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A COMBINED CONVECTION COOKING AND SALMONELLA INACTIVATION
MODEL FOR GROUND MEAT AND POULTRY PRODUCTS

by

Adam Edward Watkins

A DISSERTATION
Submitted to
Michigan State University

In partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Biosystems and Agricultural Engineering

2004

ABSTRACT
A COMBINED CONVECTION COOKING AND SALMONELLA INACTIVATION
MODEL FOR GROUND MEAT AND POULTRY PRODUCTS

By

Adam Edward Watkins

A predictive model for moist-air impingement cooking of ground-and-formed
meat and poultry products was developed. A coupled heat and mass transfer model
incorporating the effects of fat transfer was combined with a model for Salmonella
inactivation, to produce a complete prediction tool for meat and poultry processors.
The model utilized the finite element method to numerically solve separate equations
for heat, moisture, and fat transport. These equations were coupled through boundary
conditions and interdependent thermo-physical property relationships. An enthalpy
formulation for heat transfer was utilized to avoid discontinuities related to solid-to-
liquid phase changes of water and fat within the product. Boundary conditions
unique to moist-air impinging flow were incorporated into the model. These
boundary conditions accounted for the additional heating effects of surface
condensation that are common within moist air impingement systems.

To complete the fat transport component of the model, laboratory experiments
were conducted to determine the fat holding capacity of ground beef as a function of
temperature and initial fat content. Fat holding capacities ranged from 0.05 to 0.6 g

fat/ g nonfat dry matter, and a polynomial model was parameterized to those data.

Additionally, laboratory-scale, moist-air convection cooking tests were
conducted to confirm the importance of fat transport. Species and initial fat content
significantly affected (P<0.05) cooking time, yield, and fat loss for ground beef, pork,
and turkey. The heating time required to reach 85°C varied by as much as 217
seconds between different fat contents of the same species. Differences in cooking
yield of up to 18% were measured between different fat contents. Fat transport was
responsible for up to 28% yield loss in high fat products.

Finally, cooking experiments using an industrial moist-air impingement
cooking system were used to validate the temperature, moisture, and yield predictions
of the complete cooking model. The cooking model predicted transient patty center
temperatures with a standard error of prediction of 80°C for 54 cooking tests. At
temperatures above 45°C, the standard error of the prediction was 5.8°C. The
standard error for final moisture content predictions was 2.3% wet basis. Standard
error for final cooking yield predictions was 6%. Additional comparisons were
conducted between the cooking model and published data collected from moist-air
impingement ovens. Data from published sources were used to perform a verification
of the Salmonella inactivation predictions of the model. The standard error of

prediction for Salmonella inactivation was 1.3 logs (CFU/ g).

ACKNOWLEDGEMENTS

Funding for this project was provided by the United States Department of
Agriculture National Needs Graduate Fellowship Program. Additional assistance was
provided by the United States Department of Agriculture CSREES National Food Safety
Initiative. Access to the J SO-IV impingement oven was provided by FMC FoodTech of
Sandusky, OH. I would like to thank Mr. Bob Swackhamer, Dr. Nahed Kotrola, and Mr.
Todd Gerold for assistance in arranging for and conducting tests with the JSO-IV oven.
Of course this project would not have been possible without the support of my major
professor, Dr. Brad Marks, the members of my committee, the members of our research
team, and all of the people that have helped me here or there along the way. Special

thanks go out to my family and friends who stood by me through the past four years.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................................................................... viii
LIST OF FIGURES .................................................................................... x
KEY TO SYMBOLS .............................................................................. xviii
1 INTRODUCTION AND OBJECTIVES ............................................................. 1
1.1 Background .......................................................................................................... 1

1 .2 Objectives ............................................................................................................ 4

2 REVIEW OF LITERATURE .............................................................................. 5
2. 1 Introduction .......................................................................................................... 5
2.2 Introduction to cooking ........................................................................................ 5
2.3 Heat and mass transfer in meat during cooking ................................................... 6
2.4 Modeling the cooking process ............................................................................. 8
2.4. 1 Empirical models ...................................................................................... 10
2.4.2 Models based upon heat and mass transfer principles .............................. 12
2.4.3 Contact cooking ........................................................................................ 14
2.4.4 Frying ........................................................................................................ 20
2.4.5 Convection cooking .................................................................................. 23
2.4.6 Other types of cooking .............................................................................. 28

2.5 Thermal and physical properties required for modeling .................................... 31
2.6 Microbial models ............................................................................................... 33
2.6.1 Primary models ......................................................................................... 34
2.6.2 Secondary models ..................................................................................... 35
2.6.3 Tertiary models ......................................................................................... 37

2.7 Combined models .............................................................................................. 38
2.8 Limitations of models ........................................................................................ 40
2.9 Summary ............................................................................................................ 41

3 MATERIALS AND EXPERIMENTAL METHODS ....................................... 42
3.1 Overview ............................................................................................................ 42
3.2 Laboratory oven cooking tests ........................................................................... 42

3.2.1 Experimental procedure ............................................................................ 42
3.2.2 Statistical analysis ..................................................................................... 47
3.3 Measurement of fat holding capacity ................................................................. 48
3.3.1 Experimental procedure ............................................................................ 48
3.3.2 Statistical analysis ..................................................................................... 51
3.4 Industrial oven cooking tests ............................................................................. 52
3.4.1 Experimental procedure ............................................................................ 52
3.4.2 Statistical analysis ..................................................................................... 56
3.5 Cooking model validation .................................................................................. 56
3.5.1 Cooking model .......................................................................................... 56
3.5.2 Experimental data ..................................................................................... 57
3.5.3 Comparisons with literature data .............................................................. 58

3. 6 Salmonella inactivation model validation .......................................................... 59
4 MODEL DEVELOPMENT ............................................................................... 61
4. 1 Introduction ........................................................................................................ 61
4.2 Heat and mass transfer model ............................................................................ 64
4.2.1 Heat transfer solution ................................................................................ 64
4.2.2 Moisture transfer solution ......................................................................... 67
4.2.3 Fat transfer solution .................................................................................. 70
4.2.4 Heat and mass transfer coefficients - Array of slot nozzles ..................... 71
4.2.5 Heat and mass transfer coefficients — array of round nozzles .................. 74
4.3 Microbial inactivation model ............................................................................. 76
4.4 Finite element formulation ................................................................................. 77
4.4. 1 Introduction ............................................................................................... 77
4.4.2 Finite element basics ................................................................................. 77
4.4.3 Governing equations ................................................................................. 79
4.4.4 Finite difference time solution .................................................................. 86
4.4.5 Application of FEM solution .................................................................... 87
4.4.6 User interface ............................................................................................ 88

5 RESULTS AND DISCUSSION ........................................................................ 89
5.1 Overview ............................................................................................................ 89
5.2 Laboratory oven cooking tests ........................................................................... 90

vi

5.2.1 Cooking time ............................................................................................. 90

5.2.2 Cooking yield ............................................................................................ 95
5.2.3 Fat loss ...................................................................................................... 99
5.2.4 Volume change ....................................................................................... 103
5.3 Fat holding capacity experiments .................................................................... 107
5.3.1 Low fat samples (5.6% initial fat wet basis) ........................................... 108
5.3.2 High fat samples (15% initial fat wet basis) ........................................... 114
5.3.3 Summary ................................................................................................. 123
5.4 Industrial cooking tests .................................................................................... 124
5.4.1 Cooking time ........................................................................................... 124
5.4.2 Cooking yield .......................................................................................... 129
5.4.3 Fat loss .................................................................................................... 130
5.4.4 Volume change ....................................................................................... 132
5.5 Cooking model validation ................................................................................ 134
5.5.1 Finite element mesh ................................................................................ 134
5.5.2 Temperature profile-experimental data ................................................... 135
5.5.3 Temperature profile-published data ........................................................ 143
5.5.4 Moisture content- experimental data ...................................................... 145
5.5.5 Cooking yield — experimental data ......................................................... 146
5.5.6 Cooking yield - published data ............................................................... 148
5.6 Lethality model validation ............................................................................... 148
5.7 Illustration of model utility .............................................................................. 151
CONCLUSIONS .............................................................................................. 155
FUTURE WORK ............................................................................................. 158
APPENDICES ................................................................................................. 161

8.1 Model and experimental temperature versus time curves for moist-air
impingement cooking of ground beef patties ................................................... 162
8.2 Derivation of cooking-air thermo-physical property equations ....................... 189
8.3 Product thermo-physical properties ................................................................. 199
8.4 Screen shots from Visual Basic cooking model user interface ........................ 201
8.5 Visual Basic model code .................................................................................. 203
BIBLIOGRAPHY ............................................................................................ 255

vii

LIST OF TABLES

Table 3.1. Treatment conditions utilized for model validation experiments ............... 54

Table 5.1 — Analysis of variance for cooking time of ground turkey patties as affected
by temperature and fat content ...................................................................... 92

Table 5.2 - Analysis of variance for cooking time of ground beef patties as affected
by temperature and fat content ..................................................................... 93

Table 5.3 - Analysis of variance for cooking time of ground pork patties as affected
by temperature and fat content ..................................................................... 93

Table 5.4 — Analysis of variance of yield as a function of center temperature and
fat content for ground turkey patties .............................................................. 95

Table 5.5 - Analysis of variance of yield as a function of center temperature and fat
content for ground beef patties .................................................................... 96

Table 5.6 - Analysis of variance of yield as a function of center temperature and fat
content for ground pork patties .................................................................... 98

Table 5.7 — Analysis of variance for fat loss as functions of center temperature and
fat content for ground turkey patties .............................................................. 100

Table 5.8 — Analysis of variance for fat loss as functions of center temperature and
fat content for ground beef patties ................................................................ 101

Table 5.9 — Analysis of variance for fat loss as functions of center temperature and
fat content for ground pork patties ............................................................... 101

Table 5.10 — Analysis of variance for effects of temperature and initial fat content on
volume change for ground turkey patties cooked in a laboratory convection oven. . 103

Table 5.11 - Analysis of variance for effects of temperature and initial fat content on
volume change for ground beef patties cooked in a laboratory convection oven ......... 104

Table 5.12 — Analysis of variance for effects of temperature and initial fat content on
volume change for ground pork patties cooked in a laboratory convection oven. . . ..106

Table 5.13 — Results from regression of fat holding capacity as fimctions of time
and holding temperature for 5.6% fat ground beef. ........................................... 1 11

viii

Table 5.14— Results from regression of fat holding capacity versus holding
temperature for 5. 6% fat ground beef ............................................................ 112

Table 5.15 — Results from regression of fat holding versus time and holding
temperature for 15% fat ground beef ............................................................ 117

Table 5.16 - Results of regression of fat holding capacity versus holding
temperature for 15% fat ground beef ............................................................. 118

Table 5.17 — Linear regression of fat holding capacity as functions of heating
temperature, holding time, and initial fat content .............................................. 121

Table 5.18 - Linear regression of holding capacity as a function of holding
temperature and fat content ....................................................................... 122

Table 5.19 — Linear regression of patty center temperature as functions of oven
temperature, steam content, cooking time, and oven airflow ................................ 128

Table 5.20 — Regression parameters for cooking yield as a function of oven
temperature, steam content, cooking time, and airflow ....................................... 130

Table 5.21 — Regression parameters for cooking yield as a function of oven
temperature, steam content, cooking time, and airflow ...................................... 131

Table 5.22 - Standard error of prediction for the entire trial (SEP) and for data
above 45°C (SEP'D450C) for center temperature of beef patties .............................. 142

Table 5.23. Standard error of prediction for transient center temperature of ground
chicken breast patties predicted by the model and by the regression equation of
Murphy et al. (2001a) ............................................................................. 143

Table 5.24 — Difference between measured and predicted moisture content for each
oven condition. Experiment numbers correspond to the conditions listed in Table 3.1..145

Table 5.25 — Difference between measured and predicted cooking yields for each
oven condition. Experiment numbers correspond to the conditions in Table 3.1 . . . . . 146

ix

LIST OF FIGURES

Figure 3.1 — General arrangement of the laboratory convection oven showing
directions of steam and airflow .................................................................... 43

Figure 3.2 - Interior of laboratory oven showing conditioning chamber ..................... 43

Figure 3.3 - Interior of laboratory oven showing fan, ducts, and sample chamber

(located at left) ....................................................................................... 44
Figure 3.4 — (a) Picture and (b) schematic of jig used to place thermocouples into

meat patties during cooking experiments ........................................................ 46
Figure 3.5 — Schematic of tube setup used for centrifuging ................................... 50

Figure 3.6 — Schematic of the centrifuge tube after centrifuging showing the two
distinct layers that formed within the tube. In this schematic the heating tube has

been removed ......................................................................................... 51
Figure 3.7 — Stein model J SO-IV moist-air impingement oven .............................. 53
Figure 4.1 - Illustration of the airflow within an impingement oven in relation to

the product ............................................................................................ 62
Figure 4.2 — Geometry of a patty illustrating radial coordinate system ...................... 65
Figure 4.3 - Geometry of a ground meat patty with the modeled region and element

mesh indicated ....................................................................................... 78
Figure 4.4 - Illustration of the finite element mesh utilized for the model .................. 78

Figure 4.5 — Illustration of a triangular element showing counterclockwise node
numbering ............................................................................................ 81

Figure 5.1 - Center temperature as a function of cooking time for ground turkey
patties cooked in a laboratory convection oven: means of 5 replicates ....................... 90

Figure 5.2 — Center temperature as a function of cooking time for ground beef
patties cooked in a laboratory convection oven: means of 5 replicates..... . . . . . . . . . . . . ...91

Figure 5.3 — Center temperature as a function of cooking time for ground beef
patties cooked in a laboratory convection oven: means of 5 replicates..... . . . . . . . . . . . . . ......91

Figure 5.4 — Yield as a function of endpoint center temperature for ground turkey
patties of two fat contents: means of 5 replicates ................................................ 96

Figure 5.5 - Yield as a function of endpoint center temperature for ground beef
patties of two fat contents: means of 5 replicates ................................................ 97

Figure 5.6 - Yield as a function of endpoint center temperature for ground pork
patties of two fat contents: means of 5 replicates ................................................ 98

Figure 5.7 — Fat loss as a function of temperature for ground turkey patties of two
fat contents: means of 5 replicates .................................................................. 99

Figure 5.8 - Fat loss as a function of temperature for ground beef patties of two
fat contents: means of 5 replicates ............................................................... 100

Figure 5.9 - Fat loss as a function of temperature for ground pork patties of two
fat contents: means of 5 replicates ............................................................... 102

Figure 5.10 — Relationship between product temperature and volume change for
ground turkey patties cooked in a laboratory convection oven: means of 5
replicates ............................................................................................. 104

Figure 5.11 — Relationship between product temperature and volume change for
ground beef patties cooked in a laboratory convection oven: means of 5
replicates ............................................................................................. 105

Figure 5.12 - Relationship between product temperature and volume change for
ground pork patties cooked in a laboratory convection oven: means of 5
replicates ............................................................................................. 106

Figure 5.13 — Fat holding capacity as a function of temperature for 5.6% fat ground
beef heated for 2 minutes at temperatures from 30 to 90°C: means of 5
replicates ............................................................................................. 109

Figure 5.14 - Fat holding capacity as a function of temperature for 5.6% fat gr0und
beef heated for 5 minutes at temperatures from 30 to 90°C: means of 5
replicates ............................................................................................. 1 10

Figure 5.15 — Fat holding capacity as a function of temperature for 5.6% fat ground
beef heated for 10 minutes at temperatures from 30 to 90°C: means of 5
replicates ............................................................................................. 1 10

Figure 5.16 - Fat holding capacity as a function of temperature for 5.6% fat ground
beef heated for 15 minutes at temperatures from 30 to 90°C: means of 5
replicates ............................................................................................. 1 1 1

xi

Figure 5.17 — Comparison of fat holding capacity calculated from regression
model versus experimental values: means of 5 replicates ..................................... 114

Figure 5.18 — Fat holding capacity as a function of temperature for 15% fat ground
beef heated for 2 minutes at temperatures from 30 to 90°C: means of 5
replicates ............................................................................................ 1 15

Figure 5.19 - Fat holding capacity as a function of temperature for 15% fat ground
beef heated for 5 minutes at temperatures from 30 to 90°C: means of 5
replicates ............................................................................................. 1 -1 5

Figure 5.20 — Fat holding capacity as a function of temperature for 15% fat ground
beef heated for 10 minutes at temperatures from 30 to 90°C: means of 5
replicates ............................................................................................. 1 16

Figure 5.21 — Fat holding capacity as a function of temperature for 15% fat ground
beef heated for 15 minutes at temperatures from 30 to 90°C: means of 5
replicates ............................................................................................. 1 16

Figure 5.22 - Comparison of fat holding capacity calculated from regression
model versus experimental values (15% fat): means of 5 replicates ........................ 119

Figure 5.23 — Comparison of fat holding capacity calculated from regression model
versus experimental values: means of 5 replicates ............................................. 123

Figure 5.24 —— Bottom and center temperature versus time for ground beef patties

cooked at (a) oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 11.4 m/s and (b) oven temperature: 232°C, oven steam content: 50% by

volume, oven airflow: 16.8 m/s .................................................................. 125

Figure 5.25 - Cooking yield as a function of endpoint center temperature for ground
beef patties cooked in a Stein J SO-IV industrial moist air impingement oven. . . . . . . 129

Figure 5.26 - Yield loss not accounted for by moisture loss as a function of endpoint
temperature for ground beef patties cooked in a Stein J SO-IV industrial moist-air
impingement oven ................................................................................. 132

Figure 5.27 — Reduction in diameter during cooking as a function of cooking yield
for ground beef patties cooked in a Stein J SO-IV industrial moist-air impingement
oven .................................................................................................. 133

Figure 5.28 -— Example comparison of experimental temperature data with data

generated by the model (oven temperature=121°C, steam content=50%, air velocity
=11.4 m/s) ........................................................................................... 135

xii

Figure 5.29 — Example comparison of experimental temperature data with data
generated by the model (oven temperature=121°C, steam content=70%, air velocity
=11.4 m/s) ........................................................................................... 136

Figure 5.30 - Example comparison of experimental temperature data with data
generated by the model (oven temperature=121°C, steam content=8 8%, air velocity
=11.4 m/s) ........................................................................................... 136

Figure 5.31 — Comparison between model Salmonella Senfienberg lethality
predictions and data points published by Murphy et a1. (2002) .............................. 148

Figure 5.32 — Comparison between model Listeria innocua lethality predictions and
data points published by Murphy et al. (2002) ................................................. 149

Figure 8.1 - Oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 11.4 m/s (Experiment la) ............................................................... 162

Figure 8.2 — Oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 11.4 m/s (Experiment lb) ............................................................... 162

Figure 8.3 — Oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 16.8 m/s (Experiment 6a) ............................................................... 163

Figure 8.4 — Oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 16.8 m/s (Experiment 6b) ............................................................... 163

Figure 8.5 — Oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 21.8 m/s (Experiment 8a) ............................................................... 164

Figure 8.6 - Oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 21.8 m/s (Experiment 8b) ............................................................... 164

Figure 8.7 — Oven temperature: 121°C, oven steam content 70% by volume, oven
airflow: 11.4 m/s (Experiment 11a) .............................................................. 165

Figure 8.8 — Oven temperature: 121°C, oven steam content 70% by volume, oven
airflow: 11.4 m/s (Experiment 11b) .............................................................. 165

Figure 8.9 — Oven temperature: 121°C, oven steam content: 70% by volume, oven
airflow: 16.8 m/s (Experiment 13a) .............................................................. 166

Figure 8.10 - Oven temperature: 121°C, oven steam content: 70% by volume, oven
airflow: 16.8 m/s (Experiment 13b) .............................................................. 166

Figure 8.11 — Oven temperature: 121°C, oven steam content: 70% steam volume,
oven airflow: 21.8 m/s (Experiment 18a) ...................................................... 167

xiii

Figure 8.12 — Oven temperature: 121°C, oven steam content: 70% steam volume,
oven airflow: 21.8 m/s (Experiment 18b) ....................................................... 167

Figure 8.13 - Oven temperature: 121°C, oven steam content: 88% by volume, oven
airflow: 11.4 m/s (Experiment 21a) .............................................................. 168

Figure 8.14 - Oven temperature: 121°C, oven steam content:88% by volume, oven
airflow: 11.4 m/s (Experiment 21b) .............................................................. 168

Figure 8.15 - Oven temperature: 121°C, oven steam content: 78% by volume, oven
airflow: 16.8 m/s (Experiment 23a) .............................................................. 169

Figure 8.16— Oven temperature: 121°C, oven steam content: 78% by volume, oven
airflow: 16. 8 m/s (Experiment 23b) .............................................................. 169

Figure 8.17 -— Oven temperature: 121°C, oven steam content: 78% by volume, oven
airflow: 21.8 m/s (Experiment 25a) .............................................................. 170

Figure 8.18 — Oven temperature: 121°C, oven steam content: 78% by volume, oven
airflow: 21.8 m/s (Experiment 25b) .............................................................. 170

Figure 8.19 - Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 11.4 m/s (Experiment 30a) .............................................................. 171

Figure 8.20 - Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 11.4 m/s (Experiment 30b) .............................................................. 171

Figure 8.21 — Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 16.8 m/s (Experiment
32a) .............................. 172

Figure 8.22 — Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 16.8 m/s (Experiment 32b) .............................................................. 172

Figure 8.23 - Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 21.8 m/s (Experiment 34a) .............................................................. 173

Figure 8.24 - Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 21.8 m/s (Experiment 34b) .............................................................. 173

Figure 8.25 - Oven temperature: 177°C, oven steam content: 70% by volume, oven
airflow: 11.4 m/s (Experiment 37a) .............................................................. 174

Figure 8.26 — Oven temperature: 177°C, oven steam content: 70% by volume, oven
airflow: 11.4 m/s (Experiment 37b) .............................................................. 174

xiv

Figure 8.27 - Oven temperature: 177°C, oven steam content: 83% by volume, oven
airflow: 11.4 m/s (Experiment 47a) ............................................................. 175

Figure 8.28 - Oven temperature: 177°C, oven steam content: 83% by volume, oven
airflow: 11.4 m/s (Experiment 47b) ............................................................. 175

Figure 8.29 - Oven temperature: 177°C, oven moisture content: 84% by volume,
oven airflow: 16.8 m/s (Experiment 49a) ....................................................... 176

Figure 8.30 - Oven temperature: 177°C, oven moisture content: 84% by volume,
oven airflow: 16.8 m/s (Experiment 49b) ...................................................... 176

Figure 8.31 - Oven temperature: 177°C, oven steam content: 86% by volume, oven
airflow: 16.8 m/s (Experiment 50a) .............................................................. 177

Figure 8.32 - Oven temperature: 177°C, oven steam content: 86% by volume, oven
airflow: 16.8 m/s (Experiment 50b) .............................................................. 177

Figure 8.33 - Oven temperature: 177°C, oven steam content: 86% by volume, oven
airflow: 21.8 m/s (Experiment 54a) .............................................................. 178

Figure 8.34 - Oven temperature: 177°C, oven steam content: 86% by volume, oven
airflow: 21.8 m/s (Experiment 54b) .............................................................. 178

Figure 8.35 — Oven temperature: 232°C, oven steam content: 50% by volume, oven
airflow: 11.4 m/s (Experiment 56a) .............................................................. 179

Figure 8.36 - Oven temperature: 232°C, oven steam content: 50% by volume, oven
airflow: 11.4 m/s (Experiment 56b) .............................................................. 179

Figure 8.37 - Oven temperature: 232°C, oven steam content: 50% by volume, oven
airflow: 16.8 m/s (Experiment 58a) ............................................................. 180

Figure 8.38 - Oven temperature: 232°C, oven steam content: 50% by volume, oven
airflow: 16.8 m/s (Experiment 58b) .............................................................. 180

Figure 8.39 - Oven temperature: 232°C, oven steam content: 50% by volume, oven
airflow: 21.8 m/s (Experiment 63a) .............................................................. 181

Figure 8.40 - Oven temperature: 232°C, oven steam content: 50% by volume, oven
airflow: 21.8 m/s (Experiment 63b) .............................................................. 181

Figure 8.41 - Oven temperature: 232°C, oven steam content: 70% by volume, oven
airflow: 11.4 m/s (Experiment 66a) .............................................................. 182

XV

Figure 8.42 - Oven temperature: 232°C, oven steam content: 70% by volume, oven
airflow: 11.4 m/s (Experiment 66b) .............................................................. 182

Figure 8.43 - Oven temperature: 232°C, oven steam content: 70% by volume, oven
airflow: 16.8 m/s (Experiment 68a) .............................................................. 183

Figure 8.44 - Oven temperature: 232°C, oven steam content: 70% by volume, oven
airflow: 16.8 m/s (Experiment 68b) .............................................................. 183

Figure 8.45 - Oven temperature: 232°C, oven steam content: 70% by volume, oven
airflow: 21.8 m/s (Experiment 70a) ............................................................. 184

Figure 8.46 - Oven temperature: 232°C, oven steam content: 70% by volume, oven
airflow: 21.8 m/s (Experiment 70b) .............................................................. 184

Figure 8.47 — Oven temperature: 232°C, oven steam content: 82% by volume, oven
airflow: 11.4 m/s (Experiment 73a) .............................................................. 185

Figure 8.48 - Oven temperature: 232°C, oven steam content: 82% by volume, oven
airflow: 11.4 m/s (Experiment 73b) .............................................................. 185

Figure 8.49 — Oven temperature: 232°C, oven steam content: 82% by volume, oven
airflow: 11.4 m/s (Experiment 75a) .............................................................. 186

Figure 8.50 - Oven temperature: 232°C, oven steam content: 82% by volume, oven
airflow: 11.4 m/s (Experiment 75b) .............................................................. 186

Figure 8.51 — Oven temperature: 232°C, oven steam content: 82% by volume, oven
airflow: 16.8 m/s (Experiment 78a) .............................................................. 187

Figure 8.52 - Oven temperature: 232°C, oven steam content: 82% by volume, oven
airflow: 16.8 m/s (Experiment 78b) .............................................................. 187

Figure 8.53 — Oven temperature: 232°C, oven steam content: 82% by volume, oven
airflow: 21.8 m/s (Experiment 80a) .............................................................. 188

Figure 8.54 — Oven temperature: 232°C, oven steam content: 82% by volume, oven
airflow: 21.8 m/s (Experiment 80b) .............................................................. 188

Figure 8.55 - Latent heat of vaporization for water as a function of temperature
(From tabular data: Geankoplis, 1993) ......................................................... 189

Figure 8.56 - Viscosity of air as a function of temperature (From tabular data:
Geankoplis, 1993) .................................................................................. 190

xvi

Figure 8.57 - Viscosity of steam as a function of temperature (From tabular data:
Geankoplis, 1993) ................................................................................. 191

Figure 8.58 - Density of air as a function of temperature (from tabular data:
Geankoplis, 1993) ................................................................................. 193

Figure 8.59 - Density of saturated steam as a function of temperature (From tabular
data: Geankoplis, 1993) ........................................................................... 194

Figure 8.60 - Density of steam at 101.35 kPa as a function of temperature (From
tabular data: Geankoplis, 1993) .................................................................. 195

Figure 8.61 - Thermal conductivity of air as a function of temperature (From tabular

data: Geankoplis, 1993) ........................................................................... 196
Figure 8.62 - Thermal conductivity of steam as a firnction of temperature (From

tabular data: Geankoplis, 1993) .................................................................. 197
Figure 8.63 — Input screen of cooking model user interface .................................. 201
Figure 8.64 — Output screen of cooking model user interface ............................... 202

xvii

KEY TO SYMBOLS

Lower-case letters

“030068)

D‘FWG QC-
q

'-1

3

xxx-wt:-
“'1

.1

NN~<><g<~m~uo==3~

Equation constant

Equation constant

Equation constant

Mass capacity (g/g)

Heat capacity (J/g°°C)

Equation constant (Equation [22])
Diameter (cm)

Equation constant

Equation constant

Effective heat transfer coefficient (W/cm2-°C)
Mass transfer coefficient (cm/s)

Heat transfer coefficient (W/cm2'°C)
Inactivation rate constant (l/s)

Fat conductivity (g/scm)

Moisture conductivity (g/s'cm)
Thermal conductivity (W/cm'K)
Length (cm)

Moisture content — decimal dry basis
Mass flux (g/s-cmz)

Normal direction (cm) (Equation [44])
Heat flux (W/cmz)

Length, r-direction (radial coordinates) (cm)
Position (cm)

Time (8)

Air velocity (cm/s)

Slot width (cm)

Length, x-direction (cm)

Length, y-direction (cm)

Length, z-direction (cm)

Microbial z-value (°C)

Upper-case letters

A

C

D
DAB
Dcap,fat
Dm
Dref
Ea

Area (cmz)

Concentration (g/cm3)

Decimal reduction time (s)

Diffusivity ofA in B (cmZ/s)

Fat capillary diffusivity (cmz/s)

Mass diffusivity (cmZ/s)

Decimal reduction value at reference temperature
Activation energy (J/gmol)

xviii

é-lv-lgmogu

-1.-.-1

E3.

><2<

Dry-basis fat content (g fat/ g dry matter)
Initial dry-basis fat content (g fat/ g dry matter)
F atzprotein ratio (g fat/g protein)
Enthalpy (J/ g)

Slot height (cm)

Mass (g)

Number of microorganisms (CFU/ g)
Initial number of microorganisms (CFU/ g)
Pressure (Pa)

Saturation pressure (Pa)

Partial pressure of water vapor (Pa)
Ideal gas constant (cm3-Pa/g mol-K)
N/No

Relative humidity (%)

Temperature (°C)

Melting temperature (°C)

Reference temperature (°C)

Saturation temperature (°C)

Volume (cm3)

Slot width (cm)

Mass fi'action (decimal)

Dimensionless groups

Nu
Pr
Re
Sc
Sh

Greek letters

ttQDPme

Nusselt number (th/kr)
Prandtl number (Cp’lJ/k'l‘)
Reynolds number (D-v-p/u)
Schmidt number (u/p-DAB)
Sherwood number (hm'X/DAB)

Thermal diffusivity (cmz/s)

Emissivity (-)

Error term

Latent heat (J/ g)

Density (g/cm3)

Stefan-Boltzmann constant (5.676-10‘12 W/cmz-K4)
Specific growth rate (l/s)

Viscosity (Pa-s)

xix

1 INTRODUCTION AND OBJECTIVES

1.1 Background

Impingement cooking is used for manufacturing many fully and partially cooked
meat and poultry products. Impingement ovens utilize a type of convection in which
cooking air is directed normal to the product surface at high velocities (>10 m/s). This is
generally accomplished by forcing the cooking air through an array of slots or nozzles.
Impinging airflow allows for heat transfer rates an order of magnitude higher than those
that occur in conventional convection ovens (Gardon and Akfirat, 1966).

Moist-air impingement ovens combine impinging airflow with high humidity
cooking air. In moist-air ovens, humidity can exceed 90% moisture by volume. The use
of high moisture cooking air creates a condensing condition on the surface of the product.
Condensing conditions dramatically increase the rate of cooking by taking advantage of
the release of latent heat from steam in the cooking air. High moisture cooking air also
suppresses product moisture loss during cooking. This results in cooking yields higher
than those achieved when cooking with dry air. In meat cooking systems, a second
cooking section with lower air moisture content is often utilized to aid in surface
browning of the product.

Development of a computerized model for moist-air impingement cooking of
meat products is of interest for two major reasons. First, cooking models can be used as
tools for optimizing cooking processes. Heat and mass transfer models can be utilized to
optimize processes in terms of product temperature, yield, and quality. Optimizing the

cooking process can have major economic significance. In the United States alone, there

are 76 manufacturing facilities producing ground beef and poultry patties. The value of
production from these plants exceeds $520 million per year (F818, 2001). Even modest
increases in cooking yield stand to increase profits significantly. A one percent increase
in yield could potentially increase annual profits by over $5 million. Reducing energy
costs and wasted line capacity due to overcooking stands to improve profits even further.
Reducing overcooking also stands to improve product quality characteristics such as
color, flavor, and texture.

The second and more important reason for modeling the cooking process is to
ensure microbial safety of the cooked product. The Food Safety and Inspection Service
(FSIS) has proposed regulations that would shift the emphasis of cooking regulations for
ground meat and poultry products from a system of time-temperature standards to a
system of performance standards (F818, 2001). Such regulations have already been
enacted for whole muscle meat products (F818, 1999). The new performance standards
are based upon a 7-10g10 reduction in viable Salmonella for ready-to-eat (RTE) poultry
products and a 6.5-logro reduction in Salmonella for RTE beef, roast beef, and corned
beef products.

The new standards are designed to give processors the flexibility to develop
customized, science-based processing strategies. However, it will be the duty of the
processor to demonstrate that the process meets the required reductions in Salmonella.
Unfortunately, adequate scientific tools are not currently available for processors to
reliably and confidently verify compliance with the Salmonella performance standards.

Processors need a tool that will allow them to determine the effects of processing

on the reduction of Salmonella. Due to safety concerns, it is neither feasible nor

desirable for manufacturers to conduct microbial challenge studies in the factory
environment. Experiments can be conducted in the laboratory using microbial pathogens
including Salmonella and Listeria monocytogenes in model food systems, but these
results ofien do not correspond well with actual processing conditions. The use of
mathematical models is an alternative to conducting microbial chellenge studies.
Although mathematical models are not a substitute for the judgment of an experienced
microbiologist, F SIS has indicated that mathematical models based on heat transfer
equations can be used to demonstrate the effects of processing on bacterial inactivation
(F818, 2002).

Descriptive mathematical models for cooking processes would be of great value
to processors and would allow for continuing innovation in the processed meat industry.
Unfortunately, the development of cooking models is limited, with current models not
accounting for the unique surface heat and mass transfer conditions present in moist-air
impingement ovens (Chapter 2). Few existing models account for variations in product
composition which can have significant effects on heat transfer and cooking yield. In
most cases, yield loss is modeled solely as a function of moisture changes in the product
during cooking. Although exceptions exist, most of the available cooking models have
not been coupled with models for microbial inactivation.

To be of maximum utility, cooking models should include all variables that affect
microbial lethality, including product size, density, specific heat, thermal conductivity,
product composition, humidity, and strain of the organism (F818, 2001). Models should
be flexible and applicable to various cooking conditions without the need for generation

of empirical data sets. Mass losses due to both moisture and fat loss during cooking

should be considered. Finally, models should be validated against actual process data,

including microbial inactivation data from inoculated challenge studies.

1.2 Objectives

The overall goal of this study was to develop a cooking model that would meet
the above criteria, as well as provide a graphical tool for illustrating the effects of
processing to manufacturing plant personnel. The specific objectives of this study were

to:

1. Develop a coupled heat and mass transfer model incorporating the transient effects of
moisture and fat transfer during moist-air impingement cooking of ground beef
patties.

2. Incorporate Salmonella inactivation models into the heat and mass transfer model and
combine with a user interface to produce a tertiary cooking model.

3. Validate the heat and mass transfer model using data collected from an industrial
moist-air impingement oven.

4. Validate the Salmonella inactivation models using data collected from inoculated
challenge studies in a pilot-scale moist-air impingement oven.

5. Develop the user interface as a tool than can be used to illustrate the effects of
cooking parameters on yield and microbial safety to non-technical personnel,

including oven operators.

2 REVIEW OF LITERATURE

2.1 Introduction

From the perspective of the processor, the most important output variables of
cooking are temperature profile, cooking yield, and microbial inactivation. Cooking
models can be used as a powerful tool for estimating these parameters. To model the
reduction in pathogens such as Salmonella during cooking, it is first necessary to model
the cooking process itself. The temperature history of the meat during cooking is the
most important factor for modeling microbial reduction. As a result, heat transfer has
been the focus of many models for cooking processes. However, in terms of profitability,
the most important factor for processors to consider is cooking yield. Therefore, mass
transfer is also an important factor to consider when modeling cooking processes.

The first portion of this chapter summarizes previous research in the area of
cooking models. The thermo-physical properties required for modeling are also
discussed. The final portion of the chapter addresses microbial modeling and previous

work combining microbial models with models for meat cooking.

2.2 Introduction to cooking

Cooking and other heat treatments are among the most important unit operations
in the food processing industry. Heat treatments during processing vary widely,
depending upon the type of product. Foods may be fully or partially cooked, blanched to
inactivate enzymes, dried to extend shelf life, pasteurized to kill unwanted

microorganisms, or simply heated as part of the manufacturing process (Fellows, 1988).

The types of devices used for cooking and heating are nearly as numerous as the number
of products that can be produced. Examples include heat exchangers, deep-fat fryers,
contact-cooking systems, convection ovens, microwaves, and infrared ovens.

As a result of their ubiquity in the food industry, heating and cooking systems are
of great interest to the food engineer. Ovens and other heating systems must be properly
designed to ensure safe foods of the highest quality. Under-cooking may result in foods
that are not safe to eat due to surviving pathogenic microorganisms. Over-cooked
products may not have the quality characteristics demanded by customers. In addition to
affecting product quality, overcooking can also be wasteful to the processor in terms of
energy consumption and reduced product yield.

Most engineering analyses of cooking processes are aimed at determining the
temperature and/or moisture profiles of the product undergoing cooking. Meat products
are generally cooked prior to packaging. During cooking of unpackaged products, heat is
transferred into the product, resulting in increased temperatures and thermal gradients
within the meat. At the same time, moisture is transported out of the product, resulting in
moisture gradients and yield loss (with the exception of boiling, in which case moisture
may be added to the product). Fat losses during cooking can also be significant (Young

et al., 1991; Pan and Singh, 2001; Badiani et al., 2002).

2.3 Heat and mass transfer in meat during cooking
Numerous studies have described heat transfer during cooking of meat
(Dagerskog and Bengtsson, 1974; Dagerskog, 1979a and b; Skjoldebrand, 1980;

Skjoldebrand and Hallstrom, 1980; Housova and Topinka, 1985; Thorvaldsson and

Skjoldebrand, 1995; Pan and Singh, 2001; Shilton et al., 2002). Virtually every analysis
presumes that the dominant mechanism of heat transfer within meat products is
conduction. However, Shilton et al. (2002) proposed that for high fat ground meat, heat
transfer may be related to both conduction and internal convection within the melted fat
phase.

Mass transport of water within meat products has also received considerable
attention (Dagerskog and Bengtsson, 1974; Hung et al., 1978; Dagerskog, 1979a and b;
Skjoldebrand, 1980; Skjoldebrand and Hallstrom, 1980; Mittal et al., 1982; Mittal et al.,
1983; Hallstrom, 1990; Thorvaldsson and Skjoldebrand, 1995; Pan and Singh, 2001).
Most cooking models assume that mass transport of water is primarily the result of
molecular diffusion to a drying product surface (Huang and Mittal, 1995; Ngadi et al.,
1997; Zanoni et al., 1997; Chen et al., 1999; Mittal and Zhang, 2000; Mittal and Zhang,
2001; Shilton et al., 2002).

Hung et al. (1978) proposed a different mechanism for water transport during
convection cooking of bovine semitendinosus muscles in which it was determined water
loss during cooking of these muscles was most likely due to pressure forces caused by
fiber shrinkage. Furthermore, water loss was only weakly dependent on oven temperature
and appeared to depend primarily on the amount of muscle shortening during cooking.
For samples cooked directly from the frozen state, an initial period of drip loss was
observed. The water lost during this period was believed to be due to the melting of ice
crystals within the muscle. During this initial drip period, sample orientation affected the
amount of loss; with samples having muscle fibers oriented vertically producing the

highest losses. In this period, gravity was believed to contribute to the water loss. For

samples thawed before cooking, no such initial drip period existed. Thorvaldsson and
Skjoldebrand (1995) also noted that water transport was up to 25% faster in the direction
parallel to muscle fibers than across muscle fibers during oven roasting of bovine
semitendinosus muscles.

Some authors have taken to expressing moisture transport in meat in terms of
"water holding capacity" (Dagerskog, 1979a and b; Pan et al., 2000). This represents the
amount of water that a type of meat will contain, given a certain temperature history.
Upon heating, the water holding capacity of meat products generally decreases, resulting
in water losses. Although this type of analysis does not truly describe a mechanistic
process of water transport, it coincides with the knowledge that protein releases water
upon denaturing (Bodwell and McClain, 1978).

Combined with the fiber squeezing mechanism described by Hung et al. (1978),
the water holding capacity model provides a reasonable approximation of water transport
during some cooking processes. However, this technique does not account for resistance
to moisture transport within the product. This a significant weakness for most processes,
since moisture diffusivity within meats is known to be low (Zanoni et al., 1997). Models
that do not account for moisture diffusivity must be inherently empirical, as the

mechanisms for transport within the product are not described based upon first principles.

2.4 Modeling the cooking process
Models for cooking processes can be divided into two categories. They are either
based solely on data from experimental studies (empirical models) or derived from

theoretical formulas of heat and mass transfer (Hayakawa, 1970). Empirical models can

be developed for cooking processes using data collected under the conditions of interest.
These models require less rigorous mathematical analysis and can provide good
predictive capability for a limited range of conditions. Additionally, empirical models do
not generally require the researcher to have accurate thermal property data to get good
results. The weakness of empirical models is that they are not generally applicable to
situations that are different from the conditions for which the model was developed.
Slight changes in the system, such as changes in product size or geometry, require an
entirely new set of experimental tests to generate new model parameters. Although it is
theoretically possible to construct empirical models with an unlimited number of input
parameters, the number of inputs that can be meaningfully included in the model is
limited in practice.

Models based on theoretical principles are generally more robust than empirical
models. Fundamental heat and mass transfer equations can be used to produce models
that are much more general in nature than can be accomplished through experimental
methods. By changing the boundary conditions of the model equations, a single model
can often be used for several different types of cooking processes. However, models
based on theoretical principles have their own drawbacks. Differential equations for heat
and mass transfer quickly become too cumbersome to solve using analytical techniques.
Thus, numerical methods are typically applied to cooking models. In order to produce
models based upon transport equations, accurate knowledge of thermal and physical
properties of the product must be known. In situations where properties are not

accurately known, theoretical models end up being semi-empirical in nature, as the model

constants must be adjusted to fit experimental process data. A number of different

empirical and theoretical models are described in the following sections.

2.4.1 Empirical models

Empirical models can be effective for predicting the temperature and moisture
history of products during cooking. Bengtsson et al. (1976) developed a model for heat
and mass transfer during oven roasting of meat. In the cooking process that was
modeled, the meat was placed on a metal rack inside of an oven chamber with walls
maintained at a fixed temperature. The mechanisms of heat transfer in this type of
cooking are primarily natural convection and radiation. A plot consisting of a
dimensionless temperature term (Tair-chnm)/(Tau-Tim,”) as a function of a dimensionless
time term (kt)/(pcp12) on a semi-log plot was used to develop simulation equations. From
the diagram, a prediction equation for meat temperature as a function of time was
developed. Results of the comparison between predicted and experimental results were
good; however, quantitative results were not given.

Sarkin (1978) used a technique nearly identical to that of Bengtsson et al. (1976)
to produce a computer cooking simulation for meat products. An equation for the linear

portion of the heating curve was developed (Equation [2.1]).

 

T - -T
log[ 3" )=a+b-t [2.1]
Tair - Tinitial

10

Coefficients a and b are constants and were determined by regression. The model had
errors between the predicted and experimental temperatures of 02°C or less for cooking
times of up to 10 hours. These results show the close predictions that are possible with
empirical models.

Erdogdu et al. (1999) developed an empirical model for predicting the yield loss
of shrimp during immersion cooking. Shrimp were cooked isothermally for varying
periods in water at temperatures between 65 and 95°C. Afier heating, the yield losses
were measured, and an equation for yield loss was determined by multiple linear

regression (Equation [2.2]).

%YieldLoss=a+b-T+c-T2+d-t2+e-t+f-T-t [2.2]

The terms a, b, c, d, e, and f were regression constants. The error between experimental
and predicted yield losses ranged from 0.15 to 2.6% yield. This again shows the potential
for empirical models to produce very close approximations of experimental data.

The primary drawback to each of the empirical models is that they are only valid
for a specific set of cooking conditions. Although the error of the models is very low,
this is to be expected, as the same data that were used for model development were used
for error calculation. The models are not applicable to conditions other than those used in
the regression. The cooking models of Bengtsson et al. (1976) and Sarkin (1978) only
allow for meat temperature to be predicted as a function of cooking time. No provision is
made for predicting yield, and processing conditions are not considered, thereby limiting

the usefulness of the models as tools for processors. The yield model of Erdogdu (1999)

11

allows yield to be calculated as a function of time and temperature, but temperature
profile is neglected. All three models ignore the effects of product composition. In
addition, the models are not readily adaptable to products of different sizes or geometries
from those used for model development. The weaknesses illustrated by these models
show the underlying reason that empirical models cannot be utilized as general models

for the varying processing conditions that occur in industry.

2.4.2 Models based upon heat and mass transfer principles

Most cooking models reported in the literature are based on theoretical heat and
mass transfer equations. Models based on transport principles are generally more flexible
than those developed through purely empirical methods. The equations that govern heat
and mass transfer are not product specific and are generally accurate for a wide range of
conditions.

During cooking of most meat products, the predominant mechanism of heat
transfer within the product is conduction. The governing equation for conductive heat

transfer in rectangular coordinates is derived from an energy balance (Bird et al., 1960).

6T 5 0T a BI a BI
O O —— = — k 0 — + _ k . — + — k . — 203

Various cooking processes can be simulated using the conduction equation. Most
cooking processes transfer heat to the product via the food surface. As a result, different

cooking types can be modeled by changing the surface boundary conditions as

12

appropriate for the type of cooking to be modeled. Models for conductive, convective,
and radiative cooking can be constructed by including the appropriate heat flux boundary
conditions. The flux equations for conduction, convection, and radiation heat transfer are

well known and are given by Equations [2.4], [2.5], and [2.6] respectively.

q 6T

X = kT ' a surface [ 2'4]
%= 111‘ '(Too — Tsurface) [ 2'5 ]
q 4 4

X = 8 ' O ' (Tsurface '- Theat source ) I 2-6 I

Most mass transfer models are based upon transport by diffusion. The governing

equation for diffusion is similar in form to the equation for conduction (Bird et al., 1960).

99:3(Dm .§)+_6_[Dm .§]+£(Dm “QC—j [ 27]
at 6x 6x 6y 8y 62 62

Treatments of surface mass transfer vary, but are typically modeled using a

convective mass transfer boundary condition and/or a term for evaporation. The equation

13

for mass transfer by convection is analogous to convective heat transfer and is usually

given by Equation [2.8].

n = hm '(Cequilibrium " C) I 28]

Numerous authors have published theoretical models for meat cooking. Models

have been developed for contact cooking (Dagerskog, 1979a and b; Ikediala et al., 1996;
Pan et al., 2000), immersion frying (N gadi et al., 1997; Farkas et al., 1995a and b;
Vijayan and Singh, 1997; Mittal and Zhang, 2001), convection cooking (Mittal and
Blaisedell, 1982; Mittal et al., 1983; Holtz and Skjoldebrand, 1986; Huang and Mittal,
1995; Zanoni et a1, 1997; Chen et al., 1999; Mittal and Zhang, 2000), oven roasting
(Singh et al., 1984), microwave cooking (Mallikarjunan et al., 1996), and infrared
cooking (Shilton et al., 2002). The following sections describe models for each of these

types of cooking.

2. 4.3 Contact cooking

Contact cooking is one of the simplest mechanisms of cooking. During contact
cooking, products are placed in direct contact with a heating surface, resulting in heat
transfer by conduction. The rate of cooking is limited by the temperature gradient
between the product and heating surface and by the surface resistance of the
product/heating element interface. This type of cooking is common in the fast food

industry, particularly for ground beef patties.

14

Dagerskog (1979a) developed an early model for contact cooking of meat patties.
The model was based on a one-dimensional formulation of the conduction equation with
a term added to incorporate the latent heat of evaporation of water. This equation was
solved numerically using a finite difference technique.

Transfer of moisture out of the meat was calculated based upon experimental
measurements of water holding capacity. The capacity of meat to store water under
varying times and temperatures was determined experimentally. Experimental
determination of water holding capacity was conducted by heating 10 g samples enclosed
in sealed plastic pouches in an isothermal water bath. Water content of the samples was
measured afler they had been heated and allowed to drain for 1 minute on absorbent
paper. Water holding capacities were then plotted as functions of temperature and time.
For each time step in the model, it was assumed that the free water released due to the
change in water holding capacity was transported out of the meat. Heat and mass
transport equations were used to determine the amount of water that exited the meat due
to evaporation. A second water loss equation was formulated using a mass balance based
on changing water holding capacity. The difference in water loss due to changing water
holding capacity and water lost by evaporation was attributed to drip loss. Differences
between the experimental and simulated water losses ranged from 0.2 to 2.7% loss for a
pan temperature of 140°C and from 0.3 to 2.9% loss for a pan temperature of 180°C.
Predictions of the center temperature ranged from 0 to 2°C of the experimental values at
cooking times of up to 6 minutes.

The water holding capacity model has an advantage over other methods, in that it

accounts for the varying water binding capacities of different products. It also has the

15

benefit of easily describing the drip loss phenomenon. Weaknesses of the water holding
capacity model are that it is highly empirical and does not account for internal resistance
to moisture transfer. Because it does not account for internal resistance to moisture
transfer, the water holding capacity model is expected to decrease in accuracy with
increasing product thickness. The water holding capacity model is also product specific,
as different products have much different water binding properties.

Ikediala et al. (1996) developed a model for single-side pan-frying of meat patties
that was not based on the water holding capacity model, but rather a two-dimensional
Fourier conduction equation formulated in radial coordinates with an added term for the

evaporation of water.

a -c -T

M=£~§~£nkT~fl)+—a—(kT-§]+Np-M [2.9]
r

A contact heat transfer coefficient was used to describe heat transfer at the patty/ grill
interface. The same thermal properties reported by Dagerskog (1979a) were used for
calculations. The primary difference between the heat transfer component of this model
and the model of Dagerskog (1979a) was the expansion of the conduction equation into
two dimensions. This allowed for the temperature distribution in the patty during
cooking to be modeled in two dimensions, resulting in a more complete description of the
temperature profile. Deviations between experimental and predicted center temperatures

were less than 4°C.

16

Ikediala et al. (1996) modeled moisture loss in the patty slightly differently than
Dagerskog (1979a). The average moisture content of the patties was modeled using an

exponential decay model.
mave : mave,initial ' CXP(- (3 + b ' T) ' t) I 210]

The constants a and b for the mass transfer equation were determined using
regression of experimental values. The use of an exponential decay function for moisture
transfer is completely empirical and represents a total deviation from first-principles. The
result is that the model is specific for the product and conditions tested and lacks
robustness.

Pan et al. (2000) used a technique similar to that of Dagerskog (1979a) and
Ikediala et al. (1996) to model cooking of hamburgers by two-sided contact cooking.
Heat transfer was modeled using a one-dimensional formulation of the Fourier
conduction equation. However, the conduction equation was written in terms of

enthalpy, rather than temperature.

fi=£[kT(H).flaifl) [ 2.11]

Use of the enthalpy formulation allowed cooking from the frozen state to be modeled
without discontinuities related to the phase change from ice to liquid water. Relationships

between enthalpy and temperature were used to produce temperature history curves.

l7

Enthalpy of frozen hamburger was calculated using a food property computer program

(Singh and Mannapperuma, 1994). Enthalpy of unfrozen hamburger was calculated

using an empirical equation (Equation [2.12]).

Hnonfrozen =Href +p'I(l6OO+2600'XW +15'Xf 'T)'(T—Tref)} I 2-12]

The changes in moisture and fat contents, independent of position, were

determined using Equations [2.13] and [2.14].

= Thm ' (m - mequilibrium) I 2-13 I

9|?

[2.14]

12L 2 —hf ' (F - 1:equilibrium)

The model assumed that no transport of water or fat occurred below specified
threshold temperatures. The temperature dependence of the water and fat equilibrium

concentrations was given by Equations [2.15] and [2.16].

( 6w(T-Tinma1)) [ 2.15]

mequilibrium = minitial '6

18

Fequilibrium = Finitial 'e(_6F(T_Ti“m°l» [ 2-16 l

The previous equations were solved at each node for every time step, giving
moisture and fat contents as functions of temperature. A finite difference technique was
used for solving the heat transfer and mass equations. As in the model by Dagerskog
(1979a), this method treats water and fat content as "state" variables (i.e., functions of
just temperature rather than results of transport processes). The 5“, and Stems are
related to the water and fat holding capacities of the meat and were determined
experimentally with whole patties.

This model (Pan et al., 2000) is one of the most complete models for cooking
available in the literature. Differences between predicted and center temperatures were
small, although differences of up to 10°C occurred in the temperature range between 0
and 40°C. Differences between measured and predicted yields were less than 3%.
Inclusion of fat transport as a separate mechanism sets the model apart from any of the
prior contact-cooking models. In addition, utilization of the enthalpy formulation for heat
transfer represents an important step in the simulation of temperature profiles for
products that may originate in the frozen state. However, the empirical nature of the
water and fat holding capacity models limits robustness. Water and fat holding capacity
were determined using whole patties. In this way, internal resistance to moisture and fat
transfer was accounted for indirectly. However, this approach limits the utility of the
model to products of the same composition and geometry that was used to develop the

model parameters.

19

2.4.4 F tying

Frying is another operation that is commonly used to cook meat products.
Immersion (deep fat) frying is typically modeled as a moving boundary problem (Singh,
2000). Products undergoing frying are described as consisting of an inner core region
surrounded by a dry crust region. As cooking proceeds, the boundary between the crust
and core regions moves towards the product center.

Vijayan and Singh (1997) developed a model for heat transfer during flying of
frozen foods. Frying was modeled as a moving boundary problem. Separate transport
equations were utilized for the crust and core regions. Heat transfer in the crust was
modeled using a one-dimensional formulation of the Fourier conduction equation. A
convective boundary condition was used to describe heat transfer at the oil-product
interface. The position of the crust-core interface at a given time was given by Equation

[2.17].

A-t

Sj+1=3j'*'chrust —CIcore]'r_

[2.17]
pm)

In the core of the product, heat transfer was modeled using an enthalpy
formulation of the conduction equation (see Equation [2.11]). Differences between
simulated and measured center temperatures ranged from 1 to 6°C over a range of 0 to
75°C. The model did not account for mass transfer. Therefore, although useful for

predicting temperature, this model cannot be used for yield predictions.

20

A model for moisture transport during deep fat frying of chicken drums was
developed by Ngadi et al.(1997). A two-dimensional diffusion equation formulated in

radial coordinates was used to describe moisture transfer in the product (Equation [2.18]).

6K: 1 6 ac: a at:
___=._ _. .[, .__. +2.. I) .___ :218
at r arIr m ar] 62[ m 62) [ 1

The two-dimensional formulation of the conduction equation allows for modeling of non-
spherical products. This is useful not only for chicken drums, but for other products such
as chicken strips.

An exponential function was used for the surface mass transfer boundary
condition. This resulted in an empirical relationship similar to that used by Ikediala et al.

(1996)
C = Cinitial 'CXP((-a + b ° Ton ) ' t) l 2-19 1

The constants a and b were determined by regression of experimental data and were equal
to -0.045 and 4.167-10“ respectively. The utilization of an empirical boundary condition
greatly limits the robustness of the model.

The finite element grid was broken up into elements representing the bone, bone
marrow, cartilage, and muscle portions of the chicken drumstick. The moisture
diffusivity of the muscle portion of the drumstick was modeled as a function of cooking

oil temperature and moisture content using an equation developed in separate

21

experiments (Equation [220]); (Ngadi and Correia, 1995). Constant diffusivities were

used for the other components.

_ 2930

oil

 

1)m =8.35-IO'° ~exp[ —0.56l-C+O.092-C2] [ 2.20]

This model represents a good tool for determining moisture distributions in a
complex system undergoing cooking, although moisture predictions were off by up to
30% for some conditions. Unfortunately, this was not combined with a model for heat
transfer. Nonetheless, the usefulness of the finite element method for modeling systems
with multiple physical properties and complex geometries was demonstrated.

Mittal and Zhang (2001) developed a model for deep fat frying using a different
approach. An artificial neural network (ANN) was used to predict temperature, moisture,
and fat content in meatballs during deep fat frying. Input parameters included fat
diffusivity, moisture diffusivity, thermal diffusivity, heat transfer coefficient, fat
conductivity, and oil temperature, as well as frying time, meatball radius, and initial
temperature. The data used to train the artificial neural network were generated from
validated mathematical models. Heat, moisture, and fat concentration were modeled
using one-dimensional transport equations formulated in radial coordinates. The
maximum errors between modeled and experimental temperature, moisture, and fat
content were 1.9°C, 0.004% dry basis, and 0.016% dry basis, respectively, for the

optimum ANN design.

22

Artificial neural networks have several advantages over other types of modeling
for predicting temperature and moisture distributions during cooking. ANN ’s can "learn"
from new data to increase accuracy of prediction. This allows experimental data to be
combined with simulation data from models. Neural networks can often generate results
faster than mathematical models, which is useful in optimization studies where many data
sets must be analyzed. However, in many cases where artificial neural networks are
generated from simulation data, the question exists whether the actual simulation data
would be more accurate than the data generated from the ANN. In this case, it would
seem to make more sense to use the data of the original simulation rather than the ANN.

In addition, theoretical models can be more readily adapted to new conditions.

2.4.5 Convection cooking

Forced-air convection cooking is an important method for commercial cooking of
meat products. Huang and Mittal (1995) developed a computer model for forced
convection cooking, broiling, and boiling of meatballs. Heat and moisture transfer were
modeled using one-dimensional conduction and diffusion equations formulated in radial
coordinates.

An energy balance at the surface of the meatball was used for the heat transfer
boundary condition. This equation accounted for convective heat gain, latent heat of

evaporating water, and conduction at the meatball surface.

81‘ am
kT :5; surface =hT °(Tair “Tsurface)+ Dm 'p'l'glsurface I 2-21]

23

Boundary conditions for the moisture transport equation were based upon
moisture content equilibrium between the surface and the environment. The transport
equations were solved using the finite difference method. Constant transport properties
were used in the simulation. Surface heat transfer coefficients for each type of cooking
were determined using analysis of the center temperatures of aluminum spheres heated
under the conditions of interest. The thermal and mass diffusivity values were estimated
by minimizing the root-mean-square deviations between the observed and predicted
temperature and moisture histories. This was conducted using only one set of cooking
data and validated using four other data sets. The average root mean square errors for
temperature and mass prediction versus experimental results were 3-5.1°C and 0.04-0.19
g respectively. The strength of this model is that it was derived entirely from transport
equations. Although an empirical technique was used to fit thermal and mass diffusivity
values, a similar technique is often used to determine the published values that are

available for those constants.

Chen et al. (1999) developed a model for convection cooking of chicken patties.
The finite element method was used to solve heat and mass transfer equations based on
transport principles. Cooking was modeled using two-dimensional transport equations
formulated in radial coordinates. The use of two-dimensional formulations allowed for

complete description of the temperature and moisture distributions in the patties.
9.91: 1.9. r.k_T.§T_ +2 k_T.§_T_ [2.22]
at r 6r c 8r 62 62

24

9.92: 1.13. r.l‘_m_.@ +3 5&2“. [2.23]
at r 6r cm (it 62 dz

The two-dimensional formulation of the heat and mass transfer equations made it
possible to model cylindrical patties as axis-symmetric bodies. The boundary condition

for the heat transfer equation consisted of both a convective heat transfer term and a

latent heat term for surface evaporation.
kT- =hT-(Tair—T)+Dm-p-K-—— [2.24]

The boundary condition for the mass transfer equation was based upon convective

mass transfer at the surface.

= h... -(m... -m) I 2251

The equilibrium moisture term used in the mass transfer boundary condition was

calculated using the equation developed by Huang and Mittal (1995).

- 5222.47 -1 .0983
RH = ”W _— ' mequilibrium I 2-26 I
R g ' Tair

25

Based on previous research, heat capacity and thermal conductivity were modeled
as functions of temperature and moisture content respectively (Murphy et al., 1998;

Murphy and Marks, 1999). This decreased standard error of prediction for center

temperature from 5.5 to 37°C, as compared to using constant values for Cl, and k.

cp =3017.2+2.05-T+0.24-T2 +0.002r3 [ 2.27]

kT =0.194+0.436-m [ 2.28]

Singh et al. (1984) developed a heat and mass transfer model for oven roasting of

meat. Heat transfer within the meat was modeled using a two-dimensional conduction

equation.
2 2
flza. £4.22 [2.29]

A convective boundary condition with a term for evaporation of water was used at

the product surface.

6T
h ' (Tair — T): k ' hm '(Psurface " Pair )+ kT ' El surface I 2-30]

26

The finite difference method was used to solve the heat transfer equation. No attempt
was made to describe the transport of water within the product. A linear relationship was

used for the value of Psurfacc (Equation [2.31]).

I)surface =3 +b'T [ 2.31]

Although a term for evaporation was used to improve the heat transfer model, this model
is not suitable for predicting yield during cooking.

A different technique, was utilized by Mittal and Zhang (2000) to develop a
model for convective thermal processing of fiankfurters. This model was based upon an
artificial neural network. Input variables for the ANN were fat-protein ratio, initial
temperature, initial moisture content, frankfurter radius, ambient temperature, relative
humidity, and process time. The artificial neural network was trained using data from
validated mathematical models. The models were based upon one-dimensional heat and
mass transfer equations. Moisture diffusivity and equilibrium moisture content as
functions of temperature, fat protein ration, and relative humidity were used in the model

following the work of Mittal and Blaisdell (1982):

Dm = Exp(— 8.6787+0.08468.FP—0.3614-RH~FP—4341.5/Tabs +8.55-C) [ 2.32]

mequilibrium = —0.102- 1n(- Rg oFP - (T + 5.665) ln(RH)/1.132 .107) [ 2.33 ]

27

The ANN was trained using 13,500 data points generated from the mathematical
models. Fat-protein ratio was found not to be an important factor in predicting moisture
content or temperature. As discussed earlier, artificial neural networks pose some
advantages over purely numerical models, and may be effective for use in optimization
experiments. The primary advantage of artificial neural networks is that they can be
utilized to produce results much faster than models based upon the finite difference or
finite element methods. As a result, ANN ’s have promise for optimization studies, real-
time control, and other situations where fast calculations are essential. However, ANN ’s
are still limited in that they must be “trained” with data either from experimental or
model sources. The theoretical basis of numerical models means that they can be more

easily adapted to changing processing conditions without the need for “training”.

2. 4. 6 Other types of cooking
Models are available for several other types of cooking. Mallikarjunan et al.
(1996) developed a model for microwave cooking of shrimp. Heat transfer was modeled

using a two-dimensional transport equation in axial coordinates.

 

2 2
Earl—a. .1..£+_6_I+6_T + Q [234]
1' a!- arz 2 p.cp

where Q is the heat generated by the microwaves and is given

28

Q = Q0 .[exp[‘_(15£'_r)] + exp[:(_§_:i)fl [ 2.35 ]

P P

Moisture loss during cooking was calculated using the following equation:
am 9 ' V ' cp 5T
-—=hm-A-(P.at—Pw)+——- — [2.361

Heat transfer at the surface was modeled using a convective heat transfer equation
with a term for water evaporation. Equation [2.36] is somewhat unusual in that the
evaporation term is generally included in the heat transfer equation. The result is that
Equation [2.36] does not account for internal resistance to moisture transport.

Equations were solved using the finite difference method. Constant thermal
properties were used. The surface heat transfer coefficient was calculated using the

equation for natural convection over a horizontal cylinder:

0.25
AT] [ 2.37]

h =1.3196- —
T (Ad

Simulation results were validated by collecting transient temperature and mass

data during cooking in a household microwave oven. The model temperature predictions
were within 6°C of the experimental values. Mass losses were within 0.8% of the

predicted values.

29

Another type of cooking uses far-infrared radiation for heating. Shilton et al.
(2002) developed a model for far-infrared cooking of beef patties. The model accounted
for heat transfer and evaporative mass losses during cooking. Heat transfer was modeled
using a one-dimensional Fourier equation with a term for the latent heat associated with
water evaporation. Mass transport was modeled by a diffusion equation. The boundary

condition for heat transfer at the surface was given by:

(Yr 4 4
kT ' E‘— = 0" Theating element _ Tsurface I 238 I

A convective boundary condition was used for the mass transfer equation.

Dm 'E=hm '(Cair ‘Csurface) I 239]

The mass diffusion coefficient was calculated using an equation by Maroulis et al.

(1998).
1),, = a.exp(b/T)-exp(c/c) [ 2.40]

Thermal conductivity and density were modeled as a function of temperature

using equations by Choi and Okos (1985).

30

k=a+b-T—CoT2 [2.41]

p=a—b-T—c-T2 [2.42]

The authors noted that the heat transfer model was not accurate for patties
containing high levels of fat. To more accurately model heat transfer in high-fat patties,

an effective thermal conductivity was calculated.
2
k=(a+b-T—c-T )+(heffl+heff2T) [2.43]

This technique allowed more accurate modeling of heat transfer for meat
containing high levels of fat. However, it did not actually relate heat transfer to fat
content or describe the transfer of fat out of the product. Addition of a more detailed fat

transfer component would increase the utility of the model.

2.5 Thermal and physical properties required for modeling

Before it is possible to create cooking models based on heat and mass transfer
principles, accurate values for the thermal and physical properties of meat products must
be known. Since these values may vary widely with temperature and composition, it is
necessary to quantify these effects. Thermal and physical properties of a wide range of
food products are available from food engineering textbooks and handbooks, such as

those published by ASHRAE (Stroshine and Hamann, 1996; ASHRAE, 1998). Although

31

the properties of interest can often be obtained, care must be taken to scrutinize the
conditions for which they are accurate.

Numerous articles have been published on the thermal properties of meat
products. These properties were reviewed extensively by Sanz et al. (1987). Experimental
values for thermal conductivity, enthalpy, apparent heat capacity, and density of beef,
pork, mutton, poultry, and fish were reported. Most of the property values listed were in
the -30 to 30°C temperature range. Other articles related to thermal properties of meat
and meat products have been written by McProud and Lund (1983), Perez and Calvelo
(1984), Dincer (1996), and Tsai et al. (1998). Moisture diffusivity data for a wide range
of foods were compiled by Zogzas et al. (1996). Enthalpy data for meat products have
been published by Levy (1979), Skala et a1. (1989), and Tocci and Mascheroni (1998).

The rate of moisture transport in meat products is considerably lower than the rate
of heat transfer. Zanoni et al. (1997) reported that moisture diffusion affected only a 3
mm deep layer of the product surface. Chen et al. (1999) reported similar results. These
results give insight into the development of element meshes for future cooking models.
The region of highest activity for moisture transfer is confined to a layer close to the
surface of the patty, and thus a fine mesh should be used in that area.

Accurate values for thermal and physical properties are critical to development of
cooking models based on transport equations. Small deviations in pr0perty values often
result in large differences in model performance. Thus, care must be used when selecting
property data from published sources. The product composition can often have a
dramatic effect on physical properties. Thus, composition-dependent property equations,

such as those developed by Choi and Okos (1986), are often utilized.

32

2.6 Microbial models

Mathematical microbial models can provide important tools for predicting the
growth or reduction of microorganisms in foods. When properly utilized, models can
provide an initial estimate of microbial behavior without the investment in time and
materials required for microbial challenge studies. The use of models is also less
expensive. Although microbial challenge studies may still be required to verify model
results, the model may be used to more efficiently design such studies by selecting the
conditions to be tested. Models can be used to quickly determine the effects of process
changes on microbial food safety. These capabilities are invaluable for the planning of
hazard analysis critical control programs. Graphical depictions of changes in microbial
counts can serve as educational tools. This is especially valuable for showing non-
microbiologists the effects of processing on microbial safety. However, the limitations of
the model must be considered whenever a model is used to predict microbial activity in
food products (F818, 2002).

Mathematical modeling of microbial activity in food products has been
extensively reviewed in the literature (Whiting and Buchanan, 1994; Whiting, 1995;
Roberts, 1997). Whiting (1995) classified microbial models based upon a three-level
scheme-as primary, secondary, or tertiary. Primary models describe the changes in a
microbial population as a function of time. Secondary models describe the primary model
parameters as functions of environmental parameters. Tertiary models combine primary

and secondary models with a user interface to produce a complete simulation tool.

33

2. 6.1 Primary models

Primary models describe the number of bacteria in a population as a function of
time. Mathematical equations are used to describe the growth or inactivation curve using
a set of parameter values. Ideally, these parameters relate to descriptive terms such as lag
time or generation time. Examples of primary models are linear models, exponential
models, and models based on the Gompertz function (Whiting, 1995). A log-linear

inactivation model is given by Equation [2.44].
logN=logN0+—Il)--(t—t0) [2.44]
Equation [2.45] is an inactivation model based on first-order kinetics.

=e ' [2.45]

A primary model for microbial inactivation based on a Gompertz equation was

parameterized by Van Impe et al. (1995).

y 2: a - exp[— upturn—2"“: - (k — t)+ 1)] I 2-46]

34

Peleg and Cole (1998) suggested a primary inactivation model based on a Weibull

distribution (Equation [2.47]).

log10 3(1) = —b(T) . 1"”) [ 2.47 ]

Each type of primary model has its own merits and inherent weaknesses. The
log-linear model is the most commonly used inactivation model in both industry and
academia. Large quantities of data have been amassed using the model parameters for
numerous microorganisms and processing conditions. This makes implementation of the
log-linear model the easiest of any of the primary models. Additionally, log-linear
models have become ingrained as the “standar ” for microbial inactivation and are thus
widely accepted in industry. However, microbial inactivation does not always follow
log-linear kinetics (Peleg, 1997). Other types of inactivation models may be more
suitable for modeling inactivation in these cases. Unfortunately, these equations are often
more cumbersome to use, and suitable experimental data for model constants are not as
widely available as for log-linear models. To develop modeling software that is both
flexible and as widely acceptable as possible in nature, log-linear inactivation kinetics
should be utilized. However, to develop the most accurate model for specific cases, other

types of primary models may be superior.

2. 6.2 Secondary models
Secondary models describe the changes in primary model parameters as a

function of environmental conditions. These models can show the effects of variables

35

such as temperature, pH, water activity, and substrate composition. Examples of
secondary models are Arrhenius relationship models, response surface models, and
square root models (Whiting, 1995).

An Arrhenius relationship for the inactivation constant, k, was given by

Geankoplis (1993).

-133
k=a-e R-T [2.48]

Equation [2.49] is a square-root relationship for k (Whiting, 1995).

JE=a(T—T0) [2.49]

Mattick et al. (2001) parameterized the following log-logistic secondary models for

Salmonella. The models are designed to be used with the primary model in Equation

[2.47].
b(T) = 6.841/{1 + exp[(76. 14 — T)/4.204]} [ 2.50 ]
n(T) = 0.670/{1+ exp[(T - 80.14)/3.785]} . [ 2.51 ]

36

Juneja and Eblen (1999) developed a response surface model for inactivation of
Listeria monocytogenes (Equation [252]). The model described decimal reduction time

as a function of temperature, pH, salt content, and sodium pyrophosphate content.

ln(D) = c1+ c2 - (T) + c3 - (pH) + c4 - (salt) + c5 ~(phos) + c6 - (T) - (pH)
+ c7 - (T) - (salt) + cs - (T) - (phos) + c9 - (pH) - (salt) + 010 - (pH) - (phos)

+c11-(salt)-(phos)+ en -(T)2 +c13 -(pH)2 +c14 -(sa1t)2 +c15 -(phos)2

[ 2.52]

2. 6.3 Tertiary models

Tertiary models combine primary and secondary models with a user-interface to
to produce a complete simulation tool. Two widely used tertiary models in the United
States are the American Meat Institute Process Lethality Determination Spreadsheet
(AMI, 2003) and the United States Department of Agriculture's Pathogen Modeling
Program (USDA, 2004).

The AMI model is based upon a Microsoft Excel spreadsheet (AMI, 2003). The
spreadsheet uses a log-linear thermal death time model to calculate process lethality
based on time/temperature data inputted by the user. The user must also input 2 and Tm]-
values for the thermal death time models. A table of z and Trey values for common meat
microorganisms is supplied to aid the user in choosing input values. Unfortunately, 2
values may be influenced by numerous conditions that are not accounted for by the chart

of suggested values. Thus many users will likely choose z and Tref values that are

37

inappropriate for the cooking process of interest and therefore generate lethality data that
are highly suspect.

The Pathogen Modeling Program (version 7.0) is a menu-based program that is
based upon a suite of models (USDA, 2004). The program provides tools for assessing
microbial inactivation of Clostridium botulinum, Escherichia coli 0157:H7, and Listeria
monocytogenes, but not Salmonella. The models for microbial inactivation and survival

are based upon a logistic inactivation model (Equation [2.5 3]).

+ 5131+”) (17 all + e—bzflj

N (1
log-N?)- = log 12+ 3131041)) + (1+ 61320-111) [ 2.53 ]

 

2.7 Combined models

Several authors have attempted to combine cooking models with microbial
inactivation models. Zanoni et al. (1997) linked inactivation models for Enterococcus
faecium to a cooking model for Bologna sausage. Two different inactivation models

were utilized. The first was a simple first order inactivation model.

—=—k-N [2.54]

The second was a model developed by Whiting and Buchanan (1994).

38

a-(1+exp(-b1-t1)) (1-a)°(1+°"p(-b2 OW] I 2.55]

The model was validated by inoculated challenge studies. At log reductions below 6,
both the first-order inactivation model and the Whiting and Buchanan model predicted
reductions within 1- log of the experimental values. At reductions above 6-log, the first
order model greatly overestimated inactivation. However, at log reductions above 6, the
Whiting and Buchanan model was within l-log of the experimental data. Unlike the first-
order model, the Whiting and Buchanan model incorporates a tailing effect at high levels
of inactivation, which provided a better fit to the experimental data. This clearly
illustrates how certain inactivation models may be more accurate than others under some
conditions.

The contact-cooking model developed by Pan et al. (2000) included a first-order

inactivation model for E. coli OlS7:H7.

dN - 2.303

a“ = Dre, .lorrref-Tonz ' [ 2'56]

 

Temperature data from each time step were utilized to obtain the number of
microbes at each nodal location. The total surviving population at each time step was
then determined. However, the inactivation model was not validated using experimental

data.

39

Mallikarjunan et al. (1996) included an inactivation model for Listeria
monocytogenes with a model for microwave cooking of cocktail shrimp. The

inactivation model assumed first-order reaction kinetics.

 

.N [2.57]

The inactivation model was validated by injecting a liquid inoculum into the
geometric center of each shrimp. After cooking, the shrimp were tested for surviving
bacteria. This methodology made it impossible to track the actual numbers of surviving
Listeria as a function of time. Although useful for predicting the worst-case scenario, this
technique probably overestimates the cooking required for most products where

microorganisms are present either at the surface or dispersed throughout the product.

2.8 Limitations of models

Although microbial models can be valuable tools, there are limitations inherent in
any model. A model is generally only accurate for the range of conditions under which it
was developed. Extrapolating outside of the ranges used for the model development may
give misleading results. Models are generally microbe specific, and a model for one
microorganism cannot be expected to produce accurate results for another

microorganism, or even for a different product.

40

2.9 Summary

The models contained in the previous sections describe cooking processes with
varying degrees of complexity. Although each of the models discussed has its own
merits, none of the published models provide a “complete” description of the cooking
process. Ideally, a cooking model should be based entirely upon engineering first-
principles, be flexible for a wide range of products and product conditions, and describe
the interrelationships between all of the components of the cooking process including
heat transfer, moisture transfer, and the transfer of fat. The goal of the following sections

was to develop such a model.

41

3 MATERIALS AND EXPERIMENTAL METHODS

3.1 Overview

The study was broken up into three major groups of experiments. In the first
group of experiments, ground beef, ground pork, and ground turkey patties were cooked
in a laboratory convection oven. The results of these experiments were used to illustrate
and quantify the differences in cooking characteristics between species and fat content.
In the second group of experiments, laboratory studies were conducted to determine the
fat holding capacity of ground beef. The data generated in these experiments were later
used to develop the fat transfer portion of the cooking model (Chapter 4). The third set of
experiments involved cooking ground beef patties in an industrial moist-air impingement
oven. These data were then analyzed to determine the effects of processing conditions on
yield and temperature profiles. In addition, results from this group of experiments were

utilized to validate the computer-cooking model that is described in Chapter 4.

3.2 Laboratory oven cooking tests
3. 2. 1 Experimental procedure

A series of laboratory experiments was conducted to investigate the effect of meat
species and initial fat content on cooking characteristics during convection cooking of
ground beef, ground pork, and ground turkey patties. All cooking tests were conducted

using a custom built, laboratory convection oven.

42

The laboratory convection oven consisted of three chambers (Figure 3.1). These
included a large conditioning chamber (Figure 3.2), a steam generator, and the cooking

chamber itself.

 

Steam
generator

 

 

Conditioning chamber with air
heaters

 

 

 

 

 

 

 

 

Sample chamber with
suspended sample

Figure 3.1 - General arrangement of the laboratory convection oven showing
directions of steam and airflow.

 

 

Figure 3.2 - Interior of laboratory oven conditioning chamber.

43

The conditioning chamber was the largest portion of the oven and was used to
heat and condition the cooking air to the desired temperature and moisture content. The
dimensions of the conditioning chamber were approximately 83 cm in length by 56 cm in
width by 51 cm in height. The conditioning chamber contained four 350-watt strip
heaters (McMaster-Carr: Cleveland, OH). Steam was injected into the conditioning
chamber from the steam generation unit. The steam generation unit contained water
heated by a 750-watt immersion heater (Tempco: Wood Dale, IL). An electronically
activated solenoid valve was utilized to inject steam into the conditioning chamber. The
sample chamber was a small container located at the edge of the conditioning chamber.
The dimensions of the cooking chamber were approximately 10 cm by 10 cm by 10 cm.
Cooking air was drawn from the conditioning chamber by a 6-watt centrifugal fan
(Dayton model 4C440: Niles, IL) and passed through the sample chamber by means of

tubular ducts (Figure 3.3).

 

Figure 3.3 - Interior of laboratory oven showing fan, ducts, and sample chamber
(located at left).

The oven was connected to a computer interface that continuously monitored and
controlled oven temperature and moisture content. During cooking, oven dry bulb and
wet bulb temperature were controlled within i 0.2°C. Dry bulb and wet bulb temperature
within the conditioning chamber was monitored using a high temperature dry bulb/wet
bulb humidity probe (V iasala model DMP 246: Viasala, Woburn, MA). The airflow in
the cooking chamber was 1.3 m/s.

Ground turkey, ground beef, and ground pork were purchased from a local
grocery store. Additional ground pork was provided from the Michigan State University
Meat Laboratory. Two fat contents of each meat species were utilized: 1.4 and 8.6% for
ground turkey, 7.2 and 17.5% for ground beef, and 15.7 and 41.9% for ground pork. Fat
contents were determined in triplicate by solvent extraction (AOAC method 991.36:
AOAC, 2000). The moisture content of each species was determined in triplicate by
oven drying (AOAC method 950.46: AOAC, 2000). Moisture contents were 74.8 and
73.0% wet basis for the 1.4 and 8.6% fat ground turkey, 71.5 and 63.3% wet basis for the
7.2 and 17.5% fat ground beef, and 64.1 and 43.6% wet basis for the 15.7 and 41.9% fat
ground pork.

Each meat type was formed into uniform patties by pressing into plastic petri
dishes (52 mm diameter; 13 mm height) . The patties were then frozen and removed
from the dishes prior to use in the cooking experiments. Before cooking, patties were
tempered to 4°C by placing in a refrigerator for 2-3 h.

Prior to heating, each patty was weighed to the nearest 0.01 g. The radius and

thickness of each patty were then measured, to the nearest 0.1 m, using a digital caliper.

45

A 24-gauge type-K thermocouple (Omega: Stamford, CT) was then inserted into the

geometric center of each patty using a placement jig (Figure 3.4).

(a) (b)

  

\ AA \ if Thermocouples
Frame Screen

 

Figure 3.4 — (a) Picture and (b) schematic of jig used to place thermocouples into
meat patties during cooking experiments.

Patties were then placed into the cooking chamber of the oven, where they were
supported on a wire mesh screen. This allowed for airflow on all sides of the product.
The patties were cooked one at a time in the convection oven to center temperatures of
45, 55, 65, 75, and 85°C at an oven temperature of 177°C and a wet bulb temperature of
82°C. Five patties were cooked at each condition. Alter cooking, the patties were
removed from the oven and weighed to the nearest 0.01 g. The new thickness and
diameter of each patty were measured using the caliper, and the patties were frozen at -
5°C pending further analysis. After 24 h, the patties were removed from the freezer, and
the moisture content was measured by oven drying (AOAC Method 950.46: AOAC,

2000)

46

The amount of water lost during cooking was calculated for each patty using the

initial and final moisture contents (Equation [3.1]).

ANImoisture = Minitial 'Xw,initial “Mfinal 'Xw,final I 3-1 I

where Minitial and MW. are the initial and final mass of the patty and XW is the wet basis
moisture content of the meat. These data were used to calculate the component of yield
loss attributed to moisture. For many of the samples, the mass lost due to moisture loss
was considerably less than the total mass loss. The mass loss not accounted for by
moisture was attributed to fat loss during cooking. These losses were calculated using

Equation [3.2].

AM fat = AM total _ AM moisture I 3-2 I

3.2.2 Statistical analysis

Analysis of variance (ANOVA) was used to evaluate the effects (or=0.05) of
initial fat content, cooking time, and time-fat interaction on center temperature, cooking
yield, fat loss, and volume change within each meat species. AN OVA was conducted
using the Microsoft Excel Data Analysis Package (Microsoft Excel Version

2000:Redmond, WA).

47

3.3 Measurement of fat holding capacity
3.3.1 Experimental procedure

Laboratory experiments were conducted in an isothermal water bath to determine
the fat holding capacity of ground beef. The fat holding capacity of the meat is the
maximum amount of fat bound in the meat after a given heat treatment. The fat holding
capacity of the meat was determined as a function of temperature and time using a so-
called “net test” (Barbut, 1996).

Two lots of ground beef were acquired from the Michigan State University Meat
Laboratory. The fat contents of the two lots were 5.6 and 15.0% fat by mass, determined
in triplicate using solvent extraction (AOAC method 991.36: AOAC, 2000).

Brass tubes were used to contain the meat samples during cooking and were
chosen due to their high thermal conductivity. The tubes were cut from 0.36 mm thick
brass tube stock (K&S Engineering: Chicago, IL). The tubes had an inside diameter of
7.9 mm and a length of approximately 122 mm.

Prior to each heating test, a silicone stopper was placed in one end of each heating
tube. Teflon tape (13mm width) was then wrapped around the silicone stopper to ensure
that the closed end of the tube was watertight. The weight of the combined tube, stopper,
and tape was then measured to the nearest 0.1 mg using an electronic balance.
Approximately 3.5g of meat was then loaded into each tube by hand. During loading, the
meat was packed into the closed end of the brass tube by firmly tapping the tube on the
bench top. After loading, the total weight of the tube and meat was measured to the

nearest 0.1 mg.

48

A 30 um nylon mesh (Spectrum Labs: Rancho Dominguez, CA) was then
attached over the open end of each tube using laboratory tape. A type-T thermocouple
probe (Cole Parmer: Vernon Hills, IL) was inserted through the silicone stopper and into
the approximate center of the meat sample within the tube. Tubes were then placed into
an isothermal water bath (N eslab: Newington, NH) for one of four holding times. The
open end of each tube was held above the surface of the water by a test tube rack. The
entire meat sample was located in the submerged portion of the brass tube. The time
measurement was started when the temperature of the center of the sample was within
1.0°C of the water bath temperature. Samples were heated at water temperatures of 30,
35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, and 90°C and heating times of 2, 5, 10, and 15
minutes. Five tubes were heated simultaneously for each time-temperature combination.

After heating, the tubes were removed from the water bath, and the surface
moisture was removed with a paper towel. The tubes were then placed, mesh-side down
in centrifiige tubes (length: 102 mm, diameter: 14.7 mm: Fisher: Pittsburgh, PA). A brass
spacer fabricated from the same stock used to produce the heating tubes was placed in the
bottom of each centrifirge tube (Figure 3.5). The length of the brace spacers was
approximately 27 mm. Prior to centrifuging, the silicone stoppers and Teflon tape were

removed from each tube. The tubes were centrifuged at 25°C and 1000g for 15 minutes.

49

\I

‘— centrifuge tube

‘— brass tube containing meat

 

 

 

‘__. tape
*— 30 um nylon mesh

‘— brass spacer

 

Figure 3.5 — Schematic of tube setup used for centrifuging.

After centrifuging, two distinct layers were observed in each tube. A solid layer
of extracted fat was located above a liquid layer (Figure 3.6). For the purposes of this
study, only the fat located in the solid layer was of interest, as this is the component most
often overlooked by cooking models. To separate the solid fat from the liquid layer, the
brass tube containing the meat sample was first removed from each tube. A syringe with
an 18-gauge needle (BD: Franklin Lakes, NJ) was then used to withdraw the liquid layer
from beneath the solid fat layer. The centrifuge tube containing the spacer and the fat
layer was then heated in a 102°C convection oven for 24 hours to drive off any remaining
water. After heating, the tube was removed from the oven and allowed to cool in a
dessicator for one hour. The tube was then weighed to the nearest 0.1 mg, and the weight
of fat in the tube determined by subtracting the weight of the empty tube and spacer from

the weight of the tube containing the spacer and extracted fat.

50

solid layer

 

 

 

liquid layer

Figure 3.6 — Schematic of the centrifuge tube after centrifuging showing the two
distinct layers that formed within the tube. In this schematic the heating tube has
been removed. The liquid layer contains water and soluble proteins. The solid
layer is composed of fat.

The fat content remaining in each sample after centrifuging was determined by
subtracting the mass of the extracted fat from the initial mass of fat in the sample prior to
heating. It was assumed that no protein was present in the extracted fat. The fat content
was expressed in terms of dry basis fat content by dividing the mass of fat remaining in
the sample by the mass of non-fat dry matter in the sample. This fat content was

considered the fat holding capacity of the meat under the specific conditions of

temperature and time.

3.3.2 Statistical analysis
Quadratic response-surface models were used to describe the fat holding capacity

of each batch of meat as functions of time and temperature (Equation 3.3).

F=BO+Bl-T+[32-T2+B3-t+B4-t2+BS-t-T+a [3.3]

51

The statistical significance (or=0.05) of each regression parameter was determined using
t-statistics. The models were then adjusted accordingly to eliminate non-significant
terms. A regression model was also fit to the combined FHC results of the low and high

fat ground beef (Equation [3.4]).

F=B0 +B1'T+B2°T2+B3't+B4't2+B5'F+B6't'T+B7'T'FO +B8'I'F0 [ 34]

The statistical significance (or=0.05) of each regression parameter was determined using

t-statistics, and the model was again adjusted to eliminate non-significant terms.

3.4 Industrial oven cooking tests
3. 4. 1 Experimental procedure

A series of cooking tests were conducted using a commercial continuous-feed
moist-air impingement oven (Stein model J SO-IV2FMC FoodTech, Sandusky, OH)
located at the FMC FoodTech Technical Center (Sandusky, OH). The JSO-IV oven
consisted of two 3.65 m long cooking sections with a 0.8 m intermediate section. The
belt width of the oven was 1.02 m. The oven was equipped with 1.07 m long in-feed and

discharge sections, which were not utilized during this study.

52

 

Figure 3.7 - Stein model JSO-IV moist-air impingement oven.

Commercially produced, pre-formed, frozen hamburger patties (Gordon Foods,
Grand Rapids, M1) were used for the cooking tests. The patties had a thickness of 1 cm
and average diameters of ~12 cm. The average uncooked patty weight was 105 g. The
labeled fat content was 10%. This fat content was confirmed using solvent extraction
(AOAC method 991.36: AOAC, 2000).

Tests were conducted using a 1/3 partial factorial design (Table 3.1) comprised of
three cooking temperatures (121, 177, and 232°C), three oven steam contents (50, 70, and
90% steam by volume), and three oven fan speeds (50, 75, and 100% of full). For each
cooking condition, three oven belt speeds were chosen to produce varying degrees of
doneness (undercooked, fully cooked, and over cooked). Belt speeds were chosen based
upon the experience of a trained operator and adjusted as required to achieve the desired
patty temperatures. Two hamburger patties were cooked at each belt speed, for a total of

54 patties.

53

Table 3.1. Treatment conditions utilized for model validation experiments.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Experiment Temp. (°C) Steam by Volume Air Flow Time (Min.)
m/s

1 121 50 % 11.4 5
121 50 % 16.8 11

8 121 50 % 21.8 8
11 121 70 % 11.4 8
13 121 70 % 16.8 5
18 121 70 % 21.8 11
21 121 88 % 11.4 11
23 121 78 % 16.8 8
25 121 78 % 21.8 5
30 177 50 % 11.4 8
32 177 50 % 16.8 6
34 177 50 % 21.8 3
37 177 70 % 11.4 3
47 177 83 % 11.4 6
49 177 84 % 16.8 3
50 177 86 % 16.8 6
54 177 86 % 21.8 8
56 232 50 % 11.4 5
58 232 50 % 16.8 2
63 232 50 % 21.8 7
66 232 70 % 11.4 7
68 232 70 % 16.8 5
70 232 70 % 21.8 2
73 232 82 % 11.4 2
75 232 82 % 11.4 7
78 232 82 % 16.8 7
80 232 82 % 21.8 5

 

Prior to cooking, the frozen patties were tempered in a -3°C cooler (> 3 h) to

provide a uniform initial temperature distribution. Before each cooking run, the patties
were weighed to the nearest 0.1 g. The radius and diameter of each patty were measured
to the nearest 0.1 m using a digital caliper. A 24-gauge type-K thermocouple (Omega:

Stamford, CT) was then placed and held at the center of each patty using a placement jig

54

 

(Figure 3.4). Additional thermocouples were positioned at the top and bottom surfaces of
the patties and held in place by the placement jig. The thermocouples were attached to an
oven data logger (Datapaq model 9000: Datapaq, Wilmington, MA). that was able to
pass through the oven with the patties during cooking. During the cooking process, patty
and air temperatures were recorded every 1 second.

Patties were placed on the center of the oven belt and allowed to cook for the
designated time. The patties were then removed from the oven and immediately weighed
to the nearest 0.1 g to determine post-cooking weight. The thickness and diameter of
each patty was then re-measured to the nearest 0.1 mm. Afler weighing and measuring,
the patties were frozen (-20°C) using an impingement freezer and placed into sealed
plastic bags pending firrther analysis. Dwell time in the impingement freezer was 2
minutes.

After each cooking run, the data were uploaded to a personal computer, and
temperature versus time curves were constructed. The bulk moisture content of each
patty was determined using oven drying (AOAC Method 950.46: AOAC, 2000). The
cooking yield was calculated for each patty by dividing the mass of the cooked patty by
the mass of the patty prior to cooking. Using the moisture content of the raw and cooked

meat, the mass of water that exited the patty during cooking was calculated (Equation

[31]).

55

3.4.2 Statistical analysis
Multiple linear regression was conducted to quantify the effects of the cooking
variables on the center temperature achieved in the patties during cooking. Equation

[3.5] was used as the basis for the regression.

Tcenter = '30 + Bl ' Toven + I32 ' Xsteam + B3 ' time + B4 ° Airflow + 8 I 3-5 I

A second regression was performed in an attempt to quantify the effects of cooking

parameters on cooking yield. Regression was performed using Equation [3.6] as a basis.

Yield = [30 + [31 woven + [32 -xS,,,m + [33 . time + [34 - Airflow + e [ 3.6]

3.5 Cooking model validation
3.5.1 Cooking model

A computer model was developed to simulate moist-air impingement cooking of
ground-and-formed meat and poultry patties. Development of this model is detailed in
Chapter 4. The model was based upon heat and mass transfer principles and did not
require experimental data for its development, with the exception of the relationships
utilized for the fat holding capacity. However, it was necessary to validate the model
using experimental data to validate its performance. This was conducted using data
generated in the industrial cooking tests (Section 3.4) and additional published data from

a pilot-scale impingement oven.

56

3.5.2 Experimental data

The finished computer model was validated using the data generated in the
industrial impingement cooking tests described in Section 3.4. The computer model
requires user input of oven temperature, steam fraction, air velocity, and cooking time.
Additional inputs required are product initial temperature, product moisture content,
product fat content, and meat species. The computer model was executed using the 27
sets of process conditions listed in Table 3.1. The input conditions for the ground beef
were an initial temperature of —2 to 0°C, initial moisture content of 66% wet basis, and
initial fat content of 10% wet basis. The element mesh was set up for a patty diameter of
12 cm and thickness of 1 cm (see Chapter 4 for element mesh).

Output data were generated for the transient patty center and surface
temperatures, final bulk moisture content, and final cooking yield. The output were
compared to the temperature profiles generated during the cooking experiments as well as
to the final yield and moisture values measured after the cooking tests. A standard error
of prediction (SEP) between the transient model and experimental temperature profiles

was calculated for each run (Equation [3.7]).

 

 

SEP = J: (Tpredictecrll : Fll‘measured)2 [ 3.7 ]

The differences between the predicted and measured yield, moisture content, and

fat loss were determined for each model run. The aggregate standard error of prediction

57

was also calculated for the endpoint yield, moisture content, and fat loss for all of the

model runs.

3. 5.3 Comparisons with literature data

In addition to the model validation using experimental data from the industrial
cooking tests, the cooking model was compared to published data for ground chicken
breast patties (Murphy et al, 2001a and b). The purpose of this comparison was to
demonstrate the versatility of the model. The published data were collected using a pilot-
scale moist-air impingement oven (Stein model 102: Stein, Sandusky, OH). Murphy et
al. (2001a) developed a regression model for cooking time as a firnction of center

temperature, oven air velocity, and oven wet bulb temperature (Equation [3.8]).

ln(t) = 8.8678 + 0.0278 - T — 2.0410 - ln(wa) — 0.2306 - Vair + 0.0481 - ln(wa °Vair)

[3.8]

The regression for cooking time had an R2 of 0.95. Murphy et al. (2001b) developed a
second regression model for yield of ground chicken patties as a function of patty center
temperature, oven air velocity, and oven air wet bulb temperature (Equation [3.9]). The

regression model for yield had an R2 of 0.89.

Yield = 75.4031 + 0.9309 - T — 1.2443 . Va], + 0.1047 . vai,2 - 0.0121 -r2
— 0.0102 - M .v,,, + 0.0027 - wa .r

[3.9]

58

The geometry of the Stein 102 oven used by Murphy et al. (2001a and b) was
different from the geometry of the Stein JSO-IV. The Stein 102 oven used an array of
round nozzles rather than the slot nozzles utilized in the J SO-IV oven (Chapter 4).
Therefore, it was necessary to modify the cooking model with an equation for an array of
round nozzles as discussed in Chapter 4.

Model simulations were conducted using the process conditions of Murphy et al.
(2001a and b). These conditions were a dry bulb cooking temperature of 149°C, wet bulb
air temperatures of 40, 70, 85, and 95°C, and air velocities of 1.53, 2.13 and 2.73 m/s.
The product settings for the model were chicken at an initial moisture content of 80% wet
basis, initial fat content of 0.2% dry basis, and initial temperature of 4°C. The element
mesh was adjusted for a patty diameter of 127 mm and a thickness of 12.7 mm. Cooking
model tests were run using each combination of dry bulb temperature, wet bulb
temperature, and airflow giving a total of 12 cooking conditions. The cooking time of
each simulation was adjusted to give a final patty temperature of 80°C.

For each model run, the temperature profile from the model was compared to the
temperature profile generated by the regression model at 1 second intervals for the
temperature range between 55 and 80°C (the calibration range of the regression model).
Transient standard error of prediction was calculated for each run using Equation [3.7].

The same procedure was used to calculate the SEP for transient yield value for each run.

3.6 Salmonella inactivation model validation
The location of the impingement oven utilized for our experiments did not allow

for inoculated challenge studies. Therefore, simulated Salmonella Senftenberg

S9

inactivation results were compared to literature values for ground meat patties cooked in
a pilot-scale moist-air impingement oven (Murphy et al., 2002). Simulated results for
Listeria innocua were also compared to literature values (Murphy et al., 2002) to
demonstrate the flexibility of the model.

The model was run using the equations for the geometry of the Stein 102 oven
(array of round nozzles). The cooking air conditions utilized for the model runs were an
oven air temperature of 288°C, a steam content of 25% by volume, and an air velocity of
4 m/s. Although the patties used by Murphy et al. (2002) were composed of a mixture of
ground beef and ground turkey, the ground beef setting was utilized for the model runs.
This setting was chosen based upon the fairly high (20% wet basis) fat content of the
meat. The product properties were an initial moisture content of 58% wet basis, initial fat
content of 20% wet basis, and initial temperature of -2°C.

The model was run once for inactivation of S. Senftenberg and once for
inactivation of L. innocua. The D and z-values for the model runs were set to 312
seconds and 623°C for S. Senftenberg and 1,508 seconds and 490°C for L. innocua, at a

reference temperature of 60°C (Murphy et al., 2003). The reference temperature for both

sets of D and 2 values was 60°C. The experiments of Murphy et al. (2002) utilized
ground meat with an initial inoculum of 107 CFU/ g. Therefore, the model was set to run
with a limit of 7-log total reduction. Transient inactivation curves generated by the
model for each organism were plotted on the same graphs as data from the published

study and SEP’s calculated between the model and experimental data.

60

4 MODEL DEVELOPMENT

4.1 Introduction

A computer model was developed for moist-air impingement cooking of ground
and formed meat and poultry products. The computer model is composed of a coupled
heat and mass transfer mode] combined with a model for Salmonella inactivation. The
finite element method (FEM) was used to numerically solve the differential equations
associated with the heat and mass transfer model. The model was programmed using
Microsoft Visual Basic (Version 6.0 professional edition: Microsoft, Redmond, WA).

A typical impingement oven has impinging jets located above and below the
product (Figure 4.1). The cooking air enters the ductwork and is forced through slots
normal to the product. Not shown in Figure 4.1 is the perforated belt that the product
travels on through the oven. A typical impingement oven would contain many pairs of
impingement jets arranged in series along the length of the belt.

The computer model was developed for an oven with the following controllable
variables: airflow, air temperature, steam content by volume, and cooking time. The
distance between the impinging jets and the product surface must also be specified. For
the purpose of modeling, it was assumed that the upper and lower impinging jets were

located an identical distance from the product surface.

61

Input airflow

   

Impinging airflow fed .
through slots __

Product suspended on -
perforated belt (belt not 4

shown) '1‘ \
— I.
Input airflow _ ,t

Figure 4.1 - Illustration of the airflow within an impingement oven in relation to the
product.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The coupled heat and mass transfer model was based upon three transport
components. The three components were heat transfer, moisture transfer, and fat transfer.
Incorporation of the fat transfer component represents one of the unique aspects of the
cooking model as compared to most previous models that have been published. The
three transport solutions were coupled through the boundary conditions and
interdependent thermo-physical properties.

The boundary conditions associated with moist-air impingement ovens are
somewhat different than those in conventional convection ovens. One of the unique
aspects of moist-air impingement ovens is the very high oven air moisture content that is
often used for meat product cooking. The high fraction of water in the air results in a
condensing boundary condition during the period in which the surface of the patty is

below the dew point of the cooking air. This condensing condition increases the rate of

62

heat transfer and reduces the rate of moisture loss from the product. Inclusion of the
condensing boundary condition is another unique aspect of this cooking model.

Two different sets of boundary conditions were used to model the cooking
process — one for the impinging flows on the horizontal surface of the patties and one for
the vertical edges of the patties. The first set of boundary conditions was used to describe
the impinging flow that is the predominant mechanism for heat and mass transfer in
moist-air impingement ovens. The boundary conditions used to model impinging flow
were based on an expansion of the technique of Millsap and Marks (2002) and depended
on transport correlations published by Martin (1977). The second set of boundary
conditions was used to describe the convection that occurs along the vertical surfaces
(edges) of the patty. These conditions were based upon transport correlations for
turbulent flow past a flat plate (Bejan, 1995).

The following set of basic assumptions was utilized when developing the heat

and mass transfer model.

1. Heat transfer occurs by conduction within the patty and by a combination of
convection, condensation, and evaporation at the patty surface.

2. Moisture and fat transfer within the patty occur primarily by diffusion and
capillary flow, respectively.

3. When the temperature of the patty is below the dew point, moisture condenses at
the patty surface, and the moisture content of the patty surface does not change.
When the surface temperature of the patty is above the dew point, water

evaporates at the surface.

63

4. The phase change from solid to liquid water occurs at a single temperature. A
phase change from solid to liquid fat occurs at a single fat melting point.

5. Mass transport of water within the patty does not occur at temperatures below the
freezing temperature. Mass transport of fat does not occur below the melting
point of the fat.

6. Fat transport within the patty is driven by the gradient between the local fat
content and the value at the surface. Surface fat content is a function of time,
temperature, and initial fat content.

7. The size and shape of the patty does not change during cooking.

8. Boundary conditions were assumed to be the same for the top and bottom of each

patty.

4.2 Heat and mass transfer model
4. 2. 1 Heat transfer solution

The heat transfer simulation was formulated using the Fourier equation written in
terms of enthalpy. The enthalpy formulation was utilized to produce a model free from
discontinuities caused by the phase changes of fat and water. Similar techniques for
modeling heat transfer have been utilized by Volner and Cross (1981), Pan (1998), and
Pan et al. (2000).

Most ground beef and poultry patties are shaped like short cylinders (Figure 4.2).
This geometry allows heat and mass transfer in formed patties to be modeled using two-
dimensional axisymmetric solutions. Heat transfer by conduction within the patties was

modeled using Equation [4. 1].

 

 

17
'1

 

 

 

 

 

Figure 4.2 - Geometry of a patty illustrating radial coordinate system.

 

 

$1.43,. kT 2! ,3 RT 95 [4.1]
at rar cp-pdr dch-pdz

Heat capacity (cp), thermal conductivity (k7), and density (p) were modeled using
transient values based on product composition and temperature (Section 9.3). Patties
were assumed to be at a uniform internal temperature prior to cooking. The initial
product temperature was converted to enthalpy using an equation based on the technique

of Voller and Cross (1981): (Equation [4.2]).

 

rCPsfmzen O T T < Tfreezing

H =< cp.frozen ' Tfreezing T = Tfreezin g
cp,frozen ° Tfreezing + )‘w + cp ' (T ' Tfreezing) Tfreezing < T <= Tmelting,fat
[cp,frozen ' Tfreezing + Aw + cp ' (T — Tfreezing )+ A'f T > Tmelting,fat

[4.2]

65

Equation [4.2] converts temperature values into enthalpy based upon an absolute
zero reference temperature. Provision was made for two phase changes within the
product during cooking. The first was the phase change between ice and liquid water,
and the second was the phase change between solid and liquid fat. An initial temperature
equal to the freezing temperature of the meat was assumed to imply a completely frozen
sample.

A similar technique was used to convert enthalpy values back to temperature for
display purposes (Equation [4.3]). The variables op; and cm in Equation [4.3] are the heat
capacities of the frozen and thawed meat, Tf and Tan are the freezing temperature of water
and the melting temperature of fat, and Agw and M; are the latent heats of fusion for water
and fat respectively. Based on the results of the fat holding capacity experiments

described in Section 3.3, a value of 45°C was used for the fat melting temperature.

H S Cp,f °Tf
Cp,f -Tf S H

rH/c
Tr

pf

Scpf 'Tf +kw
(H—kw —273‘Cp,f)/Cp,t Cp’f 'Tf +1“, < H
Scpf °Tf +7tf +Tf 'cp,t
Tm cp,f-Tf+}tf+Tf-cp,t <H
Scpx ‘Tf +}\.f +Tf -cp,t +1tf

 

1(H —)“w -273.CP-f —)"W )/Cp,t H > Cp,f -Tf +Af +Tm °cp,t +Af

[4.3]

66

The boundary conditions for heat transfer were composed of a convection term

and a moisture transport term (Equation [4.4]).

6T
kT ' 5n- = hT ' (Tair - Tsurface ) + h m,water ° 9“vaporization ° (C air " Csurface)

[4.4]

During the period in which the surface temperature of the meat is below the dew
point temperature of the cooking air, heat transfer due to condensation occurs, causing
additional heat transfer into the patty. When the surface temperature of the meat exceeds
the dew point temperature, evaporation occurs at the patty surface, limiting sensible
temperature increases in the product. This has the effect of limiting the maximum
surface temperature reached at the product surface as a function of oven conditions and
surface moisture content. Until the surface of the patty reaches low moisture contents
due to drying, the surface temperature never exceeds the wet bulb temperature.

At the vertical and radial centerlines of the product, heat transfer was assumed to

be zero due to product symmetry (Equations [4.5] and [4.6]).

6T

_=0 :0 4.5
6r r l ]
§I=0 Z=0 [46]
dz

4. 2.2 Moisture transfer solution

Moisture transfer within the product was modeled using a two-dimensional

equation for diffusion in radial coordinates (Equation [4.7]).

67

am 1 a k m,water 6m 5 k m,water am
—: —..— r.————o-——— +———- —————--—— [4.7]
at r at cm,water ' p 51' 62 cm,water ' P 62

The units for mm... are decimal dry basis moisture content. Mass transfer at the product

surface was modeled using a convective boundary condition (Equation [4.8]).

8’

[4.8]

k m,water an = h m,water ' (C air _ Csurface)

The concentration of moisture in the cooking air was calculated using Equation [4.9].

[4.9]

Cair : psteam ' Xair

X3], is the molar fraction of steam in the cooking air. This is equal to the percentage of
steam by volume. The density of steam (psteam) was calculated using an equation derived

from tabular data (see Section 9.2).
pmam = —0.0002 - log(T)+ 0.0015 [ 4.10]
The concentration of moisture at the patty surface was calculated using Equation [4.11].

68

Csurface : ERH ' Csat I 4-11 I

Equilibrium relative humidity (ERH) was calculated using an equation by Huang and

Mittal (1995).

 

_ . -l.0983
ERH =ex 5222.47 111 [ 4.12]
1.9818-T

In Equation [4.12], m is dry basis moisture content, T is absolute temperature, and ERH
is decimal equilibrium relative humidity. The saturation concentration (Csat) was
calculated using Equation [4.13]. Equation [4.13] was developed using regression of

steam table data as shown in Section 9.2.

csat = 8.121 - 10'10 .r3 —3.520.10'8 -'r2 41.320.10‘6 .r +6.215-10-7 [ 4.13]

At the radial and vertical centerlines of the patty, mass transfer was assumed to be zero

due to product symmetry (Equations [4.14] and [4.15]).

—=0 r=0 [4.14]

69

92:0 2:0 [415]
dz

4. 2.3 Fat transfer solution

Fat transfer was modeled using a two-dimensional formulation of Darcy’s law for

diffusion of liquids through porous media (Datta, 2002).

6F 1 6 BF 6 6F
—=["‘a—r'(r'Dcap,fat '_)+—Z‘(Dcap.fat “an I 4'16]

Fat content was written in terms of dry basis fat content. Dcapfat is the capillary

diffusivity of fat in the product.

The fat content at the surface of the patty was modeled using an equation
generated from experimental values as detailed in Section 3. Equation [4.17] was utilized
to set the values of the fat content at each boundary node as a function of temperature and

product composition. This equation was derived from experimental data (Section 5.3.)

F = 0.7062 — 0.0193 - T + 0.0001 - T2 + 0.0069 . Emma, + 0.0002 - T . PM,l [ 4.17 ]

At the radial and vertical centerlines of the patty, fat transfer was assumed to be

zero due to product symmetry (Equations [4.18] and [4.19]).

70

——=0 r=0 4.18

&_ l ]
§E=0 2:0 [4J9]
dz

4. 2.4 Heat and mass transfer coefficients — Array of slot nozzles

Heat and mass transfer coefficients utilized in Equations 4.4 through 4.8 were
modeled as functions of instantaneous process conditions using the technique of Millsap
and Marks (2002). For the surfaces of the patty subject to impingement conditions (the
top and bottom), correlations developed by Martin (1977) for an array of slot nozzles
were used to determine heat and mass transfer coefficients. Prandtl number, Schmidt

number, and Reynolds number were first calculated using Equations [4.20], [4.21], and

 

 

[4.22] respectively.
c . . .
Pr: p,m1x “mix [4.20]
kmix
Se=——“ii’£— [ 4.21]
Pmn'DAB
v- . . .w
Re: J“ pm“ [4.22]
“mix

71

Values for the heat capacity (emu), viscosity (um), thermal conductivity (kmix),

density (pm), and diffusivity (D5,) of the air-steam mixture were calculated using the
relationships in Section 9.2. The (W) term in Equation [4.22] is the slot width of the

impinging jets. The term, v, in Equation [4.22] is the jet exit velocity.

Nusselt and Sherwood numbers were calculated using Equations [4.23] and

[4.24], respectively (Martin, 1977).

Z

3 3
Nu=3-Pr°-42-f0:i- Lif— [ 4.23]

3 % + 1%
0

Z

3 3
Sh=—:—-Sc0'42-f02- Z'Re [4.24]

%.+f%

where f0 is a filnction of the slot geometry, and is given by Equation [4.25].

f0 =(60+4-(%—2)2)_1/2 [ 4.25]

72

In Equation [4.25], H represents the spacing between the impinging jets and the product
surface, and W represents the slot width of the jets. Heat and mass transfer coefficients
were calculated from the Nusselt and Sherwood numbers using Equations [4.26] and

[4.27], respectively.

 

Nu-k -
h =——-——-m1x 4.26
T W l ]
Sh-D
hm,water = w 5“ [ 4.27]

For the vertical surfaces of the patty (the patty edges), correlations for turbulent
flow past a flat plate (Bejan, 1995) were used to calculate heat and mass transfer
coefficients. Turbulent flow was assumed due to the flow regime created by
configuration of the oven ductwork and turbulent air coming off of the top and bottom
surfaces of the patty. Reynolds number was calculated using Equation [4.28] where (E)

is the edge height of the patty.

Rezw [4.23]

”mix

73

Prandtl and Schmidt numbers were calculated using Equations [4.20] and [4.21],

respectively. Nusselt and Sherwood numbers were calculated using Equations [4.29] and

[4.30].
4 1

Nu = 0.037 ~ Re/S-Pr/3 [ 4.29]
4 l

Sh = 0.037 - Re/5-Sc/3 [ 4.30]

Heat and mass transfer coefficients were then calculated using Equations [4.31] and

 

[4.32], respectively.
hT 2M [ 4.31 ]
E
Sh . D
hm,water = E sa [ 4.32]

4. 2.5 Heat and mass transfer coeflicients — array of round nozzles

In addition to the correlations for an array of slot nozzles, additional correlations
were utilized to predict the heat and mass transfer coefficients for an array of round
nozzles (Martin, 1977). Nusselt and Sherwood numbers were calculated using equations

[4.33] and [4.34], respectively.

74

Nu = Pro-42. K A (HS,0t /D, r)- G(Hslot /D, f)- F(Re) [ 4.33 ]

Sh = Sc0‘42 -K A (Hslot /D, f)- G(Hslot /D, f)- F(Re) [ 4.34 ]

KA(Hs[m/D, f) is the array correction function and is given by Equation [4.35].

6 —0.05
H /1)
K H D,f=1+—S'°—t— [4.35]
A( slot/ ) [0.6/J?
where:
f=0.785-(D/L)2 [ 4.36]

In the above equations, D and L represent the nozzle diameter and nozzle spacing,
respectively. G(f, H/D) is a geometric function (Equation [4.37]), and F(Re) is a function

of the Reynolds number (Equation [4.38]).

1—2.2-~/f [4.37]

G(f’HSIOt/D)= ZW/F. I+0.2'(Hslot/D-6)"/E

 

75

F(Re): 0.5-Re2/3 [ 4.38]

From the values of Nusselt and Sherwood numbers, values for heat and mass

transfer coefficients were calculated using Equations [4.26] and [4.27] respectively.

4.3 Microbial inactivation model
A simple first-order inactivation model (Equation [4.39]) was combined with the

heat and mass transfer model.

N0 AI
10 — : — 4.39
810( N ] D l l
where:
D = Dmf .10(Tref‘T)/Z [ 4.401

The reduction in the number of Salmonella at each node was determined by
calculating the fraction of survivors after each time step. The number of survivors after
each time step was calculated by inserting the temperature data generated by the cooking

model into Equation [4.41]. The overall reduction at the center of the patty was then

76

determined for the entire cooking process. The aggregate microbial reduction for the

entire patty was calculated by calculating the volume average of all the nodes.

At
N=N0/IOD [4.41]

4.4 Finite element formulation
4. 4.1 Introduction

The finite element method (FEM) was utilized to solve the cooking model
equations. The finite element is a common numerical method for solving differential
equations that are difficult to solve using analytical techniques. The finite element
method has several advantages over other numerical techniques that make it very suitable
for formulating cooking models. FEM is readily adaptable to irregular geometries, it can
be applied to systems containing more that one set of physical properties, and it is readily
applied using computers.

The following sections briefly describe the finite element method and the
techniques used to convert the model equations into computerized form. A much more

detailed description of the finite element method was given by Segerlind (1984).

4. 4.2 Finite element basics
To apply the finite element method, the geometry of the system must first be
broken down into a finite number of discrete regions referred to as elements. Elements

are typically triangular or quadrilateral, although other geometries can also be utilized.

77

The comers of each element are referred to as nodes. Adjacent elements share the nodes
at the corners they have in common. When the finite element method is performed, a
numerical solution is generated for each node in the element mesh. Increasing or
decreasing the number of nodes controls the resolution of the model. Models with a large
number of nodes produce finer resolution at the expense of increased computing time.
Figure 4.3 shows the geometry of the system used for the computer model as well
as the element mesh. Figure 4.4 is a view of the element mesh with the coordinate axis

indicated.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.3 - Geometry of a ground meat patty with the modeled region and element
mesh indicated.

  
  

  

7.1.5474: '51 '51 P.‘ P." '4 PA PA PA '5. IL: '3‘ '1'.‘ P.- Df.‘ IF.- '4 'A
umaaaauauaaaaaaaaanu:

mmmmmmmmmm:

 
       

Figure 4.4 - Illustration of the finite element mesh utilized for the model.

The radial symmetry of ground meat and poultry patties allowed patties to be
modeled as two-dimensional axisymmetric bodies. For the purpose of the cooking
model, the modeled region was broken down into 117 triangular elements. This resulted

in 82 nodes. The density of the element grid was higher at the patty surface to better

78

model the larger moisture, fat, and temperature gradients present near the product
surface. The 117-node element mesh was chosen to balance solution accuracy with
computing time (see Section 5 .5.1).
4. 4.3 Governing equations

As discussed in the previous sections, the cooking model was formulated using
three sets of differential equations. The basic solution of the heat transfer, moisture
transfer, and fat transfer equations using the finite element method is very similar.
Therefore, solution of a generic transport equation will be shown with emphasis given to
the differences between the three solutions.

The basis for each transport solution was the time-dependent two-dimensional

field equation expressed in radial coordinates (Equation [4.42]).

2
i§$=1[nri[r@)]+D,9—Il [ 4.42]
at r or ar 522

Equation [4.41] is a generic form of the governing equations given by Equations
[4.1], [4.7], and [4.16]. In Equation [4.42], the term (of) is the unknown variable and
corresponds to the enthalpy, moisture content, and fat-protein ratio in Equations [4.1],
[4.7], and [4.16], respectively. The terms (A) and (D) are combinations of thermo-
physical property values. For the heat transfer solution, the term D is equal to the thermal
conductivity, and the term 7). is equal to the product of the heat capacity and the density of
the meat. For the moisture transfer solution, the term D is equal to the moisture

conductivity, and the term it is equal to the product of the specific moisture holding

79

capacity and the density of the meat. For the fat transfer solution, the D term is equal to
the capillary diffusivity, and the A term is not utilized.

The finite element method is a weighted residual technique. An approximate
solution is substituted into the governing equations, and the error term calculated. The
product of this term and a weighting function is then reduced to zero to produce a
numerical solution. The weighting function may take many forms. The method chosen
was Galerkin’s method. In this method, the weighting function uses the same functions

that were used for the approximate solution (Segerlind, 1984).

Using Galerkin’s method for the axisymmetric field problem, the weighted

residual equation is Equation [4.43].

{11(6)}: —][N]T[%§(r—;ij+ozg+x%]dv [ 4.43]

The term [N], is a vector of shape functions. Triangular elements were used for
the cooking model. This allowed the density of the element mesh to be increased near
the boundaries without dramatic change in the aspect ratio of the elements. Proper aspect
ratio is desirable, as elements with greatly uneven lengths and widths can contribute to
loss of solution accuracy (Segerlind, 1984). Figure 4.5 is an illustration of a typical

triangular element. The element contains three nodes that are labeled as nodes i, j, and k.

80

 

Figure 4.5 - Illustration of a triangular element showing counterclockwise node

numbering.

The shape functions for a triangular element are given by Equations [4.44]

through [4.46].

l
Nk = ‘2TA—(ak + bkr + ckz)

where:

ai = RjZk -Rij

bi=Zj—Zk

81

Ci =Rk-R'

[ 4.44]

[ 4.45]

[ 4.46 ]

=RkZi—RiZk bj=Zk—Zi °j=Ri“Rk
ak=RiZj—RJ-Zi bk =Zi—Zj Ck=Rj"Ri

The terms R" and Z“ are the r and z coordinates of the nth node, respectively.
After applying the product rule for differentiation and considerable manipulation

(Segerlind, 1984), Equation [4.43] can be converted to the form of Equation [4.47].

or:[ineavfiaelvjal

[[[N]T D, grim“) + 112 gsin 0de + {[[NF (A gijdv

F

[ 4.47]

The first integral of Equation [4.47] is related to transport within the product and

can be written as Equation [4.48], where [km] is the element stiffness matrix.

ik”-—Ii VII) Qg—W a—[N]+ wz°[—:23%]dv [4.48]

The second integral in Equation [4.47] is related to the derivative boundary condition and

can be written as Equation [4.49], where “(0} is the inter-element vector.

{1(6)}: [[N]T(D EcosfHDz gsin0]d [4.49]

82

The third integral in Equation [4.47] is a time-dependent capacitance term.

Using Equations [4.48] and [4.49], Equation [4.47] can be re-written in the form:

{1(0)}: {102)}, lk(°)]i¢(e)i+ [C(e)J[¢(e)} [ 4.50]

where [Cm] is called the capacitance matrix and is given by Equation [4.51], and {(1)}

represents the nodal values of 6CD/6t.

[CM]: [3.[N]T[N]dv [ 4.51 ]

Inserting the shape functions into the element stiffness matrix and integrating

gives the form of the stiffness matrix shown in Equation [4.52].

 

 

' 2 ‘ ’ 2 7
) Zm-‘D bi bisz bibk Zm'D Ci CiCZj CiCk
[1((e ]= 4A r bibj bj bjbk + 4A 2 CiCj Cj CjCk [ 4.52]
2

 

 

 

 

2
bibk bjbk bk CiCk CjCk Ck

‘

Expanding the capacitance matrix using a lumped solution gives the form of the
capacitance matrix given by Equation [4.53]. The lumped solution assumes that the
variation art/at is constant across each element. This is the solution that provides for the

maximum operating range of the final model.

83

0 0
[C(e)]=-’f§0 1 0 [4.53]
o 1

Treatment of the inter-element vector is somewhat more involved. A derivative

boundary condition over a boundary F can be specified using Equation [4.54].

Drgc080+ngsm0=—M¢b +S [ 4.54]

The right hand side of Equation [4.54] can be substituted into Equation [4.49] yielding

Equation [4.55].
{19}: I[INIT(1\4[N]{<I>“’)1- Sill“ I 4551

Upon expansion, Equation [4.55] produces two terms given by Equations [4.56] and

[4.57].

IkM(c)I= lMlNlrlNldl" [4.56]
r13C

a): and [4,7,

r13c

84

Equations [4.56] and [4.57] can be expanded into the form of Equation [4.58] and [4.59].

(3Ri+RJ-) (Ri+RJ-) 0

 

 

[kM(e)]=2”11‘2’IL (Ri+RJ-) (Ri+3Rj) 0 [4.58]
0 0 0
{fs(e)}= “6 Ri+2RJ~ [4-59]
0

For the convective heat transfer boundary condition, the term M is equal to the
heat transfer coefficient h. The term S is equal to the heat transfer coefficient multiplied
by the temperature of the bulk fluid. Since the heat transfer solution was formulated in
terms of enthalpy, the equilibrium enthalpy related to the oven air temperature was
calculated using Equation [4.2] and substituted in place of the oven air temperature in
calculating S.

For the convective moisture transfer boundary condition, the term M is equal to
the convective mass transfer coefficient hm. The S term is equal to the convective mass
transfer coefficient multiplied by the equilibrium moisture content for the meat under the
given oven air temperature and moisture conditions. Equilibrium moisture content was

calculated using Equation [4.60] derived from the work of Huang and Mittal (1995).

85

 

—l/l.0983
EM C = 1.9818 Tsurface ln(RH) [ 4.60 1
— 5222.47
where:
RH =100 ' (Coven /Csat) I 4°61 I

4. 4.4 Finite diflerence time solution
A finite difference time-solution was used to provide a FEM solution over a
number of time steps. This resulted in an equation with a time-step term (Equation

[462]).
([C]+ 0At[k]){<l>}b = ([c]— (1 — 0)At[l<]){cr>}a + At((1 — 0){F}b + 0{F}a) [ 4.62 ]

Setting the value of 0 equal to 1/2 (central difference method), results in Equation [4.63].
At At At
[[c]+ 70.0415), = [[c]_ ?[K]]{<D}a + 761?}. + [14,) r 4.63 1

Equation [4.63] can be reduced to Equation [4.64].

[A]{(I>}b = [P]{<r>}a + {F *} [ 4.64]

86

where:

[A] = [[c]+ 323M] [ 4.65 ]

lpl=[lcl—é‘,llkl) [ 4.66]

and

{w}: 4‘0}. +114.) 1 4.67]

4.4.5 Application of FEM solution

Equation [4.64] has a form that can be readily programmed into a personal
computer. For the cooking model, three sets of element solutions were constructed. The
first set of equations was related to the heat transfer portion of the model (Equation
[4.1]). The second set of equations was related to mass transport due to moisture
migration (Equation [4.7]). The third set of equations was related to fat transfer
(Equation [4.16]). The finite element solution was programmed using Microsoft Visual

Basic (Version 6.0 professional edition: Microsoft, Redmond, WA); (Section 9.5).

87

4. 4.6 User interface

A Windows-based user interface for the model was developed using Microsoft
Visual Basic. The user interface consisted of two primary window screens controlled by
a menu toolbar. Illustrations of each interface screen are shown in Section 9.4.

The first of the windows is the input screen. On this screen, the user inputs the
oven air temperature, the oven steam content or wet bulb temperature, the air velocity,
and the cooking time. The user also has the option of inputing a set of transient oven
conditions using a file input. The input screen also requires the user to specify the initial
temperature, fat content, and moisture content of the meat. Buttons are available for
selecting between beef, pork, and turkey. The final component of the interface screen is
the microbial inactivation input section. Default D, 2, and reference temperature values
for a seven-strain Salmonella cocktail in beef (Smith et al., 2001) are incorporated into
the model. However, these defaults may be replaced with values specified by the user.

The second screen of the user interface is the output screen. On this screen, the
output temperature, moisture, yield, and microbial inactivation are displayed in graphical
form. The final temperature, moisture, yield, and lethality values are also given in digital
form. A message is displayed in the inactivation output section indicating whether or not

the required level of lethality was achieved for the product selected.

88

5 RESULTS AND DISCUSSION

5.1 Overview

The experimental results for this study are grouped into three main sections. The
first section (Section 5.2) describes the results of a set of experiments conducted using a
laboratory convection oven. Patties of ground turkey, ground beef, and ground pork were
cooked to various endpoint temperatures to determine the effects of meat species and
initial fat content on cooking yield, heating rate, and shrinkage. These experiments were
intended to provide insight on differences in cooking behavior between meat species and
fat content and to evaluate the need for including fat transfer in meat cooking models.

Section 5.3 describes the results of a series of experiments conducted to determine
the fat holding capacity of ground beef as affected by isothermal heating. These data
were used to develop an expression for fat holding capacity as a function of initial fat
content and temperature. This expression was then included as a portion of the cooking
model (Section 4.2.3).

Section 5.4 describes the results of experiments conducted using an industrial
moist-air impingement cooking system (Stein model JSO-IV). Ground beef patties were
cooked in the oven using various cooking conditions. The effects of oven conditions on
cooking time, yield, and volume change were determined. Additionally, data from this
set of experiments were used to validate the heat and mass transfer components of the

cooking model.

89

l

 

5.2 Laboratory oven cooking tests

In these tests, small meat patties were cooked in a laboratory convection oven to
determine the effect of species and initial fat content on fat loss for ground turkey, ground
beef, and ground pork (Section 3.2). The effects of species and fat content on cooking

time, yield, and volume change were also determined.

5. 2.] Cooking time

For each cooking test, the time required to reach the target endpoint center
temperature was measured. Using this information, temperature versus time plots were

constructed for each species and fat content (Figures 5 .1-5.3)

 

 

 

 

100,
904}

6 E O a

b 805

E i 0 :1

g 70E

O. i 0 Cl

5 60,

t— : U.

Q) .

‘a 50-: -_

5 E a 01.4%fat
40; 08.6%fat
30PJJJLLFALLJ#JIII%IL11%L141

 

3.00 5.00 7.00 9.00 1 1.00 13.00

Time (minutes)

Figure 5.1 — Center temperature as a function of cooking time for ground turkey
patties cooked in a laboratory convection oven: means of 5 replicates.

90

 

 

 

 

 

 

100 .
A 90
a D 0
e 80 “F D
E 70 -:—
8. : 1:1 9
8 60 «e
t : 1: o
’a’ 50 E __
8 E 0 07.2% fat
40 l p l7._5_j/._ fat
30“ =W‘
3.00 5.00 7.00 9.00 11.00 13.00

Figure 5.2 — Center temperature as a function of cooking time for ground beef

Time (minutes)

patties cooked in a laboratory convection oven: means of 5 replicates.

 

 

 

 

 

 

 

 

100 _

A 90

a g :1 o

o 80 e:

a 2 n .

£3 70 .

8. E n o

E, 601:

a E 1:1 0

3;; 50 t

8 L , 015.7% fat
40 ":8 041.9% fat
30_ igmlrilrIllr larlilk

3.00 5.00 7.00 9.00 11.00 13.00

Time (minutes)

Figure 5.3 — Center temperature as a function of cooking time for ground pork
patties cooked in a laboratory convection oven: means of 5 replicates.

91

The cooking rates of turkey differed between the 1.4% and 8.6% fat samples
(Figure 5.1). The temperature of both the low and high fat patties increased as a function
of time. However, the heating rate of the 1.4% fat samples was slightly higher. The 1.4%
fat patties took 254 seconds to reach 45°C compared to 264 seconds for the 8.6% fat
patties. The difference in heating time increased as the patties reached higher
temperatures. The 1.4% fat patties reached 85°C in 513 seconds compared to 684
seconds for the 8.6% fat patties, a difference of almost 3 minutes. Analysis of variance
confirmed the statistical significance (P<0.05) of differences in heating times between the

two fat contents (Table 5.1).

Table 5.1 — Analysis of variance for cooking time of ground turkey patties as
affected by temperature and initial fat content.

 

 

Factor Sum of Degrees of Mean F-Value P-Value
Squares Freedom Square
Temperature 6618213 4 1654553 48.2258 <0.001
Fat Content 59305.68 1 59305.68 17.2860 <0.001
Interaction 57447.52 4 14361.88 4.1861 0.006

 

With ground beef, the heating rate was significantly higher for the 17.5% fat
patties than for the 7.2% fat patties (Figure 5.2). The average time required to heat the
17.5% fat patties to 45°C was 260 seconds compared to 361 seconds for the 7.2% fat
patties. The heating time increased with increasing patty temperature. The time required
to reach 85°C was 619 seconds for the 17.5% fat patties compared to 836 seconds for the
7.2% fat patties. Analysis of variance showed both center point temperature and fat

content to affect (P<0.05) heating time for ground beef patties (Table 5.2).

92

Table 5.2 - Analysis of variance for cooking time of ground beef patties as affected
by temperature and initial fat content.

 

Factor Sum of Degrees of Mean F-Value P-Value
Squares Freedom Square

 

Temperature 7420335 4 185508.4 19.7617 <0.001
Fat Content 1208353 1 1208353 12.8723 <0.001
Interaction 29187.52 4 7296.88 0.7773 0.547

 

As with the beef patties, the higher fat content pork patties took less time to heat
than the lower fat patties (Figure 5.3). The average time required for the 41 .9% fat
patties to reach 45°C was 236 seconds compared to 285 seconds for the 15.7% fat patties.
Heating temperature increased with increasing time. The time required for the 41 .9% fat
patties for each 85°C was 608 seconds compared to 710 seconds for the 15.7% fat patties.
Analysis of variance showed temperature and fat content to be significant (P<0.05) with

respect to heating time (Table 5.3).

Table 5.3 - Analysis of variance for cooking time of ground pork patties as affected
by temperature and initial fat content.

 

Factor Sum of Degrees of Mean F-Value P-Value
Squares Freedom Square

 

Temperature 1036732 4 2591829 59.6434 <0.001
Fat Content 59512.5 1 59512.5 13.6951 <0.001
Interaction 7689.6 4 1922.4 0.4424 0.777

 

In the case of the ground beef and ground pork patties, samples containing higher
levels of fat achieved higher rates of heating. This was most likely due to the lower

moisture content associated with the higher fat products. As water contributes the

93

greatest source of heat capacity in the meat matrix, lowering the water content would be
expected to increase the heating rate. It has also been hypothesized that inclusion of high
fat contents could result in convective heating within the meat, thereby increasing the
heating rate (Shilton et al., 2002).

Unlike the ground pork and ground beef patties, the heating rate of the ground
turkey patties was higher for the lower fat product. It should be noted that the
compositions of the two ground turkey samples were more similar than those of the
ground pork and ground beef samples. Specifically, the moisture content of the 8.6% fat
ground turkey was only 1.8 w.b. moisture points lower than the moisture content of the
1.4% fat patties. This compares with differences in moisture content of 8.2 and 20.5 w.b.
moisture points for the ground beef and ground pork, respectively. The smaller
differences in moisture content for the ground turkey mean that the heat capacity of the
two samples was nearly the same. Thus, moisture probably played a less significant role
in differences in heating rate for the ground turkey patties.

The difference in heating rate between different fat levels of ground turkey patties
was largest at temperatures above 55°C. Interaction between fat content and temperature
was significant. This differs from ground beef and ground pork, which had consistent
differences (i.e., no significant fat-temperature interaction). It is therefore likely that the
different heating characteristics of the turkey were related to other compositional
characteristics of the meat. The 8% initial fat ground turkey patties included ground skin
as a method for increasing fat content. This may have contributed to the different heating

characteristics of the ground turkey as compared to the ground beef and ground pork.

94

5. 2.2 Cooking yield

For each test, the cooking yield was calculated at each endpoint temperature
(Figures 5.4-5.6). Large differences in yield were evident between the 1.5% fat and 8.6%
fat ground turkey patties (Figure 5.4). The yields of the 1.4% fat patties were
consistently higher than the yields of the higher fat patties. The yield of the 1.4% fat
patties ranged from 90% at 45°C down to 81% at 85°C compared to yields of about 83%
to 63% for the 8.6% fat patties. The decrease in yield was approximately linear with
respect to center temperature for both fat contents. Analysis of variance confirmed the
significance (P<0.05) of both center temperature and fat content on cooking yield (Table
5.4). Higher yield losses for the 8.6% fat patties were presumably due to fat loss during
cooking. However, the results presented in the next section do not support the conclusion
that fat transport was solely responsible for yield differences between the two fat contents
of turkey. It is possible that interaction between the fat and water within the patty
resulted in lower water binding capacity for the higher fat patties, resulting in higher rates
of moisture loss during cooking. Further experimentation is needed to determine the

mechanisms for fat and water binding during cooking.

Table 5.4 — Analysis of variance of yield as a function of center temperature and
initial fat content for ground turkey patties.

 

Factor Sum of Degrees of Mean F-Value P-Value
Squares Freedom Square
Temperature 1359.399 4 339.8498 10.5351 <0.001
Fat Content 1751.823 1 1751.823 54.3051 <0.001
Interaction 203.2676 4 50.8169 1.5753 0.200

 

 

95

 

 

 

 

 

100.0
95.0 o 1.4% fat
90-0 ° , Efiéf’éfflj
% 233 1:; ° . . .
>~ ’ E n
E 75.0 “E
8 E '3
5 70.0 "E
°‘ 650-; D a
60.0
55.0
50.0:5.......r,...,,....,....
40 50 60 70 80 90
Center temperature (° C)

Figure 5.4 — Yield as a function of endpoint center temperature for ground turkey
patties of two fat contents: means of 5 replicates.

For the ground beef, only minor differences in yield were detectable between the
7.2% and 17.5% fat samples at each cooking temperature (Figure 5.5). The yield of both
products decreased from about 84% at 45°C to between 63 and 66% at 85°C. Analysis of
variance indicated that yield was related (P<0.05) to center temperature but not to initial

fat content for these data..

Table 5.5 - Analysis of variance of yield as a function of center temperature and
initial fat content for ground beef patties.

 

Factor Sum of Degrees of Mean F-Value P-Value
Squares Freedom Square
Temperature 2135.567 4 533.8917 37.9595 <0.001
Fat Content 9.6439 1 9.6439 0.6857 0.413
Interaction 21.9899 4 5.4975 0.3909 0.814

 

 

96

 

100.0 :
95-0 '0 7.2% fat
90.0 i n 17.5% fat
85.0 e r ”“m
80.0 4; g
75.0 -E III
70.0 {
65.0 —E
60.0w
55.0—
50.0“wielxlxwllwwlwin»

40 50 60 70 80 90

Center temperature (° C)

 

 

D

Percent yield

OCI

‘r—T—TT TTT'TTlTT

 

 

 

Figure 5.5 - Yield as a function of endpoint center temperature for ground beef
patties of two fat contents: means of 5 replicates.

For ground pork, there were large differences in yield between the 15.7 and
41.9% fat patties (Figure 5.6). The yield of the lower fat patties was much higher at each
center temperature. Cooking yield for the 15.7% fat patties ranged from about 90% at
45°C to about 71% at 85°C. Cooking yield for the 41.9% fat patties ranged from about
74% at 45°C to about 54% at 85°C. Cooking yield decreased linearly as a function of
temperature for patties of both fat contents. Analysis of variance showed differences

(P<0.05) in yield as a function of temperature, fat content, and temperature-fat interaction

(Table 5.6).

97

 

100.0 :

95.0

90.0 o

85.0 , ,

80.0 ,

75.0 D

70.0 0

65.0 «.

60.0 D 0

55.0

50.0:“11wil
40 50 60 70 80 90

 

:isfifit
La 4.13%!“

Percent yield

f T777
0

D

 

 

 

Center temperature (°C)

Figure 5.6 - Yield as a function of endpoint center temperature for ground pork
patties of two fat contents: means of 5 replicates.

Table 5.6 - Analysis of variance of yield as a function of center temperature and
initial fat content for ground pork patties.

 

Factor Sum of Degrees of Mean F-Value P-Value
Squares Freedom Square

 

Temperature 2132.37 4 533.0926 96.4285 <0.001
Fat Content 4153.887 1 4153.887 751.376 <0.001
Interaction 79.8724 4 19.9681 3.61 19 0.013

 

The differences in cooking yield appear to be affected by both meat species and

fat content. No significant correlation between initial fat content and cooking yield can

be made across the three meat species. Although the yield at 85°C was lowest for the

highest fat product (41 .9% fat ground pork) and highest for the lowest fat product (1.4%

fat ground turkey), the behavior of samples of in-between fat contents was not consistent.

This could be caused by differences in protein conformation and fat composition of the

98

three meat species. It may also be due to differences in water and fat binding capacity of
the meat proteins. Differences in particle size and porosity of the meat may also affect

the cooking yields by changing the moisture and fat transport dynamics.

5.2.3 Fat loss

The amount of fat lost during cooking as a function of temperature for 1.4% and
8.6% fat ground turkey patties is shown in Figure 5.7. The amount of fat lost during
cooking of ground turkey patties was below 2% for every temperature. Analysis of
variance indicated that neither temperature nor initial fat content had a significant

(P<0.05) effect on fat loss during cooking (Table 5.7).

 

 

30.0 ,

25.0 i

 

: 1.4% fat
0 8.6% fat

 

20.0 {-
15.0

10.0 7

Fat loss (% of initial mass)

.U'
o

 

 

l
tTfi—r, r
,.

inL—HIHQIJTIJIGII
4O 50 60 70 80 90

 

 

.9
o
I

 

Center temperature (° C)

Figure 5.7 - Fat loss as a function of temperature for ground turkey patties of two
fat contents: means of 5 replicates.

99

Table 5.7 — Analysis of variance for fat loss as functions of center temperature and
initial fat content for ground turkey patties.

 

 

Factor Sum of Degrees of Mean F-Value P-value
Squares Freedom Square

Temperature 29.9082 4 7.4770 1.2317 0.3184

Fat Content 4.4131 1 4.4131 0.7270 0.4006

Interaction 17.8231 4 4.4558 0.7340 0.5760

 

The amount of fat lost during cooking of ground beef patties was much larger
than the amounts lost during cooking of ground turkey (Figure 5.8). Fat losses were
higher for the 17.5% fat patties, ranging up to 6% of the initial mass of the meat.
Analysis of variance showed that initial fat content had a significant (P<0.05) effect on
the amount of fat lost during cooking (Table 5.8). A significant temperature effect on the

amount of fat lost during cooking could not be shown.

 

 

 

 

 

 

 

30.0 ,
A : 0
g 25.0 07.2/6 fat
8 E D 17.5% fat
:9: 20.0 E
E. :
«5 115.0 :—
°\° E
1,; 10.0 --
.2 1
L3 5.0 .. a D

00 l’lwm‘: ’:1°1%

40 50 60 7O 80 90
Center temperature (°C)

Figure 5.8 - Fat loss as a function of temperature for ground beef patties of two fat
contents: means of 5 replicates.

100

Table 5.8 — Analysis of variance for fat loss as functions of center temperature and
initial fat content for ground beef patties.

 

 

Factor Sum of Degrees of Mean F-Value P-value
Squares Freedom Square
Temperature 12.8007 4 3.2002 0.9060 0.4730
F at Content 240.3399 1 240.3399 68.0423 <0.001
Interaction 35.0635 4 8.7659 2.4817 0.065

 

Fat losses for ground pork were the highest of the three meat species tested
(Figure 5.9). For the 15.7% fat pork patties, the amount of fat lost during cooking was
low and never exceeded 2% of the initial patty mass. However, for the 41 .9% fat patties,
the amount of fat lost during cooking was much larger. The amount of fat lost during
cooking was about 20.5% of the initial patty mass at 45°C. The amount of fat lost
increased with increasing patty temperature up to about 28% at 85°C. Analysis of
variance showed cooking temperature, fat content, and temperature-fat interaction to

affect (P<0.05) fat loss (Table 5.9).

Table 5.9 — Analysis of variance for fat loss as functions of center temperature and
initial fat content for ground pork patties.

 

 

Factor Sum of Degrees of Mean F-Value P-value
Squares Freedom Square
Temperature 68.6303 4 17.1576 14.3904 <0.001
Fat Content 5660.502 1 5660.502 4747.583 <0.001

Interaction 56.5569 4 14.1392 1 1.8589 <0.001

 

101

 

 

 

 

 

 

 

30.0
_ a
7,; f D D
3 25.0 _ D
E
3‘ 20.0 D
E E o 15.7% fat
«5 15.0 ;
c ~ 041.9% fat
8) .- __.2_.___
a 10.0 9;
2 .
*-' i
63 5.0
o
00 H‘HTHLtMT ° W494
40 50 60 70 80 90
Center temperature (° C)

Figure 5.9 - Fat loss as a function of temperature for ground pork patties of two fat
contents: means of 5 replicates.

The quantities of fat lost by the 17.5% fat ground beef patties and 41 .9% fat
ground pork patties during cooking are of great significance for modeling. Cooking
models that do not take fat loss into account will likely over predict cooking yields. For
high fat products such a sausage, these errors in yield prediction could exceed 25%.
Many models may account for fat losses indirectly by over predicting mass transfer
coefficients. However, this method may restrict the utility of the models to products of
similar composition. To produce robust cooking models, transport of both moisture and

fat components should be considered.

102

5.2.4 Volume change

Plots of volume change as a function of product temperature were plotted for each
meat species (Figures 5. 10-5. 12). The volume of ground turkey patties as a function of
temperature is shown in Figure 5.10. Volume of the 1.4% fat patties ranged between 74
and 70% of the initial volume at center temperatures between 45 and 85°C. Volume of
the 8.6% fat patties decreased more dramatically as a fiinction of center temperature and
ranged from 62% to 56% of the original volume at center temperatures of 45 and 85°C
respectively. Analysis of variance showed that initial fat content had a significant effect
on volume change during cooking (P<0.05) (Table 5.10). However, the relationship
between center temperature and volume change was not statistically significant.
Although the change in volume was significantly different over the measured temperature
range, both products exhibited major decreases in volume between the raw state and
45°C. This shows that major volume changes occur during the initial stages of cooking,

and thus may be impossible to avoid during convection cooking.

Table 5.10 — Analysis of variance for effects of temperature and initial fat content on
volume change for ground turkey patties cooked in a laboratory convection oven.

 

Factor Sum of Degrees of Mean F-Value P-Value
Squares Freedom Square

 

Temperature 253.4806 4 63.3702 1.0724 0.383
Fat Content 2910.062 1 2910.062 49.2471 <0.001
Interaction 122.8181 4 30.7045 0.5196 0.722

 

103

100.0

 

 

 

 

 

 

 

5 o

E 90,0 —: o 1.4/o fat
'3 r Layman
3. 80.0 ‘5
a f"
.E : o .
E0 70.0 "' . . Q
0 C
“-0 -
g 60.0 {- D n
8 : U U D
E 50.0 E

40.0 lAA‘iJ"‘+“‘#Tlltlt:thih

4O 50 6O 7O 80 90
Center temperature (° C)

Figure 5.10 — Relationship between product temperature and volume change for
ground turkey patties cooked in a laboratory convection oven: means of 5 replicates.
The ground beef patties exhibited a similar decrease in volume during cooking
(Figure 5.11). Analysis of variance showed both center temperature and fat content to
significantly affect volume (Table 5.11). Unlike the ground turkey and pork patties, the

higher fat beef patties had less shrinkage during cooking than did the lower fat patties.

Table 5.11 — Analysis of variance for effects of temperature and initial fat content on
volume change for ground beef patties cooked in a laboratory convection oven.

 

Factor Sum of Degrees of Mean F-Value P-Value
Squares Freedom Square
Temperature 1424.392 4 356.0981 11.5257 <0.001
Fat Content 631.248 1 63 l .248 20.43 14 <0.001
Interaction 133.973 4 33 .4933 1.0847 0.377

104

 

 

 

 

 

 

 

100.0 ,
: 0
o 90.0 ; 07.2/0 fat
E : arts/oat
g 80.0 ~;
"g t
'30 70.0 f El
’5 t , a
.8 . o
3 60.0 . 0 CI
5 .
8 50.0 1
° . o
a. -
40.0F**‘*#r*4%;illi+gain:.rr.
40 50 60 70 80 90
Center temperature (° C)

Figure 5.11 — Relationship between product temperature and volume change for
ground beef patties cooked in a laboratory convection oven: means of 5 replicates.

The largest changes in volume during cooking occurred for ground pork patties
(Figure 5.12). Volume decreased as a function of cooking temperature for both the
15.7% and 41 .9% fat patties. The 15.7% fat patties ranged from about 74% to 60% of
their original volumes at 45°C and 85°C, respectively. The 41.9% fat patties ranged from
70% to 42% of their original volumes at 45°C and 85°C, respectively. The large volume
changes correlate with high yield losses during cooking. Analysis of variance indicated

that both center temperature and fat content significantly (P<0.05) affected volume

change (Table 5.12).

105

Table 5.12 - Analysis of variance for effects of temperature and initial fat content on
volume change for ground pork patties cooked in a laboratory convection oven.

 

Factor Sum of Degrees of Mean F-Value P-Value
Smres Freedom Sjuare

 

 

 

 

 

 

 

 

 

Temperature 2125.544 4 531.386 16.7989 <0.001
Fat Content 1506.957 1 1506.957 47.6400 <0.001
Interaction 451.4145 4 112.8536 3.5677 0.014
100.0 .
t
GS) 900 - O 15.7% fat
% : 041.9% fat
3. 80.0
2 E .
é" 70.0 -} 6
.2 : o ’
° 600 n o
E ' : D
“2’ : 9
a: 50.0 ‘
40.0'1~w~~w~x+~wwpll
40 50 60 70 80 90
Center temperature (°C)

Figure 5.12 — Relationship between product temperature and volume change for
ground pork patties cooked in a laboratory convection oven: means of 5 replicates.

Changes in volume are a major concern in the development of cooking models.
Slight changes in thickness greatly affect predictions of center temperature and may also
have large effects on yield predictions. Due to the geometry of the products tested, most
heat and mass transfer occurs in the axial direction. Therefore, changes in product

thickness are of greater concern than changes in product radius.

106

Although the volume of all samples decreased with increasing product
temperature, the same was not true for patty thickness. Most of the reduction in patty
volume was caused by a reduction in patty diameter. The thicknesses of the 1.4 and 8.6%
fat ground turkey patties, when cooked to a temperature of 85°C, were 120 and 110% of
the initial values, respectively. The final thicknesses of the 7.2 and 17.5% fat ground
beef patties were 97 and 120% of their initial values, respectively. The thicknesses of the
15.7 and 41.9% fat ground pork patties were 120 and 110% of their initial values,
respectively. Therefore, in all but one case, patty thickness actually increased, while the
radius decreased.

These results indicate that changes in volume cannot be modeled by simply
reducing the size of the element mesh in proportion to the reduction in product mass.
Further information is needed to determine the mechanisms for patty shrinkage. This

would be an excellent area for further study.

5.3 Fat holding capacity experiments

Fat holding capacity (FHC) was determined as a function of temperature and initial
fat content for two lots of ground beef with initial fat contents of 5.6 and 15% by mass
(Section 3.3). The fat holding capacity of the meat was defined as the amount of fat
remaining in the meat after heating and centrifuging for 15 minutes at 1000 g. The fat
content of the centrifuged meat was expressed in terms of dry basis fat content using

Equation [5. 1].

107

F: Mfat /Mdry matter [ 5-1]

where M dry mane, refers to the mass of meat that is neither water nor fat.

The mass of dry matter in each sample was calculated using Equation [5.2].

Mdry matter : I\dtotal _ (M fat + Mwater) [ 5-2 ]

Mm and Mwatc, were calculated from the initial wet basis moisture and fat contents of the

meat. The amount of dry matter was assumed to be constant during heating and

centrifugation. During cooking, small quantities of dry matter are released in the forms

of soluble proteins. Future studies should be conducted to describe the relationships

between meat proteins and moisture and fat holding capacities in a more fundamental

manner. However, inclusion of such relationships was beyond the scope of this study.
The fat content, P, was plotted as a function of heating temperature for each

heating time (Figures 5.13-5.16, 5.18-5.21).

5.3.] Low fat samples (5. 6% initial fat wet basis)

The relationship between fat holding capacity and temperature for heating times
of 2, 5, 10, and 15 minutes can be seen in Figures 5.13-5.16. The initial fat content of the
ground beef was 0.14 g fat/g dry matter. For each heating time, the fat holding capacity
was markedly lower at 55°C. The fat holding capacity of samples heated to 90°C ranged

from 0.03 to 0.05 g fat/g dry matter.

108

Linear regression was used to fit a model for fat/dry matter ratio as a function of
heating temperature and holding time for the 5.6% fat (wet basis) ground beef samples.

A quadratic response surface was chosen for the initial regression model (Equation [5.3]).

F=BO+BI~T+Bz~T2+B3-t+B4-t2+BS-t-T+e [5.3]

Regression coefficients for the response surface model are given in Table 5.13.

.o .o
\l 00
+ l
I
1

9’
o\

/ g dry matter)
53 .o
A u.

 

 

H T
£3(l3 T
3 a
[4., 0.2 i
L O O O Q Q
0 1 + o .
- » o
: e o o , .
0 _ _.L_L_JHL I 1.,1__.i_ l__+_1__.t__l _r_+_l_i__4__t_l,4__i__i__i_+_i_;t_i_+.1.4_t_t_+ 1 .L_J._L_T

20 30 40 50 60 70 80 90 100
Temperature (° C)

Figure 5.13 - Fat holding capacity as a function of temperature for 5.6% fat ground
beef heated for 2 minutes at temperatures from 30 to 90°C: means of 5 replicates.

109

0.6

 

0.4

(g fat / g dry matter)

Temperature (° C)

Figure 5.14 - Fat holding capacity as a function of temperature for 5.6% fat ground
beef heated for 5 minutes at temperatures from 30 to 90°C: means of 5 replicates.

 

 

 

0.8 I *4 - _ - ;

0.7 I
75 0.6 7t
E I
E 0.5 |
w 0.4 it»
a 0.3 i
39
u. 02 if 0 o o

o
0.1 j . O . Q 0 O Q .
O _1_L_.1_L_+,_,1 1.0411 1 l 1 1 L1. 1 1% 1 1 1 l +_1___L_1._L_+_J.__1_L_J,+_JW1__L_1
20 3O 40 50 60 70 8O 90 100
Temperature (° C)

Figure 5.15 — Fat holding capacity as a function of temperature for 5.6% fat ground
beef heated for 10 minutes at temperatures from 30 to 90°C: means of 5 replicates.

110

9
DJ
1*

F (g fat / g dry matter)

.C
N
”—F—fi'

00......

O ,l 1 Ln 1 T ; . 1 I 1.,h_i 1 191,4 L._1__l_+_ L..L_L_..l.. 1.4.4 1.,1._+.1,n__1__1 +44 1 ,LJ

20 30 40 50 60 70 80 90 l 00
Temperature (° C)

 

Figure 5.14 — Fat holding capacity as a function of temperature for 5.6% fat ground
beef heated for 5 minutes at temperatures from 30 to 90°C: means of 5 replicates.

0.8 g
0.7 I
I»

.39
MON

 

t / g dry matter)
9
.5
r——'—r—'—rT-'Tt r—r—r-r

 

 

.53 0.3
39
a. 0.2 .r
9 O O .
0'1 - o 9 . o o O o o
O 14; 1.1 1 l 1-1-J,._*_LL_1_L_+_L_L_L..L+_J__J_J.J_+..LNL_1_4+L.J_A_J_+__1_L.L..L_1
20 30 40 50 60 70 80 90 100

Temperature (° C)

Figure 5.15 - Fat holding capacity as a function of temperature for 5.6% fat ground
beef heated for 10 minutes at temperatures from 30 to 90°C: means of 5 replicates.

110

.9
00
I
I
I
I
I
I
I
I
I

.0
\l

._—

WI ._T.. .._1_

v :
Tno- on? T - - ‘7 v v v ‘ ‘-T—Tw‘d. T . T «-<
I

F (g fat / g dry matter)
O .0 .O .O .0 .O
N w A u. as

O

5 O
O

O

O

O

I
I
I
I
I
I
I

N
O
b)
O
A
O
LI!
0
O\
O
\l
O
00
O
\O
O
p—n
O
0

Temperature (° C)

Figure 5.16 — Fat holding capacity as a function of temperature for 5.6% fat ground
beef heated for 15 minutes at temperatures from 30 to 90°C: means of 5 replicates.

Table 5.13 - Results from regression of fat holding capacity as functions of time and
holding temperature for 5.6% fat ground beef.

(F=[3o +3, 'T+13¢ -T’ +0, ~t+04 -t’ +13, -t-T+e)

 

Degrees of Sum of Mean F-Value P-Value
Freedom Squares Square

 

 

 

 

Regression 5 0.0703 0.0141 37.8124 <0.001
Residual 46 0.0171 0.0004
Total 51 0.0875
Factor Coefficient Standard t-Statistic P-Value
Error
[30 (g fat/g dry matter) 0.3459 0.0339 10.2141 <0.001
B1 (g fat/g dry matter)-°C'1 -0.0060 0.0011 -5.6101 <0.001
02 (g fat/g dry matter)-°C'2 3.28E-05 8.67E-06 3.7842 <0.001
83 (g fat/g dry matter)'s" -00057 0.0032 -1.8057 0.078
[34 (g fat/g dry matter)-s'2 0.0002 0.0002 1.3904 0.171

05 (gfat/gdrymatter)°°C’l-s'l 2.96E-05 2.86E-05 1.0368 0.305

111

Heating temperature was found to affect the fat holding capacity of the meat
(P<0.05). The heating time and time-temperature interaction terms of the response
surface model were not significant. A second regression was performed, neglecting the

time and time-temperature interaction terms (Equation [5.4]).
F=BO+BI.T+8,.T2+2 [5.4]

The regression coefficients of the modified response surface model are given in Table

5.14.

Table 5.14 — Results from linear regression of fat holding capacity versus holding
temperature for 5.6% fat ground beef.

(F=[i,,+[ll -T+[12-'l‘z +8)

 

Degrees of Sum of Mean F-Value P-Value
Freedom Squares Square

 

 

 

 

Regression 2 0.0691 0.0346 92.2654 2.43E-l7
Residual 49 0.0184 0.0004
Total 51 0.0875
Factor Coefficient Standard t-Statistic P-value
Error
I30 (g fat/gdrymatter) 0.3178 0.0295 10.7649 <0.001

B: (g fat/gdrymatter)-°C" -0.0057 0.0011 -5.4687 <0.001
[32 (gfat/gdrymatter)-°C'2 3.24E-05 8.69E—06 3.7254 <0.001

The regression model shows the fat holding capacity of the meat to be inversely
proportional to the heating temperature. The drop off in fat holding capacity between 50

and 55°C is likely related to melting of fat globules within the meat matrix. The melting

112

temperatures of the most prevalent saturated fatty acids in ground beef occur near this
range, with the melting temperatures of myristic, palmitic, and stearic acids being 54, 63,
and 70°C, respectively (Bodwell and McClain, 1978). Ground beef with a fat content of
only 5.6% contains only small quantities of extra-muscular fat and thus should be
expected to contain lower percentages of saturated fatty acids. This could explain the
lack of further losses in F HC at temperatures above 60°C.

The lack of time-significance seems to support the idea that fat holding capacity is
largely related to the physical state of the fat itself. Since fat melting occurs over a short
period, once the fat has melted, further heating should not affect the amount of liquid fat.
If fat holding capacity changes were driven primarily by changes in protein
conformation, denaturing, etc., longer heating times would be expected to produce
increasingly lower levels of fat holding capacity.

A comparison of the fat holding capacity values predicted by the regression
equation and the experimental data resulted in an R2 of 0.79 (Figure 5.17). The root

mean square error (RMSE) of the regression was 0.02 g fat/g dry matter.

113

 

 

 

 

0.25

E 0.2

(U

E C o

S 0.15 -2 °
~ 9 o

g” ’ o o 90

d3 01 00 o .. .

£9 ' ’00.. .0

g “ «A: 0.

*5 0.05 --

'6 .

g I

an O I I I I
0 0.05 0.1 0.15 0.2 0.25

Experimental F (g fat/ g dry matter)

Figure 5.17 — Comparison of fat holding capacity calculated from regression model
versus experimental values: means of 5 replicates.

5.3.2 High fat samples (15% initial fat wet basis)

The effect of temperature on fat holding capacity was much more pronounced for
the 15% fat meat than for the 5.6% fat meat (Figures 5.18-5.21). The initial fat content of
the 15% fat meat was 0.6 g fat/g dry matter. Fat holding capacity decreased with
temperature and ranged between 0.07 and 0.17 g fat/g dry matter for samples heated to
90°C.

Multiple linear regression was again used to model fat content as functions of
heating temperature and holding time, using a quadratic response surface equation

(Equation [5.3]). The regression coefficients for the response surface equation are given

in Table 5.15.

114

.o .o
\l 00
I—f‘r—r r-v

I

F (g fat / g dry matter)

95:99.09
Nw-hUIO

.—
rrI-Tr

 

 

o
o
o o
o 9 ° , o
. o
o
1___L._+___L. __ _1__+_ J.__1..__L___L__..+_ _._ 1___L

40 60 80 100

Temperature (°C)

Figure 5.18 — Fat holding capacity as a function of temperature for 15% fat ground
beef heated for 2 minutes at temperatures from 30 to 90°C: means of 5 replicates.

 

F (g fat / g dry matter)
O O O O O O O

 

 

 

0 o

o

9 ° .
Q 0 O .

o
0 o
.1 .+_ 4 Al___.L___.l_ _1_ i 1 1___L.__l
40 60 80 100

Temperature (0 C)

Figure 5.19 - Fat holding capacity as a function of temperature for 15% fat ground
beef heated for 5 minutes at temperatures from 30 to 90°C: means of 5 replicates.

115

I
I
I
I
I
I
I
I
I
I
I
I
I
J

.9
ox

 

I
c I
Q) [r .
E I- ’ I
s I
3" 0.4 I .
E 0.3 I 9
89 r ’
LL. 0.2 5* 9 .
I . . .
0.1 I
“ o o
0 HI. 1‘; 4 ' __.L 4 4+. .1.______L_ 1__1 +4- «1._ __L I
20 40 60 80 100
Temperature (° C)

Figure 5.20 — Fat holding capacity as a function of temperature for 15% fat ground
beef heated for 10 minutes at temperatures from 30 to 90°C: means of 5 replicates.

 

 

0.8 I . - - - hi -~— I
0.7 I

I .

I ’ .

99
L110
+44

F (g fat / g dry matter)

iota):-

p O O O
O
O
O
O
O
0

fl

 

 

 

0 .f ____1__L_L__J-__ +._____1_.__1____1__-L_ + 1 1 I 1 1 1 1 ___I

20 40 60 80 l 00
Temperature (°C)

 

Figure 5.21 - Fat holding capacity as a function of temperature for 15% fat ground
beef heated for 15 minutes at temperatures from 30 to 90°C: means of 5 replicates.

116

Table 5.15 — Results from regression of fat holding capacity versus time and holding
temperature for 15% fat ground beef.

(F=Bo+B1'T+B2 'T2 +33 't+B4’t2 +35 -t-T+£)

 

 

 

 

 

Degrees of Sum of Mean F-Value P-Value

Freedom Squares Square
Regression 5 1.4071 0.2814 54.8295 <0.001
Residual 47 0.2412 0.0051
Total 52 1.6483

Factor Coefficient Standard t-Statistic P-value
Error

[30 (g fat/g dry matter) 1.3497 0.1237 10.9131 <0.001
01 (g fat/g dry matter)-°C'l -0.0322 0.0039 -8.1558 <0.001
[3; (g fat/g dry matter)-°C‘2 0.0002 3.19E-05 6.5391 <0.001
[3. (g fat/g dry matter)°s" 0.0155 0.0117 1.3221 0.193
[34 (g fat/g dry matter)°s'2 -0.0006 0.0006 -1 .0612 0.294
135 (g fat/g dry matter)-°C"-s" -0.0001 0.0001 -1.0881 0.282

 

Heating temperature again affected fat holding capacity (P<0.05).

However, as

with the lower fat beef, holding time and time-temperature interaction were not

statistically significant. A modified regression was performed omitting the terms for

holding time and time-temperature interaction (Equation [5.5]).

[5.5]

The regression coefficients of the modified regression are shown in Table 5.16.

117

Table 5.16 — Results of regression of fat holding capacity versus holding
temperature for 15% fat ground beef.

(11:13o +13, -T+p, -T’ +2)

 

Degrees of Sum of Mean F-Value P-Value
Freedom Squares Sguare

 

 

 

Regression 2 1.3922 0.6961 135.9025 <0.001
Residual 50 0.2561 0.0051
Total 52 1.6483
Factor Coefficient Standar t-Statistic P-Value
d Error
[30 (g fat/gdrymatter) 1.4163 0.1085 13.0474 <0.001

8. (g fat/gdrymatter)'°C" -00329 0.0039 -8.5302 <0.001
112 (gfat/gdrymatter)-°C'2 0.0002 3.191305 6.4820 <0.001

The fat holding capacity of the 15% fat ground beef was inversely proportional
to the heating temperature. The effect of heating temperature was much more
pronounced for the 15% fat ground beef than for the 5.6% fat product. The coefficient
for the linear temperature term in the regression equation was —0.0329 (g fat/g dry
matter)/°C for 15% fat ground beef, compared to —0.0057 (g fat/g dry matter)/°C for
5.6% fat ground beef.

Like the low fat samples, the largest change in fat holding capacity for the 15%
fat samples occurred between 40 and 55°C. However, unlike the lower fat samples, fat
holding capacity of the 15% fat samples continued to decrease at temperatures above
55°C. This was likely due to the inclusion of higher quantities of long chain saturated
fatty acids.

The lack of time-dependence of the fat holding capacity was similar to that

exhibited by the lower fat samples. This further supports the idea that fat holding

118

capacity, as measured in this study, is governed primarily by the physical state of the fat

globules within the meat matrix. This is of special interest when formulating models for

mass transfer, as the physical state of the fat must be considered. Clearly, solid fat does

not have the transportability of liquid fat. However, it may be possible to develop highly

advanced fat transport models based upon knowledge of the individual fat constituents of

a given species and cut of meat. This would allow for maximum flexibility of advanced

processing models.

A comparison of the fat holding capacity values predicted by the regression model

(Equation [5.5]) to the experimental data resulted in an R2 of 0.84 (Figure 5.22). The

RMSE of the regression model was 0.071 g fat/g dry matter.

 

 

 

 

l
33‘
£08
000.6 .
E I 0
39 __ o «o
1.1.. 0.4 00 .
E m e
o O O O
:5 0.2—— o
g I“... ”r
0* II ““+I““I
0 0.2 0.4 0.6 0.8 1

Experimental F (g fat/ g dry matter)

Figure 5.22 - Comparison of fat holding capacity calculated from regression model
versus experimental values (15% fat): means of 5 replicates.

119

After completing separate regression analyses for the 5.6% and 15% fat ground
beef samples, a combined regression was performed to create a model for fat holding
capacity as a function of heating temperature, holding time, and initial fat content. A
response surface model was chosen for the regression (Equation [5.6]). No second- order
terms for fat content were utilized in the regression equation, because data were only
available from two initial fat contents. The coefficients for the regression model are

listed in Table 5.17.

F=Bo+131°T+Bz°T24433444344205'F0+I36't'T+B7'T'FO 15.61
+Bg°t'F+8

120

Table 5.17 — Linear regression of fat holding capacity as functions of heating
temperature, holding time, and initial fat content.

F=B0+Bl “T7432 'T2 +53 't'I'B4 't2 +65 'Fo +56 't'T'I‘B-I °T'Fo
+Bs-t-Fo+a

 

 

 

 

 

Degrees of Sum of Mean F-Value P-Value
Freedom Squares Square
Regression 9 2.1005 0.2339 65.0322 <0.001
Residual 95 0.3409 0.0036
Total 104 2.4414
Factor Coefficient Standard t-Statistic P-value
Error
[30 (g fat/g dry matter) 0.3334 0.1104 3.0193 0.003
B! (g fat/g dry matter)°C'l -0.0136 0.0027 -5.037 <0.001
[3; (g fat/g dry matter)°C'2 0.0001 1.90E-05 6.3768 <0.001
[3, (g fat/g dry matter)°s" -0.0042 0.0111 -0.3804 0.704
[34 (g fat/g dry matter)-s'2 -0.0002 0.0003 -0.4541 0.651
[35 (none) 0.0503 0.0080 6.3074 <0.001
8. (g fat/g dry matter)-°C"°s" 0.0001 0.0002 0.7967 0.428
[37 (°C") -0.0005 0.0001 -4.2171 <0.001
Ba (8") 0.0008 0.0008 0.9458 0.347

 

None of the terms containing holding time were significant (P<0.05). A modified

regression was performed excluding the time terms (Equation [5.7]).
[ 5.7 ]

F=Bo+131°T+I32'T2+I33°F0+I34'T°F0+8

The coefficients of the modified regression are shown in Table 5.18.

121

Table 5.18 - Linear regression of fat holding capacity as a function of holding
temperature and fat content.

(F=11.+9. -T+11. -T’ +11. ~F.+11. -T-F.+e)

 

Degrees of Sum of Mean F-Value P-Value
Freedom Squares Square

 

 

 

 

Regression 4 1.9767 0.4942 106.330 <0.001
Residual 100 0.4647 0.0046
Total 104 2.4414
Factor Coefficient Standard t-Statistic P-value
Error
[30 (g fat/g dry matter) 0.7062 0.0746 9.4719 0.006

B. (g fat/g dry matter)-°C'l -0.0193 0.0026 -7.3996 <0.001
[3; (g fat/g dry matter)-°C'2 0.0001 2.16E-05 5.0990 <0.001
[3; (none) 0.0069 0.0020 3.4355 <0.001
[3.. (°C") 0.0002 3.41E-05 7.1495 <0.001

A comparison of the fat holding capacity values predicted by the regression and
the experimental data produced an R2 of 0.81 (Figure 5 .23). The RMSE for the

regression was 0.068 g fat/g dry matter.

122

 

.6 .0
\l 00
1

0.6 I

lllTlI-T‘Irfi

.9
LII
1

0.3 I Q 9
: O
0.2 I; 9

Predicted F (g fat/ g dry matter)
O
.b
o

O
p—A
1
1

 

 

1 4 g 1 r I
I

0.2 0.4 0.6 0.8
Experimental F (g fat/ g dry matter)

0
O'I

Figure 5.23 - Comparison of fat holding capacity calculated from regression model
versus experimental values: means of 5 repetitions.

5.3.3 Summary

The regression models presented for fat holding capacity provide a tool that can
be utilized as a component of the cooking model. However, it should be noted that this
results in an empirical fat transfer model. The equations developed for fat holding
capacity are specific for ground beef. Further studies should be conducted to measure the
FHC of other meat species. In addition, mechanisms of fat transport should be firrther
studied to produce a theoretical model for FHC.

The methods used to determine fat holding capacity were considered adequate for
the purposes of this study. However, several factors should be taken into consideration
for further experiments. The most probable source of error for the FHC experiments was
in the centrifugation procedure. The centrifuge utilized for the study was held at 25°C.

Although the centrifuge tubes were insulated from the rotor by plastic inserts, some re-

123

solidification of fat took place during centrifugation. To eliminate this potential source of
error, a method of separating the free fat while maintaining the sample temperature
should be determined. It may also be desirable to measure the fat content of the meat
directly using a method such as solvent extraction rather than the method of mass

balances utilized in this study.

5.4 Industrial cooking tests

A set of cooking experiments was conducted using an industrial moist-air
impingement oven (Stein Model .1 SO-IV:F MC Foodtech, Sandusky, OH). Ground beef
patties were cooked under different cooking conditions to quantify the effects of process
conditions on heating rate and cooking yield. The results of this set of experiments were

also used to validate the computer-cooking model.

5. 4.] Cooking time

For each cooking experiment, temperature versus time was plotted for both the
center and surface temperatures of the patty (Figure 5.24). The complete collection of
temperature-time plots can be found in Chapter 9. The general form of each time-
temperature plot was similar. After entering the oven, the surface temperature of the
patty quickly rose to a semi-equilibrium level. Calculations showed that this level was
equal to the wet bulb temperature of the meat surface (Section 4). The temperature at the
meat surface was limited by a number of factors, including oven temperature, oven steam

content, and the surface moisture content of the meat.

124

(a)

 

  

100 .
9O Surface
805% W
70 I
60 If.
f1

   

Center

Temperature (°C)
KI!
o

 

 

 

 

20 I
10 .
0 I ‘ ‘ . 1 ' . II I 1
0 50 100 150 200
Time (s)
(b)
100 .
90 Surface

\100

OO
1

ttr

ox
o
it

00

1
TYTTWYI'TIIII

  

Temperature (° C)
W -b M

   

Center

 

 

 

20 g
10
0 F 1 I 1 . . . I
0 50 100 150 200
Time (s)

Figure 5.24 - Example surface and center temperature versus time for ground beef
patties cooked at (a) oven temperature: 121°C, oven steam content: 50% by volume,
oven airflow: 11.4 m/s and (b) oven temperature: 232°C, oven steam content: 50%
by volume, oven airflow: 16.8 m/s.

125

The center temperature of the meat increased at a much slower rate than the
surface and approached the surface temperature asymptotically. For most cooking
experiments, the center temperature of the patty was at least 10°C below the surface
temperature at the end cooking.

Several revelations about the cooking process can be inferred from the
experimental data. The first is that increasing the oven temperature does not
proportionally increase the heating rate of the patties. The wet bulb temperature always
limits the surface temperature of the patty. With the exception of extremely low surface
moisture contents, the temperature of the patty surface cannot exceed the wet bulb
temperature of the cooking air. The moisture content at which surface temperature will
exceed the wet bulb temperature is a function of the equilibrium relative humidity of the
meat. Under the conditions tested, this moisture content was between 5 and 7% wet
basis. No observations of surface temperature exceeding 100°C were made for any of the

54 experimental cooking trials.

126

Because the heating rate at the center of the patty is controlled by the thermal
gradients between the center of the patty and the patty surface, the patty surface
temperature is the limiting factor for cooking time. Multiple linear regression was used
to describe center temperature as a function of oven temperature, steam content, cooking
time, and airflow. Table 5.19 shows the results of the regression. As expected, steam
content, and cooking time were significant (P<0.05). A positive correlation existed
between center temperature and steam content and cooking time. Not-surprisingly,
cooking time had the largest effect on center temperature. More interesting however, is
that oven steam content had a much more pronounced effect on center temperature than
did oven temperature. The effect of steam content resulted in a potential temperature
difference of up to 9.8°C over the range of steam contents tested, as compared to a non-
significant effect of oven temperature. This compares to a difference of only 3.4°C
related to oven temperature, indicating that oven steam content has a much greater effect
on the surface wet bulb temperature of the patty than oven temperature. This observation

should be taken into account when working to optimize oven settings.

127

Table 5.19 - Linear regression of patty center temperature as functions of oven
temperature, steam content, cooking time, and oven airflow.

(T=BO+BI .Toven +32 .M+B3 't+B4.Valr)

 

Degrees of Sum of Mean F-Value P-Value
Freedom Squares Square

 

Regression 4 6548.757 1637.189 35.6310 <0.001
Residual 50 2297.425 45.9485
Total 54 8846. 182

 

 

Factor Coefficient Standard t-Statistic P-value

 

Error
Bo °C 30.3786 8.0882 3.7559 <0.001
01°C/°C 0.0306 0.0228 1.3391 0.187
B2 °C°%Steam" 0.2583 0.0641 4.0275 <0.001
B3 °C-s" 4.1720 0.4146 10.0639 <0.001
8. °C-s-m" -0.0218 0.2194 -0.0993 0.921

 

Oven air velocity did not significantly affect the patty center temperature. The
temperature gradients that drive conduction within the patty result from increases in the
patty surface temperature. Although increasing the airflow has the effect of increasing
the heat transfer coefficient at the patty surface, corresponding increases in the mass
transfer coefficient result in increased evaporative cooling in the later stages of cooling.
The result was a zero net gain in the heating rate of the patty for the conditions tested.

During the early stages of cooking, increases in airflow may increase the rate of
condensation at the patty surface, thereby temporarily increasing the heating rate.
However, as illustrated in the heating curves, the surface temperature reaches equilibrium

within about 10 seconds for the range of oven conditions tested. The marginal gains in

128

surface temperature heating rate caused by increasing airflow did not significantly

increase the rate of heating at the patty center.

5. 4.2 Cooking yield
Figure 5.25 shows the relationship between patty center temperature and cooking
yield for ground beef patties cooked in the J SO-IV impingement oven. Cooking yields

decreased as a function of patty center temperature.

 

1
I

100.0 2' ,- mam"- _- i“,
95.0
90.0
85.0 .
80.0
75.0 3: .00
70.0 .
65.0 a: o
60.0 7E . .9 .8
55.0 o
50.0 -

40 50 60 70 80 90 100 110

Y T

Cooking yield (percent)

 

 

 

Center temperature (°C)

Figure 5.25 - Cooking yield as a function of endpoint center temperature for ground
beef patties cooked in a Stein JSO-IV industrial moist air impingement oven.

The results of linear regression are shown in Table 5.20. Oven temperature,
cooking time, and oven steam content were each found to significantly affect patty yield
(P<0.05). Cooking time, oven temperature, and oven steam content each had a negative

effect on cooking yield. Cooking time had the strongest effect on yield. Oven

129

temperature and steam content each had an effect approximately one order of magnitude
below the effect of cooking time. Oven temperature created a potential difference in
cooking yield of up to 5.6% over the range of temperatures used. Steam content had an

effect equal to 4.8% yield over the range of steam contents tested.

Table 5.20 — Regression parameters for cooking yield as a function of oven
temperature, steam content, cooking time, and airflow.

(Y=Bo +Bl .Tovel +B2 .M+B3 .t+p4 .valr)

 

 

 

 

 

Degrees of Sum of Mean F-Value P-Value
Freedom Squares Square
Regression 4 3836.668 959.1669 50.0733 <0.001
Residual 50 957.7628 19.1553
Total 54 4794.43
Factor Coefficient Standard t-Statistic P-value
Error
[30 109.6251 5.2223 20.9917 <0.001
[31 °C'l -0.0503 0.0147 -3.4133 0.001
B; °o Steam’l -0.1259 0.0414 -3.0397 0.004
[33 s'2 -3.4932 0.2677 .13.0507 <0.001
8. s-m" -00453 0.1416 -0.3201 0.750

 

5.4.3 Fat loss

For each patty, the cooking loss not accounted for by moisture loss was calculated
and assumed to be entirely due to fat loss. The amount of fat lost increased with center
temperature (Figure 5.26) and ranged from less than 1% of the initial mass at 50°C up to
almost 10% of the initial mass at 95°C. This accounts for nearly all of the fat initially

present in the patty. The results of linear regression for fat loss as a function of oven

130

parameters are shown in Table 5.21. Only cooking time significantly affected the yield

lost due to fat. Clearly, the effects of fat loss must be taken into account to accurately

model yield losses during cooking.

Table 5.21 - Regression parameters for fat loss as a function of oven temperature,
steam content, cooking time, and airflow.

(AYfat =fi0 +51 °Toven +32 'M+B3 't+B4 'valr)

 

Degrees of Sum of Mean F-Value P-Value
Freedom Squares Smiare

 

 

 

 

Regression 4 88.1928 22.0482 7.7278 <0.001
Residual 22 62.7679 2.8531
Total 26 150.9607

Factor Coefficient Standard t-Statistic P-value

Error

130 -3.0612 2.9207 -l.0481 0.306
13l (°C") 0.0070 0.0081 0.8722 0.393
132 (% Steam") 0.0464 0.0230 2.0188 0.056
B3 (s") 0.0118 0.0025 4.7997 <0.001
134 (s-m'l) -0.0161 0.0771 -0.2085 0.837

 

131

 

10.0 -

 

 

 

>. r
.o 9 9.0 7:: .
H T.
‘2 T3 8.0 7E: .0 .
1:: >4 : 0
{=3 :9 70 a . o o 09.

c f

o
8 3 5 0 f O .
(U a . ..
H o 4:.
g T. 4.0 . .
"’ H J-
E a 3.0 *7 . 9
2 5 2.0 1‘ 0
0 :.
i; E 10 ‘? . .0
0.0 7' . . . . a 4 - 4
40 60 80 100
Center temperature (°C)

Figure 5.26 - Yield loss not accounted for by moisture loss as a function of endpoint
temperature for ground beef patties cooked in a Stein JSO—IV industrial moist-air
impingement oven.

5. 4.4 Volume change

Patties exhibited a reduction in volume during cooking. The volume change was
primarily due to dramatic reductions in the diameter of the patties during cooking. The
diameter reduction as a function of endpoint center temperature is shown in Figure 5.27.
The diameter of the patties decreased linearly as a function of center temperature. The
thickness of the patties did not exhibit the same behavior. The thickness of the patties
remained relatively unchanged during cooking, with approximately half of the patties
undergoing slight decreases in thickness, and the other half undergoing slight increases.
The average absolute change in thickness was 0.76 mm. The mean change in thickness

was —0.14 mm.

132

40

 

35 o
30 7 9’ O
z ‘ . °
25 ‘. .0. ~ 0

g . ‘2'.
20 o

15_ ...

Percent reduction in diameter

fl r

10- o
_ o

 

 

 

a
_ 1 1 1 1 1 1 1 1 1 l 1 - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
l 1 I I . l

40 50 6O 70 80 90 100 1 10

Center temperature (° C)

Figure 5.27 - Reduction in diameter during cooking as a function of cooking yield
for ground beef patties cooked in a stein J SO-IV industrial moist-air impingement
oven.

The fact that the diameter of the patties changes so drastically during cooking,
while the thickness stays relatively unchanged, presents a challenge for modeling
cooking. To model cooking as completely as possible, volume changes should be
incorporated into the model. However, for patties such as those utilized in these
experiments, heat and mass transfer occur primarily in the axial direction, due to the
height/diameter ratio of the patties. Since patty thickness remains relatively unchanged,
changes in volume do not affect heat and mass transfer dramatically from the standpoint

of modeling. However, further research in this area could shed light on the changing

physical conditions that occur within meat during cooking.

133

5.5 Cooking model validation

Validation of the cooking model was conducted using the data collected from
cooking experiments with a Stein J SO-IV moist-air impingement oven (Sections 3.4 and
5.4). Validation was conducted for temperature, moisture, and yield. Additional
comparisons were made using yield and temperature data from published sources

(Murphy et al., 2001a and b).

5. 5. 1 Finite element mesh

The finite element mesh utilized for cooking model validation consisted of 116
triangular elements with a total of 78 nodes. The element mesh was shown graphically in
Chapter 4. A total of 28 nodes were located along the convective boundaries of the patty
geometry. Mesh density was lowest at the center of the patty and increased near the
surface where the temperature and moisture gradients were expected to be highest.

The element mesh utilized was chosenoto balance solution accuracy with
computing time. Increasing the number of elements from 48 to 116 lowered the transient
standard error of prediction for the center temperature of a test experiment from 11.6°C
to 55°C. This increase in accuracy came at the cost of computing time. The 48 element
model only took 17 seconds to complete, compared to 75 seconds for the 116 element
model. Increasing the number of elements from 116 to 234 did not improve the accuracy
of the temperature prediction. However, the computing time required to run each test
was increased to 295 seconds. This clearly illustrates that increasing the mesh density

comes at a major cost in computing time. This is of importance to users utilizing the

134

model to simulate large numbers of conditions for applications such as process
optimization.

The value of the time step selected for the model was 1 second. This value was
utilized to aid in comparisons with experimental data, which was recorded at one-second
intervals. Unfortunately in certain cases, the one-second value for the time step may
result in numerical oscillations if a constant element mesh size is utilized. Automatic
routines for mesh generation and time step optimization could eliminate this potential

problem.

5 . 5 .2 Temperature profile-experimental data

Validation of temperature profiles was conducted by comparing transient center
temperature data collected during the experimental tests to corresponding predictions
generated by the model. Example center temperature profiles are shown in Figures 5.28-

5.30. All experimental and predicted temperature results are graphed in Chapter 9.

135

 

100

1 1
Ittrrt—r—I—rrW—v—rTrt—rrrvv—T—r

 
  

 

Temperature (° C)
(I.
o

 

 

 

 

30 ~—-- . 1
20 lg: I Expenmenta
10 1E —— Model
0 _ t 4 t ' 1 ‘ 1 1 . 1’7.
0 50 100 150 200 250

Figure 5.28 — Example comparison of experimental temperature data with data
generated by the cooking model (oven temperature=121°C, steam content=50%, air
velocity=11.4 m/s).

 

100 :
90
80
70
6O
50
40

  
  
 
  
 
 

 

Temperature (° C)

I Experimental

L".— Model

 

 

 

 

Time (s)

Figure 5.29 - Example comparison of experimental temperature data with data
generated by the cooking model (oven temperature=121°C, steam content=70%, air
velocity=11.4 m/s).

136

 

100
90
80
70
60
50
40
3O
20
10

   

Temperature (° C )

 

 

I Experimental
—- Model

0 100 200 300 400
Time (s)

 

 

 

 

Figure 5.30 — Example comparison of experimental temperature data with data
generated by the cooking model (oven temperature=121°C, steam content=88%, air
velocity=11.4 m/s).

Figure 5.28 is typical of most comparisons in several respects. Noticeable
deviation between the model and experimental data occurred during the early stages of
cooking. During the later stages of cooking, profiles for the model and experimental data
were very close. Although the level of agreement between the model and experimental
data varied between cooking runs, the phenomenon of the predicted center temperature
lagging behind the measured temperature during the early stages of cooking was common
to all cooking experiments. These results were similar to those reported by Pan et al.
(2000) who found that temperature values lagged below predicted values for contact
cooking of hamburger patties. These deviations were largest between 0 and 40°C.
Disparities in the sample cooking curve provided by Pan et al. (2000) were very similar

to the predicted error shown in Figure 5.28.

137

Based on Figure 5.28, several conclusions can be made. First, it is possible that
the deviation between the temperature profile predicted by the model and the
experimental data was caused by inaccurate predictions of thermo-physical properties in
the model. The thermal conductivity may have been underestimated at low temperatures.
The model also may have overestimated the heat capacity of the product at low
temperatures.

A second possibility for the deviation between the model and the experimental
temperature profiles is the effect of volume change. The model did not take into account
volume change during cooking. If the thickness of the patty changed significantly during
cooking, the rate of heat transfer to the center of the patty may have changed, even under
conditions of constant thermal properties. However, as discussed in Section 5.4.4, the
thickness of the patties remained constant during cooking, with significant shrinkage
occurring only in the radial direction. Because heat transfer occurred primarily in the
vertical direction, it is unlikely that shape changes contributed significantly to error in the
model for the product tested.

Figures 5.29 and 5.30 illustrate a more pronounced deviation between the model
and experimental temperatures during the first minute of cooking. This type of deviation
was seen in about half of the 54 cooking trials. Typically, the deviation consisted of an
increase in measured temperature above the predicted value followed by a drop in
temperature to a level consistent with the predicted value. This deviation typically
occurred over a temperature range of 0 to 40°C. At temperatures above 40°C, the
experimental temperature profiles then closely matched the values predicted by the

cooking model.

138

One possibility for the disparity between the model and experimental data in the
early stages of cooking is that the thermocouple measuring center temperature may have
moved as the meat began to cook. Changes in product texture, along with diameter
changes occurring during cooking, would explain why the thermocouple could move
during the early portion of cooking. However, this explanation does not account for the
resolution between the measured and predicted temperature values at higher
temperatures.

The possibility that the author considers most likely is that fat-related effects were
the cause of these deviations. From the fat holding capacity experiments in Section 3.3,
large changes in fat holding capacitywere seen in the temperature range between 40 and
50°C. The laboratory experiments in Section 3.2 also show that yield loss is roughly
linear at temperatures above 45°C. Most fat is in a form that is available for transport at
temperatures above 45°C.

Prior to heating, ground beef consists of a mixture of ground fat and muscle
particles. Unlike whole muscle products, the fat particles are dispersed throughout the
meat, rather than in their naturally occurring structures. In this sense, the fat particles
exist as a dispersed phase within a matrix of lean meat. When the fat particles are in
solid form, physical forces serve to keep them in suspension within the meat. However,
upon melting, the fat is free to exit the meat through capillary mechanisms.

It is proposed that during the initial phases of cooking, changes in the meat
structure occurred that resulted in the thermocouple not giving an accurate representation
of the average temperature in the center of the patty. Due to the near-frozen condition of

the meat before cooking, the meat had a very firm initial texture. As a result, when the

139

thermocouple was inserted into the patty, a minute void space may have been created
around the thermocouple, which would have provided a channel for condensing water to
penetrate to the center of the patty, and thereby increase the thermocouple temperature
above that of the meat. At temperatures above 40-50°C, constriction of the meat and
filling of internal voids by melting fat would result in more intimate contact between the
thermocouple and the meat matrix, thereby resulting in more accurate temperature
readings. The vertical orientation of the thermocouple may have contributed to this type
of effect by providing an uninterrupted vertical channel between the bottom and center of
the patty.

This theory would explain the deviations seen in Figures 5.29 and 5.30. It is also
highly likely that the deviation in Figure 5.28, although less pronounced than the others,
was caused by a similar effect. In the temperature curves where significant deviations
occur between the measured and predicted center temperature values, the measured
center temperature value appears to temporarily move in the direction of the surface
temperature of the patty, before dropping back down to a level more consistent with the
predicted value. This seems to support the hypothesis that condensing water at the
surface temperature may be penetrating the patty along the thermocouple “channel”.

Unfortunately, accurate surface temperature values were not available for many of
the cooking trials due to the difficulty in maintaining uninterrupted contact between the
thermocouples and the patty surface during cooking. The high airflow of the oven,
combined with changes in patty geometry, often separated the thermocouples from the
patty surface. For this reason, surface temperature data are missing from many of the

figures in Section 9.1. Due to inconsistentcies in the surface temperature data, it was not

140

considered meaningful to calculate transient SEP for the surface temperature of each
model run. Instead, general observations of the predicted and measured surface
temperatures were made.

For each simulated cooking run, the surface temperature of the meat quickly rose
to a semi-equilibrium value. This temperature reflects the wet bulb temperature of the
oven air. This temperature never exceeded the boiling point of water for any of the
patties tested. For patties in which accurate surface temperatures are available, the
predicted surface temperatures corresponded closely with the experimental values. In
many cases, the measured surface temperature values were considerably different from
the predicted values. However, it is the opinion of the author that these cases represent
situations in which the thermocouple was not in contact with the actual surface of the
patty. In some cases the thermocouple may have been lodged slightly below the surface
of the patty. These cases illustrate the difficulty inherent in measuring the surface
temperature of meat patties within commercial convection ovens.

Standard error of prediction was calculated for the center temperature data of each
cooking trial, using temperature data collected at 1 second intervals from both the model
and experiments. The SEP of the transient center temperatures for individual trials ranged

from 2.1 to 139°C (Table 5.21). The overall SEP for all of the cooking trials was 80°C.

The SEP for the final center temperature of all of the patties was also 80°C. The error in
the temperature curves was generally concentrated in the early portion of the curves. The
cooking experiments that were run for short times had the largest errors. As the patty
center reached high temperatures, the differences between the model and experimental

data became small. A second set of SEP data was generated for the portion of each

141

cooking run above 45°C (Table 5.22). These SEP’s ranged from 0.5 to 108°C, a large
improvement from the values taken over the entire temperature range. The overall SEP
for all data points above 45°C was 58°C. The trials with large SEP values for
temperature were generally the experiments with the shortest cooking times. In these
tn’als, deviations between the model and experimental data at low temperatures affected a
larger percentage of the total cooking time, thus resulting in larger SEP between

experimental and predicted data.

142

Table 5.22 — Standard error of prediction for the entire trial (SEP) and for data
above 45°C (SEPNsoc) for center temperature of beef paties.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EXP- SEP (°C) SEP1>45°C (°C) EXP- SEP (°C) SEPT>45°C (°C)
# (data points) (data points) # (data points) (data points)
la 5.5 (193) 1.2 (128) lb 4.7 (157) 5.1Q3)
6a 3.9 (416) 4.1 (330) 6b 3.3 (418) 3.1 (324)
8a 2.4 (300) 1.2 (214) 8b 6.9 (303) 3.4 (247)
11a 6.9 (294) 2.8 (220) 11b 8.9 (295) 3.6 (244)
13a 4.0 (187) 3.1(115) 13b 12.5 (189) 9.6 (152)
18a 4.1 (3%) 3.6 (340) 18b 7.6 (396) 1.9 (341)

21a 5.5 (368) 2.4 (298) 21b 7.1 (369) 1.3 Q94)
23a 3.0 (320) 3.g229) 23b 5.1 (301) 4.3 (214)
25a 5.1 (L41) 4.1 (60) 25b 8.6 (117) 6.3 (60)
30a 3.2 (276) 2.1 (177) 30b 6.8 (280) 3.3 (206)
32a 4.2 (224) 1.7 (126) 32b 14.7 (224) 12.3 (196)
34a 5.3 (125) 2.5 (29) 34b 12.5 (116) 9.1 (68)
37a 7.1 (105) 0.5 (25) 37b 23.8 (108) 21.5 (69)
47a 2.1 (264) 1.7 (187) 47b 9.74273) 5.9 (206)
49a 8.0 (114) 9.5 (26) 49b 16.5 (120) 8.0 (27L
50a 9.5 (240) 7.7 (125) 50b 3.6 (240) 2.5 (129)
54a 13.5 (304) 10.8 (198) 54b 7.3 (313) 6.9 (254)
56a 6.3 (188) 5.5 (112) 56b 17.8 (186) 12.4 (119)
58a 13.1 (118) 7.1 (21) 58b 5.5 (116) 8.0 (23)
63a 7.2 (259) 5.1 (190) 63b 4.5 (262) 2.1 (169)
66a 4.0 (256) 1.8 (177) 66b 5.4 (260) 5.7 (161)
68a 2.8 (186) 0.8 (112) 68b 7.5 (190) 8.1 (83)
70a 7.1 (75) * 70b 12.0 (70) 9.9 (11)
73a 10.7 (77) * 73b 8.8 (73) 2.4 (5)
75a 8.0 (259) 2.3 (195) 75b 13.2 (259) 12.4 (235)
78a 2.2 (191) 0.9 (98) 78b 2.7 (192) 2.2 (109)
80a 8.3 (191) 2.5 (114) 80b 11.4 (192) 8.9 (161)

 

* Center temperatures did not exceed 45°C

5 .5 .3 Temperature profile-published data

 

A total of 12 model runs were conducted to simulate the conditions utilized in the
experiments of Murphy et al. (2001a). The center temperature profiles of each model run
were compared to temperature predictions from the regression equation developed by

Murphy et al. (see Section 3.5.3). Transient comparisons were made between the center

143

temperatures predicted by the model and the Murphy et al. regression equation for the
temperature range between 55 and 80°C. The model and regression transient temperature
data were compared at 1 second intervals. Standard error of prediction for transient

center temperature was calculated for each model run (Table 5.23).

Table 5.23. Standard error of prediction for transient center temperature of ground
chicken breast patties predicted by the model and by the regression equation of
Murphy et a1. (2001 a).

 

 

 

 

 

 

 

 

 

 

 

 

 

Dry Bulb Steam by Airflow SEP Center
Temperature (°C) Volume (m/s) Temperature (°C)
149 6 1.53 10.3
149 6 2.13 12.4
149 6 2.73 13.5
149 25 1.53 1.4
149 25 2.13 1.6
149 25 2.73 2.4
149 60 1.53 2.6
149 60 2.13 1.6
149 60 2.73 1.8
149 91 1.53 3.7
149 91 2.13 2.8
149 91 2.73 3.2

 

 

 

 

 

 

For the wet bulb temperatures of 70 to 95°C, the SEP ranged from 1.1 to 61°C.
This was comparable to the SEP of the experimental data collected from the Stein J SO-
IV oven. However, for the driest cooking air condition (wa=40°C), the SEP ranged from
13.3 to 157°C. This indicates that the cooking model may not be reliable for extremely
dry air conditions. This was probably due to the equation for equilibrium relative

humidity. However, the cooking model was designed for high moisture impingement

144

ovens, so deviations at extremely dry conditions are not critical for the intended use of

the model.

5. 5.4 Moisture content- experimental data

For each experimental cooking run in the Stein J SO-IV oven, the endpoint
moisture content predicted by the model was compared with the experimental values
(Section 5.4.2). The deviations between the model and experimental moisture contents
ranged from —5.2% to 3.6% wet basis moisture, with an average deviation of —0.04% wet
basis (Table 5.24). This indicates that the model moisture predictions are centered
around the measured values with little bias. Standard error of prediction for the complete

set of final moisture contents was 2.3% moisture w.b.

145

Table 5.24 — Difference between measured and predicted moisture content for each
oven condition. Experiment numbers correspond to the conditions listed in Table
3.1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Experiment Difference between measured and
Number predicted moisture content (% wet basis)

1 1.6
6 -4.7
8 -2.5
11 0.8
13 2.4
18 -1.2
21 -0.5
23 1.7
25 3.6
30 -5.2
32 -3.7
34 -0.3
37 0.0
47 0.0
49 -l.2
50 1.5
54 1.2
56 2.0
58 -O.8
63 -l.7
66 -l .3
68 0.6
70 0.0
73 1.8
75 1.2
78 3.5
80 4.0

 

 

 

 

5. 5.5 Cooking yield — experimental data
For each cooking run in the Stein J SO-IV oven, the cooking yield predicted by the
model was compared to the experimental value. Standard error of prediction for the

complete set of cooking yields was 5.9%. The deviation in predicted yield ranged from —

146

10.2 to 10.5% with an average deviation of -—1.2% (Table 5.25), indicating that the model

had a slight bias towards overpredicting yield loss.

Table 5.25 — Difference between measured and predicted cooking yields for each
oven condition. Experiment numbers correspond to the conditions in Table 3.1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Experiment Difference between predicted and
Number measured cookingxield (% yield)
1 3.7
6 10.5
8 6.3
11 -3.4
13 -6.5
18 4.1
21 -2.3
23 -2.6
25 -5.7
30 9.3
32 0.9
34 4.1
37 -5.3
47 -3.6
49 -7.6
50 -10.2
54 -6.0
56 8.9
58 -4.7
63 -l.1
66 2.1
68 -0.4
70 2.4
73 -1.4
75 -4.3
78 -10.2
80 -8.4

 

 

 

 

147

5. 5. 6 Cooking yield — published data
Yield predictions of the model were also compared to yield predictions from the
literature (Murphy et al., 2001b). The published paper presented a regression model for

the yield of ground chicken patties cooked in a Stein model 102 impingement oven.

The model and published yield equation were compared using the same procedures

described in Section 5.4.1.1. Like the temperature predictions, the yield data was a much
closer fit for wet bulb temperatures between 70 and 95°C. The SEP in that temperature
range varied from 1.1 to 6.1% yield. At a wet bulb temperature of 40°C, the SEP ranged

from 13.3 to 15.7% yield.

5.6 Lethality model validation

For each run of the computer model, two sets of Salmonella lethality data were
generated. The first set of data was the inactivation profile at the center point of the
patty, generated using a log-linear equation. For the second data set, total inactivation of
Salmonella within the patty was determined. A volume averaging procedure was used to
determine total inactivation from inactivation at all of the nodal points. This overall
reduction in Salmonella is the value that would be determined experimentally when
counting the number of surviving organisms in a whole ground beef patty.

Due to facility constraints, it was not possible to perform inoculated challenge
studies for Salmonella in the moist-air impingement oven used in this study. However,
data exist in the literature for tests that were conducted in a pilot-scale impingement oven

(Murphy et al., 2002).

148

In order to minimally validate the combined cooking/ inactivation model,
simulations were conducted to compare values of microbial inactivation predicted by the
model with those from the literature. The cooking model was run using the cooking
conditions used by Murphy et al. (2002), as described in Section 3.6. The resulting
inactivation profiles were plotted on the same graphs as data points reported in the

published study (Figures 5.31 and 5.32).

 

 

 

 

10
t
9t
8 7
1:1
7 5 _ __._ ___._ ______
a: 6 t D ,/"/ B E E
a 5 £5 /0/ D D
£2” 41; ’ n B
3 If”
2~;
1 4“
0‘»1...1..1,....;...-i...-
150 160 170 180 190 200
Time(s)

Figure 5.31 - Comparison between model Salmonella Senftenberg lethality
predictions and data points published by Murphy et al. (2002).

149

 

 

 

 

10
9 .Z
8 L
- 1:1 :1
A 7 5 "~6- -- —
€ 6 7? Q E
a 5 t; B E /
8° 4 4E a
2 !/
IE
1 4:.
0: 4 1H+1~fii~w
150 160 170 180 190 200
Time(s)

Figure 5.32 — Comparison between model Listeria innocua lethality predictions and
data points published by Murphy et al. (2002).

The Salmonella Senftenberg inactivation curve predicted by the model was based
on D and z-values for Salmonella Senftenberg heated in turkey (Murphy et al., 2003).
For heating times of 150 and 160 seconds, the inactivation curve predicted by the model
is within the bounds of the experimental data reported by Murphy et al. (2002).
However, the model reaches a maximum reduction of 7-log (set by the initial inoculum
level) approximately 15 seconds before the experimental data. At times above 180
seconds, both the model and experimental data indicate a reduction of 7—logs. Standard
error of prediction for the Salmonella inactivation curve was 1.3 logs (n=21).

A comparison of the Listeria inactivation curve predicted by the model and the

experimental data of Murphy et al. (2002) indicated a close fit between the experimental

150

and predicted inactivation’s. The model inactivation curve was generated using D and 2-
values for Listeria innocua in ground turkey (Murphy et al., 2003). At a heating time of
150 seconds, the model slightly underestimated the level of lethality, although the model
predictions were within l-log of the lower end of the experimental data range. As
heating time increased, the inactivation curve predicted by the model closely estimated
the experimental data. Standard error of prediction for the L. innocua inactivation curve
was 1.1 logs (n=21).

These comparisons only give a rough verification of the ability of the cooking
model to predict microbial inactivation. However, the temperature prediction capability
of the model has been verified much more extensively. Combining the temperature
prediction capabilities of the model with experimentally derived kinetic parameters for
microbial estimation (D and z-values) should enable the cooking model to produce an
acceptable first estimate of microbial lethality during cooking. In addition, due to the
design of the cooking model, the effects of process changes on corresponding changes in
microbial lethality are readily visible. This makes the combined cooking and Salmonella
inactivation model a valuable tool for predicting the effects of processing parameters on

microbial safety.

5.7 Illustration of model utility

The utility of a combined cooking and inactivation model is several-fold. Some

examples are listed below.

151

1. A processor uses a moist-air impingement oven to produce ready-to-eat ground
beef patties. The oven is currently operated at a dry bulb temperature of 177°C, a
steam content of 75% by volume, and an air velocity of 18 m/s. The dwell time in
the oven is 3 minutes. The processor wishes to increase the throughput of the
oven by adjusting the moisture content of the cooking air from 75 to 85%
moisture by volume. The processor wants to know what the new oven dwell time
will be to reach the same patty center temperature that was achieved by the

previous process.

In this case, determining the new dwell time in the oven could be determined by
running the cooking model under the new set of process conditions. Running the model
under the original cooking conditions, the processor finds that the original final center
temperature was 787°C. The processor then runs the model under the new conditions
and sees that the dwell time that will achieve 787°C is 170 seconds, a ten second
improvement over the original conditions, which would translate into a 5.8% increase in

throughput.

2. In the situation above, the processor is concerned that the faster cooking time will

not be adequate to achieve the desired level of microbial inactivation.
In this case, the processor can use the lethality prediction function of the computer

model. Running the cooking model, the processor sees that the cooking process greatly

exceeds the required lethality in both cases. However, based on a 6.5-log target

152

reduction, the higher steam content cooking air achieves the desired lethality 14 seconds

earlier than the original condition.

3. The processor is not convinced that the current operating conditions are making
the most efficient use of the oven system. Experience indicates that increasing the
steam content of the oven will decrease cooking time. However, quantitative data
are not available. The processor would like to develop a set of experiments to

optimize the oven settings.

Taking the oven system offline to perform optimization experiments is time
consuming and expensive. In addition to the time and manpower spent conducting the
experiments, the oven must be taken out of production for the time period of the
experiment. To determine the effects of changing process conditions, the processor runs
simulations using the cooking model. Examples 1 and 2 show how the model can be
used to illustrate changes in process conditions. By running the model under many more
sets of conditions, the processor can determine the ideal settings for the desired cooking

results.

4. Product output from oven systems sometimes falls behind due to problems with
equipment and backups during previous unit operations. Operators might try to
catch up with production quotas by increasing the oven belt speed. The manager

is concerned that the desired level of microbial lethality is not being achieved due

153

to increases in belt speed. It is difficult for the manager to illustrate to the

operators that their actions may result in food that is potentially unsafe.

This instance illustrates the benefit of the graphical temperature and lethality
outputs of the model. The graphical outputs can be used to illustrate the effects of
process changes to personnel with no knowledge of heat transfer or microbial inactivation
kinetics. If the oven conditions utilized in Example 2 were in use, the oven would
achieve a 6.5-log reduction in Salmonella in 129 seconds. Decreasing the dwell time by
ten seconds (119 seconds cooking time) would decrease the predicted lethality to only
0.8-log. In this case, a very small adjustment in cooking time would result in a product

that is potentially unsafe.

154

6 CONCLUSIONS

The main conclusions of this study were:

1. Fat content had a significant effect on the cooking time for ground turkey, ground
beef, and ground pork. Differences in the time required to reach 85°C between
different fat contents of each meat species were 171, 217, and 102 seconds for
ground turkey, ground beef, and ground pork patties, respectively. Higher fat
samples of ground beef and ground pork cooked faster than the lower fat samples.
The opposite was true for ground turkey. Fat content significantly affected the
yields of ground turkey with initial fat contents of 1.4% and 8.6%, and ground
pork of initial fat contents of 15.7% and 41.9% fat. Differences in yield between
fat contents of ground turkey and ground pork were 18 and 17%, respectively,
when cooked to 85°C. No significant differences in yield were measured between
ground beef with initial fat contents of 7.2 and 17.5%. Fat transfer contributed up
to 6% of the yield loss of ground beef and up to 28% of the yield loss of ground

pork patties.

2. Fat holding capacity of ground beef was modeled as a function of initial fat
content and heating time using multiple linear regression. Holding time did not
significantly affect fat holding capacity. The regression equation for fat holding
capacity had a coefficient of variation of 0.81 and a standard error of prediction of

0.068 g fat/g dry matter. This equation was an important step towards developing

155

a model for heat and mass transfer of ground beef products that includes the

effects of fat transport.

The effects of oven temperature, steam content, and airflow on heating time,
yield, fat loss, and volume change of ground beef patties cooked in an industrial
moist-air impingement oven were quantified. Multiple linear regression showed
that oven steam content had the largest effect on both heating rate and cooking
yield. Neither oven temperature nor air velocity had a significant effect on patty
center temperature. Oven temperature, steam content, and cooking time all had
significant affects on cooking yield. Fat loss increased roughly linearly, and patty
diameter decreased roughly linearly with patty temperature. Patty thickness

remained fairly constant during cooking.

A finite-element method-based cooking and Salmonella inactivation model was
developed. The model was an improvement over previously published models for
the following reasons:

a) The heat and moisture transport portions of the model were based
completely on heat and mass transfer principles. Empirical correlations
were only used for thermal property relationships. This allowed for
maximum flexibility of the model over different cooking conditions.

b) Fat transport was incorporated in the model. The incorporation of fat
transport created a model that could predict yield losses for products

containing a wide range of fat levels.

156

c) The model incorporated the transient effects of heat and mass transfer
related to moist-air impingement, including the effects of condensation as
a surface mass transfer process (rather than as an “effective heat transfer”

effect).

d) The model was validated with experimental data from an industrial moist-

air impingement oven.

The cooking model was validated using data collected from an industrial moist-air
impingement oven. The transient standard error of prediction for temperature was
8°C. The model more accurately predicted temperatures above 45°C. At
temperatures above 45°C, the SEP was reduced to 57°C. Predictions of moisture
content had errors ranging from O to 5.2% wet basis. Predictions of cooking yield
had errors ranging from 0.1 to 15.4% with an average deviation of 5.9%.
Comparisons with temperature and yield data compiled by other researchers were
also favorable. Standard error of prediction for center temperature of chicken patties
at cooking air wet bulb temperatures between 70 and 95°C ranged from 1.4 to 3.7°C.
Higher errors occurred at an air wet bulb temperature of 40°C. Standard errors of

prediction for yield ranged from 1.1 to 15.7%.

The microbial lethality prediction of the model was compared to published data.
Standard error of prediction for inactivation of Salmonella Senftenberg in ground
beef patties was 1.3 logs (CFU/g). Standard error of prediction for Listeria innocua

was 1.1 logs (CFU/g).

157

7 FUTURE WORK

There are a number of studies that should be conducted in order to further develop
our understanding of the dynamics of meat and poultry cooking. The results of these
studies could be utilized for the development of more precise and more robust cooking
models. In addition, further understanding of the mechanics of cooking could improve
our general understanding of the effects of processing on meat and poultry quality

attributes. Recommendations for further studies include:

1. Experiments designed to determine the effects of protein and fat chemistry on fat
and moisture transport should be conducted. Specifically, the effects of fat
composition on transport properties should be investigated. Models for fat
viscosity as functions of lipid composition and temperature could contribute
greatly to understanding of fat transport mechanisms. Understanding of these
mechanisms is needed to maximize the effectiveness of cooking models for a

wide range of product compositions.

2. An analysis of the changes in meat microstructure during cooking should be
performed. Such studies should include analysis of the changes in porosity during
cooking as well as the relationships between meat microstructure and volume

change. The dynamics of volume change during cooking should be investigated.

158

Studies should be designed to determine the driving forces for fat transport and
moisture transport via drip loss. These phenomena are generally attributed to
changes in water and fat holding capacities, but there is little understanding of the
physical mechanisms behind such changes. Further knowledge of these driving
forces is needed for the development of mechanistic models for water and fat

holding capacities and transport.

Extensive validation of microbial inactivation kinetics during impingement
cooking of meat and poultry products should be performed. This should be done

utilizing inoculated challenge studies with an actual moist-air impingement oven.

Automatic mesh generation subroutines should be added to the cooking model.
This would allow for much easier modeling of cooking for various product
geometries. It would also allow for the cooking model to account for volume

change during cooking.

The model should be used for optimization of impingment cooking processes, in

terms of safety, cooking yield, and oven throughput.

The model should be adapted in the future to simulate cooking of whole-muscle

products.

159

8 APPENDICES

160

8.1 Model and experimental temperature versus time curves for moist-air

impingement cooking of ground beef patties.

 

Surface

Center

Temperature (’ C)

 

Time (s)

   

 

200 250

Figure 8.1 - Oven temperature: 121°C, oven steam content: 50% by volume, oven

airflow: 11.4 m/s (Experiment 1a).

 

 

 

100 a:
35:2, Surface
80 x )p’iw" ~ -
o . ‘
my
3 60
a
g 0
4
g 7' Center
P“ 20
/ — Ode]
O . . . 4 . 4 + L4. 4 . 1
0 50 100 150 200
Time (s)

Figure 8.2 - Oven temperature: 121°C, oven steam content: 50% by volume, oven

airflow: 11.4 m/s (Experiment lb).

161

 

100

 

 

 

 

G 80
Q/
2
3 60
g 40
E“
D
E" 20
O 1
O 100 200 300 400 500
Time (s)

Figure 8.3 — Oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 16.8 m/s (Experiment 6a).

 

100

 

00
0

Temperature (’ C)
4s Ox
o o

N
O

 

 

 

500

 

Time (s)

Figure 8.4 - Oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 16.8 m/s (Experiment 6b).

162

 

 

40

Temperature (’ C)
0‘
O

N
O

 

 

 

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

Figure 8.5 - Oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 21.8 m/s (Experiment 8a).

 

p—a
O
O

 

00
O

Temperature (J C)
O\
o

20

 

 

 

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

Figure 8.6 - Oven temperature: 121°C, oven steam content: 50% by volume, oven
airflow: 21.8 m/s (Experiment 8b).

163

 

100 ,

 

 

  

 

 

 

, Surface
6‘ 80 1
3 t
E 60 37 Center
g 40
8‘
13
20 * —Model
0 , 44 4 4 4
0 50 100 150 200 250 300 350
Time (s)

Figure 8.7 — Oven temperature: 121°C, oven steam content: 70% by volume, oven

airflow: 11.4 m/s (Experiment 11a).

 

 

40

Temperature (’ C )

N
C

Figure 8.8 - Oven temperature: 121
airflow: 11.4 m/s (Experiment 11b).

  

Center

   

 

150 200
Time (s)

°C, oven steam content: 70% by volume, oven

164

 

 

100 ,

 

 

00
O

   
 

Center

   

 

1

Temperature (’ C)
4:. O\
o o

— Model

 

1 J___l_1 141 1
l

O 50 100 150 200
Time (s)

Figure 8.9 — Oven temperature: 121°C, oven steam content 70% by volume, oven
airflow: 16.8 m/s (Experiment 13a).

 

100 _

 

 

80 ~:

  

60 at

   

Center

 

Temperature (’ C)

 

 

40*
20 '
——Model
0 4 4 4 4 4 - 4%
0 50 100 150 200

Time (s)

Figure 8.10 — Oven temperature: 121°C, oven steam content 70% by volume, oven
airflow: 16.8 m/s (Experiment 13b).

165

 

31—.
O
O

 

OO
O

05
0

Temperature (’ C)
h
o

 

 

0 100 200 300 400 500
Time (s)

Figure 8.11 - Oven temperature: 121°C, oven steam content: 70% steam volume,
oven airflow: 21.8 m/s (Experiment 18a).

 

—
00 O
O O
11*r
a
O
(D

   

1

  

Temperature (’ C)
0‘
o

 

 

 

40
20 8 —Model
0 4 4 4 4 4 4 44 4 4 4 4
O 100 200 300 400 500

Time (s)

Figure 8.12 - Oven temperature: 121°C, oven steam content: 70% steam volume,
oven airflow: 21.8 m/s (Experiment 18b).

166

 

100

 

Temperature C C)

 

   

 

 

O 1 1 1 1 1r 1 1 1 1 i 1
O 100 200 300 400

Tirm (s)

Figure 8.13 — Oven temperature: 121°C, oven steam content: 88% by volume, oven
airflow: 11.4 m/s (Experiment 21a).

 

100

 

80

60

 

Temperature (’ C)

 

 

 

0 100 200 300 400
Time (s)

Figure 8.14 — Oven temperature: 121°C, oven steam content: 88% by volume, oven
airflow: 11.4 m/s (Experiment 21b).

167

 

 

 
    
 
 
 

 

 

100 _

a 80 7:
a.’ _
E 60 ,5 Center
8
g. 40 4,
,2 E

20 if —Model

0 P 1 % 1 L

 

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

Figure 8.15 — Oven temperature: 121°C, oven steam content: 78% by volume, oven
airflow: 16.8 m/s (Experiment 23a).

 

 

 

 

         

 

100

6 80

w

8 60

a

8.40

E

o .

‘" 20

——Model
0 4444444444444444444fi44444444444444
0 50 100 150 200 250 300 350
Time(s)

Figure 8.16 - Oven temperature: 121°C, oven steam content: 78% by volume, oven
airflow: 16.8 m/s (Experiment 23b).

168

 

fl
0
O

 

OO
O

  

Center

    

Temperature C C)
A O\
o O

—— Model

N
O
l

 

 

   

0 50 100 150
Time (s)

C

Figure 8.17 - Oven temperature: 121°C, oven steam content: 78% by volume, oven
airflow: 21.8 m/s (Experiment 25a).

 

p—
O
O

 

OO
O

   
 

Center

Temperature C C)
A 04
o o

N
O

— Model

lLllllll
T I I

0 20 40 60 80 100 120 140
Time (s)

 

Figure 8.18 - Oven temperature: 121°C, oven steam content:78% by volume, oven
airflow: 21.8 m/s (Experiment 25b).

169

 

.—-0
O
O

E Surface

 

m
C
1

Temperature (’ C)
A O\
o o

N

O
l

tit

 

 

 

O
.L
T I

 

Time (s)

Figure 8.19 — Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 11.43 mls (Experiment 30a).

 

 

 

 

100
Surface __

6 80 -.
b ,-
2 5
g 60 g. 4 ' Center
8. 40 '
E x
43

20 ' ——Model

0 " 4 4 44 4 4 4 4 4 4 4 4
0 50 100 150 200 250 300

Figure 8.20 - Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 11.43 mls (Experimen 30b).

170

 

._-
O
O

OO
0

Temperature C C)
A ON
C O

N
O

 

 

 

 

0 50 100 150 200 250
Time (s)

Figure 8.21 - Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 16.8 mls (Experiment 32a).

 

 

  
 

Temperature C C)

N
O

 

 

 

250

Time (s)

Figure 8.22 - Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 16.8 mls (Experiment 32b).

171

 

 

 

Temperature (D C)

 

 

 

140

 

Time (s)

Figure 8.23 - Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 21.8 m/s (Experiment 34a).

 

 

 

6‘
0.x

E 4*"

g Center

E

d)

{-41

— —— Model

11111111111111111411
l T T l

60 80 100 120 140
Time (s)

 

 

 

 

Figure 8.24 - Oven temperature: 177°C, oven steam content: 50% by volume, oven
airflow: 21.8 m/s (Experiment 34b).

172

 

.—-s
O
O

 

 

OO
O

Temperature C C )
A as
o o

N
O

 

 

O

120

 

Time (s)

Figure 8.25 - Oven temperature: 177°C, oven steam content: 70% by volume,
oven airflow: 11.4 m/s (Experiment 37a).

 

H
O
O

 

 

00
O

Temperature (’ C )
A as
o o

N
O

 

 

 

Time (s)

Figure 8.26 - Oven temperature: 177°C, oven steam content: 70% by volume,
oven airflow: 11.4 mls (Experiment 37b).

173

 

p.—
O
O

OO
O

.L
VIITI

 

    
   

Center

Temperature (’ C)
O‘
o

 

 

 

40
20 E
—Model
0 44444444444 44
0 50 100 150 200 250 300
Time(s)

Figure 8.27 - Oven temperature: 177°C, oven moisture content: 83% by volume,
oven airflow: 11.4 mls (Experiment 47a).

 

 

    
 

 

 

 

G
1.x
E Center
4;
g: :
o _
{-4 :
E! —Model
0 4 . . . 1 4 1 .1.
0 50 100 150 200 250 300
Tine (s)

Figure 8.28 - Oven temperature: 177°C, oven moisture content: 83% by volume,
oven airflow: 11.4 mls (Experiment 47b).

174

 

p—l
O
O

 

OO
O

    
 

Center

Temperature (’ C)
O'\
O

-— Model

OJ: LlrllljiliiliTl 1711111T t
O 20 40 60 80 100 120 140

Tine (s)

 

 

Figure 8.29 - Oven temperature: 177°C, oven steam content: 84% by volume,
oven airflow: 16.8 mls (Experiment 49a).

 

~
O
O

 

OO
O

Temperature (’ C)
A ox
o o

N
O

 

 

Time (s)

Figure 8.30 - Oven temperature: 177°C, oven steam content: 84% by volume,
oven airflow: 16.8 mls (Experiment 49b).

175

 

.—
O
O

 

OO
O

Surface

   
 

Center

Temperature C C)
04
o

 

 

 

40 2'
29 ": ——Model
0 i A T I L T JT
0 50 100 150 200 250 300

Time (s)

Figure 8.31 - Oven temperature: 177°C, oven steam content: 86% by volume,
oven airflow: 16.8 mls (Experiment 50a).

 

Temperature (’ C)

 

 

 

 

Titre (s)

Figure 8.32 - Oven temperature: 177°C, oven steam content: 86% by volume,
oven airflow: 16.8 mls (Experiment 50b).

176

 

p—a
O
O

 

00
Q
g
(D

   
 

Temperature C C)
A ox
o o

N
O
l

 

 

O
A
III
~44—
.4

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

Figure 8.33 - Oven temperature: 177°C, oven steam content: 86% by volume,
oven airflow: 21.8 mls (Experiment 54a).

 

5
o

 

 

0‘ m
o O
1
O 4
0
5'

N
O
44 4

 

— Model

11111111111114111111
I I I

 

O
.4—
_4...
_.

Temperature (’ C)
3
-\ 1““ .

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

Figure 8.34 - Oven temperature: 177°C, oven steam content: 86% by volume,
oven airflow: 21.8 m/s (Experiment 54b).

177

 

1—0
0
O

 

OO
O

 

Temperature (’ C )
Ox
o

 

 

 

40 Center
20 44"
i: —Model
0 ’ 1 ,_ . 1 1 1 1 1 1 1
0 50 100 150 200
Tine (s)

Figure 8.35 - Oven temperature: 232°C, oven steam content: 50% by volume,
oven airflow: 11.4 m/s (Experiment 56a).

 

H
O
O

 

 

4;
O

   

N
0

Temperature 6 C)
O\ 00
o o

 

 

Model

0 50 100 150 200
Time (s)

Figure 8.36 - Oven temperature: 232°C, oven steam content: 50% by volume,
oven airflow: 11.4 mls (Experiment 56b).

178

 

#
O
O

 

 

 

OO
O

1
1 1 1

Temperature C C)
A O\
o o

N
O

 

 

 

O

 

Time (s)

Figure 8.37 — Oven temperature: 232°C, oven steam content: 50% by volume,
oven airflow: 16.8 m/s (Experiment 58a).

 

p—ua
O
O

 

 

 

00
O

  

A
O

   

Center

Temperature (’ C)
O\
o

20
— Mode

 

lllllLlllll
I

0 20 40 60 80 100 120 140
Time (s)

Figure 8.38 — Oven temperature: 232°C, oven steam content: 50% by volume,
oven airflow: 16.8 m/s (Experiment 58b).

179

 

y—n
O
O
I

 

 

 

 

ESIn’face
A 80 4K
0
Lu
5 6O 4
£3 E Center
g. 40 7
Q) at
E“ 20 _
—Model
O11411111)111111111+L141411111
0 50 100 150 200 250 300
Tirrre(s)

Figure 8.39 - Oven temperature: 232°C, oven steam content: 50% by volume,
oven airflow: 21.8 m/s (Experiment 63a).

 

 

    

 

 

 

 

100 ,
E Surface ‘
G 80 .4%/{r
K1
93- 60 44
a -
E. 40 k
E
Q)
l" 20 4
—Model
0 I— 1 L4 + 1 1 1 1 1 1 1 1 1 l 1 J #1 % 1 1 1 l l J 1 1 1
0 50 100 150 200 250 300

Time (s)

Figure 8.40 - Oven temperature: 232°C, oven steam content: 50% by volume,
oven airflow: 21.8 mls (Experiment 63b).

180

 

.—e
O
0

Surface 4 -

00
O
1

Center

Ch
O

 

Temperature (3 C )
h
o

— Model

N
O

 

11m111111
l I

0 50 100 150 200 250 300
Time (s)

 

—I>—

Figure 8.41 - Oven temperature: 232°C, oven steam content: 70% by volume,
oven airflow: 11.4 mls (Experiment 66a).

 

)—
O
O

 

OO
O

  

Center

   

Temperature C C )
as
o

 

 

 

40
20 ~ —— Model
0 44444444 4 4 e 4 4
0 50 100 150 200 250 300

Tim: (8)

Figure 8.42 - Oven temperature: 232°C, oven steam content: 70% by volume,
oven airflow: 11.4 mls (Experiment 66b).

181

 

 

Temperature (’ C )

 

 

 

Time (s)

Figure 8.43 - Oven temperature: 232°C, oven steam content: 70% by volume,
oven airflow: 16.8 mls (Experiment 68a).

 

 

4o . Center

Temperature (J C)

— Model
1 1 1 4L 1 1r 1 1 1

0 50 100 150 200
Tine (s)

4

 

 

O
l
1-

Figure 8.44 - Oven temperature: 232°C, oven steam content: 70% by volume,
oven airflow: 16.8 m/s (Experiment 68b).

182

 

Surface

Temperature C C)

 

 

 

 

80

 

Time (s)

Figure 8.45 - Oven temperature: 232°C, oven steam content: 70% by volume,
oven airflow: 21.8 m/s (Experiment 70a).

 

100 ,
F Ice

 

V
..........

80 7:

 

 

Temperature C C)

 

 

80

 

Tine (s)

Figure 8.46 - Oven temperature: 232°C, oven steam content: 70% by volume,
oven airflow: 21.8 mls (Experiment 70b).

183

 

H
O
O

  

Temperature C C)
A O\ 00
o o o

N
O

 

 

 

O

1 00
Tina (s)

Figure 8.47 - Oven temperature: 232°C, oven steam content: 82% by volume,
oven airflow: 11.4 m/s (Experiment 73a).

 

 

 

 

 

 

 

100

a 80 _ “““ “
1.x a

e _

B 60 I

§ 46:

E

O ,

E" 20 44

O 4 A
0 20 40 6O 80
Time(s)

Figure 8.48 - Oven temperature: 232°C, oven steam content: 82% by volume,
oven airflow: 11.4 mls (Experiment 73b).

184

 

H
O
O

 

Temperature 6 C)
A ON 00
o o o

N
O

 

 

 

0 50 100 150 200 250 300
Titre (s)

Figure 8.49 — Oven temperature: 232°C, oven steam content: 82% by volume,
oven airflow: 11.4 mls (Experiment 75a).

 

 

   
 

 

 

 

100
6 80 ~
Ru
2 a
a 60
8 40
E
,2
20 1 ——Model
0 4 44 444444444 444 4 4
0 50 100 150 200 250 300

Tine (s)

Figure 8.50 — Oven temperature: 232°C, oven steam content: 82% by volume,
oven airflow: 11.4 mls (Experiment 75b).

185

 

H
O
O

 

OO
O
1

  

O\
O

Center

   
   

Temperature (’ C)
A
o

N
O

— Model

1 1 1 1 1 1
I

0 50 100 150 200
Time (s)

 

 

 

III

Figure 8.51 — Oven temperature: 232°C, oven steam content: 82% by volume,
oven airflow: 16.8 mls (Experiment 78a).

 

y—a
O
O

 

OO
O

    
 

Center

Temperature 6 C)
0‘
o

 

 

 

40
20 i

. ——Model
0 4 4 4 4 4 4 4 4 4

0 50 100 150 200 250

Time (s)

Figure 8.52 — Oven temperature: 232°C, oven steam content: 82% by volume,
oven airflow: 16.8 mls (Experiment 78b).

186

 

 

 

   

Temperature C) C)

N
O

 

 

 

O 50 100 150 200
Time (s)

Figure 8.53 - Oven temperature: 232°C, oven steam content: 82% by volume,
oven airflow: 21.8 mls (Experiment 80a).

 

_
O
O

 

   

Temperature (’ C)
A as oo
o o o

N
O

 

 

l 1 1 1 1

04
0 50 100 150 200 250

Time (s)

c—lh—

 

Figure 8.54 - Oven temperature: 232°C, oven steam content: 82% by volume,
oven airflow: 21.8 m/s (Experiment 80b).

187

8.2 Derivation of cooking-air thermo-physical property equations
The oven-air thermo-physical properties used in the cooking model were
modeled using non-steady-state relationships. The latent heat of vaporization for

water as a function of temperature was modeled using linear regression of tabular

data (Geankoplis, 1993); (Equation [8.1]).

,1 = -—2.429 - T + 2502.8 [ 8.1 ]

The resulting regression had an R2 value of 0.998 (Figure 8.55).

 

2550 1

N
U!
C
O

2450 f
2400 4

    

2350 f

2300 g 3.“, = -2.429T + 2502.8

Latent heat of vaporization (J/ g)

 

 

 

2250 1 R2 = 0.9998
2200 :44444 14.44 11.41.11 .. .. ,. 1. ., .. ,1
0 20 40 60 80 100 120
Temperature (° C)

Figure 8.55 - Latent heat of vaporization for water as a function of temperature
(From tabular data: Geankoplis, 1993).

188

The viscosity of air was modeled as a function of temperature using regression of

tabular data (Geankoplis, 1993); (Equation [8.2]).

14144 = —0.0000000002 - T2 + 0.0000005 -T + 0.0002 [ 8.2 ]

The resulting regression equation had an R2 value of 0.9998 (Figure 8.56).

0.0003 _
0.00028 43

A 0.00026
.3 0.00024

 

l

   
  
  

g/c
.9
o
o
o
N
N

W

.8, 0-00013 11,, = 213101“2 + 513-071~ + 0.0002
g 0.00016 4:

E R2 = 0.9998
0.00014 g

0
00001 P111111L11111m1111111111111111
. l T I I l

0 50 100 150 200 250 300
Temperature (° C)

 

 

 

Figure 8.56 - Viscosity of air as a function of temperature (From tabular data:
Geankoplis, 1993).

The viscosity of steam was modeled as a function of temperature using regression of

tabular data (Geankoplis, 1993); (Equation [8.3]).

41.4.... = —0.00000001-T2 + 0.00004 - T + 0.0089 [ 8.3 ]

189

The resulting regression had an R2 value of (Figure 8.57).

 

0.02 4
0.019 4;_
0.018 1;

7g 0.017
g, 0.016 g
34 0.015 4;
0.014 f
0.013 ;
0.012 45
0.011
0.01:‘*“4“‘4444444441111...

50 100 150 200 250 300

Temperature (° C)

Viscosi

11.4.4444 = are-081° + 413051“ + 0.0089
R2 = 0.9997

 

 

 

Figure 8.57 - Viscosity of steam as a function of temperature (From tabular
data: Geankoplis, 1993).

A mixture equation developed by Burrneister (1983) was utilized to calculate the

viscosity of the air-steam mixture (Equation [8.4]).

H air + k1 steam

“mix =
1+¢,,-’%( 1+¢.-x%(

where:

 

 

[8.4]

190

. [44444-4 (44 4/ J l

 

and
. 4444,1444 /.,,..°'”ll

 

4 4 44w

The density of air was modeled as a function temperature using regression of tabular

data (Geankoplis, 1983); (Equation [87]).

pair = -0.00000000002 ~ T3 + 0.00000001-T2 — 0.000004 . T + 0.0013 [ 8.7 ]

The regression equation had an R2 value of 1 (Figure 8.58).

191

 

 

1-

t _ 3 - - -
0.0005 pa1,—-2E—11T +11+3418T2 41+: 06T+0,()o13

 

 

 

R2=1
0.0003 ruralwrrUllelHL
0 50 100 150 200 250 300
Temperature (°C)

Figure 8.58 - Density of air as a function of temperature (from tabular data:
Geankoplis, 1993).

The density of water vapor at temperatures below 100°C was modeled as a function

of temperature using tabular data (Geankoplis, 1993); (Equation [8.8]).

P vapor

= 0.00000000003 T“ + 0.00000000025 - T3 - 0.0000000039 - T2 + 0.00000085 . T

[8.8]

The regression equation had an R2 value of 1 (Figure 8.59).

192

 

0.0007

 

 

 

 

_ 3 2
A 0.0006 A psteam— 8.12E-10T - 3.52E-08T +1.32E-06T+
"e 6.22E-07
3;; 0.0004
5
Q 0.0003 a
O
u:
'5 0.0002 +
0
£‘
0.0001 —
0 . 1 . .
0 20 40 60 80 100 120
Temperature (° C)

Figure 8.59 - Density of saturated steam as a function of temperature (From
tabular data: Geankoplis, 1993).

The density of steam at atmospheric pressure and temperatures above 100°C was

modeled as a function of temperature using regression of tabular data (Geankoplis,

1993); (Equation [8.9]).

= —0.0002 - log(T)+ 0.0015 [ 8.9 ]

p steam

The regression equation had an R2 value of 0.999 (Figure 8.60).

193

 

0.0007

 

 

 

MA 0.0006 *

E

g 0.0005

.5 0.0004

5

0 0.0003 1

a psteam = -0.0002Ln(T) + 0.0015

m 0.0001 1

O l l l
90 140 190 240
Temperature (°C)

Figure 8.60 - Density of steam at 101.35 kPa as a function of temperature (From
tabular data: Geankoplis, 1993).

The density of the air-steam mixture was assumed given by Equation [8.10] which

assumes that the molar fraction of each component is equal to the volume fraction.
pmix = pair .Xair + psteam . Xstcam I: 810]

The heat capacity of the air-steam mixture was calculated as functions of temperature

and composition using an equation given by Millsap (2002); (Equation [8.11]).

. : (pair 'Xair 'cpair + psteam .xsteam .cpstcam) [ 811 ]
P-m'x (pair . Xair + psteam . xstcam)

 

C

194

The thermal conductivity of air was modeled as a function of temperature using

regression of tabular data (Geankoplis, 1993); (Equation [8.12]).

kair = 0.000000003 - T2 + 0.0000008 - T + 0.002 [ 8.12 ]

The regression had an R2 value of 0.9999 (Figure 8.61).

 

 

Thermal conductivity (W/cm K)

 

 

2 1 k = -7E-09T3 - 513-0n2 + 0.0076T + 2.4187
1'5 R2 = 0.9999

1’:
0.5:

0 UL“.11111.11.511.............

 

O 50 100 150 200 250 300
Temperature (° C)

Figure 8.61 - Thermal conductivity of air as a function of temperature (From
tabular data: Geankoplis, 1993).

The thermal conductivity of steam was modeled as a function of temperature using

regression of tabular data (Geankoplis, 1993); (Equation [813]).

195

k = 0.0000009 - T + 0.0002 [ 8.13 ]

steam

The regression had an R2 value of (Figure 8.62).

 

   

 

 

 

A 0.00045 '
M
3 0.0004
5
E)
E 0.00035
8
+3
5 0.0003 -
“7.: km, = 9E-07T + 0.0002
-:
i-'
0.0002 . 1 1 T
50 100 150 200 250 300
Temperature (° C)

Figure 8.62 - Thermal conductivity of steam as a function of temperature (From
tabular data: Geankoplis, 1993).

Thermal conductivity of the air-steam mix was modeled using an equation given by

Burmeister (1983); (Equation [8.14]).

k p = kair + ksteam [ 814]

mm Xstcam xair
1+Aas.( Aair) 1+Asa ( /Xstcam)

 

196

where:

 

Aas =-1-<
4
and:
r
A33. :11

 

1+ [
“steam

1+[

p air

steam

 

l’l steam

1

M

air

air

 

’1 air

)4

M

steam

1

0.75 1 + 8%
air.K

 

Sscam
1+ ! Aairx

0.75 l + Ss‘“%airx

 

Ssair
1 + flairx

0.5 ‘

. 1 + [0.733 - (S... 6...... )°" l/TM

 

 

0.5 ‘

Sair
1+ fiairx

[ 8.15]

> . 1 + [0.733 . (sWm . 3.1, )0-51/TM

 

 

3.
1+ m Tair.K

[ 8.16]

The diffusivity of steam in air was modeled using an equation from Vargaflik (1966);

(Equation [8. 17]).

Dw=02

 

T
16.[

air,K

273

I.

197

[ 8.17]

8.3 Product thermo-physical properties

Composition dependent values of product thermo-physical properties were
used for the model calculations (Equations [8.18] and [8.19]). The heat capacity of
meat was modeled as a function of the mass fractions of water, protein, and fat (Choi
and Okos, 1986). Separate equations were utilized for the temperature ranges below

(Equation [8.18]) and above (Equation [8.19]) freezing.

Cp.frozcn = Cp.ice .xW +cp.protein .XP +cp.fat .XF [ 8'18]

c = c -XW +Cp.protein -XP +cp‘fat ~XF [ 8.19]

Thermal conductivity of the meat was calculated using a series model based
upon the volume fraction and thermal conductivity of the water, protein, and meat

fractions of the meat (Choi and Okos, 1986); (Equation [820]).

kT=k ~X +k ~X +k X [8.20]

water water protein protein fat . fat

The density of the meat was calculated using a parallel model based upon the
density of the water, fat, and protein components of the meat (Choi and Okos, 1986);

(Equation [8.21]).

198

1

mwater/p water + mfat /p fat + mprotein /pprotein

 

p: [8.21]

The moisture diffusivity of the meat was modeled as a function of fat-protein
ratio and temperature using an equation developed by Mittal and Blaisdell (1984);

(Equation [822]).

4829.7

 

km = 0.003~exp(— 0.442+FP— +1155] [ 8.22]

A constant value of 64.4 J/g was utilized for the latent heat a fusion of fat
(Skala etal., 1989). The latent heat of fusion for water was set at 337.8 J/g
(Geankoplis, 1993). A value of 0.003 g/g was utilized for the value of moisture
capacity, cm (Chen et al., 1999). A constant value of 0.0008 g/s'cm2 was utilized for

the fat conductivity, kf.

199

8.4 Screen shots from Visual Basic cooking model user interface

kawmymm

MSU Biosystems Engineering Convection Cooking Model

Oven Conditions
0 Input Steady Conditions Oven Conditions

Ewen Temperature [€039] '1)

Owen Steam Comer” 1'13 100% h, xolun‘vel a

- lemrev ah ne I2

I 0R I)ve" \Nel Bulb Tempélalute [C] u

A11 Flow R11»: (U 530 W's-,1

Tempermurer 1

Cooking T1me (11; . a)
0 input Translenl Conditions From File

Product Salmonella Kinetics

Product "1““ Conditions -— O Uh? Orr-.4311 Model F’avarrwii—«S

_ - . Input Model F‘Jvamelevs
o 5.354 Temperature (259:4 I.) u

. pmk Moisture Conteni 0.135%) a ‘ III » Value a Reference
. . Temperature

0 Tuvkey Fat Content (075%) 1 ' Value

 

Figure 8.63 — Input screen of cooking model user interface.

200

He Mm 01w Geometry Heb [that
MSU Biosystems Engineering Convection Cooking Model

Temperature Profile Yield Iloisture Content

1111— 11 ’ Mustang (Lumen:

Fusl 1.11434'0l9rhp'1371l1" ‘

hm] genial 'Qfl'tCIx‘lalgle r41: 4 L

E ‘+ Log ledm’lfin no! 311191.941]

‘Mwmun L Stunted log veduchm - i

 

”6:121:73:~

Figure 8.64 — Output screen of cooking model user interface.

201

8.5 Visual Basic model code

Module — Main

Version 1.0 - July 2004

By Adam E. Watkins and Dr. Bradley P. Marks
Department of Biosystems and Agricultural Engineering
Michigan State University

Sub Main()

'Many of the modules associated with the basic FEM architecture were provided by or modified
' from modules provided by Dr. Larry Segerlind - Michigan State University. These modules
' have been identified in the module comment statements.

‘ FEM cooking and microbial inactivation model - Main Module

' This module controls the calculations for a two-dimensional heat and mass transfer problem.
'Open the main input file. Contains node data, element data, boundary nodes, etc.

Open "c:\Tdeield47.txt" For Input As #1

'Open the main output file

Open "c:\modeloutput.txt" For Output As #3

'Input setup information

Input #1, NumNodes, NumEle, NudeyNodes, NumMatlSets, NumDeridey

'Set solution timestep to l s

timestep = 1

'Format display

Form2.MSChart1.Visible = True 'Show chart 1
Form2.MSChart2.Visible = True 'Show chart 2
Form2.MSChart3.Visible = True 'Show chart 3

'Input oven conditions from a file or from the input screen

If Forml .Option2 = True Then
Open "c:\OvenConditions.txt" For Input As #2 'Open oven conditions input file
Input #2, NumTimeSteps 'Get cooking time from file
ReDim OvenConditions(NumTimeSteps, 3) 'Dimension oven conditions matrix
Call InputOvenConditions(OvenConditions(), NumTimeSteps) 'Get oven conditions from file
Forrnl .Frame3.Visible = True 'Show oven conditions graph

202

Forml.MSChart1.Visible = True

OvenConditions(O, 1) = ”Temperature C"
OvenConditions(O, 2) = "Steam Content”
OvenConditions(O, 3) = "Air Velocity m/s"
Forml.MSChart1.ChartData = OvenConditions

'Close oven conditions file
'Oven conditions from form

Close #2
Else
NumTimeSteps = Forml .Text4.Text

ReDim OvenConditions(NumTimeSteps, 3)

For I = 1 To NumTimeSteps

OvenConditions(I, l) = Forml .Textl .Text
OvenConditions(I, 2) = Forml .Text2.Text

'Show oven conditions graph
'Labels on graph
'Labels on graph
'Labels on graph
'Plot oven conditions on graph

'Input cooking time
'Dimension oven conditions matrix

'Oven temperature C
'Oven steam content % by volume

If OvenConditions(I, 2) = 0 Then OvenConditions(I, 2) = 1

OvenConditions(I, 3) = For‘ml .Text3.Text

Next I
Forml .Frame3.Visible = True
Forml .MSChartl .Visible = True

OvenConditions(O, l) = "Temperature C"
OvenConditions(O, 2) = "Steam Content"
OvenConditions(O, 3) = "Air Velocity m/s"
Forml.MSChart1 .ChartData = OvenConditions

End If

ReDim Coord(NumNodes, 2)

ReDim EleMatlData(NumEle)

ReDim EleNodeData(NumEle, 4)

ReDim SubDerivBC(NumDefidey + l, 2)
ReDim BoundaryType(NumDeridey + 1)
ReDim deNode(NudeyNodes)

'Input node,element, and derivative boundary data

'Oven air velocity cm/s

'Show oven conditions graph
'Show oven conditions graph
'Labels on graph
'Labels on graph
'Labels on graph
'Plot oven conditions on graph

'Dimension matrix of coordinate data
' Coord( ,1) X coordinate
' Coord( ,2) Y coordinate

'Dimension element material set

'Dimension matrix of element node data
' EleNodeData%( ,l) = Node I
' EleNodeData%( ,2) = Node J
' EleNodeData%( ,3) = Node K
' EleNodeData%( ,4) = Node M

'Subscripts of the derivative
' boundary condition
'Convection or impingement at each
' derivative boundary
'Vector of boundary nodes

Call InputBasicData(NumMatlSets, NumNodes, NumEle, Coord(), EleMatlDataO, EleNodeDataO)
Call InputDerivBC(NumDeridey, NudeyNodes, deNode(), SubDerivBC(), BoundaryTypeO)

Close #1 'Close input data file
'Input type of product
If Forml .Option3 = True Then

Meat = 1 'Beef

TR = 6.5 'Target reduction

Form2.Labe12.Caption = "6.5"
Elself Forml .Option4 = True Then

'Label on Salmonella reduction graph

203

Meat = 2 'Pork

TR = 6.5 'Target reduction

Form2.Label2.Caption = "6.5" 'Label on Salmonella reduction graph
Elself Forml .OptionS = True Then

Meat = 3 ' Turkey
TR = 7 'Target reduction
Forrn2.Labe12.Caption = "7" 'Label on Salmonella reduction graph

End If

'Input meat temperature, moisture, and fat content

InitialT = Forml .Text5.Text 'Initial meat temperature

InitialM = Forml .Text6.Text 'lnitial meat moisture content

If InitialM = 0 Then InitialM = 0.1 'Avoid zero moisture condition not allowed
' by model equations

InitialF = Forml .Text7.Text 'Initial meat fat content

'Input Salmonella reduction equation coefficients

Dvalue = Form1.Text9.Text 'D-value Salmonella
Z = Forml .Text10.Text 'z-value Salmonella
Tref = Forml.Text11.Text ’T-ref Salmonella

'Dimension vectors for calculation of inactivation

ReDim AverageN(NumTimeSteps) 'Volume average of microbial concentration
Dim Reduction As Variant 'Overall microbial reduction
ReDim logreduction(NumNodes) 'Center microbial reduction
Ninitial = 1000000000 'Value used for log-reduction calculation
ReDim No(NumN odes) 'Used for log—reduction calculation
ReDim NW(NumNodes) 'Used for log-reduction calculation
ReDim Nnew(NumNodes) 'Used forlog-reduction calculation
ReDim NnewW(NumNodes) 'Used for log-reduction calculation
For X = 1 To NumNodes
No(X) = Ninitial 'Create a nodal microbial concentration vector
NW(X) = Ninitial

Next X

'Dimension global solution vector

ReDim GSV(NumNodes) 'Enthalpy
ReDim GSV_M(NumNodes) 'Moisture
ReDim GSV_F (NumN odes) 'Fat

'Vector of temperature values

ReDim temperature(NumNodes) Temperature vector

'Calculate solid content of the product

Solid = 100 - lnitialM - InitialF 'Initial solid material in meat (NOT fat or water)

'Load vectors of moisture and F? values

204

For K = 1 To NumNodes

GSV_M(K) = InitialM 'Load vector of nodal moisture values
GSV_F(K) = InitialF / Solid 'Load vector of nodal fat values
temperature(K) = InitialT 'Load vector of initial temperature values

Next K
’Calculate the initial therrnophysical properties of the product

Call CalculatelnitialProperties(heat_capacity_f, Meat, Solid, InitialT, InitialM, InitialF, heat_capacity,
latentW, frozenH, latentF)

'Formulate enthalpy table for reconversion to temperature

LevelOne = frozenI-I

LevelTwo = frozenI-I + latentW

LevelFive = frozenH + latentW + (2 "' 3.35)

LevelThree = frozenH + latentW + (45 * 3.35) + (2 * 3.35)
LevelFour = frozenl-I + latentW + (45 * 3.35) + latentF + (2 * 3.45)

'Percent of water thawed at the exact value of freezing temperature
PercentThawed = 1
'Convert product temperature to enthalpy

If InitialT < -2 Then
For K = 1 To NumNodes
GSV(K) = (InitialT + 273) " heat_capacity_f
Next K
Elself InitialT = -2 Then
For K = 1 To NumNodes
GSV(K) = frozenH + PercentThawed * latentW
Next K
Elself InitialT > -2 And lnitialT <= 45 Then
For K = 1 To NumNodes
GSV(K) = ((2 + InitialT) " 3.35) + (frozenH + latentW)
Next K
Elself InitialT > 45 Then
For K = I To NumNodes
GSV(K) = ((2 + InitialT) "' 3.35) + (frozenH + latentW + latentF)
Next K
End If

'Calculate oven air moisture content from wet bulb temperature if specified in that manner

If Form1.Checkl.Value = 1 Then

wa = Forml .Text8.Text

Tov = Forrn1.Text1.Text

latent = (-2.429 "‘ Forml.Text1.Text + 2502.8)

Cwb = 8.12078904E-10 "' wa " 3 - 0.000000035203247 "' wa " 2 + 0.00000131977474 "'
wa

+ 0.000000621530732

ConcOvenAir = Cwb - ((0.000731 / latent) "' (Tov - wa))

End If

205

'Dimension output matrices

ReDim Output(NumTimeSteps, 2) Temperature output

ReDim Output_S(NumTimeSteps, 3) 'Salmonella linear lethality output
ReDim Output_SW(NumTimeSteps, 2) 'Salmonella logistic lethality output
ReDim AverageOutput(NumTimeSteps, 2) 'Yield and moisture output

ReDim AverageFat(NumTimeSteps) 'Element average fat content

'Dimension matrix of Element Physical Data

ReDim ElePhyData(NumMatlSets, 15) 'ElePhyData( ,1) Equation coef, Dx
'ElePhyData( ,2) Equation coef, Dy
'ElePhyData( ,3) Equation coef, G
'ElePhyData( ,4) Equation coef, Q
'ElePhyData( ,5) Equation coef, lamda
'ElePhyData( ,6) Equation coef, Dx_M
'ElePhyData( ,7) Equation coef, Dy_M
'ElePhyData( ,8) Equation coef, G_M
'ElePhyData( ,9) Equation coef, Q_M
'ElePhyData( ,10) Equation coef, lamda_M
'ElePhyData( ,l 1) Equation coef, Dx_F
'ElePhyData( ,12) Equation coef, Dy_F
'ElePhyData( ,13) Equation coef, G_F
'ElePhyData( ,14) Equation coef, Q_F
'ElePhyData( ,15) Equation coef, lamda_F

'Calculate Bandwidth
Call Cachandwidth(NumEle, EleNodeDataO, BandWidth)

'Dimension element matrices
'Element force vector

Dim EFQ(4) 'Enthalpy
Dim EFQ_M(4) 'Moisture
Dim EFQ_F(4) 'Fat

'Element force vector, derivative BC

Dim EFS(2) 'Enthalpy
Dim EFS_M(2) 'Moisture
Dim EFS_F(2) 'Fat

'Element stiffness matrix

Dim ESM(4, 4) 'Enthalpy
Dim ESM_M(4, 4) 'Moisture
Dim ESM_F(4, 4) 'Fat

'Element stiffness matrix, derivative BC

Dim EKM(2, 2) 'Enthalpy
Dim EKM_M(2, 2) 'Moisture
Dim EKM_F (2, 2) 'Fat

'Element subscript values

206

Dim EleSub(4)

'Element Phi values (nodal unknown values)

Dim ElePhiVal(4) 'Enthalpy
Dim ElePhiVal_M(4) 'Moisture
Dim ElePhiVal_F (4) 'Fat

'Element capacitance matrix

Dim ECM(4, 4) 'Enthalpy
Dim ECM_M(4, 4) 'Moisture
Dim ECM_F(4, 4) 'F at

'Element capacitance matrix, derivative BC

Dim ECQ(4, 4) 'Enthalpy
Dim ECQ_M(4, 4) 'Moisture
Dim ECQ_F (4, 4) 'F at

'Dimension the arrays for the global system of equations

NumEleSub = 4
NumEleNodes = 4

'Vectors used in finite difference time solution

Temporary solution vector

ReDim Temp(NumNodes) 'Enthalpy
ReDim Temp_M(N umN odes) 'Moisture
ReDim Temp_F(NumNodes) 'F at

'Global force vector

ReDim GFV(NumNodes) 'Enthalpy
ReDim GFV_M(NumNodes) 'Moisture
ReDim GF V_F(N umN odes) 'Fat
ReDim GFV_A(NumNodes) 'Enthalpy
ReDim GFV_M_A(NumNodes) 'Moisture
ReDim GFV_F_A(NumNodes) 'Fat
ReDim GF V_Star(NumN odes) 'Enthalpy
ReDim GFV_M_Star(NumN odes) 'Moisture
ReDim GFV_F_Star(NumNodes) 'Fat

'Global stiffness matrix

ReDim GSM(NumNodes, BandWidth) 'Enthalpy
ReDim GSM_M(NumNodes, BandWidth) 'Moisture
ReDim GSM_F(NumNodes, BandWidth) 'Fat

'Global capacitance matrix

207

ReDim GCM(NumNodes, BandWidth) 'Enthalpy
ReDim GCM_M(NumN odes, BandWidth) 'Moisture
ReDim GCM_F(NumNodes, BandWidth) 'Fat

'Global A matrix used for FD time solution

ReDim GAM(NumNodes, BandWidth) 'Enthalpy
ReDim GAM_M(NumNodes, BandWidth) 'Moisture
ReDim GAM_F(NumNodes, BandWidth) 'Fat

'Global P matrix used for FD time solution

ReDim GPM(NumNodes, BandWidth) 'Enthalpy
ReDim GPM_M(NumNodes, BandWidth) 'Moisture
ReDim GPM_F(NumNodes, BandWidth) 'Fat

'Vector used for volume averaging

ReDim Volume(NumEle)

 

' FEM / Finite difference time solution

 

'Zero the global matrices

Call ZeroGlobalMatrices(NumNodes, BandWidth, GFV(), GSM(), GFV_M(), GSM_M(), GFV_F(),
GSM_FO. GCMO. GCM_MO, GCM_F())

'Show message indicating program is running

Form6.Show

'Initialize the global matrices for the first time step of the time solution
T = 1

'Build the banded system of equations

For K = 1 To NumEle
Call CalculateProperties(Meat, Solid, NumEle, EleNodeData(), E1ePhyData(), temperature(),
GSV_M(), GSV_F(), moisture_capacity, density)
Call ESMatrix2DField(EleMatlData(), EleNodeData(), Coord(), E1ePhyData(), EFQ(),
ESM(),
EFQ_M(), ESM_M(), EFQ_F(), ESM_F(), Volume(l. KK)
Call EleCapMatrix(ElePhyData(), EleNodeData(), ECM(), ECM_M(), ECM_F(), Coord(),
KK)
Call EleSubscriptValues(EleNodeData(), KK, NumEleSub, EleSub())
Call BuildBandedSystem(NumEleSub, EleSub(), EFQ(), ESM(), GFV(), GSM(), ECM(),
GCMO,
EFQ_M(), ESM_M(), GFV_M(), GSM_M(), ECM_M(), (ECM_M(), EFQ_F(),
ESM_F(),
GFV_F(), GSM_F(), ECM_F(), GCM_F())
Next K

'Add in the derivative boundary conditions when they occur - direct stiffness method

208

NumEleSub = 2

For I = 1 To NumDeridey
Call CalDerivBC(heat_capacity_f, frozenH, latentW, latentF, I, Coord(), SubDerivBC(),
EleSub(),
EKM(), EFS(), GSVO, EKM_MO, EFS_MO, GSV_M(), EKM_F(), EFS_FO,
GSV_F(),
OvenConditions(), T, BoundaryTypeO, InitialM, InitialF, moisture_capacity, density,
temperature(), ConcOvenAir)
Call BuildBandedSystem(NumEleSub, EleSub(), EFS(), EKM(), GFV(), GSM(), ECQ(),
GCM(),
EFS_MO, EKM_MO. GFV_M(), GSM_M(), ECQ_MO. GCM_M(), EFS_FO,
EKM_F(),
GFV_F(), GSM_F(), ECQ_FO, GCM_F())
Next I

'Set force vector values for next time step

For X = 1 To NumNodes
GFV_A(X) = GFV(X)
GFV_M_A(X) = GFV_M(X)
GFV_F_A(X) = GFV_F(X)
Next X

'Begin time stepping solution
For T = 1 To NumTimeSteps
'Construct global matrices using direct stiffness method
For K = 1 To NumEle
Call CalculateProperties(Meat, Solid, NumEle, EleNodeData(), E1ePhyData(),

temperature(), GSV_M(), GSV_F(), moisture_capacity, density)
Call ESMatrix2DField(EleMatlData(), EleNodeData(), Coord(), E1ePhyData(),

EFQ(),
ESM(), EFQ_M(), ESM_M(), EFQFO, ESM_F(), Volume(), KK)
Call EleCapMatrix(ElePhyData(), EleNodeData(), ECM(), ECM_M(), ECM_F(),
Coord(), KK)
Call EleSubscriptValues(EleNodeData(), KK, NumEleSub, EleSub())
Call BuildBandedSystem(NumEleSub, EleSub(), EFQ(), ESM(), GFV(), GSM(),
ECM(),
GCM(), EFQ_M(), ESM_M(), GFV_M(), GSM_M(), ECM_M(),
GCM_M(),
EFQFO, ESM_F(), GFV_F(), GSM_F(), ECM_F(), GCM_F())
Next K
'Add in the derivative boundary conditions when they occur - direct stiffness method
NumEleSub = 2
For I = 1 To NumDeridey
Call CalDerivBC(heat_capacity_f, frozenH, latentW, latentF, I, Coord(),
SubDerivBC(),

209

EleSub(), EKM(), EFS(), GSV(), EKM_MO, EFS_M(), GSV_M(),

EKM_F(), EFS_F(), GSV_F(), OvenConditions(), T, BoundaryTypeO,

InitialM, InitialF, moisture_capacity, density, temperature(), ConcOvenAir)
Call BuildBandedSystem(NumEleSub, EleSub(), EFS(), EKM(), GFV(), GSM(),

GCM(), EFS_M(), EKM_MO, GFV_M(), GSM_M(), ECQ_M(), '
GCM_M(),
EFS_F(), EKM_F(), GFV_F(), GSM_F(), ECQ_FO, GCM_F())
Next I

'Build matrices for finite difference (time) solution

Call BuildGAM(BandWidth, GAM(), GCM(), GSM(), GAM_MO, GCM_M(), GSM_M(),
GAM_F(), GCM_F(), GSM_F(), timestep, NumNodes)

Call BuildGPM(BandWidth, GPM(), GCM(), GSM(), GPM_M(), GCM_M(), GSM_M(),
GPM_F(), GCM_F(), GSM_F(), timestep, NumNodes)

Call ModifyGFV(GFV(), GFV_A(), GFV_StarO, GFV_M(), GFV_M_AO, GFV_M_StarO,
GFV_F_A(), GFV_F_StarO, GFV_F(), timestep, NumNodes)

'Solution of equations

Call MultpyBandMatrix(NumNodes, BandWidth, GPM(), GSV(), Temp())
Call MultpyBandMatrix(NumNodes, BandWidth, GPM_M(), GSV_M(), Temp_M())
Call MultpyBandMatrix(NumNodes, BandWidth, GPM_F(), GSV_F(), Temp_F())

For I = 1 To NumNodes
Temp(l) = Temp(I) + GFV_Star(l)
Temp_M(I) = Temp_M(l) + GFV_M_Star(I)
Temp_F (I) = Temp_F(I) + GFV_F_Star(I)
GFV_A(I) = GFV(I)
GFV_M_A(I) = GFV_M(I)
GFV_F_A(I) = GFV_F(I)

Next I

Call DecompBandMatrix(NumNodes, BandWidth, GAM())

Call DecompBandMatrix(NumNodes, BandWidth, GAM_MO)

Call DecompBandMatrix(NumNodes, BandWidth, GAM_F())

Call SolveBandMatrix(NumNodes, BandWidth, GSV(), Temp(), GAM())

Call SolveBandMatrix(NumNodes, BandWidth, GSV_M(), Temp_M(), GAM_MO)

Call SolveBandMatrix(NumNodes, BandWidth, GSV_F(), Temp_F(), GAM_F())

'Convert enthalpy data back to temperature for display

Call ConverthoTemp(heat_capacity__f, GSV(), GSV_M, GSV_F, temperature(), NumNodes,
InitialM, Solid, frozenH, latentW, latentF, heat_capacity, LevelOne, LevelTwo,
LevelThree, LevelF our, LevelFive)

'Calculate surface fat for next time step

Call CalculateSurfaceFatContent(InitialF, T, GSV_F(), NudeyNodes, deNodeO,
temperature(), T)

'Calculate Salmonella reduction

Call CalculateSurvivors(NumNodes, timestep, temperature(), Ninitial, No(), NW(), Nnew(),

210

NnewW(), logreduction(), logreductionW, Dvalue, Tref, Z, TimeToLimit, TR)
'Volume averaging of Salmonella reduction

Call CalculateElementAverage(Volume(), NumEle, EleNodeData(), Nnew(),
WeightedAverage)

AverageN(T) = WeightedAverage

Reduction = -Log(AverageN(T) / Ninitial) / Log(10)

'Calculate the average moisture and FF of each element for overall yield and bulk moisture
determination

'Moisture

Call CalculateElementAverage(Volume(), NumEle, EleNodeData(), GSV_M(),
WeightedAverage)

'F at

Call CalculateElementAverage(Volume(), NumEle, EleNodeData(), GSV_F(),
WeightedAverage)

AverageFat(T) = WeightedAverage

F = AverageFat(T)

'Calculate yield from moisture and fat percentages

 

 

Fat = F " Solid ‘Mass fat based on 100 g initial
Water = M * (Fat + Solid) / (1 - M) 'Mass water based on 100 g initial
Total = Fat + Water + Solid Total mass

' OUTPUT SECTION

'Graph yield

AverageOutput(T, 2) = Total 'Yield

AverageOutput(O, 2) = "Yield" 'Graph label

'Graph moisture

AverageOutput(T, 1) = WeightedAverage 'Moisture
AverageOutput(O, 1) = "M" 'Graph label
M = AverageOutput(T, 1)/ 100

'Graph temperature output

Output(O, l) = "Center Temp" 'Graph labels
Output(O, 2) = "Surface Temp" 'Graph labels
Output(T, l) =temperature(1) 'Center temperature
Output(T, 2) = temperature(4) 'Surface temperature

'Graph Salmonella reduction

Output_S(0, l) = "Log reduction" 'Graph label

211

Next T

Output_S(O, 2) = "Target reduction" 'Graph label

Output_S(T, l) = logreduction(l) 'Graph Salmonella reduction from linear eqn.
Output_S(T, 2) = TR 'Graph target reduction

Output_S(T, 3) = logreductionW ‘Graph Salmonella reduction from logistic eqn.

'Write data to output file

Write #3, temperature(l), temperature(Z), temperature(4), Total, M, logreduction( 1 ),
logreductionW, Reduction

'Labels for Salmonella reduction graph

If logreduction(l) < TR Then
Form2.Labe13.Caption = "Log reduction not acheived!"
Form2.Label4.Visible = False
F onn2.Labe15.Visible = False

Else
F orm2.Label3.Caption = "Log reduction acheived atz"
Form2.Label4.Visible = True
Form2.Labe15.Visible = True
Form2.Label4.Caption = TimeToLimit
Form2.LabelS.Caption = "s"

End If

'End time stepping

'Digital display of endpoint data

Form2.Label8.Caption = Round(temperature(4), 1) 'Display surface temperature
F orm2.Label9.Caption = Round(temperature( 1 ), 1) 'Display center temperature
F orm2.Labe112.Caption = Round(Total, 1) 'Display yield
Form2.Labell3.Caption = Round((M "‘ 100), 1) 'Display moisture content

'Remove

run message box

F orm6.Hide

'Display graphs

Form2.MSChartl .ChartData = Output Temperature graph
F orrn2.MSChart2.ChartData = AverageOutput 'Yield/Moisture graph
Form2.MSChart3.ChartData = Output_S 'Salmonella graph

Close #3

End Sub

212

Module — CacherivBC

Sub CalDerivBC(heat_capacity_f, frozenH, latentW, latentF, I, Coord(), SubDerivBC(), EleSub(),
EKM(),
EFS(), GSV(), EKM_MO. EFS_M(), GSV_M(), EKM_F(), EFS_F(), GSV_F(),
OvenConditions(), T, BoundaryTypeO, InitialM, InitialF, moisture_capacity, density,
temperature(), ConcOvenAir)

This subroutine calculates the element contributions resulting from
'derivative boundary conditions

'Assign the node numbers of the side to the array EleSub()

EleSub(l) = SubDerivBC(I, 1)
EleSub(Z) = SubDerivBC(I, 2)

'Grab oven conditions

OvenAirT = OvenConditions(T, 1)
OvenAirM = OvenConditions(T, 2)
OvenAirV = OvenConditions(T, 3) "' 100

'Evaluate the coefficients in the element matrices

XLength = Coord(SubDerivBC(I, 1), l) - Coord(SubDerivBC(I, 2), l)
YLength = Coord(SubDerivBC(I, l), 2) - Coord(SubDerivBC(I, 2), 2)

Ri = Coord(SubDerivBC(I, l), l)
Rj = Coord(SubDerivBC(I, 2), 1)

SideLength = Sqr(XLength " 2 + YLength A 2)
'Get temperature and moisture for each boundary

' T1 = temperature at node 1 of boundary
' T2 = temperature at node 2 of boundary

T1 = l "' temperature(SubDerivBC(I, 1))
T2 = l * temperature(SubDerivBC(I, 2))
Tave = (Tl + T2) / 2

' M1 = moisture at node 1 of boundary (decimal)
' M2 = moisture at node 2 of boundary (decimal)

M] = GSV_M(SubDerivBC(I, 1))
M2 = GSV_M(SubDerivBC(I, 2))
Mave=(M1 ’1+M2*1)/2
If Mave < 0 Then Mave = 0.01

'Convert moisture content to dry basis

MDB = (100 "‘ Mave) / (100 - Mave)

213

 

'Oven geometry

Height = 6.35 'cm
S = 4.6482 'cm

E = 1.17 'cm
W = 0.635 'cm
F = W / (2 "‘ S) 'Martin Equation parameter

fo = (60 + 4 * (Height / W - 2) A 2) A -0.5 'Martin Equation parameter

 

'Calculate air physical properties as functions of temperature and steam content
'Steam concentration g/cmA3

If OvenAirT <= 100 Then
DensitySatSteam = 8.12078904E-10 “ OvenAirT A 3 - 0.000000035203247 * OvenAirT A 2 +
0.00000131977474 * OvenAirT + 0.000000621530732
Elself OvenAirT > 100 Then
DensitySatSteam = -0.0002 * Log(OvenAirT) + 0.0015 'g/cmA3
End If

If Forml.Checkl.Value = 1 Then
OvenAirM = ConcOvenAir
density_air = -0.00000000002 * OvenAirT A 3 + 0.00000001 "‘ OvenAirT A 2 - 0.000004 "‘
OvenAirT + 0.0013 'g/cmA3
X_S = OvenAirM / (OvenAirM + density_air)
Else
X_S = OvenAirM/ 100
End If

'Mass fraction air
X_A = 1 - X_S

'Molar weight of steam and air

M_steam = 18
M_air = 28

S_a = 79 'K
S_s = 559.5 'K
'Latent heat

latent = 1.2 * (-2.429 * Tave + 2502.8) 'J/g
'Air and steam viscosity

viscosity_air = -0.0000000002 "‘ OvenAirT A 2 + 0.0000005 "‘ OvenAirT + 0.0002 'g/cm s
viscosity_steam = -0.00000001 "’ OvenAirT A 2 + 0.00004 * OvenAirT + 0.0089 'g/cm 3

phi_as = 1 / 8 A 0.5 * ((1 + (viscosity_air / viscosity_steam) A 0.5 * (M_steam / M_air) A 0.25) A 2 / (1

+
(M_air / M_steam)) A 0.5) '-

214

phi_sa = l / 8 A 0.5 * ((1 + (viscosity_steam / viscosity_air) A 0.5 * (M_air / M_steam) A 0.25) A 2 / (l
+

(M_steam / M_air)) A 0.5) '-
’Mixture viscosity

viscosity_mix = viscosity_air / (1 + phi_as * X_S / X_A) + viscosity_steam / (1 + phi_sa * X_A/
X_S) 'g/cm s

'Air and mixture density

density_air = 000000000002 * OvenAirT A 3 + 0.0000000] * OvenAirT A 2 - 0.000004 * OvenAirT
+

0.0013 'g/cmA3
density_mix = density_air "' X_A + DensitySatSteam * X_S 'g/cmA3
'Heat capacity
cpair = 1.01

cpsteam = 0.00000087429 * OvenAirT A 2 + 0.00018055 * OvenAirT + 1.8616 '1.888 '4.2
cpmix = (density_air * X_A "' cpair + DensitySatSteam * X_S * cpsteam) / (density_air "‘ X_A +
DensitySatSteam * X_S)

'Diffusivity
Dsa = (2.16 * 10 A -5 * ((OvenAirT + 2731/273) A 1.8) * 10000 'cmA2/s

A_as = 0.25 "' (1 + ((viscosity_air * viscosity_steam) "‘ (M_steam / M_air) A 0.75 + (1 + S_a/
(OvenAirT +

273)) / (l + S_s / (OvenAirT + 273))) A 0.5) A 2 "' (1 + 0.733 * (S_a "‘ S_s) A 0.5 / (OvenAirT
+

273)) / (l + S_a / (OvenAirT + 273)) '-
A_sa = 0.25 * (1 + ((viscosity_steam / viscosity_air) "' (M_air / M_steam) A 0.75 * (1 + S_s/
(OvenAirT +

273)) / (1 + S_a / (OvenAirT + 273))) A 0.5) A 2 * (l + 0.733 "‘ (S_a " S_s) A 0.5 / (OvenAirT
+

273)) / (1 + S_s / (OvenAirT + 273)) '-

'Thermal conductivity

k_air = 00000000003 "' OvenAirT A 2 + 0.0000008 "' OvenAirT + 0.002 'W/cm C

k_steam = 0.000000009 * OvenAirT + 0.0002 'W/cm C

k_mix = (k_air / (1 + A_as " (X_S / X_A)) + k_steam / (1 + A_sa "' (X_A / X_S))) 'W/cm
C

'Calculate Prandtl and Schmidt numbers

Pr_mix = cpmix * viscosity_mix / k_mix -
Sc_mix = viscosity_mix / (density_mix "' Dsa) '-

 

'Calculate oven steam moisture concentration
If Forml .Check1.Value = 1 Then

ConcOvenAir = ConcOvenAir
Else

215

ConcOvenAir = DensitySatSteam * OvenAirM / 100 'g/cmA3
End If

 

'Calculate model parameters
'Convection condition
If BoundaryType(l) = 1 Then
'Calculate transport coefi'rcients
Re = OvenAirV "' density_mix "' E / viscosity_mix

Nu = 0.037 " Re A 0.8 " Pr_mix A (l /3)
SH = 0.037 "' Re A 0.8 * Sc_mix A (1 / 3)

H_M=SH"'Dsa/E
h=Nu*k_mix/E

'Calculate surface moisture concentration

If Tave < 0 Then
Csat = 0.000000621530732
Else
Csat = 8.12078904E-10 "' Tave A 3 - 0.000000035203247 "' Tave A 2 +
0.00000131977474 * Tave + 0.000000621530732
End If

'Calculate effective relative humidity
RH = 100 "‘ (ConcOvenAir / Csat)

MDB = (Mave * 100) / (100 - Mave)
ERH = Exp((-5222.47 “ (MDB A -1.0983)) / (1.9818 * (Tave + 273)))
Cs = ERH * Csat

'Calculate energy fluxes
Q_conv = h * (OvenAirT - Tave)

If ConcOvenAir > Csat Then
Q_cond = H_M * latent "‘ (ConcOvenAir - Cs)
Q_evap = 0

Else
Q_cond = 0

End If

If Cs > ConcOvenAir Then
Q_evap = H_M * latent * (Cs - ConcOvenAir)
Q_cond = 0

Else
Q_evap = 0

Endlf

216

Q_total = Qficonv + Q_cond - Q_evap
'Effective heat transfer coefficient based on total flux (convection, condensing, evaporation)
heff = Q_total / (OvenAirT - Tave)
'Calculate equilibrium values under current conditions
'Enthalpy
Hequilibrium = OvenAirT * 3.35 + frozenH + latentW + latentF + (2 * 3.35)
'Moisture
If RH >= 97 Then
EMCWB = InitialM
Else
EMC = ((1.9818 ‘ (Tave + 273) "' Log(RH/ 100)) / ~5222.47) A (l / -l.0983)
EMCWB = (100 * EMC) / (100 + EMC)
End If
X = 1 "' InitialM
If X <= (EMCWB * 1) Then
EMCWB = X
End If
'FEM boundary coefficients

'Heat transfer

CoetM = heff "‘ SideLength / 6
CoefS = heff "' Hequilibrium "‘ SideLength / 2

'Moisture transfer
CoetM_M = H_M "' density " moisture_capacity " SideLength / 6 'Moisture transfer
CoefS_M = H_M * density "' moisture_capacity * EMCWB “ SideLength / 2
'Stein JSO-IV impingement condition
Elself BoundaryType(I) = 2 Then 'Impingement
'Calculate transport coefficients
Re = OvenAirV "‘ W "' density_mix / viscosity_mix
Nu=(2/3)*Pr_mixA0.42* foA0.75 ‘((2"'Re)/(F/fo+fo/F))A(2/3)
SI-I=(2/3)* Sc_mixA0.42 * foA0.75 * ((2 * Re)/(F/fo+fo/F))A(2/3)

H_M=SH"'Dsa/W
h=Nu"'k_mix/W

'Calculate surface moisture concentration

If Tave < 0 Then
Csat = 0.000000621530732

217

Else
Csat = 8.12078904E-10 " Tave A 3 - 0.000000035203247 * Tave A 2 +
0.00000131977474 * Tave + 0.000000621530732
End If

'Calculate effective relative humidity
RH = 100 * (ConcOvenAir / Csat)

MDB = (Mave * 100) / (100 - Mave)
ERH = Exp((-5222.47 * (MDB A -1.0983)) / (1.9818 * (Tave + 273)))
Cs = ERH "' Csat

'Calculate energy fluxes
Q_conv = h " (OvenAirT - Tave)

If ConcOvenAir > Cs Then
(Lcond = H_M * latent * (ConcOvenAir - Cs)
Q_evap = 0

Else
Q_cond = 0

End If

If Cs > ConcOvenAir Then
(Levap = H_M “ latent * (Cs - ConcOvenAir)
Q_cond = 0

Else
Q_evap = 0

End If

Q_total = (Lconv + Q_cond - Q_evap
'Effective heat transfer coefficient based on total flux (convection, condensing, evaporation)
heff = Qtotal / (OvenAirT - Tave)
'Calculate equilibrium values under current conditions
'Enthalpy
Hequilibrium = OvenAirT "‘ 3.35 + frozenH + latentW + latentF + (2 * 3.35) '792.64 + 6.44
'Moisture
If RH >= 97 Then
EMCWB =Initia1M
Else
EMC = ((1.9818 "‘ (Tave + 273) "' Log(RH/ 100)) / -5222.47) A (1 /-1.0983)
EMCWB = (100 * EMC) / (100 + EMC)
End If
X = 1 "' InitialM

IfX <= (EMCWB * 1) Then
EMCWB = x

218

End If
'FEM boundary coefficients
'Heat transfer

CoefM = heff "' SideLength / 6
CoefS = heff "‘ Hequilibrium "' SideLength / 2

'Moisture transfer

CoetM__M = H_M "' density * moisture_capacity * SideLength / 6 'Moisture transfer
CoetS_M = H_M * density * moisture_capacity * EMCWB * SideLength / 2

Elself BoundaryType(I) = 3 Then 'Small oven
W = 2.6
'Calculate transport coefficients
Re = OvenAirV * density_mix * W / viscosity_mix

Nu = 0.037 "' Re A 0.8 * Pr_mix A (1 /3)
SH = 0.037 * Re A 0.8 "‘ Sc_mix A (1 /3) '0.037 * Re A 0.8 * Sc_mix A (1 l3)

H_M=SH*Dsa/W
h=Nu"'k_mix/W

'Calculate surface moisture concentration

If Tave < 0 Then
Csat = 0.000000621530732
Else
Csat = 8.12078904E-10 "' Tave A 3 - 0.000000035203247 "‘ Tave A 2 +
0.00000131977474 "' Tave + 0.000000621530732
EndIf

'Calculate effective relative humidity

RH = 100 * (ConcOvenAir / Csat)

MDB = (Mave "' 100) / (100 - Mave)

ERH = Exp((-5222.47 * (MDB A -1.0983)) / (1.9818 " (Tave + 273)))
Cs = ERH * Csat

'Calculate energy fluxes

Q_conv = h * (OvenAirT - Tave)

If ConcOvenAir > Csat Then
Q_cond = H_M * latent "‘ (ConcOvenAir - Cs)
Q_evap = 0

Else
Q_cond = 0

219

End If

If Cs > ConcOvenAir Then
Q_evap = H_M * latent * (Cs - ConcOvenAir)
Q_cond = 0

Else
Q_evap = 0

Endlf

Q_total = Q_conv + Q_cond - Q_evap

'Effective heat transfer coefficient based on total flux (convection, condensing, evaporation)
heff = Q_total / (OvenAirT - Tave)

'Calculate equilibrium values under current conditions

'Enthalpy

Hequilibrium = OvenAirT "‘ 3.35 + frozenH + latentW + latentF + (2 "‘ 3.35)
'Moisture

If RH >= 97 Then
EMCWB = InitialM

Else
EMC = ((1.9818 "' (Tave + 273) "' Log(RH/ 100)) / -5222.47) A (1 /-l.0983)
EMCWB = (100 "' EMC) / (100 + EMC)

End If

X = 1 "' InitialM

IfX <= (EMCWB " 1) Then
EMCWB = X

End If

'FEM boundary coefficients

'Heat transfer

CoefM = heff "' SideLength / 6
CoefS = heff "‘ Hequilibrium * SideLength / 2

'Moisture transfer

CoefM_M = H_M * density "' moisture_capacity "' SideLength / 6 'Moisture transfer
CoefS_M = H_M * density * moisture_capacity * EMCWB * SideLength / 2

Elself BoundaryType(l) = 4 Then 'Stein 102 oven

8 'height above belt
1.27 'nozzle diameter
6

h
d
L 'nozzle spacing

220

'F = 0.906 * (d / l) A 2 'offset rows configuration
' o o o o

o o o

F = 0.785 * (d / L) A 2 'even rows configuration
' o o o o

o o o 0

'Calculate transport coefficients
Re = OvenAirV * d "' density_mix / viscosity_mix

KHDF = (1 + ((h / d) / (0.6 / Sqr(F))) A 6) A -0.05
GFHD = ((1 - (2.2 * Sqr(F))) / (1 + 0.2 * (h / d - 6) * Sqr(F))) * 2 * Sqr(F)
FREARN = 0.5 * (Re A 0.66)

Nu = (Pr_mix A 0.42) “ KHDF "' GFHD * FREARN
SH = (Sc_mix A 0.42) * KHDF " GFHD "' FREARN

H_M=SH*Dsa/d
h=Nu"'k_mix/d

'Calculate surface moisture concentration

If Tave < 0 Then
Csat = 0.000000621530732
Else
Csat = 8.12078904E-10 * Tave A 3 - 0.000000035203247 * Tave A 2 +
0.00000131977474 * Tave + 0.000000621530732
End If

'Calculate effective relative humidity

RH = 100 * (ConcOvenAir/ Csat)

MDB = (Mave "‘ 100) / (100 — Mave)

ERH = Exp((-5222.47 "‘ (MDB A -1.0983)) / (1.9818 * (Tave + 273)))
Cs = ERH * Csat

'Calculate energy fluxes

Q_conv = h * (OvenAirT - Tave)

If ConcOvenAir > Cs Then
Q_cond = H_M * latent * (ConcOvenAir - Cs)

Q_evap = 0

Q_cond = 0

Else
End If

If Cs > ConcOvenAir Then
Q_evap = H_M * latent * (Cs - ConcOvenAir)
Q_cond = 0

Else
Q_evap = 0

End If

221

Endlf

Q_total = Qconv + Q_cond - Q_evap

'Effective heat transfer coefficient based on total flux (convection, condensing, evaporation)
heff = Q_total / (OvenAirT - Tave)

'Calculate equilibrium values under current conditions

'Enthalpy

Hequilibrium = OvenAirT * 3.35 + frozenH + latentW + latentF '792.64 + 6.44
'Moisture

If RH >= 97 Then
EMCWB = InitialM

Else
EMC = ((1.9818 " (Tave + 273) "' Log(RH/ 100)) / -5222.47) A (l / -l .0983)
EMCWB = (100 * EMC) / (100 + EMC)

End If

X = 1 * InitialM
If X <= (EMCWB * 1) Then
EMCWB = X
Endlf
'F EM boundary coefficients

'Heat transfer

CoefM = heff * SideLength / 6
CoefS = heff "' Hequilibrium * SideLength / 2

'Moisture transfer

CoefM_M = H_M * density * moisture_capacity "' SideLength / 6 'Moisture transfer
CoefS_M = H_M * density " moisture_capacity * EMCWB "' SideLength / 2

 

'Evaluate the element matrices

EKM(1,1)=2"‘ 3.14/2 ‘CoefM*(3 * Ri+Rj)
EKM(1,2)=2 * 3.14/2 * CoeflVl *(R1+Rj)
EKM(2,1)=2 * 3.14/2 * CoefM . (Ri +Rj)
EKM(2,2)=2 * 3.14/2 * CoefM *(Ri+3 * Rj)

EKM_M(1,1)= 2 * 3.14/2 * CoefM_M * (3 * Ri +Rj)
EKM_M(I, 2) = 2 * 3.14/2 * CoefM_M * (Ri + Rj)
EKM_M(2, 1) = 2 * 3.14/2 * CoefM_M * (R1 + Rj)
EKM_M(2, 2) = 2 * 3.14 / 2 * CoefM_M * (R1 + 3 * Rj)

222

EFS(1)=2*3.14/3‘CoetS*(2*Ri+Rj)
EFS(2)=2*3.14/3‘CoefS*(Ri+2"'Rj)

EFS_M(I) =2 * 3.14/3 * CoefS_M * (2 * R1 + Rj)
EFS_M(2) = 2 * 3.14 / 3 * CoefS_M * (R1 + 2 * Rj)

End Sub

Module — MultplyBandMatrix

Sub MultpyBandMatrix(NumNodes, BandWidth, GSM(), GFV(), ProdVectorO)
This module provided by Dr. Larry Segerlind - Michigan State University

This subprogram multiplies a symmetric banded matrix and a column vector
' The banded matrix is stored as a rectangular array and only the
' coefficients on and above the main diagonal are stored in the array.

For I = 1 To NumNodes
Sum=0!
K=I-l

For J = 2 To BandWidth
M = J +1 - 1
If (M <= NumNodes) Then
Sum = Sum + GSM(I, J) * GFV(M)
End If
If (K > 0) Then
Sum = Sum + GSM(K, J) * GFV(K)
K = K - 1
End If
Next J
ProdVector(I) = Sum + GSM(I, 1) * GFV(I)
Next I

End Sub

223

Module - EleSubscriptValues

Sub EleSubscriptValues(EleNodeData(), KK, NumEleSub, EleSub())
'This module provided by Dr. Larry Segerlind - Michigan State University
' This subprogram calculates the subscripts associated with the

' element. The subprogram allows the element to have one, two or

' three unknown values at a node.

NumEleSub = 4
If (EleNodeData(KK, 4) = 0) Then NumEleSub = 3

EleSub(l) = EleNodeData(KK, 1)
EleSub(2) = EleNodeData(KK, 2)
EleSub(3) = EleNodeData(KK, 3)
EleSub(4) = EleNodeData(KK, 4)

End Sub

224

Module - ESMatrix2DField

Sub ESMatrix2DField(E1eMatlData(), EleNodeData(), Coord(), E1ePhyData(), EFV(), ESM(),
EFV_M(), ESM_M(), EFV__F(), ESM_F(), Volume(), KK)

' This module heavily modified from a module by Dr. Larry Segerlind - Michigan State University

This subprogram calculates the element stiffness matrix and element
' force vector for the three node triangular and the four node
' bilinear rectangular element

'Evaluation of the parameters in the element stiffness matrix
MatlSet = EleMatlData(KK)
Thermal properties

Dx = ElePhyData(MatlSet, l)
Dy = ElePhyData(MatlSet, 2)
G = ElePhyData(MatlSet, 3)
Q = ElePhyData(MatlSet, 4)

'Moisture transfer properties

Dx_M = ElePhyData(MatlSet, 6)
Dy_M = ElePhyData(MatlSet, 7)
G_M = ElePhyData(MatlSet, 8)
QM = ElePhyData(MatlSet, 9)

'Fat transfer properties

Dx_F = ElePhyData(MatlSet, ll)
Dy_F = ElePhyData(MatlSet, 12)
G_F = ElePhyData(MatlSet, 13)
Q_F = ElePhyData(MatlSet, 14)

'Evaluate the element stiffness matrix

If (EleNodeData(KK, 4) = 0) Then Triangular Element
XI = Coord(EleNodeData(KK, 1), 1)
Y1 = Coord(EleNodeData(KK, 1), 2)
Xj = Coord(EleNodeData(KK, 2), l)
Yj = Coord(EleNodeData(KK, 2), 2)
Xk = Coord(EleNodeData(KK, 3), 1)
Yk = Coord(EleNodeData(KK, 3), 2)

Ai=Xj*Yk-Xk"Yj
Aj=Xk"'Yl-XI*Yk
Ak=XI*Yj-Xj"‘YI

Bi=Yj-Yk

Bj=Yk-Yl
Bk=YI~Yj

225

Rbar *

Rbar *

Rbar *

Rbar "'

Rbar *

Rbar

3.14

3.14

c1=xrc-x1
Q=XLXk
Ck=Xj-Xl

TwiceArea = Ai + Aj + Ak
Rbar=(XI+Xj +Xk)/3
Volume(KK) = TwiceArea "' 3.14 * Rbar

DxOver4A = Dx / (2 "' TwiceArea)
DyOver4A = Dy / (2 "' TwiceArea)
GAreaOver12 = G "' TwiceArea / 24

Dx_MOver4A = Dx_M / (2 " TwiceArea)
Dy_MOver4A = Dy_M / (2 " TwiceArea)
G_MAreaOverl2 = G_M " TwiceArea / 24

Dx_FOver4A = Dx_F / (2 "' TwiceArea)
Dy_FOver4A = Dy_F / (2 "‘ TwiceArea)
G_FAreaOverlZ = G_F "‘ TwiceArea / 24

'Calculate the element stiffness matrix - enthalpy
ESM(1,1)= 2 * 3.14 “ Rbar "‘ (DxOver4A "' Bi "‘ Bi + DyOver4A * Ci * Ci) + 2 "‘ 3.14 *

GAreaOver12 " 2
ESM(I, 2) = 2 "‘ 3.14 " Rbar "‘ (DxOver4A * Bi " Bj + DyOver4A * Ci * Cj) + 2 * 3.14 "'

GAreaOver12
ESM(l, 3) = 2 "' 3.14 * Rbar “ (DxOver4A * Bi * Bk + DyOver4A * Ci * Ck) + 2 * 3.14 "'

GAreaOver12
ESM(2, l) = ESM(l, 2)
ESM(2, 2) = 2 " 3.14 " Rbar * (DxOver4A "' Bj * Bj + DyOver4A * Cj "' Cj) + 2 "' 3.14 *

GAreaOver12 * 2
ESM(2, 3) = 2 "' 3.14 "' Rbar "' (DxOver4A * Bj * Bk + DyOver4A * Cj * Ck) + 2 * 3.14 "

GAreaOver12

ESM(3, l) = ESM(l, 3)

ESM(3, 2) = ESM(2, 3)

ESM(3, 3) = 2 "' 3.14 "' Rbar "‘ (DxOver4A " Bk * Bk + DyOver4A * Ck * Ck) + 2 "' 3.14 *
" GAreaOver12 "' 2

'Calculate the element stiffness matrix - moisture

ESM_M(l, l) = 2 "' 3.14 * Rbar "' (Dx_MOver4A "' Bi "‘ Bi + Dy_MOver4A " Ci "' Ci) + 2 "'

* Rbar * G_MAreaOverlZ * 2
ESM_M(l, 2) = 2 * 3.14 * Rbar * (Dx_MOver4A * 131 r 131 + Dy_MOver4A * Ci * Cj) + 2 *

"' Rbar * G_MAreaOverlZ

ESM_M(l, 3) = 2 * 3.14 "' Rbar "' (Dx_MOver4A * Bi * Bk + Dy_MOver4A "‘ Ci " Ck) + 2"I
3.14 "' Rbar * G_MAreaOverlZ

ESM_M(Z, 1) = ESM_M(l, 2)

226

3.14

3.14*

3.14*

3.14*

3.14*

3.14*

3.14

Else

ESM_M(2, 2) = 2 * 3.14 * Rbar * (Dx_MOver4A * Bj * Bj + Dy_MOver4A * Cj * Cj) + 2 *

* Rbar * G_MAreaOver12 * 2
ESM_M(2, 3) = 2 * 3.14 "‘ Rbar " (Dx_MOver4A * Bj * Bk + Dy_MOver4A * Cj " Ck) + 2

3.14 * Rbar * G_MAreaOver12

ESM_M(3, l) = ESM_M(l, 3)

ESM_M(3, 2) = ESM_M(2, 3)

ESM_M(3, 3) = 2 "' 3.14 * Rbar * (Dx_MOver4A "‘ Bk * Bk + Dy_MOver4A * Ck * Ck) + 2
3.14 * Rbar "' G_MAreaOver12 " 2

'Calculate the element stiffness matrix - fat

ESM_F(I, 1) = 2 * 3.14 * Rbar * (Dx_FOver4A * Bi "' Bi + Dy_FOver4A * Ci * Ci) + 2 *

Rbar * G_FAreaOver12 * 2
ESM_F(l, 2) = 2 * 3.14 * Rbar " (Dx_FOver4A "' Bi * Bj + Dy_FOver4A * Ci * Cj) + 2 *

Rbar * G_FAreaOver12
ESM_F(l, 3) = 2 "' 3.14 "' Rbar * (Dx_FOver4A * Bi * Bk + Dy_FOver4A * Ci * Ck) + 2 *

Rbar * G_FAreaOver12
ESM_F(2, l) = ESM_F(I, 2)
ESM_F(2, 2) = 2 * 3.14 " Rbar * (Dx_FOver4A * Bj * B] + Dy_FOver4A * Cj * Cj) + 2 "'

Rbar "' G_FAreaOver12 "‘ 2
ESM_F(2, 3) = 2 "’ 3.14 " Rbar * (Dx_FOver4A * Bj * Bk + Dy_FOver4A * Cj * Ck) + 2 "

Rbar "' G_FAreaOver12

ESM_F(3, 1) = ESM_F(I, 3)

ESM_F(3, 2) = ESM_F(2, 3)

ESM_F(3, 3) = 2 * 3.14 * Rbar "' (Dx_FOver4A * Bk * Bk + Dy_FOver4A * Ck * Ck) + 2 *
"' Rbar * G_FAreaOver12 * 2

'Calculate the element force vector - enthalpy

EFV(l) = (2 * 3.14 / 4) * (Q * TwiceArea/6) "' (2 * XI + Xj + Xk)
EFV(2) = (2 "‘ 3.14 / 4) "‘ (Q * TwiceArea / 6) "' (XI + 2 * Xj + Xk)
EFV(3) = (2 "‘ 3.14 / 4) "' (Q "' TwiceArea / 6) * (XI + Xj + 2 "‘ Xk)
'Calculate the element force vector - moisture

EFV_M(1)=(2 * 3.14 / 4) "‘ (QM "' TwiceArea / 6) “ (2 "' XI + Xj + Xk)
EFV_M(2) = (2 * 3.14 / 4) “ (QM "‘ TwiceArea / 6) "' (XI + 2 * Xj + Xk)
EFV_M(3) = (2 * 3.14 / 4) * (QM "‘ TwiceArea / 6) "‘ (XI + Xj + 2 * Xk)
'Calculate the element force vector - fat

EFV_F(l) = (2 * 3.14 / 4) "‘ (QF "' TwiceArea / 6) * (2 "' XI + Xj + Xk)
EFV_F(2) = (2 * 3.14 / 4) "' (QF * TwiceArea / 6) * (XI + 2 "‘ Xj + Xk)
EFV_F(3) = (2 "' 3.14 / 4) "‘ (QF * TwiceArea / 6) * (XI + Xj + 2 "' Xk)

'Rectangular element

227

XLength = Coord(EleNodeData(KK, 2), 1) - Coord(EleNodeData(KK, l), l)

YLength = Coord(EleNodeData(KK, 4), 2) - Coord(EleNodeData(KK, 1), 2)

Area = XLength * YLength

Rbar = (Coord(EleNodeData(KK, l), 1) + Coord(EleNodeData(KK, 2), 1) +
Coord(EleNodeData(KK, 3), l) + Coord(EleNodeData(KK, 4), 1)) / 4

Volume(KK) = Area " 2 * Rbar * 3.14

Donver6B = Dx "‘ YLength / (6 "' XLength)
DyBover6A = Dy "' XLength / (6 * YLength)
GAreaover36 = G "‘ Areal 36

Dx_Moner63 = Dx_M * YLength / (6 "‘ XLength)
Dy_MBover6A = Dy_M "' XLength / (6 “ YLength)
G_MAreaover36 = G_M * Area / 36

Dx_Foner63 = Dx_F * YLength / (6 "' XLength)
Dy_FBover6A = Dy_F "' XLength / (6 * YLength)
G_FAreaover36 = G_F "' Area / 36

'Calculate element stiffness matrix - enthalpy

ESM(l, 1) = 2 * Donver68 + 2 * DyBover6A + 4 * GAreaover36
ESM(l, 2) = -2 * Donver63 + DyBover6A + 2 * GAreaover36
ESM(l, 3) = -Donver68 - DyBover6A + GAreaover36
ESM(l, 4) = Donver6B - 2 * DyBover6A + 2 "' GAreaover36
ESM(2, 1) = ESM(l, 2)

ESM(2, 2) = ESM(l, l)

ESM(2, 3) = ESM(l, 4)

ESM(2, 4) = ESM(l, 3)

ESM(3, 1) = ESM(l, 3)

ESM(3, 2) = ESM(2, 3)

ESM(3, 3) = ESM(l, 1)

ESM(3, 4) = ESM(l, 2)

ESM(4, 1) = ESM(l, 4)

ESM(4, 2) = ESM(2, 4)

ESM(4, 3) = ESM(3, 4)

ESM(4, 4) = ESM(l, 1)

'Calculate element stiffness matrix - moisture

ESM_M(], l) = 2 * Dx_Moner68 + 2 "' Dy_MBover6A + 4 * G_MAreaover36
ESM_M(l, 2) = -2 * Dx_Moner68 + Dy_MBover6A + 2 "' G_MAreaover36
ESM_M(], 3) = -Dx_Moner68 - Dy_MBover6A + G_MAreaover36
ESM_M(l , 4) = Dx_Moner68 - 2 * Dy_MBover6A + 2 * G_MAreaover36
ESM_M(2, 1) = ESM_M(], 2)

ESM_M(2, 2) = ESM_M(], l)

ESM_M(2, 3) = ESM_M(], 4)

ESM_M(2, 4) = ESM_M( 1, 3)

ESM_M(3, 1)= ESM_M(l, 3)

ESM_M(3, 2) = ESM_M(2, 3)

ESM_M(3, 3) = ESM_M(l, l)

ESM_M(3, 4) = ESM_M(], 2)

ESM_M(4, l) = ESM_M(l, 4)

ESM_M(4, 2) = ESM_M(2, 4)

ESM_M(4, 3) = ESM_M(3, 4)

ESM_M(4, 4) = ESM_M(l, l)

228

End If

End Sub

'Calculate element stiffness matrix - fat

ESM_F(l, 1) = 2 * Dx_Foner63 + 2 "' Dy_FBover6A + 4 "' G_FAreaover36
ESM_F( 1 , 2) = -2 "' Dx_Foner68 + Dy_FBover6A + 2 * G_FAreaover36
ESM_F( l , 3) = -Dx_Foner63 - Dy_FBover6A + G_FAreaover36
ESM_F( 1, 4) = Dx_Foner6B - 2 "‘ Dy_FBover6A + 2 " G_FAreaover36
ESM_F(2, l) = ESM_F(], 2)

ESM_F(2, 2) = ESM_F(], l)

ESM_F(2, 3) = ESM_F(], 4)

ESM_F(2, 4) = ESM_F(l, 3)

ESM_F(3, 1) = ESM_F(I, 3)

ESM_F(3, 2) = ESM_F(2, 3)

ESM_F(3, 3) = ESM_F(l, l)

ESM_F(3, 4) = ESM_F(l, 2)

ESM_F(4, l) = ESM_F(], 4)

ESM_F(4, 2) = ESM_F(2, 4)

ESM_F(4, 3) = ESM_F(3, 4)

ESM_F(4, 4) = ESM_F(], 1)

'Calculate element force vector - enthalpy

EFV(1)= Q * Area/4
EFV(2) = EFV(l)
EFV(3) = EFV(l)
EFV(4) = EFV( 1)

'Calculate element force vector - moisture

EFV_M(1)= Q_M * Area / 4
EFV_M(2) = EFV_M(l)
EFV_M(3) = EFV_M(l)
EFV_M(4) = EFV_M(l)

'Calculate element force vector - fat
EFV_F(1)= Q_F "' Area / 4
EFV_F(2) = EFV_F( l)

EFV_F(3) = EFV_F( 1)
EFV_F(4) = EFV_F(l )

229

Module - InputBasicData

Sub InputBasicData(NumMatlSets, NumNodes, NumEle, Coord(), EleMatlData(), EleNodeData())
This module slightly modified from a module by Dr. Larry Segerlind - Michigan State University

This subprogram inputs the nodal coordinates, the element material
' index, the element node numbers and the equation coefficients Dx, Dy,
' G and Q

For I = 1 To NumNodes
Input #1, Coord(I, 1), Coord(I, 2)
Next I

For I = 1 To NumEle
Input #1, EleMatlData(I), EleNodeData(I, l), EleNodeData(I, 2), EleNodeData(I, 3),
EleNodeData(l, 4)

Next I

End Sub

230

Module — InputDerivBC

Sub InputDerivBC(NumDerivBC, NudeyNodes, deNodeO, SubDerivBC(), BoundaryTypeO)
'This module slightly modified from a module by Dr. Larry Segerlind - Michigan State University

This subprogram inputs the subscripts and coefficients for the
' derivative boundary condition

For I = 1 To NumDerivBC

Input #1, SubDerivBC(l, l), SubDerivBC(l, 2), BoundaryType(I)
Next I
For I = 1 To NudeyNodes

Input #1, deNode(I)
Next I

End Sub

231

Module — BuildGAM

Sub BuildGAM(BandWidth, GAM(), GCM(), GSM(), GAM_MO, GCM_M(), GSM_M(), GAM_F(),
GCM_F(), GSM_F(), timestep, NumNodes)

'This module calculates the A matrix for the finite difference time solution using the central
' difference method

For X = 1 To NumNodes

For Y = 1 To BandWidth
GAM(X, Y) = GCM(X, Y) + (GSM(X, Y) "‘ (timestep / 2))
GAM_M(X, Y) = GCM_M(X, Y) + (GSM_M(X, Y) "' (timestep / 2))
GAM_F(X, Y) = GCM_F(X, Y) + (GSM_F(X, Y) "‘ (timestep / 2))
Next Y
Next X

End Sub

232

Module - SolveBandMatrix

Sub SolveBandMatrix(NumNodes, BandWidth, GSV(), GFV(), GSM())
'This module provide by Dr. Larry Segerlind - Michigan State University

This subprogram solves the system of banded equations using the method

' of Gaussian elimination. The stiffness matrix has been decomposed

' prior to entering this subprogram using DecomposeBandMatrix.

' This subprogram decomposes the column vector GFV(NumNodalVal%) before
' solving the system using backward substitution

'Decompose the global force vector

ForI= 1 To (NumNodes- 1)
MJ =1 + BandWidth - 1
If (MJ > NumNodes) Then
M] = NumNodes
Endlf
NJ = I + l
L = 1

For J = N] To M]
L = L + 1
GFV(J) = GFV(J) - GSM(I, L) * GFV(I) / GSM(I, 1)
Next 1
Next I

' Solution by backward substitution
GSV(NumNodes) = GFV(NumNodes) / GSM(NumNodes, 1)

For K = 1 To (NumNodes - l)
I = NumNodes - K
MJ = BandWidth
If ((I + BandWidth - 1) > NumNodes) Then
MJ=NumNodes-I+l
End If
Sum = 0!

For J = 2 To MJ
' n=I+J-1
Sum = Sum + GSM(I, J) "' GSV(n)
Next J
GSV(I) = (GFV(I) - Sum) / GSM(I, 1)
Next K

End Sub

233

Module — BuildBandedSystem

Sub BuildBandedSystem(NumEleSub, EleSub(), EFV(), ESM(), GFV(), GSM(), ECM(), GCM(),
EFV_MO, ESM_M(), GFV_M(), GSM_M(), ECM_M(), GCM_M(), EFV_FO. ESM_F(),
GFV_F(), GSM_F(), ECM_F(), GCM_F())

This module modified from a module provided by Dr. Larry Segerlind - Michigan State University

This subprogram incorporates the element force vector and the element
' stiffness matrix into the global force matrix and the global stiffness
' matrix.

'This subprogram ASSUMES that the global stiffness matrix is symmetrical
' and stored in a rectangular format.

The direct stiffness method for a banded system of equations

For I = 1 To NumEleSub
II = EleSub(I)

'Force vector

GFV(II) = GFV(II) + EFV(I)
GFV_M(II) = GFV_M(II) + EFV_M(I)
GFV_F(II) = GFV_F(II) + EFV_F(I)

For J = 1 To NumEleSub
JJ = EleSub(J)
JJ = JJ - II + 1
If (JJ > 0) Then

'Stiffness matrix

GSM(II, JJ) = GSM(II, JJ) + ESM(I, J)
GSM_M(II, JJ) = GSM_M(II, JJ) + ESM_M(I, J)
GSM_F(II, JJ) = GSM_F(II, JJ) + ESM_F(I, J)

'Capacitance matrix
GCM(II, JJ) = GCM(II, JJ) + ECM(I, J)
GCM_M(II, JJ) = GCM_M(II, JJ) + ECM_M(I, J)
GCM_F(II, JJ) = GCM_F(II, JJ) + ECM_F(I, J)
End If
Next J
Next 1

End Sub

234

Module — BuildGPM

Sub BuildGPM(BandWidth, GPM(), GCM(), GSM(), GPM_M(), GCM_M(), GSM_M(), GPM_F(),
GCM_F(), GSM_F(), timestep, NumNodes)

'This module calculates the P matrix for the finite difference time solution using the central
' difference method

For X = 1 To NumNodes

For Y = 1 To BandWidth
GPM(X, Y) = GCM(X, Y) - (GSM(X, Y) " (timestep / 2))
GPM_M(X, Y) = GCM_M(X, Y) - (GSM_M(X, Y) ‘ (timestep / 2))
GPM_F(X, Y) = GCM_F(X, Y) - (GSM_F(X, Y) "' (timestep / 2))
Next Y
Next X

End Sub

235

Module — ModifyGFV

Sub ModifyGFV(GFV(). GFV_A(), GFV_StarO, GFV_M(), GFV_M_AO. GFV_M_StarO,
GFV_F_A(),
GFV_F_StarO, GFV_F(), timestep, NumNodes)
'Modify the global force vector for the finite difference time solution
For X = 1 To NumNodes
GFV_Star(X) = (GFV_A(X) + GFV(X)) * (timestep / 2)
GFV_M_Star(X) = (GFV_M(X) + GFV_M(X)) * (timestep / 2)
GFV_F_Star(X) = (GFV_F_A(X) + GFV_F(X)) * (timestep / 2)
Next X

End Sub

236

Module — EleCapMatrix

Sub EleCapMatrix(ElePhyData(), EleNodeData(), ECM(), ECM_M(), ECM_F(), Coord(), KK)
' Compose the element capacitance matrix
If (EleNodeData(KK, 4) = 0) Then

' Triangular Element

' [c]=lamda*A/3*[ l 0 0
' 0 1 0
' 0 l 1 ]

EleNodel = EleNodeData(KK, 1)
EleNode2 = EleNodeData(KK, 2)
EleNode3 = EleNodeData(KK, 3)

' Calculate the element area

Ri = Coord(EleNodel, 1)
Rj = Coord(EleNodeZ, 1)
Rk = Coord(EleNode3, 1)

Rbar=(Ri+Rj+Rk)/3

XLengthSidel = Coord(EleNodeZ, l) — Coord(EleNodel, l)
YLengthSidel = Coord(EleNodeZ, 2) - Coord(EleNodel , 2)
XLengthSide2 = Coord(EleNode3, l) - Coord(EleNodeZ, 1)
YLengthSide2 = Coord(EleNode3, 2) - Coord(EleNodeZ, 2)
XLengthSide3 = Coord(EleNode3, l) - Coord(EleNodel, l)
YLengthSide3 = Coord(EleNode3, 2) - Coord(EleNodel, 2)

SidelLength = Sqr(XLengthSidel A 2 + YLengthSidel A 2)
Side2Length = Sqr(XLengthSideZ A 2 + YLengthSide2 A 2)
Side3Length = Sqr(XLengthSide3 A 2 + YLengthSide3 A 2)
Half? = (SidelLength + Side2Length + Side3Length) / 2
Area = Sqr(HalfP * (HalfP - SidelLength) * (HalfP - Side2Length) * (HalfP - Side3Length»
Lamda = ElePhyData(l, 5)

Lamda_M = ElePhyData(l, 10)

Lamda_F = ElePhyData(l, 15)

ECM(1,1)= 2 * 3.14 "' Rbar "' Lamda * Area/3

ECM(2, 2) = 2 "' 3.14 * Rbar * Lamda * Area/3

ECM(3, 3) = 2 * 3.14 " Rbar * Lamda "' Area/3
ECM_M(1,1)= 2 * 3.14 "' Rbar * Lamda_M * Area/3
ECM_M(Z, 2) = 2 "' 3.14 * Rbar "' Lamda_M * Area / 3
ECM_M(3, 3) = 2 * 3.14 * Rbar "' Lamda_M * Area / 3

ECM_F(1,1)= 2 * 3.14 "‘ Rbar * Lamda_F "' Area / 3

237

ECM_F(Z, 2) = 2 "' 3.14 " Rbar * Lamda_F "‘ Area / 3
ECM_F(3, 3) = 2 "' 3.14 * Rbar "' Lamda_F " Area / 3

Else

' Rectangular element

' [c]=1amda*A/4"'[ l 0 0 0
' 0100
' 0010
' 0001]

EleNodel = EleNodeData(KK, l)
EleNode2 = EleNodeData(KK, 2)
EleNode3 = EleNodeData(KK, 3)
EleNode4 = EleNodeData(KK, 4)

' Calculate the element area

Sidel = Coord(EleNode4, 2) - Coord(EleNodel, 2)
Side2 = Coord(EleNodeZ, 1) - Coord(EleNodel, 1)

Area = Sidel "' Side2

Lamda = ElePhyData(l, 5)

Lamda_M = ElePhyData(l, 10)

Lamda_F = ElePhyData(l, 15)

For J = 1 To 4
ECM(J, J) = Lamda * Area / 4
ECM_M(J, J) = Lamda_M "' Area / 4
ECM_F(J, J) = Lamda_F * Area / 4

Next J

Endlf

End Sub

238

Module - ConverthoTemp

Sub ConverthoTemp(heat_capacity_f, GSV(), GSV_M(), GSV_F(), temperature(), NumNodes,
InitialM,

Solid, frozenH, latentW, latentF, heat_capacity, LevelOne, LevelTwo, LevelThree,
LevelFour,

LevelFive)

'Convert enthalpy to temperature data

For K = 1 To NumNodes
If GSV(K) < LevelOne Then
temperature(K) = GSV(K) / heat_capacity_f - 273
Elself GSV(K) >= LevelOne And GSV(K) <= LevelTwo Then
temperature(K) = -2
Elself GSV(K) > LevelTwo And GSV(K) <= LevelThree Then
temperature(K) = ((GSV(K) - (frozenH + latentW)) / 3.35) - 2
Elself GSV(K) > LevelThree And GSV(K) <= LevelFour Then
temperature(K) = 45
Elself GSV(K) > LevelFour Then
temperature(K) = ((GSV(K) - (frozenH + latentW + 1atentF))/ 3.35) - 2
End If
Next K

End Sub

239

Module — CalculateProperties

Sub CalculateProperties(Meat, Solid, NumEle, EleNodeData(), E1ePhyData(), temperature(), GSV_M,
GSV_F(), moisture_capacity, density)

ForI = 1 To NumEle
'Calculate element average temperature, moisture, and fat

If EleNodeData(I, 4) = 0 Then
ElementAverageT = (1 "' temperature(EleNodeData(I, 1)) + 1 *
temperature(EleNodeData(I, 2)) + 1 "' temperature(EleNodeData(I, 3))) / 3
ElementAverageM = (1 * GSV_M(EleNodeData(I, 1)) + 1 *
GSV_M(EleNodeData(I, 2))
+ 1 " GSV_M(EleNodeData(I, 3))) / 3
ElementAverageF P = (1 "' GSV_F(EleNodeData(I, 1)) + 1 * GSV_F(EleNodeData(I,
2))
+ 1 * GSV_F(EleNodeData(I, 3))) / 3
Else
ElementAverageT = (l "‘ temperature(EleNodeData(I, 1)) + 1 *
temperature(EleNodeData(I, 2)) + l * temperature(EleNodeData(I, 3)) + 1
*
temperature(EleNodeData(I, 4))) / 4
ElementAverageM = (1 " GSV_M(EleNodeData(I, 1)) + 1 *
GSV_M(EleNodeData(I, 2))
+ 1 "' GSV_M(EleNodeData(I, 3)) + 1 "' GSV_M(EleNodeData(I, 4))) / 4
ElementAverageFP = (l "' GSV_F(EleNodeData(I, 1)) + l * GSV_F(EleNodeData(I,
2))
+ 1 * GSV_F(EleNodeData(I, 3)) + l "' GSV_F(EleNodeData(I, 4))) / 4
End If

moisture = ElementAverageM/ 100 'Element average moisture, decimal wb

MP = Solid 'Mass protein

MF = ElementAverageF P * Solid 'Mass fat

MM = (moisture * MF + moisture * MP) / (1 - moisture) 'Mass moisture
Total = MP + MM + MF 'Mass total

Fat = MF / Total 'Percent fat
Protein = MP / Total 'Percent protein

'Beef
If Meat = 1 Then

If ElementAverageT <= -2 Then
heat_capacity = 1.9 "‘ moisture + 1.711 * Protein + 1.298 "' Fat
Else
heat_capacity = 0.9 * (4.18 "' moisture + 1.711 " Protein + 1.298 "' Fat)
'(4.l8 *
moisture + 1.711 * Protein + 1.298 "' Fat)
End If

moisture_capacity = 0.003 'g/g

240

capacity

capacity

melting

melting

density = 1 / (moisture + Fat / 0.9 + Protein / 1.4) 'g/cmA3

Vmoisture = moisture / 1 'Volume moisture

Vprotein = Protein / 1.4 'Volume protein

Vfat = Fat / 0.9 'Volume fat

Vtotal = Vmoisture + Vprotein + Vfat Total volume

VFm = Vmoisture / Vtotal 'Volume fraction moisture

VFp = Vprotein / Vtotal 'Volume fraction protein

VFf = Vfat / Vtotal 'Volume fraction fat

'thermal conductivity

ElePhyData(l, 1) = ((0.602 * VFm + 0.18 * VFf + 0.2 "' VFp)/ 100)

ElePhyData(l, 2) = ((0.602 ’ VFm + 0.18 " VFf + 0.2 * VFp)/ 100)

ElePhyData( 1, 5) = heat_capacity " density 'density‘heat

'Coefficient for Mittal moisture diffusivity equation

A = 0.003

ElePhyData(], 6) = 10000 * A * (0.003 "' Exp(-0.442 * ElementAverageFP - 4829.7
(ElementAverageT + 273) + 11.55 * moisture)) * moisture_capacity "'
density 'moisture conductivity

E1ePhyData(l, 7) = 10000 "‘ A * (0.003 * Exp(-0.442 * ElementAverageFP - 4829.7
(ElementAverageT + 273) + 11.55 "' moisture)) " moisture_capacity *

density 'moisture conductivity
ElePhyData(], 10) = density "' moisture_capacity 'density‘moisture

If ElementAverageT <= 45 Then

ElePhyData(] , 11) = 0 'fat diffusivity r-direction zero below
ElePhyData(], 12) = 0 'fat diffusivity z-direction zero below
Else
ElePhyData(], 11) = 0.0008 'fat diffusivity r-direction
ElePhyData(], 12) = 0.0008 'fat diffusivity z-direction
End If
E1ePhyData(l , 15) = 1 'Variable not used for fat trasfer equations

Elself Meat = 2 Then

If ElementAverageT <= -2 Then
heat_capacity = 1.9 " moisture + 1.711 * Protein + 1.298 "‘ Fat
Else
heat_capacity = (4.18 * moisture + 1.711 "' Protein + 1.298 * Fat) '(4.18 "'
moisture + 1.711 "' Protein + 1.298 "' Fat)
End If

241

capacity

capacity

melting

melting

Turkey

moisture_capacity = 0.003 'g/ g
density = l / (moisture + Fat / 0.9 + Protein / 1.4) 'g/cmA3

Vmoisture = moisture / 1 'Volume moisture
Vprotein = Protein / 1.4 'Volume protein
Vfat = Fat / 0.9 'Volume fat

Vtotal = Vmoisture + Vprotein + Vfat Total volume

VFm = Vmoisture / Vtotal 'Volume fraction moisture

VFp = Vprotein / Vtotal 'Volume fraction protein

VF f = Vfat / Vtotal 'Volume fraction fat

'thermal conductivity

ElePhyData(l, 1) = ((0.602 " VFm + 0.18 "' VFf+ 0.2 * VFp)/ 100)
ElePhyData(], 2) = ((0.602 * VFm + 0.18 "' VFf + 0.2 * VFp)/ 100)
ElePhyData(l, 5) = heat_capacity "' density 'density‘heat
'Coefficient for Mittal moisture diffusivity equation

A = 0.003

ElePhyData(l, 6) = 10000 "‘ A * (0.003 "‘ Exp(-0.442 "' ElementAverageFP - 4829.7

(ElementAverageT + 273) + 11.55 " moisture)) "' moisture_capacity "'
density

'moisture conductivity 0.0000000129 '
ElePhyData(l, 7) = 10000 "' A * (0.003 "' Exp(-0.442 " ElementAverageFP - 4829.7

(ElementAverageT + 273) + 11.55 "' moisture)) * moisture_capacity *
density
‘moisture conductivity 0.0000000129 '

ElePhyData(], 10) = density "' moisture_capacity 'density*moisture

If ElementAverageT <= 45 Then

ElePhyData(l, l 1) = 0 'fat diffusivity r-direction zero below
ElePhyData(], 12) = 0 'fat diffusivity z-direction zero below
Else
ElePhyData(l, 11) = 0.00125 'fat diffusivity r-direction
ElePhyData( l , 12) = 0.00125 'fat diffusivity z-direction
End If
ElePhyData( 1 , 15) = 1 'Variable not used for fat trasfer equations

Elself Meat = 3 Then

If ElementAverageT <= -2 Then

242

capacity

capacity

melting

melting

End If

heat_capacity = 1.9 * moisture + 1.711 * Protein + 1.298 "' Fat

Else
heat_capacity = (4.18 * moisture + 1.711 "' Protein + 1.298 * Fat) '(4.18 *
moisture + 1.711 " Protein + 1.298 "‘ Fat)

Endlf

moisture_capacity = 0.003 'g/g
density = l / (moisture + Fat / 0.9 + Protein / 1.4) 'g/cmA3

Vmoisture = moisture / 1 'Volume of moisture
Vprotein = Protein / 1.4 'Volume of protein
Vfat = Fat / 0.9 'Volume of fat

Vtotal = Vmoisture + Vprotein + Vfat Total volume

VFm = Vmoisture / Vtotal 'Volume fraction moisture
VFp = Vprotein / Vtotal 'Volume fraction protein
VFf = Vfat / Vtotal 'Volume fi'action fat

'thermal conductivity

ElePhyData(l, 1) = ((0.602 "‘ VFm + 0.18 "‘ VFf+ 0.2 "' VFp)/ 100)
ElePhyData(] , 2) = ((0.602 " VFm + 0.18 " VFf+ 0.2 "' VFp)/ 100)
ElePhyData(l, 5) = heat_capacity " density 'density‘heat

'Coefficient for Mittal moisture diffusivity equation

A = 0.003

ElePhyData(], 6) = 10000 * A " (0.003 * Exp(-0.442 * ElementAverageFP - 4829.7
(ElementAverageT + 273) + 11.55 "' moisture)) "‘ moisture_capacity *

density
'moisture conductivity

ElePhyData(l, 7) = 10000 * A * (0.003 * Exp(-0.442 * ElementAverageFP - 4829.7
(ElementAverageT + 273) + 11.55 * moisture)) "' moisture_capacity *

density

'moisture conductivity

ElePhyData(] , 10) = density "' moisture_capacity 'density‘moisture

If ElementAverageT <= 45 Then

ElePhyData(], 11) = 0 'fat diffusivity r-direction zero below
ElePhyData( 1 , 12) = 0 'fat diffusivity z-direction zero below
Else
ElePhyData(], 11) = 0.00125 'fat diffusivity r-direction
ElePhyData(l , 12) = 0.00125 'fat diffusivity z-direction
End If
ElePhyData(l, 15) = 1 'Variable not used for fat trasfer equations

243

Next I

End Sub

244

Module — CalculateElementAverage

Sub CalculateElementAverage(Volume(), NumEle, EleNodeData(), Phi(), WeightedAverage)
This module calculates a weighted volume average of the nodal values.

ReDim VolumeRatio(NumEle)
ReDim ElementAverage(NumEle)

WeightedAverage = 0

For K = 1 To NumEle
TotalVolume = TotalVolume + Volume(KK)

'Calculate the element average of the nodal values

If (EleNodeData(KK, 4) = 0) Then Triangular
element
Nodel = EleNodeData(KK, l)
Node2 = EleNodeData(KK, 2)
Node3 = EleNodeData(KK, 3)
ElementAverage(KK) = (Phi(Nodel) + Phi(Node2) + Phi(Node3)) / 3
Else 'Rectangular
element
Nodel = EleNodeData(KK, 1)
Node2 = EleNodeData(KK, 2)
Node3 = EleNodeData(KK, 3)
Node4 = EleNodeData(KK, 4)
ElementAverage(KK) =(Phi(Node1) + Phi(Node2) + Phi(Node3) + Phi(Node4)) / 4
Endlf
Next KK

'Calculate the volume weighted average
For K = 1 To NumEle
VolumeRatio(KK) = Volume(KK) / TotalVolume

ElementAverage(KK) = ElementAverage(KK) " VolumeRatio(KK)
Next K

For I = 1 To NumEle
WeightedAverage = Wei ghtedAverage + ElementAverage(I)
Next I

End Sub

245

Module — InputOvenConditions

Sub InputOvenConditions(OvenConditions(), NumTimeSteps)

'Input oven conditions from file

For I = 1 To NumTimeSteps
Input #2, OvenConditions(I, 1), OvenConditions(I, 2), OvenConditions(I, 3)
If OvenConditions(I, 2) = 0 Then OvenConditions(I, 2) = 1

Next I

End Sub

246

Module - CalculatesurfaceFatContent

Sub CalculatesurfaceFatContent(InitialF, T, GSV_F(), NudeyNodes, deNode(), temperature(),
timestep)

’This soubroutine calculates the fat content at the surface of the meat as functions

' of time and temperature. Experimental fat holding data was utilized to determine

' the relationship between time, temperature and fat content. Fat content is expressed
' in terms of Fat/Protein ratio.

'Calculate FP at each boundary node

For I = 1 To NudeyNodes
X = 0.23 - (0.0127 "' temperamre(deNode(I))) + (0.0001 "‘ (temperamre(deNode(I)) A 2)) +
(0.0617 "' InitialF) - (0.0007 * InitialF "‘ temperamre(deNode(I)))
If X > GSV_F(deNode(I)) Then
GSV_F(deNode(I)) = GSV_F(deNode(I))
Else
GSV_F(deNode(I)) = X
End If
If GSV_F(deNode(I)) < 0 Then GSV_F(deNode(I)) = 0
Next I

End Sub

247

Module - CalculateSurvivors

Sub CalculateSurvivors(NumNodes, timestep, temperature(), Ninitial, No(), NW(), Nnew(), NnewW(),
logreduction(), logreductionW, Dvalue, Tref, Z, TimeToLimit, TR)

'Calculate the number of surviving microorganisms for eanch node at each time step

'Log-linear inactivation equation

For X = 1 To NumNodes
If logreduction(X) < 9 Then 'limit solution to 9 log reduction in order to avoid
' computer overload from low numbers as well as because a
'9-log reduction is well above the legal requirement
If ((Tref - temperature(X)) / Z) > -2 Then
(1 = Dvalue "‘ 10 A ((Tref - temperature(X)) / Z)
Else
d = 0.05
End If
Nnew(X) = No(X) / (10 A (timestep / d))
If Nnew(X) < 1 Then Nnew(X) = l
No(X) = Nnew(X)
logreduction(X) = -Log(No(X) / Ninitial) / Log(10)
Else
logreduction(X) = 9
End If

If logreduction(X) > 9 Then
logreduction(X) = 9
End If

'Weibull inactivation equation

If logreductionW < 9 Then
b = 0.000000000011047 "‘ Exp(0.4l758 * temperature(l)) '0.03 * (temperature(l)) A
2 _
(2.7 " temperature(l)) + 72.19
11 = 1.12
NnewW(X) = NW(X) "‘ (10 A (-b "' ((timestep / 60) A n)))
If Nnew(X) < 1 Then Nnew(X) = 1
NW(X) = NnewW(X)
logreductionW = -Log(NW(X) / Ninitial) / Log(10)
Else
logreductionW = 9
End If

Next X
If logreduction( 1) <= TR Then
TimeToLimit = TimeToLimit + timestep

End If

End Sub

248

Module — CalculateInitialProperties

Sub CalculateInitialProperties(heat_capacity_f, Meat, Solid, InitialT, InitialM, InitialF, heat_capacity,
latentW, frozenH, latentF)

'Calculate the initial thermal properties of the product before time-stepping.

moisture = InitialM / 100 'Moisture, decimal wb

Fat = InitialF / 100 'Fat content, decimal wb
Protein = Solid / 100 'Protein content, decimal wb
'Beef

If Meat = I Then

If InitialT <= -2 Then
heat_capacity_f = 1.9 "‘ moisture + 1.711 "' Protein + 1.298 "' Fat
frozenH = (1.9 * moisture + 1.711 * Protein + 1.298 "‘ Fat) "‘ 271 'Enthalpy tied up
'in frozen meat

latentW = 337.78 * moisture 'Latent heat of
water
latentF = 64.4 "' Fat 'Latent heat of fat
Else
heat_capacity_f = 1.9 * moisture + 1.711 * Protein + 1.298 * Fat
heat_capacity = (4.18 "' moisture + 1.711 * Protein + 1.298 * Fat)
frozenH = (1.9 * moisture + 1.711 * Protein + 1.298 " Fat) * 271 'Enthalpy tied up
‘in frozen meat
latentW = 337.78 "' moisture 'Latent heat of
water
latentF = 64.4 "' Fat
End If
’Pork

Elself Meat = 2 Then

If InitialT <= -2 Then
heat_capacity_f = 1.9 * moisture + 1.711 "' Protein + 1.298 "' Fat
frozenH = (1.9 "‘ moisture + 1.711 * Protein + 1.298 "' Fat) * 271 'Enthalpy tied up
'in frozen meat

latentW = 337.78 * moisture 'Latent heat of
water
latentF = 64.4 "' Fat 'Latent heat of fat
Else
heat_capacity_f = 1.9 * moisture + 1.711 * Protein + 1.298 "' Fat
heat_capacity = (4.18 * moisture + 1.711 * Protein + 1.298 * Fat)
frozenH = (1.9 * moisture + 1.711 "' Protein + 1.298 "' Fat) * 271 'Enthalpy tied up
'in frozen meat
latentW = 337.78 * moisture 'Latent heat of
water
latentF = 64.4 * Fat 'Latent heat of fat
End If
Turkey

249

Elself Meat = 3 Then

If InitialT <= -2 Then
heat_capacity_f = 1.9 " moisture + 1.711 "' Protein + 1.298 "' Fat
frozenH = (1.9 "' moisture + 1.711 "' Protein + 1.298 "' Fat) * 271 'Enthalpy tied up
'in frozen meat

latentW = 337.78 "‘ moisture 'Latent heat of
water
latentF = 64.4 * Fat 'Latent heat of fat
Else
heat_capacity_f = 1.9 * moisture + 1.711 "' Protein + 1.298 * Fat
heat_capacity = (4.18 "' moisture + 1.711 ‘ Protein + 1.298 "‘ Fat)
frozenH = (1.9 * moisture + 1.711 "' Protein + 1.298 " Fat) * 271 'Enthalpy tied up
'in frozen meat
latentW = 337.78 " moisture 'Latent heat of
water
latentF = 64.4 "' Fat 'Latent heat of fat
End If
End If
End Sub

250

Module — Cachandwidth

Sub Cachandwidth(NumEle, EleNodeData(), BandWidth)
This module provided by Dr. Larry Segerlind - Michigan State University

'This subprogram evaluates the bandwidth for any group of elements.
' The subprogram assumes that triangular elements have as many data
' values as rectangular elements. The extra data values are zeros.

'Evaluate the bandwidth
NumEleNodes = 4
MaxDiff = 0

For I = 1 To NumEle

For J = 1 To (NumEleNodes - 1)
JJ = EleNodeData(l, J)
If (JJ = 0) Then Exit For

For K = (J + 1) To NumEleNodes
K = EleNodeData(I, K)
If (K = 0) Then Exit For
Diff = Abs(JJ - KK)
If (Diff > MaxDiff) Then
MaxDiff = Diff
Element = I
Endlf
Next K
Next J
Next I
BandWidth = (MaxDiff + 1)

End Sub

251

Module — DecompBandMatrix

Sub DecompBandMatrix(NumNodes, BandWidth, GAM())
'This module provided by Dr. Larry Segerlind - Michigan State University

This subprogram decomposes a symmetric banded matrix into an upper

' triangular form using the method of Gaussian elemination. The

' matrix is stored in the rectangular array GSM(NumNodalVal%,BandWidth%).
' Only the upper part of the banded matrix is stored.

'Decompose the global stiffness matrix stored in a rectangular format
NumNodalVal = NumNodes * 1

For I = 1 To (NumNodalVal - 1)

MJ = I + BandWidth - 1

If (MJ > NumNodalVal) Then
MJ = NumNodalVal

End If

NJ = I + l

MK = BandWidth

If ((NumNodalVal - I + l) < BandWidth) Then
MK = NumNodalVal - I +1

End If

ND = 0

ForJ=NJToMJ
MK=MK-1
ND=ND+1
NL=ND+1

For K = 1 To MK
NK = ND + K
GAM(J, K) = GAM(J, K) - GAM(I, NL) * GAM(I, NK) / GAM(I, 1)
Next K
Next J
Next 1

End Sub

252

Module - ZeroGlobalMatrices

Sub ZeroGlobalMatrices(NumNodalVal, BandWidth, GFV(), GSM(), GFV_M(), GSM_M(),
GFV_F(),
GSM_F(), GCM(), GCM_M(), GCM_F())

'This module modified from a module by Dr. Segerlind - Michigan State University
'This subprogram fills the global stiffness matrix and force vector
' with zero values. This subprogram assumes that the global stiffness

' matrix is symmetrical and stored in a rectangular format.

For I = 1 To NumNodalVal

GFV(I) = 0
GFV_M(I) = 0
GFV_F(I) = 0
For J = 1 To BandWidth
GSM(I, J) = 0
GSM_M(I, J) = 0
GSM_F(I, J) = 0
GCM(I, J) = 0
GCM_M(I, J) = 0
GCM_F(I, J) = 0
Next J
Next I
End Sub

253

9 BIBLIOGRAPHY

AMI. 2003. AMI Process Lethality Determination Spreadsheet. American Meat
Institute. Arlington VA.

AOAC. 2000. Oflicial Methods of Analysis of A OAC International. AOAC
International. Arlington, VA.

ASHRAE. 1998. ASHRAE Handbook: Fundamentals. American Society of
Heating, Refrigerating, and Air-conditioning Engineers. Atlanta, GA.

Badiani, A., Stipa, S., Bitossi, F ., Gatta, P.P., Vignola, G., and R. Chizzolini. 2002.
Lipid composition, retention, and oxidation in fresh and completely trimmed
beef muscles as affected by common culinary practices. Meat Science.
60:169-186.

Barbut, S. 1996. Determining water and fat holding. Methods of Testing Protein
Functionality. Blackie Academic and Professional. New York.

Bejan, A. 1995. Convection Heat Transfer 2nd Ed. John Wiley and Sons. New York.

Bengtsson, N.E., Jakobsson, B., and M. Dagerskog. 1976. Cooking of beef by oven
roasting: a study of heat and mass transfer. Journal of Food Science. 41:1047-
1053.

Bimbenet, J .J ., Loncin, M., and H. Brusset. 1971. Heat and mass transfer during air
drying of solids. Canadian Journal of Chemical Engineering. 49:860-865.

Bird, R.B., Stewart, W.E., and EN. Lightfoot. 1960. Transport Phenomena. John
Wiley and Sons. New York

Bodwell, CE. and P.E. McClain. 1978. Proteins. The Science of Meat and Meat
Proteins. Food and Nutrition Press. Westport, CT.

Chen, H., Marks, BR, and R.Y. Murphy. 1999. Modeling coupled heat and mass
transfer for convection cooking of chicken patties. Journal of Food
Engineering. 42: 139-146.

Choi, Y., and M. Okos. 1985. Effects of temperature and composition on the thermal
properties of foods. Food Engineering and Process Applications, Transport

Phenomena vol. 1. Elsevier. London.

Dagerskog, M. 1979 a. Pan frying of meat patties I. A study of heat and mass transfer.
Lebensmittel- Wissenschaft and T echnologie. 12:217-224.

254

Dagerskog, M. 1979 b. Pan frying of meat patties 11. Influence of processing
conditions on heat transfer, crust color formation, cooking losses, and sensory
quality. Lebensmittel- Wissenschaft and T echnologie. 12:225-230.

Dagerskog, M. and NE. Bengtsson. 1974. Pan frying of meat patties - relationship
among crust formation, yield, composition, and processing condition.
Lebensmittel- Wissenschaft and T echnologie. 72202-207.

Datta, A. 2002. Biological and Bioenvironmental Heat and Mass Transfer. Marcel
Dekker. New York. pp. 199-215.

Dincer, I. 1996. Modeling of thermal and moisture diffusions in cylindrically shaped
sausages during frying. Journal of Food Engineering. 28:35-43.

Erdogdu, F., Balaban, M.O., and K.V. Chau. 1999. Mathematical model to predict
yield loss of medium and large tiger shrimp (Penaues monodon) during
cooking. Journal of Food Process Engineering. 22:383-394.

Farkas, B.E., Singh, RP, and T.R. Rumsey. 1996 a. Modeling heat and mass transfer
in immersion frying. 1., model development. Journal of Food Engineering.
29:211-226.

Farkas, B.E., Singh, RP, and T.R. Rumsey. 1996 b. Modeling heat and mass transfer
in immersion frying. 11, model solution and verification. Journal of Food
Engineering. 29:227-248.

Fellows, P. 1988. Food Processing Technology: Principals and Practice. Ellis
Horwood. New York.

FSIS. 1999. Performance standards for the production of certain meat and poultry
products. Federal Register. January 6, 1999. 732-749.

F SIS. 2001. Performance standards for the production of processed meat and poultry
products; proposed rule. Federal Register. February 27, 2001. 12590-12636.

F SIS. 2002. Use of microbial pathogen computer modeling in HACCP plans. FSIS
Notice 55-02. December 2, 2002.

Gardon, R., and J .C. Akfirat. 1966. Heat transfer characteristics of impinging two-
dimensional air jets. Journal of Heat Transfer. 88:101-108.

Geankopolis, C.J. 1993. Transport Processes and Unit Operations 3rd Ed. Prentice
Hall. Englewood Cliffs, NJ.

255

Hallstrbm, B. 1990. Mass transport of water in foods - a consideration of the
engineering aspects. Journal of Food Engineering. 12:45-52.

Hayakawa, K. 1970. Experimental formulas for accurate estimation of transient
temperature of food and their application to thermal process evaluation. Food
Technology. 24:89- 100.

Holtz, E. and C. Skjoldebrand. 1986. Simulation of the temperature of a meat loaf
during the cooking process. Journal of Food Engineering. 5:109-121.

Housova, J. and P. Topinka. 1985. Heat transfer during contact cooking of minced
meat patties. Journal of Food Engineering. 4: 169-188.

Huang, E. and GS. Mittal. 1995. meatball cooking - modeling and simulation.
Journal of Food Engineering. 24:87-100.

Hung, C.C., Davis, E.A., Gordon, J ., and HT. Davis. 1978. Mechanisms of water loss
of bovine semitendinosus muscle dry cooked from the frozen state. Journal of
Food Science. 43:1191-1 195.

Ikediala, J .N., Correia, L.R., Fenton, GA, and N. Ben-Abdallah. 1996. Finite
element modeling of heat transfer in meat patties during single-sided pan-
frying. Journal of Food Science. 61:796-802.

J uneja, V.K. and BS. Eblen. 1999. Predictive thermal inactivation model for Listeria
monocytogenes with temperature, pH, NaCl, and sodium pyrophosphate as
controlling factors. Journal of Food Protection. 62:986-993.

Levy, FL. 1979. Enthalpy and specific heat of meat and fish in the freezing range.
Journal of Food Technology. 14:549-560.

Mallikarjunan, P., Hung, Y.C., and S. Gundavarapu. 1996. Modeling microwave
cooking of cocktail shrimp. Journal of Food Process Engineering. 19:97-1 l 1.

Maroulis, Z.B., Kiranoudis, C.T., and D. Marinos-Kouris. 1998. Heat and mass
transfer modeling in air-drying foods. Journal of Food Engineering. 26:113-
130.

Martin, H. 197 7. Heat and mass transfer between impinging gas jets and solid
surfaces. Advances in Heat Transfer. 13:1-60.

Mattick, K.L., Legan, J .D., Humphrey, T.J., and M. Peleg. 2001. Calculating
Salmonella inactivation in non-isothermal heat treatments from isothermal
nonlinear survival curves. Journal of Food Protection. 64:606-613.

256

McProud, L.M. and DB. Lund. 1983. Thermal properties of beef loaf produced in
foodservice systems. Journal of Food Science. 48:677-680.

Millsap, SC. and RP. Marks. 2002. Modeling condensing/convective heat transfer to
food products in moist air impingement ovens. ASAE Paper No. 026045.

Mannapperuma, J .D. and RF. Singh. 1989. A computer-aided method for the
Prediction of properties and freezing/thawing times of foods. Journal of Food
Engineering. 9:275-304.

Mittal, GS. and J .L. Blaisdell. 1982. Moisture mobility in fiankfurter during thermal
processing: analysis of moisture profile. Journal of Food Processing and
Preservation. 6:1 11-126.

Mittal, GS. and J .L. Blaisdell. 1984. Heat and mass transfer properties of meat
emulsion. Lebensmittel- Wissenschaft and T echnologie. 17:94-98.

Mittal, GS. and J. Zhang. 2000. Prediction of temperature and moisture content of
frankfurters during thermal processing using neural network. Meat Science.
55:13-24.

Mittal, GS. and J. Zhang. 2001. Artificial neural network for the prediction of
temperature, moisture, and fat contents in meatballs during deep-fat frying.
International Journal of Food Science and Technology. 36:489-497.

Mittal, G.S., Blaisdell, J.L., and FL. Herum. 1982. Moisture mobility in meat
emulsion during cooking. Journal of Food Technology. 17:709-717.

Mittal, G.S., Blaisdell, J .L., and FL. Herum. 1983. Moisture mobility in meat
emulsion during thermal processing: analysis of slab moisture profile. Meat
Science. 8:15-32.

Murphy, R.Y., Marks, B.P., and J .A. Marcy. 1998. Apparent specific heat of chicken
breast patties and their constituent proteins by differential scanning
calorimetry. Journal of Food Science. 63:88-91.

Murphy, R.Y., Johnson, L.K., Duncan, L.K., Davis, M.D., Johnson, M.G., and J.A.
Marcy. 2001a. Thermal inactivation of Salmonella spp. and Listeria innocua
in chicken breast patties processed in a pilot-scale air-convection oven.

Journal of Food Science. 66:734-741.
Murphy, R.Y., Duncan, L.K., Johnson, ER, and MD. David. 2001b. Process

lethality and product yield for chicken patties processed in a pilot-scale air-
steam impingement oven. Journal of Food Protection. 64: 1549-1555.

257

Murphy, R.T., Duncan, L.K., Johnson, E.R., Davis, M.D., and J .A. Marcy. 2002.
thermal inactivation of Salmonella seftenberg and Listeria innocua in
beef/turkey blended patties cooked via fryer and/or air convection oven.
Journal of Food Science. 67: 1879-1885.

Murphy, R.Y., Duncan, L.K., Beard, B.L., and K.H. Driscoll. 2003. D and 2 values of
Salmonella, Listeria innocua, and Listeria monocytogenes in fully cooked
poultry products. Journal of Food Science. 68: 1443-1447.

Ngadi, MD. and LR. Correia. 1995. Moisture diffusivity during deep-fat frying of
chicken drum muscle. J. CSAE. 37:339-344.

Ngadi, M.O., Watts, KC, and LR. Correia. 1997. Finite element method modeling
Of moisture transfer in chicken drum during deep fat frying. Journal of Food
Engineering. 32:1 1-20.

Pan, Z. 1998. Predictive modeling and optimization of hamburger patty contact-
cooking process. PhD. Dissertation. University of California Davis. Davis,
CA.

Pan, 2., Singh, RR, and T.R. Rumsey. 2000. Predictive modeling of contact heating
process for cooking a hamburger patty. Journal of Food Engineering. 4629-19.

Pan, Z. and RP. Singh. 2001. Physical and thermal properties of ground beef during
cooking. Lebensmittel- Wissenschaft and T echnologi. 34:437-444.

Peleg, M. and MB. Cole. 1998. Reinterpretation of microbial survival curves.
Critical Reviews in Food Science. 38:353-380.

Perez, M.G.R. and A. Calvelo. 1984. Modeling the thermal conductivity of cooked
meat. Journal of Food Science. 49: 152-156.

Roberts, T.A. 1997. Microbial grth and survival: developments in predictive
modeling. Food Technology. 51:88-99.

Sanz, P.D., Alonso, M.D., and RH. Mascheroni. 1987. Thermophysical properties of
meat products: general bibliography and experimental values. Transactions of
the ASAE. 30:283-296.

Sarkin, RS. 1978. Computerized cooking simulation of meat products. Journal of
Food Science. 43:1 140-1 143.

Segerlind, L]. 1984. Applied Finite Element Analysis. John Wiley and Sons. New
York.

258

Sereno, A.M. and G.L. Medeiros. 1990. A simplified model for the prediction of
drying rates for foods. Journal of Food Engineering. 12:1-11.

Shilton, N., Mallikarjunan, P., and P. Sheridan. 2002. Modeling of heat transfer and
evaporative mass losses during the cooking of beef patties using far-infrared
radiation. Journal of Food Engineering..55:217-222.

Singh, RP. 2000. Moving boundaries in food engineering. Food Technology. 54:44-
53.

Singh, N., Akins, R.G., and LE. Erickson. 1984. Modeling heat and mass transfer
during the oven roasting of meat. Journal of Food Process Engineering.
7:205-220.

Skala, D., Bastic, M., Bastic, L.J., Remberg, G., and J. Jovanovic. 1989. Thermal
behavior of different hog and cattle tissue lipids by DSC analysis.
Proceedings. 35th International Congress of Meat Science and Technology.
Roskilde, Denmark. Danish Meat Research Institute.

Skjoldebrand, C. 1980. Convection oven fiying: heat and mass transfer between air
and product. Journal of Food Science. 45: 1354-1362.

Skjoldebrand, C. and B. Hallstrom. 1980. Convection oven fiying: heat and mass
transport in the product. Journal of Food Science. 45:1347-1353.

Smith, SE, Maurer, J.L., Orta-Ramirez, A., Ryser, ET, and D.M. Smith. 2001.
thermal inactivation of Salmonella spp., Salmonella typhimurium DT104, and
Escherichia coli 0157:H7 in ground beef. Journal of Food Science. 66:1164-
1168.

Stroshine, R. and D. Hamann. 1996. Physical Properties of Agricultural Materials
And Food Products.

Thorvaldsson, K. and C. Skjoldebrand. 1995. Water transport in meat during
reheating. Journal of Food Engineering. 29: 13-21 .

Tocci, A.M. and R.MH. Mascheroni. 1998. Characteristics of differential scanning
calorimetry determination of therrnophysical properties of meats.
Lebensmittel- Wissenschaft und T echnologie. 31:418-426.

Tsai, S.J., Unklesbay, N., Unklesbay, K., and A. Clarke. 1998. Thermal properties of
restructured beef products at different isothermal temperatures. Journal of
Food Science. 63:481-484.

USDA. 2002.Pathogen Modeling Program Version 5.1. United States Department of
Agriculture

259

Van Impe, J .F ., Nicoli, B.M., Schellekens, M., Martens, T., and J. DeBaerdemaeker.
1995. Predictive microbiology in a dynamic environment: a system theory
approach. International Journal of Food Microbiology. 25:227—249.

Vijayan, J. and RP. Singh. 1997. Heat transfer during immersion frying of frozen
foods. Journal of Food Engineering. 34:293-314.

Voller, V. and M. Cross. 1981. Accurate solutions of moving boundary problems
using the enthalpy method. International Journal of Heat and Mass Transfer.
24:545-556.

Whiting, RC. 1995. Microbial modeling in foods. Critical Reviews in Food Science
and Nutrition. 35:467-494.

Whiting, RC. and R.L. Buchanan. 1994. Scientific status summary: microbial
modeling. Food Technology. 48:1 13-120.

Young, L.L., Garcia, J.M., Lillard, H.S., Lyon, CE, and CM. Papa. 1991. Fat
content effects of yield, quality, and microbial characteristics of chicken
patties. Journal of Food Science. 56:1527-1528, 1541.

Zanoni, B., Peri, C., Garzaroli, C., and S. Pierucci. 1997. A dynamic mathematical
model of the thermal inactivation of Enterococcusfaecium during bologna

sausage cooking. Lebensmittel- Wissenschaft und T echnologie. 30:727-734.

Zogzas, N.P., Maroulis, 2.8., and D. Marinos-Kouris. 1996. Moisture diffusivity
data compilation in foodstuffs. Drying Technology. 14:2225-2253.

260