 3.69 4200 4.85 4200 4.96

14400 3.52 4800 4.75 4800 4.86

16200 3.36 5400 4.66 5400 4.77

18000 3.21 6000 4.57 6000 4.67

19800 3.07 6600 4.48 6600 4.59

21600 2.93 7200 4.40 7200 4.50

23400 2.81 7800 4.33 7800 4.43

25200 2.68 8400 4.25 8400 4.35

27000 2.57 9000 4.18 9000 4.28

28800 2.45 9600 4.11 9600 4.20

30600 2.35 10200 4.04 10200 4.13

32400 2.25 10800 3.98 10800 4.07

34200 2.15 11400 3.91 11400 4.00

36000 2.06 12000 3.85 12000 3.94

37800 1.97 12600 3.79 12600 3.88

39600 1.88 13200 3.73 13200 3.82

13800 3.67 13800 3.76

14400 3.62 14400 3.70

15000 3.56 15000 3.64

15600 3.51 15600 3.59

16200 3.45 16200 3.53

16800 3.40 16800 3.48

17400 3.35 17400 3.43

18000 3.30 18000 3.38

18600 3.25 18600 3.33

19200 3.20 19200 3.28

19800 3.16 19800 3.23

20400 3.11 20400 3.18

21000 3.06 21000 3.13

21600 3.02 21600 3.09

22200 2.97 22200 3.04

107


22800 2.93 22800 3.00

23400 2.89 23400 2.95

24000 2.85 24000 2.91

24600 2.80 24600 2.87

25200 2.76 25200 2.83

25800 2.72 25800 2.79

26400 2.68 26400 2.74

27000 2.64 27000 2.70

27600 2.60 27600 2.67

28200 2.57 28200 2.63

28800 2.53 28800 2.59

29400 2.49 29400 2.55

30000 2.46 30000 2.51

30600 2.42 30600 2.48

31200 2.39 31200 2.44

31800 2.35 31800 2.41

32400 2.32 32400 2.37

33000 2.28 33000 2.34

33600 2.25 33600 2.30

34200 2.22 34200 2.27

34800 2.19 34800 2.24

35400 2.15 35400 2.20

36000 2.12 36000 2.17

36600 2.09 36600 2.14

37200 2.06 37200 2.11

37800 2.03 37800 2.08

38400 2.00 38400 2.05

39000 1.97 39000 2.02

39600 1.94 39600 1.99

108



time (s)

pX

(g/g bds)

time (s)

pX

(g/g bds)

time (s)

pX

(g/g bds) 1200 5.24 600 5.55 600 5.63

2400 4.78 1200 5.28 1200 5.37

3600 4.50 1800 5.08 1800 5.17

4800 4.26 2400 4.91 2400 5.00

6000 4.06 3000 4.76 3000 4.85

7200 3.88 3600 4.63 3600 4.72

8400 3.71 4200 4.51 4200 4.60

9600 3.56 4800 4.39 4800 4.48

10800 3.41 5400 4.29 5400 4.37

12000 3.27 6000 4.19 6000 4.27

13200 3.14 6600 4.09 6600 4.18

14400 3.02 7200 4.00 7200 4.08

15600 2.90 7800 3.92 7800 3.99

16800 2.79 8400 3.83 8400 3.91

18000 2.68 9000 3.75 9000 3.83

19200 2.58 9600 3.67 9600 3.75

20400 2.48 10200 3.60 10200 3.67

21600 2.38 10800 3.52 10800 3.60

22800 2.29 11400 3.45 11400 3.52

24000 2.20 12000 3.38 12000 3.45

25200 2.12 12600 3.32 12600 3.38

26400 2.04 13200 3.25 13200 3.32

27600 1.96 13800 3.19 13800 3.25

28800 1.88 14400 3.12 14400 3.19

30000 1.81 15000 3.06 15000 3.13

31200 1.74 15600 3.00 15600 3.07

32400 1.68 16200 2.94 16200 3.01

33600 1.61 16800 2.89 16800 2.95

17400 2.83 17400 2.89

18000 2.78 18000 2.83

18600 2.72 18600 2.78

19200 2.67 19200 2.73

19800 2.62 19800 2.67

20400 2.57 20400 2.62

21000 2.52 21000 2.57

21600 2.47 21600 2.52

22200 2.42 22200 2.47

109


22800 2.37 22800 2.43

23400 2.33 23400 2.38

24000 2.28 24000 2.33

24600 2.24 24600 2.29

25200 2.20 25200 2.24

25800 2.16 25800 2.20

26400 2.11 26400 2.16

27000 2.07 27000 2.12

27600 2.03 27600 2.08

28200 1.99 28200 2.04

28800 1.96 28800 2.00

29400 1.92 29400 1.96

30000 1.88 30000 1.92

30600 1.85 30600 1.89

31200 1.81 31200 1.85

31800 1.78 31800 1.81

32400 1.74 32400 1.78

33000 1.71 33000 1.75

33600 1.68 33600 1.71

110



time (s)

pX

(g/g bds)

time (s)

pX

(g/g bds)

time (s)

pX

(g/g bds) 600 5.22 600 5.36 600 5.43

1200 4.85 1200 5.02 1200 5.10

1800 4.59 1800 4.76 1800 4.84

2400 4.38 2400 4.54 2400 4.63

3000 4.20 3000 4.36 3000 4.44

3600 4.03 3600 4.19 3600 4.27

4200 3.88 4200 4.03 4200 4.11

4800 3.74 4800 3.89 4800 3.96

5400 3.61 5400 3.75 5400 3.83

6000 3.49 6000 3.62 6000 3.70

6600 3.37 6600 3.50 6600 3.57

7200 3.26 7200 3.39 7200 3.45

7800 3.15 7800 3.28 7800 3.34

8400 3.05 8400 3.17 8400 3.23

9000 2.95 9000 3.07 9000 3.13

9600 2.85 9600 2.97 9600 3.03

10200 2.76 10200 2.87 10200 2.93

10800 2.67 10800 2.78 10800 2.84

11400 2.59 11400 2.69 11400 2.75

12000 2.50 12000 2.61 12000 2.66

12600 2.42 12600 2.52 12600 2.58

13200 2.35 13200 2.44 13200 2.49

13800 2.27 13800 2.37 13800 2.42

14400 2.20 14400 2.29 14400 2.34

15000 2.13 15000 2.22 15000 2.26

15600 2.06 15600 2.15 15600 2.19

16200 2.00 16200 2.08 16200 2.12

16800 1.93 16800 2.01 16800 2.06

17400 1.87 17400 1.95 17400 1.99

18000 1.81 18000 1.89 18000 1.93

18600 1.75 18600 1.83 18600 1.87

19200 1.70 19200 1.77 19200 1.81

19800 1.64 19800 1.72 19800 1.75

111

Table D.3 Error and Time Dependent Correction Factor Analysis Table D.3.1 Root Mean Square Error (RMSE) and Sum Square Error (SSE) values

between predicted and experimental average moisture content data with respect to

node numbers (m) for apple slab drying at 55oC and 65oC.

Ts (

oC) 55oC 65oC ∆z (m) 0.002 0.001 0.0005 0.002 0.001 0.0005

m 11 21 41 11 21 41 SSE 0.83203 0.64533 0.53525 0.73527 0.60417 0.48881

RMSE 0.91215 0.80332 0.73161 0.85748 0.77728 0.69915 Table D.3.2 Root Mean Square Error (RMSE) and Sum Square Error (SSE) values

between predicted and experimental average moisture content data with respect to

node numbers (m) for apple slab drying at 75oC and 96oC.

Ts (

oC) 75oC 96oC ∆z (m) 0.002 0.001 0.0005 0.002 0.001 0.0005

m 11 21 41 11 21 41 SSE 0.81661 0.65435 0.55530 0.92084 0.70497 0.60691

RMSE 0.90367 0.80892 0.74518 0.95960 0.83962 0.77904 Table D.3.3 Model constants found by nonlinear regression analysis according to

Equation 2.19 (for node-41).

sT (oC) 55 65 75 96

a 1.1060 1.1419 1.1259 1.1223 b 0.1204 0.1105 0.1326 0.1448

-510c ⋅ 8.5091 0.0001 0.0002 0.0004

R2 0.94 0.95 0.95 0.98

112

Table D.4 Final Values of the Model Predicted Average Moisture Content, f-pX

Table D.4.1 Predicted data found for apple slab drying at 55oC (for node-41).

time (s)

f-pX

(g/g bds)

time (s)

f-pX

(g/g bds)

time (s)

f-pX

(g/g bds) 600 7.16 24600 4.49 48600 3.00 1200 6.96 25200 4.44 49200 2.97 1800 6.81 25800 4.40 49800 2.94 2400 6.68 26400 4.36 50400 2.92 3000 6.58 27000 4.31 51000 2.89 3600 6.48 27600 4.27 51600 2.86 4200 6.39 28200 4.23 52200 2.83 4800 6.31 28800 4.19 52800 2.80 5400 6.23 29400 4.14 53400 2.77 6000 6.16 30000 4.10 54000 2.74 6600 6.09 30600 4.06 7200 6.02 31200 4.02 7800 5.95 31800 3.98 8400 5.89 32400 3.94 9000 5.82 33000 3.90 9600 5.76 33600 3.86 10200 5.70 34200 3.83 10800 5.64 34800 3.79 11400 5.59 35400 3.75 12000 5.53 36000 3.71 12600 5.47 36600 3.68 13200 5.42 37200 3.64 13800 5.37 37800 3.60 14400 5.31 38400 3.57 15000 5.26 39000 3.53 15600 5.21 39600 3.50 16200 5.16 40200 3.46 16800 5.10 40800 3.43 17400 5.05 41400 3.39 18000 5.00 42000 3.36 18600 4.95 42600 3.32 19200 4.91 43200 3.29 19800 4.86 43800 3.26 20400 4.81 44400 3.22 21000 4.76 45000 3.19 21600 4.72 45600 3.16 22200 4.67 46200 3.13 22800 4.62 46800 3.10 23400 4.58 47400 3.07 24000 4.53 48000 3.04

113


time (s)

f-pX

(g/g bds)

time (s)

f-pX

(g/g bds) 600 6.78 26400 3.42 1200 6.54 27000 3.37 1800 6.37 27600 3.33 2400 6.23 28200 3.28 3000 6.10 28800 3.23 3600 5.99 29400 3.18 4200 5.88 30000 3.14 4800 5.78 30600 3.09 5400 5.69 31200 3.05 6000 5.60 31800 3.01 6600 5.51 32400 2.96 7200 5.43 33000 2.92 7800 5.35 33600 2.88 8400 5.27 34200 2.84 9000 5.19 34800 2.80 9600 5.12 35400 2.76 10200 5.04 36000 2.72 10800 4.97 36600 2.68 11400 4.90 37200 2.64 12000 4.83 37800 2.60 12600 4.76 38400 2.56 13200 4.69 39000 2.53 13800 4.63 39600 2.49 14400 4.56 40200 2.45 15000 4.50 15600 4.43 16200 4.37 16800 4.31 17400 4.25 18000 4.19 18600 4.13 19200 4.07 19800 4.01 20400 3.95 21000 3.90 21600 3.84 22200 3.79 22800 3.73 23400 3.68 24000 3.63 24600 3.57 25200 3.52 25800 3.47

114


time (s)

f-pX

(g/g bds)

time (s)

f-pX

(g/g bds) 600 6.43 27000 2.66 1200 6.21 27600 2.61 1800 6.04 28200 2.56 2400 5.90 28800 2.51 3000 5.77 29400 2.47 3600 5.65 30000 2.42 4200 5.53 30600 2.37 4800 5.43 31200 2.33 5400 5.32 31800 2.28 6000 5.22 32400 2.24 6600 5.12 33000 2.20 7200 5.02 7800 4.93 8400 4.83 9000 4.74 9600 4.65 10200 4.57 10800 4.48 11400 4.39 12000 4.31 12600 4.23 13200 4.15 13800 4.07 14400 3.99 15000 3.92 15600 3.84 16200 3.77 16800 3.70 17400 3.63 18000 3.56 18600 3.49 19200 3.42 19800 3.36 20400 3.29 21000 3.23 21600 3.17 22200 3.11 22800 3.05 23400 2.99 24000 2.93 24600 2.88 25200 2.82 25800 2.77 26400 2.72

115


time (s)

f-pX

(g/g bds) 600 6.26 1200 6.00 1800 5.80 2400 5.61 3000 5.43 3600 5.26 4200 5.10 4800 4.94 5400 4.78 6000 4.63 6600 4.49 7200 4.35 7800 4.21 8400 4.08 9000 3.95 9600 3.83 10200 3.71 10800 3.59 11400 3.48 12000 3.37 12600 3.26 13200 3.16 13800 3.06 14400 2.96 15000 2.87 15600 2.78 16200 2.69 16800 2.61 17400 2.52 18000 2.44 18600 2.37 19200 2.29 19800 2.22

116

Table D.5 Final Values of the Model Predicted Drying Rate, f-fR Table D.5.1 Predicted data found for apple slab drying at 55oC (for node-41).

time (s)

f-fR

(g/g bds.s)

time (s)

f-fR

(g/g bds.s)

time (s)

f-fR

(g/g bds.s) 600 0.000342 25200 0.000073 49800 0.000049 1200 0.000250 25800 0.000072 50400 0.000049 1800 0.000205 26400 0.000072 51000 0.000048 2400 0.000179 27000 0.000071 51600 0.000048 3000 0.000160 27600 0.000070 52200 0.000047 3600 0.000147 28200 0.000070 52800 0.000047 4200 0.000138 28800 0.000069 53400 0.000047 4800 0.000130 29400 0.000069 5400 0.000124 30000 0.000068 6000 0.000118 30600 0.000067 6600 0.000114 31200 0.000067 7200 0.000110 31800 0.000066 7800 0.000107 32400 0.000065 8400 0.000104 33000 0.000065 9000 0.000102 33600 0.000064 9600 0.000100 34200 0.000064 10200 0.000098 34800 0.000063 10800 0.000096 35400 0.000062 11400 0.000095 36000 0.000062 12000 0.000093 36600 0.000061 12600 0.000092 37200 0.000061 13200 0.000090 37800 0.000060 13800 0.000089 38400 0.000060 14400 0.000088 39000 0.000059 15000 0.000087 39600 0.000058 15600 0.000086 40200 0.000058 16200 0.000085 40800 0.000057 16800 0.000084 41400 0.000057 17400 0.000083 42000 0.000056 18000 0.000082 42600 0.000056 18600 0.000081 43200 0.000055 19200 0.000081 43800 0.000054 19800 0.000080 44400 0.000054 20400 0.000079 45000 0.000053 21000 0.000078 45600 0.000053 21600 0.000077 46200 0.000052 22200 0.000077 46800 0.000052 22800 0.000076 47400 0.000051 23400 0.000075 48000 0.000051 24000 0.000074 48600 0.000050 24600 0.000074 49200 0.000050

117


time (s)

f-fR

(g/g bds.s)

time (s)

f-fR

(g/g bds.s) 600 0.000388 27000 0.000081 1200 0.000287 27600 0.000080 1800 0.000239 28200 0.000079 2400 0.000210 28800 0.000077 3000 0.000190 29400 0.000076 3600 0.000176 30000 0.000075 4200 0.000165 30600 0.000074 4800 0.000157 31200 0.000073 5400 0.000150 31800 0.000072 6000 0.000144 32400 0.000071 6600 0.000140 33000 0.000070 7200 0.000136 33600 0.000069 7800 0.000132 34200 0.000068 8400 0.000129 34800 0.000067 9000 0.000126 35400 0.000066 9600 0.000123 36000 0.000065 10200 0.000121 36600 0.000064 10800 0.000119 37200 0.000063 11400 0.000117 37800 0.000062 12000 0.000115 38400 0.000061 12600 0.000113 39000 0.000061 13200 0.000111 39600 0.000060 13800 0.000109 14400 0.000108 15000 0.000106 15600 0.000105 16200 0.000103 16800 0.000102 17400 0.000101 18000 0.000099 18600 0.000098 19200 0.000096 19800 0.000095 20400 0.000094 21000 0.000093 21600 0.000091 22200 0.000090 22800 0.000089 23400 0.000088 24000 0.000086 24600 0.000085 25200 0.000084 25800 0.000083 26400 0.000082

118


time (s)

f-fR

(g/g bds.s)

time (s)

f-fR

(g/g bds.s) 600 0.000373 27000 0.000085 1200 0.000279 27600 0.000083 1800 0.000237 28200 0.000082 2400 0.000214 28800 0.000080 3000 0.000199 29400 0.000079 3600 0.000189 30000 0.000077 4200 0.000181 30600 0.000076 4800 0.000175 31200 0.000074 5400 0.000170 31800 0.000073 6000 0.000166 32400 0.000071 6600 0.000162 33000 0.000070 7200 0.000159 7800 0.000155 8400 0.000152 9000 0.000149 9600 0.000147 10200 0.000144 10800 0.000141 11400 0.000139 12000 0.000136 12600 0.000134 13200 0.000131 13800 0.000129 14400 0.000126 15000 0.000124 15600 0.000122 16200 0.000120 16800 0.000117 17400 0.000115 18000 0.000113 18600 0.000111 19200 0.000109 19800 0.000107 20400 0.000105 21000 0.000103 21600 0.000101 22200 0.000099 22800 0.000097 23400 0.000095 24000 0.000094 24600 0.000092 25200 0.000090 25800 0.000088 26400 0.000087

119


time (s)

f-fR

(g/g bds.s) 600 0.000433 1200 0.000346 1800 0.000313 2400 0.000295 3000 0.000283 3600 0.000274 4200 0.000265 4800 0.000257 5400 0.000249 6000 0.000242 6600 0.000235 7200 0.000227 7800 0.000220 8400 0.000214 9000 0.000207 9600 0.000201 10200 0.000195 10800 0.000189 11400 0.000183 12000 0.000177 12600 0.000171 13200 0.000166 13800 0.000161 14400 0.000156 15000 0.000151 15600 0.000146 16200 0.000142 16800 0.000137 17400 0.000133 18000 0.000129 18600 0.000125 19200 0.000121

120

APPENDIX E

APPLE SLAB DRYING UNDER CHANGE IN THE SAMPLE POINT (DRYING) TEMPERATURE

Table E.1 Drying Data Table E.1.1 Drying rate of apple slab under change in sample point temperature

from 55oC to 96oC after step change in the inlet air temperature from 66oC to 135oC

at 1800th s. ( av = 0.04m/s)

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 63.1 54.8 66.98 58.98 88.06 7.37 0.000158

600 66.4 54.6 66.22 58.22 87.92 7.28 0.000167 1200 67.7 54.6 65.42 57.42 87.77 7.18 0.000163 1800 67 54.7 64.64 56.64 87.62 7.08 0.000135 2400 137 96.5 63.99 55.99 87.50 7.00 0.000150 3000 140 96.6 63.27 55.27 87.36 6.91 0.000244 3600 134 95.7 62.10 54.10 87.12 6.76 0.000292 4200 134 95.5 60.70 52.70 86.82 6.59 0.000304 4800 140 95.7 59.24 51.24 86.50 6.41 0.000317 5400 139 96.2 57.72 49.72 86.14 6.22 0.000306 6000 138 95.4 56.25 48.25 85.78 6.03 0.000300 6600 137 95.4 54.81 46.81 85.40 5.85 0.000290 7200 138 95.8 53.42 45.42 85.02 5.68 0.000281 7800 137 96.1 52.07 44.07 84.64 5.51 0.000263 8400 134 95.4 50.81 42.81 84.26 5.35 0.000260 9000 139 96 49.56 41.56 83.86 5.20 0.000252 9600 136 96.1 48.35 40.35 83.45 5.04 0.000242 10200 139 96.3 47.19 39.19 83.05 4.90 0.000240 10800 137 96.6 46.04 38.04 82.62 4.76

121

Table E.1.2 Drying rate of apple slab under change in sample point temperature


at 10200th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 64.2 55.3 67.78 59.48 87.75 7.17 0.000102

600 63.4 55.1 67.27 58.97 87.66 7.10 0.000141 1200 63.2 55.0 66.57 58.27 87.53 7.02 0.000157 1800 63.4 54.6 65.79 57.49 87.38 6.93 0.000157 2400 63.2 54.9 65.01 56.71 87.23 6.83 0.000159 3000 64.3 55.2 64.22 55.92 87.08 6.74 0.000153 3600 63.9 55.1 63.46 55.16 86.92 6.65 0.000153 4200 63.9 54.9 62.70 54.40 86.76 6.55 0.000151 4800 63.4 55.0 61.95 53.65 86.60 6.46 0.000143 5400 63.3 55.1 61.24 52.94 86.45 6.38 0.000139 6000 63.7 55.3 60.55 52.25 86.29 6.30 0.000137 6600 63.2 54.9 59.87 51.57 86.14 6.21 0.000133 7200 63.8 55.1 59.21 50.91 85.98 6.13 0.000133 7800 63.4 54.9 58.55 50.25 85.82 6.05 0.000129 8400 63.1 54.8 57.91 49.61 85.67 5.98 0.000120 9000 63.1 55.2 57.31 49.01 85.52 5.90 0.000124 9600 64.2 54.8 56.69 48.39 85.36 5.83 0.000112 10200 62.7 55.0 56.13 47.83 85.21 5.76 0.000076 10800 126.8 96.1 55.75 47.45 85.11 5.72 0.000086 11400 127.5 96.5 55.32 47.02 85.00 5.67 0.000159 12000 126.6 96.4 54.53 46.23 84.78 5.57 0.000207 12600 125.9 96.7 53.50 45.20 84.49 5.45 0.000247 13200 128.0 95.8 52.27 43.97 84.12 5.30 0.000249 13800 126.7 95.6 51.03 42.73 83.74 5.15 0.000243 14400 127.8 95.7 49.82 41.52 83.34 5.00 0.000227 15000 125.4 96.5 48.69 40.39 82.95 4.87 0.000233 15600 127.0 96.3 47.53 39.23 82.54 4.73 0.000231 16200 129.3 96.4 46.38 38.08 82.10 4.59 0.000219 16800 127.6 96.2 45.29 36.99 81.67 4.46 0.000221 17400 125.4 95.8 44.19 35.89 81.22 4.32 0.000213 18000 126.7 95.9 43.13 34.83 80.76 4.20 0.000209 18600 125.8 95.6 42.09 33.79 80.28 4.07 0.000203 19200 126.4 95.5 41.08 32.78 79.80 3.95 0.000193 19800 127.0 95.7 40.12 31.82 79.31 3.83 0.000189 20400 125.2 95.5 39.18 30.88 78.82 3.72 0.000193 21000 125.8 95.5 38.22 29.92 78.28 3.60 0.000183 21600 126.6 95.7 37.31 29.01 77.75 3.50 0.000177 22200 126.3 95.4 36.43 28.13 77.22 3.39 0.000173 22800 127.3 95.5 35.57 27.27 76.67 3.29 0.000167 23400 126.4 95.6 34.74 26.44 76.11 3.19 0.000171 24000 125.6 95.4 33.89 25.59 75.51 3.08 0.000171

122

Table E.1.2 (continued)

24600 127.3 95.3 33.04 24.74 74.88 2.98 0.000159 25200 126.0 95.7 32.25 23.95 74.26 2.89 0.000161 25800 125.3 95.7 31.45 23.15 73.61 2.79 0.000141 26400 127.4 95.2 30.75 22.45 73.01 2.70 0.000155 27000 127.0 95.2 29.98 21.68 72.31 2.61 0.000145 27600 125.2 95.2 29.26 20.96 71.63 2.53 0.000153 28200 125.3 95.7 28.50 20.20 70.88 2.43 0.000141 28800 127.3 95.5 27.80 19.50 70.14 2.35 0.000137 29400 126.2 96.0 27.12 18.82 69.40 2.27

123



at 16200th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 69.0 55.0 67.53 58.93 87.26 6.85 0.000112

600 70.5 55.3 66.95 58.35 87.15 6.78 0.000140 1200 69.4 55.5 66.23 57.63 87.01 6.70 0.000143 1800 69.1 54.8 65.49 56.89 86.87 6.62 0.000145 2400 69.7 55.1 64.74 56.14 86.72 6.53 0.000145 3000 70.1 55.5 63.99 55.39 86.56 6.44 0.000143 3600 69.0 54.9 63.25 54.65 86.40 6.35 0.000138 4200 69.9 55.4 62.54 53.94 86.25 6.27 0.000138 4800 68.4 54.7 61.83 53.23 86.09 6.19 0.000136 5400 69.9 55.0 61.13 52.53 85.93 6.11 0.000128 6000 70.0 55.1 60.47 51.87 85.78 6.03 0.000124 6600 70.2 55.2 59.83 51.23 85.63 5.96 0.000124 7200 70.1 55.4 59.19 50.59 85.47 5.88 0.000122 7800 70.3 55.2 58.56 49.96 85.31 5.81 0.000120 8400 69.9 55.3 57.94 49.34 85.16 5.74 0.000114 9000 69.7 55.1 57.35 48.75 85.00 5.67 0.000112 9600 70.0 55.4 56.77 48.17 84.85 5.60 0.000103 10200 71.1 55.2 56.24 47.64 84.71 5.54 0.000105 10800 70.0 55.1 55.70 47.10 84.56 5.48 0.000107 11400 69.0 55.0 55.15 46.55 84.41 5.41 0.000103 12000 69.8 55.1 54.62 46.02 84.25 5.35 0.000103 12600 69.6 54.8 54.09 45.49 84.10 5.29 0.000095 13200 68.6 54.7 53.60 45.00 83.96 5.23 0.000093 13800 68.0 54.9 53.12 44.52 83.81 5.18 0.000097 14400 69.0 55.2 52.62 44.02 83.66 5.12 0.000091 15000 69.2 55.0 52.15 43.55 83.51 5.06 0.000089 15600 68.9 54.8 51.69 43.09 83.36 5.01 0.000087 16200 68.7 55.0 51.24 42.64 83.22 4.96 0.000031 16800 137.5 96.2 51.08 42.48 83.16 4.94 0.000027 17400 138.6 96.5 50.94 42.34 83.12 4.92 0.000110 18000 136.6 95.8 50.37 41.77 82.93 4.86 0.000172 18600 137.6 96.2 49.48 40.88 82.62 4.75 0.000203 19200 135.2 95.7 48.43 39.83 82.24 4.63 0.000217 19800 135.6 96.1 47.31 38.71 81.82 4.50 0.000221 20400 135.3 96.6 46.17 37.57 81.37 4.37 0.000223 21000 134.5 95.5 45.02 36.42 80.90 4.23 0.000209 21600 134.7 96.0 43.94 35.34 80.43 4.11 0.000213 22200 136.8 95.6 42.84 34.24 79.93 3.98 0.000205 22800 138.7 96.2 41.78 33.18 79.42 3.86 0.000198 23400 138.2 96.1 40.76 32.16 78.90 3.74 0.000196 24000 135.3 96.4 39.75 31.15 78.36 3.62 0.000180

124


24600 135.0 95.6 38.82 30.22 77.85 3.51 0.000186 25200 136.0 95.7 37.86 29.26 77.28 3.40 0.000180 25800 133.5 95.6 36.93 28.33 76.71 3.29 0.000169 26400 134.0 95.6 36.06 27.46 76.15 3.19 0.000165 27000 135.5 95.9 35.21 26.61 75.58 3.09 0.000163 27600 136.1 96.2 34.37 25.77 74.98 3.00

125



at 19200th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 64.7 55.1 67.97 59.52 87.57 7.04 0.000148

600 64.4 55.0 67.22 58.77 87.43 6.96 0.000146 1200 65.4 55.0 66.48 58.03 87.29 6.87 0.000162 1800 65.4 55.3 65.66 57.21 87.13 6.77 0.000158 2400 64.9 55.2 64.86 56.41 86.97 6.68 0.000156 3000 66.6 55.8 64.07 55.62 86.81 6.58 0.000146 3600 65.4 55.1 63.33 54.88 86.66 6.49 0.000144 4200 66.3 55.8 62.60 54.15 86.50 6.41 0.000136 4800 65.1 55.1 61.91 53.46 86.35 6.33 0.000138 5400 67.0 56.0 61.21 52.76 86.20 6.24 0.000126 6000 64.9 55.0 60.57 52.12 86.05 6.17 0.000128 6600 65.3 55.1 59.92 51.47 85.90 6.09 0.000120 7200 65.8 55.1 59.31 50.86 85.75 6.02 0.000114 7800 64.7 54.7 58.73 50.28 85.61 5.95 0.000122 8400 64.8 54.8 58.11 49.66 85.46 5.88 0.000112 9000 65.1 55.1 57.54 49.09 85.31 5.81 0.000110 9600 66.1 55.2 56.98 48.53 85.17 5.74 0.000105 10200 66.0 55.0 56.45 48.00 85.03 5.68 0.000103 10800 65.0 54.8 55.93 47.48 84.89 5.62 0.000103 11400 65.7 54.7 55.41 46.96 84.75 5.56 0.000091 12000 65.2 54.7 54.95 46.50 84.62 5.50 0.000099 12600 65.6 54.9 54.45 46.00 84.48 5.44 0.000091 13200 66.3 54.8 53.99 45.54 84.35 5.39 0.000093 13800 64.4 54.8 53.52 45.07 84.21 5.33 0.000087 14400 65.9 55.3 53.08 44.63 84.08 5.28 0.000089 15000 64.8 55.0 52.63 44.18 83.94 5.23 0.000085 15600 65.0 54.4 52.20 43.75 83.81 5.18 0.000091 16200 65.3 54.7 51.74 43.29 83.67 5.12 0.000081 16800 65.9 54.9 51.33 42.88 83.54 5.07 0.000085 17400 65.6 55.0 50.90 42.45 83.40 5.02 0.000083 18000 66.2 55.1 50.48 42.03 83.26 4.97 0.000081 18600 66.4 55.1 50.07 41.62 83.12 4.93 0.000081 19200 65.9 54.9 49.66 41.21 82.98 4.88 0.000016 19800 144.9 96.7 49.58 41.13 82.96 4.87 0.000020 20400 147.0 96.1 49.48 41.03 82.92 4.86 0.000101 21000 144.2 96.8 48.97 40.52 82.74 4.80 0.000156 21600 143.6 96.3 48.18 39.73 82.46 4.70 0.000187 22200 146.0 96.7 47.23 38.78 82.11 4.59 0.000197 22800 146.6 96.0 46.23 37.78 81.72 4.47 0.000201 23400 143.5 95.6 45.21 36.76 81.31 4.35 0.000207 24000 145.5 95.7 44.16 35.71 80.87 4.23 0.000195

126


24600 145.5 95.6 43.17 34.72 80.43 4.11 0.000197 25200 146.1 95.5 42.17 33.72 79.96 3.99 0.000191 25800 144.6 96.2 41.20 32.75 79.49 3.88 0.000178 26400 144.9 95.5 40.30 31.85 79.03 3.77 0.000179 27000 144.4 96.2 39.39 30.94 78.55 3.66 0.000174 27600 145.2 95.7 38.51 30.06 78.06 3.56 0.000166 28200 143.3 95.6 37.67 29.22 77.57 3.46 0.000164 28800 146.8 96.2 36.84 28.39 77.06 3.36 0.000162 29400 146.6 96.8 36.02 27.57 76.54 3.26 0.000158 30000 144.1 95.1 35.22 26.77 76.01 3.17 0.000152 30600 144.9 96.1 34.45 26.00 75.47 3.08 0.000146 31200 145.5 95.9 33.71 25.26 74.93 2.99 0.000148 31800 143.6 95.6 32.96 24.51 74.36 2.90 0.000136 32400 143.3 96.1 32.27 23.82 73.81 2.82 0.000136 33000 146.2 96.1 31.58 23.13 73.24 2.74 0.000130 33600 145.7 96.5 30.92 22.47 72.67 2.66 0.000128 34200 145.0 96.2 30.27 21.82 72.08 2.58 0.000128 34800 143.4 95.8 29.62 21.17 71.47 2.51 0.000124 35400 142.7 95.3 28.99 20.54 70.85 2.43 0.000124 36000 143.0 95.2 28.36 19.91 70.20 2.36 0.000116 36600 143.6 96.3 27.77 19.32 69.57 2.29 0.000110 37200 143.0 95.5 27.21 18.76 68.95 2.22 0.000114 37800 142.5 95.9 26.63 18.18 68.27 2.15 0.000103 38400 143.0 96.0 26.11 17.66 67.64 2.09 0.000103 39000 142.0 96.1 25.59 17.14 66.98 2.03 0.000097 39600 142.7 95.6 25.10 16.65 66.33 1.97 0.000101 40200 143.7 95.3 24.59 16.14 65.64 1.91

127


from 65oC to 96oC after step change in the inlet air temperature from 86.5oC to

135oC at 3600th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 87.1 64.7 71.60 62.44 87.21 6.82 0.000113

600 86 65.2 70.98 61.82 87.09 6.75 0.000142 1200 87.4 64.8 70.20 61.04 86.95 6.66 0.000167 1800 86.8 65.1 69.28 60.12 86.78 6.56 0.000173 2400 86.6 65.3 68.33 59.17 86.59 6.46 0.000180 3000 86.8 64.9 67.34 58.18 86.40 6.35 0.000182 3600 86.2 64.9 66.34 57.18 86.19 6.24 0.000167 4200 136 95.1 65.42 56.26 86.00 6.14 0.000180 4800 137 96.1 64.43 55.27 85.78 6.03 0.000224 5400 140 96.6 63.20 54.04 85.51 5.90 0.000260 6000 137 96.1 61.77 52.61 85.17 5.74 0.000278 6600 138 96.3 60.24 51.08 84.79 5.58 0.000273 7200 137 96.5 58.74 49.58 84.41 5.41 0.000269 7800 137 95.8 57.26 48.10 84.00 5.25 0.000260 8400 135 95.4 55.83 46.67 83.59 5.09 0.000249 9000 135 96.2 54.46 45.30 83.18 4.95 0.000240 9600 136 96.8 53.14 43.98 82.76 4.80 0.000229 10200 135 95.9 51.88 42.72 82.34 4.66 0.000224 10800 139 96 50.65 41.49 81.92 4.53 0.000218 11400 139 96 49.45 40.29 81.48 4.40 0.000211 12000 134 96.1 48.29 39.13 81.03 4.27

128



135oC at 7800th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 75.3 65.2 66.03 58.23 88.19 7.47 0.000205

600 75.0 65.0 65.07 57.27 88.01 7.34 0.000214 1200 75.6 65.2 64.07 56.27 87.83 7.21 0.000203 1800 74.0 64.6 63.12 55.32 87.64 7.09 0.000197 2400 74.9 64.4 62.20 54.40 87.46 6.97 0.000186 3000 74.4 64.9 61.33 53.53 87.28 6.86 0.000182 3600 74.2 64.8 60.48 52.68 87.10 6.75 0.000173 4200 75.5 65.2 59.67 51.87 86.93 6.65 0.000169 4800 75.1 65.0 58.88 51.08 86.75 6.55 0.000165 5400 74.8 64.8 58.11 50.31 86.58 6.45 0.000156 6000 74.4 65.1 57.38 49.58 86.41 6.36 0.000156 6600 74.7 64.8 56.65 48.85 86.23 6.26 0.000115 7200 74.8 65.0 56.11 48.31 86.10 6.19 0.000118 7800 74.3 65.1 55.56 47.76 85.96 6.12 0.000173 8400 124.6 95.7 54.75 46.95 85.75 6.02 0.000218 9000 123.6 66.3 53.73 45.93 85.48 5.89 0.000248 9600 122.7 96.0 52.57 44.77 85.16 5.74 0.000263 10200 122.0 95.9 51.34 43.54 84.81 5.58 0.000256 10800 123.1 96.4 50.14 42.34 84.44 5.43 0.000252 11400 122.2 96.1 48.96 41.16 84.07 5.28 0.000250 12000 121.9 95.8 47.79 39.99 83.68 5.13 0.000237 12600 121.9 95.5 46.68 38.88 83.29 4.98 0.000237 13200 123.2 96.3 45.57 37.77 82.88 4.84 0.000222 13800 121.2 95.6 44.53 36.73 82.48 4.71 0.000222 14400 123.4 96.1 43.49 35.69 82.06 4.58 0.000222 15000 121.7 96.2 42.45 34.65 81.63 4.44 0.000214 15600 120.3 96.1 41.45 33.65 81.18 4.31 0.000205 16200 123.6 96.3 40.49 32.69 80.74 4.19 0.000199 16800 121.6 95.8 39.56 31.76 80.28 4.07 0.000197 17400 122.3 96.4 38.64 30.84 79.81 3.95 0.000188 18000 122.5 96.6 37.76 29.96 79.34 3.84 0.000177 18600 121.6 95.9 36.93 29.13 78.88 3.73 0.000173 19200 123.0 95.7 36.12 28.32 78.41 3.63 0.000169 19800 122.4 96.1 35.33 27.53 77.92 3.53 0.000167 20400 122.3 95.6 34.55 26.75 77.42 3.43 0.000160 21000 121.7 95.7 33.80 26.00 76.92 3.33 0.000156 21600 121.2 95.3 33.07 25.27 76.41 3.24 0.000150 22200 120.1 95.8 32.37 24.57 75.90 3.15 0.000147 22800 123.1 95.7 31.68 23.88 75.38 3.06 0.000145 23400 121.1 96.2 31.00 23.20 74.84 2.97

129



135oC at 14400th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 83.0 65.1 71.78 62.55 87.14 6.78 0.000177

600 82.8 64.8 70.80 61.57 86.96 6.67 0.000193 1200 84.2 64.5 69.73 60.50 86.76 6.55 0.000200 1800 86.6 65.0 68.62 59.39 86.55 6.43 0.000198 2400 85.5 65.5 67.53 58.30 86.33 6.32 0.000187 3000 85.0 65.3 66.49 57.26 86.12 6.20 0.000183 3600 85.9 65.1 65.47 56.24 85.90 6.09 0.000180 4200 86.3 65.0 64.48 55.25 85.69 5.99 0.000173 4800 87.2 65.0 63.52 54.29 85.47 5.88 0.000169 5400 86.4 64.9 62.59 53.36 85.25 5.78 0.000162 6000 85.0 64.7 61.69 52.46 85.04 5.68 0.000158 6600 86.5 64.6 60.81 51.58 84.82 5.59 0.000150 7200 86.0 64.8 59.98 50.75 84.61 5.50 0.000144 7800 85.1 64.8 59.18 49.95 84.40 5.41 0.000138 8400 86.8 64.9 58.42 49.19 84.20 5.33 0.000130 9000 87.5 65.0 57.70 48.47 84.00 5.25 0.000125 9600 86.1 65.3 57.01 47.78 83.81 5.18 0.000122 10200 85.8 64.8 56.33 47.10 83.62 5.10 0.000119 10800 86.3 64.9 55.68 46.45 83.42 5.03 0.000117 11400 86.4 64.5 55.03 45.80 83.23 4.96 0.000114 12000 87.1 65.2 54.40 45.17 83.03 4.89 0.000110 12600 87.4 64.7 53.79 44.56 82.84 4.83 0.000108 13200 86.5 64.9 53.19 43.96 82.65 4.76 0.000107 13800 86.7 64.7 52.60 43.37 82.45 4.70 0.000103 14400 86.1 65.1 52.03 42.80 82.26 4.64 0.000076 15000 139.0 96.1 51.61 42.38 82.11 4.59 0.000065 15600 136.2 95.3 51.25 42.02 81.99 4.55 0.000128 16200 134.4 95.4 50.54 41.31 81.74 4.48 0.000166 16800 138.8 96.2 49.62 40.39 81.40 4.38 0.000179 17400 137.0 96.2 48.63 39.40 81.02 4.27 0.000188 18000 138.1 96.4 47.59 38.36 80.60 4.16 0.000186 18600 140.7 96.3 46.56 37.33 80.17 4.04 0.000175 19200 138.0 96.1 45.59 36.36 79.75 3.94 0.000172 19800 137.2 96.2 44.64 35.41 79.32 3.84 0.000173 20400 139.8 96.6 43.68 34.45 78.87 3.73 0.000166 21000 138.1 96.1 42.76 33.53 78.41 3.63 0.000168 21600 136.6 95.6 41.83 32.60 77.93 3.53 0.000161 22200 137.3 95.2 40.94 31.71 77.45 3.44 0.000155 22800 137.0 95.6 40.08 30.85 76.97 3.34 0.000155 23400 137.0 95.4 39.22 29.99 76.46 3.25 0.000153 24000 137.6 95.8 38.37 29.14 75.94 3.16

130



135oC at 18600th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 87.4 64.6 68.05 59.05 86.77 6.56 0.000109

600 86.0 65.1 67.46 58.46 86.66 6.50 0.000157 1200 84.3 65.1 66.61 57.61 86.49 6.40 0.000174 1800 84.7 63.9 65.67 56.67 86.30 6.30 0.000189 2400 85.9 65.2 64.65 55.65 86.08 6.18 0.000181 3000 86.9 65.7 63.67 54.67 85.86 6.07 0.000180 3600 86.1 65.2 62.70 53.70 85.65 5.97 0.000180 4200 87.3 64.6 61.73 52.73 85.42 5.86 0.000170 4800 86.1 65.2 60.81 51.81 85.20 5.76 0.000169 5400 85.8 65.1 59.90 50.90 84.97 5.66 0.000163 6000 86.5 65.1 59.02 50.02 84.75 5.56 0.000159 6600 85.0 64.7 58.16 49.16 84.53 5.46 0.000157 7200 85.9 64.4 57.31 48.31 84.30 5.37 0.000148 7800 87.0 64.9 56.51 47.51 84.07 5.28 0.000148 8400 88.0 65.0 55.71 46.71 83.84 5.19 0.000135 9000 87.4 65.5 54.98 45.98 83.63 5.11 0.000130 9600 85.8 64.9 54.28 45.28 83.42 5.03 0.000124 10200 86.8 64.9 53.61 44.61 83.21 4.96 0.000126 10800 86.5 65.2 52.93 43.93 83.00 4.88 0.000122 11400 86.5 65.1 52.27 43.27 82.78 4.81 0.000119 12000 85.7 65.0 51.63 42.63 82.57 4.74 0.000115 12600 84.6 65.5 51.01 42.01 82.36 4.67 0.000111 13200 87.1 64.6 50.41 41.41 82.15 4.60 0.000111 13800 87.2 65.2 49.81 40.81 81.93 4.53 0.000104 14400 85.3 65.0 49.25 40.25 81.73 4.47 0.000102 15000 86.0 65.0 48.70 39.70 81.52 4.41 0.000098 15600 87.6 65.7 48.17 39.17 81.32 4.35 0.000100 16200 85.0 65.0 47.63 38.63 81.10 4.29 0.000098 16800 86.5 64.9 47.10 38.10 80.89 4.23 0.000096 17400 86.3 65.0 46.58 37.58 80.68 4.18 0.000093 18000 85.9 64.4 46.08 37.08 80.47 4.12 0.000093 18600 87.7 65.3 45.58 36.58 80.25 4.06 0.000037 19200 137.7 96.0 45.38 36.38 80.17 4.04 0.000037 19800 139.1 96.6 45.18 36.18 80.08 4.02 0.000113 20400 137.1 95.6 44.57 35.57 79.81 3.95 0.000161 21000 138.4 95.3 43.70 34.70 79.41 3.86 0.000165 21600 139.8 96.3 42.81 33.81 78.98 3.76 0.000169 22200 140.7 96.4 41.90 32.90 78.52 3.66 0.000178 22800 138.6 96.1 40.94 31.94 78.02 3.55 0.000169 23400 139.1 96.2 40.03 31.03 77.52 3.45 0.000174 24000 138.3 96.6 39.09 30.09 76.98 3.34 0.000170

131


24600 136.3 95.7 38.17 29.17 76.42 3.24 0.000167 25200 135.6 96.0 37.27 28.27 75.85 3.14 0.000165 25800 139.4 95.2 36.38 27.38 75.26 3.04 0.000163 26400 136.7 96.1 35.50 26.50 74.65 2.94 0.000154 27000 134.8 96.2 34.67 25.67 74.04 2.85 0.000156 27600 136.6 95.0 33.83 24.83 73.40 2.76 0.000150 28200 136.9 95.4 33.02 24.02 72.74 2.67 0.000148 28800 137.0 96.0 32.22 23.22 72.07 2.58 0.000143 29400 140.0 97.2 31.45 22.45 71.38 2.49 0.000143 30000 139.0 95.1 30.68 21.68 70.66 2.41 0.000135 30600 137.4 95.3 29.95 20.95 69.95 2.33 0.000137 31200 136.4 95.2 29.21 20.21 69.19 2.25 0.000135 31800 137.6 95.3 28.48 19.48 68.40 2.16 0.000135 32400 138.5 96.1 27.75 18.75 67.57 2.08 0.000133 33000 137.8 96.0 27.03 18.03 66.70 2.00

132



135oC at 3600th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 101 74.7 69.44 59.82 86.15 6.22 0.000139

600 99.1 74.6 68.64 59.02 85.98 6.14 0.000177 1200 99.8 74.5 67.62 58.00 85.77 6.03 0.000203 1800 101 74.6 66.45 56.83 85.52 5.91 0.000210 2400 102 75.1 65.24 55.62 85.25 5.78 0.000217 3000 101 74.5 63.99 54.37 84.97 5.65 0.000208 3600 101 74.6 62.79 53.17 84.68 5.53 0.000196 4200 134 95.5 61.66 52.04 84.40 5.41 0.000230 4800 136 96.3 60.33 50.71 84.05 5.27 0.000260 5400 134 95.9 58.83 49.21 83.65 5.12 0.000275 6000 135 95.9 57.24 47.62 83.19 4.95 0.000272 6600 136 96.1 55.67 46.05 82.72 4.79 0.000263 7200 137 95.6 54.15 44.53 82.23 4.63 0.000256 7800 137 96.4 52.67 43.05 81.74 4.48 0.000246 8400 135 96.1 51.25 41.63 81.23 4.33 0.000236 9000 134 96.1 49.89 40.27 80.72 4.19 0.000225 9600 133 95.9 48.59 38.97 80.20 4.05 0.000217 10200 135 96.2 47.34 37.72 79.68 3.92 0.000206 10800 133 95.8 46.15 36.53 79.15 3.80 0.000196 11400 136 96.3 45.02 35.40 78.63 3.68 0.000192 12000 135 95.8 43.91 34.29 78.09 3.56 0.000189 12600 133 95.6 42.82 33.20 77.53 3.45

133



135oC at 6600th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 89.4 75.1 64.11 56.11 87.52 7.01 0.000164

600 88.2 75.1 63.32 55.32 87.37 6.92 0.000208 1200 89.4 75.3 62.32 54.32 87.16 6.79 0.000223 1800 88.4 75.3 61.25 53.25 86.94 6.66 0.000237 2400 90.0 75.6 60.12 52.12 86.69 6.51 0.000241 3000 89.7 75.7 58.96 50.96 86.43 6.37 0.000231 3600 88.8 75.1 57.85 49.85 86.17 6.23 0.000221 4200 89.4 75.3 56.79 48.79 85.91 6.10 0.000208 4800 88.5 75.1 55.79 47.79 85.66 5.97 0.000198 5400 89.6 74.9 54.84 46.84 85.41 5.86 0.000189 6000 89.3 75.3 53.94 45.94 85.17 5.74 0.000177 6600 88.2 74.7 53.09 45.09 84.93 5.64 0.000168 7200 122.0 96.1 52.28 44.28 84.70 5.53 0.000166 7800 122.2 96.3 51.48 43.48 84.46 5.44 0.000156 8400 123.4 96.6 50.73 42.73 84.23 5.34 0.000137 9000 122.4 96.5 50.07 42.07 84.02 5.26 0.000135 9600 122.5 96.1 49.43 41.43 83.81 5.18 0.000164 10200 120.9 95.7 48.64 40.64 83.55 5.08 0.000208 10800 121.8 96.2 47.64 39.64 83.21 4.95 0.000227 11400 120.0 95.7 46.55 38.55 82.81 4.82 0.000235 12000 121.0 96.3 45.42 37.42 82.39 4.68 0.000231 12600 121.1 95.7 44.31 36.31 81.95 4.54 0.000227 13200 120.7 95.8 43.23 35.23 81.49 4.40 0.000216 13800 122.3 96.0 42.19 34.19 81.04 4.27 0.000206 14400 123.6 96.2 41.20 33.20 80.58 4.15 0.000202 15000 121.5 96.6 40.23 32.23 80.11 4.03 0.000191 15600 120.2 95.9 39.31 31.31 79.65 3.91 0.000189 16200 121.5 96.0 38.40 30.40 79.17 3.80 0.000185 16800 120.2 95.8 37.51 29.51 78.68 3.69 0.000181 17400 122.1 95.7 36.65 28.65 78.17 3.58 0.000175 18000 121.0 95.4 35.81 27.81 77.66 3.48 0.000173 18600 121.0 95.9 34.98 26.98 77.13 3.37 0.000171 19200 123.6 95.9 34.16 26.16 76.58 3.27 0.000160 19800 121.6 96.2 33.39 25.39 76.04 3.17 0.000158 20400 121.7 96.3 32.63 24.63 75.49 3.08 0.000154 21000 122.7 96.5 31.90 23.90 74.92 2.99 0.000152 21600 122.3 96.1 31.17 23.17 74.33 2.90 0.000152 22200 122.2 96.2 30.44 22.44 73.72 2.80 0.000148 22800 121.0 95.7 29.73 21.73 73.09 2.72 0.000145 23400 120.8 96.4 29.03 21.03 72.44 2.63 0.000141 24000 121.8 96.0 28.35 20.35 71.79 2.54

134



135oC at 11400th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 106.0 74.9 71.16 61.96 87.07 6.73 0.000170

600 103.5 75.0 70.22 61.02 86.90 6.63 0.000210 1200 104.1 75.6 69.06 59.86 86.68 6.51 0.000230 1800 101.7 74.6 67.79 58.59 86.43 6.37 0.000239 2400 104.8 75.3 66.47 57.27 86.16 6.23 0.000236 3000 101.7 74.8 65.17 55.97 85.88 6.08 0.000228 3600 103.3 75.7 63.91 54.71 85.60 5.95 0.000216 4200 102.5 75.3 62.72 53.52 85.33 5.82 0.000210 4800 102.0 75.4 61.56 52.36 85.06 5.69 0.000197 5400 101.5 75.0 60.47 51.27 84.79 5.57 0.000194 6000 103.2 74.8 59.40 50.20 84.51 5.46 0.000184 6600 101.8 75.2 58.39 49.19 84.24 5.35 0.000178 7200 102.5 75.0 57.41 48.21 83.97 5.24 0.000176 7800 103.1 74.5 56.44 47.24 83.70 5.13 0.000159 8400 102.6 74.3 55.56 46.36 83.44 5.04 0.000156 9000 102.4 74.2 54.70 45.50 83.18 4.95 0.000149 9600 102.5 74.8 53.88 44.68 82.92 4.86 0.000145 10200 102.0 74.8 53.08 43.88 82.67 4.77 0.000143 10800 101.5 75.2 52.29 43.09 82.40 4.68 0.000138 11400 102.5 75.0 51.53 42.33 82.15 4.60 0.000114 12000 136 95.5 50.90 41.70 81.92 4.53 0.000111 12600 135.8 95.7 50.29 41.09 81.70 4.47 0.000147 13200 136.7 96 49.48 40.28 81.41 4.38 0.000163 13800 136.4 96.1 48.58 39.38 81.06 4.28 0.000181 14400 136.6 95.7 47.58 38.38 80.66 4.17 0.000197 15000 137.0 95.5 46.49 37.29 80.21 4.05 0.000201 15600 138.0 96.6 45.38 36.18 79.73 3.93 0.000196 16200 137.0 96.8 44.30 35.10 79.23 3.81 0.000187 16800 137.4 95.9 43.27 34.07 78.74 3.70 0.000187 17400 136.4 96.2 42.24 33.04 78.22 3.59 0.000183 18000 135.8 96.1 41.23 32.03 77.68 3.48 0.000174 18600 136.7 96.7 40.27 31.07 77.15 3.38 0.000174 19200 134.7 96.0 39.31 30.11 76.59 3.27 0.000169 19800 135.4 95.6 38.37 29.17 76.03 3.17 0.000161 20400 136.8 96.0 37.49 28.29 75.46 3.07 0.000160 21000 134.0 95.7 36.60 27.40 74.86 2.98 0.000158 21600 135.8 95.7 35.73 26.53 74.25 2.88

135



135oC at 19200th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 104.1 74.5 72.91 63.61 87.24 6.84 0.000130

600 103.4 75.0 72.18 62.88 87.21 6.76 0.000162 1200 105.5 75.0 71.28 61.98 87.04 6.66 0.000196 1800 101.2 75.1 70.19 60.89 86.83 6.55 0.000225 2400 103.0 75.6 68.93 59.63 86.58 6.41 0.000223 3000 102.8 75.2 67.69 58.39 86.33 6.28 0.000221 3600 102.0 75.4 66.45 57.15 86.06 6.15 0.000216 4200 102.7 75.2 65.25 55.95 85.79 6.02 0.000207 4800 101.8 74.6 64.10 54.80 85.53 5.89 0.000198 5400 103.3 75.7 62.99 53.69 85.26 5.77 0.000193 6000 100.8 75.0 61.92 52.62 84.99 5.66 0.000189 6600 102.7 75.1 60.86 51.56 84.72 5.54 0.000182 7200 100.4 75.3 59.85 50.55 84.44 5.44 0.000173 7800 101.9 75.4 58.88 49.58 84.17 5.33 0.000169 8400 103.2 74.6 57.94 48.64 83.90 5.23 0.000160 9000 102.0 74.9 57.05 47.75 83.63 5.13 0.000153 9600 103.2 75.5 56.19 46.89 83.36 5.04 0.000148 10200 101.9 75.0 55.37 46.07 83.10 4.95 0.000144 10800 101.7 75.5 54.56 45.26 82.83 4.87 0.000146 11400 101.9 74.9 53.75 44.45 82.55 4.78 0.000141 12000 100.9 74.5 52.96 43.66 82.27 4.70 0.000133 12600 102.2 74.4 52.22 42.92 81.99 4.62 0.000132 13200 103.0 75.1 51.49 42.19 81.71 4.54 0.000126 13800 101.1 75.2 50.78 41.48 81.44 4.46 0.000130 14400 101.5 74.9 50.06 40.76 81.14 4.38 0.000123 15000 101.6 75.1 49.37 40.07 80.86 4.31 0.000121 15600 101.3 75.3 48.70 39.40 80.56 4.24 0.000115 16200 103.1 75.8 48.06 38.76 80.28 4.17 0.000115 16800 103.4 75.3 47.41 38.11 79.98 4.10 0.000117 17400 101.5 74.9 46.76 37.46 79.67 4.03 0.000110 18000 101.4 75.2 46.14 36.84 79.37 3.96 0.000106 18600 102.6 74.9 45.55 36.25 79.06 3.90 0.000106 19200 101.2 75.1 44.95 35.65 78.75 3.83 0.000071 19800 137.2 95.9 44.56 35.26 78.53 3.79 0.000083 20400 136.8 96.8 44.10 34.80 78.27 3.74 0.000117 21000 135.7 96.8 43.44 34.14 77.90 3.67 0.000139 21600 137.9 95.6 42.67 33.37 77.46 3.59 0.000153 22200 136.1 96.3 41.81 32.51 76.95 3.50 0.000155 22800 136.0 96.4 40.95 31.65 76.41 3.40 0.000160 23400 134.6 95.9 40.06 30.76 75.83 3.31 0.000157 24000 137.0 96.7 39.18 29.88 75.23 3.21 0.000151

136


24600 133.0 96.1 38.34 29.04 74.62 3.12 0.000142 25200 135.0 95.0 37.54 28.24 74.02 3.04 0.000141 25800 136.9 96.7 36.76 27.46 73.40 2.95 0.000133 26400 135.5 95.6 36.01 26.71 72.78 2.87 0.000126 27000 134.0 95.3 35.31 26.01 72.16 2.80 0.000121 27600 136.0 96.0 34.64 25.34 71.53 2.72 0.000119 28200 136.3 95.7 33.97 24.67 70.89 2.65 0.000115 28800 136.0 96.0 33.33 24.03 70.24 2.58 0.000115 29400 135.3 96.2 32.68 23.38 69.56 2.51 0.000114 30000 137.0 96.0 32.05 22.75 68.85 2.45 0.000106 30600 137.4 96.4 31.45 22.15 68.16 2.38 0.000108 31200 134.4 96.0 30.85 21.55 67.43 2.32 0.000103 31800 133.4 95.3 30.28 20.98 66.69 2.26 0.000101 32400 135.2 96.0 29.71 20.41 65.93 2.19 0.000098 33000 135.7 95.6 29.17 19.87 65.17 2.14 0.000099 33600 134.5 95.8 28.61 19.31 64.35 2.08 0.000090 34200 133.4 95.8 28.11 18.81 63.57 2.02 0.000094 34800 134.0 95.4 27.59 18.29 62.73 1.97 0.000094 35400 134.4 95.4 27.06 17.76 61.84 1.91 0.000085 36000 133.5 96.0 26.59 17.29 60.99 1.86 0.000083 36600 136.3 95.7 26.12 16.82 60.12 1.81 0.000092 37200 134.2 95.6 25.61 16.31 59.12 1.75 0.000076 37800 135.5 96.3 25.18 15.88 58.24 1.71 0.000080 38400 134.0 96.0 24.74 15.44 57.28 1.66 0.000076 39000 134.4 95.5 24.32 15.02 56.32 1.61 0.000072 39600 134.0 96.2 23.91 14.61 55.35 1.57

137



to at 3000th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 142 96.4 69.27 59.52 85.92 6.10 0.000123

600 145 96.1 68.55 58.80 85.78 6.03 0.000256 1200 143 96.5 67.05 57.30 85.46 5.88 0.000308 1800 141 96.3 65.25 55.50 85.06 5.69 0.000328 2400 140 95.4 63.33 53.58 84.60 5.50 0.000323 3000 142 95.7 61.44 51.69 84.13 5.30 0.000335 3600 70.4 58.3 59.48 49.73 83.61 5.10 0.000313 4200 69.6 55.2 57.65 47.90 83.09 4.91 0.000244 4800 69.3 54.4 56.22 46.47 82.66 4.77 0.000193 5400 69.8 54.4 55.09 45.34 82.30 4.65 0.000156 6000 70 54.3 54.18 44.43 82.00 4.56 0.000130 6600 69.2 54.9 53.42 43.67 81.75 4.48 0.000120 7200 70.4 54.9 52.72 42.97 81.51 4.41 0.000109 7800 68.2 54.5 52.08 42.33 81.28 4.34 0.000103 8400 69.7 54.9 51.48 41.73 81.06 4.28 0.000101 9000 69.2 54.7 50.89 41.14 80.84 4.22 0.000097 9600 70.4 54.9 50.32 40.57 80.62 4.16 0.000091 10200 70.8 54.6 49.79 40.04 80.42 4.11 0.000094 10800 70.2 54.9 49.24 39.49 80.20 4.05 0.000091 11400 70.5 55 48.71 38.96 79.98 4.00 0.000089 12000 69.1 54.7 48.19 38.44 79.77 3.94 0.000091 12600 70.8 54.8 47.66 37.91 79.54 3.89 0.000085 13200 70.3 54.8 47.16 37.41 79.33 3.84 0.000084 13800 69.7 54.9 46.67 36.92 79.11 3.79 0.000084 14400 70.5 55 46.18 36.43 78.89 3.74 0.000080 15000 69.1 55 45.71 35.96 78.67 3.69 0.000077 15600 69.7 54.7 45.26 35.51 78.46 3.64 0.000077 16200 70.8 54.8 44.81 35.06 78.24 3.60 0.000079 16800 70.5 54.8 44.35 34.60 78.02 3.55 0.000075 17400 69.7 54.6 43.91 34.16 77.80 3.50 0.000075 18000 70.1 54.5 43.47 33.72 77.57 3.46 0.000074 18600 69.1 54.9 43.04 33.29 77.35 3.41 0.000072 19200 69.7 54.6 42.62 32.87 77.12 3.37

138



to at 7200th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 128.2 96.1 64.29 55.38 86.14 6.22 0.000275

600 128.3 95.6 62.82 53.91 85.82 6.05 0.000317 1200 126.4 95.7 61.12 52.21 85.42 5.86 0.000341 1800 128.8 96.2 59.30 50.39 84.98 5.66 0.000347 2400 126.4 95.8 57.44 48.53 84.49 5.45 0.000334 3000 125.4 96.0 55.66 46.75 83.99 5.25 0.000304 3600 127.7 96.0 54.03 45.12 83.51 5.06 0.000302 4200 127.7 95.5 52.42 43.51 83.00 4.88 0.000285 4800 126.1 96.2 50.89 41.98 82.49 4.71 0.000265 5400 127.8 96.3 49.48 40.57 81.99 4.55 0.000250 6000 126.3 95.9 48.14 39.23 81.49 4.40 0.000239 6600 124.9 96.2 46.86 37.95 80.99 4.26 0.000231 7200 127.7 96.0 45.63 36.72 80.47 4.12 0.000250 7800 66.6 58.9 44.29 35.38 79.88 3.97 0.000237 8400 65.3 56.6 43.02 34.11 79.29 3.83 0.000180 9000 64.3 56.4 42.06 33.15 78.82 3.72 0.000146 9600 64.2 55.4 41.28 32.37 78.42 3.63 0.000113 10200 64.8 55.4 40.68 31.77 78.10 3.57 0.000094 10800 63.9 55.3 40.17 31.26 77.82 3.51 0.000083 11400 63.8 55.3 39.73 30.82 77.57 3.46 0.000074 12000 63.9 55.0 39.33 30.42 77.35 3.41 0.000068 12600 63.7 55.3 38.97 30.06 77.14 3.37 0.000066 13200 63.8 55.4 38.62 29.71 76.93 3.33 0.000061 13800 64.2 55.0 38.29 29.38 76.73 3.30 0.000061 14400 63.5 55.1 37.96 29.05 76.53 3.26 0.000057 15000 63.4 55.0 37.66 28.75 76.34 3.23 0.000059 15600 63.5 54.9 37.34 28.43 76.14 3.19 0.000061 16200 63.4 55.0 37.01 28.10 75.93 3.15 0.000055 16800 64.0 55.4 36.72 27.81 75.73 3.12 0.000061 17400 64.8 55.4 36.39 27.48 75.51 3.08 0.000048 18000 63.2 55.0 36.13 27.22 75.34 3.05 0.000057 18600 63.6 55.2 35.82 26.91 75.13 3.02 0.000055 19200 63.7 55.1 35.53 26.62 74.92 2.99 0.000053 19800 62.9 55.0 35.25 26.34 74.72 2.96 0.000055 20400 63.0 55.1 34.95 26.04 74.51 2.92 0.000051 21000 63.4 55.1 34.68 25.77 74.31 2.89 0.000051 21600 63.9 55.2 34.41 25.50 74.11 2.86 0.000053 22200 63.7 55.1 34.13 25.22 73.90 2.83 0.000053 22800 63.5 55.1 33.85 24.94 73.68 2.80 0.000053 23400 64.1 55.3 33.57 24.66 73.46 2.77 0.000053 24000 64.3 55.4 33.29 24.38 73.23 2.74 0.000055

139


24600 65 55.7 32.99 24.08 72.99 2.70 0.000051 25200 64.7 55.6 32.72 23.81 72.77 2.67 0.000048 25800 64.2 55.4 32.46 23.55 72.55 2.64 0.000048 26400 64 55.3 32.20 23.29 72.33 2.61 0.000051 27000 64.1 55.3 31.93 23.02 72.10 2.58 0.000048 27600 64.3 55.3 31.67 22.76 71.87 2.55 0.000051 28200 64 5534. 31.40 22.49 71.63 2.52 0.000048 28800 64.4 55.5 31.14 22.23 71.39 2.50 0.000046 29400 64.2 55.1 30.90 21.99 71.16 2.47

140



to at 12600th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 138.0 96.5 66.60 57.58 86.46 6.38 0.000155

600 137.0 96.2 65.76 56.74 86.28 6.29 0.000283 1200 135.0 95.4 64.23 55.21 85.96 6.12 0.000336 1800 134.8 95.7 62.41 53.39 85.55 5.92 0.000318 2400 138.0 96.4 60.69 51.67 85.14 5.73 0.000305 3000 134.0 96.3 59.04 50.02 84.72 5.55 0.000296 3600 136.3 96.0 57.44 48.42 84.30 5.37 0.000281 4200 136.1 96.0 55.92 46.90 83.87 5.20 0.000262 4800 137.9 96.1 54.50 45.48 83.45 5.04 0.000259 5400 134.7 95.7 53.10 44.08 83.01 4.89 0.000235 6000 135.5 95.9 51.83 42.81 82.60 4.75 0.000231 6600 134.2 96.3 50.58 41.56 82.17 4.61 0.000222 7200 136.2 96.2 49.38 40.36 81.73 4.47 0.000209 7800 135.8 96.7 48.25 39.23 81.31 4.35 0.000203 8400 133.9 95.9 47.15 38.13 80.87 4.23 0.000211 9000 135.6 95.9 46.01 36.99 80.40 4.10 0.000190 9600 135.0 96.6 44.98 35.96 79.95 3.99 0.000187 10200 134.0 96.0 43.97 34.95 79.49 3.87 0.000183 10800 133.6 95.9 42.98 33.96 79.01 3.76 0.000177 11400 136.6 95.8 42.02 33.00 78.53 3.66 0.000172 12000 133.5 96.1 41.09 32.07 78.05 3.56 0.000163 12600 134.0 96.0 40.21 31.19 77.57 3.46 0.000209 13200 67.0 55.8 39.08 30.06 76.92 3.33 0.000212 13800 67.3 55.6 37.93 28.91 76.22 3.21 0.000166 14400 66.8 55.9 37.03 28.01 75.64 3.11 0.000122 15000 66.4 55.9 36.37 27.35 75.20 3.03 0.000094 15600 65.9 54.5 35.86 26.84 74.85 2.98 0.000076 16200 66.3 54.7 35.45 26.43 74.56 2.93 0.000061 16800 66.5 54.3 35.12 26.10 74.32 2.89 0.000054 17400 67.4 54.8 34.83 25.81 74.10 2.86 0.000046 18000 68.1 55.2 34.58 25.56 73.92 2.83 0.000044 18600 67.0 54.9 34.34 25.32 73.73 2.81 0.000044 19200 67.2 54.8 34.10 25.08 73.55 2.78 0.000041 19800 67.9 54.6 33.88 24.86 73.38 2.76 0.000041 20400 66.7 54.3 33.66 24.64 73.20 2.73 0.000037 21000 66.0 55.0 33.46 24.44 73.04 2.71 0.000039 21600 66.0 54.5 33.25 24.23 72.87 2.69 0.000035 22200 66.0 54.7 33.06 24.04 72.72 2.67 0.000037 22800 67.4 55.0 32.86 23.84 72.55 2.64 0.000035 23400 68.1 55.0 32.67 23.65 72.39 2.62 0.000037 24000 67.0 54.8 32.47 23.45 72.22 2.60 0.000035

141


24600 67.0 55.2 32.28 23.26 72.06 2.58 0.000035 25200 67.2 54.9 32.09 23.07 71.89 2.56 0.000033 25800 65.9 54.7 31.91 22.89 71.73 2.54 0.000035 26400 66.3 54.7 31.72 22.70 71.56 2.52 0.000033 27000 67.2 54.3 31.54 22.52 71.40 2.50

142



to at 18600th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 138.7 95.2 72.60 62.65 86.29 6.30 0.000079

600 137.8 95.3 72.13 62.18 86.21 6.25 0.000199 1200 140.7 96.3 70.94 60.99 85.97 6.13 0.000253 1800 137.1 95.9 69.43 59.48 85.67 5.98 0.000293 2400 135.1 96.5 67.68 57.73 85.30 5.80 0.000305 3000 137.5 95.8 65.86 55.91 84.89 5.62 0.000295 3600 133.4 95.4 64.10 54.15 84.48 5.44 0.000288 4200 133.2 96.1 62.38 52.43 84.05 5.27 0.000286 4800 136.0 95.5 60.67 50.72 83.60 5.10 0.000276 5400 133.6 95.3 59.02 49.07 83.14 4.93 0.000255 6000 136.1 95.7 57.50 47.55 82.70 4.78 0.000235 6600 133.0 95.4 56.10 46.15 82.26 4.64 0.000223 7200 135.0 95.6 54.77 44.82 81.83 4.50 0.000206 7800 137.0 95.4 53.54 43.59 81.42 4.38 0.000206 8400 135.5 96.1 52.31 42.36 80.98 4.26 0.000196 9000 132.7 96.5 51.14 41.19 80.54 4.14 0.000196 9600 135.4 96.3 49.97 40.02 80.09 4.02 0.000191 10200 137.3 95.3 48.83 38.88 79.62 3.91 0.000178 10800 133.2 96.5 47.77 37.82 79.17 3.80 0.000174 11400 131.2 96.4 46.73 36.78 78.71 3.70 0.000168 12000 134.3 95.6 45.73 35.78 78.24 3.60 0.000168 12600 134.0 96.4 44.73 34.78 77.76 3.50 0.000152 13200 134.3 95.6 43.82 33.87 77.29 3.40 0.000139 13800 135.2 96.5 42.99 33.04 76.86 3.32 0.000139 14400 133.3 96.1 42.16 32.21 76.40 3.24 0.000136 15000 134.4 96.4 41.35 31.40 75.94 3.16 0.000141 15600 133.1 96.0 40.51 30.56 75.44 3.07 0.000131 16200 134.6 95.4 39.73 29.78 74.96 2.99 0.000117 16800 133.3 95.8 39.03 29.08 74.51 2.92 0.000112 17400 134.8 95.8 38.36 28.41 74.06 2.86 0.000117 18000 132.8 95.6 37.66 27.71 73.58 2.78 0.000112 18600 132.9 95.2 36.99 27.04 73.10 2.72 0.000146 19200 64.7 55.0 36.12 26.17 72.45 2.63 0.000156 19800 62.8 54.7 35.19 25.24 71.72 2.54 0.000112 20400 64.0 54.9 34.52 24.57 71.18 2.47 0.000069 21000 66.6 54.6 34.11 24.16 70.83 2.43 0.000047 21600 69.7 56.1 33.83 23.88 70.59 2.40 0.000037 22200 67.1 55.0 33.61 23.66 70.40 2.38 0.000028 22800 67.1 54.6 33.44 23.49 70.25 2.36 0.000025 23400 68.4 55.1 33.29 23.34 70.11 2.35 0.000022 24000 67.8 55.3 33.16 23.21 69.99 2.33 0.000020

143


24600 69.1 55.5 33.04 23.09 69.88 2.32 0.000018 25200 69.8 55.7 32.93 22.98 69.78 2.31 0.000017 25800 68.8 54.9 32.83 22.88 69.69 2.30 0.000018 26400 68.5 55.3 32.72 22.77 69.59 2.29 0.000015 27000 67.1 54.9 32.63 22.68 69.51 2.28 0.000015 27600 68.5 55.3 32.54 22.59 69.42 2.27 0.000013 28200 69.8 55.7 32.46 22.51 69.35 2.26 0.000015 28800 68.4 54.9 32.37 22.42 69.26 2.25 0.000013 29400 67.1 55.7 32.29 22.34 69.19 2.25 0.000012 30000 68.4 54.9 32.22 22.27 69.12 2.24 0.000013 30600 68.8 55.7 32.14 22.19 69.04 2.23 0.000012 31200 68.5 54.9 32.07 22.12 68.97 2.22 0.000012 31800 68.5 55.7 32.00 22.05 68.91 2.22 0.000012 32400 67.1 54.9 31.93 21.98 68.84 2.21 0.000013 33000 69.8 55.3 31.85 21.90 68.76 2.20 0.000010 33600 68.4 55.3 31.79 21.84 68.70 2.19

144



86.5oC to at 3000th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 141 96.2 72.20 62.97 87.22 6.82 0.000143

600 142 96.5 71.41 62.18 87.04 6.74 0.000261 1200 142 96.7 69.96 60.73 86.74 6.58 0.000308 1800 140 95.4 68.26 59.03 86.38 6.40 0.000322 2400 140 95.8 66.47 57.24 85.97 6.20 0.000319 3000 141 96.2 64.71 55.48 85.55 6.01 0.000324 3600 84 65 62.92 53.69 85.09 5.82 0.000302 4200 83 64.4 61.24 52.01 84.64 5.64 0.000265 4800 83 64 59.77 50.54 84.25 5.48 0.000217 5400 83.9 64.3 58.57 49.34 83.92 5.35 0.000181 6000 84.3 64.6 57.57 48.34 83.64 5.24 0.000159 6600 84.4 64.6 56.69 47.46 83.38 5.14 0.000146 7200 86 64 55.88 46.65 83.13 5.05 0.000135 7800 86.5 64.2 55.13 45.90 82.89 4.97 0.000128 8400 88.7 64.6 54.43 45.20 82.66 4.90 0.000126 9000 88.4 65.4 53.73 44.50 82.43 4.82 0.000123 9600 87.6 64.8 53.05 43.82 82.20 4.75 0.000119 10200 88.1 64.5 52.39 43.16 81.96 4.68 0.000117 10800 86.3 64.6 51.74 42.51 81.73 4.61 0.000114 11400 85.7 64.4 51.11 41.88 81.49 4.54 0.000114 12000 85.7 64.6 50.48 41.25 81.25 4.47 0.000112 12600 88.4 64.6 49.86 40.63 81.01 4.40 0.000108 13200 87.6 65 49.26 40.03 80.77 4.34 0.000105 13800 88.1 64.5 48.68 39.45 80.53 4.27 0.000106 14400 87.5 64.5 48.10 38.87 80.28 4.21 0.000103 15000 88 65.2 47.53 38.30 80.03 4.15 0.000099 15600 87.2 65.1 46.98 37.75 79.79 4.09 0.000096 16200 87.1 65 46.45 37.22 79.54 4.03 0.000094 16800 87 65 45.93 36.70 79.30 3.98 0.000094 17400 87.2 65.3 45.41 36.18 79.05 3.92 0.000090 18000 85.7 64.9 44.91 35.68 78.81 3.87 0.000090 18600 88.5 65 44.41 35.18 78.55 3.81 0.000086 19200 88.1 65 43.93 34.70 78.31 3.76 0.000085 19800 87.5 65.4 43.47 34.24 78.06 3.71 0.000083 20400 88 65.1 43.01 33.78 77.81 3.66 0.000083 21000 87.2 64.8 42.55 33.32 77.56 3.61 0.000083 21600 87 64.6 42.09 32.86 77.30 3.56

145



86.5oC to at 6600th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 121.5 96.4 66.84 57.76 86.42 6.36 0.000163

600 122.4 96.2 65.95 56.87 86.23 6.26 0.000275 1200 121.0 95.8 64.45 55.37 85.91 6.10 0.000307 1800 122.2 96.1 62.78 53.70 85.54 5.91 0.000325 2400 122.5 95.9 61.01 51.93 85.12 5.72 0.000305 3000 120.2 95.8 59.35 50.27 84.70 5.54 0.000288 3600 121.9 96.0 57.78 48.70 84.29 5.36 0.000272 4200 119.9 96.1 56.30 47.22 83.87 5.20 0.000268 4800 120.6 96.3 54.84 45.76 83.44 5.04 0.000255 5400 122.8 96.5 53.45 44.37 83.01 4.89 0.000250 6000 119.3 96.2 52.09 43.01 82.57 4.74 0.000237 6600 120.1 95.9 50.80 41.72 82.13 4.59 0.000248 7200 74.4 66.5 49.45 40.37 81.64 4.45 0.000235 7800 74.5 64.9 48.17 39.09 81.15 4.31 0.000191 8400 72.0 64.2 47.13 38.05 80.73 4.19 0.000149 9000 73.5 65.1 46.32 37.24 80.40 4.10 0.000123 9600 74.2 65.2 45.65 36.57 80.11 4.03 0.000110 10200 74.3 65.2 45.05 35.97 79.84 3.96 0.000099 10800 74.2 65.0 44.51 35.43 79.60 3.90 0.000095 11400 74.9 65.2 43.99 34.91 79.36 3.84 0.000092 12000 74.1 64.6 43.49 34.41 79.12 3.79 0.000088 12600 74.7 65.1 43.01 33.93 78.89 3.74 0.000086 13200 75.1 65.5 42.54 33.46 78.66 3.69 0.000084 13800 75.6 64.9 42.08 33.00 78.42 3.63 0.000086 14400 74.5 64.9 41.61 32.53 78.18 3.58 0.000081 15000 74.4 65.2 41.17 32.09 77.95 3.53 0.000084 15600 75.3 65.2 40.71 31.63 77.70 3.48 0.000081 16200 74.1 65.2 40.27 31.19 77.45 3.44 0.000083 16800 74.3 64.9 39.82 30.74 77.20 3.39 0.000079 17400 75.0 65.1 39.39 30.31 76.95 3.34 0.000079 18000 74.2 65.3 38.96 29.88 76.69 3.29 0.000077 18600 73.6 65.1 38.54 29.46 76.44 3.24 0.000077 19200 73.7 64.6 38.12 29.04 76.18 3.20 0.000075 19800 74.9 65.1 37.71 28.63 75.92 3.15 0.000079 20400 75.0 65.3 37.28 28.20 75.64 3.11 0.000073 21000 75.6 66.0 36.88 27.80 75.38 3.06 0.000075 21600 75.5 65.5 36.47 27.39 75.10 3.02 0.000077 22200 74.6 64.9 36.05 26.97 74.81 2.97 0.000073 22800 73.7 64.4 35.65 26.57 74.53 2.93 0.000072 23400 75.2 65.2 35.26 26.18 74.25 2.88 0.000072 24000 74.4 64.9 34.87 25.79 73.96 2.84 0.000073

146


24600 74.0 65.0 34.47 25.39 73.66 2.80 0.000070 25200 74.6 65.5 34.09 25.01 73.36 2.75 0.000068 25800 73.2 64.5 33.72 24.64 73.07 2.71 0.000068 26400 75.0 64.9 33.35 24.27 72.77 2.67 0.000070 27000 74.0 64.9 32.97 23.89 72.46 2.63 0.000068 27600 74.5 64.8 32.60 23.52 72.15 2.59 0.000066 28200 74.0 65.2 32.24 23.16 71.84 2.55

147



86.5oC to at 12600th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 135.9 95.4 70.94 60.93 85.89 6.09 0.000080

600 139.6 96.0 70.46 60.45 85.79 6.04 0.000223 1200 137.8 96.6 69.12 59.11 85.52 5.91 0.000258 1800 135.9 96.6 67.57 57.56 85.19 5.75 0.000305 2400 134.2 96.2 65.74 55.73 84.77 5.57 0.000290 3000 136.0 95.6 64.00 53.99 84.36 5.39 0.000283 3600 133.6 95.2 62.30 52.29 83.93 5.22 0.000263 4200 132.6 95.9 60.72 50.71 83.51 5.07 0.000278 4800 133.3 95.3 59.05 49.04 83.05 4.90 0.000236 5400 134.8 95.6 57.63 47.62 82.63 4.76 0.000225 6000 134.7 96.0 56.28 46.27 82.21 4.62 0.000228 6600 136.5 96.2 54.91 44.90 81.77 4.49 0.000208 7200 132.0 96.3 53.66 43.65 81.35 4.36 0.000200 7800 134.7 95.8 52.46 42.45 80.92 4.24 0.000200 8400 133.3 95.7 51.26 41.25 80.47 4.12 0.000186 9000 132.9 96.2 50.14 40.13 80.04 4.01 0.000190 9600 134.7 96.3 49.00 38.99 79.57 3.90 0.000171 10200 133.3 95.8 47.97 37.96 79.13 3.79 0.000162 10800 132.2 96.6 47.00 36.99 78.70 3.70 0.000175 11400 133.1 95.7 45.95 35.94 78.22 3.59 0.000158 12000 133.7 95.7 45.00 34.99 77.76 3.50 0.000158 12600 133.4 96.2 44.05 34.04 77.28 3.40 0.000193 13200 82.8 65 42.89 32.88 76.66 3.28 0.000188 13800 82.0 64.6 41.76 31.75 76.03 3.17 0.000145 14400 80.1 64.3 40.89 30.88 75.52 3.08 0.000112 15000 81.7 64.4 40.22 30.21 75.11 3.02 0.000085 15600 83.1 64.7 39.71 29.70 74.79 2.97 0.000072 16200 82.0 64.2 39.28 29.27 74.52 2.92 0.000068 16800 82.7 64.7 38.87 28.86 74.25 2.88 0.000063 17400 83.0 64.3 38.49 28.48 73.99 2.85 0.000058 18000 81.6 64.4 38.14 28.13 73.75 2.81 0.000060 18600 82.9 64.3 37.78 27.77 73.50 2.77 0.000053 19200 83.6 64.8 37.46 27.45 73.28 2.74 0.000058 19800 82.5 64.6 37.11 27.10 73.03 2.71 0.000055 20400 82.0 64.2 36.78 26.77 72.78 2.67 0.000057 21000 82.7 64.7 36.44 26.43 72.53 2.64 0.000055 21600 82.9 64.3 36.11 26.10 72.28 2.61 0.000052 22200 83.1 64.7 35.80 25.79 72.04 2.58 0.000053 22800 82.9 64.7 35.48 25.47 71.79 2.54 0.000052 23400 82.5 65.1 35.17 25.16 71.54 2.51 0.000050 24000 83.6 65.3 34.87 24.86 71.29 2.48 0.000048

148


24600 83.6 65.0 34.58 24.57 71.05 2.45 0.000052 25200 83.6 64.2 34.27 24.26 70.79 2.42 0.000050 25800 82.5 64.2 33.97 23.96 70.53 2.39 0.000050 26400 81.7 64.8 33.67 23.66 70.27 2.36 0.000048 27000 81.9 64.0 33.38 23.37 70.01 2.33 0.000047 27600 82.0 64.0 33.10 23.09 69.76 2.31 0.000047 28200 82.1 64.5 32.82 22.81 69.50 2.28 0.000047 28800 82.9 64.2 32.54 22.53 69.24 2.25

149



86.5oC to at 18600th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 137.1 95.8 71.82 61.93 86.23 6.26 0.000174

600 135.6 95.6 70.79 60.90 86.03 6.16 0.000270 1200 135.9 95.8 69.19 59.30 85.71 6.00 0.000298 1800 134.7 96.0 67.42 57.53 85.33 5.82 0.000308 2400 137.0 97.1 65.59 55.70 84.92 5.63 0.000298 3000 135.5 96.2 63.82 53.93 84.50 5.45 0.000290 3600 134.6 96.5 62.10 52.21 84.07 5.28 0.000276 4200 136.5 96.7 60.46 50.57 83.64 5.11 0.000266 4800 137.9 96.4 58.88 48.99 83.20 4.95 0.000265 5400 134.7 95.6 57.31 47.42 82.74 4.79 0.000239 6000 135.3 96.7 55.89 46.00 82.30 4.65 0.000217 6600 132.1 95.3 54.60 44.71 81.89 4.52 0.000219 7200 134.6 95.7 53.30 43.41 81.44 4.39 0.000209 7800 135.8 96.4 52.06 42.17 81.00 4.26 0.000201 8400 134.0 95.6 50.87 40.98 80.56 4.14 0.000192 9000 135.3 95.4 49.73 39.84 80.11 4.03 0.000187 9600 135.2 96.1 48.62 38.73 79.66 3.92 0.000185 10200 134.4 95.5 47.52 37.63 79.19 3.80 0.000172 10800 133.5 95.6 46.50 36.61 78.73 3.70 0.000167 11400 132.0 96.0 45.51 35.62 78.27 3.60 0.000163 12000 133.5 95.0 44.54 34.65 77.80 3.50 0.000163 12600 134.0 95.3 43.57 33.68 77.30 3.41 0.000158 13200 131.6 96.2 42.63 32.74 76.80 3.31 0.000150 13800 135.2 95.9 41.74 31.85 76.31 3.22 0.000152 14400 135.4 96.2 40.84 30.95 75.78 3.13 0.000150 15000 134.5 95.5 39.95 30.06 75.24 3.04 0.000150 15600 134.0 95.1 39.06 29.17 74.68 2.95 0.000143 16200 134.0 95.0 38.21 28.32 74.12 2.86 0.000140 16800 131.0 95.8 37.38 27.49 73.54 2.78 0.000135 17400 134.8 95.4 36.58 26.69 72.96 2.70 0.000131 18000 132.2 96.0 35.80 25.91 72.37 2.62 0.000125 18600 136.0 96.1 35.06 25.17 71.79 2.54 0.000145 19200 83.1 65.1 34.20 24.31 71.08 2.46 0.000150 19800 84.0 64.0 33.31 23.42 70.31 2.37 0.000126 20400 83.1 64.1 32.56 22.67 69.63 2.29 0.000094 21000 84.6 64.6 32.00 22.11 69.09 2.24 0.000071 21600 84.1 64.9 31.58 21.69 68.68 2.19 0.000061 22200 84.7 64.9 31.22 21.33 68.32 2.16 0.000057 22800 84.5 65.2 30.88 20.99 67.97 2.12 0.000054 23400 84.4 64.4 30.56 20.67 67.64 2.09 0.000047 24000 84.4 65.0 30.28 20.39 67.34 2.06 0.000051

150


24600 83.4 64.8 29.98 20.09 67.01 2.03 0.000046 25200 84.5 65.1 29.71 19.82 66.71 2.00 0.000047 25800 84.2 64.9 29.43 19.54 66.39 1.98 0.000044 26400 84.1 65.0 29.17 19.28 66.10 1.95 0.000042 27000 84.0 64.8 28.92 19.03 65.80 1.92 0.000042 27600 84.0 64.7 28.67 18.78 65.50 1.90 0.000042 28200 84.3 64.5 28.42 18.53 65.20 1.87 0.000040 28800 84.2 64.6 28.18 18.29 64.90 1.85 0.000039 29400 84.5 64.9 27.95 18.06 64.62 1.83 0.000039 30000 84.6 65.0 27.72 17.83 64.32 1.80 0.000037 30600 84.5 65.2 27.50 17.61 64.04 1.78 0.000039 31200 84.4 64.8 27.27 17.38 63.73 1.76 0.000039 31800 84.1 64.8 27.04 17.15 63.42 1.73 0.000037 32400 84.3 64.8 26.82 16.93 63.12 1.71 0.000035 33000 84.0 65.0 26.61 16.72 62.83 1.69 0.000037 33600 84.7 65.0 26.39 16.50 62.52 1.67 0.000037 34200 84.5 65.1 26.17 16.28 62.21 1.65 0.000034 34800 84.6 64.9 25.97 16.08 61.92 1.63 0.000035 35400 84.3 65.0 25.76 15.87 61.61 1.60 0.000032 36000 84.8 64.8 25.57 15.68 61.32 1.59 0.000034 36600 83.9 65.0 25.37 15.48 61.02 1.57 0.000032 37200 84.2 64.8 25.18 15.29 60.72 1.55

151



100.5oC to at 2400th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 136.7 95.8 73.77 64.46 87.38 6.92 0.000140

600 134.8 96.3 72.99 63.68 87.24 6.84 0.000254 1200 134.8 96.1 71.57 62.26 86.99 6.69 0.000308 1800 136.6 96.8 69.85 60.54 86.67 6.50 0.000306 2400 133.1 95.7 68.14 58.83 86.34 6.32 0.000315 3000 97.3 75.5 66.38 57.07 85.97 6.13 0.000308 3600 97.9 75.0 64.66 55.35 85.60 5.95 0.000263 4200 99.0 74.3 63.19 53.88 85.27 5.79 0.000220 4800 100.8 74.7 61.96 52.65 84.97 5.66 0.000193 5400 99.6 75.0 60.88 51.57 84.71 5.54 0.000184 6000 101.8 75.4 59.85 50.54 84.44 5.43 0.000172 6600 101.9 74.9 58.89 49.58 84.19 5.33 0.000166 7200 99.1 74.3 57.96 48.65 83.94 5.23 0.000163 7800 100.2 75.0 57.05 47.74 83.68 5.13 0.000150 8400 101.6 75.1 56.21 46.90 83.44 5.04 0.000158 9000 102.2 75.2 55.33 46.02 83.17 4.94 0.000150 9600 102.8 75.4 54.49 45.18 82.91 4.85 0.000145 10200 102.5 75.0 53.68 44.37 82.66 4.77 0.000140 10800 102.5 75.2 52.90 43.59 82.40 4.68 0.000138 11400 101.9 75.1 52.13 42.82 82.14 4.60 0.000131 12000 102.6 75.2 51.40 42.09 81.89 4.52 0.000129 12600 102.3 75.2 50.68 41.37 81.63 4.44 0.000127 13200 102.1 74.8 49.97 40.66 81.37 4.37 0.000122 13800 101.8 74.7 49.29 39.98 81.11 4.29 0.000120 14400 102.2 74.8 48.62 39.31 80.85 4.22 0.000118 15000 101.9 75.0 47.96 38.65 80.59 4.15 0.000116 15600 102.6 74.7 47.31 38.00 80.32 4.08 0.000111 16200 102.3 74.6 46.69 37.38 80.06 4.02 0.000109 16800 101.0 74.8 46.08 36.77 79.80 3.95 0.000106 17400 101.8 74.9 45.49 36.18 79.53 3.89 0.000102 18000 101.7 74.8 44.92 35.61 79.27 3.82 0.000100 18600 102.2 74.9 44.36 35.05 79.01 3.76 0.000100 19200 101.8 74.7 43.80 34.49 78.74 3.70

152



100.5oC to at 6000th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 118.0 95.7 61.10 52.80 86.42 6.58 0.000268

600 121.7 96.1 59.81 51.51 86.12 6.41 0.000326 1200 120.2 96.2 58.24 49.94 85.75 6.22 0.000347 1800 121.9 96.4 56.57 48.27 85.33 6.01 0.000336 2400 121.2 96.3 54.95 46.65 84.90 5.81 0.000330 3000 119.3 95.6 53.36 45.06 84.45 5.61 0.000309 3600 118.9 95.8 51.87 43.57 84.00 5.43 0.000291 4200 120.9 96.3 50.47 42.17 83.55 5.25 0.000284 4800 120.3 95.9 49.10 40.80 83.10 5.08 0.000272 5400 120.9 96.3 47.79 39.49 82.63 4.92 0.000264 6000 118.8 96.0 46.52 38.22 82.16 4.76 0.000266 6600 89.8 76.1 45.24 36.94 81.65 4.60 0.000264 7200 87.6 75.3 43.97 35.67 81.12 4.44 0.000197 7800 88.8 75.2 43.02 34.72 80.71 4.32 0.000174 8400 88.4 74.8 42.18 33.88 80.32 4.22 0.000154 9000 88.2 75.2 41.44 33.14 79.97 4.13 0.000143 9600 89.4 75.4 40.75 32.45 79.63 4.04 0.000135 10200 88.3 74.7 40.10 31.80 79.30 3.96 0.000131 10800 88.8 75.1 39.47 31.17 78.97 3.88 0.000127 11400 88.5 75.5 38.86 30.56 78.64 3.81 0.000127 12000 87.9 74.8 38.25 29.95 78.30 3.73 0.000122 12600 88.8 75.4 37.66 29.36 77.96 3.66 0.000118 13200 87.5 75.9 37.09 28.79 77.62 3.59 0.000116 13800 88.2 75.1 36.53 28.23 77.28 3.52 0.000114 14400 88.7 75.0 35.98 27.68 76.93 3.45 0.000112 15000 87.9 75.4 35.44 27.14 76.58 3.38 0.000108 15600 88.7 75.0 34.92 26.62 76.23 3.32 0.000110 16200 87.4 74.9 34.39 26.09 75.87 3.25 0.000106 16800 87.8 75.4 33.88 25.58 75.50 3.19 0.000102 17400 88.3 75.1 33.39 25.09 75.14 3.12 0.000104 18000 88.6 75.3 32.89 24.59 74.76 3.06 0.000102 18600 88.5 75.1 32.40 24.10 74.38 3.00 0.000098 19200 88.2 75.3 31.93 23.63 74.01 2.94 0.000098 19800 87.5 75.0 31.46 23.16 73.62 2.88 0.000095 20400 87.5 74.6 31.00 22.70 73.23 2.83 0.000093 21000 88.0 75.0 30.55 22.25 72.83 2.77 0.000095 21600 87.4 74.4 30.09 21.79 72.42 2.71 0.000089 22200 87.0 74.9 29.66 21.36 72.02 2.66 0.000089 22800 88.0 74.5 29.23 20.93 71.60 2.61 0.000089 23400 88.4 75.0 28.80 20.50 71.18 2.55 0.000087 24000 88.8 75.3 28.38 20.08 70.75 2.50 0.000083

153

Table E.1.22 (continued) 24600 88.7 75.0 27.98 19.68 70.34 2.45 0.000085 25200 87.4 74.9 27.57 19.27 69.89 2.40

154



100.5oC to at 12000th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 136.3 95.6 65.75 56.89 86.52 6.42 0.000190

600 136.9 96.2 64.74 55.88 86.31 6.31 0.000290 1200 137.3 94.8 63.20 54.34 85.98 6.13 0.000322 1800 139.0 96.7 61.49 52.63 85.59 5.94 0.000327 2400 134.9 95.3 59.75 50.89 85.17 5.74 0.000327 3000 135.0 95.6 58.01 49.15 84.73 5.55 0.000318 3600 136.5 95.5 56.32 47.46 84.27 5.36 0.000303 4200 139.5 96.2 54.71 45.85 83.81 5.17 0.000292 4800 138.0 96.8 53.16 44.30 83.33 5.00 0.000278 5400 134.2 95.3 51.68 42.82 82.86 4.83 0.000275 6000 133.3 95.7 50.22 41.36 82.36 4.67 0.000250 6600 137.6 95.6 48.89 40.03 81.88 4.52 0.000250 7200 131.6 95.7 47.56 38.70 81.37 4.37 0.000229 7800 134.5 95.9 46.34 37.48 80.88 4.23 0.000233 8400 133.0 95.5 45.10 36.24 80.35 4.09 0.000211 9000 134.3 95.7 43.98 35.12 79.85 3.96 0.000205 9600 132.0 95.0 42.89 34.03 79.34 3.84 0.000203 10200 133.1 96.0 41.81 32.95 78.81 3.72 0.000194 10800 134.4 95.7 40.78 31.92 78.27 3.60 0.000188 11400 136.3 96.1 39.78 30.92 77.73 3.49 0.000181 12000 132.7 96.3 38.82 29.96 77.18 3.38 0.000209 12600 102.3 75.5 37.71 28.85 76.50 3.26 0.000188 13200 100.1 75.2 36.71 27.85 75.86 3.14 0.000150 13800 99.7 74.9 35.91 27.05 75.33 3.05 0.000132 14400 101.3 75.4 35.21 26.35 74.84 2.97 0.000109 15000 102.0 75.4 34.63 25.77 74.42 2.91 0.000103 15600 100.9 75.2 34.08 25.22 74.00 2.85 0.000098 16200 100.9 75.4 33.56 24.70 73.60 2.79 0.000092 16800 100.3 75.2 33.07 24.21 73.21 2.73 0.000083 17400 101.5 75.5 32.63 23.77 72.85 2.68 0.000087 18000 101.0 75.8 32.17 23.31 72.46 2.63 0.000087 18600 102.0 75.4 31.71 22.85 72.06 2.58 0.000085 19200 101.0 75.1 31.26 22.40 71.66 2.53 0.000085 19800 101.3 75.4 30.81 21.95 71.24 2.48 0.000077 20400 101.8 75.0 30.40 21.54 70.86 2.43 0.000081 21000 101.0 75.7 29.97 21.11 70.44 2.38 0.000079 21600 101.2 75.1 29.55 20.69 70.02 2.34 0.000079 22200 101.5 75.5 29.13 20.27 69.58 2.29 0.000081 22800 99.7 74.8 28.70 19.84 69.13 2.24 0.000077 23400 99.8 74.9 28.29 19.43 68.68 2.19 0.000075 24000 101.2 75.2 27.89 19.03 68.23 2.15 0.000077

155

Table E.1.23 (continued) 24600 101.0 75.3 27.48 18.62 67.76 2.10 0.000075 25200 101.5 75.0 27.08 18.22 67.28 2.06 0.000075 25800 101.4 74.6 26.68 17.82 66.79 2.01 0.000073 26400 101.0 74.9 26.29 17.43 66.30 1.97 0.000073 27000 99.7 75.0 25.90 17.04 65.79 1.92

156



100.5oC to at 18600th s ( av = 0.04m/s).

time (s)

iT

(oC) sT

(oC) aW

(g) w-aW

(g) a%X aX

(g/g bds) dR

(g/g bds.s) 0 136.6 96.3 65.05 56.16 86.33 6.32 0.000184

600 137.2 95.7 64.07 55.18 86.12 6.21 0.000292 1200 135.5 95.7 62.51 53.62 85.78 6.03 0.000330 1800 137.8 96.0 60.75 51.86 85.37 5.83 0.000358 2400 135.5 96.3 58.84 49.95 84.89 5.62 0.000347 3000 136.3 96.2 56.99 48.10 84.40 5.41 0.000315 3600 137.6 96.8 55.31 46.42 83.93 5.22 0.000307 4200 134.4 95.4 53.67 44.78 83.44 5.04 0.000279 4800 135.5 95.3 52.18 43.29 82.96 4.87 0.000262 5400 136.6 96.6 50.78 41.89 82.49 4.71 0.000253 6000 135.0 96.7 49.43 40.54 82.01 4.56 0.000253 6600 136.0 96.0 48.08 39.19 81.51 4.41 0.000232 7200 135.8 95.9 46.84 37.95 81.02 4.27 0.000229 7800 133.8 96.5 45.62 36.73 80.51 4.13 0.000216 8400 132.9 95.7 44.47 35.58 80.01 4.00 0.000214 9000 133.2 95.7 43.33 34.44 79.48 3.87 0.000210 9600 133.7 95.3 42.21 33.32 78.94 3.75 0.000195 10200 134.8 96.0 41.17 32.28 78.41 3.63 0.000191 10800 134.4 96.1 40.15 31.26 77.86 3.52 0.000184 11400 135.6 96.7 39.17 30.28 77.30 3.41 0.000171 12000 134.7 95.4 38.26 29.37 76.76 3.30 0.000169 12600 135.5 95.8 37.36 28.47 76.20 3.20 0.000167 13200 134.2 96.0 36.47 27.58 75.62 3.10 0.000159 13800 133.9 95.9 35.62 26.73 75.04 3.01 0.000159 14400 132.7 96.2 34.77 25.88 74.43 2.91 0.000154 15000 133.3 96.0 33.95 25.06 73.81 2.82 0.000152 15600 132.5 95.8 33.14 24.25 73.17 2.73 0.000142 16200 134.1 96.0 32.38 23.49 72.54 2.64 0.000150 16800 133.0 96.0 31.58 22.69 71.85 2.55 0.000144 17400 133.2 95.7 30.81 21.92 71.15 2.47 0.000144 18000 134.5 96.3 30.04 21.15 70.41 2.38 0.000139 18600 136.0 96.7 29.30 20.41 69.66 2.30 0.000156 19200 98.7 76.5 28.47 19.58 68.77 2.20 0.000148 19800 98.2 74.7 27.68 18.79 67.88 2.11 0.000116 20400 100.0 75.0 27.06 18.17 67.15 2.04 0.000086 21000 100.2 75.0 26.60 17.71 66.58 1.99 0.000077 21600 99.7 75.3 26.19 17.30 66.06 1.95 0.000073 22200 100.4 74.8 25.80 16.91 65.54 1.90 0.000064 22800 98.1 74.4 25.46 16.57 65.08 1.86 0.000060 23400 100.1 75.0 25.14 16.25 64.64 1.83 0.000058 24000 102.1 74.9 24.83 15.94 64.20 1.79 0.000054

157

Table E.1.24 (continued) 24600 98.5 74.8 24.54 15.65 63.77 1.76 0.000054 25200 99.9 74.1 24.25 15.36 63.34 1.73 0.000054 25800 101.1 74.8 23.96 15.07 62.90 1.70 0.000051 26400 100.7 75.0 23.69 14.80 62.47 1.66 0.000054 27000 99.8 74.8 23.40 14.51 62.01 1.63 0.000052 27600 99.0 74.2 23.12 14.23 61.55 1.60 0.000052 28200 101.4 74.3 22.84 13.95 61.08 1.57 0.000051 28800 100.6 74.6 22.57 13.68 60.61 1.54 0.000047 29400 99.2 74.4 22.32 13.43 60.17 1.51 0.000047 30000 99.9 74.5 22.07 13.18 59.72 1.48 0.000045 30600 101.2 75.1 21.83 12.94 59.28 1.46 0.000047 31200 100.0 75.0 21.58 12.69 58.80 1.43 0.000045 31800 101.3 74.8 21.34 12.45 58.34 1.40 0.000043 32400 101.0 74.6 21.11 12.22 57.89 1.37 0.000041 33000 100.5 74.5 20.89 12.00 57.44 1.35 0.000043 33600 100.3 74.9 20.66 11.77 56.97 1.32 0.000043 34200 100.2 75.1 20.43 11.54 56.49 1.30 0.000041 34800 100.6 75.0 20.21 11.32 56.01 1.27 0.000039 35400 100.0 75.0 20.00 11.11 55.55 1.25 0.000041 36000 101.3 74.8 19.78 10.89 55.06 1.22

158

E.2 Nonlinear Regression Analysis for the Last 10 Drying Rate Data of the

Apple Slab Obtained Under New Steady-state External Conditions After

Change in Sample Point Temperature

Table E.2.1 Model constants found by nonlinear regression analysis for the change

in sample point temperature from 96oC to 55oC.

Step change time (s) 3000 7200 12600 18600

a ⋅7.2176 10-5 ⋅4.7218 10-5 ⋅3.1043 10-5 ⋅8850.5 10-6 b 0.0004 0.0883 0.0002 ⋅2086.7 10-5 c 0.0002 0.0004 0.0002 ⋅8821.7 10-5

R2 0.95 0.59 0.58 0.45 Table E.2.2 Model constants found by nonlinear regression analysis for the change

in sample point temperature from 96oC to 65oC


a ⋅5.7821 10-5 ⋅2.5260 10-5 ⋅3080.1- 10-5 ⋅9080.6- 10-5 b 0.0002 0.0003 0.1839 ⋅4940.1 10-4 c ⋅0.0396 10-5 ⋅9.0000 10-5 ⋅3280.3 10-6 ⋅1.0600 10-5

R2 0.98 0.82 0.90 0.75 Table E.2.3 Model constants found by nonlinear regression analysis for the change

in sample point temperature from 96oC to 75oC


a 0549.0- ⋅3564.1 10-5 ⋅1720.8 10-5 ⋅7792.3 10-5 b 1642.0 0.0002 1.904e-4 0.0033 c 0013.0 ⋅2725.3 10-5 ⋅7970.7 10-5 0.0002

R2 0.99 0.90 0.79 0.86

159

E.3 Process Dynamic Parameters of the Slab Drying Under Change in the

Sample Point Temperature

Table E.3.1 Dynamic parameters for apple slab drying under change in sample point



sT∆ (oC) 41 41 41 41

1K (g/s. oC) ⋅− 23.52 10-6 ⋅− 51.62 10-6 ⋅− 37.58 10-6 ⋅− 79.57 10-6

2K (g/s. oC) ⋅70.50 10-6 ⋅96.66 10-6 ⋅12.66 10-6 ⋅19.66 10-6

K (g/s. oC) ⋅47.4 10-6 ⋅45.4 10-6 ⋅74.3 10-6 ⋅40.3 10-6

1τ (min) 12.29 14.38 12.70 12.35

2τ (min) 14.61 16.71 15.18 14.76

3τ (min) -14.88 -18.35 -25.89 -28.72

Absolute dif. ⋅81.2 10-11 ⋅58.1 10-11 ⋅57.3 10-11 ⋅5.2 10-11


temperature from 65oC to 96oC


sT∆ (oC) 31 31 31 31

1K (g/s. oC) ⋅− 40.69 10-6 ⋅− 47.61 10-6 ⋅− 66.33 10-6 ⋅− 18.63 10-6

2K (g/s. oC) ⋅17.74 10-6 ⋅47.65 10-6 ⋅70.36 10-6 ⋅36.66 10-6

K (g/s. oC) ⋅77.4 10-6 ⋅00.4 10-6 ⋅10.3 10-6 ⋅18.3 10-6

1τ (min) 16.15 12.57 10.98 11.64

2τ (min) 18.19 15.01 14.17 13.79

3τ (min) -13.56 -24.81 -23.58 -31.03

Absolute dif. ⋅74.3 10-11 ⋅97.1 10-11 ⋅50.3 10-11 ⋅01.4 10-11

160




sT∆ (oC) 21 21 21 21

1K (g/s. oC) ⋅− 51.24 10-6 ⋅− 37.93 10-6 ⋅− 41.66 10-6 ⋅− 17.48 10-6

2K (g/s. oC) ⋅45.28 10-6 ⋅84.98 10-6 ⋅94.69 10-6 ⋅05.51 10-6

K (g/s. oC) ⋅94.3 10-6 ⋅47.5 10-6 ⋅53.3 10-6 ⋅88.2 10-6

1τ (min) 8.52 16.69 14.67 11.67

2τ (min) 11.70 18.86 16.86 14.14

3τ (min) -11.29 -20.30 -26.46 -29.63

Absolute dif. ⋅03.0 10-11 ⋅69.1 10-11 ⋅91.0 10-11 ⋅31.0 10-11




sT∆ (oC) -41 -41 -41 -41

1K (g/s. oC) ⋅− 50.50 10-6 ⋅− 30.42 10-6 ⋅− 19.61 10-6 ⋅− 08.42 10-6

2K (g/s. oC) ⋅11.56 10-6 ⋅49.46 10-6 ⋅25.64 10-6 ⋅75.44 10-6

K (g/s. oC) ⋅62.5 10-6 ⋅19.4 10-6 ⋅36.3 10-6 ⋅62.2 10-6

1τ (min) 14.84 15.72 15.79 14.80

2τ (min) 17.75 18.55 18.06 17.18

3τ (min) -11.25 -12.89 -29.69 -22.76

Absolute dif. ⋅− 86.0 10-11 ⋅− 09.0 10-11 ⋅− 75.0 10-11 ⋅− 61.1 10-11

161




sT∆ (oC) -31 -31 -31 -31

1K (g/s. oC) ⋅− 86.64 10-6 ⋅− 26.56 10-6 ⋅− 71.40 10-6 ⋅− 88.46 10-6

2K (g/s. oC) ⋅43.71 10-6 ⋅27.61 10-6 ⋅94.43 10-6 ⋅58.49 10-6

K (g/s. oC) ⋅58.6 10-6 ⋅01.5 10-6 ⋅22.3 10-6 ⋅20.2 10-6

1τ (min) 17.24 15.21 12.44 15.70

2τ (min) 20.01 17.66 15.43 18.05

3τ (min) -10.08 -12.34 -25.31 -25.00

Absolute dif. ⋅− 76.0 10-11 ⋅− 78.0 10-11 ⋅− 05.1 10-11 ⋅− 00.1 10-11




sT∆ (oC) -21 -21 -21 -21

1K (g/s. oC) ⋅− 39.80 10-6 ⋅− 85.59 10-6 ⋅− 30.47 10-6 ⋅− 81.31 10-6

2K (g/s. oC) ⋅82.87 10-6 ⋅62.66 10-6 ⋅80.51 10-6 ⋅34.35 10-6

K (g/s. oC) ⋅42.7 10-6 ⋅77.6 10-6 ⋅50.4 10-6 ⋅53.3 10-6

1τ (min) 15.61 13.89 12.57 11.35

2τ (min) 18.24 16.63 15.61 14.32

3τ (min) -12.88 -10.35 -19.46 -15.39

Absolute dif. ⋅− 93.1 10-11 ⋅− 89.2 10-11 ⋅− 45.0 10-12 ⋅− 92.4 10-12

162

APPENDIX F

COMPUTER CODES IN MATLAB

F.1 Flow Chart for the Determination of the Average Moisture Content in

Finite Difference Method by Matlab Computer Code

Input data

Node number “n” determination

Time interval “∆t” determination by stability criteria

Initial condition: X = Xi at t = 0

Calculation of the moisture contents at air exposed surfaces and within the nodes

t = t + ∆t

t < td

Calculation of the average moisture content

Comparision of the predicted and

experimental data Stop

Yes

No

Start

163

F.2 Computer Code Developed for the Solution of the One Dimensional

Transient Analysis of Moisture Distribution in Apple Slab Drying

clear all

2L=0.02; %(m)

td=….; %(s)

Xi=….; %(g/g)

Xe=….; %(g/g)

Deff=….; %(m2/s)

kair=…..; %(m/s)

∆z=….; %(m)

∆t=(∆z^2)/(2*Deff); %(s)

∆tF=….;%(s)

N=td-mod

Fo=(∆tF*Deff)/dz^2;

NSh=(∆z*kair)/Deff;

m=(2*L/∆z)+1;

n=(N/∆tF);

%For experimental data

if (∆tF < 600)

a=1;

b=N/600;

c=600/∆tF;

d=N/∆tF;

else

a=(∆tF/600);%

b=N/600;

c=1;

e=N/∆tF;

if int32(e)>e;

d=int32(e)-1;

164

F.2 (continued) else

d=int32(e);

end

end

%For experimental data

EXP=[….];

EXPF=EXP';

EXPFF=EXPF(a:a:b);

%For predicted data

t=0; %second

XP(1:m)=Xi;

for i=1:d

XPN(2:m-1)=(1-2*Fo)*XP(2:m-1)+(Fo)*XP(3:m)+(Fo)*XP(1:m-2);

XPN(m)=(XPN(m-1)+ NSh*Xe)/(1+ NSh);

XPN(1)=(XPN(2)+ NSh*Xe)/(1+ NSh);

XP(1:m)=XPN(1:m);

XPF(i,:)=XP;

XPA(i)=(norm(XPF(i,:),1))/m;

XPAF=XPA';

end

XPAFF(1:(d/c))=XPAF(c:c:d)

XPAFFF=XPAFF'

end

plot(EXPFF)

hold all

plot(XPAFFF)

where ∆tF represents the time interval predicted by stability criteria, EXPFF

represents the experimental values of the average moisture contents, XP represents

the predicted moisture contents at the nodes, XPAFFF represents the predicted

average moisture contents at the nodes, N, a, b, c, d represent the coefficients.

165

F.3 Computer Code Developed for the Nonlinear Regression Analysis for the

Inverse Response Dynamic Behavior of the Apple Slab Drying

PART-1 function [ absdiffrence ] = mymodel( coeff )

y_exp=[…];

t=[…:…:…];

M=…;

K=coeff(1);tao1=coeff(2);tao2=coeff(3);tao3=coeff(4);

y_calc=K*M*(1+((tao3-tao1)/(tao1-tao2)).*exp(-t/tao1) +

((tao3-tao2)/(tao2-tao1)).*exp(-t/tao2));

absdiffrence=abs(y_calc-y_exp);

PART-2 function [ output_args ] = optmodel( input_args )

M=…;

% initial parameter values

% coeff=[ K tao1 tao2 tao3]

initialcoeff=[1.5 0.2 5 -10];

options = optimset('Display','iter','TolFun',1e-11,'TolX',1e-11,...

'MaxFunEvals',1000,'MaxIter',500);

[sonuc,resnorm] = lsqnonlin('mymodel',initialcoeff,[],[],options );

[sonuc]

[resnorm/M]

format long

% Plotting

y_exp=[…];

t=[…:…:…];

t2=[0:0.1:…];

K=sonuc(1);tao1=sonuc(2);tao2=sonuc(3);tao3=sonuc(4);

166

F.3 (continued) y_calc=K*M*(1+((tao3-tao1)/(tao1-tao2)).*exp(-t2/tao1) +

((tao3-tao2)/(tao2-tao1)).*exp(-t2/tao2));

plot(t,y_exp,'x',t2,y_calc,'-r');

legend('Experimental','Model');

[t2',y_calc'];

where M refers to the change in sample point temperature, tao1 refers to 1τ , tao2

refers to 2τ , tao3 refers to 3τ and K is the overall gain of the system.

SIMULATION OF A BATCH DRYER BY THE FINITE DIFFERENCE ... · I wish to thank Suzan Tireki, Halil...

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