Title: Pressure and velocity distribution for air flow through fruits packed in shipping containers using porous media flow analysis /
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 Material Information
Title: Pressure and velocity distribution for air flow through fruits packed in shipping containers using porous media flow analysis /
Physical Description: xxix, 449 leaves : ill. ; 28 cm.
Language: English
Creator: Talbot, Michael T ( Michael Thomas ), 1948-
Publisher: s.n.
Publication Date: 1987
Copyright Date: 1987
 Subjects
Subject: Air flow -- Cooling   ( lcsh )
Fruit -- Preservation   ( lcsh )
Permeability   ( lcsh )
Mechanical Engineering thesis Ph. D
Dissertations, Academic -- Mechanical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1987.
Bibliography: Bibliography: leaves 437-448.
Statement of Responsibility: by Michael Thomas Talbot.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00099076
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000968177
oclc - 17394368
notis - AEU3389

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PRESSURE AND VELOCITY DISTRIBUTION FOR AIR FLOW
THROUGH FRUITS PACKED IN SHIPPING CONTAINERS
USING POROUS MEDIA FLOW ANALYSIS
















By

MICHAEL THOMAS TALBOT


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1987


u~a:- i
















ACKNOWLEDGMENT


The author wishes to express his sincere appreciation

to Dr. Calvin C. Oliver who served as chairman of his

supervisory committee. His guidance, encouragement, and

assistance were invaluable throughout the course of this

study.

For the leadership, interest, and assistance, they

exhibited, appreciation is extended to the other members of

his supervisory committee: Mr. J. J. Gaffney, USDA

Agricultural Engineer; Dr. H. A. Ingley III, Dr. T. I. Shih,

Mechanical Engineering; and Dr. R. F. Matthews, Food Science

and Human Nutrition. Special thanks are also expressed for

the early contributions of Dr. R. K. Irey, an original

committee member who is presently Chairman of the Mechanical

Engineering Department at the University of Toledo, Ohio.

Appreciation is expressed to Dr. R. B. Gaither,

Chairman of the Mechanical Engineering Department, and to

the faculty members who contributed to his graduate program.

A special word of thanks is necessary for the

invaluable assistance of Dean M. M. Lockhart, Dean J. L.

Woeste and Dean G. L. Zachariah, as well as the Graduate

Council, without which pursuit of this course of study would

not have been possible.










He is indebted to the Department of Agricultural

Engineering for financial support through full-time

employment during the part-time pursuit of this course of

study and to Dr. G. W. Isaacs, chairman, for his leadership,

encouragement, and untiring willingness to cope with

administrative problems involved. Thanks are also due to

each of the many staff and faculty members who assisted in

numerous ways, especially to Dr. C. D. Baird and Dr. K. V.

Chau who actively supported his research as project leaders.

For their cheerful sacrifices and loving patience, the

author dedicates this dissertation to his wife, Anita, his

son, Michael, and his father, James Vernon (Buck) Talbot,

who waited so eagerly and proudly but who could not be here

for the completion.

Most of all, the author wishes to acknowledge the

source of all knowledge and wisdom--the very God of the

universe through His Son the Lord Jesus Christ. To God be

the glory, for great things He hath done.

















TABLE OF CONTENTS



Page

ACKNOWLEDGMENT ........................................ ii

LIST OF TABLES ........................................ vi

LIST OF FIGURES .................... ... ................ vii

ABSTRACT .......... ....................................xxvii

INTRODUCTION .......................................... 1

OBJECTIVES ............................................ 5

REVIEW OF LITERATURE ................................... 6

Background of Research .................... ........ .... 6
Fluid Flow Through Porous Media ..................... 12

PROCEDURE ........................................... 32

Commercial Finite Element Package.................... 43
Verification ........................................ 44
Two-Dimensional Grain Bin ......................... 49
Three-Dimensional Grain Bin ....................... 52
Solution for Oranges Packed in Cartons .............. 57
Three-Dimensional Bulk Oranges .................... 60
Three-Dimensional Orange Carton ................... 78
Experimental Temperature Measurement ................ 88
Description of Experimental Facilities ............ 88
Basic Construction of Precooler ................... 88
Refrigeration Components and Controls ............. 89
Air Temperature and Humidity ...................... 90
Air Flow .......................................... 95
Reheat Section .................................... 97
Experimental Test Procedures ...................... 98
Heat Transfer Model ................................... 101
Selection of Model .................................. 102
Convective Heat Transfer Coefficient .............. 112
Heat Transfer Model Modifications ................. 116
ANSYS Output Data Reorganization .................. 124
Heat Transfer Program ...... ....................... 126
Temperature Response Data Reorganization ......... 128










RESULTS AND DISCUSSION ............................... 131

Two-Dimensional Grain Bin............................ 131
Three-Dimensional Grain Bin........................... 139
Three-Dimensional Bulk Oranges....................... 154
Three-Dimensional Orange Carton.................... 161
Experimental Temperature Measurement................. 165
Heat Transfer Model.................................... 175
Experimental Error .................................... 179
Convective Heat Transfer Coefficient................. 181
Variable Porosity................................... 182
Approximate Model Thermocouple Location.............. 183
Experimental Versus Predicted Temperature Response... 183

SUMMARY...... ........ .................................. 197

SUGGESTIONS FOR FUTURE WORK............................ 200

APPLICATION OF RESULTS ................................. 202

CONCLUSIONS...... ........ .............................. 204

APPENDIX A METHOD FOR ACCURATELY POSITIONING
THERMOCOUPLES IN FRUITS AND VEGETABLES................. 206

APPENDIX B FORTRAN PROGRAMS AND DATA FILES............ 210

APPENDIX C PLOT INFORMATION........................... 234

APPENDIX D ADDITIONAL DISCUSSION OF RESULTS FOR TWO-
AND THREE-DIMENSIONAL GRAIN BIN MODELS................. 236

APPENDIX E ADDITIONAL EXPERIMENTAL TEMPERATURE RESPONSE
GRAPHS................................................. 290

APPENDIX F ADDITIONAL PREDICTED TEMPERATURE RESPONSE
GRAPHS................................................. 324

APPENDIX G ADDITIONAL PREDICTED VERSUS EXPERIMENTAL
TEMPERATURE RESPONSE GRAPHS............................ 358

APPENDIX H ADDITIONAL PREDICTED VERSUS EXPERIMENTAL
TEMPERATURE REGRESSION GRAPHS.......................... 425

BIBLIOGRAPHY..... ......... ............................. 437

BIOGRAPHICAL SKETCH. .................................. 449
















LIST OF TABLES


Table Page

1. Summary of variable porosity and ANSYS input data. 85

2. Inlet and outlet air vent locations as indicated
by the numbers illustrated in Figure 11........... 87

3. Boundary conditions and air flow rates (2 flow
rates per venting arrangement) for tests
conducted with an experimental orange carton
packed with size 100 oranges ..................... 87

4. Physical and thermal properties of size 100
Valencia oranges, Gaffney and Baird (1980)........ 114

5. Pressure values (inches of water) at specific
points for the linear flow problem............... 132

6. Three-dimensional pressure loss data for oranges
in bulk and orange carton sides................... 155
















LIST OF FIGURES


Figure Page

1. Region divided into finite element................ 36

2. Various types of finite elements.................. 39

3. The node and element grid employed by Segerlind
(1982) ............................................. 50

4. Boundary conditions used by Segerlind (1982)...... 51

5. The node and element locations used in current
study.............................................. 53

6. Four different grainbin aeration systems studied
by Khompis et al. (1984) .......................... 55

7 Finite element model used by Khompis et al.(1984). 56

8. Tne node and element locations used in current
study.............................................. 58

9. The experimental setup used by Chau et al. (1983)
or measuring air flow resistance in orange......... 64

10. The node and element locations used to model 3-D
oranges packed in bulk and in cartons ............. 71

11. The experimental orange carton..................... 80

12. The node and element locations used to model a
3-D orange carton.................................. 81

13. Forced-air precooler components and air
circulation diagram, Baird et al. (1975) .......... 91

14. Modification of product bin to accommodate
experimental carton............................... 92

15. Precooler refrigeration components and controls,
Baird et al. (1975) ............................... 93

16. Fruit and thermocouple locations in experimental
carton............................................. 100










17. Diagram indicating a heat balance on the three
types of nodes used in a sphere, Baird and Gaffney
(1976) ............. ... .................... 104

18. Flow diagram for the computer program for heat
conduction in a sphere with arbitrarily varying
air temperature, Baird and Gaffney (1976)......... 107

19. Control volume for heat balance on bed, Baird and
Gaffney (1976) ................................... 109

20. Flow diagram for the computer program to solve
deep bed heat transfer equations, Baird and
Gaffney (1976) .................................. 113

21. Typical three-dimensional element showing velocity
at centroid and air flow across element faces.... 118

22. Flow diagram for the computer program to solve
orange carton heat transfer equations............. 123

23. Points of comparison for linear airflow
calculations ..................................... 132

24. Plot of experimental isobaric lines from Brooker
(1969) ............................................ 133

25. 85 percent isobaric contour line from Segerlind
(1982) ............................................. 134

26. Isobaric contour lines calculated using ANSYS
for a boundary condition of 1 inch of water...... 135

27. Isobaric contour lines calculated using ANSYS
for a boundary condition of 2 inches of water.... 136

28. Isobaric contour lines calculated using ANSYS
for a boundary condition of 3 inches of water.... 137

29. Isobaric distribution in percent of total
pressure loss on a vertical cutting plane from
Khompis (1983) .................................. 141

30. Isobaric distribution in percent of total
pressure loss on horizontal cutting plane at 5
percent of bin height from Khompis (1983)......... 142

31. Isobaric distribution calculated using ANSYS in
percent of total pressure loss on the vertical
cutting plane.................................... 143

32. Isobaric distribution calculated using ANSYS in
percent of total pressure loss on the horizontal
cutting plane at 5 percent of bin height.......... 144


viii










33. Velocity distribution in meters per minute on a
vertical cutting plane from Khompis (1983) ....... 145

34. Velocity distribution in meters per minute on a
horizontal cutting plane at 5 percent of bin
height from Khompis (1983) ....................... 146

35. Velocity distribution calculated using ANSYS in
meters per minute on a verical cutting plane..... 147

36. Velocity distribution calculated using ANSYS in
meters per minute on a horizontal cutting plane
at 5 percent of bin height.... ................... 148

37. 70 percent of total pressure loss isobaric
surface from Khompis (1983), view 2.............. 149

38. 70 percent of total pressure loss isobaric
surface produced by ANSYS, view 2................ 150

39. Constant velocity surface of 2.4 meters per
minute from Khompis (1983), view 2................ 152

40. Constant velocity surface of 2.8 meters per
minute calculated using ANSYS, view 2............. 153

41. Nodal pressure printout for ANSYS solution........ 162

42. Element flow rate printout for ANSYS solution.... 163

43. Example of the data printout produced by the data
acquisition system for one experimental cooling
test............................................. 166

44. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 1-1,
thermocouple locations 10, 13, 66, and 69)....... 168

45. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 1-1,
3rd layer thermocouple locations 31, 35, 36, 44,
and 45) .......................................... 171

46. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 1-1,
2nd layer thermocouple locations 23, 24, 26, 27,
and 29).......................................... 172










47. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 1-1,
1st layer thermocouple locations 1, 9, 10, 14,
and 15)........................................... 173

48. Graphical representation of three-dimensional
color graphics outpur of cooling response for
oranges within an experimental carton............. 174

49. Predicted center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 1-1,
3rd layer thermocouple locations 31, 35, 36, 44,
and 45) ........................................... 176

50. Predicted center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 1-1,
2nd layer thermocouple locations 23, 24, 26, 27,
and 29).......................................... 177

51. Predicted center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 1-1,
1st layer thermocouple locations 1, 9, 10, 14,
and 15)........................................... 178

52. Predicted versus experimental data for Boundary
Condition 1-1, 3rd layer thermocouples 31 and
35................................................. 185

53. Predicted versus experimental data for Boundary
Condition 1-1, 3rd layer thermocouples 44 and
45. ............................................... 186

54. Predicted versus experimental data for Boundary
Condition 1-1, 2nd layer thermocouples 23 and
24................................................. 187

55. Predicted versus experimental data for Boundary
Condition 1-1, 2nd layer thermocouples 26 and
29................................................. 188

56. Predicted versus experimental data for Boundary
Condition 1-1, 1st layer thermocouples 1 and 9... 189

57. Predicted versus experimental data for Boundary
Condition 1-1, 1st layer thermocouples 14 and
15................................................. 190










58. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 1-1 and test times of 3, 7, and 11
hours ............................................ 191

A-i. Thermocouple positioning, part 1.................. 208

A-2. Thermocouple positioning, part 2.................. 209

B-1. Portion of input data file "Dimensions" which
lists for each element from left to right, the
number, volume, x, y, and z dimensions, and the
porosity ......................................... 211

B-2. Portion of input data file "Data" which lists
for each element from left to right, the number,
total velocity and x, y, and z component
velocities....................................... 212

B-3. Fortran program "Data-Reduction" which was used
to calculate convective heat transfer coefficient
and x, y, and z component mass flow rates at the
centroid.......................................... 213

B-4. Portion of input/output data file "Heat-Mass"
which lists for each element, the number, the
convective heat transfer coefficient and x, y,
and z component mass flow rates at the centroid.. 215

B-5. Fortran program "Flow-Wall" which was used to
calculate x, y, and z component mass flow rates
across the face of each element.................. 216

B-6. Portion of input/output data file "Mass-Flow"
which lists for each element, the number, the x,
y, and z component mass flow rates across the
face of each element............................. 218

B-7. Fortran program "Temp-ID" which was used to
calculate mass flow rates entering each element.. 219

B-8. Portion of input/output data file "Mass-Temp"
which lists for each element, the number, the
mass flow rates entering the x (left and right),
y (bottom and top), and z (back and front) faces
of each element along with the corresponding
temperature identification label.................. 222

B-9. Fortran program "Heat-Do" which was used to solve
the heat transfer model .......................... 223










B-10. Input data file "Data-In-1" which lists the
product thermal diffusivity, conductivity, and
radius, the specific heat of the air and product
and product density............................... 229

B-ll. Input data file "Data-In-2" which lists number
of product nodal points, the initial product
temperature, the time increment, total test
cooling time, data output print frequency,
entering air temperature, and number of elements. 229

B-12. Portion of input/output data file "Temp-
Response" which lists for each element, the
number, time (seconds),the air temperature, the
product temperature at each of the nodes, and the
product mass-average temperature................. 230

B-13. Fortran program "Plot-Data" which was used to
reorganize the temperature response to give 78
thermocouple readings............................ 231

D-l. Numerical pressure pattern for linear air flow
through shelled corn from Brooker (1969), for 2
inches of water duct pressure.................... 237

D-2. Numerical pressure pattern for nonlinear air
flow through shelled corn from Brooker (1969).
The same pattern was obtained for duct pressures
of 1, 2, and 3 inches of water.................... 238

D-3. Numerical velocity distribution for nonlinear
air flow through shelled corn from Brooker
(1969) ........................................... 239

D-4. Plot of experimental isobaric lines from
Brooker (1969) .................................. 239

D-5. 85 percent isobaric contour line from Segerlind
(1982) ............................................ 240

D-6. Isobaric contour lines calculated using ANSYS
for a boundary condition of 1 inch of water...... 242

D-7. Isobaric contour lines calculated using ANSYS
for a boundary condition of 2 inches of water.... 243

D-8. Isobaric contour lines calculated using ANSYS
for a boundary condition of 3 inches of water.... 244

D-9. Modified boundary conditions used by Segerlind
(1982) ........................................... 246

D-10. 85 percent isobaric lines from Segerlind (1982).. 246










D-11. Isobaric pattern for a duct pressure of 3 inches
of water from Segerlind (1982) ................... 247

D-12. Isobaric contour lines calculated using ANSYS
with a modified boundary condition of 1 inch of
water ............................................. 248

D-13. Isobaric contour lines calculated using ANSYS
with a modified boundary condition of 2 inches
of water ......................................... 249

D-14. Isobaric contour lines calculated using ANSYS
with a modified boundary condition of 3 inches
of water ......................................... 250

D-15. Vertical and Horizontal Cuttings Planes used
by Khompis (1983) ................................ 252

D-16. Location of the elements on the vertical
cutting plane from Khompis (1983) ................ 253

D-17. Location of the elements on the horizontal
cutting plane from Khompis (1983) ................ 254

D-18. Isobaric distribution in percent of total
pressure loss on a vertical cutting plane from
Khompis (1983) .................................. 256

D-19. Isobaric distribution in percent of total
pressure loss on horizontal cutting plane at 5
percent of bin height from Khompis (1983)......... 257

D-20. Isobaric distribution in percent of total
pressure loss on a horizontal cutting plane at
50 percent of bin height from Khompis (1983)..... 258

D-21. 70 percent of total pressure loss isobaric
surface from Khompis (1983) ..................... 260

D-22. 70 percent of total pressure loss isobaric
surface from Khompis (1983)....................... 261

D-23. 70 percent of total pressure loss isobaric
surface from Khompis (1983) ..................... 262

D-24. Velocity distribution in meters per minute on
a vertical cutting plane from Khompis (1983)..... 263

D-25. Velocity distribution in meters per minute on
a horizontal cutting plane at 5 percent of bin
height from Khompis (1983) ....................... 264


xiii










D-26. Velocity distribution in meters per minute on
a vertical cutting plane at 50 percent of bin
height from Khompis (1983)........................ 265

D-27. Minimum velocity regions for the square duct
system from Khompis (1983) ....................... 266

D-28. Constant velocity surface of 2.4 meters per
minute from Khompis (1983) ....................... 267

D-29. Constant velocity surface of 2.4 meters per
minute from Khompis (1983)........................ 268

D-30. Constant velocity surface of 2.4 meters per
minute from Khompis (1983) ....................... 269

D-31. Isobaric distribution calculated using ANSYS in
percent of total pressure loss on the vertical
cutting plane.................................... 271

D-32. Isobaric distribution calculated using ANSYS in
percent of total pressure loss on the horizontal
cutting plane at 5 percent of bin height.......... 272

D-33. Isobaric distribution calculated using ANSYS in
percent of total pressure loss on the horizontal
cutting plane at 50 percent of bin height......... 273

D-34. 70 percent of total pressure loss isobaric
surface produced by ANSYS, view 1................. 276

D-35. 70 percent of total pressure loss isobaric
surface produced by ANSYS, view 2................. 277

D-36. 70 percent of total pressure loss isobaric
surface produced by ANSYS, view 3................. 278

D-37. Velocity distribution calculated using ANSYS in
meters per minute on a verical cutting plane..... 281

D-38. Velocity distribution calculated using ANSYS in
meters per minute on a horizontal cutting plane
at 5 percent of bin height....................... 282

D-39. Velocity distribution calculated using ANSYS in
meters per minute on a horizontal cutting plane
at 50 percent of bin height ...................... 283

D-40. Constant velocity surface of 2.8 meters per
minute calculated using ANSYS, view 1............. 287

D-41. Constant velocity surface of 2.8 meters per
minute calculated using ANSYS, view 2............. 288










D-42. Constant velocity surface of 2.8 meters per
minute calculated using ANSYS, view 3............ 289

E-l. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 1-2, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45)...................................... ......... 291

E-2. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Conurtion 1-2, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29)............................................... 292

E-3. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 1-2, 1st
layer thermocouple locations 1, 9, 10, 14, and
15)... ........................................... 293

E-4. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 2-1, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45).............................................. 294

E-5. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 2-1, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29)... ........................................... 295

E-6. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 2-1, 1st
layer thermocouple locations 1, 9, 10, 14, and
15).... ........................................... 296

E-7. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 2-2, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45) ............................................. 297

E-8. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 2-2, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29)...................................... ......... 298










E-9. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 2-2, 1st
layer thermocouple locations 1, 9, 10, 14, and
15).... ........................................... 299

E-10. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 3-1, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45)... ........................................... 300

E-ll. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 3-1, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29)... ........................................... 301

E-12. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 3-1, 1st
layer thermocouple locations 1, 9, 10, 14, and
15)... ........................................... 302

E-13. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 3-2, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45)... ........................................... 303

E-14. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 3-2, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29)... ........................................... 304

E-15. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 3-2, 1st
layer thermocouple locations 1, 9, 10, 14, and
15)...................................... ......... 305

E-16. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 4-1, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45).... ........................................... 306

E-17. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 4-1, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29).... ........................................... 307


xvi










E-18. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 4-1, 1st
layer thermocouple locations 1, 9, 10, 14, and
15)..... .................... ...................... 308

E-19. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 4-2, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45).............................................. 309

E-20. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 4-2, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29)............................................... 310

E-21. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 4-2, 1st
layer thermocouple locations 1, 9, 10, 14, and
15).............................................. 311

E-22. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 5-1, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45).............................................. 312

E-23. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 5-1, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29)... ........................................... 313

E-24. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 5-1, 1st
layer thermocouple locations 1, 9, 10, 14, and
15).............................................. 314

E-25. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 5-2, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45)... ........................................... 315

E-26. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 5-2, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29)... ........................................... 316


xvii










E-27. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 5-2, Ist
layer thermocouple locations 1, 9, 10, 14, and
15)...................................... ......... 317

E-28. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 6-1, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45)... ........................................... 318

E-29. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 6-1, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29).............................................. 319

E-30. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 6-1, 1st
layer thermocouple locations 1, 9, 10, 14, and
15)... ........................................... 320

E-31. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 6-2, 3rd
layer thermocouple locations 31, 35, 36, 44, and
45).... ........................................... 321

E-32. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 6-2, 2nd
layer thermocouple locations 23, 24, 26, 27, and
29)...................................... ......... 322

E-33. Experimental center temperature response of
oranges in an experimental orange carton for
forced-air cooling (Boundary Condition 6-2, 1st
layer thermocouple locations 1, 9, 10, 14, and
15).... .............................. ............. 323

F-l. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 1-2, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45)... 325

F-2. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 1-2, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29)... 326


xviii










F-3. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 1-2, Ist layer
thermocouple locations 1, 9, 10, 14, and 15) ..... 327

F-4. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 2-1, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45)... 328

F-5. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 2-1, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29)... 329

F-6. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 2-1, Ist layer
thermocouple locations 1, 9, 10, 14, and 15) ..... 330

F-7. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 2-2, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45)... 331

F-8. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 2-2, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29)... 332

F-9. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 2-2, 1st layer
thermocouple locations 1, 9, 10, 14, and 15)..... 333

F-10. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 3-1, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45)... 334

F-ll. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 3-1, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29)... 335

F-12. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 3-1, 1st layer
thermocouple locations 1, 9, 10, 14, and 15) ..... 336

F-13. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 3-2, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45)... 337










F-14. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 3-2, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29).. 338

F-15. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 3-2, 1st layer
thermocouple locations 1, 9, 10, 14, and 15).... 339

F-16. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 4-1, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45).. 340

F-17. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 4-1, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29)... 341

F-18. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 4-1, 1st layer
thermocouple locations 1, 9, 10, 14, and 15)..... 342

F-19. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 4-2, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45)... 343

F-20. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 4-2, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29)... 344

F-21. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 4-2, Ist layer
thermocouple locations 1, 9, 10, 14, and 15)..... 345

F-22. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 5-1, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45)... 346

F-23. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 5-1, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29)... 347

F-24. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 5-1, Ist layer
thermocouple locations 1, 9, 10, 14, and 15)..... 348










F-25. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 5-2, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45)... 349

F-26. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 5-2, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29)... 350

F-27. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 5-2, 1st layer
thermocouple locations 1, 9, 10, 14, and 15) ..... 351

F-28. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 6-1, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45)... 352

F-29. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 6-1, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29)... 353

F-30. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 6-1, 1st layer
thermocouple locations 1, 9, 10, 14, and 15) ..... 354

F-31. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 6-2, 3rd layer
thermocouple locations 31, 35, 36, 44, and 45)... 355

F-32. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 6-2, 2nd layer
thermocouple locations 23, 24, 26, 27, and 29)... 356

F-33. Predicted center temperature response of oranges
in an experimental orange carton for forced-air
cooling (Boundary Condition 6-2, 1st layer
thermocouple locations 1, 9, 10, 14, and 15) ..... 357

G-l. Predicted versus experimental data for Boundary
Condition 1-2, 3rd layer thermocouples 31 and 35. 359

G-2. Predicted versus experimental data for Boundary
Condition 1-2, 3rd layer thermocouples 44 and 45. 360

G-3. Predicted versus experimental data for Boundary
Condition 1-2, 2nd layer thermocouples 23 and 24. 361










G-4. Predicted
Condition

G-5. Predicted
Condition

G-6. Predicted
Condition

G-7. Predicted
Condition

G-8. Predicted
Condition

G-9. Predicted
Condition

G-10. Predicted
Condition

G-ll. Predicted
Condition

G-12. Predicted
Condition

G-13. Predicted
Condition

G-14. Predicted
Condition

G-15. Predicted
Condition

G-16. Predicted
Condition

G-17. Predicted
Condition

G-18. Predicted
Condition

G-19. Predicted
Condition

G-20. Predicted
Condition

G-21. Predicted
Condition


versus experimental data for
1-2, 2nd layer thermocouples

versus experimental data for
1-2, 1st layer thermocouples

versus experimental data for
1-2, 1st layer thermocouples

versus experimental data for
2-1, 3rd layer thermocouples

versus experimental data for
2-1, 3rd layer thermocouples

versus experimental data for
2-1, 2nd layer thermocouples

versus experimental data for
2-1, 2nd layer thermocouples

versus experimental data for
2-1, Ist layer thermocouples

versus experimental data for
2-1, 1st layer thermocouples

versus experimental data for
2-2, 3rd layer thermocouples

versus experimental data for
2-2, 3rd layer thermocouples

versus experimental data for
2-2, 2nd layer thermocouples

versus experimental data for
2-2, 2nd layer thermocouples

versus experimental data for
2-2, 1st layer thermocouples

versus experimental data for
2-2, 1st layer thermocouples

versus experimental data for
3-1, 3rd layer thermocouples

versus experimental data for
3-1, 3rd layer thermocouples

versus experimental data for
3-1, 2nd layer thermocouples


Boundary
26 and 29. 362

Boundary
1 and 9... 363

Boundary
14 and 15. 364

Boundary
31 and 35. 365

Boundary
44 and 45. 366

Boundary
23 and 24. 367

Boundary
26 and 29. 368

Boundary
1 and 9... 369

Boundary
14 and 15. 370

Boundary
31 and 35. 371

Boundary
44 and 45. 372

Boundary
23 and 24. 373

Boundary
26 and 29. 374

Boundary
1 and 9... 375

Boundary
14 and 15. 376

Boundary
31 and 35. 377

Boundary
44 and 45. 378

Boundary
23 and 24. 379


xxii










G-22. Predicted versus experimental data for Boundary
Condition 3-1, 2nd layer thermocouples 26 and 29. 380


G-23. Predicted versus experimental data for Boundary
Condition 3-1, 1st layer thermocouples 1 and 9...


381


G-24. Predicted versus experimental data for
Condition 3-1, 1st layer thermocouples

G-25. Predicted versus experimental data for
Condition 3-2, 3rd layer thermocouples

G-26. Predicted versus experimental data for
Condition 3-2, 3rd layer thermocouples


Boundary
14 and 15. 382

Boundary
31 and 35. 383

Boundary
44 and 45. 384


G-27. Predicted versus experimental data for Boundary
Condition 3-2, 2nd layer thermocouples 23 and 24.


G-28. Predicted versus experimental data for
Condition 3-2, 2nd layer thermocouples

G-29. Predicted versus experimental data for
Condition 3-2, 1st layer thermocouples

G-30. Predicted versus experimental data for
Condition 3-2, 1st layer thermocouples

G-31. Predicted versus experimental data for
Condition 4-1, 3rd layer thermocouples


Boundary
26 and 29.

Boundary
1 and 9...

Boundary
14 and 15.

Boundary
31 and 35.


G-32. Predicted versus experimental data for Boundary
Condition 4-1, 3rd layer thermocouples 44 and 45.

G-33. Predicted versus experimental data for Boundary
Condition 4-1, 2nd layer thermocouples 23 and 24.


G-34. Predicted versus experimental data for
Condition 4-1, 2nd layer thermocouples

G-35. Predicted versus experimental data for
Condition 4-1, 1st layer thermocouples

G-36. Predicted versus experimental data for
Condition 4-1, 1st layer thermocouples

G-37. Predicted versus experimental data for
Condition 4-2, 3rd layer thermocouples

G-38. Predicted versus experimental data for
Condition 4-2, 3rd layer thermocouples


Boundary
26 and 29.

Boundary
1 and 9...

Boundary
14 and 15.

Boundary
31 and 35.

Boundary
44 and 45.


G-39. Predicted versus experimental data for Boundary
Condition 4-2, 2nd layer thermocouples 23 and 24.


xxiii










G-40. Predicted
Condition

G-41. Predicted
Condition

G-42. Predicted
Condition

G-43. Predicted
Condition

G-44. Predicted
Condition

G-45. Predicted
Condition

G-46. Predicted
Condition

G-47. Predicted
Condition

G-48. Predicted
Condition

G-49. Predicted
Condition

G-50. Predicted
Condition

G-51. Predicted
Condition

G-52. Predicted
Condition

G-53. Predicted
Condition

G-54. Predicted
Condition

G-55. Predicted
Condition

G-56. Predicted
Condition

G-57. Predicted
Condition


versus experimental data for Boundary
4-2, 2nd layer thermocouples 26 and 29. 398

versus experimental data for Boundary
4-2, 1st layer thermocouples 1 and 9... 399

versus experimental data for Boundary
4-2, 1st layer thermocouples 14 and 15. 400

versus experimental data for Boundary
5-1, 3rd layer thermocouples 31 and 35. 401

versus experimental data for Boundary
5-1, 3rd layer thermocouples 44 and 45. 402

versus experimental data for Boundary
5-1, 2nd layer thermocouples 23 and 24. 403

versus experimental data for Boundary
5-1, 2nd layer thermocouples 26 and 29. 404

versus experimental data for Boundary
5-1, 1st layer thermocouples 1 and 9... 405

versus experimental data for Boundary
5-1, 1st layer thermocouples 14 and 15. 406

versus experimental data for Boundary
5-2, 3rd layer thermocouples 31 and 35. 407

versus experimental data for Boundary
5-2, 3rd layer thermocouples 44 and 45. 408

versus experimental data for Boundary
5-2, 2nd layer thermocouples 23 and 24. 409

versus experimental data for Boundary
5-2, 2nd layer thermocouples 26 and 29. 410

versus experimental data for Boundary
5-2, 1st layer thermocouples 1 and 9... 411

versus experimental data for Boundary
5-2, 1st layer thermocouples 14 and 15. 412

versus experimental data for Boundary
6-1, 3rd layer thermocouples 31 and 35. 413

versus experimental data for Boundary
6-1, 3rd layer thermocouples 44 and 45. 414

versus experimental data for Boundary
6-1, 2nd layer thermocouples 23 and 24. 415


xxiv










G-58. Predicted versus experimental data for Boundary
Condition 6-1, 2nd layer thermocouples 26 and 29. 416

G-59. Predicted versus experimental data for Boundary
Condition 6-1, 1st layer thermocouples 1 and 9... 417

G-60. Predicted versus experimental data for Boundary
Condition 6-1, 1st layer thermocouples 14 and 15. 418

G-61. Predicted versus experimental data for Boundary
Condition 6-2, 3rd layer thermocouples 31 and 35. 419

G-62. Predicted versus experimental data for Boundary
Condition 6-2, 3rd layer thermocouples 44 and 45. 420

G-63. Predicted versus experimental data for Boundary
Condition 6-2, 2nd layer thermocouples 23 and 24. 421

G-64. Predicted versus experimental data for Boundary
Condition 6-2, 2nd layer thermocouples 26 and 29. 422

G-65. Predicted versus experimental data for Boundary
Condition 6-2, 1st layer thermocouples 1 and 9... 423

G-66. Predicted versus experimental data for Boundary
Condition 6-2, 1st layer thermocouples 14 and 15. 424

H-1. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 1-2 and test times of 0.75, 1.75, and
2.75 hours ....................................... 426

H-2. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 2-1 and test times of 2, 5, and 8
hours ............................................ 427

H-3. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 2-2 and test times of 1, 4, and 7
hours ........................................... 428

H-4. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 3-1 and test times of 2, 4, and 6
hours . .......................................... 429

H-5. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 3-2 and test times of 0.4, 1.2, and 2.0
hours ........................................... 430


xxv










H-6. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 4-1 and test times of 1, 4, and 7
hours ........................................... 431

H-7. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 4-2 and test times of 1, 2.5, and 4
hours ............................................ 432

H-8. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 5-1 and test times of 2, 5, and 8
hours ............................................ 433

H-9. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 5-2 and test times of 1, 2.5, and 4
hours ............................................ 434

H-10. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 6-1 and test times of 2, 6, and 10
hours ............................................ 435

H-ll. Regression plot of the predicted temperature
versus the experimental temperature for Boundary
Condition 6-2 and test times of 1, 2, and 5
hours ............................................ 436


xxvi
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


PRESSURE AND VELOCITY DISTRIBUTION FOR AIR FLOW
THROUGH FRUITS PACKED IN SHIPPING CONTAINERS
USING POROUS MEDIA FLOW ANALYSIS

By

Michael Thomas Talbot

August, 1987


Chairman: Dr. C. C. Oliver
Major Department: Mechanical Engineering


Detailed pressure and velocity fields for low air flow

rates used for forced cooling of oranges packed in

containers cannot be established experimentally. The

superficial velocity and overall pressure drop can be

determined for oranges packed in bulk and the temperature

response during cooling of oranges packed in containers can

be measured.

Flow analysis requires the consideration of at least a

second order velocity term to account for friction and

dynamic effects. The theory of fluid flow through porous

media incorporates such an analysis and has been

successfully applied to determine the pressure and velocity

fields for air flow through semi-infinite bulk agricultural

products. The objective of this study was to evaluate the


xxvii











applicability of porous media flow theory for pressure and

velocity flow field predictions for the finite boundary

conditions of air flow through oranges packed in shipping

containers.

A commercial finite element solution package was

adapted to determine the pressure and velocity distribution

and was verified by comparison with existing detailed

studies of bulk systems. Porous media input parameters and

boundary condition specifications for the current study were

obtained from studies of pressure loss through oranges in

bulk and in simulated orange cartons. An existing heat

transfer program was modified to incorporate predicted

velocity inputs and calculate temperature response.

Twelve tests were conducted using an experimental

orange carton with 88 Valencia oranges (size 100) packed in

a face-centered cubic packing arrangement using air flow
-3 -2 3
rates ranging from 1.5 x 10 to 2.0 x 10 m /s. The tests

involved six different venting arrangements. Temperature

responses were measured for 58 orange centers, 10 orange

surfaces, and 10 air spaces.

The measured temperature responses for the twelve tests

were compared to the predicted temperature responses and

provided indirect evaluation of the input flow predictions.

Variable porosity to account for edge effects was used to

improve initial results.

Although several areas for improvement were noted, the

porous media flow analysis was found to provide necessary


xxviii











flow field information to calculate the thermal response for

various flow boundary conditions.


xxix
















INTRODUCTION


Losses of fresh fruits and vegetables and other

horticultural commodities from decay and shriveling as a

result of poor temperature management during postharvest

handling, transportation, and marketing are substantial in

the United States, as well as in other production and

consumption areas of the world.

Reduction of such losses is perhaps of more immediate

importance than increasing yields through improved

production practices. This is especially apparent when one

considers that all of the cost and energy inputs for

production, harvesting, packing, transportation, etc., are

wasted for that portion of the supply for which losses occur

near the point of consumption. For most fruits and

vegetables, the total cost for harvesting, packing,

transportation, storage, selling, etc., greatly exceeds the

cost of production. Therefore, the direct costs which can

be attributed to losses during marketing are magnified

greatly when compared with the alternative of increased

production to effect increased supply.

Respiration is the overall process by which stored

organic materials (carbohydrates, proteins, fats) are broken

into simple end products with a release of energy. Oxygen










(02) is used in this process and carbon dioxide (C02) is

produced by the commodity. The loss of the stored organic

materials from the commodity during respiration means

hastened senescence as the energy for maintaining the living

status of the commodity is exhausted, loss of food value

(energy value) for the consumer, reduced flavor quality,

especially sweetness, and loss of salable dry weight. The

rate of deterioration (perishability) of harvested

commodities is generally proportional to their respiration

rate. Product temperature is a major determinant of the

rate of respiratory activity.

Water loss can be one of the main causes of

deterioration since it results in not only direct

quantitative losses, but also causes losses in appearance

(due to wilting and shriveling), textural quality

(softening, flaccidity, limpness, loss of crispness, and

juiciness), and nutritional quality. The rate of water loss

is controlled by the vapor-pressure difference between the

product and its environment, which is governed by the

temperature and relative humidity.

Ethylene (C2H4) is the simplest organic compound that

has an affect on physiological processes of plants. It is a

natural product of plant metabolism and is considered the

natural aging and ripening hormone. Generally, ethylene

production increases with maturity at harvest, physical

injuries, disease incidence, increased temperatures up to

30 C (86 F), and water stress. On the other hand, ethylene










production rates of fresh horticultural crops are reduced by

storage at the lowest safe temperature of each commodity.

Temperature affects both the rate of ethylene production and

the sensitivity of products to ethylene.

After harvest, fresh horticultural commodities are

susceptible to attack and damage by microbial pathogens.

Lower temperature affects the rate of growth and spread of

these microorganisms. Physical injuries can result from

abuses to fresh commodities at any temperature, but low

temperatures reduce the results of injuries.

Temperature is the most important environmental factor

that influences the deterioration rate of harvested

commodities. It is a well known fact that for each increase

of 10 C (18 F) above the optimum temperature, the rate of

deterioration increases two- to threefold.

Thus improved cooling of fruits and vegetables before

or during shipment, along with proper temperature

maintenance throughout the marketing channels, has the

potential of greatly reducing these losses.

Nearly all fresh fruits and vegetables are now marketed

in corrugated fiberboard shipping containers. These

containers provide a barrier to proper air flow and

efficient heat transfer required for cooling. Although

cooling of fruits and vegetables before packing is much more

desirable from a heat transfer standpoint, most products are

not cooled until after packing because of other

relationships in the overall handling system. Cooling of










fruits and vegetables during transit is a particular problem

because of inherent restrictions on the size of the cooling

units and on the design of the air circulation system in

transit vehicles. Such systems are often inadequate for

cooling of fresh produce and problems of overheating, over-

ripening, and losses due to decay are common. These problems

are further aggravated by increased use of palletized

shipping, introduction of new types of produce containers

and moves toward maximum utilization of vehicle volume

because of increased transportation costs. The need for

additional research on air movement as related to cooling of

fruits and vegetables in palletized shipments has been

widely recognized by industry sources.

During the last thirty years, numerous researchers and

commercial shipping companies have conducted research aimed

at improved cooling of fruits and vegetables packed in

shipping containers. Although substantial improvements have

been made, the current level of losses during marketing

indicates that much work must be done. Significant further

improvements can come only from complete design of the

containers, vent arrangements, stacking and loading

patterns, air circulation systems and refrigeration

controls, based on sound engineering principles. Before

this can be accomplished, however, basic research is needed

on the mechanisms of heat and mass transfer from the packed

product to the surrounding air and on the air flow

relationships involved. The objectives of this study follow.















OBJECTIVES


The objectives of this study were to


1. Study the basic principles related to air flow through

fruits (and vegetables) packed in fiberboard packing

containers.


2. Determine the feasibility of the theory of flow through

porous media analysis for this problem.


3. Develop a numerical model to predict air pressure and

velocity distribution for air flow through fruits (and

vegetables) packed in fiberboard packing containers.


4. Evaluate the experimental cooling response using the

mathematical model for flow in conjunction with an existing

heat and mass transfer model.















REVIEW OF LITERATURE


Background of Research


A considerable amount of information has been published

related to basic theory and experimental research on heat

and mass transfer of fruits and vegetables and on various

aspects of air cooling produce packed in shipping

containers. A number of publications (Anthony 1969, Camp

1979, Chalutz et al. 1974, Harris et al. 1969, Harris and

Harvey 1976, Hinds and Robertson 1965, Hinsch et al. 1978,

McDonald and Camp 1973, Mitchell et al. 1968, Pentzer et al.

1945, Redit and Hamer 1959, Rij 1977, Winston et al. 1959)

provide information on product cooling response during test

shipments of fruits and vegetables in refrigerated vehicles

as related to loading patterns and air flow arrangement.

Such tests give general information on whether a particular

system is better than another, but yield little of the

basic engineering information needed to design improved

transit cooling systems. It is obvious from most of these

tests that contact of the cooling air with the product

(rather than the capacity of the refrigeration system) was a

limited factor in the rate of cooling because of the rather

large cooling times in relation to the product heat loads.

Biales et al. (1973) and Kindya and Bongers (1979) conducted










shipping tests on cooling of grapefruit with outside ambient

air on non-refrigerated ships. They showed that moderate

amounts of product cooling can be obtained in this manner in

certain regions during times of year when the outside air

temperature is relatively low. Such practice has

implications relative to cost savings and energy reduction

and points out the even greater importance of basic

engineering design of the system for optimum performance

when the source of refrigeration is limited.

Some researchers (Felsenstein and Zafrir 1975, Hale et

al. 1973, Hinsch et al. 1978, McDonald and Camp 1973,

McDonald et al. 1979, Olsen et al. 1960, Patchen 1969)

have developed new carton designs or have studied the

effect of different carton venting arrangements or air flow

directions through stacks of cartons as an aid to improved

cooling. For the most part, these were all comparative

tests and little basic engineering information on air flow

rates, air distribution and other important parameters was

obtained. Additional studies (Fountain and Hovey 1970,

Hale et al. 1969, Hale and Risse 1974, Olsen et al. 1960)

have involved packing fruits in shipping containers with

pulp or plastic tray packs. However, little information was

given to the direct influence of the tray packs on cooling

rates other than total cooling times or final temperature at

the end of test shipments.

A large amount of experimental research aimed directly

at studying cooling response of fruits and vegetables packed










in shipping containers as influenced by different variables,

has been conducted by various researchers: Chalutz et al.

(1974), Gentry and Nelson (1964), Grierson et al. (1970),

Grierson (1975), Guillou (1960), Haas and Felsenstein

(1975), Kasmire (1977), Kushman and Ballinger (1962),

Leggett and Sutton (1951), Lindsay et al. (1976), Mitchell

et al. (1972), Parson et al. (1972), Patchen and Schomer

(1971), Pentzer et al. (1945), Perry et al. (1971),

Richardson et al. (1955), Sainsbury (1951) and (1961),

Sainsbury and Schomer (1967), and Soule et al. (1969).

Results of such tests provide data on product temperature

response as related to product size, type of container,

venting and stacking arrangement and, in most cases, air

temperature and air flow rate. Most of these tests were

conducted in a valid manner and provide excellent general

information on the influence of different variables as they

affect cooling rates. Also, they provide some specific

engineering data useful in the design of cooling systems but

only for the particular product size, the particular

containers, stacking arrangements, vent designs, size of

product load, air flow rates and temperatures used in the

experiments. In most of these tests, the range of variables

studied was quite limited. In certain tests, the mass

transfer effects on product loss and evaporative cooling,

other than the overall product weight loss measurements

under the particular conditions of the test, were not

obtained.










In order to provide the necessary information for

design and analysis of alternate cooling systems,

experiments must be conducted in a much more basic manner

than those described above. The interaction of the cooling

air with the product, the individual containers, and stacks

of containers as related to the basics of heat and mass

transfer should be evaluated for different product sizes,

shapes, and packing configurations. More basic studies of

air flow patterns and pressure drops in and around cartons

and stacks of cartons are needed. A limited amount of this

basic type of research was conducted. Van Beek (1975)

studied the effective thermal conductivity of packed

spherical products and evaluated the effect of natural

convection for different heat flow directions. Meffert and

Van Vliet (1971) developed a test system for evaluation of

air circulation and product temperature in transit vehicles

and validated the results with experiments on model cargo.

Wang and Tunpun (1969) conducted basic studies on forced air

cooling of tomatoes in cartons. They presented information

on temperature response of fruits at 36 positions within two

cartons stacked in register at three different air flow

rates. They also conducted some basic studies of pressure

drop versus air flow relationships for bulk fruit and

cartons, packed and empty. Haas et al. (1976) studied

pressure drop relationships for oranges, and cartons,

packed and empty. They also studied the effect of vent hole

size, total area, position, and number on carton pressure










drop. Chau et al. (1983) studied pressure drop, as a

function of air flow and packing porosity, for four

different sizes of oranges with four different stacking

patterns. They also studied the effect the shape and number

of carton vent openings had on carton pressure drop.

Ramaker (1974) investigated the thermal resistance of

corrugated fiber board as used in shipping containers.

Some basic work has been done on analytical methods for

evaluating heat transfer in produce packed in cartons. Hicks

(1955) and Sainsbury (1961) used basic conduction equations

to evaluate theoretical cooling rates as a function of the

convective heat transfer coefficient at the container

surfaces. Meffert et al. (1971) and Meffert and Van Beek

(1975) used analytical solutions for transient heat transfer

to predict temperature response in stacks of produce

including heat generation from product respiration.

However, such solutions are applicable only to certain

limited cases. Information given in other publications

(Breakiron 1974, Gaffney 1977, Goddard 1972, Ryall and

Pentzer 1967, Stewart 1977) has pointed out the need for

additional basic research so that cooling systems may be

designed based upon sound engineering principles.

Researchers at the University of Florida have extensive

background in research related to heat and mass transfer in

fruits and vegetables. Most of the experimental work has

involved basic studies on cooling of oranges, grapefruits,

avocados and bell peppers in bulk loads (Baird and Gaffney










1974a, 1974b, and 1976; Chau et al. 1983, 1984; Gaffney

1977; Gaffney and Baird 1977, 1980). Certain methodologies

and techniques for measurement of product temperature, air

temperature, flow rates and humidity have been developed and

refined during the course of this research (Baird et al.

1975; Gaffney et al. 1980, 1985a, 1985b). In addition to

the experimental work, a computer based mathematical model

has been developed to analyze heat and mass transfer during

cooling of fruits and vegetables in bulk loads (Baird and

Gaffney 1974a, 1976; Eshleman et al. 1976; Chau et al.

1984). Currently experimental work in this area involves

measurement of thermal and physical properties and

transpiration coefficients of oranges and grapefruits and

theoretical studies of heat and mass transfer during surface

moisture drying of citrus.

This study was a subcomponent of a project concerning

heat and mass transfer relationships of fruit and vegetables

packed in fiberboard shipping containers. To understand and

model the heat transfer during cooling and storage of fresh

fruits and vegetables packed in shipping containers, the

pressure and velocity field characteristics within the

container must be established. Both distributions are

difficult to establish experimentally. There are many

interrelated variables involved in air cooling of fruits

and vegetables. These include thermal properties, physical

properties, size, and shape of the product and temperature,

flow rate and relative humidity of the cooling air. When










cooling products in containers, additional variables of

importance are container size, shape and wall thickness,

venting arrangements, stacking arrangements, product packing

configurations, and the direction of air flow.

The actual process that occurs between the individual

fruits (or vegetables) and the flowing air is complex and

not well understood. Previous researchers have studied the

cooling, heating and drying of semi-infinite systems of bulk

piled agricultural products such as fruits, grains,

vegetables, nuts, and root crops using the theory of fluid

flow through a porous media to determine the pressure and

velocity fields. The validity of porous media flow theory

for finite boundary conditions of air flow through fresh

fruits and vegetables packed in shipping containers will be

evaluated. The theory of porous media flow will be

considered before looking at previous work related to other

agricultural products.



Fluid Flow Through Porous Media


In addition to the interest received from a purely

scientific point of view, the theory of flow through porous

media has become important in many fields of scientific and

engineering applications.

Such diversified fields as ground water hydrology,

petroleum engineering, civil engineering, agricultural

engineering, chemical engineering, soil mechanics (physics),

water purification, industrial filtration, ceramic











engineering, powder metallurgy, medicine and bioengineering,

all rely heavily upon porous media flow theory as

fundamental to their individual applications.

Perhaps the most important of these areas of technology

that depend on the properties of porous media is hydrology,

which relates to water movement in earth and sand

structures, like dams, flow to walls from water-bearing

formation, intrusion of sea water in coastal areas, filter

beds for purification of drinking water and sewage, etc.,

and petroleum engineering which is primarily concerned with

petroleum and natural gas production, exploration, well

drilling, and logging, etc. In the introductory chapter of

the book by Muskat (1946), R.D. Wyckoff points out that

despite the great similarity of the physical systems and the

processes in these two fields of technology, each has

produced distinct technical literature and terminology.

Therefore, it is no surprise that as each branch of science

and engineering has addressed specific problems, each has

contributed to the vast amount of literature available

related to porous media flow theory. Unfortunately, this

literature is often difficult to correlate.

Porous media was defined by Collins (1961) as a solid

containing pores or voids of sufficient number, either

connected or nonconnected, dispersed within it in either a

regular or random manner. The interconnected pore space is

termed effective pore space. Fluid can flow through a

porous material only through the effective pore space.











Muskat (1946) represented an ideal porous media as a body of

unconsolidated sand in which all the voids are

interconnected, which means total pore space is actually

effective pore space.

By the above definition, a system of agricultural

products, in bulk piles or in storage structures, which are

of non-uniform shape and size, such as fruits, vegetables,

grains, beans, peanuts and root crops, can be considered as

an ideal porous media. A homogenous fluid, such as air or

water, used as a medium for cooling, heating or drying, can

flow through an assembly of interconnected channels of

varying sizes and shapes in this system of agricultural

products.

The actual process that occurs between individual

product particles and the flowing fluid is complex and not

well understood. For this reason, the only feasible method

of analysis is a macroscopic approach, which implies that

the values of variables under consideration are averages of

indeterminate instantaneous values.

Muskat (1946), Scheidegger (1960), and others indicated

that the earliest recorded investigation of the flow through

porous media was that of Darcy in 1856. Darcy was interested

in the flow characteristics of sand filters. Because of the

analytical difficulties in describing this type of flow,

Darcy had to resort to an experimental study of the problem.

His classic experiments led to the real foundation of the

quantitative theory of the flow of homogeneous fluids











through a porous media. He ran experiments in water flowing

through a vertical pipe filled with sand. From his

investigations, Darcy concluded that the flow rate was

proportional to the pressure loss and the cross-sectional

area and inversely proportional to the length of the flow

path. The famous Darcy's Law is written as follows:


(1) Q = k A Ah/ L


Dividing both sides by the cross-sectional area, A, yields


(2) V = k Ah/ L
s


where

Q = flow rate in volume per unit time

k = permeability or hydraulic conductivity

A = the cross-sectional area

Ah = pressure loss or head

AL = the length of the flow path

V = the superficial velocity
s


The study of the validity of Darcy's Law has been the

subject of numerous investigations. These investigations

have been of essentially two types: those with the

objectives of either verifying Equation 1 or establishing an

appropriate modification of this equation and those

concerned with the nature of the constant, k, as determined

by the properties of the sand or porous media. The bases of









nearly all engineering calculations for porous media flow

problems have originated from Darcy's Law and/or purely

empirical determinations.

Later investigators found that Darcy's Law is limited

in its application only to very low velocity (creep) flow

and becomes invalid when inertial forces become effective.

Since then, several related theories and approaches have

been developed to approximate the flow of fluid through

porous media. Scheidegger (1960) recorded six different

theories, namely 1. heuristic correlation (curve fitting);

2. dimensional analysis; 3. capillaric model; 4. tube bundle

theory (sometimes known as Kozeny Theory); 5. drag theory;

and 6. statistical theory. Bird et al. (1960) recognized

two general theories, namely tube bundle and drag theories.

The most acceptable one and the one which has been pursued

by other workers is the tube bundle theory. This theory

assumes that the packed column is a bundle of tangled tubes

of irregular cross-section and that the established

principles of flow through a single tube are applicable.

Using the tube bundle approach in introducing the

parameter's porosity, hydraulic radius and mean particle

diameter into the Hagen-Poiseuille equation, Bird et al.

(1960) presented the following equation for laminar flow

through porous media:


2 2 3
(3) AP/AL = 150v V (1-c) /D C
s p










where

= porosity

= void volume/total volume

p = fluid viscosity

D = mean diameter of particles of the porous material
P

AP = pressure loss

and the rest of the symbols are as previously defined.


This is the Blake-Kozeny equation (first derived by

Kozeny and modified by Blake), which is valid for modified

Reynolds number equivalent to V D p/w(l-E) less than 10.

This equation is identical to Darcy's equation where the

permeability, k, is expressed in terms of the characteristic

properties of the material and the fluid.

Similarly, for turbulent flow through a porous media,

the relationship is


2 3
(4) AP/AL = 3.50 (pV /2)(1-e)/D
s p


where is the fluid density and the other symbols are as

previously defined. The kinetic energy term appears inside

the first parenthesis. This implies that the pressure drop

is proportional to the loss of kinetic energy which becomes

appreciable when the fluid velocity is large. This is the

Burke-Plummer equation (first derived by Burke and Plummer),

which is valid for modified Reynolds number greater than

1,000.











Based on the theory of Reynolds for resistance to fluid

flow, Ergun (1952) illustrated that a pressure drop through

a porous media is caused by the simultaneous viscous and

kinetic energy losses. Pursuing this, he developed a

general equation of fluid flow through porous media, which

combines the Blake-Kozeny equation for laminar flow and the

Burke-Plummer equation for turbulent flow. This equation

(the Ergun equation) is


2 2 3 2 3
(5) AP/AL = 150P V (1-E) /D E + 1.75 P V (l-E)/D E
s p s p


The first term on the right side represents the viscous

energy loss and the second term on the right side represents

the kinetic energy loss.

In terms of dimensionless groups, the Ergun equation

can be written as follows:


3 2
(6) APD e / LpV (1-e) = 150w(l-E)/pV D + 1.75
p s sp


In this form, one can see that if the velocity becomes

very large, the first term on the right becomes negligible

and the Ergun equation reduces to the Burke-Plummer

equation. On the other hand, if the velocity is very small,

the second term on the right is negligible compared to the

first and the Ergun equation reduces to the Blake-Kozeny

equation.










In three-dimensional form, the Ergun equation, Equation

(5), may be written as follows:


2
(7a) -9P/9x = f U + f U
1 2


and


2
(7b) -3P/ay = f V + f V
1 2


and


2
(7c) -9P/az = f W + f W
1 2


where

x,y,z = Rectangular coordinates

DP/3x = Pressure gradient decreasing along positive x
direction

DP/3y = Pressure gradient decreasing along positive y
direction

BP/8z = Pressure gradient decreasing along positive z
direction


2 2 3
f = 150l(l-E) /D
1 p

3
f = 1.75p(l-E)/D
2 p

U,V,W = Velocity in x,y, and z directions, respectively.

Bakker-Arkema et al. (1969), after an extensive review

of static pressure/air flow relationships in packed beds of










granular biological products, adopted the Ergun equation in

their experiments on one-dimensional cooling of cherry pits.

They indicated that the Ergun equation was the most general

among the available equations.

Stanek and Szekely (1972, 1973) applied the Ergun

equation in two-dimensional flow in both isothermal and non-

isothermal packed beds. They found that non-uniform bed

porosity could cause flow maldistribution.

Lai and Haque (1976) applied the Ergun equation in

analyzing the three-dimensional flow of air through non-

uniform grain beds by adding a third coordinate axis, z, as

indicated in Equation 7. The equations were first written

in vector notation as follows:


(8) -(grad P) = V (f + f IVI)
1 2

where

grad = gradient of a scalar function

V = Velocity in vector form

IVI = Magnitude of the velocity

and the rest of the symbols are as previously defined.

This, together with the continuity equation, was simplified

by transforming the equation into dimensionless forms and by

using a stream function. The resulting nonlinear partial

differential equations were solved numerically by a line by

line method. Non-uniform porosity was indicated to cause

different velocity and pressure distribution.









Lai (1980) used Ergun's equation to calculate the

three-dimensional airflow and pressure distributions

through a circular grain bin using the method of lines,

coupled with a sophisticated ordinary differential equation

integrator. This method is similar to the finite difference

method.

Hague et al. (1980) modified Ergun's equation to

calculate the pressure and velocity fields for airflow

through conical-top grain beds containing corn and fines

distributed nonuniformly. A two-dimensional (cylindrical

coordinate) finite difference technique was used.

Geertsma (1974) reported it necessary to apply a more

general flow law for the prediction or analysis of the

production behavior of gas wells. These wells produce high

single-phase fluid flow rates through the surrounding

reservoir rocks. He indicated the appropriate formula was

given in 1901 by Forchheimer and is written as



(9) -(grad P) = ajV + 8p JV V


where

a = coefficient of viscous flow resistance

= 1/k (approximately)

S= coefficient of inertial flow resistance.

This equation is noted to be of the same form as Equation 8.

Geertsma (1974) and Fircozabadi and Katz (1979)

discussed the widely differing views used to describe the

mechanisms that consume energy at more than a linear rate











with velocity. The 8 term has been given various names

depending on the users interpretation of the flow mechanism.

For example Fircozabadi and Katz (1979) indicated 8 should

be called "velocity coefficient" while Geertsma (1974)

defined B as indicated above as the "inertial coefficient."

Also noted was the difficulty relating the 8 term with the

porosity, permeability and other measurable porous media

characteristics.

Geertsma (1974) reported that the validity of Equation

9 could be checked by plotting the ratio pV/p as a function

of (grad P)/pV. Hie indicated that this plot must yield a

straignr line. Tne value of coefficient a will then be

determined by the intercept of the straight line plotted and

the (grad P)/pV axis while coefficient ( is found from the

slope of the line with respect to the x-axis. This

procedure is applicable for liquids only, since density p is

nearly independent of the pressure P.

For a gas, Equation 9 cannot be applied directly.

However, the equation is applicable for a small distance dx

in the flow direction. By introducing the mass flow rate

G = pV of the gas and using the ideal gas law, p = P/c in

which

c = RT/M

where

R = universal gas constant,

T = absolute temperature and

M = molecular weight,











Equation 9 can be rewritten as


2
(10) -p dP = apGc+BG c/dx.


By performing integration between x = L and x = 0, Equation

10 becomes


2 2
(11) -(P P )/2LcGy = a + B (G/p).
L 0


Therefore for gases, Geertsma (1974) reported that the

validity of Equation 11 could be checked by plotting G/p as

a function of the term on the left side of Equation 11. He

indicated that this plot must yield a straight line. The

value of coefficient a will then be determined by the

intercept of the straight line plotted and the axis of the

term on the left side of Equation 11, while coefficient P is

found from the slope of the line with respect to the x-axis.

The Agricultural Engineering Department, in addition to

other departments of the University of Florida, leases a

university version of the ANSYS*** general purpose finite

element computer program offered by Swanson Analysis Systems

(SASI) (DeSalvo and Swanson 1983).




***The information given herein is supplied with the
understanding that no discrimination is intended and no
endorsement by the Institute of Food and Agricultural
Sciences of the University of Florida is implied. The
listing of specific trade names herein does not constitute
endorsement of these products in preference to other
products which have equivalent capabilities.










During system verification and preparation for student

laboratory use, many of the practical structural, thermal

and flow engineering analysis techniques were tested. An

option exists that allows both 2-D and 3-D isoparametric

thermal solid elements to be used to model nonlinear

steady-state fluid flow through a porous medium (DeSalvo and

Swanson 1983).

The porous media flow problem is formulated in a

manner identical to that used for the thermal analysis,

requiring only a change of variables to use thermal analysis

to obtain a solution. Pressure is the variable rather than

temperature. The momentum equation is simplified to



(12) -(grad P) = Reff V


where

grad = gradient of a scalar function

P = pressure

V = seepage velocity vector


and


(13) Reff = i/K + Spl1v


where

p = gas viscosity

K = absolute permeability of porous medias

B = visco-inertial parameter

p = density.










The components of Equations 13 are


(14) -aP/3x = Reff Vx


(15) -2P/2y = Reff Vy


(16) -aP/az = Reff Vz.


The continuity equation is


(17) div (pV) = 0


where div = divergence of a vector field, or


(18) 3(pVx)/3x + 3(pVy)/3y + a(pVz)/3z = 0.


Combining Equations 14, 15, 16, and 18 yields


(19) a(k3P/ax)/ax + 3(k3P/ay)/3y + a(kRP/az)/az = 0


where k =p /Reff. Equation 19 is nonlinear because Reff is

a function of velocity. The coefficients of permeability,

k, are (kx, ky, kz) internally calculated for each

coordinate direction as


(20) k = Kp/(p+Kpp6VI)


To handle the flow of power law fluids, an exponent can be


applied to IVI as an additional analysis input variable.

Combining Equation 12 and 13 yields


-(grad P) = IV/K + BpIVj V,


~











which is the same as Equation 9. In fact, personal inquiry

to SASI revealed that the porous media option was based on

the work of Geertsma (1974) and Fircozabadi and Katz (1979).

Another modification of Darcy's Law that has received

considerable attention is a simplified, semi-empirical

approach to account for nonlinear flow patterns of air

through porous media. This is accomplished by assuming that

the air velocity is proportional to the pressure gradient

raised to a power. Shedd (1953) introduced the following

equation in the study of air flow through grain storage:


B
(22) V = A(DP/3n)


where


V = interstitial fluid velocity

DP/an = pressure gradient along any direction

A,B = experimentally determined constants.


Equation 22 is sometimes rearranged in terms of


b
(23) 3P/an = aV ,


where


a,b = experimentally determined constants


and referred to as the Ramsin Equation.

One major difference between Equation 22 and the Ergun

Equation, Equation 6, is the absence of porosity. It is











assumed to be lumped with other characteristic properties of

the fluid and porous medium into constants a and b.

Shedd (1953) stated that A and B were not constant over

a wide variation in pressure gradients. Equation 22 is a

straight line on a log-log plot; however most of the

experimental data plots did not produce straight lines.

By generalizing information given by Shedd (1953),

Brooker (1961) obtained the two-dimensional velocity

component equations


2 2 (B-1)/2
(24) Vx = A [(SP/9x) + (OP/3y) ] P/9x


and

2 2 (B-l)/2
(25) Vy = A [(DP/3x) + (2P/Dy) ] 3P/3y


where Vx and Vy are the velocity components and the rest of

the symbols are as previously defined.

Brooker (1961) substituted Equations 24 and 25 into the

continuity equation, Equation 18 assuming constant density

and considering only the x and y directions, to obtain the

partial differential equation


2 2 2 2 2 2
(26) [(SP/3x) + (8P/8y) ] [ P/Dx + 3 P/y ]


2 2 2 2
2m[(8P/3x) 3 P/3x + 2(DP/x) (3P/3y)(3 P/3x8y)


2 2 2
+ (3P/3y) 3 P/iy ] = 0










where

m = (1-B)/2.


The finite difference method was used to solve this

differential equation. The numerical solution agreed

favorably with experimental pressure values measured in a

rectangular wheat bin.

Brooker (1969) proposed that the log-log plots of

velocity versus pressure gradient be modeled by a series of

straight line segments using Equation 22 and reported

several values of A and B for a range of pressure gradients.

The finite difference method was used to solve Equation 26

with the value of B adjusted at each node corresponding to

the value of the gradient P/3n. Brooker (1969) used this

model to calculate pressure and velocity distributions for

two-dimensional nonlinear airflow through a rectangular corn

bin. The numerical results did not agree well with

experimental findings.

Jindal and Thompson (1972) applied Brooker's (1969)

model to calculate the two-dimensional pressure distribution

for air flow through conical-shaped piles of grain sorghum.

Pierce and Thompson (1974) also applied Brooker's

(1969) model by modifying the Jindal and Thompson (1972)

model for the prediction of airflow characteristics in

conical piles of corn or sorghum.

Marchant (1976a) applied Shedd's equation and used the

finite difference method to establish the total airflow

through large round hay bales. Three-dimensional airflow











was discussed briefly. Marchant (1976b) also used the

finite element method to solve linear airflow problems in

two dimensions. The experiments of Brooker (1958) and

Borrowman and Boyce (1966) were used in verifying the model.

Segerlind (1982) developed a model for two-dimensional

nonlinear airflow and used the finite element technique to

solve for the pressure distribution in rectangular grain

bins. The model was based on a generalization of Shedd's

equation and calculated constant pressure lines through

shelled corn similar to the lines measured by Brooker (1969).

Smith (1982) applied the two dimensional finite element

method used by Marchant (1976b) to estimate the pressure and

velocity distributions in a three-dimensional field of a

rectangular heap of grain with an on-floor monoduct system.

A comparison was made between experimental values of

pressure and velocity of Marchant (1976a) and the calculated

results using the finite element method. Smith (1982) found

that the pressure could be calculated more accurately than

velocity but that both could be calculated with reasonable

accuracy in practical problems.

Khompis (1983) applied Shedd's equation and used the

finite element method to solve the three-dimensional

nonlinear pressure and velocity distribution within

cylindrical grain storage. Four types of perforated floor

systems were studied using corn as the grain media.

Extensive graphics were demonstrated to display the three-

dimensional pressure and velocity distributions.











From the above discussion it is evident that the theory

of porous media flow has been used extensively in the study

of air flow through agricultural commodities. This previous

work can be generally classified into three areas of study.

The first area of study is a development of equations

for prediction of pressure drop as a result of air flow

through bulk fruits and vegetables. In addition to the

previous work cited, several workers applied the Shedd or

Ergun equation to determine the relationship of pressure

drop to velocity of air flow through the various

commodities. Patterson et al. (1971) studied cherry pits,

shell corn, and navy beans using both the Shedd and Ergun

equations. Steele (1974) studied peanuts using Shedd's

equation. Akritidis and Siatras (1979) used Shedd's

equation in their study of pumpkin seeds. Neale and Messer

studied root and bulb vegetables (1976) and leafy vegetables

(1978), also using Shedd's equation. Gaffney and Baird

(1977) studied bell peppers using Shedd's equation. Wilhelm

et al. studied snap beans (1978) and lima beans and southern

peas (1981) using Shedd's equation. Abrams and Fish (1978)

applied Shedd's equation in their study of sweet potatoes.

Rumsey (1981) used the Ergun equation in the study of

English walnuts.

The second area of study is the prediction of pressure

and velocity field distributions for airflow through bulk

agricultural products. These are sufficiently outlined

above. It is noted that each of these individual studies











has several common aspects. They all use some modification

of Darcy's approach to porous media flow. They all involve

a large bulk of porous media (semi-infinite). The air

entrance into the porous media was through a line or area

source. The air exit from the porous media was through a

large exit area. Finally, these various studies used

several modeling analysis techniques which were specifically

developed for the particular situation being studied. This

factor is emphasized as a result of difficulty in using the

various models for other than the specific situation for

which they were intended.

The third area of study and most limited is the

development of equations to predict the pressure drop for

airflow through fruits and vegetables packed in shipping

containers.

As mentioned above, Wang and Tunpun (1969) studied the

pressure drop versus airflow relationships for tomatoes in-

bulk and in-cartons. Haas et al. (1976) used a modified

Shedd equation to predict equations relating pressure drop

to air velocity for oranges in-bulk and in-cartons. Chau

et al. (1983) used both the Shedd and Ergun equation to

predict the pressure drop as a function of airflow for

oranges in-bulk and in-cartons.

No studies exist which predict the pressure and

velocity fields for air flow through fruits or vegetables

packed in shipping containers.
















PROCEDURE


In order to better understand the heat transfer and

the product temperature change which occurs when forced air

cooling is used to reduce the temperature of fruits and

vegetables packed in fiberboard containers, the velocity and

pressure fields within the carton must be analyzed.

Intuitively it was anticipated that the velocity

characteristic of the air flowing through packed cartons of

fruits and vegetables would vary in three dimensions. As

mentioned previously, direct measurement of the pressure and

velocity within and throughout the packed carton is not

feasible with current instrumentation. The measurement of

temperature is the only direct method which can be used as

an indication of the velocity pattern of the air passing

through the carton as the product is cooled.

Extensive temperature response experiments were

conducted with oranges packed in an experimental packing

container of a size similar to regular packing cartons. The

number and location of vent holes could be changed and

several flow rates were used. Eighty 36-gage thermocouples

were used to measure air temperature entering and leaving

the box, air temperature at various locations inside the










box, and surface and interior temperatures of oranges at

various locations within the box.

The temperature response data were analyzed graphically

and by computer simulation which indicated changes in

temperature within the carton through various cutting planes

by use of color variation for various temperature ranges.

From this temperature response data, no symmetrical

patterns could be identified. This leads to the assumption

that the airflow within the orange carton varies in three

dimensions.

In order to evaluate the velocity and pressure fields,

an analysis technique had to be selected. The two principal

numerical solution methods employed by previous researchers

are the finite difference and the finite element techniques.

Each method has advantages and disadvantages.

The flow of air through a porous media may be described

in terms of partial differential equations. In real

situations, the geometry of the system to be modeled is

usually not simple and these equations cannot be solved by

analytical solution. In these cases the use of some

numerical solution is necessary. Brooker (1961) and others

proposed a finite difference method of solution. In this

method the porous medium is divided into a regular grid.

The partial differential equations of static pressure at

each grid point are approximated by truncated Taylor Series

expansion in terms of the static pressure at the surrounding

points. In this way algebraic equations can be formed for










each grid point therefore setting up a system of equations

to be solved simultaneously. While the finite difference

method can give good results, it has several disadvantages:


1. The method is not easily adapted to regions which have

boundaries not lying on the grid lines.

2. It is difficult to write a computer program to take

account of a general geometric shape.

3. A grid size which gives acceptable accuracy in areas

with rapidly varying pressure will be far too small in the

areas of slight variation, thus leading to a solution of a

large number of superfluous equations.

4. It is difficult to apply the method to media which are

not homogeneous and isotropic.

5. For each equation relating air velocity to pressure

gradient, a completely different partial differential

equation has to be solved thus requiring a new set of

equations to be set up in each case.

The finite element method is especially suited for the

study of airflow through fruit and vegetable crops because,

as well as overcoming the general points 1 to 3, it can also

deal easily with 4 and 5 which are very pertinent.

In the past ten years, much work has been done to

develop the numerical solution of partial differential

equations using the finite element method. The finite

element approach has its origins in the field of solid

mechanics (elasticity, plasticity, and structural analysis),

but its application for use with a much wider range of











problems was soon realized. Finite element methods have

replaced finite difference methods in many areas of solid

mechanics and are making inroads into fluid mechanics, heat

transfer, and other fields. More recently, considerable

interest has been shown in its application to nonlinear

problems, such as the problem under consideration.

The finite element method is a technique that

approximates a continuous quantity, such as a partial

differential equation, by a discrete model composed of a set

of piecewise continuous functions. This discretization is

done on a small but finite portion of the component. As its

name suggests, the finite element method consists of

dividing the region into a number of smaller elements which

can be of any shape (Figure 1). Nodal points are

distributed around the boundary of the element and

occasionally inside it. These nodes are arbitrarily

numbered. The overall model is comprised of a finite number

of nodes. At each node the value of the variable (i.e.,

pressure, flow, temperature, deflections) for a differential

equation is either known or unknown. Between the nodes are

subdomains called elements. Mathematics converts the

partial differential equations into matrix equations that

describe each element. The individual elements (in matrix

form) are joined together to form an overall model. The

results of the finite element simulation output are given in

terms of the variable at each nodal point. After the known

variables are obtained, internal quantities can be































































Figure 1. Region divided into finite element.










estimated. An example structural problem would solve for

the deflection on a loaded component and then evaluate the

stresses inside each element.

With the finite element approach, the partial

differential equations describing the desired quantity (such

as pressure, flow, temperature, displacement, deflections)

in the continuum often are not dealt with directly.

Instead, the continuum is divided into a number of finite

elements, which are assumed to be joined at a discrete

number of points along their boundaries. A functional form

is then chosen to represent the variation of desired

quantity over each element in terms of the values of this

quantity at the discrete boundary points of the element. By

using the physical properties of the continuum and the

appropriate physical laws (usually involving some sort of

minimization principal), a set of simultaneous equations in

the unknown quantities at the element boundary points can be

obtained. This set of equations is in general quite large,

but the matrix is banded.

For those situations where the finite element

technology has been developed, there are three primary

advantages of the finite element approach over the finite

difference methods. These are


1. Irregularly shaped regions can be handled easily,

without the special treatment usually required by finite

difference methods.










2. The size of the finite element can easily be varied over

the region, permitting the use of small elements where

strong variations occur and large elements where only

slight variations are expected. With finite difference

methods, at least in their conventional form, the use of

several different mesh sizes can cause bookkeeping

difficulty.

3. For comparable accuracy, the finite elements can usually

be considerably larger than the mesh elements of a finite

difference grid. As a result, when elliptic problems are

involved the band matrix referred to earlier is usually

small enough to be solved directly without recourse to the

iterative methods which are usually necessary in finite

difference methods.

The main advantage of the finite element method is that

a general computer program can be written to create the

overall model matrix. For different applications (e.g.,

different shape geometry) input is easily changed and a new

solution obtained with the same computer program. Other

important advantages include the following:


1. Material properties can be different between each

element.

2. Irregular shaped geometries can be modeled.

3. The size of elements can be varied.

4. Mixed boundary conditions can be handled (e.g., loads

and temperatures).







The number of nodes in an element can vary with the
addition of internal nodes that provide a means to curve the
element or specify a nonlinear variable. Many types of
elements are available (Figure 2).


POINT (MASS)


LINE
(SPRING, BEAM, SPAR, GAP)


---


AREA
(THIN SHELL, 2D SOLID,
AXISYMMETRIC SOLID)


VOLUME
(3D SOLID)





THICK SHELL


Figure 2. Various types of finite elements.


N2


E-: 40










The two methods (finite element and finite difference)

are similar in that an algebraic equation is set up for each

nodal point, but the way in which they are set up differs.

Compared with finite difference programs, for example, they

are easier to use and require fewer computer resources. For

modeling nonlinear material properties, finite element

programs require less than 20 iterations compared with the

200 or 300 required for finite difference programs. With

recently developed finite element software, the calculated

distributions can be shown graphically on a computer screen,

which makes it relatively easy to see areas of high and low

pressure distribution. Working with a geometrical model of

a particular size, an engineer can vary the input

specifications such as material properties and inlet and

outlet locations and see the effect of these variations on

the resulting pressure distributions. The program also

calculates performance parameters. With this information,

the engineer can upgrade the design geometry and change the

material requirements and modify the energy losses, and

thereby achieve the best design performance with the least

amount of material or achieve the optimum cooling. The

procedure is quick and relatively inexpensive when compared

with conditional procedures such as building and testing

prototypes.

After review of the above literature and analysis of

the temperature response data, application of a 3-D finite









element nonlinear porous media flow analysis for the problem

under consideration was thought feasible.

As mentioned above, previous modeling involved a porous

media which was of much larger scale and without the more

significant boundary conditions for the situation of air

flow through a packing container. Also the finite element

models developed by previous workers were developed

specifically for the problem which they were addressing.

Although the 3-D finite element porous media flow analysis

appeared feasible, in the early stages of this study it was

not deemed appropriate to devote a considerable amount of

time in developing a specific model for this situation

before feasibility of the approach was confirmed.

Finite element programs are available and are becoming

the most utilized engineering tool for design. These large

and powerful tools have been written by engineering graduate

students and professors with advanced technical knowledge in

their areas of expertise and therefore might pose a problem

to the majority of engineers who have little background in

finite elements and other dynamic simulation computer

programs. This leads the average engineer relying on the

program written by others and thus using the analysis

techniques as a black box: entering input and receiving

output. The degree of success in using a finite element

program is related to the assumptions and models developed

by the program originator. The user of the finite element

program must realize that the actual results from the finite










element model are a function of engineering judgement used

to select the element type and size, and a function of the

computer program with its associated mathematical

assumptions.

Also mentioned above was the discovery that the SASI

finite element analysis package, DeSalvo and Swanson (1983),

contained an option which allows the modeling of nonlinear

steady state fluid flow through a porous medium. This

solution technique was studied and appeared to be applicable

to the case at hand. A limited review was unable to

identify other commercial solution packages which offered

the needed analysis option and capabilities.

If the SASI package or a similar general finite element

computer program was readily accessible to the workers

interested in porous media flow analysis applied to

agricultural crops, a tool would be available which would

allow for a uniform solution technique and easy comparison

of the results of various workers. Therefore, in addition

to evaluating the feasibility of the finite element porous

media flow analysis, this study is also evaluating the

feasibility of applying a commercial general computer

program. If it is established that the analysis technique

is feasible, as well as, the general computer program, then

the industry which provides such software can be encouraged

and solicited to provide the needed software for widespread

adoption and application.










Commercial Finite Element Package


The ANSYS finite element computer program is based on

classical engineering concepts and documented finite element

and numerical analysis techniques (DeSalvo and Swanson

1983). This self-contained general purpose program uses

efficient solution techniques to solve a large class of

engineering problems. It is user oriented, command driven,

has extensive graphic capabilities, and is documented,

benchmarked, and verified.

A typical analysis consists of three phases: Pre-

Processing (Analysis Definition), Solution, and Post-

Processing (Interpretation of Results).

The Pre-Processing phase is very important since the

accuracy of the solution depends directly upon the degree of

accuracy of the problem description. Input data prepared in

the analysis definition would include the model description,

boundary conditions, and the analysis type and options.

The model description involves creating the desired

geometry, selections) from the element library,

specification of geometric (real) constants describing

properties of elements (e.g., area, moment of inertia, and

height of a beam), and identification of material properties

(e.g., viscosity, conductivity, and density). The user must

ensure dimensional homogenity.

The analysis is performed in the solution phase. For

the nonlinear porous media flow case, this involves the

solution of the matrix equation










(27) [K] {P} = {Q}

where

[K] = the transmissivity matrix, defined by Reff in
equation 13.

{P} = pressure vector (unknown)

{Q} = mass flow rate vector


and the calculations of the pressure and mass flow

distributions.

The solution results are evaluated in the third phase.

The user should determine if the objective of the analysis

was met. Tools to use in making this decision include

graphics (contour maps, distorted shape plots, and graphs of

one variable versus another); scans of results; and

combinations of results.

Therefore, the use of ANSYS is straightforward, but

careful engineering judgment must be exercised during the

analysis definition phase in selecting the geometry and

elements, as well as specifying the input parameters and

boundary conditions.


Verification


In order to verify the ANSYS calculated solutions, the

solutions must be compared with known theoretical solutions,

experimental results, and/or with other calculated

solutions. The extensive verification problem manual,

DeSalvo (1982), provided by SASI does contain a ground water

flow verification problem. However, it does not contain a










porous media flow problem of the type under consideration

with which to compare the ANSYS output to a theoretical

solution. Therefore, before attempting to solve the problem

under consideration, ANSYS was used to solve similar porous

media flow problems which have been solved by other workers.

ANSYS was used to solve the two-dimensional rectangular

grain bin problem solved by Segerlind (1982) and one of the

three-dimensional grain bin problems solved by Khompis et

al. (1984). As noted above, ANSYS is based on an Ergun

Equation analysis while the work of Segerlind (1982) and

Khompis et al. (1984) is based on a Shedd Equation analysis.

Patterson (1969) determined the coefficients for pressure

drops through grain beds using both the Shedd Equation and

Ergun Equation. It is assumed that appropriate coefficients

can be selected for the Ergun Equation, that correspond to

the coefficients used in the Shedd Equation by Segerlind

(1982) and Khompis et al. (1984). It is further assumed,

that ANSYS provides a satisfactory solution if the

calculated results are within reasonable accuracy of the

results determined by Segerlind (1982) and Khompis et al.

(1984).

The input coefficients needed to use ANSYS are

indicated in Equation 13. The gas viscosity and the density

are for air and are taken from standard data tables. The

absolute permeability of the porous media and the visco-

inertial parameter are derived from experimental results.










The work of Patterson (1969) is used to obtain values for

these last two parameters.

Patterson (1969) used a modified Ergun Equation


2 2 3 2 3
(28) AP/AL = Ke [150 PV (1-E) /D + 1.75 p V (l-E)/D ]
s p s p


where

Ke = modified Ergun product constant.


Equation 28 is identical to Equation 5 with the

addition of the product multiplication factor, Ke.

Comparing Equation 28 and Equations 12 and 13, the absolute

permeability of the porous media, K, and the visco-inertial

parameter, 0, needed for ANSYS input can be equated as

follows:

2 2 3
(29) 1/K = Ke [150 (1-E) /D E ]
p


and

3
(30) A = Ke [1.75 (l-E)/D E ].
p


Patterson (1969) reported all the parameters required

to solve Equations 29 and 30, for shelled corn and various

other biological materials. For shelled corn at a moisture

content of 16.01 %, a temperature of 87 F, and a normal

fill, Patterson (1969) reported the following values:


D = 0.032 feet; E = 0.43; and Ke = 4.5











The conditions of the corn are similar to that used in

the work of Segerlind (1982) and Khompis et al. (1984).

Both studies used coefficients which were reported in the

research of Shedd (1953). This research was based on normal

fill, shelled yellow dent corn at a moisture content of

12.4 %. However, Shedd (1953) reported only a 1 % variation

in the pressure drop for the same corn at a moisture content

of 15.8 %.

Solving Equations 29 and 30 using the values given

above results in the following:


6 -2 -7 2
1/K = 2.69 x 10 ft ; or, K = 3.71 x 10 ft


3 -1
and 8 = 1.76 x 10 ft The following viscosity and

density for the air were used:


-5 -2 3
S= 1.24 x 10 Ibm/ft-sec ; and p = 7.25 x 10 lbm/ft


The English Engineering system of units was used by

Segerlind (1982) and the SI system of units was used by

Khompis et al. (1984) and Chau et al. (1983). Conversion

factors were used to modify the units of the viscosity and

density to insure dimensional homogenity. When the desired

units for the pressure are inches of water and the desired

units for the velocity are feet per minute, the unit

conversion results in the following values for viscosity and

density for input to ANSYS:







48



-9
p/g = 1.24 x 10 in. water-min;
c


-7 2 2
p/g = 1.20 x 10 in. water-min /ft ;
c


where the Newton's Law gravitational constant is


2
g = 32.2 ft-lbm/lbf-sec
c


When the desired units for the pressure are Pascals and

the desired units for the velocity are meters per second,

the unit conversion results in the following values:


-5 3
p = 1.846 x 10 kg/m-sec; and p = 1.1614 kg/m.


After conversion of the units, the other input

parameters for ANSYS become


7 -2
1/K = 2.9 x 10 m ; or,


3 -1
and B = 5.77 x 10 m


-8 2
K = 3.45 x 10 m


Now all the necessary input data to use the ANSYS

finite element program are known.









Two-Dimensional Grain Bin


The physical problem selected for solution by Segerlind

(1982) was the same rectangular grain bin that was analyzed

by Brooker (1961) and (1969). The bin was 121.92 cm wide (48

inches) and 243.8 cm high (96 inches). The duct was 20.32 cm

high and 40.64 cm wide (8 inches by 16 inches).

Segerlind (1982) utilized the six node quadratic

triangular element in his finite element method solution.

The bin was divided into 32 elements with 85 nodes. The

horizontal node spacings was 10.16 cm (4 inches) while the

vertical spacing was 5.08 cm (2 inches) in the region of the

inlet duct. The upper 76.2 cm (30 inches) was modeled with

two elements because all of Brooker's (1969) results

indicated that a linear pressure gradient existed in this

region. The element and node locations used by Segerlind

(1982) are shown in Figure 3 and the boundary conditions

are shown in Figure 4. Segerlind (1982) noted that the

known pressure values at the duct inlet and at the free

surface are easily incorporated into all finite element

programs. The no-flow or impermeable boundary condition,

3P/3n = 0, is automatically enforced on the boundary when

the pressure values are not specified. This differs

significantly from the finite difference technique used by

Brooker (1969) where extra rows of nodes had to be added

outside the region to satisfy the no flow boundary

condition.











---- 24" -


96"


























8"

Figure 3. The node and element grid employed by Segerlind
(1982).
















P= 0


p= 0
aP
aP o / ax
ax








aP

ayESSURE SPECIFIED

PRESSURE SPECIFIED


Figure 4. Boundary conditions used by Segerlind (1982).










The ANSYS analysis option to be used did not contain

an element type identical to the type used by Segerlind

(1982). A four node 2-D isoparametric thermal solid element

was used. The element option which allowed modeling of

nonlinear steady-state fluid flow through a porous media

was used. This element could be formed into a triangular

(three node) shape by the duplication of two of the four

nodes. The node and element locations used are shown in

Figure 5. The model consisted of 85 nodes and 71 elements.

All but nine nodes were placed identical to the locations

used in Segerlind's (1982) model. Over twice as many

elements were required because smaller elements were used

which did not contain the midpoint nodes like the elements

used by Segerlind (1982). Although this resulted in

additional computer solution time, the results were as

refined or more refined when compared to Segerlind's (1982)

solution. The boundary conditions were specified in a

manner similar to Segerlind's (1982) model.


Three-Dimensional Grain Bin


Khompis et al. (1984) extended the procedures of

Segerlind (1982) to account for three dimensional pressure

and velocity fields in cylindrical grain storage. Four

different aeration systems were studied. The four systems

were differentiated by the size and and shape of the

perforated floor. The system to be considered in the

present study consisted of the square perforated type floor




































































Figure 5. The node and element locations used in current
study.









for a bin 12.8 meters high and 18.3 meters in diameter. The

perforated floor was 9.15 x 9.15 meters square. Specific

details of these sizes and shapes are shown in Figure 6.

Flush floor ducts were assumed.

Khompis et al. (1984) used a twenty node, quadratic,

three dimensional element for the finite element model. The

cylindrical grain storage investigated were symmetrical and

the finite element analysis was performed using only half of

the grain bin. The limitation of computer central memory

and desire for reasonable computing restricted the finite

element model to five layers with 12 elements per layer.

The 60 elements had 406 nodes. The height of the elements

at each layer depended on the distance from the perforated

floor and height of the grain storage. The shortest

elements were located in the layer in contact with the floor

and had a height of about 5 percent of the total grain

storage height. The element grid is shown in Figure 7.

The boundary conditions consisted of a pressure of 500

Pascals (2 inches of water) specified at the square

perforated floor and atmospheric pressure at the free

surface on the top of the grain bin. Again, the no-flow or

impermeable boundary condition, 2P/8n = 0, is automatically

enforced on the boundary when the pressure values are not

specified.

Again, the identical type element which Khompis et al.

(1984) used was not available for use with the ANSYS solution

package. An eight node 3-D isoparametric thermal solid

































FULL PERFORATED FLOOR


SQUARE DUCT


Y DUCT


Figure 6. Four different grainbin aeration systems studied
by Khompis et al. (1984).


STRAIGHT DUCT

































x


Cross-section of the elements
in the X-Y plane.


Y


Division into the elements
of a half cylindrical
grain storage.


Figure 7. Finite element model used by Khompis et al.(1984).










element, which allows modeling of nonlinear steady-state

flow through a porous media, was used. This element could

be formed into prism or tetrahedron shaped elements by

defining duplicate node numbers. Although ANSYS did have

twenty node elements like those used by Khompis et al.

(1984), these elements did not have the porous media flow

option capability. The node and element locations used with

ANSYS are shown in Figure 8. The model consisted of 741

nodes and 552 elements.

The comparison of results with those of Segerlind

(1982) and Khompis et al. (1984) received detailed coverage

in the Results and Discussion section, below. However, the

results supported the use of ANSYS to study the problem

under consideration. The procedures for this application

follow.


Solution for Oranges Packed in Containers


The use of the ANSYS porous media analysis to study the

pressure and velocity distributions as air flows through a

container of oranges presented several questions that were

not significant problems for previous workers. A primary

concern was the overall scale of the porous media used in

this case. This scale was finite when compared to the semi-

infinite cases studied by others. Because of the small

dimension of the packing container, boundary (wall) effects

could be significant. The wall contact with the fruit or

vegetable presented two possible difficulties. The drag


















































E

N


















3-0 #.m am( SIATIC P~U~ft-AWInfl .ATrmUW


Figure 8. The node and element locations used in current
study.










caused as the air passes the wall was one consideration.

The second concern, which has been reported by other

workers (Pillai 1977, Ridgway and Tarbuck 1968, Stanke and

Eckert 1979) relates to the variance of the voidage or

porosity adjacent to the walls when compared to the central

portion of the porous media.

Other researchers have reported the significant

pressure drops produced by the air inlets of packing

containers. This point was another possible significant

difficulty when comparing the pressure drop of the porous

media (fruit or vegetable within the cartons) to that of the

pressure drop across the inlet(s) and exit(s).

Another consideration in terms of fruits and vegetables

was the compaction possible during packing and subsequent

shrinkage with time due to physiological changes and

moisture loss. The porosity may change, for example, if the

product become more compact due to shrinkage or handling.

The porosity next to the inside top of the carton could

increase allowing more air to pass through this area.

The impact of some or all of these points of interest

is addressed in the Results and Discussion section.

Before verifying the applicability of the ANSYS finite

element program modeling air flow through a carton of fruit,

the work of Chau et al. (1983) was modeled in order to

better understand how the model of a commercial orange

carton should be developed.










Three-Dimensional Bulk Citrus


As previously mentioned, Chau et al. (1983) used both

the Shedd and Ergun equations to predict pressure drop as a

function of air flow for oranges in bulk and packed in

cartons. This work involved the determination of pressure

drop though empty cartons (for a single carton wall and for

both walls of an empty carton) and for oranges packed in the

simulated carton. Fruit stacking pattern (bed porosity),

fruit size, vent hole shape, vent hole number, and air flow

rate were some of the variables studied.

The experimental equipment employed by Chau et al.

(1983) is shown in Figure 9. The cross-section of the

product bin was 55 cm x 55 cm. The actual cross-section

for each test varied slightly depending on fruit size and

stacking pattern.

There were 16 tests involving oranges in bulk, which

consisted of four orange sizes and four stacking patterns.

Since Chau et al. (1983) presented more data for Florida

size 80 (8.1 cm) waxed Valencia oranges stacked in a square

staggered pattern than other combinations of size and

stacking pattern, the present study considered only these

data. In the square staggered pattern each layer had a

square pattern (every 4 adjacent fruit forming a square)

with the next layer offset so each fruit on the subsequent

layer rested on 4 fruit from the previous layer. The air

flow rate was varied for each test to cover a range of

approach velocities from 0.1 to 1.0 meters per second.











Chau et al. (1983) fitted the experimental data using

the Ramsin Equation (Equation 22) in terms of pressure

change per bed depth of fruit:


b
(31) AP/h = a V


For the size 80 oranges stacked in a square staggered

pattern, with a bulk depth of 0.33 meters and a porosity of

0.435, the constants in Equation 31 were found to be

a = 278; and b = 1.937.

Chau et al. (1983) presented a variation of Ergun's

equations as


2 2 3 2 3
(32) AP/h = K P V (1-E) /D E g + K P V (l-E)/D E g
1 s p c 2 s p c


where


h = AL in Equation 5

K = 150 in Equation 5
1

K = 1.75 in Equation 5
2

g = Newton's Law gravitational constant.
c


This work did not employ a modified Ergun product

coefficient, Ke, like that used by Patterson (1969).

Therefore, eliminating Ke and substituting K and K
1 2
Equations 29 and 30 can be rewritten











2 2 3
(33) 1/K = K [ (1-E) /D E ]
1 p


and

3
(34) = K [ (1-e)/D e ].
2 p


Chau et al. (1983) determined K and K by fitting
1 2
the experimental data and reported all the parameters

required to solve Equations 33 and 34 for bulk packed

oranges. For size 80 oranges arranged in a square staggered

stacking pattern, the following values were reported:


D = 0.081 meters; E = 0.435; K = 890; and K = 2.73.
P 1 2


Solving Equations 33 and 34 using the values given

above resulted in the following:


5 -2 -6 2
1/K = 5.2608 x 10 m ; or, K = 1.9009 x 10 m


-1
and 8 = 231.3 m


Since the SI system of units was used by Chau et al.

(1983), the desired units for the pressure are Pascals and

the desired units for the velocity are meters per second.

The inlet air condition used by Chau et al. (1983) are

approximately the same as those used by Segerlind (1982) and

Khompis et al. (1984). The air viscosity and density











specified above as input for the ANSYS finite element

program to solve the problem reported by Khompis et al.

(1984) were used again,


-5 3
V = 1.846 x 10 kg/m-sec; and p = 1.1614 kg/m .


To determine the pressure drop across rectangular and

round vent holes, Chau et al. (1983) conducted tests using

the experimental set-up shown in Figure 9, for one layer

of corrugated fiberboard (simulating one side of a carton)

with 4, 6, and 8 vent holes. The tests were repeated with 2

layers of fiberboard separated by 30 centimeters (simulating

2 sides of a carton). Each fiberboard side had 4, 6, or 8

vent holes placed uniformly in the fiberboard cross-section.

The cross-sectional area of each vent hole was 12.94 square

centimeters. The thickness of the fiberboard used to make

the cartons was 3.2 millimeters.

The experimental data were normalized on the basis of

actual air velocities through the vent holes (calculated as

total flow rate divided by total vent area), and the data

points for the tests with 4, 6, and 8 vent holes fitted the

same straight line on a log-log plot.

The equations of best fit for the experimental data

were presented by Chau et al. (1983) as pressure drop

across one side of carton with rectangular vent holes

2
(35) AP = 1.59 V ;















































PRESSURE
TAPS










R


BLOWER


FLOW ELEMENT7

I,


DUCT
EXTENSION






: FRUIT






A-I-- R--------
L AIR
STRAIGHTENER



L SLIDING
DAMPER


Figure 9. The experimental setup used by Chau et al. (1983)
for measuring air flow resistance in orange.


1.


I


AI











pressure drop across two side of carton with rectangular

vent holes

2
(36) Ap = 2.78 V ;
1


pressure drop across one side of carton with round vent

holes


2
(37) AP = 1.76 V ; and
i

pressure drop across two side of carton with round vent

holes


2
(38) AP = 2.89 V
i


Chau et al. (1983) noted that the pressure drop across

two sides of the carton was less than twice the pressure

drop across one side of the carton alone. The explanation

for this difference given by Chau et al. (1983) was that the

loss for one carton side is similar to an exit loss. For

two sides of the carton, the loss is more like a duct

enlargement loss as the air travels between the sides of the

carton. The enlargement loss is smaller than an abrupt exit

loss.

The explanation for the slightly higher pressure drop

for the round vent holes was attributed to roughness of the

edges of the holes. The rectangular holes were machine

stamped while the round holes were cut using a hole saw.











It is interesting to note that Kaminski (1986a)

presented the following equation for static pressure

differences:

2
(39) AP = C p V / 2g
s D m c
where


Ap = Static pressure difference across a given
s configuration


C = Static pressure coefficient based on highest the
D highest average velocity for a given configuration

V = Highest average velocity
m

and the other terms are as previously defined.


For turbulent flow, Reynolds Number greater than

10,000, through a square-edged orifice the static pressure

coefficient was reported as C = 2.80.
D


If Equations 35 and 37 are rearranged into the form of

Equation 39, the coefficients of static pressure are

respectively, 2.74 and 3.03. Therefore, the pressure loss

across a single fiberboard can be approximated by the

pressure loss across an orifice or orifice plate.

No such comparison of the pressure loss across two

fiberboards, separated by a fixed distance, could be made

with the typically presented expansion and contraction

configurations reported in various fluid flow text and data

books.










Chau et al. (1983) experimentally evaluated the

cumulative effect of air flow resistance of the vent holes

and the oranges. Measurements were made of the pressure

drop across oranges stacked on fiberboard with 4 rectangular

vent holes which simulated the fruit with one side of a

carton. The pressure drop was also measured across oranges

stacked between 2 fiberboards with 4 rectangular vent

holes of the same size and location on each fiberboard.

All the tests were repeated using 4 different stacking

patterns and 4 different orange sizes.

Chau et al. (1983) fitted the experimental data using

a modification of Equation 31 which did not normalize the

pressure drop to a unit depth of fruit


b
(40) AP = a' V


This was necessary because the measured pressure drop was a

combination of the pressure change across the oranges, which

was a function of fruit depth, and the pressure change across

the vent holes, which was not a function of fruit depth.

For the size 80 oranges stacked in a square staggered

pattern, with a bulk depth of 0.33 meters and a porosity of

0.430, the constants in Equation 40 for oranges with one

carton side were found to be

4
a' = 1.85 x 10 and b = 1.98, while for oranges with

two carton sides the constants were found to be

4


a' = 2.46 x 10


and b = 1.95.











Chau et al. (1983) noted that most of the pressure loss

was caused by the resistance of the vent holes. The

pressure drop across a carton of fruit was always greater

than the sum of the pressure drops across each side of the

carton plus the pressure drop across the fruit. Wang and

Tunpun (1969) reported similar results.

According to Chau et al. (1983), the pressure drop

across the sides of the carton was expected to be higher

than the sum of the pressure drops across the sides of the

carton and the fruit. The fruit disturbed the flow past the

vent holes since there was always an amount of obstruction

by the fruit. Further, the air flow pattern through the

oranges in a carton with vent holes was much different from

the uniform flow through bulk loads of oranges even for the

same approach velocity. Near the area of the vent holes for

flow through oranges in cartons, high localized velocities

occurred. Since the pressure loss through the vent holes

was proportional to the velocity squared, approximately, the

high localized velocities produced a higher overall pressure

drop across the vent holes and through the fruit near the

vent holes.

All the necessary input data to use the ANSYS finite

element program to model the experimental work of Chau et

al. (1983) are known.

Extensive modeling using two-dimensional elements,

like that employed previously in modeling the work of

Segerlind (1982) and Khompis (1983), was carried out but











was not be reported here since the work of Chau et al.

(1983) was three-dimensional. The two-dimensional work was

very helpful initially in providing necessary direction

without the additional computer time required when working

with three-dimensions. The two-dimensional modeling results

were essentially the same as the three-dimensional modeling,

which is discussed in detail below.

Modeling the experimental work of Chau et al. (1983)

involved more than one application of the capabilities of

the ANSYS finite element solution package. First, the

flow through the oranges in bulk was easily modeled as

illustrated previously for the grain bin problems of

Segerlind (1982) and Khompis (1983). The boundary

conditions at the cross-section of the entrance and exit

were easily applied for this uniform flow situation. The

second application dealt with the reduced boundary

conditions as a result of the entrance and exit vents of the

oranges packaged in cartons. The third application involved

modeling the orifice-type loss through the vents of the

carton walls. Finally, the combination of the second and

third applications was applied to study the combination

losses which were reported by Chau et al. (1983) and Wang

and Tunpun (1969).

The physical problem selected for solution was the

experimental set-up shown in Figure 9 and described above.

An eight node 3-D isoparametric thermal solid element,

which allowed modeling of nonlinear steady-state flow











through a porous media, was used to model the pressure loss

through oranges in bulk and oranges in cartons, without

consideration of the pressure loss through the vent holes.

This element could be formed into prism or tetrahedron

shaped elements by defining duplicate node numbers.

However, only 3-D rectangular elements were needed. The

node and element locations are shown in Figure 10. The

model consisted of 320 nodes and 196 elements. The nodes

were placed every 8.1 cm. The boundary conditions were

specified at the inlets and outlets corresponding to data

reported by Chau et al. (1983). For the situation with

oranges in bulk, the pressure or flow rate across the inlet

and outlet cross-sections was specified. For the situation

with oranges in cartons, the pressure or flow was specified

at nodal locations corresponding to the four inlet and exit

vent hole locations of the experimental set-up used by Chau

et al. (1983). The inlet and outlet locations were point

sources and the velocity and pressure varied in the x-,

y- and z-axis. As noted in previous models, the no-flow

or impermeable boundary conditions are automatically

enforced on the boundary when no pressure values were

specified.

In order to use ANSYS to model the pressure loss

through the vent holes in the fiberboard, a new element type

was used. The element was the hydraulic conductance element

(DeSalvo and Swanson 1983). This uniaxial two node element

had the ability to transmit flow between its nodal points.





































I 4






4





















56.7cm



3-o AIR rNou RSIST4NC rat amReS IN LUK
2 3


Figure 10. The node and element locations used to model 3-D
oranges packed in bulk and in cartons.




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