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SIMULATION OF A BATCH DRYER BY THE FINITE DIFFERENCE ... · I wish to thank Suzan Tireki, Halil...
Transcript of 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.
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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
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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
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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.
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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
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. ……………………. 85
A.2.3. Data for sample point temperature change from 75oC to 96oC after step
change in the inlet air temperature from 100.5oC to 135oC. …………………... 86
A.2.4. Data for sample point temperature change from 96oC to 55oC after step
change in the inlet air temperature from 135oC to 66oC……….……………..… 87
A.2.5. Data for sample point temperature change from 96oC to 65oC after step
change in the inlet air temperature from 135oC to 86.5oC……………………... 88
A.2.6. Data for sample point temperature change from 96oC to 75oC after step
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
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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
from 97oC to 56oC after step change in the inlet air temperature from 135oC to
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
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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
E.1.2. 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 10200th s ( av = 0.04m/s) ……..……..……..……..……..……..……..…….. 121
E.1.3. 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 16200th s ( av = 0.04m/s) ……..……..……..……..……..……..……..…….. 123
E.1.4. 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 19200th s ( av = 0.04m/s) ……..……..……..……..……..……..……..…….. 125
E.1.5. Drying rate of apple slab under change in sample point temperature from
65oC to 96oC after step change in the inlet air temperature from 86.5oC to 135oC
at 3600th s ( av = 0.04m/s) ……..……..………....……..……..……..………… 127
E.1.6. Drying rate of apple slab under change in sample point temperature from
65oC to 96oC after step change in the inlet air temperature from 86.5oC to 135oC
at 7800th s ( av = 0.04m/s) ……..……..……..…..……..……..……..………… 128
E.1.7. Drying rate of apple slab under change in sample point temperature from
65oC to 96oC after step change in the inlet air temperature from 86.5oC to 135oC
at 14400th s ( av = 0.04m/s) ……..……..…..……..……..……..……..……….. 129
E.1.8. Drying rate of apple slab under change in sample point temperature from
65oC to 96oC after step change in the inlet air temperature from 86.5oC to 135oC
at 18600th s ( av = 0.04m/s) ……..……..…..……..……..……..……..……….. 130
E.1.9. Drying rate of apple slab under change in sample point temperature from
75oC to 96oC after step change in the inlet air temperature from 100.5oC to 135oC
at 3600th s ( av = 0.04m/s) ……..…..……..……..……..……..……..………… 132
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E.1.10. Drying rate of apple slab under change in sample point temperature from
75oC to 96oC after step change in the inlet air temperature from 100.5oC to 135oC
at 6600th s ( av = 0.04m/s) ……..……..……..……..……..……..……..………. 133
E.1.11. Drying rate of apple slab under change in sample point temperature from
75oC to 96oC after step change in the inlet air temperature from 100.5oC to 135oC
at 11400th s ( av = 0.04m/s) ……..……..……..……..……..……..……..…….. 134
E.1.12. Drying rate of apple slab under change in sample point temperature from
75oC to 96oC after step change in the inlet air temperature from 100.5oC to 135oC
at 19200th s ( av = 0.04m/s) ……..……..……..……..……..……..……………. 135
E.1.13. Drying rate of apple slab under change in sample point temperature from
96oC to 55oC after step change in the inlet air temperature from 135oC to 66oC to
at 3000th s ( av = 0.04m/s) ……..……..……..……..……..……..……..……… 137
E.1.14. Drying rate of apple slab under change in sample point temperature from
96oC to 55oC after step change in the inlet air temperature from 135oC to 66oC to
at 7200th s ( av = 0.04m/s) ……..……..……..……..……..……..……..……… 138
E.1.15. Drying rate of apple slab under change in sample point temperature from
96oC to 55oC after step change in the inlet air temperature from 135oC to 66oC to
at 12600th s ( av = 0.04m/s) ……..………..………..………..………..……….. 140
E.1.16. Drying rate of apple slab under change in sample point temperature from
96oC to 55oC after step change in the inlet air temperature from 135oC to 66oC to
at 18600th s ( av = 0.04m/s) ……..………..………..………..………..……….. 142
E.1.17. Drying rate of apple slab under change in sample point temperature from
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
E.1.18. Drying rate of apple slab under change in sample point temperature from
96oC to 65oC after step change in the inlet air temperature from 135oC to 86.5oC
to at 6600th s ( av = 0.04m/s) ……..………..………..………..………..……… 145
E.1.19. Drying rate of apple slab under change in sample point temperature from
96oC to 65oC after step change in the inlet air temperature from 135oC to 86.5oC
to at 12600th s ( av = 0.04m/s) ……..………..………..………..………..…….. 147
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E.1.20. Drying rate of apple slab under change in sample point temperature from
96oC to 65oC after step change in the inlet air temperature from 135oC to 86.5oC
to at 18600th s ( av = 0.04m/s) ……..………..………..………..………..….…. 149
E.1.21. Drying rate of apple slab under change in sample point temperature from
96oC to 75oC after step change in the inlet air temperature from 135oC to 100.5oC
to at 2400th s ( av = 0.04m/s) ……..………..………..………..………..……… 151
E.1.22. Drying rate of apple slab under change in sample point temperature from
96oC to 75oC after step change in the inlet air temperature from 135oC to 100.5oC
to at 6000th s ( av = 0.04m/s) ……..………..………..………..………..……… 152
E.1.23. Drying rate of apple slab under change in sample point temperature from
96oC to 75oC after step change in the inlet air temperature from 135oC to 100.5oC
to at 12000th s ( av = 0.04m/s) ……..………..………..………..………..…….. 154
E.1.24. Drying rate of apple slab under change in sample point temperature from
96oC to 75oC after step change in the inlet air temperature from 135oC to 100.5oC
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
E.2.2. Model constants found by nonlinear regression analysis for the change in
sample point temperature from 96oC to 65oC……..………..………..……….. 158
E.2.3. Model constants found by nonlinear regression analysis for the change in
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
E.3.2. Dynamic parameters for apple slab drying under change in sample point
temperature from 65oC to 96oC ……..………..………..………..………..…… 159
E.3.3. Dynamic parameters for apple slab drying under change in sample point
temperature from 75oC to 96oC ……..………..………..………..………..…… 160
E.3.4. Dynamic parameters for apple slab drying under change in sample point
temperature from 96oC to 55oC ……..………..………..………..………..…… 160
E.3.5. Dynamic parameters for apple slab drying under change in sample point
temperature from 96oC to 65oC ……..………..………..………..………..…… 161
xix
E.3.6. Dynamic parameters for apple slab drying under change in sample point
temperature from 96oC to 75oC ……..………..………..………..………..…… 161
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
3.3. Change in the sample point temperature from 65oC to 96oC after step change in
the inlet temperature from 86.5oC to 135oC…..……..……..……..……..……… 47
3.4. Change in the sample point temperature from 75oC to 96oC after step change in
the inlet temperature from 100.5oC to 135oC…..………..………..………..…… 48
3.5. Change in the sample point temperature from 96oC to 55oC after step change in
the inlet temperature from 135oC to 66oC…..……..…………..………..……… 48
3.6. Change in the sample point temperature from 96oC to 65oC after step change in
the inlet temperature from 135oC to 86.5oC……………………………………. 49
3.7. Change in the sample point temperature from 96oC to 75oC after step change in
the inlet temperature from 135oC to 100.5oC…………………………………… 49
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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
3.13. Comparison of the experimental and predicted average moisture contents for
apple slab drying at 65oC……………………………………………………….. 56
3.14. Comparison of the experimental and predicted average moisture contents for
apple slab drying at 75oC……………………………………………………….. 57
3.15. Comparison of the experimental and predicted average moisture contents for
apple slab drying at 96oC……………………………………………………….. 57
3.16. Nonlinear regression analysis of the ( )tC vs. time plot for apple slab drying at
55oC…………………………………………………………………………….. 59
3.17. Nonlinear regression analysis of the ( )tC vs. time plot for apple slab drying at
65oC…………………………………………………………………………….. 59
3.18. Nonlinear regression analysis of the ( )tC vs. time plot for apple slab drying at
75oC…………………………………………………………………………….. 60
3.19. Nonlinear regression analysis of the ( )tC vs. time plot for apple slab drying at
96oC…………………………………………………………………………….. 60
3.20. Drying rate of apple slab under change in the sample point (drying) temperature
from 55oC to 96oC………………………………………………………………. 61
3.21. Drying rate of apple slab under change in the sample point (drying) temperature
from 65oC to 96oC………………………………………………………………. 62
xxii
3.22. Drying rate of apple slab under change in the sample point (drying) temperature
from 75oC to 96oC………………………………………………………………. 62
3.23. Drying rate of apple slab under change in the sample point (drying) temperature
from 96oC to 55oC………………………………………………………………. 63
3.24. Drying rate of apple slab under change in the sample point (drying) temperature
from 96oC to 65oC………………………………………………………………. 63
3.25. Drying rate of apple slab under change in the sample point (drying) temperature
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
3.27. 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 7200th s………………………….…………. 65
3.28. 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 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
3.30. Response part of the apple slab drying rate versus time curve under change in
the sample point (drying) temperature from 65oC to 96oC ……………………. 69
3.31. Response part of the apple slab drying rate versus time curve under change in
the sample point (drying) temperature from 75oC to 96oC.……………………. 69
3.32. Response part of the apple slab drying rate versus time curve under change in
the sample point (drying) temperature from 96oC to55oC ……………………... 70
3.33. Response part of the apple slab drying rate versus time curve under change in
the sample point (drying) temperature from 96oC to 65oC....…………………... 70
3.34. Response part of the apple slab drying rate versus time curve under change in
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
3.36. Predicted and experimental drying rate data under change in the sample point
(drying) temperature from 65oC to 96oC at 14400th s ……………..……...…..... 73
3.37. Predicted and experimental drying rate data under change in the sample point
(drying) temperature from 75oC to 96oC at 11400th s ..…………………..…...... 73
3.38. Predicted and experimental drying rate data under change in the sample point
(drying) temperature from 96oC to 55oC at 7200th s …...………………………. 74
3.39. Predicted and experimental drying rate data under change in the sample point
(drying) temperature from 96oC to 65oC at 6600th s .…...……………………… 74
3.40. Predicted and experimental drying rate data under change in the sample point
(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
experimentalpredicted by Eq. 1.18
Figure 3.3 Change in the sample point temperature from 65oC to 96oC after step
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)
experimentalpredicted by Eq. 1.18
Figure 3.4 Change in the sample point temperature from 75oC to 96oC after step
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
experimentalpredicted by Eq. 1.18
Figure 3.5 Change in the sample point temperature from 96oC to 55oC after step
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
experimentalpredicted by Eq. 1.18
Figure 3.6 Change in the sample point temperature from 96oC to 65oC after step
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
experimentalpredicted by Eq. 1.18
Figure 3.7 Change in the sample point temperature from 96oC to 75oC after step
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)
node 11 node 21 node 41 experimental
65oC
Figure 3.13 Comparison of the experimental and predicted average moisture
contents for apple slab drying at 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)
node 11 node 21 node 41 experimental
75oC
Figure 3.14 Comparison of the experimental and predicted average moisture
contents for apple slab drying at 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)
node 11 node 21 node 41 experimental
96oC
Figure 3.15 Comparison of the experimental and predicted average moisture
contents for apple slab drying at 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
Figure 3.17 Nonlinear regression analysis of the time vs. ( )tC plot for apple slab
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
Figure 3.18 Nonlinear regression analysis of the time vs. ( )tC plot for apple slab
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
Figure 3.19 Nonlinear regression analysis of the time vs. ( )tC plot for apple slab
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)
at 96C at 65C change in 3600th s
change in 8400th s change in 14400th s change in 18600th s
65oC to 96oC
Figure 3.21 Drying rate of apple slab under change in the sample point (drying)
temperature from 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)
at 96C at 75C change in 3600th s
change in 6600th s change in 11400th s change in 19200th s
75oC to 96oC
Figure 3.22 Drying rate of apple slab under change in the sample point (drying)
temperature from 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)
at 96C at 55C change in 3000th s
change in 7200th s change in 12600th s change in 18600th s
96oC to 55oC
Figure 3.23 Drying rate of apple slab under change in the sample point (drying)
temperature from 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
)
at 96C at 65C change in 3000th s
change in 6600th s change in 12600th s change in 18600th s
96oC to 65oC
Figure 3.24 Drying rate of apple slab under change in the sample point (drying)
temperature from 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)
at 96C at 75C change in 2400th s
change in 6000th s change in 12000th s change in 18600th s
96oC to 75oC
Figure 3.25 Drying rate of apple slab under change in the sample point (drying)
temperature from 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
experimentalpredicted
Figure 3.27 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 65oC at 13200th s.
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
experimentalpredicted
Figure 3.28 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 75oC at 12600th s.
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
Figure 3.30 Response part of the apple slab drying rate versus time curve under
change in the sample point (drying) temperature from 65oC to 96oC.
0 1200 2400 3600 4800-2
-1
0
1
2x 10
-4
time (s)
dry
ing
rate
(g/
g b
ds.
s)
Experimentalat 3600th sExperimentalat 6600th sExperimentalat 11400th sExperimentalat 19200th s
Figure 3.31 Response part of the apple slab drying rate versus time curve under
change in the sample point (drying) temperature from 75oC to 96oC.
70
0 2400 4800 7200-3
-2
-1
0
1
2
3x 10
-4
time (s)
dry
ing
rate
(g/
g b
ds.
s)
Experimentalat 3000th sExperimentalat 7200th sExperimentalat 12600th sExperimentalat 18600th s
Figure 3.32 Response part of the apple slab drying rate versus time curve under
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)
Experimentalat 3000th sExperimentalat 6600th sExperimentalat 12600th sExperimentalat 18600th s
Figure 3.33 Response part of the apple slab drying rate versus time curve under
change in the sample point (drying) temperature from 96oC to 65oC.
71
0 2400 4800 7200-3
-1-1
0
1
2
3x 10
-4
time (s)
dry
ing
rate
(g/
g b
ds.
s)
Experimentalat 2400th sExperimentalat 6000th sExperimentalat 12000th sExperimentalat 18600th s
Figure 3.34 Response part of the apple slab drying rate versus time curve under
change in the sample point (drying) temperature from 96oC to 75oC.
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
point (drying) temperature from 55oC to 96oC at 16200th s.
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)
experimental predicted
Figure 3.36 Predicted and experimental drying rate data under change in the sample
point (drying) temperature from 65oC to 96oC at 14400th 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)
experimental predicted
Figure 3.37 Predicted and experimental drying rate data under change in the sample
point (drying) temperature from 75oC to 96oC at 11400th 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)
experimental predicted
Figure 3.38 Predicted and experimental drying rate data under change in the sample
point (drying) temperature from 96oC to 55oC at 7200th 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)
experimental predicted
Figure 3.39 Predicted and experimental drying rate data under change in the sample
point (drying) temperature from 96oC to 65oC at 6600th 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)
experimental predicted
Figure 3.40 Predicted and experimental drying rate data under change in the sample
point (drying) temperature from 96oC to 75oC at 6000th 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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Analytical study of cyclic temperature drying: effect on drying kinetics and product
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Hussain M.M., Dincer I., 2003. Two-dimensional heat and moisture transfer analysis
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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
Table A.2.3 Data for sample point temperature change from 75oC to 96oC after step
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
Table A.2.4 Data for sample point temperature change from 96oC to 55oC after step
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
Table A.2.5 Data for sample point temperature change from 96oC to 65oC after step
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
Table A.2.6 Data for sample point temperature change from 96oC to 75oC after step
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
Table C.1.1 (continued)
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
Table C.1.2 (continued)
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
Table C.1.3 Drying data for apple slab at 75oC (%RH = 3.3, 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 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
Table C.1.3 (continued)
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
Table C.1.4 Drying data for apple slab at 96oC (%RH = 1.5, 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
−
−
0 132.3 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
Table D.2.1 (continued)
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
Table D.2.2 Predicted data found for apple slab drying at 65oC.
node-11 node-21 node-41
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
Table D.2.2 (continued)
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
Table D.2.3 Predicted data found for apple slab drying at 75oC.
node-11 node-21 node-41
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
Table D.2.3 (continued)
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
Table D.2.4 Predicted data found for apple slab drying at 96oC.
node-11 node-21 node-41
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
Table D.4.2 Predicted data found for apple slab drying at 65oC (for node-41).
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
Table D.4.3 Predicted data found for apple slab drying at 75oC (for node-41).
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
Table D.4.4 Predicted data found for apple slab drying at 96oC (for node-41).
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
Table D.5.2 Predicted data found for apple slab drying at 65oC (for node-41).
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
Table D.5.3 Predicted data found for apple slab drying at 75oC (for node-41).
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
Table D.5.4 Predicted data found for apple slab drying at 96oC (for node-41).
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
from 55oC to 96oC after step change in the inlet air temperature from 66oC to 135oC
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
Table E.1.3 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 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
Table E.1.3 (continued)
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
Table E.1.4 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 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
Table E.1.4 (continued)
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
Table E.1.5 Drying rate of apple slab under change in sample point temperature
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
Table E.1.6 Drying rate of apple slab under change in sample point temperature
from 65oC to 96oC after step change in the inlet air temperature from 86.5oC to
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
Table E.1.7 Drying rate of apple slab under change in sample point temperature
from 65oC to 96oC after step change in the inlet air temperature from 86.5oC to
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
Table E.1.8 Drying rate of apple slab under change in sample point temperature
from 65oC to 96oC after step change in the inlet air temperature from 86.5oC to
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
Table E.1.8 (continued)
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
Table E.1.9 Drying rate of apple slab under change in sample point temperature
from 75oC to 96oC after step change in the inlet air temperature from 100.5oC 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) 0 101 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
Table E.1.10 Drying rate of apple slab under change in sample point temperature
from 75oC to 96oC after step change in the inlet air temperature from 100.5oC to
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
Table E.1.11 Drying rate of apple slab under change in sample point temperature
from 75oC to 96oC after step change in the inlet air temperature from 100.5oC to
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
Table E.1.12 Drying rate of apple slab under change in sample point temperature
from 75oC to 96oC after step change in the inlet air temperature from 100.5oC to
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
Table E.1.12 (continued)
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
Table E.1.13 Drying rate of apple slab under change in sample point temperature
from 96oC to 55oC after step change in the inlet air temperature from 135oC to 66oC
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
Table E.1.14 Drying rate of apple slab under change in sample point temperature
from 96oC to 55oC after step change in the inlet air temperature from 135oC to 66oC
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
Table E.1.14 (continued)
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
Table E.1.15 Drying rate of apple slab under change in sample point temperature
from 96oC to 55oC after step change in the inlet air temperature from 135oC to 66oC
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
Table E.1.15 (continued)
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
Table E.1.16 Drying rate of apple slab under change in sample point temperature
from 96oC to 55oC after step change in the inlet air temperature from 135oC to 66oC
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
Table E.1.16 (continued)
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
Table E.1.17 Drying rate of apple slab under change in sample point temperature
from 96oC to 65oC after step change in the inlet air temperature from 135oC to
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
Table E.1.18 Drying rate of apple slab under change in sample point temperature
from 96oC to 65oC after step change in the inlet air temperature from 135oC to
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
Table E.1.18 (continued)
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
Table E.1.19 Drying rate of apple slab under change in sample point temperature
from 96oC to 65oC after step change in the inlet air temperature from 135oC to
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
Table E.1.19 (continued)
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
Table E.1.20 Drying rate of apple slab under change in sample point temperature
from 96oC to 65oC after step change in the inlet air temperature from 135oC to
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
Table E.1.20 (continued)
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
Table E.1.21 Drying rate of apple slab under change in sample point temperature
from 96oC to 75oC after step change in the inlet air temperature from 135oC to
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
Table E.1.22 Drying rate of apple slab under change in sample point temperature
from 96oC to 75oC after step change in the inlet air temperature from 135oC to
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
Table E.1.23 Drying rate of apple slab under change in sample point temperature
from 96oC to 75oC after step change in the inlet air temperature from 135oC to
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
Table E.1.24 Drying rate of apple slab under change in sample point temperature
from 96oC to 75oC after step change in the inlet air temperature from 135oC to
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
Step change time (s) 3000 6600 12600 18600
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
Step change time (s) 2400 6000 12600 18600
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
temperature from 55oC to 96oC.
Step change time (s) 1800 10200 16200 19200
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
Table E.3.2 Dynamic parameters for apple slab drying under change in sample point
temperature from 65oC to 96oC
Step change time (s) 3600 7800 14400 18600
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
Table E.3.3 Dynamic parameters for apple slab drying under change in sample point
temperature from 75oC to 96oC.
Step change time (s) 3600 6600 11400 19200
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
Table E.3.4 Dynamic parameters for apple slab drying under change in sample point
temperature from 96oC to 55oC.
Step change time (s) 3000 7200 12600 18600
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
Table E.3.5 Dynamic parameters for apple slab drying under change in sample point
temperature from 96oC to 65oC.
Step change time (s) 3000 6600 12600 18600
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
Table E.3.6 Dynamic parameters for apple slab drying under change in sample point
temperature from 96oC to 75oC.
Step change time (s) 2400 6000 12000 18600
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.