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Modeling the Transport of Salmonella Into Whole-Muscle Meat
Products During Marination

presented by
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6/07 p./ClRC/DateDueindd~p.1

MODELING THE TRANSPORT OF SALMONELLA INTO WHOLE-
MUSCLE MEAT PRODUCTS DURING MARINATION

By

Julie A. Rochowiak

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
For the degree of

MASTER OF SCIENCE
Department of Biosystems and Agricultural Engineering

2007

ABSTRACT

MODELING THE TRANSPORT OF SALMONELLA INTO WHOLE-MUSCLE
MEAT PRODUCTS DURING MARIN ATION

By
Julie A. Rochowiak
Salmonella is a harmful bacterium that can cause serious illness if contaminated
meat is not properly processed. Recent research has shown that the assumption that the
interior of intact whole-muscle meat products is sterile is not necessarily true for
marinated products exposed to pathogens. Therefore, the objectives of this project were
to: (1) develop a mechanistic mathematical model to describe the transport of Salmonella
into intact whole-muscle meat products during marination, (2) estimate model parameters
from laboratory trials, and (3) validate the model using independent data. The proposed
model represents one-dimensional transport of Salmonella into whole-muscle meat
products, assuming capillary diffusion of marinade and capsule-like transport of the
Salmonella cells along with the marinade. The model solution was stable and yielded
reasonable relationships between input and output variables (i.e., moisture uptake and
Salmonella migration). In experimental trials, the total mass uptake was very low, < 1%,
but significant numbers of Salmonella entered the product (counts > 101 at 4.5 cm from
the surface in contact with marinade). However, the model did not predict the
distribution of Salmonella in whole—muscle meat products during marination with

sufficient accuracy to support the hypothesis regarding the underlying mechanism.

This thesis is dedicated to my parents for the endless love and support that they

have given me.

iii

ACKNOWLEDGMENTS
I would like to express my sincere appreciation to Dr. Bradley Marks for the vast
amount of guidance that he has given to me throughout my college career, as well as the
many hours, that he has spent with me working on all aspects of this study. I would also
like to thank my committee members: Dr. Alicia Orta-Ramirez, Dr. Alden Booren, Dr.
Kirk Dolan, and Dr. Elliot Ryser for the support and advice that they have given to me
throughout my project. Finally, I would like to thank my family and friends for their love

and support.

iv

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... viii
LIST OF FIGURES ........................................................................................................... ix
KEY TO SYMBOLS ......................................................................................................... xi
1. INTRODUCTION .......................................................................................................... 1
2. LITERATURE REVIEW ............................................................................................... 4
2.1 Overview ................................................................................................................... 4
2.2 Salmonella ................................................................................................................ 4
2.2.1 Description ......................................................................................................... 4
2.2.2 Salmonellosis ..................................................................................................... 5

2.3 Marination ................................................................................................................. 6
2.4 Effects of Vacuum and Tumbling ............................................................................. 7
2.5 Surface Penetration and Bacteria Transport into Whole-Muscle ............................. 8
2.6 Flow through a Fibrous Porous Medium ................................................................ 11
2.6.1 Fluid Flow ........................................................................................................ 11
2.6.2 Particle Flow .................................................................................................... 12

2.7 Capillary Flow Models ........................................................................................... 13
2.8 Pipe Flow ................................................................................................................ 13
2.9 Capsule Flow Models ............................................................................................. 15
2.10 Summary ............................................................................................................... 18

3. METHODS AND MATERIALS .................................................................................. 20
3.1 Overview ................................................................................................................. 20
3.3 Moisture Concentration .......................................................................................... 30
3.4 Effective Gap Radius .............................................................................................. 31
3.5 Darcy’s Velocity ..................................................................................................... 31
3.6 Marinade Velocity .................................................................................................. 32
3.7 Salmonella Velocity ................................................................................................ 32

3.8 Salmonella Location ............................................................................................... 35

3.9 Salmonella Concentration Profile ........................................................................... 36
3.10 Model Inputs ......................................................................................................... 36
3.10.1 Meat Properties .............................................................................................. 36
3.10.2 Marinade Properties ....................................................................................... 37
3.10.3 Salmonella Properties .................................................................................... 37
3.10.4 Calculated Initial Values ................................................................................ 38

3.11 Validation .............................................................................................................. 38
3.11.1 Moisture content ............................................................................................ 38
3.11.2 Salmonella concentration ............................................................................... 39
3.12 Limitations ............................................................................................................ 41

4. RESULTS ..................................................................................................................... 42
4.1 Overview ................................................................................................................. 42
4.2 Model Outputs ........................................................................................................ 42
4.2.1 Moisture Concentration ................................................................................... 43
4.2.2 Effective Gap Radius ....................................................................................... 44
4.2.3 Darcy’s Velocity .............................................................................................. 45
4.2.4 True Marinade Velocity ................................................................................... 46
4.2.5 Salmonella Velocity ......................................................................................... 47
4.2.6 Salmonella Location ........................................................................................ 48
4.2.7 Salmonella Concentration Profile .................................................................... 49

4.3 Sensitivity Analysis ................................................................................................ 50
4.3.1 Saturation Moisture Concentration .................................................................. 51
4.3.2 Tortuosity ......................................................................................................... 52
4.3.3 Initial Effective Gap Radius ............................................................................. 54
4.3.4 Effective Length of Flagella ............................................................................ 55

4.4 Attempted Model Validation .................................................................................. 57
4.12 Summary ............................................................................................................... 60

5. CONCLUSION ............................................................................................................. 61
6. FUTURE RESEARCH ................................................................................................. 62

vi

7. APPENDICES .............................................................................................................. 64

Appendix 7.1: Derivative of Capillary Potential with Respect to Dry Basis Moisture

Concentration ................................................................................................................ 65
Appendix 7.3: Moisture Concentration Test Results .................................................... 69
Appendix 7.4: Marinade Percent Uptake Test Results ................................................. 71
Appendix 7.5: Salmonella Concentration Experimental Results .................................. 72

Appendix 7.6: Calculation of Required Marinade Uptake to Achieve given

Experimental Salmonella Log Counts .......................................................................... 74
Appendix 7.7: Master Input Page for Model ................................................................ 76
Appendix 7.8: Model Output for Moisture Concentration ........................................... 77
Appendix 7.9: Model Output for Effective Gap Radius ............................................... 82
Appendix 7.10: Mode] Output for Darcy’s Velocity .................................................... 87
Appendix 7.11: Model Output for Salmonella Velocity ............................................... 92
Appendix 7.12: Model Output for Salmonella Concentration Profile .......................... 97
Appendix 7.13: Three-Dimensional Graphs of Model Output ................................... 102
8. REFERENCES ........................................................................................................... 105

vii

LIST OF TABLES

Table 4.1: Input variable values used in the evaluation of the model. Starred (*)
variables are analyzed in Section 4.3. ....................................................................... 43

Table 7.1: Moisture concentration results using the coring method described in section

3.11.1 ......................................................................................................................... 69
Table 7.2: Moisture concentration results using the slicing method described in section

3.11.1 ......................................................................................................................... 70
Table 7 .3: Marinade percent uptake test results ............................................................... 71
Table 7.4: Salmonella concentration experimental results. ............................................. 72
Table 7.5: Salmonella concentration average of all three replicates. .............................. 73
Table 7.6: Adjustable model input variables as they appear in the Excel Model ............ 76

Table 7.7: Predicted wet basis moisture concentration profile generated by Excel, given
the input variables listed in Table 4.1. ...................................................................... 77

Table 7.8: Predicted effective gap radius profile generated by Excel, given the input
variables listed in Table 4.1. ..................................................................................... 82

Table 7 .9: Predicted Darcy’s velocity profile generated by Excel, given the input
variables listed in Table 4.1. ..................................................................................... 87

Table 7.10: Predicted Salmonella velocity profile generated by Excel, given the input
variables listed in Table 4.1. ..................................................................................... 92

Table 7.11: Predicted Salmonella concentration profile generated by Excel, given the
input variables listed in Table 4.1. ............................................................................ 97

viii

LIST OF FIGURES

Figure 2.1: Types of solid-liquid flow: (a) homogeneous flow, (b) heterogeneous flow,
(c) intermediate flow, (d) saltation flow, and (e) capsule flow. This figure was

replicated from Sastry and Zuritz’s (1987) figure 3. ................................................ 15
Figure 3.1: Conceptual image for the structure of meat as a bundle of small capillary

tubes. ......................................................................................................................... 21
Figure 3.3: Steps and conceptual framework of the marination/migration model. ......... 23

Figure 3.5: Micrograph of non-marinated irradiated turkey breast (Photo courtesy of V.

Tuntivanich, Michigan State University) .................................................................. 26
Figure 3.7: MATLAB’s threshold image of Figure 3.3. .................................................. 27
Figure 3.9: Schematic of Salmonella in a vertical pipe. .................................................. 34
Figure 4.1: Predicted wet basis moisture concentration profile ....................................... 44
Figure 4.3: Predicted effective gap radius profile ............................................................ 45
Figure 4.5: Darcy’s velocity profile. ................................................................................ 46
Figure 4.7: True marinade velocity profile. ..................................................................... 47
Figure 4.9: Salmonella velocity profile. .......................................................................... 48
Figure 4.11: Salmonella location wave profile. ............................................................... 49

Figure 4.13: Comparison of predicted Salmonella concentration at t = 4.35 min versus
Warsow’s (2003) experimental results at t = 20 min. ............................................... 50

Figure 4.15: Wet basis moisture concentration with varying saturation moisture
concentrations at t = 120 s. ....................................................................................... 51

Figure 4.17: Salmonella concentration profile with varying saturation moisture

concentrations at t = 120 s. ....................................................................................... 52
Figure 4.19: Wet basis moisture concentration with varying tortuosity at t = 120 s. ...... 53
Figure 4.21: Salmonella concentration profile with varying r at t = 120 s ..................... 54

Figure 4.23: Salmonella concentration profile with varying initial effective gap radius at
t= 120 s ..................................................................................................................... 55

ix

Figure 4.25: Salmonella velocity with varying effective lengths of flagella at t = 120 s. 56

Figure 4.27: Salmonella concentration profile with varying effective length of flagella at
t = 120 s ..................................................................................................................... 57

Figure 4.29: Comparison of predicted and experimental results for the Salmonella
concentration profile of turkey breast. ...................................................................... 58

Figure 4.31: Comparison of smoothed (iZAx) and non-smoothed predicted Salmonella

concentration profile at t = 261 s. ............................................................................. 59
Figure 7.1: Three-dimensional graph of the dry basis moisture content. ....................... 102
Figure 7.3: Three-dimensional graph of effective gap radius. ........................................ 103
Figure 7.5: Three-dimensional graph of Darcy’s velocity. ............................................. 103
Figure 7.7: Three-dimensional graph of marinade velocity ............................................ 104
Figure 7.9: Three-dimensional graph of Salmonella velocity. ....................................... 104

KEY TO SYMBOLS

Lower-case Letters

effective length of flagella, measured from centerline of the cell (m)
diameter of Salmonella (m)

gravitational force (msz)

water potential (m)

capsule to pipe diameter ratio, d/Dp
empirical parameter

Darcy ’s velocity (m3 '5" °m'2)

initial effective gap radius (m)

effective gap radius at time t (m)

radius of pores in the ith pore size class (m)
distance along flow (m)

time (s)

Upper-case Letters

A
ADM

Asap. 0
Agap. t

Amuscle,0

B

Cdb, 0
Cdb, t
Cs
Cwb, 0
Cwb, t

2<2<,<..<<..<mozr7<.,cu
: 3 <

total cross-sectional area (m2)
percent area of dry matter

initial percent area of the gap
percent area of the gap at time t
initial percent area of muscle

empirical parameter
initial dry basis moisture concentration (decimal)

dry basis moisture concentration at time t (decimal)
Salmonella concentration of the marinade (cells/ml)
initial wet basis moisture concentration (decimal)

wet basis moisture concentration at time t (decimal)
capillary diffusivity (mz-s")

pipe diameter (m)

capillary conductivity (m/s)

length of individual Salmonella cell (m)

number of Salmonella cells entering the meat (cells-cm'3)

volumetric flow rate per pore (m3°s")
Salmonella relative density

average water velocity (m-s'l)

hypothetical average marinade velocity (m-s")
capsule velocity (ms'l)

threshold velocity (m-s")

location of Salmonella at time t
location of Salmonella at the previous time step

xi

Greek Letters

ABi volume fraction of pores with radius ri
ACdb change in dry basis moisture concentration
At time step (3)

Ax layer thickness (m)

pmfinade density of marinade (kg-m")

Pmeat density of meat (kg-m’3)

r tortuosity of the meat

u marinade viscosity (N -s-m’2)

v marinade velocity (m-s’l

(oi number of pores in the i pore size class

marinade surface tension (N -m'l)
time distance ratio
capillary potential (N m'z)

€>’-<

xii

1. INTRODUCTION

Salmonella, a group of bacteria found in the intestinal tract of humans and other
animals, can contaminate the outside of meat and poultry products through contact with
feces or other possible sources of contamination. Salmonella is the second leading
bacterial cause of food-bome illness in the US, after Campylobacter, and is responsible
for an estimated 1.4 million cases of salmonellosis annually in the United States,
resulting in more than 500 fatalities (CDC, 2005).

Salmonella is the target organism in the United States Department of Agriculture
(USDA) Food Safety Inspection Service (F SIS) lethality performance standards for
ready-to-eat meat and poultry products (FSIS, 1999). Ready-to-eat products have been
increasing in popularity, and marination is one means that is used to increase product
flavor, quality, and value. Value-added processing, such as vacuum tumbling, is used to
improve marinade uptake in whole-muscle meat products and can result in an increased
number of Salmonella being transferred to the interior of the meat if the exterior of the
meat or the marinade is contaminated (Warsow, 2003). Thus, marinated whole-muscle
meat products might be at risk for Salmonella contamination both in the interior and on
the exterior surfaces.

Food processors producing marinated whole-muscle meat products are operating
under the common assumption that the interiors of intact undamaged whole-muscle
products are pathogen free (Elmossalami and Wasef, 1971). This belief is also reflected
in federal regulations. Prior to 2005, F818 distinguished between intact beef cuts (e.g.,

steaks and roasts) and non-intact cuts when establishing policies regarding E. coli

0157:H7 contamination (F SIS, 2002). An intact piece of beef is defined as “a cut of
whole muscle that has not been injected, mechanically tenderized, or reconstructed”
(FSIS, 1999). The F SIS justified this distinction by stating, “. .. the interior of intact
products remains essentially protected from pathogens migrating below the exterior.
Consequently, customary cooking of intact products will destroy any E. coli 0157:H7”
(FSIS, 2002). However, due to growing concern regarding the sterility of marinated
whole-muscle meat products, FSIS issued a notice requiring all whole-muscle meat
products “injected with marinade” to be treated as mechanically tenderized product
(F SIS, 2005). Even though this notice states products “injected with marinade”, many
inspectors are applying this rule to all marinated products (Booren, 2006).
Recommendations of using known intervention techniques in these products is prudent.

Also, recent research at Michigan State University has shown that the assumption
of interior sterility may not be accurate (Warsow, 2003; Velasquez, 2006). These studies
indicate that pathogens can indeed migrate into the interior of intact products, particularly
if the product has been vacuum tumbled (Warsow, 2003). This interior contamination
can be especially dangerous, because other tests have also shown that pathogens found
inside whole-muscle products have a higher thermal resistance than those in ground
products (Orta Ramirez et al., 2005; Tuntivanich et al., 2005; Velasquez et al., 2005;
Velasquez, 2006). Thus, current cooking models may be ineffective at accurately
predicting the time and temperature required to ensure adequate thermal processing.

The limited knowledge and tools currently available makes it difficult for
companies to perform reliable lethality predictions and process validations for whole-

muscle products. No current model exists that predicts how pathogens can migrate into

whole muscle meat products. A model would allow processors to understand better how
far and fast Salmonella can contaminate the interior of whole-muscle meat products.
Thus, it is important to develop a model that the industry can use to verify process
effectiveness and ensure the microbial safety of whole-muscle products.

The type of model influences the range and usefulness of the model. Empirical
models, based entirely on experimental data, can give a very good representation of the
process they represent. However, if any of the parameters in the process change, then the
original empirical model cannot be presumed valid. Models based solely on theoretical
principles provide a much broader range of use, but it can be difficult to describe
complex materials and processes completely in theoretical terms. Therefore, the
proposed model in this study was based on mechanistic principles, such as capillary flow,
rather than solely statistical results, with the hope of giving it a broader range of
applicability than an empirical model.

The objective of this project was to develop a mechanistic mathematical model to
describe the transport of Salmonella into intact whole-muscle meat products during
marination, estimate model parameters from laboratory trials, and validate the model

using independent data.

2. LITERATURE REVIEW

2.1 Overview

This literature review is divided into nine sections that describe risks associated
with Salmonella, the type of product at risk, the problem with bacterial transport into
whole-muscle, and different types of models that could be used to represent this process.
Section 2.2 describes Salmonella and the illness that it can cause to show why it is
important to ensure that Salmonella is not found in cooked whole-muscle meat products.
Sections 2.3 and 2.4 describe the processes of marination and vacuum tumbling to show
how Salmonella may be transported into meat during processing. Section 2.5 describes
surface penetration and bacterial transport of Salmonella and marinade into whole-muscle
meat products. The remaining sections, 2.6 through 2.9, describe various models based

on flow through a porous medium, capillary flow, pipe flow, and capsule flow.

2.2 Salmonella

2.2.1 Description

Salmonella are enterobacteriaceae of the genus Salmonella. Salmonella are
Gram-negative, non-sporeforming, rod shaped bacilli found in the intestinal tract of
animals. The bacilli are straight rods that are approximately 0.5 pm wide and 2 pm long
(Adams and Moss, 2000). They exist as a single, in pairs, or in short chains, and move
by means of 1-5 uniformly distributed flagella. These flagella assist in the invasion of

human cells (CDC, 2005).

Salmonella is a facultatively anaerobic bacterium; it can survive and grow under
conditions of reduced oxygen (Adams and Moss, 2000). Salmonella can survive and
grow in temperature between 5 and 47 °C, with an optimum temperature for growth of 37
°C (Adams and Moss, 2000). The minimum water activity for growth of Salmonella is
0.93, but it can survive in foods with much lower water activities (Adams and Moss,
2000). The minimum pH for Salmonella growth is ~4.5 , with an optimum of 6.5 to 7.5
(Doyle et al., 1997). Salmonella is heat sensitive and can be killed through adequate
pasteurization or cooking processes; therefore, it is critical that food processors know

about Salmonella and the treatment required to eliminate it from contaminated foods.

2. 2.2 Salmonellosis

Salmonellosis is the disease caused by ingesting Salmonella. Two thousand of
the 2,500 Salmonella serotypes identified cause illness in humans (CDC, 2005). The
serotypes S. Enteritidis and S. Typhimurium are responsible for half of all reported cases
of salmonellosis (CDC, 2005). Because Salmonella is found in the intestinal tract of
animals, the most common sources for contaminated food include meat, milk, poultry,
and eggs. These contaminated foods cause illness in humans when they are consumed
without proper cooking or if the food is cross-contaminated without further cooking
(CDC, 2005). The typical infectious dose is as low as 1 to 10 cells, but varies depending
on food source, susceptibility of individual, and the virulence of the serotype (Doyle et
al., 1997). Symptoms of salmonellosis include fever, abdominal cramps, and diarrhea
(sometimes bloody), which usually appear 12-72 h after infection (CDC, 2005).

Everyone exposed to Salmonella is at risk of contracting salmonellosis; however, these

symptoms are more severe in the elderly, young, and immune compromised (CDC,

2005).

2.3 Marination

Marination is a process in which a solution is added to meat products, in order to
improve water-holding capacity and thereby increase the juiciness, tenderness, and
weight of the meat. A typical marinade is composed of water, salt, and phosphate. The
effects of salt and phosphate vary depending on how much is added to the meat. High
levels of salt (IO-15%) lower the water activity in the meat and act as a preservative to
reduce the risk of microbial growth (Barbut, 2002). Low levels of salt have the reverse
effect and increase the water holding capacity of meat (Barbut, 2002). High levels of
phosphate can result in a decrease in the rate and depth of penetration compared to lower
percentages (Xiong and Kupski, 1999a). The rate of phosphate transport into meat also
depends on the type of phosphate, with larger phosphates diffusing more slowly than
smaller phosphate molecules (Xiong and Kupski, 1999a). The combination of low levels
of salt and phosphates creates a synergistic effect and results in decreased cooking loss
and thereby greater yield than adding salt or phosphate alone (Froning and Sackett, 1985;
Sheard and Tali, 2004). However, combining higher levels of salt (8%) and phosphate
eliminates the benefits of phosphate (Xiong and Kupski, 1999a; Xiong and Kupski,

1999b).

During marination, there is an initial outward flow of water and soluble proteins
from the muscle to the marinade, due to the lower osmotic pressure of the marinade

(Lawrie, 1985). However, as the salt diffuses into the meat and binds with the proteins,

the osmotic pressure of the meat becomes less than that of the marinade, and the
marinade begins to flow into the meat (Lawrie, 1985; Xiong and Kupski, 1999). The rate
of diffusion depends on the strength of the marinade and the microstructure of the muscle

tissue (Lawrie, 1985).

It is clear that adding salt and phosphate can improve marinade uptake; however,
no references were found that described the fundamental mechanism responsible for

transporting plain water into whole-muscle meat products.

2.4 Effects of Vacuum and Tumbling

Vacuum and tumbling are methods used by processors of marinated whole-
muscle meat products to improve marinade uptake. Tumbling occurs in a rotating drum
equipped with different ribbon designs on the sides (Barbut, 2002). Most commercial
tumblers have vacuum capabilities, which help remove air bubbles from the exudates and
assist in protein extraction (Barbut, 2002). The purpose of tumbling is to improve
marinade distribution and protein extraction from whole and chunked muscle products
(Barbut, 2002). The meat is subjected to a certain degree of agitation that assists in
distributing the salt and myofibrillar proteins. Marinade absorption during tumbling is
time dependent, because the marinade must overcome physical barriers in muscle in
order to diffuse into muscle fibers and myofibril matrices (Xiong and Kupski, 1999). The
effects of vacuum and tumbling were not considered in the current model, but if they

were included, they would likely increase the rate of Salmonella uptake in the meat.

2.5 Surface Penetration and Bacteria Transport into Whole-Muscle

Researchers have studied the effects of tenderization and marination treatments in
which the surface of the whole muscle is penetrated by a needle, blade, or other
mechanical device (Boyd et al., 1978; Johnston, 1978; Raccach and Henrickson, 1979).
Boyd et al (1978) tested the sensory characteristics and microbial counts of mechanically
tenderized beef and found that bacterial concentrations were higher in samples that had
been tenderized four times compared to those tenderized once or twice or the
untenderized control. Johnston (1978) summarized various aspects of a petition by the
Community Nutrition Institute concerning the mechanical tenderization and packing of
meat. The petition claimed that mechanical tenderization increases the microbial counts
in meat by forcing bacteria, including Salmonella, into the interior of meat. Johnston
(1979) also gave examples of outbreaks that occurred because of severe undercooking of
meat that had surface Salmonella injected into the interior of the meat. Raccach and
Henrickson (1979) showed that Salmonella and other bacteria can contaminate the
interior of beef rounds and ribeyes during mechanical tenderization, but can be greatly
reduced through proper sanitation of equipment. This research has shown that pathogens
can migrate to the center of whole-muscle meat products if the surface of the meat is
broken (Boyd et al., 1978; Johnston, 1979; Raccach and Henrickson, 1979).

A few previous studies have measured the effects of various bacteria and meat
characteristics during unidirectional migration of bacteria into surface-inoculated whole
muscle meat under atmospheric conditions (Gill and Penney, 1977; Gill and Penney,

1982; Gupta et al., 1983; Maxcy, 1981; Sikes and Maxcy, 1980; Thomas et al., 1987).

These studies have shown that the transport of pathogens into meat is affected by
proteolytic activity, bacteria motility, water availability, and fiber orientation.

The proteolytic ability of bacteria, (the ability to break down protein molecules),
has been shown to increase the depth of bacterial penetration (Gill and Penney, 1977; Gill
and Penney, 1982; Gupta et al., 1983; Thomas et al., 1987). Proteolysis is presumed to
speed the process of penetration by breaking down the connective tissue between muscle
fibers (Gill and Penney, 1977; Thomas et al., 1987).

There have been some conflicting studies as to the effect of bacteria motility. Gill
and Penny ( 1977) concluded that both motile and non-motile proteolytic bacteria could
penetrate the surface of the meat. However, Thomas et al. (1987) found that motile
bacterial strains were able to penetrate whole-muscle while non-motile strains remained
on the muscle surface, regardless of the bacteria’s ability to produce proteolytic enzymes.

The water-holding capacity of meat proteins also appears to have a significant
effect on bacterial penetration into meat (Maxcy, 1981; Sikes and Maxcy, 1980; Thomas
et al., 1987). Some researchers showed that freezing and thawing meat can lead to
increased bacterial penetration (Maxcy, 1981; Sikes and Maxcy, 1980). They suggested
that the loss of water molecules during freezing and thawing and the collapsed state of
the muscle proteins result in the creation of larger pores for bacteria to move through
(Maxcy, 1981; Sikes and Maxcy, 1980). Thomas et al. (1987) showed that high water
contents increased the rates of penetration, presumably by increasing the inter-fiber
distance in muscle and thereby decreasing the resistance to microbial movement within

the tissue.

The rate of bacterial invasion into intact whole-muscle pork was greater in
samples where the inoculum pathway was parallel to the fiber orientation than in samples
with the inoculum perpendicular to the fiber orientation (Maxcy, 1981; Sikes and Maxcy,
1980)

Recent unidirectional marination trials with whole-muscle turkey breasts at
Michigan State University indicated that Salmonella counts decreased with distance
below the contaminated surface, increased with application of a vacuum, and increased
with time (Warsow, 2003). Warsow’s Salmonella log counts from unidirectional
marination tests were used to estimate parameters for the proposed model in this study;
his procedure and results are given in sections 3.10.1 and 4.10, respectively.
Multidirectional tests were also conducted with whole—muscle pork and turkey exposed to
Salmonella-inoculated marinade, which entered the product from all directions. These
experiments resulted in the same general conclusions as the unidirectional tests
(Tuntivanich et al., 2006; Velasquez, 2006; Warsow, 2003).

If Salmonella enters a whole-muscle product, thermal inactivation tests have
shown that Salmonella exhibits greater heat resistance than in ground products (Orta-
Ramirez et al., 2005; Tuntivanich et al., 2005 ; Velasquez et al., 2005; Velasquez, 2006).
Those studies involved immersing core samples of irradiated whole-muscle meat in
Salmonella-inoculated marinade for 20 min. The percent uptake was determined for the
whole-muscle samples, so that the same amount of inoculated marinade could be mixed
into the ground samples. There was no difference between the initial log counts of the
whole and ground muscle samples. However, the thermal inactivation rate constants in

ground muscle were double those in whole muscle at a given temperature. The

10

composition, bacterial load, and thermal history were identical among sample types;
therefore, the difference in Salmonella thermal resistance was attributed to the different
physical state of meat components in the sample. Thus, it was concluded that the ground
meat thermal inactivation studies used to set performance standards for meat and poultry
products may be insufficient for whole-muscle products (Orta—Ramirez et al., 2005;

Tuntivanich et al., 2005; Velasquez et al., 2005; Velasquez, 2006).

2.6 Flow through a Fibrous Porous Medium

2.6.] Fluid Flow

The process of marinade absorption into meat can be viewed as the flow of a
liquid through a fibrous material. Several researchers have modeled various aspects of
liquid flow through a fibrous medium (Davis and James, 1996; Dhotkar et al., 1999;
Kolodziej et al., 1998; Papathanasiou, 2001). Most of these studies focused on modeling
matrix properties, such as permeability (Davis and James, 1996; Kolodziej et al., 1998;
Papathanasiou, 2001). Dhotkar et al. (1999) developed friction, pressure, and total drag
coefficient equations for non-Newtonian fluid flow through a fibrous medium. None of
these articles gives any details as to the driving force responsible for the movement of
fluid.

Zhong et al. in 2005 examined gravity driven flow of a fluid through a very
porous fibrous medium. However, all the liquid can pass all the way through very porous
media (meaning no capillary potential); therefore, this case would not accurately

represent mass transfer through muscle tissue.

11

2. 6.2 Particle Flow

Nilsson and Stenstrom (1994) investigated capillary gas diffusion of water vapor
molecules through sheets of a fibrous porous medium. Multiple researchers have
modeled the deposition of aerosol particles on fiber filters (Kirsh, 2004; Lebedev et al.,
2002). However, these models are based on gas flow and very small particles; therefore,
they do not provide the best representation of the current application.

The rate of particle migration through granular filters also has been modeled
under varying hydraulic loads (Indraratna and Vafai, 1997; Locke et al., 2001). Mauret
and Renaud (1996) compared capillary and particulate representation of the drag force on
a cylinder in a fiber bed. Li and Park (2000) modeled the flow of colloidal particles in a
liquid through fibrous and granular material by convection and Brownian diffusion. The
particles described in that paper were small colloidal particles that were usually a few
orders of magnitude smaller than the filter. Koska et al. (1996) modeled osmotically
driven flow of particles through a hollow fiber bioreactor. A hollow fiber bioreactor
resembles a tissue-capillary system; however, the equations developed rely heavily on the
specific dimensions of the bioreactor, which are unknown for the current application.

A common problem with all of the articles on particle flow through a fibrous
medium is that the particles are very small compared to the size of the pore space. In
contrast, Salmonella particles are relatively large, on the same order of magnitude as the
intercellular pore space in whole-muscle meat products. These models all represent a
similar concept of the flow of particles through a fibrous medium; however, none of them
combined the type of fluid, size of particle, and driving force needed for the present

application.

12

2.7 Capillary Flow Models

Capillary flow was assumed to be the major driving force responsible for drawing
bacteria and marinade into whole-muscle products. There are limited models that
describe capillary driven flow through a fibrous or porous medium. However, other
models based on capillary flow exist that do not involve movement through a fibrous
material.

Fink and Muller (2000) compared the penetration depth of different viscous
liquids in silicone rubber. The liquids used in the experiments were viscous solutions
containing the corrosive alkali lithium hydroxide (LiOH) and therefore do not provide a
good representation of the marinade used in the current model.

Some published models have described the flow of red blood cells in capillary
networks (Bruinsma, 1996; Schmid-Schonbein et al., 1980). Bruinsma (1996) focused on
the rheology and shape transition of red blood cells under capillary flow, showing how
particles move through a complex capillary; however, they did not model the velocity or
concentration of the particles. Schmid-Schonbein et al. (1980) modeled the distribution
of red blood cells in capillaries; however, their model is complex and does not yield a set

of equations that can easily be applied to a different situation.

2.8 Pipe Flow

Capillaries in general can be described as very small tubes; therefore, a very
simplified model could describe Salmonella movement in meat as particle flow through
bundles of tiny capillary tubes. Jean and Peddieson (2001) modeled the velocity, volume

fraction distribution, and particle segregation patterns due to buoyancy of particulate

l3

suspensions in vertical pipes. Unfortunately, this was for the steady-state flow of very
small particles. Other researchers have studied the flow of food particles through a
vertical pipe as it relates to aseptic processing (Lareo and Fryer, 1998; McCarthy et al.,
1997; Sastry and Zuritz, 1987). However, these were all for liquid particle mixtures
where the particle diameter was significantly smaller than the diameter of the pipe.
Sastry and Zuritz (1987) classified the various types of solid-liquid flow in a pipe as
homogeneous flow, heterogeneous flow, intermediate flow, saltation flow, and capsule

flow (Figure 2.1).

14

 

 

 

 

 

 

 

a. . r . _ .- 1}. I '7 '. a. . I v
.- ’ I.-.'. - '3'. ' .3 ‘5'," ’ I. ' .‘ .32 '
. .f - o ' .- 5‘. ' :z.‘ ...._fi _ ._ _ .-'.-" _'. . _ '2.
. - t-o\ .-’ at“ " .‘-’ ..'-' _' \-~ - '. _c,.- ,. '4' .-
...'\'_\ .u ' ‘ ‘ .”\ ;.: ' 0'. . '5 ' .' 0 3‘ 0' 0., :3
27- " n. ' 0'. .' . ' .'. . ~‘-' '--3. -.-... -'.~-°'.'.
Q n I .' .. ..., .u l‘, ., 5”,", \ u . . ..\ . a.
‘] _—+ ' ' "1“I.’ - , .. a ,\ .."’: \ - " f 0,: -' -'.. .FI._"JO': \.'.
. ' '0 u a: '2'..." n " . -_ 'du'-”\} .I.. . I . ‘e..’: ‘J . I
u .. - ’.. - .1 g I’ f“ \u‘ u. I". 0'." ,4 _ _‘
. ‘ .-‘ . ..-, v ... ’9 '. I" ~
. ‘ n; ..‘.: ‘ ) '3’ .- ..'.' 2 . 0.. ‘ :3". r" l.
" " '. - ,'.\.',-.‘.\‘, :9 '~ ‘ . -' ' I'i/ZI
' 4: 'n... ‘ ".~' -\I ' L A
I
. . O . 1 . Q . . C
0 0 O O
O a ' . . a
.
b __’ ' I 1' I
a a I . a
i ' l ' i
o _a
a I . .a I . a . . .0 a I .
I I O I I O ' I I 0
1"0 1"0 I"O fl'o
- . l . l
. ', . I , . I . - . I . . .. . -
p -‘I O I ’ ' 5.9 a ' . 3.? a.
' ". . .' 1 . '3 ‘ . :g 'u .
' D .. -. P . . . ’ .. . . . . 2" . .' .
" o -= ' . '3' a . o ' ID 'i .‘ ..‘-
e e e . I .- . 1'...
‘ z; ‘1'- .‘ ‘5". ‘ .q i"-- . .411"- ._
.'g I F-' .g .I l' . .1 D“ G A i 9‘

 

 

 

 

 

e—i

 

 

Figure 2.1: Types of solid-liquid flow: (a) homogeneous flow, (b) heterogeneous
flow, (c) intermediate flow, ((1) saltation flow, and (e) capsule flow. This figure was

replicated from Sastry and Zuritz’s (1987) figure 3.

Capsule flow involves the movement of an object through a pipe of roughly the

same internal diameter. Therefore, this form of flow may be a reasonable model for the

movement of Salmonella through muscle tissue.

2.9 Capsule Flow Models

Capsule flow has been studied the most as it relates to capsule pipelining.

Pipelining is a method of transporting material by enclosing it in a capsule and then

15

moving it through a pipe via the flow of fluid, usually water or air. In the 1960’s and
1970’s, the Research Council of Alberta (ARC) conducted theoretical and experimental
research on capsule pipelining and published a series of 18 articles on their findings;
however, only three of these articles included information on the velocity of a cylindrical
capsule moving though a pipe (Charles, 1963; Ellis, 1964; Hodgson and Charles, 1963).

The first paper in the series illustrated the various flow patterns of oil capsules
moving in a horizontal pipe under low and high water velocities. Testing showed that the
velocities of the oil drops exceeded the overall velocities, because the oil drops are
located in the section of pipe where the linear velocity is significantly greater than the
average velocity of the overall pipe flow (Hodgson and Charles, 1963). Unfortunately,
they did not quantify or model the velocity of the capsules.

Charles (1963) predicted the steady-state capsule velocity for a long cylindrical
capsule moving through the center of a horizontal pipe under laminar and turbulent flow
conditions. The proposed equation for capsule velocity under steady state laminar flow

was:
V =2V' (14(2) [2.1]
C av

where:
V’av = hypothetical average velocity (m/s)

k = capsule to pipe diameter ratio, d/D

This equation could be used for the current application; however, an equation

representing flow in a vertical pipe, rather than a horizontal pipe, was used instead.

16

Ellis (1964) compared Charles’ theoretical model to experimental velocity
measurements for cylindrical and spherical capsules in a horizontal pipeline. The flow of
single capsules of different sizes, shapes, and densities was tested in various liquids to
determine the effects of length and end shape of the cylindrical capsules. Dimensional
analysis showed that the velocity ratio was a function of four independent variables:
average water velocity, diameter ratio, capsule length/ diameter ratio, and capsule end
shape.

Latto and Chow (1982) modeled the steady-state velocity of a cylindrical capsule
in a vertical pipe. Experiments were conducted using a 7.6 cm inside diameter vertical
steel pipe and aluminum and nylon cylindrical capsules of 0.49, 0.65, and 0.82 diameter
ratio. The velocity of the water ranged from 0.3 to 5.5 m/s. They developed a semi-
empirical equation for capsule velocity by first plotting the experimental results for
capsule velocity against the average water velocity in a test section minus the suspension

(threshold) velocity. The equation of best fit for the data was:

V. = [(3'le (V... —V.) [2.21

where:
Vc = capsule velocity

k = capsule to pipe diameter ratio, d/D
L = capsule length (m)
d = capsule diameter (m)

Vav = average water velocity (m's'l)

v0 = threshold velocity (m-s")

17

An equation for threshold velocity was determined using dimensional analysis.

0.371
] [2.3]

Vo : 281309 *1)(1-k2)(§
where:
g = acceleration due to gravity (9.806 my?)
D = pipe diameter (m)

S = capsule reality density

Substituting equation 2.3 into equation 2.2 yielded the semi-empirical equation for

capsule velocity through a vertical pipe.

V, = ( 2017816)“ V.v — \/2gD(S—l(§-)(l—k2):l [2.4]

Because this equation is for a small particle to pipe ratio in a vertical pipe, it offers a

 

reasonable representation of the transport of Salmonella through the meat. Therefore,
this equation is used in section 3.6 to calculate the velocity of a Salmonella bacterium

moving through meat.

2.10 Summary

0 Salmonella is a harmful bacterium that can cause serious illness if contaminated
meat is not properly processed (CDC, 2005).
0 Capillary diffusion was assumed to be the driving force transporting marinade

into whole-muscle meat products.

18

Research has shown that pathogens can migrate to the center of the meat if the
surface of the meat had been damaged by blades, needles, or known process
techniques (Boyd et al., 1978; Johnston, 1979; Raccach and Henrickson, 1979).
Unidirectional (Warsow, 2003) and multidirectional (Velasquez, 2006; Warsow,
2003) marination trials at Michigan State University indicated that Salmonella can
contaminate the interior of whole-muscle meat products regardless of whether the
outside had been damaged by blades, needles, or known process techniques.
Thermal inactivation tests showed that Salmonella exhibits greater heat resistance
in whole-muscle as compared to ground products (Orta-Ramirez et al., 2005;
Tuntivanich et al., 2005; Velasquez et al., 2005; Velasquez, 2006)

The equation by Latto and Chow (1982) for capsule flow through a vertical pipe
represents a reasonable model for the velocity of Salmonella as it moves through

or into a whole-muscle product.

19

3. METHODS AND MATERIALS

3.1 Overview

The model proposed in this study represents one—dimensional transport of

Salmonella into whole-muscle meat products, assuming capillary diffusion of marinade

through small uniform tubes, and capsule-like transport of the Salmonella cells along

with the marinade.

The transport of Salmonella into whole-muscle meat during marination is a

complex process; therefore, in developing s simplified one-dimensional model, a number

of assumptions were needed. These assumptions were:

1.

Capillary diffusion is the only driving force transporting marinade into the meat
(Gravitational force was calculated to be ~ 1.2 % of capillary force for this case).
The meat is composed of a bundle of capillaries that are uniform cylindrical tubes
of the same size (Figure 3.1).

All of the marinade transported into the meat is immediately absorbed into the
cells of the meat, causing them to expand, thereby reducing intercellular space.
When the intercellular space decreases below the size of a Salmonella cell, the
migration of Salmonella within the meat stops.

The physical properties of the marinade are equal to those of water.

The density of Salmonella is equal to the density of water.

All Salmonella are non-motile inert capsules of the same size.

There are no physicochemical effects from salts and phosphates in the marinade.

20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.1: Conceptual image for the structure of meat as a bundle of small
capillary tubes.

The overall solution consisted of eight steps, which sequentially estimated and/or
solved for the following:
1. Physical and transport properties
2. Moisture concentration
3. Effective gap radius
4. Darcy 's velocity
5. True marinade velocity
6. Salmonella velocity
7. Salmonella location profile
8. Salmonella concentration profile
Figure 3.2 contains a flow chart that describes the steps and conceptual
framework of the model. The model was implemented completely in Microsofi Excel
(Excel 2003, Microsoft Corp., Richmond, WA), using several user-defined functions and
Palisade’s RISKOptimizer (RISKOptimizer trial version, Palisade Corp., Ithica, NY).
Once the model was formulated, two experimental data sets (Salmonella concentration

profiles) were used to estimate two model parameters and to validate the solution.

21

 

C START

A

 

Calculate initial diffusivitv (Eq. 3.15)

 

 

 

 

 

 

 

 

 

 

 

 

INPUTS $
p: Y, pmarinade, 15,
ADM, C0, Pmeat, Calculate At based on the parameters of the central
Felt, 0, Agap ’ 0, d5, difference solution (Eq. 3.17)
LS, S, g, AX H

t = 0
H N
Y

 

 

 

 

 

Recalculate D given the new conditions (Eq. 3.15)

END

 

 

T

 

Calculate Dry Basis Concentration at t based on the central
difference solution of l-D capillary diffusion (Eq. 3.16)

 

 

I
”

 

7

 

 

Convert to Wet Basis Concentration at t

"‘

 

i

I

 

Calculate Effective Gap Radius at t based on the expansion of
cells with marinade uptake (Eq. 3.8)

O
"
---

I
’l
I
—-:=-

 

7

\
’

 

Calculate Darcv’s Velocitv (Eq. 3.18)

k

‘
——

 

7

‘\

 

Calculate Marinade Velocity in intercellular space (Ea. 3.19)

\

‘
---

 

V

 

Calculate Salmonella Velocity based on capsule flow through
intercellular space (Eq. 3.20)

 

T

 

Calculate Salmonella Location based on the previous
Salmonella Location, the average Salmonella Velocity, and the
change in time (Eq. 3.22)

 

7

 

Calculate Salmonella Concentration by treating the
Salmonella entering the meat at each time step as separate
groups and then adding up the number of Salmonella at a given
location and time. (Eq. 3.23)

 

 

22

 

 

 

x=x+l

 

 

 

Figure 3.2: Steps and conceptual framework of the marination/migration model.

3.2 Physical and Transport Properties

Darcy ’3 Law describes the overall flow of a liquid through a porous medium:

n" =Kfi [3.1]
as

where:

nv = volumetric flux or Darcy’s velocity (m3 ~s'1-m'2)
K = hydraulic conductivity (m-s'l)

h = water potential (m)

s = distance along flow (m)

Written this way, Darcy ’3 Law is a generic equation for liquid flow through any
type of porous material. Therefore, equation 3.1 was modified to better represent
capillary flow through a bundle of small tubes. This was done in part by using the

following equation for volumetric flow rate of Poiseuille’s flow through a tube of

uniform radius (Datta, 2002):

4
7r-r. ~p'g
Q, =_ _l__ (fll) [3.2]
1 8p 65

where:
Qi = volumetric flow rate per pore (m3-s")
fl = radius of the tube (m)
p = density of (kg-m")

g = acceleration due to gravity (9.806 m-s'z)

23

l4 = viscosity of the fluid (N 'S'm'z)

The Poiseuille’s flow equation is for a single straight pipe; therefore, equation 3.2

was altered to take into account a distribution of pore sizes (Datta, 2002), such that:

A3. = —L——' [3.3]

where:

ABi = volume fraction of pores with radius ri
co; = number of pores in the ith pore size class

r; = radius of pores in the ith pore size class (m)

A = total cross-sectional area (m2)
In addition, a tortuosity factor (I) was introduced to account for a non-straight
travel path of the fluid; r is the ratio of the roundabout path along the pore to the straight

flow path (Datta, 2002). Therefore, the volumetric flux, or Darcy’s velocity was

represented by:

n"=8—_'——"#g 2 Mr 3(2—3] [3.4]

2

Given the stated assumption of uniform pore size, 2 A'Biri simplifies to r2 , so that:

efl,t

 

v pa 2 6h]
__. . . _ 3.5
n 8-p-r refit (as [ ]

Pressure is related to head as P = pgh; therefore, terms in equation 3.4 can be combined

to produce the following equation:

24

r2
n": ——eff” (31:) [3.6]
8-,u-r 6s

Thus, the conductivity of the meat is represented by the following equation:

2
_ r 617,1
K— 8m [3.7]

 

where:
u = marinade viscosity (N -s-m'2)
r = tortuosity of the meat

rem t = effective radius of the pore space at time t, representing the intercellular

gap (In)

The effective gap radius of the pore space at time t was calculated based on the

assumption that the effective gap radius is proportional to the area of the gap.

 

_ (reflfl llAgapJ) 3 8
reflfit _ A [ ' ]
gap,0
where:
refl,0 = lnltlal effective gap radlus (m)
Agap,0 = inltlal percent area of the gap
Agap, t = percent area of the gap at tlme t

The initial reff value was approximated from Figure 3.3 by measuring the area

between cells (i.e., the thickness of the gap) at various locations. The measured

25

thicknesses were averaged and divided by two to get an initial value for the effective gap

radius.

 

Figure 3.3: Micrograph of non-marinated irradiated turkey breast (Photo courtesy
of V. Tuntivanich, Michigan State University).

The initial percent area of the gap (Asap, 0) was estimated using MATLAB’s
Image Processing Toolbox (MatLab Image Processing Toolbox, The Mathworks, Natick,
MA ). A thresholding value was used to determine the cutoff point between light and
dark pixels. The meat’s pore space was represented by white pixels (pixels with a value
below the threshold), while the meat’s cellular space was represented by the pixels above

the threshold value (Figure 3.4). An optimum thresholding value (145) was

26

automatically calculated in MATLAB using the Otsu method for thesholding. The

number of white pixels was divided by the total number of pixels in the image to yield

the percentage of pore space.

 

Figure 3.4: MATLAB’s threshold image of Figure 3.3.

The percentage of pore space decreases as the moisture content of the meat

increases and was calculated using:

A
A [3.9]

Agaptfl‘ 1 C
, — wb,t

where:

27

ADM = percent area of dry matter in a piece of meat

CW = wet basis moisture concentration at time t

The percent area of dry matter in the meat was calculated from the initial wet
basis moisture concentration and the initial percent area of the gap. The amount of dry
matter in meat does not change with marinade uptake and, therefore, does not have to be

recalculated at each time step.

ADM =(i_aw,0)[i-igap0] a...
where:
C = initial wet basis moisture concentration
wb,0
Agap,0 = lnltlal percent area of the gap

The capillary-based Darcy '3 flow equation 3.6 was converted to a capillary

diffusion based flow using the definition of capillary diffusivity (Chow et al., 1988):

_ d‘I’

 

where:
K = capillary conductivity of meat (m-s")
‘1’ = capillary potential (N °m'2)

Cdb = dry basis moisture concentration

28

The capillary conductivity used in this equation was calculated using equation 3.7.

Capillary potential was/expressed as (Datta, 2002):

 

q! 2 2'7 [3.12]
refl,t

where:
y = surface tension of the liquid (N 111'!)

r = effective gap radius at time t (m)

eff,t

Based on this equation, [fl] from equation 3.11 can be rewritten as follows:
db

 

6C db 6C db

r
[—‘N’ ]=——efl’t [3.13]

The derivative of capillary pressure with respect to dry basis moisture content was solved

using the chain rule. The complete derivation is included in Appendix 7.1, yielding:

a[—27
r 2. -A -A
efl : 7 gap,0 DM [3.14]

a(M ) refl,O(MC,t -(1—ADM))2

Substituting this equation and equation 3.7 into equation 3.11 results in the completed

 

 

equation for the diffusivity of the meat at any given time step:

2 2-7-A A

D- refl,t gap,0. DM

.. [3.15]
8-u-r refl,O(MCt-(l—ADM»2

 

29

3.3 Moisture Concentration

Given D, the central difference solution for l-D unsteady-state diffusion (Merva,

1995) can be applied as follows:

CW“) = A'C(i—1,t)+(1_2"l)c(i,t)”'C(i+1,t) [3.16]

where lambda (A) is the following time and distance ratio:

_ DAt

,1—
Ax2

[3.17]

and:
At = time step (5)

Ax = solution layer thickness (m)

For solution stability, the time step and layer thickness were selected based on the
rule that (Merva, 1995):

0<11=D—A2t30.5

Ax
Following this rule, equation 3.17 was used to calculate At from the initial diffusivity and
a reasonable guess for Ax, and setting the equation equal to 0.5. The value of Ax was
adjusted until a reasonable time step value was found.
Diffusivity of the meat decreases as the moisture concentration of the meat
increases, because, as marinade diffuses into the muscle, the cells swell, and the percent
area of the gap decreases along with the effective gap radius of the pores. The effective

gap radius used in equation 3.15 is the effective gap radius at the previous time step.

30

Because it is not a fixed value, it must be recalculated at every time step. In order to do
this, a user function was created (see appendix A.2). This user function calculates a new
it for each position at each time step and enters it into equation 3.16, returning the dry
basis moisture concentration for the given location and time.
The solution for the moisture concentration profile had the following boundary
conditions:
1. At the initial time step (t = 0), moisture concentration at all locations was set
to the initial moisture concentration of the meat (Co).
2. At every other time step (t > 0) the moisture concentration in the first layer
C(x = 0) was set to a saturation value consistent with the results of the

moisture concentration tests described in section 3.11.1.

3.4 Effective Gap Radius

The effective gap radius was calculated using equations 3.8 and 3.9.

3.5 Darcy’s Velocity

Darcy ’s velocity of the marinade is the volumetric flow rate of the marinade,
relative to the cross-sectional area of the meat. Darcy ’s velocity of the marinade was

calculated using Fick’s Law (Datta, 2002):

AC
n" = —D.[—A%] [3.18]

where:

nv = Darcy 'S velocity of the marinade (mB'SJ'mQ)

D = capillary diffusivity (mZ-s'l)

31

ACdb = change in dry basis moisture concentration

Ax = layer thickness (m)

3.6 Marinade Velocity

Darcy 's velocity does not reflect the true average velocity of the marinade within
the pores, because the cross-sectional area of the pores (i.e., area available for flow) is
smaller than the total cross-sectional area of the entire piece of meat. Thus, the true
marinade velocity as it is drawn into the meat can be estimated by dividing Darcy ’s
velocity by the porosity of the meat. In this case, porosity is defined as the fraction of

cross-sectional area that is intercellular space.

 

 

 

( \
nv t
x
v . = , [3.19]
d ,t,
marina e x — A D M
K 1 — CWOJ )
where:
”V t = Darcy ’s velocity at the given location and time (m3-s'l'm'2)
x9

ADM = percent area of dry matter

CW b t = wet basis moisture concentration at time t

3.7 Salmonella Velocity

Velocity of a single Salmonella bacterium was calculated using Latto and Chow’s

(1982) equation for a cylindrical capsule in water moving through a vertical pipe:

32

 

 

Vc = [#11128 ][(_§_]0-128 Vav — ngms - 1(30 — 19)] [3.20]

where:
Vc = capsule velocity (m-s")
k = capsule to pipe diameter ratio, d/D
L = capsule length (m)
d = capsule diameter (111)

Vav = average water velocity (m-s")

Vo = threshold velocity (m'S'l)

g = acceleration due to gravity (9.806 m-s'z)
D = pipe diameter (m)

S = capsule relative density

However, equation 3.20 is for a smooth cylindrical capsule flowing through a larger

smooth pipe; therefore, the capsule never slows down or stops because of reducing pipe
diameter. A new term (dfl/I'eff, t) and two model parameters were added to equation 3.20

to take into account the flagella that could cause the Salmonella to slow down and stop in
a manner that would occur with an idealized, smooth capsule, so that equation 3.20 was

modified to become:

0.128 2-r "
yea-[galls] V..—l2gD<S-Il§l(I-k2)l—;”—’

f

 

 

where:

33

B = empirical parameter
df = effective length of flagella, measured from centerline of the cell (111)

r = effective gap radius at time t (m)

efl,t
n = empirical parameter
The number of flagella was not incorporated into this term, except to assume that
flagella were unifome distributed around the cell. A schematic of a Salmonella cell in a

vertical pipe is shown in Figure 3.5.

U
9.
‘0

a)

)e

-_.€--.Z--_(___§.-__i____

Q.
m

 

 

 

""7""2'"?"i"?'""+'""
I

—>

«a:
DJ
<

 

 

Figure 3.5: Schematic of Salmonella in a vertical pipe.

The current model assumes a capsule (Salmonella) relative density of one;
therefore, part of equation 3.21 reduces to zero, resulting in the following simplified

version of the equation:

34

n

0.128 2-r
1 L my
V =B-( ][[—j V ]— [3.22]

 

An optimization procedure (RISKOptimizer trial version, Palisade Corp., Ithica, NY) was
used to find the best-fit estimate of B and 11 based on minimization of the sum of squared
errors between the model and Warsow’s (2003) experimental data for Salmonella

concentration profile.

3.8 Salmonella Location

A wave profile of the location of Salmonella in the meat was created, where a
new group (or wave) of Salmonella enters the meat at each time step, flowing along with
marinade. The location of each existing wave is recalculated at each time step, based on
the new velocity at the current location.

Xt =Xt_ +V At [3.22]

1 c...
where:
X, = location of the Salmonella at time t
XH = location of the Salmonella at the previous time step

Vc x t = Salmonella velocity at the given location and time

At = time step

The algorithm by which VCM was determined within the model solution can be found in

Appendix Al 1.

35

3.9 Salmonella Concentration Profile

The number of cells entering the meat with each new wave was determined based
on the velocity and concentration of the marinade.

N=nv-C oAt-i [3.23]
s Ax

where:

N = number of Salmonella cells entering the meat (cells/cm3)
nv = Darcy ’s velocity of the marinade (m3's'l-m'2)

CS = Salmonella concentration of the marinade (cells/ml)

At = time step (8)

Ax = layer thickness (m)

A Salmonella concentration profile was created by summing the number of

Salmonella in each layer at each time step.
3.10 Model Inputs

3.] 0.1 Meat Properties
The initial percent area of muscle and pore spaces estimated from microscopy
image (Figure 3.3) were:

Areamuscle,0 = 0.87

Area = 0.13
gap,0

The percent area dry matter was calculated using equation 3.10

36

Area DM = 0.24
The tortuosity factor (I) introduced in equation 3.4 is an unknown variable;
however, it was assumed to be three, because the pore space of the meat is not perfectly
straight and therefore I must be greater than one.

The density of the meat were assumed to be:

_ kg

3.10.2 Marinade Properties
The viscosity ([1), surface tension (y), and density (p) of the marinade was

assumed equal to that of pure water at 4 °C:

#:1519‘3M

m2
N
= .0 4—
y 0 75 m
p=1005—’5-g3—
m

These values are input values in the model and can be changed easily should more exact

values for the marinade be found.

3.10.3 Salmonella Properties
Adams and Moss (2000) reported the length and diameter of Salmonella as:
Diameter of Salmonella = 5.E-07 111
Length of Salmonella = 2.E-06 m
The relative density of Salmonella is approximately the same as pure water; therefore, a

relative density of 1.0 was used.

37

3.10.4 Calculated Initial Values

The initial capillary diffusivity calculated using equation 3.15 was:

D = 1557151305 mz/s

The time step and the thickness of the layers was calculated using equation 3.17:
At = 0. ls

Ax = 0.005m

3.1 1 Validation

A moisture concentration test and a Salmonella log count test were conducted to
validate the proposed model. These tests were conducted in a controlled temperature

room at 4 °C.

3.11.1 Moisture content

The marination procedure used for the moisture concentration trials was the same
as the one used by Warsow (2003) for his unidirectional marination study.

For the moisture concentration test, irradiated (10 kGy) whole-muscle, boneless,
skinless turkey breast was used. The turkey breast was frozen and held in vacuum-
packed bags at -15 ° C for ~2 years and then thawed for 48 h at 4 ° C. A 10x10x5 cm
block was cut from a turkey breast and placed out side down in a 15 cm diameter Petri
dish containing a round perforated stainless steel plate and 40 ml of sterile marinade. The
marinade reached approximately 5 m up the side of the turkey breast. The marinade
was composed of 95 % water (filtered and deionized), 3.3 % NaCl, and 1.7 % mixed
phosphate solution. In order to limit the airflow around the marinating meat, the sample

was sealed in a laboratory-scale tumbler during marination. After 20 min of marinating,

38

the sample was removed from the marinade and placed on a new 15 cm diameter Petri
dish. Two different methods of removing samples from the turkey block, described
below, were tested to determine the method with the smallest degree of error in
determining moisture content:

1. Four one-inch diameter cores were removed from the center of the turkey
block and sliced into three pieces. The meat segments were cut into smaller
pieces and placed into aluminum weighing dishes.

2. The block of meat was sliced using a dry knife into three layers, and each
layer was ground separately in a small Black and Decker chopper. Four 10-15

g samples from each layer were placed into aluminum weighing dishes.

Before the meat was added to the dishes, the dishes were heated in a drying oven
(Yamato DX400, Yamato Scientific America Inc., Santa Clara, CA) for 30 min at 120 °C
to remove any moisture from the dish, placed in a desiccator to cool, and then weighed.
The raw meat in the dish was weighed before being dried overnight (16 - 18 h) at 100 °C.
After drying, the samples were allowed to cool in a desiccator before the final weight was

taken.

3.11.2 Salmonella concentration

The procedure used for the log count test was the same as that used by Warsow
(2003) for his unidirectional Salmonella log count tests. The inoculum was a Salmonella
cocktail that consisted of eight strains of Salmonella (S. Thompson FSIS 120, S.
Enteriditis H3527 and H3502, S. Typhimurium DT 104 H3380, S. Hadar MF60404, S.

Copenhagen 8457, S. Montevideo FSIS 051, and S. Heildelberg FSO38B61), previously

39

acquired from Dr. V.K. Juneja (Agricultural Research Service, Eastern Regional
Research Center, USDA-ARS, Philadelphia, Penn, USA). The individual strains were
stored in tryptic soy broth (TSB) containing 10% glycerol in a -80 °C freezer. The
cultures were propagated by two consecutive daily transfers of one loopful of culture to 9
ml of fresh TSB in a 20 ml culture tube. The cultures were incubated 18-24 h at 37 °C
between each transfer. On the day of the experiment, 9 ml of each of the eight serovars
grown separately in TSB were combined and centrifuged at 6,000 x g for 20 min at 4 °C.
The supernatant was poured off, and the cell pellet was resuspended in approximately
520 ml of sterile marinade to give a final Salmonella concentration of ~109 CFU/ml. The
Salmonella population in the marinade was confirmed by serial dilution in 0.1% peptone
water with duplicate plating on PetrifilmTM Aerobic Count Plates (3M Corp., St. Paul,
Minn., U.S.A.).

Four cores were removed from the block using a 1.23 cm inside diameter Warner-
Bratzler hand coring device and placed in a disposable Petri dish containing a piece of
sterile cotton gauze. The cored sample was cut into five 1 cm segments, starting from the
end furtherest from the marinade, and placed in individual 4 oz Whirl-pakTM bags.
Thereafter, 4 ml of 0.1 % peptone water was added to each bag to achieve the highest
possible concentration while still adding sufficient peptone water for plating. The weight
of each sample was taken before and after adding peptone. The samples were
mechanically macerated for 180 s before being plated on PetrifilmTM Aerobic Count

Plates. The plates were incubated for 48 h at 37 °C before enumeration.

40

3.12 Limitations

Multiple assumptions were made about the structure of meat and its pore space in

order to deve10p a sufficiently simple model, which resulted in several limitations.

This model relies solely on capillary potential as the major driving force
transporting Salmonella into meat, neglecting other possible driving forces
contributing to marinade uptake, such as osmotic potential due to salt and
phosphates in the marinade.

This model calculated the initial effective gap radius and the initial area of the gap
from only one turkey microscopy image. In order to get more reliable values for
these variables, the initial effective gap radius and the initial area of the gap, a
minimum of three microscopy images should be analyzed and averaged.

This model assumes that the pore space in the meat was a bundle of uniform
straight cylindrical capillaries, even though the area between the cells is actually
irregular, interconnected interstices.

This model uses the Latto and Chow (1982) equation for Salmonella velocity;
however, their equation was based on experimental data that is outside the range
of the capsule to pipe diameter ratios of Salmonella moving through a capillary.
Latto and Chow’s equation was for capsule to pipe diameter ratios ranging from
0.49 to 0.82, while this model had capsule to pipe diameters ranging from 0.16 to

0.19.

41

4. RESULTS

4.1 Overview

This chapter is divided into three main sections. The first section (4.2) describes
the actual outputs of the model. The second section (4.3) describes the sensitivity of the
model to various critical input parameters. The third section (4.4) describes the

attempted validation of the model and the problems associated with the model.

4.2 Model Outputs

The model outputs for moisture concentration, effective gap radius, Darcy ’s
velocity, marinade velocity, Salmonella velocity, Salmonella location, and Salmonella

concentration were based on a set of standard input variables values (Table 4.1).

42

Table 4.1: Input variable values used in the evaluation of the model. Starred ('1')
variables are analyzed in Section 4.3.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Variables Value Units
Meat Properties
* feff, o 3 . 12513-06 M
Cwb, o 0.720 -
* Cwb’ sat 0.726 "'
Agap’ 0 0.13 -
* r 3 -
p] 1 120 kg/m3
ADM 0.2436 -
Marinade Properties
11 1.5191303 N-s/m2
y 7.54E-02 N/m
9 1005 kg/m3
Salmonella Properties
(18 5.E-O7 M
LS 2.E-06 M
s 1 -
* df l.E-06 M
Empirical Parameters
B 0.216 -
11 0.770 -
Misc.
Ax 0.005 M
At 1.03 S

 

 

 

4. 2.1 Moisture Concentration

The moisture concentration of the meat was calculated using equation 3.15. As
expected, the moisture concentration profile increased with time and decreased as the
distance from the marinade exposed surface increased (Figure 4.1). Given the conditions

in Table 4.1, the moisture concentration approached equilibrium (i 0.1%) at ~4 min.

43

 

0.727 -
0.726 2408
0.725 -
0.724 ~
0.723 -
0.722 .
0.721 ~ 55
0.720 . 05

0.719 1 i i i
0 1 2 3 4 5

Distance from surface in contact with marinade (cm)

 

120$

 

605
305

 

 

 

 

Wet Basis Moisture Concentration

Figure 4.1: Predicted wet basis moisture concentration profile.

4. 2.2 Effective Gap Radius

Given the conditions defined in Table 4.1 and using equations 3.8 and 3.9, the
effective gap radius decreased from the initial value of3. 125 pm to 2.671 um due to the
increase in moisture concentration. The effective gap radius is derived from the moisture
concentration profile; therefore, as expected, the effective gap radius profile is the inverse
mirror of the moisture concentration profile (Figure 4.2). At equilibrium, the gap size

shrinks to 85% of the initial gap size.

44

 

9°
to

 

3.1 ~
OS 53

5.0
O
l

303

2.9 603

I

120s

N
oo
l

2403

 

.N
N
I

 

Effective Gap Radius (pm)

 

2.6 l l T l l
0 1 2 3 4 5

Distance from surface in contact with marinade (cm)

Figure 4.2: Predicted effective gap radius profile.

4. 2.3 Darcy ’3 Velocity

Darcy ’s velocity of the marinade is the volumetric flow rate of the marinade,
relative to the cross-sectional area of the meat and was calculated using equation 3.17.
Darcy ’s velocity of the marinade was found to decrease with time and distance into the
meat. The non-smooth nature of the graph of Darcy ’s velocity at short time (Fig. 4.3) is
because the thickness of the layers is such that a new velocity is calculated only every

0.005 m.

45

 

6.E-05
5.E—05 ~
4.E-05
3.E-05 -

2.E-05 a

Darcy Velocity (ma-s'1°m'2)

1.E-05 -

 

 

 

 

 

 

0.E+00
0 1 2 3 4 5 6

Distance from surface in contact with marinade (cm)

Figure 4.3: Darcy’s velocity profile.

4. 2.4 True Marinade Velocity
Because true marinade velocity was found by dividing the Darcy ’s velocity by the
porosity of the meat, true marinade velocities had the same overall profile as the Darcy ’s

velocities, but with larger magnitude (Figure 4.4).

46

6.E-04

 

 

 

 

 

 

5.E-04 — 58
E
E, 4E 04
Q .
o
.2
g 3.E-04
o 308
B
2 1203

1.E-04 ~

2403
0.E+00 l i \l W I
0 1 2 3 4 5 6

Distance from surface in contact with marinade (cm)

Figure 4.4: True marinade velocity profile.

4.2.5 Salmonella Velocity
The Salmonella velocity profile generated in Excel using the modified Latto and
Chow equation (3.20) had the same overall trend as the true marinade velocity profile,

but the values for Salmonella velocity are higher than those for marinade velocity (Figure

4.5).

47

 

 

 

 

 

 

 

 

 

7.504
55
6.E-04 «
a
E 5.504
E
:6 4.504
g
§ 3.5.04. 305
O
2
0 603
E 2.5-04
8
1205
1.5-04 ~
2405 \
0.E+00 I i \l
o 1 2 3 4 5 6

Distance from surface In contact with marinade (cm)

Figure 4.5: Salmonella velocity profile.

4. 2. 6 Salmonella Location

The location of Salmonella at any given time step was calculated using the wave
theory described in section 3.8 (Figure 4.6). The wave profile shows that, as time
increases, the velocity of the waves entering decreases; thus, each wave enters the meat at
a slower velocity than the previous waves. The first wave of Salmonella travels the
fastest and moves 0.035 m into the meat after 4.37 min. The decreasing velocity causes
subsequent waves to get closer together, resulting in the majority of the waves being

found in the first two layers of the meat after marination.

48

 

A 0.035

.0
o
00
o

0.025 ~
0.020 .
0.015 ~
0.010 -
0.005 ~
0.000

0.00 50.00 100.00 150.00 200.00 250.00 300.00

time (s)

 

 

Distance from surface in
contact with marinade (m

 

 

Figure 4.6: Salmonella location wave profile.

4.2. 7 Salmonella Concentration Profile

As described in Section 3.9, the optimum predicted Salmonella concentration
profile for the model was found by minimizing the sum of square errors between
Warsow’s (2003) experimental log counts and the predicted values, by changing the
empirical parameters B and It found in the Salmonella velocity equation. Unfortunately,
Figure 4.7 shows that the optimized predicted profile was not able to achieve a very good

fit to the experimental results.

49

 

 

1 .E+06

Predicted \

1.E+05 .

   

1.E+04 -

1.5+03 .
Expenmental

 
 

1 .E+02

1.E+01 -

 

Salmonella Concentration (CFUIg)

 

 

1.E+00 . . . .
0 1 2 3 4 5
Distance from surface in contact with marinade (cm)

Figure 4.7: Comparison of predicted Salmonella concentration at t = 4.35 min
versus Warsow’s (2003) experimental results at t = 20 min.

4.3 Sensitivity Analysis

This model requires 18 input variables that can easily be adjusted to represent a
specific case. Many of these inputs, such as the marinade properties, are reasonably well-
known and should be very close to the actual values. However, there are other critical
inputs used in the model that have a higher degree of uncertainty; therefore, changing
them could have an unknown impact on the output of the model. This section evaluates
the model sensitivity to changes in the saturation moisture concentration, tortuosity,
initial effective gap radius, and the effective length of flagella. In order to compare the
sensitivity to these variables, all runs were compared at t = 120 s, using a standard set of

inputs (Table 4.1) and varying only one input variable at a time.

50

4.3.1 Saturation Moisture Concentration

The saturation moisture concentration used in the model is only 1% higher than
the initial moisture concentration, which coincides with the results of the moisture
concentration laboratory trials that can be found in Appendix A.3. However, this is much
lower than the expected saturation moisture concentration. Increasing the saturation
moisture concentration increases the moisture concentration throughout the meat at any

given time (Figure 4.8) and thereby results in a high Salmonella concentration deeper in

 

 

 

 

 

 

 

 

the meat (Figure 4.9).
0.755
8
‘5
§ 0.745 -
8 Sat MC = 0.74
o 0.740 4
a
.3 0.735 -
5 Sat MC = 0.73
.2 0.730 ~
10
8
; Sat MC = 0.726
0.720 i i i .
O 1 2 3 4 5

Distance from surface in contact with marinade (cm)

Figure 4.8: Wet basis moisture concentration with varying saturation moisture
concentrations at t = 120 s.

51

 

5 7.E+05

   
   
   

F”
rn
+
O
01

Sat MC = 0.75
5.E+05

4.E+05 -
3.E+05 -
Sat MC = 0.74
2.E+05 ~

1.E+05 -

 

 

Salmonella concentration (CFU

. , at MC=0.72 .
0 1 2 3 4 5 6
Distance from surface in contact with marinade (cm)

1.E+04

 

Figure 4.9: Salmonella concentration profile with varying saturation moisture
concentrations at t = 120 s.

4.3.2 T ortuosity

The tortuosity of the meat is the most uncertain of the input variables, because the
r of whole-muscle turkey breast is unknown, and the value used was simply a reasonable
guess. Therefore, it is important to determine the sensitivity of the model to a change in
r. Tortuosity represents the complexity of the path that the marinade travels; therefore,
increasing r decreases the diffusivity of the meat and increases the required time step in
the solution. This change causes the moisture concentration at a given time to decrease
with increasing t (Figure 4.10). For example, an increase in 1: from 3 to 4 resulted in a

decrease in the minimum moisture concentration from ~ 72.4 to ~ 72.3 %.

52

 

0.727
0.726 ~

0.726 a
1':
0.725 -
r=3
0.725 ~
1; %
0.724 -

0.724 r

0.723 ‘

Wet basis Moisture Concentration

 

 

 

0.723 . i i i
0 1 2 3 4 5

Distance from surface in contact with marinade (cm)

Figure 4.10: Wet basis moisture concentration with varying tortuosity at t = 120 s.

In addition, increasing I also decreases the velocity of the marinade and

Salmonella and ultimately decreases the depth of Salmonella penetration (Figure 4.11).

53

 

1.E+05

1.E+05 ~

1.E+05 -

8.E+04 -

6.E+04 ,

4.E+04

2.E+04

Salmonella Concentration (CPU/g)

 

 

 

 

0.E+00 . , ,
0 1 2 3 4 5
Distance from surface in contact with marinade (cm)

Figure 4.11: Salmonella concentration profile with varying 1: at t = 120 5.

4.3.3 Initial Effective Gap Radius

The initial effective gap radius was estimated from only one micrograph;
therefore, this parameter was also analyzed to see what effect changing this value might
have on the overall solution. Increasing the initial effective gap radius causes the
diffusivity of the meat to increase and, because the time step is calculated using the
diffusivity of the meat, the time step decreases. The penetration depth of Salmonella
decreases as the initial effective gap radius increases (Figure 4.12), due to a loss in

capillary potential.

54

 

1.4E+05
1.2E+05 4

1.0E+05 1

   

8.0E+04 -

rang = 6 rang = 3.125

     

Salmonella Concentration (CPU/g)

 

 

 

6.0E+04 -
4. E+04 -
0 reff_o=4.5
2.0E+04 ~
0.0E+00 . i . .
0 1 2 3 4 5

Distance from surface in contact with marinade (cm)

Figure 4.12: Salmonella concentration profile with varying initial effective gap
radius at t = 120 s.

4.3.4 Effective Length of F lagella

The effective length of the flagella was combined with the effective diameter of
the gap to form a dimensionless term that was added to the Salmonella velocity equation,
to act as a “stickiness” term that would slow down the Salmonella velocity as the radius
of the gap decreased. Figure 4.13 shows that, as the effective length of the flagella
increases and the ratio of the effective diameter of the gap to effective length of flagella

decreases, the slower the Salmonella move through the meat.

55

 

1 .6E-04

=7. 5 7
1.45.044 d‘ 5 4)

125'“ d.=1.05-06

1.0E-04 4
df = 1.5E-06
8.0E-05 -
6.0E-05 ~

4.0E-05 -

Salmonella velocity (mls)

20505 J

 

 

 

0.0E+OO . . . .
0 1 2 3 4 5
Distance from surface in contact with marinade (cm)

Figure 4.13: Salmonella velocity with varying effective lengths of flagella at t = 120
s.

As the effective length of the flagella increases, velocity of the Salmonella
decreases, and therefore the final penetration depth of the bacteria decreases (Figure

4.14).

56

 

1.5E+05

13505 1 df = 1.5E-06

1.1E+05 1

90504 , d,=1.0506

7.0E+04 - dr = 7.5E-07
5.0E+04

3.0E+04 ~

 

 

 

Salmonella Concentration (CFUIg)

1.0E+O4 . i
0 1 2 3 4 5
Distance from surface in contact with marinade (cm)

Figure 4.14: Salmonella concentration profile with varying effective length of
flagella at t = 120 s.

4.4 Attempted Model Validation

The Salmonella log count trials described in section 3.10.2 were used to validate
this model. These experimental results, Warsow’s (2003) results, and the predicted
Salmonella concentration profile are plotted together in Figure 4.15. Unfortunately, it is
evident from Figure 4.15 that the predicted Salmonella concentration does not provide a

good representation of the laboratory results.

57

 

1.E+08

 
  
 
 
  
  

 

 

 

g 1 E +07 - experimental
iii .
3&3 1.E+06 .
m ’
8 a 1.E+04 . \ predicted
$6“ 9 1.5+03 -
5 1.5+02 ~ W320“
E experimental
‘6 1.E+01 -
(I)

1.E+00 i

O 2 4 6

Distance from surface in contact with marinade (cm)

Figure 4.15: Comparison of predicted and experimental results for the Salmonella
concentration profile of turkey breast.

Due to the limited number of columns in Excel, the predicted curve represents
only the first 4 min of marination, while both sets of experimental data represent 20 min
of marination. However, even if the predicted model were extended to 20 min it would
still not provide an accurate representation of the Salmonella concentration, because the
model is currently predicting a higher amount of Salmonella within the turkey breast after
only 4 min of marination; therefore, extending the marination time would only increase
the log counts.

Most likely, the major reason why the predicted results do not match the
experimental results is the wave theory used to predict the location of the Salmonella at a
given time. The wave theory assumes that a new group, or wave, of Salmonella enters
the whole muscle at each new time step. The number of cells entering the meat with each

new wave was calculated by equation 3.22, which was based on the velocity of

58

Salmonella and the concentration of the marinade. This creates a problem because the
velocity of Salmonella is greater at the initial time steps before the effective gap radius
shrinks due to moisture uptake. The higher velocity results in large initial concentrations
entering the meat. The current model for marination allows for these high concentrations
of cells to move rapidly through the meat and produce an unexpected high concentration
of Salmonella in the interior of the meat. Using the wave profile approach, the cells in
each wave move together at the same rate through the meat, creating clusters of cells.
However, in reality, there would be more of a distribution of the cells. By analogy, if dye
were dropped in a river, the dye would not stay together and flow down the river, but
would spread and fade out as it moves. In an attempt to correct this problem and create a
smoother distribution of the Salmonella a moving average (i2Ax) at the final time step

was found and graphed against the original predicted results in Figure 4.16.

 

1.6E+05
1.4E+05 -
1.2E+05 -
1.0E+05 -
8.0E+04 -
6.0E+O4 .

    
  

 

 

 

 

Salmonella Concentration
(C FUIg)

 

l— — Srnoothed profili]
4-0E+04 L—fl-_—-Non-smooth profilel
2.0E+04 ~ \ \
0.0E+OO ‘ l 1 r 1—
o 1 2 3 4 5

Distance from surface in contact with marinade (cm)

Figure 4.16: Comparison of smoothed (:tZAx) and non-smoothed predicted
Salmonella concentration profile at t = 261 s.

59

Taking the moving average of the profile did smooth out the distribution of
Salmonella in the meat; however, even the smooth profile did not provide an accurate
representation of the experimental results. In addition, averaging the cells creates
additional data points and gives a false representation of the predicted distance that

Salmonella moves through whole muscle.

4.12 Summary

The model was able to run and produce a result that reflects a logical relationship
between input and output variables. In addition, the sensitivity analysis showed that the
model was robust in the range of interest. Unfortunately, there was a problem in finding
B and n values that resulted in a Salmonella concentration profile that resembled the

experimental results.

60

5. CONCLUSION

This model does not predict the distribution of Salmonella in whole-muscle meat

products during marination with sufficient accuracy to support the hypothesis regarding

the underlying mechanism. Some key problems are as follows:

The wave theory used to determine the location and concentration of
Salmonella in the meat does not allow a smooth distribution of Salmonella,
but rather a clumped, unrealistic distribution.

Exact values for input variables are unknown.

Based on current inputs and Excel limitations, the total time does not

correspond to the experimental marination time.

However, even though this model does not provide a good representation of the

experimental results for Salmonella transport into whole-muscle meat products during

marination, there was some valuable information gained through this model.

This model provides a good starting point on which to base future models.
The results suggest that some of the model assumptions are incorrect. The
moisture concentration tests performed in this study showed an immeasurable
change after 20 min of marination. Prior to conducting these experiments, it
was assumed that the meat was taking up more marinade during marination.
With so little change in moisture concentration, the assumption that capillary
diffusion is the primary driving force transporting the Salmonella into the

meat may be incorrect.

61

6. FUTURE RESEARCH

This model provides a good starting point for modeling the transport of

Salmonella through whole-muscle meat products; however, there is significant research

that could be done to improve this model. Suggestions for filture research include:

Adjust the model so that capillary diffusion is not the only driving force
transporting marinade into the meat. The moisture concentration tests conducted
during this study showed that a very small amount of moisture uptake was needed
to achieve Salmonella concentrations at the t0p of whole-muscle turkey breast.
Therefore, capillary diffusion may not be the primary force responsible for the
movement of Salmonella and marinade into meat.

Conduct laboratory tests to determine whether Salmonella ’s flagella could
transport Salmonella into meat without the aid of marinade. The model created in
this study assumes that Salmonella are non-motile; therefore, if these tests reveal
that the bacteria’s motility has significant impact on the pathogens transport into
whole muscle, then they should be incorporated into the model.

Conduct additional moisture concentration trials to verify the rate at which
marinade is being absorbed into the meat and test to see if the species of meat has
any impact on the amount of marinade absorbed.

Adjust the model to take into account the effects of salt and phosphate. The
effects of salt and phosphate were neglected in order to simplify the model;
however, to make the model more accurate, the osmotic potential caused by these

additives found in marinades should be considered.

62

Adjust the model so that it better represents the structure of meat. For simplicity,
this model assumed that the meat is composed of a bundle of capillaries that are
uniform cylindrical tubes of the same size, but instead the intercellular space in
meat is actually various sized curvy flat spaces.

Use micrographs taken after marination to compare the actual area gap change to
the predicted change. In addition, additional premarination images are needed to
find a more accurate initial effective gap radius and area of the gap.

Improve the Salmonella velocity equation, so that it is more applicable to the
range of parameters involved in marinade uptake.

Confirm the effects of freezing and thawing meat on bacteria penetration and
moisture uptake. Researchers have shown that freezing and thawing of meat can
influence bacteria penetration (Maxcy 1981; Sikes and Maxcy 1980); however, it
would be useful to fiirther quantify the effects and, if they are significant, then
account for the difference between fresh and frozen meat in the model.

The current model has limitations due to the column restrictions in Excel;
therefore, future models should be created in a program such as MATLAB to
minimize the limitations associated with program capabilities.

Finally, converting a working one-dimensional model to a two-dimensional model
would better represent the actual marination process where meat is surrounded by

marinade.

63

7. APPENDICES

The appendices are divided into the following 13 sections:

Appendix 7.1: Derivative of Capillary Potential with Respect to Dry Basis
Moisture Concentration

Appendix 7.2: User Functions

Appendix 7.3: Moisture Concentration Test Results

Appendix 7.4: Marinade Percent Uptake Test Results

Appendix 7.5: Salmonella Concentration Experimental Results

Appendix 7.6: Calculation of Required Marinade Uptake to Achieve given
Experimental Salmonella Log Counts

Appendix 7.7: Master Input Page for Model

Appendix 7.8: Mode] Output for Moisture Concentration

Appendix 7.9: Model Output for Effective Gap Radius

Appendix 7.10: Model Output for Darcy’s Velocity

Appendix 7.11: Model Output for Salmonella Velocity

Appendix 7.12: Model Output for Salmonella Concentration

Appendix 7.13: Three-Dimensional Graphs of Model Output

64

 

Appendix 7.1: Derivative of Capillary Potential with Respect to Dry Basis Moisture

Concentration

Capillary pressure is expressed as two times the surface tension divided by the

effective gap radius at time t (_2__-_}'_ ). Substituting in the equation for the effective gap

ref];

radius at time t yields the following equation:

 

 

/ l) _1
2 —l 7
2'7 2.7.[Agap0i Cdb
— . _ . _.___L__
-— l A 1 [7.1]
r r DM 1+C
efl" efi' db
t 0 t
t l

 

 

The devivative of capillary pressure with respect to moisture concentration can be found

using the chain rule.

 

 

 

 

 

 

 

 

\
K 1)
_ —3
a r2 7 A 2 c “1 7
ejft) _1 7 gapO db
=_ -l——A .1_____
\ 2 rejf DM 1+Cdb
t)
\ l
l \
—2
Cdbt {1+CdbtJ—Cdbt -(0+1)
ADM. l—TE; - 2 [7.2]
t 1+C ]
db
\ l t l

Combining terms simplifies the equation to:

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\ f )
r
a 2'7 effo l
r
eflt) = \ l ADM
) 3
_ — C
a[Caro C 1 2 db,
t) dbt 1_I_C—
l—ADM. 1— + db
1+Cdb ( t y
t
This simplifies to:
r w 1
6 2'7 . A 3 (A )
{refit ) _ 7 gapO DM
arc V f )1
2
\ dbl)
C
A db
refl. -1— DM 1— ’ -[1+Cdb]
0 Cd”: 1+Cdbt I
—1+Cdb
\ t /

 

 

Combining the terms in the denominator results in the final equation:

66

 

[7.4]

 

 

 

 

 

 

 

 

 

3
)2
ADM
c
_ dbt
1+C
dbl ) l [7.51

67

Appendix 7.2: User Functions

The user function used in equation 3.15 for calculating lambda was as follows:
Function Lambda(refft, Ct, dt, dx, vis, tor, st, Agi, DM, reffi)

D = (reffi " 2 / (vis * 8 * tor)) * (((st * Agi " 0.5 * DM)/reffi) / ((1 - DM / (1 - Ct/(1+
Ct») " (3 / 2»)

Lambda = (D * dt) / dx " 2

End Function

The user function used in equation 3.22 for calculating the Salmonella concentration at a
given location was as follows:
Function Concentration(X, dx, Row, loc, cells)
Concentration = 0
Column = l
me = loc(Row, Column)
Do While me > (X - dx)
If Xwave <= X Then Concentration = Concentration + cells(Column)
Column = Column + 1
me = loc(Row, Column)
Loop

End Function

68

Appendix 7.3: Moisture Concentration Test Results

The exact moisture concentration profile of the meat after 20 min of marination

could not be determined because the maximum change in moisture concentration was

only 1% and that difference could be attributed to error associated with the procedure.

However, these results do show that there is a lot less marinade being absorbed into the

meat than was assumed. The results from the two different moisture concentration tests

described in section 3.10.1 were:

Table 7.1: Moisture concentration results using the coring method described in
section 3.11.1.

TOP

 

Y

BOTTOM

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dish +

Core - Dish +Raw Raw Dried Dried Moisture

Slice Dish Meat Meat Meat Meat Content average_
1-1 1 .2805 24.684 23.4035 7.5207 6.2402 0.733 0.731
21 1.2682 16.8176 15.5494 5.5645 4.2963 0.724 0.730
3-1 1.271 1 14.9505 13.6794 4.8558 3.5847 0.738 0.730
4-1 1.2691 14.1723 12.9032 4.7509 3.4818 0.730
1-2 1.2716 21.4134 20.1418 6.7785 5.5069 0.727
2-2 1.2809 16.2106 14.9297 5.3005 4.0196 0.731
3-2 1 .2773 14.4263 13.149 4.8333 3.556 0.730
4-2 1.267 13.6046 12.3376 4.5664 3.2994 0.733
1-3 1.2771 20.1615 18.8844 6.499 5.2219 0.723
2-3 1.2734 20.4605 19.1871 6.3614 5.088 0.735
3-3 1.2735 12.215 10.9415 4.2368 2.9633 0.729
4-3 1.2706 10.0752 8.8046 3.6174 2.3468 0.733

Wtial l 1.2692 I 15.3803 l 14.1111 l 5.2252 1 3.956 I 0.720 I

 

69

 

Table 7.2: Moisture concentration results using the slicing method described in

section 3.11.1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dish 1-
Dish + Raw Dried Dried Moisture
Dish Raw Meat Meat Meat Meat Content average_
TOP 1-1 1.2753 10.8599 9.5846 3.8741 2.5988 0.729 0.729
2-1 1.2758 15.3639 14.0881 5.0929 3.8171 0.729 0.718
3-1 1.2716 12.195 10.9234 4.2367 2.9651 0.729 0.722
4-1 1.255 11.7325 10.4775 4.0862 2.8312 0.730
1-2 1.2723 11.6699 10.3976 4.2168 2.9445 0.717
2-2 1.277 11.0314 9.7544 4.018 2.741 0.719
3-2 1.2686 13.8091 12.5405 4.8461 3.5775 0.715
4-2 1.2735 10.4544 9.1809 3.8164 2.5429 0.723
1-3 1 .2726 9.859 8.5864 3.6732 2.4006 0.720
V 23 1 .2864 1 1.2238 9.9374 4.0358 2.7494 0.723
3-3 1.273 14.0432 12.7702 4.8589 3.5859 0.719
BOTTOM 4-3 1.2685 10.7871 9.5186 3.8799 2.6114 0.726
Linitial H2692 l 15.3803 ] 14.1111 l 5.2252] 3.956 I 0.720 ]

 

70

 

Appendix 7.4: Marinade Percent Uptake Test Results

A laboratory test was conducted in order to determine how much marinade was

being absorbed into the meat during marination. This experiment was done by placing a

piece of whole-muscle pork roast in a Petri dish containing a measured amount of

marinade and removing the meat alter a set time and taking the weight of the remaining

marinade. A new piece of meat was used for each of the time intervals. The result of this

experiment is as follows:

Table 7.3: Marinade percent uptake test results.

 

 

 

 

 

 

 

 

 

 

Meat Marinade Marinade weight
time time Initial Initial after marination Uptake Percent
sample (s) (min) Weight (g) Wight (g) (g) (g) Uptake
1 20 0.33 106.89 20.1 19.41 0.69 0.65%
2 50 0.83 92.01 20.58 20.05 0.53 0.58%
3 105 1.75 90.73 20.03 18.91 1.12 1.23%
4 225 3.75 83.63 19.54 18.65 0.89 1.06%
5 450 7.5 95.63 18.78 18.00 0.78 0.82%
6 900 15 91.68 21.63 20.87 0.76 0.83%
7 1200 20 88.78 20.32 19.72 0.60 0.68%

 

 

 

 

 

 

 

 

71

 

Appendix 7.5: Salmonella Concentration Experimental Results

The results of the Salmonella concentration tests described in section 3.11.2 are
shown in Tables 7.4 and 7.5. Three trials were conducted where four cores were
removed from each block of turkey and each core was sliced into five sections; therefore,
the sample number corresponds to the rep number, core number, and the slice number.
For example, sample 1-2-3 is rep 1 - core 2 — slice 3. The average for the second
replication was based off from the results from two cores rather than four cores due to a

problem with the initial sterility of two of the corers.

Table 7.4: Salmonella concentration experimental results.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Distance from Meat +
Sample surface in contact Meat Peptone '09 '09 Average Adjusted
with marinade (m) (g) (9) counts counts Average
marinade 1 .22E+09 1 .36E+09 1 .29E+09
1-1-1 0.045 0.5 4.31 0.00E+00 0.00E+00 0.00E+00 0.00E+00
1-1-2 0.035 0.32 4.23 0.00E+00 0.00E+00 0.00E+00 0.00E+00
1-1-3 0.025 0.36 4.25 8.00E+01 1.30E+02 1.05E+02 4.46E+02
1-1-4 0.015 0.48 4.4 8.10E+03 9.40E+03 8.75E+03 3.85E+04
1-1-5 0.005 0.45 4.3 1.81 E+06 1.92906 1.87E+06 8.02E+06
1-2-1 0.045 0.41 4.37 1.50E+02 2.305102 1.90E+02 8.30802
1-2-2 0.035 0.31 4.18 6.1 OE+02 6.40E+02 6.25E+02 2.61E+03
1-2-3 0.025 0.28 4.17 1.56E+04 1.56E+04 1.56E+04 6.51 E+04
1-2-4 0.015 0.49 6.32 3.70E+05 5.20E+05 4.4SE+05 2.81 E+06
1-2-5 0.005 0.65 4.54 4.10E+06 5.00E+06 4.55E+06 2.07E+07
1-3—1 0.045 0.55 4.48 0.00E+00 0.00E+00 0.00E+00 0.00E+00
1-3-2 0.035 0.46 4.39 0.00E+00 0.00E+00 0.00E+00 0.00E+00
1-3-3 0.025 0.53 4.44 8.60E+03 8.60E+03 8.60E+03 3.82E+04
1-3-4 0.015 0.51 4.45 7.70E+05 6.70E+05 7.20E+05 3.20E+06
1-3—5 0.005 0.6 4.53 5.50E+06 5.20E+06 5.35E+06 2.42E+07
1-4-1 0.045 0.66 4.54 0.00E+00 0.00E+00 0.00E+00 0.00E+00
1-4-2 0.035 0.47 4.44 0.00E+00 0.00E+00 0.00E+00 0.00E+00
1-4-3 0.025 0.5 4.3 1.00E+02 8.00E+01 9.00E+01 3.87E+02
1-4-4 0.015 0.55 4.38 1.98E+04 2.00E+04 1.99E+04 8.72E+04
1-4-5 0.005 0.76 4.62 2.60E+06 1.80E+06 2.20E+06 1.02E+07
marinade 1 .22E+09 1 .36E+09 1 .29E+09

2-1-1 0.045 0.71 4.67 0.00E+00 0.00E+00 0.00E+00 0.00E+00
2-1-2 0.035 0.6 4.53 0.00E+00 0.00E+00 0.00E+00 0.00E+00
2-1-3 0.025 0.6 4.53 8.60E+02 9.10E+02 8.85E+02 4.01E+03
2-1-4 0.015 0.7 4.57 8.50E+05 9.50E+05 9.00E+05 4.11E+06
2-1-5 0.005 0.65 4.66 8.7OE+06 7.10E+06 7.90E+06 3.68E+07
2-2-1 0.045 0.72 4.65 0.00E+00 0.00E+00 0.00E+00 0.00E+00
2—2-2 0.035 0.51 4.41 0.00E+00 0.00E+00 0.00E+00 0.00E+00

72

 

Table 7.4 Continued

 

 

 

 

 

 

2-2-3 0.025 0.55 4.48 7.00E+01 1 .70E+02 1 .20E+02 5.38E+02
2-2-4 0.015 0.83 4.83 9.30E+04 8.10E+04 8.70E+04 4.20E+05
2-2-5 0.005 0.73 4.66 5.20806 4.1OE+06 4.65E+06 2.17E+07
marinade 1.53E+09 1 ASE-+09 1 .48E+09
3-1-1 0.045 0.9 4.8 0.00E+00 0.00E+00 0.00E+00 0.00E+00
3-1-2 0.035 0.7 4.6 0.00E+OO 0.00E+OO 0.00E+00 0.00E+00
3-1-3 0.025 0.9 4.9 0.00E+00 1.00E+01 5.00E+00 2.45E+01
3-1-4 0.015 1.1 4.9 3.80E+05 3.90E+05 3.85E+05 1.89E+06
3-1-5 0.005 1.1 5 9.30E+06 9.00E+06 9.1SE+06 4.58E+07
3-2-1 0.045 0.8 4.7 0.00E+00 0.00E+00 0.00E+00 0.00E+00
3-2-2 0.035 0.8 4.6 2.30E+02 1 .SOE+02 1 .QOE+02 8.74E+02
3-2-3 0.025 0.9 4.8 1.12E+05 1 2554-05 1.19E+05 5.695+05
3-2-4 0.015 0.9 4.9 1.26E+06 1 .02E+06 1 .14E+06 55954-06
3-2-5 0.005 0.9 4.9 3.ZOE+06 3.10E+06 3.15E+06 1.54E+07
3—3—1 0.045 0.9 4.7 1.00E+01 0.00E+00 5.00E+00 2.35E+01
3-3-2 0.035 0.8 4.6 1.70E+02 2.60E+02 2.15E+02 9.89E+02
3-3-3 0.025 0.9 4.9 6.00E+01 7.00E+01 6.SOE+01 3.1QE+02
3-3-4 0.015 1 4.9 4.40E+03 4.60E+03 4.50E+03 2.21 E+04
3-3-5 0.005 1.1 5 1.685+06 1.34E+06 1.51 E+06 7.55E+06
3-4-1 0.045 1 4.8 1 .00E+01 0.00E+00 5.00E+00 2.4OE+01
3-4-2 0.035 1 4.8 0.00E+00 0.00E+00 0.00E+00 0.00E+00
3-4-3 0.025 0.09 4.8 8.80E+04 1 .1 1 E+05 9.95E+04 4.78E+05
3-4-4 0.015 0.8 4.7 8808-05 1 .09E+06 9.855405 4.63E+06
3-4-5 0.005 1 .2 5.1 2.20E+06 2.50E+06 2.35E+06 1 .20E+07

 

Table 7.5: Salmonella concentration average of all three replicates.

 

Combined Average

Distance from surface in contact with marinade (m)

 

 

 

7.32E+01
3.73E+02
9.67E+04
2.28E+06
2.17E+07

 

0.045
0.035
0.025
0.015
0.005

 

73

 

 

Appendix 7.6: Calculation of Required Marinade Uptake to Achieve given

Experimental Salmonella Log Counts

The moisture concentration tests resulted in a much smaller change in moisture
concentration than was expected; therefore, the following calculations were done to
detemrine how much uptake was required to achieve the experimental log count results

based on the concentration of the marinade:

Volume of meat layer = 100 cm3
Density of Turkey = 1.12 g/cm3

Based on the following experimental data from Table 7.5:

 

 

7
Slicel: 100cm3[1'12g][2'1710 CFU

J: 2.43-109CFU
g

cm

Slice 2: 100cm3 2.55-108CFU

 

 

{.1 12g] 2. 28 106CFU]_

[.1 12g] 9.67 104cm

Slice 3: 100cm3 3

 

 

]=1.08-107CFU
(cm

-31
l

1 6151312 .73 102cm}
(.31

 

 

Slice 4: 100cm3 4.18-104CFU

Slice 5: 100cm3 8.20 - 103 CFU

 

 

[1.12g] 7.32 101CFU]_

Total CFU’s per 10x10x5 cm3 turkey block = 2.70109 CFU

Marinade concentration = 1.39109 CFU/ml

74

lml
1.39-109 CFU

 

Required Salmonella Uptake = 2.70 - 109{ j = 1.94m]

75

Appendix 7.7: Master Input Page for Model

The first sheet in Excel based model is a master page that contains all of the input
variables and the graphs of the model outputs. The adjustable input variables are shown

in Table 7.9, while the graphs are shown throughout this paper.

Table 7.6: Adjustable model input variables as they appear in the Excel Model

 

 

 

Meat Properties Marinade Properties Salmonella Properties
1...... (m) = 3.125506 11(N-slm2) = 1.519503 d. (m) = 5.507
C“, o = 0.72 y (Nlm) = 0.0754 L. (m) = 2.E-06
A9.” = 0.13 p (kg/m3) = 1005 s = 1
t = 3 d, (m) = 1.0E-06

 

 

 

p0 (kg/m3) = 1 120
p0 (glcm3) = 1.12

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

An! = 0.2436
Lg (m/s’) = 9.806 Ax (m) = 0.005
a ’11::1 )
Cwb in: C “t Cab. o 6(m 1+1 ) D (MAZIS) At (5)
0.726 2.65 2.57 45212.16 1.21505 1.03

 

 

76

Appendix 7.8: Model Output for Moisture Concentration

Excel formula used to calculate the dry basis moisture concentration at t = 1.03 s

and x = 0.005 m. (i.e. cell D5):

D5 =LAMBDA('Effective Gap
Radius'!D4,D4,dt,dx,MASTER!vis,MASTER! tor,MASTER! st,MASTER!
Agi,MASTER!ADM,MASTER! reffi)*C4+( 1 -2*LAMBDA('Effective Gap
Radius'!D4,D4,dt,dx,MASTER!vis,MASTER!tor,MASTER!st,MASTER!
Agi,MASTER!ADM,MASTER!reffi))*D4+LAMBDA('Effective Gap
Radius'lD4,D4,dt,dx,MASTER!vis,MASTER!tor,MASTER!st,MASTER!

Agi,MASTER!ADM,MASTER!reffi)*E4

Conversion from dry basis to wet basis:

D5 ='Moisture Content db'!D5/(1+'Moisture Content db'!DS)

Table 7.7 : Predicted wet basis moisture concentration profile generated by Excel,
given the input variables listed in Table 4.1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Moisture Content
i => 0 1 2 3 4 5 6 7 8 9 10
j 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
0 0 0.726 0.720 0.720 0.720 0.720 0.720 0.720 0.720 0.720 0.720 0.720
1 1.03 0.726 0.723 0.720 0.720 0.720 0.720 0.720 0.720 0.720 0.720 0.720
2 2.06 0.726 0.723 0.722 0.720 0.720 0.720 0.720 0.720 0.720 0.720 0.720
3 3.10 0.726 0.724 0.722 0.721 0.720 0.720 0.720 0.720 0.720 0.720 0.720
4 4.13 0.726 0.724 0.722 0.721 0.720 0.720 0.720 0.720 0.720 0.720 0.720
5 5.16 0.726 0.724 0.722 0.721 0.720 0.720 0.720 0.720 0.720 0.720 0.720
6 6.19 0.726 0.724 0.723 0.721 0.721 0.720 0.720 0.720 0.720 0.720 0.720
7 7.22 0.726 0.724 0.723 0.722 0.721 0.720 0.720 0.720 0.720 0.720 0.720
8 8.26 0.726 0.724 0.723 0.722 0.721 0.720 0.720 0.720 0.720 0.720 0.720
9 9.29 0.726 0.725 0.723 0.722 0.721 0.721 0.720 0.720 0.720 0.720 0.720
10 10.32 0.726 0.725 0.723 0.722 0.721 0.721 0.720 0.720 0.720 0.720 0.720
11 11.35 0.726 0.725 0.723 0.722 0.721 0.721 0.720 0.720 0.720 0.720 0.720
12 12.39 0.726 0.725 0.723 0.722 0.722 0.721 0.721 0.720 0.720 0.720 0.720
13 13.42 0.726 0.725 0.724 0.723 0.722 0.721 0.721 0.720 0.720 0.720 0.720
14 14.45 0.726 0.725 0.724 0.723 0.722 0.721 0.721 0.720 0.720 0.720 0.720
15 15.48 0.726 0.725 0.724 0.723 0.722 0.721 0.721 0.720 0.720 0.720 0.720

 

 

 

 

 

 

77

 

 

 

 

 

 

 

Table 7.7 Continued

 

16 16.51 0.726 0.725 0.724 0.723 0.722 0.721 0.721 0.720 0.720 0.720 0.720

 

17 17.55 0.726 0.725 0.724 0.723 0.722 0.721 0.721 0.721 0.720 0.720 0.720

 

18 18.58 0.726 0.725 0.724 0.723 0.722 0.721 0.721 0.721 0.720 0.720 0.720

 

19 19.61 0.726 0.725 0.724 0.723 0.722 0.722 0.721 0.721 0.720 0.720 0.720

 

20 20.64 0.726 0.725 0.724 0.723 0.722 0.722 0.721 0.721 0.720 0.720 0.720

 

21 21.67 0.726 0.725 0.724 0.723 0.722 0.722 0.721 0.721 0.721 0.720 0.720

 

22 22.71 0.726 0.725 0.724 0.723 0.722 0.722 0.721 0.721 0.721 0.720 0.720

 

23 23.74 0.726 0.725 0.724 0.723 0.722 0.722 0.721 0.721 0.721 0.720 0.720

 

24 24.77 0.726 0.725 0.724 0.723 0.723 0.722 0.721 0.721 0.721 0.720 0.720

 

25 25.80 0.726 0.725 0.724 0.723 0.723 0.722 0.721 0.721 0.721 0.721 0.720

 

26 26.83 0.726 0.725 0.724 0.723 0.723 0.722 0.721 0.721 0.721 0.721 0.720

 

27 27.87 0.726 0.725 0.724 0.723 0.723 0.722 0.722 0.721 0.721 0.721 0.721

 

28 28.90 0.726 0.725 0.724 0.723 0.723 0.722 0.722 0.721 0.721 0.721 0.721

 

29 29.93 0.726 0.725 0.724 0.724 0.723 0.722 0.722 0.721 0.721 0.721 0.721

 

30 30.96 0.726 0.725 0.724 0.724 0.723 0.722 0.722 0.721 0.721 0.721 0.721

 

31 32.00 0.726 0.725 0.724 0.724 0.723 0.722 0.722 0.721 0.721 0.721 0.721

 

32 33.03 0.726 0.725 0.724 0.724 0.723 0.722 0.722 0.721 0.721 0.721 0.721

 

33 34.06 0.726 0.725 0.724 0.724 0.723 0.722 0.722 0.721 0.721 0.721 0.721

 

34 35.09 0.726 0.725 0.724 0.724 0.723 0.722 0.722 0.721 0.721 0.721 0.721

 

35 36.12 0.726 0.725 0.724 0.724 0.723 0.722 0.722 0.722 0.721 0.721 0.721

 

36 37.16 0.726 0.725 0.724 0.724 0.723 0.723 0.722 0.722 0.721 0.721 0.721

 

37 38.19 0.726 0.725 0.724 0.724 0.723 0.723 0.722 0.722 0.721 0.721 0.721

 

38 39.22 0.726 0.725 0.725 0.724 0.723 0.723 0.722 0.722 0.721 0.721 0.721

 

39 40.25 0.726 0.725 0.725 0.724 0.723 0.723 0.722 0.722 0.721 0.721 0.721

 

40 41.28 0.726 0.725 0.725 0.724 0.723 0.723 0.722 0.722 0.721 0.721 0.721

 

41 42.32 0.726 0.725 0.725 0.724 0.723 0.723 0.722 0.722 0.722 0.721 0.721

 

42 43.35 0.726 0.725 0.725 0.724 0.723 0.723 0.722 0.722 0.722 0.721 0.721

 

43 44.38 0.726 0.725 0.725 0.724 0.723 0.723 0.722 0.722 0.722 0.721 0.721

 

44 45.41 0.726 0.725 0.725 0.724 0.723 0.723 0.722 0.722 0.722 0.721 0.721

 

45 46.44 0.726 0.725 0.725 0.724 0.723 0.723 0.722 0.722 0.722 0.722 0.721

 

46 47.48 0.726 0.725 0.725 0.724 0.723 0.723 0.722 0.722 0.722 0.722 0.721

 

47 48.51 0.726 0.725 0.725 0.724 0.723 0.723 0.722 0.722 0.722 0.722 0.722

 

48 49.54 0.726 0.725 0.725 0.724 0.723 0.723 0.723 0.722 0.722 0.722 0.722

 

49 50.57 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.722 0.722 0.722 0.722

 

50 51.61 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.722 0.722 0.722 0.722

 

51 52.64 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.722 0.722 0.722 0.722

 

52 53.67 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.722 0.722 0.722 0.722

 

53 54.70 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.722 0.722 0.722 0.722

 

54 55.73 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.722 0.722 0.722 0.722

 

55 56.77 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.722 0.722 0.722 0.722

 

56 57.80 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.722 0.722 0.722 0.722

 

57 58.83 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.722 0.722 0.722

 

58 59.86 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.722 0.722 0.722

 

59 60.89 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.722 0.722 0.722

 

60 61.93 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.722 0.722 0.722

 

61 62.96 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.722 0.722 0.722

 

62 63.99 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.722 0.722 0.722

 

63 65.02 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.723 0.722 0.722

 

64 66.05 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.723 0.722 0.722

 

65 67.09 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.722 0.722

 

66 68.12 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.722 0.722

 

67 69.15 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.722 0.722

 

68 70.18 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723 0.722

 

69 71.22 0.726 0.726 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723 0.722

 

70 72.25 0.726 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.723 0.723

 

71 73.28 0.726 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.723 0.723

 

72 74.31 0.726 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.723 0.723

 

73 75.34 0.726 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.723 0.723

 

74 76.38 0.726 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.723 0.723

 

75 77.41 0.726 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.723 0.723

 

76 78.44 0.726 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.723 0.723

 

 

 

 

 

 

 

 

 

 

 

 

 

 

77 79.47 0.726 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.723 0.723 0.723

78

 

Table 7.7 Continued

 

78 80.50 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723

 

79 81.54 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723

 

80 82.57 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723

 

81 83.60 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723

 

82 84.63 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723

 

83 85.66 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723

 

84 86.70 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723

 

85 87.73 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723

 

86 88.76 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.723 0.723 0.723 0.723

 

87 89.79 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723 0.723

 

88 90.83 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723 0.723

 

89 91.86 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723 0.723

 

90 92.89 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723 0.723

 

91 93.92 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723 0.723

 

92 94.95 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723 0.723

 

93 95.99 0.726 0.726 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723 0.723

 

94 97.02 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723

 

95 98.05 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723

 

96 99.08 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723

 

97 100.11 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.723 0.723

 

98 101.15 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.723

 

99 102.18 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.723

 

100 103.21 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

101 104.24 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

102 105.27 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

103 106.31 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

104 107.34 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

105 108.37 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

106 109.40 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

107 110.44 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

108 111.47 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

109 112.50 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

110 113.53 0.726 0.726 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724 0.724

 

111 114.56 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

112 115.60 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

113 116.63 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

114 117.66 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

115 118.69 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

116 119.72 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

117 120.76 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

118 121.79 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

119 122.82 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

120 123.85 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

121 124.88 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

122 125.92 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

123 126.95 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724 0.724

 

124 127.98 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724

 

125 129.01 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724

 

126 130.05 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724

 

127 131.08 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724

 

128 132.11 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724

 

129 133.14 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724

 

130 134.17 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724

 

131 135.21 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724

 

132 136.24 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724

 

133 137.27 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.724 0.724 0.724 0.724

 

134 138.30 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724

 

135 139.33 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724

 

136 140.37 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724

 

137 141.40 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724

 

138 142.43 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724

 

 

139 143.46 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.724 0.724 0.724

 

 

 

 

 

 

 

 

 

 

 

 

79

 

Table 7.7 Continued

 

140 144.49 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.724 0.724

 

141 145.53 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.724 0.724

 

142 146.56 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.724 0.724

 

143 147.59 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.724 0.724

 

144 148.62 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.724 0.724

 

145 149.66 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.724

 

146 150.69 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.724

 

147 151.72 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

148 152.75 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

149 153.78 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

150 154.82 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

151 155.85 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

152 156.88 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

153 157.91 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

154 158.94 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

155 159.98 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

156 161.01 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

157 162.04 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

158 163.07 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

159 164.10 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

160 165.14 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

161 166.17 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

162 167.20 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

163 168.23 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

164 169.27 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

165 170.30 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

166 171.33 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

167 172.36 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

168 173.39 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

169 174.43 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

170 175.46 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

171 176.49 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

172 177.52 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

173 178.55 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

174 179.59 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

175 180.62 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

176 181.65 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

177 182.68 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

178 183.71 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

179 184.75 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

180 185.78 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

181 186.81 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

182 187.84 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

183 188.88 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

184 189.91 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

185 190.94 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

186 191.97 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

187 193.00 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

188 194.04 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

189 195.07 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

190 196.10 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

191 197.13 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

192 198.16 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725

 

193 199.20 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

194 200.23 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

195 201.26 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

196 202.29 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

197 203.32 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

198 204.36 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

199 205.39 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

200 206.42 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

 

 

 

 

 

 

 

 

 

 

 

 

 

201 207.45 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

80

 

Table 7.7 Continued

 

202 208.49 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

203 209.52 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

204 210.55 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

205 211.58 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

206 212.61 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

207 213.65 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

208 214.68 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

209 215.71 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

210 216.74 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725 0.725

 

211 217.77 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

212 218.81 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

213 219.84 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

214 220.87 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

215 221.90 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

216 222.93 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

217 223.97 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

218 225.00 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

219 226.03 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

220 227.06 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

221 228.10 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

222 229.13 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

223 230.16 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725 0.725

 

224 231.19 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725

 

225 232.22 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725

 

226 233.26 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725

 

227 234.29 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725

 

228 235.32 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725

 

229 236.35 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725

 

230 237.38 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725

 

231 238.42 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725

 

232 239.45 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725

 

233 240.48 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725 0.725

 

234 241.51 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725

 

235 242.54 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725

 

236 243.58 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725

 

237 244.61 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725

 

238 245.64 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725

 

239 246.67 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725 0.725

 

240 247.71 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725

 

241 248.74 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725

 

242 249.77 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725

 

243 250.80 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725

 

244 251.83 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725 0.725

 

245 252.87 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725

 

246 253.90 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.725

 

247 254.93 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726

 

248 255.96 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726

 

249 256.99 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726

 

250 258.03 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726

 

251 259.06 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726

 

252 260.09 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726

 

253 261.12 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726

 

 

 

 

 

 

 

 

 

 

 

 

 

 

254 262.15 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726 0.726

 

81

 

Appendix 7.9: Model Output for Effective Gap Radius

m. (i.e. cell D5):

Excel formula used to calculate the effective gap radius at t = 1.03 s and x = 0.005

D5 =(reffi*(l-MASTER!ADM/(1-'Moisture Content wb '!D5)))/Agi

Table 7.8: Predicted effective gap radius profile generated by Excel, given the input
variables listed in Table 4.1.

 

 

0

1

2

Effective Gap Radius
3 4 5 6

8

9

10

 

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

 

2.66506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

 

2.66506

2.89506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

 

2.66506

2.89506

3.01506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

 

2.66506

2.84506

3.01506

3.07506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

 

2.66506

2.84506

2.95506

3.07506

3.10506

3.13506

3.13506

3.13506

3.13506

3.13506

3.13506

 

2.66506

2.81506

2.95506

3.03506

3.10506

3.11506

3.13506

3.13506

3.13506

3.13506

3.13506

 

2.66506

2.81506

2.92506

3.02506

3.07506

3.11506

3.12506

3.13506

3.13506

3.13506

3.13506

 

2.66506

2.79506

2.92506

2.99506

3.07506

3.09506

3.12506

3.12506

3.13506

3.13506

3.13506

 

mummwa-Aow

2.66506

2.79506

2.89506

2.99506

3.04506

3.09506

3.11506

3.12506

3.12506

3.13506

3.13506

 

2.66506

2.78506

2.89506

2.97506

3.04506

3.08506

3.11506

3.12506

3.12506

3.12506

3.13506

 

2.66506

2.78506

2.88506

2.97506

3.02506

3.07506

3.10506

3.12506

3.12506

3.12506

3.12506

 

2.66506

2.77506

2.87506

2.95506

3.02506

3.06506

3.09506

3.11506

3.12506

3.12506

3.12506

 

2.66506

2.77506

2.86506

2.95506

3.00506

3.06506

3.08506

3.11506

3.11506

3.12506

3.12506

 

2.66506

2.76506

2.86506

2.93506

3.00506

3.04506

3.08506

3.10506

3.11506

3.12506

3.12506

 

2.66506

2.76506

2.85506

2.93506

2 .99506

3.04506

3.07506

3.10506

3.11506

3.12506

3.12506

 

2.66506

2.76506

2.85506

2.92506

2.99506

3.03506

3.07506

3.09506

3.11506

3.11506

3.12506

 

2.66506

2.76506

2.84506

2.92506

2.98506

3.03506

3.06506

3.09506

3.10506

3.11506

3.12506

 

2.66506

2.75506

2.84506

2.91506

2.97506

3.02506

3.06506

3.08506

3.10506

3.11506

3.12506

 

2.66506

2.75506

2.83506

2.90506

2.96506

3.02506

3.05506

3.08506

3.10506

3.11506

3.11506

 

2.66506

2.75506

2.83506

2.90506

2.96506

3.01506

3.05506

3.07506

3.09506

3.10506

3.11506

 

2.66506

2.75506

2.82506

2.89506

2.95506

3.00506

3.04506

3.07506

3.09506

3.10506

3.11506

 

2.66506

2.74506

2.82506

2.89506

2.95506

3.00506

3.04506

3.06506

3.09506

3.10506

3.11506

 

2.66506

2.74506

2.82506

2.88506

2.94506

2.99506

3.03506

3.06506

3.08506

3.10506

3.10506

 

2.66506

2.74506

2.81506

2.88506

2.94506

2.99506

3.03506

3.06506

3.08506

3.09506

3.10506

 

2.66506

2.74506

2.81506

2.88506

2.93506

2.98506

3.02506

3.05506

3.07506

3.09506

3.10506

 

2.66506

2.74506

2.81506

2 .87506

2.93506

2.98506

3.02506

3.05506

3.07506

3.08506

3.09506

 

2.66506

2.74506

2.80506

2.87506

2.92506

2.97506

3.01506

3.05506

3.07506

3.08506

3.09506

 

2.66506

2.73506

2.80506

2.86506

2.92506

2.97506

3.01506

3.04506

3.06506

3.08506

3.09506

 

2.66506

2.73506

2.80506

2.86506

2.92506

2.97506

3.00506

3.04506

3.06506

3.07506

3.08506

 

2.66506

2.73506

2.80506

2 .86506

2.91506

2.96506

3.00506

3.03506

3.06506

3.07506

3.08506

 

2.66506

2.73506

2.80506

2.86506

2.91506

2.96506

3.00506

3.03506

3.05506

3.07506

3.07506

 

2.66506

2.73506

2.79506

2 .85506

2.91506

2.95506

2.99506

3.02506

3.05506

3.06506

3.07506

 

2.66506

2.73506

2.79506

2.85506

2.90506

2.95506

2.99506

3.02506

3.04506

3.06506

3.07506

 

2.66506

2.73506

2.79506

2.85506

2.90506

2.95506

2.99506

3.02506

3.04506

3.05506

3.06506

 

2.66506

2.73506

2.79506

2.85506

2.90506

2.94506

2.98506

3.01506

3.04506

3.05506

3.06506

 

2.66506

2.73506

2.79506

2.84506

2.89506

2.94506

2.98506

3.01506

3.03506

3.05506

3.05506

 

2.66506

2.73506

2.78506

2.84506

2.89506

2.94506

2.97506

3.01506

3.03506

3.04506

3.05506

 

 

 

 

2.66506

 

2.72506

 

2.78506

 

2.84506

 

2.89506

82

 

2.93506

 

2.97506

 

3.00506

 

3.02506

 

3.04506

 

3.05506

 

Table 7.8 Continued

 

38

39.22

2.66506

2.72506

2.78506

2.84506

2.89506

2.93506

2.97506

3.00506

3.02506

3.04506

3.04506

 

39

40.25

2.66506

2.72506

2.78506

2.83506

2.88506

2.93506

2.96506

2.99506

3.02506

3.03506

3.04506

 

40

41.28

2.66506

2.72506

2.78506

2.83506

2.88506

2.92506

2.96506

2.99506

3.01506

3.03506

3.04506

 

41

42.32

2.66506

2.72506

2.78506

2.83506

2.88506

2.92506

2.96506

2.99506

3.01506

3.02506

3.03506

 

42

43.35

2.66506

2.72506

2.78506

2.83506

2.87506

2.92506

2.95506

2.98506

3.01506

3.02506

3.03506

 

43

44.38

2.66506

2.72506

2.77506

2.83506

2.87506

2.91506

2.95506

2.98506

3.00506

3.02506

3.02506

 

44

45.41

2.66506

2.72506

2.77506

2.82506

2.87506

2.91506

2.95506

2.98506

3.00506

3.01506

3.02506

 

45

46.44

2.66506

2.72506

2.77506

2.82506

2.87506

2.91506

2.94506

2.97506

2.99506

3.01506

3.02506

 

46

47.48

2.66506

2.72506

2.77506

2.82506

2.87506

2.91506

2.94506

2.97506

2.99506

3.01506

3.01506

 

47

48.51

2.66506

2.72506

2.77506

2.82506

2.86506

2.90506

2.94506

2.97506

2.99506

3.00506

3.01506

 

48

49.54

2.66506

2.72506

2.77506

2.82506

2.86506

2.90506

2.93506

2.96506

2.98506

3.00506

3.01506

 

49

50.57

2.66506

2.72506

2.77506

2.81506

2.86506

2.90506

2.93506

2.96506

2.98506

3.00506

3.00506

 

50

51.61

2.66506

2.72506

2.77506

2.81506

2.86506

2.90506

2.93506

2.96506

2.98506

2.99506

3.00506

 

51

52.64

2.66506

2.72506

2.76506

2.81506

2.85506

2.89506

2.93506

2.95506

2.97506

2 .99506

3.00506

 

52

53.67

2.66506

2.71506

2.76506

2.81506

2.85506

2.89506

2.92506

2.95506

2.97506

2.98506

2.99506

 

53

54.70

2.66506

2.71506

2.76506

2.81506

2.85506

2.89506

2.92506

2.95506

2.97506

2.98506

2.99506

 

54

55.73

2.66506

2.71506

2.76506

2.81506

2.85506

2.89506

2.92506

2.94506

2.96506

2.98506

2.98506

 

55

56.77

2.66506

2.71506

2.76506

2.80506

2.85506

2.88506

2.91506

2.94506

2.96506

2.97506

2.98506

 

56

57.80

2.66506

2.71506

2.76506

2.80506

2.84506

2.88506

2.91506

2.94506

2.96506

2.97506

2.98506

 

57

58.83

2.66506

2.71506

2.76506

2.80506

2.84506

2.88506

2.91506

2.93506

2.95506

2.97506

2.97506

 

58

59.86

2.66506

2.71506

2.76506

2.80506

2.84506

2.88506

2.91506

2.93506

2.95506

2.96506

2.97506

 

59

60.89

2.66506

2.71506

2.76506

2.80506

2.84506

2.87506

2.90506

2.93506

2.95506

2.96506

2.97506

 

60

61.93

2.66506

2.71506

2.75506

2.80506

2.84506

2.87506

2.90506

2.93506

2.95506

2.96506

2.96506

 

61

62.96

2.66506

2.71506

2.75506

2.80506

2.83506

2.87506

2.90506

2.92506

2.94506

2.96506

2.96506

 

62

63.99

2.66506

2.71506

2.75506

2.79506

2.83506

2.87506

2.90506

2.92506

2.94506

2.95506

2.96506

 

63

65.02

2.66506

2.71506

2.75506

2.79506

2.83506

2.86506

2.89506

2.92506

2.94506

2.95506

2.96506

 

64

66.05

2.66506

2.71506

2.75506

2.79506

2.83506

2.86506

2.89506

2.92506

2.93506

2.95506

2.95506

 

65

67.09

2.66506

2.71506

2.75506

2.79506

2.83506

2.86506

2.89506

2.91506

2.93506

2.94506

2.95506

 

66

68.12

2.66506

2.71506

2.75506

2.79506

2.82506

2.86506

2.89506

2.91506

2.93506

2.94506

2.95506

 

67

69.15

2.66506

2.71506

2.75506

2.79506

2.82506

2.86506

2.88506

2.91506

2.92506

2.94506

2.94506

 

68

70.18

2.66506

2.71506

2.75506

2.79506

2.82506

2.85506

2.88506

2.90506

2.92506

2.93506

2.94506

 

69

71.22

2.66506

2.71506

2 .75506

2.78506

2.82506

2.85506

2.88506

2.90506

2.92506

2.93506

2.94506

 

70

72.25

2.66506

2.71506

2.75506

2.78506

2.82506

2.85506

2.88506

2.90506

2.92506

2.93506

2.93506

 

71

73.28

2.66506

2.71506

2.74506

2.78506

2.82506

2 .85506

2.87506

2.90506

2.91506

2.93506

2.93506

 

72

74.31

2.66506

2.70506

2.74506

2.78506

2.81506

2.85506

2.87506

2.89506

2.91506

2.92506

2.93506

 

73

75.34

2.66506

2.70506

2.74506

2.78506

2.81506

2.84506

2.87506

2.89506

2.91506

2.92506

2.93506

 

74

76.38

2.66506

2.70506

2.74506

2.78506

2.81506

2.84506

2.87506

2.89506

2.91506

2.92506

2.92506

 

75

77.41

2.66506

2.70506

2.74506

2.78506

2.81506

2.84506

2.87506

2.89506

2.90506

2.91506

2.92506

 

76

78.44

2.66506

2.70506

2.74506

2 .78506

2.81506

2.84506

2.86506

2.88506

2.90506

2.91506

2.92506

 

77

79.47

2.66506

2.70506

2.74506

2.77506

2.81506

2.84506

2.86506

2.88506

2.90506

2.91506

2.91506

 

78

80.50

2.66506

2.70506

2.74506

2.77506

2.81506

2.83506

2.86506

2.88506

2.90506

2.91506

2.91506

 

79

81.54

2.66506

2.70506

2.74506

2.77506

2.80506

2.83506

2.86506

2.88506

2.89506

2.90506

2.91506

 

80

82.57

2.66506

2.70506

2.74506

2.77506

2.80506

2.83506

2.85506

2.88506

2.89506

2.90506

2.91506

 

81

83.60

2.66506

2.70506

2.74506

2.77506

2.80506

2.83506

2.85506

2.87506

2.89506

2.90506

2.90506

 

82

84.63

2.66506

2.70506

2.74506

2.77506

2.80506

2.83506

2.85506

2.87506

2.89506

2.90506

2.90506

 

83

85.66

2.66506

2.70506

2.73506

2.77506

2.80506

2.83506

2 .85506

2.87506

2.88506

2.89506

2.90506

 

84

86.70

2.66506

2.70506

2.73506

2.77506

2.80506

2.82506

2.85506

2.87506

2.88506

2.89506

2.90506

 

85

87.73

2.66506

2.70506

2.73506

2.77506

2.79506

2.82506

2.84506

2.86506

2.88506

2.89506

2.89506

 

86

88.76

2.66506

2.70506

2.73506

2.76506

2.79506

2.82506

2.84506

2.86506

2.88506

2.89506

2.89506

 

87

89.79

2.66506

2.70506

2.73506

2.76506

2.79506

2.82506

2.84506

2.86506

2.87506

2.88506

2.89506

 

88

90.83

2.66506

2.70506

2.73506

2.76506

2.79506

2.82506

2.84506

2.86506

2.87506

2.88506

2.89506

 

89

91.86

2.66506

2.70506

2.73506

2.76506

2.79506

2.82506

2.84506

2.86506

2.87506

2.88506

2.88506

 

90

92.89

2.66506

2.70506

2.73506

2.76506

2.79506

2.81506

2.84506

2.85506

2.87506

2.88506

2.88506

 

91

93.92

2.66506

2.70506

2.73506

2.76506

2.79506

2.81506

2.83506

2.85506

2.87506

2.87506

2.88506

 

92

94.95

2.66506

2.70506

2.73506

2.76506

2.79506

2.81506

2.83506

2.85506

2.86506

2.87506

2.88506

 

93

95.99

2.66506

2.70506

2.73506

2.76506

2.78506

2.81506

2.83506

2.85506

2.86506

2.87506

2.87506

 

94

97.02

2.66506

2.70506

2.73506

2.76506

2.78506

2.81506

2.83506

2.85506

2.86506

2.87506

2.87506

 

 

95

 

98.05

 

2.66506

 

2.70506

 

2.73506

 

2.75506

 

2.78506

83

 

2.81506

 

2.83506

 

2.84506

 

2.86506

 

2.87506

 

2.87506

 

Table 7.8 Continued

 

96

99.08

2.66506

2.70506

2.73506

2.75506

2.78506

2.80506

2.82506

2.84506

2.85506

2.86506

2.87506

 

97

100.11

2.66506

2.70506

2.72506

2 .75506

2.78506

2.80506

2.82506

2.84506

2.85506

2.86506

2.87506

 

98

101.15

2.66506

2.69506

2.72506

2.75506

2.78506

2.80506

2.82506

2.84506

2.85506

2.86506

2.86506

 

99

102.18

2.66506

2.69506

2.72506

2.75506

2.78506

2.80506

2.82506

2.84506

2 .85506

2.86506

2.86506

 

100

103.21

2.66506

2.69506

2.72506

2.75506

2.78506

2.80506

2.82506

2.83506

2.85506

2.86506

2.86506

 

101

104.24

2.66506

2.69506

2.72506

2.75506

2.77506

2.80506

2.82506

2.83506

2.84506

2.85506

2.86506

 

102

105.27

2.66506

2.69506

2.72506

2.75506

2.77506

2.80506

2.81506

2.83506

2.84506

2.85506

2.86506

 

103

106.31

2.66506

2.69506

2.72506

2.75506

2.77506

2.79506

2.81506

2.83506

2.84506

2.85506

2.85506

 

104

107.34

2.66506

2.69506

2.72506

2.75506

2.77506

2.79506

2.81506

2.83506

2.84506

2.85506

2.85506

 

105

108.37

2.66506

2.69506

2.72506

2.75506

2.77506

2.79506

2.81506

2.83506

2.84506

2.85506

2.85506

 

106

109.40

2.66506

2.69506

2.72506

2.74506

2.77506

2.79506

2.81506

2.82506

2.84506

2.84506

2.85506

 

107

110.44

2.66506

2.69506

2.72506

2.74506

2.77506

2.79506

2.81506

2.82506

2.83506

2.84506

2.85506

 

108

111.47

2.66506

2.69506

2.72506

2.74506

2.77506

2.79506

2.81506

2.82506

2.83506

2.84506

2.84506

 

109

112.50

2.66506

2.69506

2.72506

2.74506

2.77506

2.79506

2.80506

2.82506

2.83506

2.84506

2.84506

 

110

113.53

2.66506

2.69506

2.72506

2.74506

2.76506

2.78506

2.80506

2.82506

2.83506

2.84506

2.84506

 

111

114.56

2.66506

2.69506

2.72506

2.74506

2.76506

2.78506

2.80506

2.82506

2.83506

2.83506

2.84506

 

112

115.60

2.66506

2.69506

2.72506

2.74506

2.76506

2.78506

2.80506

2.81506

2.82506

2.83506

2.84506

 

113

116.63

2.66506

2.69506

2.72506

2.74506

2.76506

2 .78506

2.80506

2.81506

2.82506

2.83506

2.83506

 

114

117.66

2.66506

2.69506

2.71506

2.74506

2.76506

2.78506

2.80506

2.81506

2.82506

2.83506

2.83506

 

115

118.69

2.66506

2.69506

2.71506

2.74506

2.76506

2 .78506

2.80506

2.81506

2.82506

2.83506

2.83506

 

116

119.72

2.66506

2.69506

2.71506

2.74506

2.76506

2.78506

2.79506

2.81506

2.82506

2.83506

2.83506

 

117

120.76

2.66506

2.69506

2.71506

2.74506

2.76506

2.78506

2.79506

2.81506

2.82506

2.82506

2.83506

 

118

121.79

2.66506

2.69506

2.71506

2.74506

2.76506

2.77506

2.79506

2.80506

2.81506

2.82506

2.83506

 

119

122.82

2.66506

2.69506

2.71506

2.73506

2.75506

2.77506

2.79506

2.80506

2.81506

2.82506

2.82506

 

120

123.85

2.66506

2.69506

2.71506

2.73506

2.75506

2.77506

2.79506

2.80506

2.81506

2.82506

2.82506

 

121

124.88

2.66506

2.69506

2.71506

2.73506

2.75506

2.77506

2.79506

2.80506

2.81506

2.82506

2.82506

 

122

125.92

2.66506

2.69506

2.71506

2.73506

2.75506

2.77506

2.79506

2.80506

2.81506

2.82506

2.82506

 

123

126.95

2.66506

2.69506

2.71506

2.73506

2.75506

2.77506

2.78506

2.80506

2.81506

2.81506

2.82506

 

124

127.98

2.66506

2.69506

2.71506

2.73506

2.75506

2.77506

2.78506

2.80506

2.81506

2.81506

2.82506

 

125

129.01

2.66506

2.69506

2.71506

2.73506

2.75506

2.77506

2.78506

2.79506

2.80506

2.81506

2.81506

 

126

130.05

2.66506

2.69506

2.71506

2.73506

2.75506

2.77506

2.78506

2.79506

2.80506

2.81506

2.81506

 

127

131.08

2.66506

2.69506

2.71506

2.73506

2.75506

2.76506

2.78506

2.79506

2.80506

2.81506

2.81506

 

128

132.11

2.66506

2.69506

2.71506

2.73506

2.75506

2.76506

2.78506

2.79506

2.80506

2.81506

2.81506

 

129

133.14

2.66506

2.69506

2.71506

2.73506

2.75506

2.76506

2.78506

2.79506

2.80506

2.80506

2.81506

 

130

134.17

2.66506

2.69506

2.71506

2.73506

2.74506

2.76506

2.78506

2.79506

2.80506

2.80506

2.81506

 

131

135.21

2.66506

2.69506

2.71506

2.73506

2.74506

2.76506

2.77506

2.79506

2.80506

2.80506

2.80506

 

132

136.24

2.66506

2.69506

2.71506

2.73506

2.74506

2.76506

2.77506

2.79506

2.79506

2.80506

2.80506

 

133

137.27

2.66506

2.69506

2.71506

2.72506

2.74506

2.76506

2.77506

2.78506

2.79506

2.80506

2.80506

 

134

138.30

2.66506

2.69506

2.71506

2.72506

2.74506

2.76506

2.77506

2.78506

2.79506

2.80506

2.80506

 

135

139.33

2.66506

2.69506

2.70506

2.72506

2.74506

2.76506

2.77506

2.78506

2.79506

2.80506

2.80506

 

136

140.37

2.66506

2.68506

2.70506

2.72506

2.74506

2.76506

2.77506

2.78506

2.79506

2.79506

2.80506

 

137

141.40

2.66506

2.68506

2.70506

2.72506

2.74506

2.75506

2.77506

2.78506

2.79506

2.79506

2.80506

 

138

142.43

2.66506

2.68506

2.70506

2.72506

2.74506

2.75506

2.77506

2.78506

2.79506

2.79506

2.79506

 

139

143.46

2.66506

2 .68506

2.70506

2.72506

2.74506

2.75506

2.77506

2.78506

2.78506

2.79506

2.79506

 

140

144.49

2.66506

2.68506

2.70506

2.72506

2.74506

2.75506

2.76506

2.78506

2.78506

2.79506

2.79506

 

141

145.53

2.66506

2.68506

2.70506

2.72506

2.74506

2.75506

2.76506

2.77506

2.78506

2.79506

2.79506

 

142

146.56

2.66506

2.68506

2.70506

2.72506

2.74506

2.75506

2.76506

2.77506

2.78506

2.79506

2.79506

 

143

147.59

2.66506

2.68506

2.70506

2.72506

2.73506

2.75506

2.76506

2.77506

2.78506

2.78506

2.79506

 

144

148.62

2.66506

2.68506

2.70506

2.72506

2.73506

2.75506

2.76506

2.77506

2.78506

2.78506

2.79506

 

145

149.66

2.66506

2.68506

2.70506

2.72506

2.73506

2.75506

2.76506

2.77506

2.78506

2.78506

2.78506

 

146

150.69

2.66506

2.68506

2.70506

2.72506

2.73506

2.75506

2.76506

2.77506

2.78506

2.78506

2.78506

 

147

151.72

2.66506

2.68506

2.70506

2.72506

2.73506

2.75506

2.76506

2.77506

2.77506

2.78506

2.78506

 

148

152.75

2.66506

2.68506

2.70506

2.72506

2.73506

2.74506

2.76506

2.77506

2.77506

2.78506

2.78506

 

149

153.78

2.66506

2.68506

2.70506

2.72506

2.73506

2.74506

2.76506

2.77506

2.77506

2.78506

2.78506

 

150

154.82

2.66506

2.68506

2.70506

2.71506

2.73506

2.74506

2.75506

2.76506

2.77506

2.78506

2.78506

 

151

155.85

2.66506

2.68506

2.70506

2.71506

2.73506

2.74506

2.75506

2.76506

2.77506

2.78506

2.78506

 

152

156.88

2.66506

2.68506

2.70506

2.71506

2.73506

2.74506

2.75506

2.76506

2.77506

2.77506

2.78506

 

 

153

 

157.91

 

2.66506

 

2.68506

 

2.70506

 

2.71506

 

2.73506

84

 

2.74506

 

2.75506

 

2.76506

 

2.77506

 

2.77506

 

2.78506

 

Table 7.8 Continued

 

154

158.94

2.66506

2.68506

2.70506

2.71506

2.73506

2.74506

2.75506

2.76506

2.77506

2.77506

2.77506

 

155

159.98

2.66506

2.68506

2.70506

2.71506

2.73506

2.74506

2.75506

2.76506

2.77506

2.77506

2.77506

 

156

161.01

2.66506

2.68506

2.70506

2.71506

2.73506

2.74506

2.75506

2.76506

2.76506

2.77506

2.77506

 

157

162.04

2.66506

2.68506

2.70506

2.71506

2.72506

2.74506

2.75506

2.76506

2.76506

2.77506

2.77506

 

158

163.07

2.66506

2.68506

2.70506

2.71506

2.72506

2.74506

2.75506

2.76506

2.76506

2.77506

2.77506

 

159

164.10

2.66506

2.68506

2.70506

2.71506

2.72506

2.74506

2.75506

2.75506

2.76506

2.77506

2.77506

 

160

165.14

2.66506

2.68506

2.70506

2.71506

2.72506

2.74506

2.75506

2.75506

2.76506

2.77506

2.77506

 

161

166.17

2.66506

2.68506

2.70506

2.71506

2.72506

2.73506

2.74506

2.75506

2.76506

2.76506

2.77506

 

162

167.20

2.66506

2.68506

2.69506

2.71506

2.72506

2.73506

2.74506

2.75506

2.76506

2.76506

2.77506

 

163

168.23

2.66506

2.68506

2.69506

2.71506

2.72506

2.73506

2.74506

2.75506

2.76506

2.76506

2.76506

 

164

169.27

2.66506

2.68506

2.69506

2.71506

2.72506

2.73506

2.74506

2.75506

2.76506

2.76506

2.76506

 

165

170.30

2.66506

2.68506

2.69506

2.71506

2.72506

2.73506

2.74506

2.75506

2.76506

2.76506

2.76506

 

166

171.33

2.66506

2.68506

2.69506

2.71506

2.72506

2.73506

2.74506

2.75506

2.75506

2.76506

2.76506

 

167

172.36

2.66506

2.68506

2.69506

2.71506

2.72506

2.73506

2.74506

2 .75506

2.75506

2.76506

2.76506

 

168

173.39

2.66506

2.68506

2 .69506

2.71506

2.72506

2.73506

2.74506

2.75506

2.75506

2.76506

2.76506

 

169

174.43

2.66506

2.68506

2.69506

2.71506

2.72506

2.73506

2.74506

2.75506

2.75506

2.76506

2.76506

 

170

175.46

2.66506

2.68506

2.69506

2.71506

2.72506

2.73506

2.74506

2.75506

2.75506

2.75506

2.76506

 

171

176.49

2.66506

2.68506

2.69506

2.70506

2.72506

2.73506

2.74506

2.74506

2.75506

2.75506

2.76506

 

172

177.52

2.66506

2.68506

2.69506

2.70506

2.72506

2.73506

2.74506

2.74506

2.75506

2.75506

2.75506

 

173

178.55

2.66506

2.68506

2.69506

2.70506

2.72506

2.73506

2.74506

2.74506

2.75506

2.75506

2.75506

 

174

179.59

2.66506

2.68506

2.69506

2.70506

2.71506

2.73506

2.73506

2.74506

2.75506

2.75506

2.75506

 

175

180.62

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.74506

2.75506

2.75506

2.75506

 

176

181.65

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.74506

2.75506

2.75506

2.75506

 

177

182.68

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.74506

2.74506

2.75506

2.75506

 

178

183.71

2.66506

2.68506

2 .69506

2.70506

2.71506

2.72506

2.73506

2.74506

2.74506

2.75506

2.75506

 

179

184.75

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.74506

2.74506

2.75506

2.75506

 

180

185.78

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.74506

2.74506

2.75506

2.75506

 

181

186.81

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.74506

2.74506

2.74506

2.75506

 

182

187.84

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.74506

2.74506

2.74506

2.75506

 

183

188.88

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.73506

2.74506

2.74506

2.75506

 

184

189.91

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.73506

2 .74506

2.74506

2.74506

 

185

190.94

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.73506

2.74506

2.74506

2.74506

 

186

191.97

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.73506

2.74506

2.74506

2.74506

 

187

193.00

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.73506

2.73506

2.74506

2.74506

2.74506

 

188

194.04

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.72506

2.73506

2.74506

2.74506

2.74506

 

189

195.07

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.72506

2.73506

2.74506

2.74506

2.74506

 

190

196.10

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.72506

2.73506

2.73506

2.74506

2.74506

 

191

197.13

2.66506

2.68506

2.69506

2.70506

2.71506

2.72506

2.72506

2.73506

2.73506

2.74506

2.74506

 

192

198.16

2.66506

2.68506

2.69506

2.70506

2.71506

2.71506

2.72506

2.73506

2.73506

2.74506

2.74506

 

193

199.20

2.66506

2.68506

2.69506

2.70506

2.71506

2.71506

2.72506

2.73506

2.73506

2.74506

2.74506

 

194

200.23

2.66506

2.68506

2.69506

2.70506

2.71506

2.71506

2.72506

2.73506

2.73506

2.73506

2.74506

 

195

201.26

2.66506

2.68506

2.69506

2.70506

2.70506

2.71506

2.72506

2.73506

2.73506

2.73506

2.74506

 

196

202.29

2.66506

2.68506

2.69506

2.70506

2.70506

2.71506

2.72506

2 .73506

2.73506

2.73506

2.73506

 

197

203.32

2.66506

2.68506

2.69506

2.70506

2.70506

2.71506

2.72506

2 .73506

2.73506

2.73506

2.73506

 

198

204.36

2.66506

2.68506

2.69506

2.69506

2.70506

2.71506

2.72506

2.72506

2.73506

2.73506

2.73506

 

199

205.39

2.66506

2.68506

2.69506

2.69506

2.70506

2.71506

2.72506

2.72506

2.73506

2.73506

2.73506

 

200

206.42

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.72506

2 .72506

2.73506

2.73506

2.73506

 

201

207.45

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.72506

2.72506

2.73506

2.73506

2.73506

 

202

208.49

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.72506

2.72506

2.73506

2.73506

2.73506

 

203

209.52

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.72506

2.72506

2.73506

2.73506

2.73506

 

204

210.55

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.72506

2.72506

2.72506

2.73506

2.73506

 

205

211.58

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.71506

2.72506

2.72506

2.73506

2.73506

 

206

212.61

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.71506

2.72506

2.72506

2.73506

2 .73506

 

207

213.65

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.71506

2.72506

2.72506

2.73506

2.73506

 

208

214.68

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.71506

2.72506

2.72506

2.73506

2.73506

 

209

215.71

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.71506

2.72506

2.72506

2.72506

2.73506

 

210

216.74

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.71506

2.72506

2.72506

2.72506

2.73506

 

 

211

 

217.77

 

2.66506

 

2 .67506

 

2.68506

 

2.69506

 

2.70506

85

 

2.71506

 

2.71506

 

2.72506

 

2.72506

 

2.72506

 

2.72506

 

Table 7.8 Continued

 

212

218.81

2.66506

2.67506

2.68506

2.69506

2.70506

2.71506

2.71506

2.72506

2.72506

2.72506

2.72506

 

213

219.84

2.66506

2.67506

2.68506

2.69506

2.70506

2.70506

2.71506

2.72506

2.72506

2.72506

2.72506

 

214

220.87

2.66506

2.67506

2.68506

2.69506

2.70506

2.70506

2.71506

2.72506

2.72506

2.72506

2.72506

 

215

221.90

2.66506

2.67506

2.68506

2.69506

2.70506

2.70506

2.71506

2.71506

2.72506

2.72506

2.72506

 

216

222.93

2.66506

2.67506

2.68506

2.69506

2.70506

2.70506

2.71506

2.71506

2.72506

2.72506

2.72506

 

217

223.97

2.66506

2.67506

2 .68506

2.69506

2.70506

2.70506

2.71506

2.71506

2.72506

2.72506

2.72506

 

218

225.00

2.66506

2.67506

2.68506

2.69506

2.70506

2.70506

2.71506

2.71506

2.72506

2.72506

2.72506

 

219

226.03

2.66506

2.67506

2.68506

2.69506

2.70506

2.70506

2.71506

2.71506

2.72506

2.72506

2.72506

 

220

227.06

2.66506

2.67506

2.68506

2.69506

2.70506

2.70506

2.71506

2.71506

2.72506

2.72506

2.72506

 

221

228.10

2.66506

2.67506

2.68506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.72506

2.72506

 

222

229.13

2 .66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.71506

2.71506

2.71506

2.72506

2.72506

 

223

230.16

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.71506

2.71506

2.71506

2.72506

2.72506

 

224

231.19

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.71506

2.71506

2.71506

2.72506

2.72506

 

225

232.22

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.71506

2.71506

2.71506

2.72506

2.72506

 

226

233.26

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.72506

 

227

234.29

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.72506

 

228

235.32

2.66506

2.67506

2 .68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.71506

 

229

236.35

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.71506

 

230

237.38

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.71506

 

231

238.42

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.71506

 

232

239.45

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.71506

 

233

240.48

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.71506

 

234

241.51

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.71506

 

235

242.54

2.66506

2.67506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

2.71506

 

236

243.58

2.66506

2.67506

2.68506

2.68506

2.69506

2.70506

2.70506

2.70506

2.71506

2.71506

2.71506

 

237

244.61

2.66506

2.67506

2.68506

2.68506

2.69506

2.70506

2.70506

2.70506

2.71506

2.71506

2.71506

 

238

245.64

2.66506

2.67506

2.68506

2.68506

2.69506

2.70506

2.70506

2.70506

2.71506

2.71506

2.71506

 

239

246.67

2.66506

2.67506

2.68506

2.68506

2.69506

2.70506

2.70506

2.70506

2.71506

2.71506

2.71506

 

240

247.71

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

 

241

248.74

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2 .70506

2.70506

2.71506

2.71506

2.71506

 

242

249.77

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2.70506

2.70506

2.71506

2.71506

2.71506

 

243

250.80

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2.70506

2.70506

2.70506

2.71506

2.71506

 

244

251.83

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2.70506

2.70506

2.70506

2.71506

2.71506

 

245

252.87

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2.70506

2.70506

2.70506

2.71506

2.71506

 

246

253.90

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2.70506

2.70506

2.70506

2.71506

2.71506

 

247

254.93

2.66506

2.67506

2.68506

2.68506

2.69506

2 .69506

2.70506

2.70506

2.70506

2.70506

2.71506

 

248

255.96

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2 .70506

2.70506

2.70506

2.70506

2.71506

 

249

256.99

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2.70506

2.70506

2.70506

2.70506

2.70506

 

250

258.03

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2.70506

2.70506

2.70506

2.70506

2.70506

 

251

259.06

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2.70506

2.70506

2.70506

2.70506

2.70506

 

252

260.09

2.66506

2 .67506

2.68506

2.68506

2.69506

2.69506

2.70506

2.70506

2.70506

2.70506

2.70506

 

253

261.12

2.66506

2.67506

2.68506

2.68506

2.69506

2.69506

2.69506

2.70506

2.70506

2.70506

2.70506

 

 

254

 

262.15

 

2.66506

 

2.67506

 

2.68506

 

2.68506

 

 

2.69506

2 .69506

 

2.69506

 

2.70506

 

2.70506

 

2.70506

 

2.70506

 

86

 

Appendix 7.10: Model Output for Darcy’s Velocity

Excel formula used to calculate the Darcy ’s velocity at t = 1.03 s and x = 0.005

m. (i.e. cell D5):

D5 =-('Effective Gap
Radius'!DSAZ/(MASTER!vis*8*MASTER!tor)*(MASTER!st*MASTER!
Agi"0.5*MASTER!ADM)/(MASTER!reffi*(l-MASTERlADM/(l-
'Moisture Content db'!D5/(1+'Moisture Content
db'!D5)))"(3/2))*('Moisture Content db'!E5-'Moisture Content db'!D5))/dx

Table 7.9: Predicted Darcy’s velocity profile generated by Excel, given the input
variables listed in Table 4.1.

 

Darcy's Velocity

 

i=>

0

1

2

3

4

5

10

 

0

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

0.0350

0
0

0|
0

0.0500

 

1.76504

0

0

0

0

0

 

1.03

8.79505

9.16505

0

0

0

0

 

2.06

8.79505

4.58505

4.67505

0

0

0

 

3.10

6.67505

6.72505

2.33505

2.36505

0

0

 

4.13

6.59505

4.61505

4.54505

1.18505

1.18505

0

0000

 

5.16

5.58505

5.55505

2.94505

2.87505

5.92506

5.93506

0

00000

 

6.19

5.50505

4.33505

4.20505

1 .79505

1 .73505

2.97506

2.97506

0

 

7.22

4.89505

4.84505

3.10505

2.96505

1 .05505

1 .02505

1 .49506

1 .49506

0000000090

 

m‘laio'lwa-Kob-

8.26

4.81505

4.05505

3.89505

2.10505

1 .99505

6.01506

5.83506

7.43507

7.43507

00000000020

 

CD

9.29

4.41505

4.35505

3.11505

2.93505

1 .36505

1 .29505

3.38506

3.29506

3.72507

3.72507

 

.3
C

10.32

4.34505

3.79505

3.63505

2.25505

2.11505

8.52506

8.08506

1 .88506

1 .83506

1 .86507

 

A
—L

11.35

4.04505

3.98505

3.05505

2.86505

1 .56505

1 .46505

5.21506

4.95506

1 .03506

9.15507

 

.3
N

12.39

3.98505

3.57505

3.41505

2.32505

2.16505

1 .05505

9.75506

3.13506

2.94506

5.17507

 

.3
03

13.42

3.75505

3.69505

2.97505

2.78505

1 .69505

1 .56505

6.82506

6.34506

1 .82506

1.47506

 

 

.3
§

14.45

3.69505

3.38505

3.23505

2.35505

2.17505

1.19505

1.10505

4.33506

3.91506

9.13507

 

.3
0"

15.48

3.52505

3.46505

2.88505

2.69505

1 .78505

1.63505

8.16506

7.44506

2.63506

1 .95506

 

.3
a)

16.51

3.46505

3.22505

3.07505

2.35505

2.16505

1 .30505

1.19505

5.41506

4.70506

1.31506

 

—L
N

17.55

3.32505

3.27505

2.80505

2.61505

1 .83505

1.67505

9.26506

8.27506

3.37506

2.35506

 

.3
a

18.58

3.27505

3.08505

2.94505

2.33505

2.14505

1 .39505

1 .25505

6.33506

5.31506

1.68506

 

.3
(O

19.61

3.16505

3.11505

2.72505

2.53505

1.87505

1 .69505

1.01505

8.88506

4.01506

2.65506

 

N
0

20.64

3.11505

2.95505

2.82505

2.30505

2.11505

1 .44 505

1 .29505

7.09506

5.77506

2.01506

 

N
.A

21.67

3.01505

2.97505

2.64505

2.46505

1 .88505

1 .70505

1 .08505

9.31506

4.56506

2.88506

 

N
N

22.71

2.97 505

2.83505

2.71505

2.27505

2.07505

1 .49505

1 .31505

7.70506

6.10506

2.28506

 

N
0

23.74

2.89505

2.84505

2.56505

2.39505

1 .88505

1.69505

1.13505

9.60506

5.00506

3.04506

 

N
&

24.77

2.85505

2.73505

2.62505

2.23505

2.04505

1.51505

1 .32505

8.18506

6.32506

2.50506

 

N
0|

25.80

2.78505

2.73505

2.49505

2.33505

1 .88505

1 .68505

1.17505

9.78506

5.35506

3.15506

 

N
O)

26.83

2.74505

2.64505

2.53505

2.19505

2.00505

1 .53505

1.33505

8.54506

6.46506

2.67506

 

N
N

27.87

2.68505

2.64505

2.42505

2.26505

1 .87505

1 .66505

1.19505

9.86506

5.62506

3.22506

 

N
a)

28.90

2.64505

2.56505

2.45505

2.15505

1 .96505

1 .53505

1.32505

8.80506

6.54506

2.80506

 

N
(D

29.93

2.59505

2.55505

2.36505

2.21505

1 .85505

1 .64505

1 .21505

9.89506

5.81506

3.26506

 

(3)
0

30.96

2.55505

2.48505

2.38505

2.11505

1 .92 505

1 .53505

1.31505

8.98506

6.57506

2.90506

 

O)
.3

32 .00

2.50505

2.47505

2.30505

2.15505

1.83505

1.62505

1.22505

9.86506

5.95506

3.28506

 

0)
N

33.03

2.47505

2.41505

2.31505

2.07505

1 .88505

1.53505

1.30505

9.08506

6.57506

2.97506

 

(a)
(3)

 

34.06

 

2.43505

 

2.39505

 

2.24505

 

2.10505

 

1 .80505

87

 

1.59505

 

1 .22505

 

9.79506

 

6.04506

 

3.27506

 

0000000000000000000000000000000000

 

Table 7.9 Continued

 

34

35.09

2.40505

2.34505

2.25505

2.03505

1.84505

1.51505

1 .29505

9.13506

6.53506

3.01506

 

35

36.12

2.36505

2.32505

2.19505

2.04505

1 .77505

1 .56505

1 .22505

9.69506

6.08506

3.25506

 

36

37.16

2.33505

2.28505

2.19505

1 .98 505

1 .80505

1 .50505

1 .27505

9.14506

6.47506

3.03506

 

37

38.19

2.29505

2.26505

2.13505

2.00505

1 .74505

1.54505

1 .21505

9.57506

6.10506

3.22506

 

38

39.22

2.27505

2.22505

2.13505

1 .94505

1 .77505

1 .48505

1 .25505

9.10506

6.40506

3.03506

 

39

40.25

2.23505

2.20505

2.08505

1 .95 505

1.71505

1.51505

1 .20505

9.43506

6.08506

3.18506

 

40

41.28

2.20505

2.16505

2.07505

1 .90505

1 .73505

1.46505

1 .23505

9.04 506

6.31506

3.02506

 

41

42.32

2.17505

2.14505

2.03505

1 .90505

1 .68505

1 .48505

1.18505

9.28506

6.04506

3.13506

 

42

43.35

2.15505

2.11505

2.02505

1 .86 505

1 .69505

1 .43505

1 .20505

8.95506

6.21506

3.00506

 

43

44.38

2.12505

2.09505

1 .99505

1 .86 505

1 .65505

1 .45505

1.17505

9.12506

5.98506

3.08506

 

44

45.41

2.09505

2.06505

1 .97505

1 .82 505

1 .65505

1 .41505

1.18505

8.84506

6.11506

2.97506

 

45

46.44

2.07505

2.04505

1 .94505

1.81505

1 .62505

1 .42505

1.15505

8.96506

5.91506

3.03506

 

46

47.48

2.04505

2.01505

1 .93505

1 .78505

1 .62505

1 .39505

1.16505

8.71506

6.00506

2.94506

 

47

48.51

2.02505

1 .99505

1 .90505

1 .77505

1 .59505

1 .39505

1 .13505

8.79506

5.83506

2.98506

 

48

49.54

1.99505

1 .96505

1.88505

1 .74505

1 .58505

1 .36505

1.13505

8.58506

5.89506

2.89506

 

49

50.57

1.97505

1 .94505

1.85505

1 .73505

1 .55505

1 .36505

1.11505

8.62506

5.74506

2.92506

 

50

51 .61

1.95505

1.91505

1 .84505

1.71505

1.55505

1.33505

1.11505

8.43506

5.77506

2.85506

 

51

52.64

1.92505

1 .89505

1.81505

1 .69505

1.52505

1 .33505

1 .09505

8.45506

5.65506

2.86506

 

52

53.67

1 .90505

1 .87505

1 .80505

1 .67505

1.51505

1 .31505

1 .09505

8.28506

5.66506

2.80506

 

53

54.70

1 .88505

1 .85505

1 .77505

1 .66505

1 .49505

1 .30505

1.07505

8.28506

5.55506

2.80506

 

54

55.73

1 .86505

1.83505

1 .75505

1.63505

1 .48505

1 .28505

1 .07505

8.13506

5.54506

2.75506

 

55

56.77

1 .83505

1.81505

1 .73505

1 .62 505

1 .46505

1.27505

1.05505

8.11506

5.45506

2.74506

 

56

57.80

1.81505

1 .78505

1.71505

1 .60505

1 .45505

1.25505

1.04505

7.97506

5.43506

2.69506

 

57

58.83

1 .79505

1 .77505

1 .69505

1 .58505

1.43505

1.25505

1 .03505

7.94506

5.34506

2.69506

 

58

59.86

1 .77505

1 .74505

1 .68505

1.56505

1.42505

1 .23505

1 .02505

7.82506

5.32506

2.64506

 

59

60.89

1 .75505

1 .73505

1 .65505

1 .55505

1 .40505

1 .22505

1.01505

7.77506

5.24506

2.63506

 

60

61.93

1.73505

1.71505

1 .64505

1 .53505

1 .38505

1 .20505

9.99506

7.66506

5.20506

2.59506

 

61

62.96

1.71505

1 .69505

1 .62505

1.51505

1 .37505

1.19505

9.85506

7.60506

5.13506

2.57506

 

62

63.99

1 .70505

1 .67505

1 .60505

1 .49505

1 .35505

1.18505

9.77506

7.50506

5.09506

2.53506

 

63

65.02

1 .68505

1 .65505

1 .58505

1.48505

1.34505

1.17505

9.65506

7.44506

5.02506

2.51506

 

64

66.05

1 .66505

1.63505

1 .57505

1.46505

1.32505

1.15505

9.56506

7.34506

4.98506

2.48506

 

65

67.09

1 .64505

1.61505

1 .55505

1 .45505

1.31505

1.14505

9.44506

7.28506

4.92506

2.46506

 

66

68.12

1 .62505

1 .59505

1 .53505

1.43505

1 .29505

1 .13505

9.35506

7.19506

4.87506

2.43506

 

67

69.15

1 .60505

1 .58505

1.51505

1 .41 505

1 .28505

1.12505

9.24506

7.12506

4.81506

2.40506

 

68

70.18

1.59505

1.56505

1 .50505

1 .40505

1.27505

1.10505

9.15506

7.03506

4.77506

2.37506

 

69

71 .22

1.57505

1 .54505

1 .48505

1.38505

1 .25505

1 .09505

9.04506

6.96506

4.71506

2.35506

 

70

72.25

1 .55505

1 .53505

1 .46505

1.37505

1 .24505

1.08505

8.95506

6.88506

4.66506

2.32506

 

71

73.28

1 .53505

1.51505

1 .45505

1 .35505

1 .22505

1.07505

8.84506

6.81506

4.61506

2.30506

 

72

74.31

1 .52505

1 .49505

1 .43505

1 .34505

1 .21505

1.06505

8.75506

6.73506

4.56506

2.27506

 

73

75.34

1 .50505

1 .48505

1 .42505

1 .32505

1 .20505

1.04505

8.65506

6.66506

4.51506

2.25506

 

74

76.38

1 .48505

1 .46505

1 .40505

1.31505

1.18505

1 .03505

8.56506

6.59506

4.46506

2.22506

 

75

77.41

1 .47505

1 .44 505

1 .39505

1.29505

1.17505

1 .02505

8.46506

6.52506

4.41506

2.19506

 

76

78.44

1 .45505

1 .43505

1 .37505

1 .28505

1.16505

1.01505

8.37506

6.44506

4.36506

2.17506

 

77

79.47

1 .44505

1 .41505

1 .35505

1 .27505

1.15505

9.99506

8.28506

6.37506

4.31506

2.15506

 

78

80.50

1 .42505

1 .40505

1 .34505

1 .25505

1 .13505

9.88506

8.19506

6.30506

4.26506

2.12506

 

79

81.54

1 .40505

1 .38505

1 .33505

1 .24505

1.12505

9.77506

8.10506

6.23506

4.22506

2.10506

 

80

82.57

1 .39505

1 .37505

1.31505

1 .22505

1.11505

9.66506

8.01506

6.16506

4.17506

2.07506

 

81

83.60

1 .37505

1 .35505

1 .30505

1.21505

1 .10505

9.56506

7.92506

6.10506

4.12506

2.05506

 

82

84.63

1.36505

1.34505

1 .28505

1 .20505

1 .08505

9.45506

7.83506

6.03506

4.08506

2.03506

 

83

85.66

1.34505

1 .32505

1 .27505

1.18505

1 .07505

9.35506

7.75506

5.96506

4.03506

2.00506

 

84

86.70

1.33505

1.31505

1 .25505

1.17505

1 .06505

9.24506

7.66506

5.89506

3.99506

1.98506

 

85

87.73

1 .32505

1 .29505

1 .24505

1.16505

1 .05505

9.14506

7.58506

5.83506

3.94506

1.96506

 

86

88.76

1 .30505

1 .28505

1 .23505

1.15505

1.04505

9.04506

7.49506

5.77506

3.90506

1.94506

 

89.79

1 .29505

1 .27505

1 .21505

1.13505

1 .03505

8.94506

7.41 506

5.70506

3.85506

1.91506

 

90.83

1 .27505

1 .25505

1 .20505

1.12505

1.01505

8.84506

7.33506

5.64506

3.81506

1 .89506

 

91 .86

1.26505

1.24505

1.19505

1.11505

1 .00505

8.75506

7.25506

5.58506

3.77506

1 .87506

 

 

 

92.89

 

1 .25505

 

1.23505

 

1.17505

 

1.10505

 

9.93506

88

 

8.65506

 

7.17506

 

5.51506

 

3.73506

 

1 .85506

 

000000000000000000000000000000000000000000000000000000000

 

Table 7.9 Continued

 

91

93.92

1.23505

1.21505

1.16505

1 .08505

9.82506

8.56506

7.09506

5.45506

3.69506

1 .83506

 

92

94.95

1 .22505

1 .20505

1.15505

1.07505

9.71506

8.46506

7.01506

5.39506

3.64506

1.81506

 

93

95.99

1 .21505

1.19505

1.14505

1 .06505

9.60506

8.37506

6.93506

5.33506

3.60506

1 .79506

 

94

97.02

1.19505

1.17505

1.12505

1 .05505

9.50506

8.28506

6.86506

5.27506

3.56506

1.77506

 

95

98.05

1.18505

1.16505

1.11505

1.04505

9.40506

8.19506

6.78506

5.22506

3.52506

1.75506

 

96

99.08

1.17505

1.15505

1.10505

1.03505

9.29506

8.10506

6.71506

5.16506

3.49506

1.73506

 

97

100.11

1.16505

1.14505

1 .09505

1 .02505

9.19506

8.01506

6.63506

5.10506

3.45506

1.71506

 

98

101.15

1.14505

1.12505

1 .08505

1 .00505

9.09506

7.92506

6.56506

5.05506

3.41506

1.69506

 

99

102.18

1.13505

1.11505

1 .06505

9.94506

8.99506

7.83506

6.49506

4.99506

3.37 506

1.67506

 

100

103.21

1.12505

1.10505

1 .05505

9.83506

8.89506

7.75506

6.42506

4.93506

3.33506

1 .65506

 

101

104.24

1.11505

1 .09505

1.04505

9.72506

8.80506

7.66506

6.35506

4.88506

3.30506

1.63506

 

102

105.27

1 .09505

1 .08505

1 .03505

9.62506

8.70506

7.58506

6.28506

4.83506

3.26506

1.61506

 

103

106.31

1.08505

1 .06505

1 .02505

9.51506

8.61506

7.50506

6.21506

4.77506

3.22506

1 .59506

 

104

107.34

1 .07505

1 .05505

1.01505

9.41506

8.51506

7.41506

6.14506

4.72506

3.19506

1 .58506

 

105

108.37

1 .06 505

1.04505

9.97506

9.31506

8.42506

7.33506

6.07506

4.67506

3.15506

1 .56506

 

106

109.40

1 .05505

1 .03505

9.87506

9.21506

8.33506

7.25506

6.01506

4.62506

3.12506

1.54506

 

107

110.44

1.04505

1 .02505

9.76506

9.11506

8.24506

7.17506

5.94506

4.57506

3.08506

1 .52 506

 

108

111.47

1.03505

1.01505

9.65506

9.01506

8.15506

7.10506

5.88506

4.52506

3.05506

1.51506

 

109

112.50

1.02505

9.97506

9.55506

8.91506

8.06506

7.02506

5.81506

4.47506

3.01506

1 .49506

 

110

113.53

1 .00505

9.86506

9.45506

8.81506

7.97506

6.94506

5.75506

4.42506

2.98506

1 .47506

 

111

114.56

9.94506

9.75506

9.35506

8.72506

7.89506

6.87506

5.69506

4.37506

2.95506

1 .46506

 

112

115.60

9.83506

9.65506

9.24 506

8.62 506

7.80506

6.79506

5.62506

4.32506

2.92506

1 .44 506

 

113

116.63

9.73506

9.55506

9.15506

8.53506

7.71506

6.72506

5.56506

4.27506

2.88506

1 .42506

 

114

117.66

9.62506

9.44506

9.05506

8.44506

7.63506

6.64506

5.50506

4.23506

2.85506

1.41506

 

115

118.69

9.52506

9.34506

8.95506

8.35506

7.55506

6.57506

5.44506

4.18506

2.82506

1 .39506

 

116

119.72

9.42506

9.24506

8.85506

8.26506

7.47506

6.50506

5.38506

4.13506

2 .79506

1 .38506

 

117

120.76

9.32506

9.14506

8.76506

8.17506

7.39506

6.43506

5.32506

4.09506

2.76506

1 .36506

 

118

121.79

9.22506

9.04506

8.66506

8.08506

7.31506

6.36506

5.26506

4.04506

2.73506

1 .35506

 

119

122.82

9.12506

8.95506

8.57506

7.99506

7.23506

6.29506

5.21506

4.00506

2.70506

1 .33506

 

120

123.85

9.02506

8.85506

8.48506

7.91506

7.15506

6.22506

5.15506

3.96506

2.67506

1 .32506

 

121

124.88

8.92506

8.76506

8.39506

7.82506

7.07506

6.16506

5.09506

3.91506

2.64506

1 .30506

 

122

125.92

8.83506

8.66506

8.30506

7.74506

6.99506

6.09506

5.04506

3.87506

2.61506

1.29506

 

123

126.95

8.73506

8.57506

8.21506

7.65506

6.92506

6.02506

4.98506

3.83506

2.58506

1.27506

 

124

127.98

8.64506

8.48 506

8.12506

7.57506

6.84506

5.96506

4.93506

3.79506

2.55506

1.26506

 

125

129.01

8.55506

8.39506

8.03506

7.49506

6.77506

5.89506

4.88506

3.75506

2 .53506

1.24506

 

126

130.05

8.46506

8.30506

7.94506

7.41506

6.70506

5.83506

4.82506

3.70506

2.50506

1 .23506

 

127

131.08

8.37506

8.21506

7.86506

7.33506

6.62506

5.77506

4.77506

3.66506

2.47506

1 .22506

 

128

132.11

8.28506

8.12506

7.78506

7.25506

6.55506

5.70506

4.72506

3.62506

2.44506

1 .20506

 

129

133.14

8.19506

8.03506

7.69506

7.17506

6.48506

5.64506

4.67506

3.58506

2.42506

1.19506

 

130

134.17

8.10506

7.95506

7.61506

7.09506

6.41506

5.58506

4.62 506

3.55506

2.39506

1.18506

 

131

135.21

8.02506

7.86506

7.53506

7.02506

6.34506

5.52506

4.57506

3.51506

2.36506

1.16506

 

132

136.24

7.93506

7.78506

7.45506

6.94506

6.27506

5.46506

4.52506

3.47506

2.34506

1.15506

 

133

137.27

7.85506

7.70506

7.37506

6.87506

6.21506

5.40506

4.47506

3.43506

2.31506

1.14506

 

134

138.30

7.76506

7.61506

7.29506

6.79506

6.14506

5.34506

4.42506

3.39506

2.29506

1.13506

 

135

139.33

7.68506

7.53506

7.21506

6.72506

6.07506

5.28506

4.37506

3.36506

2.26506

1.11506

 

136

140.37

7.60506

7.45506

7.13506

6.65506

6.01506

5.23506

4.32506

3.32506

2.24506

1.10506

 

137

141.40

7.52506

7.37506

7.06506

6.58506

5.94506

5.17506

4.28506

3.28506

2.21506

1 .09506

 

138

142.43

7.44506

7.29506

6.98506

6.51506

5.88506

5.12506

4.23506

3.25506

2.19506

1 .08506

 

139

143.46

7.36506

7.22506

6.91506

6.44506

5.82506

5.06506

4.19506

3.21506

2.16506

1 .06506

 

140

144.49

7.28506

7.14506

6.83506

6.37506

5.75506

5.01506

4.14506

3.18506

2.14506

1.05506

 

141

145.53

7.20506

7.06506

6.76506

6.30506

5.69506

4.95506

4.10506

3.14506

2.12506

1.04506

 

142

146.56

7.13506

6.99506

6.69506

6.23506

5.63506

4.90506

4.05506

3.11506

2.09506

1.03506

 

143

147.59

7.05506

6.91506

6.61506

6.16506

5.57506

4.85506

4.01506

3.08506

2.07506

1 .02506

 

144

148.62

6.98506

6.84506

6.54506

6.10506

5.51506

4.79506

3.96506

3.04506

2.05506

1.01506

 

145

149.66

6.90506

6.77506

6.47506

6.03506

5.45506

4.74506

3.92506

3.01506

2.03506

9.96507

 

 

146

 

150.69

 

6.83506

 

6.69506

 

6.41506

 

5.97506

 

5.39506

89

 

4.69506

 

3.88506

 

2.98506

 

2.00506

 

9.85507

 

00000000000000000000000000000000000000000000000000000000

 

Table 7.9 Continued

 

147

151.72

6.76506

6.62506

6.34506

5.90506

5.33506

4.64506

3.84506

2.94506

1.98506

9.74507

 

148

152.75

6.68506

6.55506

6.27506

5.84506

5.28506

4.59506

3.80506

2.91506

1 .96506

9.64507

 

149

153.78

6.61506

6.48506

6.20506

5.78506

5.22506

4.54506

3.76506

2.88506

1.94506

9.53507

 

150

154.82

6.54506

6.41506

6.14506

5.72506

5.16506

4.49506

3.71506

2.85506

1 .92506

9.43507

 

151

155.85

6.47506

6.34506

6.07506

5.65506

5.11506

4.44 506

3.67506

2.82 506

1 .90506

9.32507

 

152

156.88

6.40506

6.28506

6.01506

5.59506

5.05506

4.40506

3.63506

2.79506

1 .88506

9.22507

 

153

157.91

6.34506

6.21506

5.94506

5.53506

5.00506

4.35506

3.60506

2.76506

1 .86506

9.12507

 

154

158.94

6.27506

6.14506

5.88506

5.48506

4.95506

4.30506

3.56506

2.73506

1 .84506

9.02507

 

155

159.98

6.20506

6.08506

5.82506

5.42506

4.89506

4.26506

3.52506

2.70506

1 .82 506

8.92507

 

156

161.01

6.14506

6.01506

5.75506

5.36506

4.84506

4.21506

3.48506

2.67506

1 .80506

8.82507

 

157

162.04

6.07506

5.95 506

5.69506

5.30506

4.79506

4.16506

3.44506

2.64506

1 .78506

8.73507

 

158

163.07

6.01506

5.89506

5.63506

5.24506

4.74 506

4.12506

3.41506

2.61506

1 .76506

8.63507

 

159

164.10

5.94506

5.82506

5.57506

5.19506

4.69506

4.08506

3.37506

2.59506

1 .74 506

8.54507

 

160

165.14

5.88506

5.76506

5.51506

5.13506

4.64506

4.03506

3.33506

2.56506

1 .72506

8.44507

 

161

166.17

5.82506

5.70506

5.45506

5.08506

4.59506

3.99506

3.30506

2.53506

1 .70506

8.35507

 

162

167.20

5.76506

5.64506

5.39506

5.02 506

4.54506

3.95506

3.26506

2.50506

1 .68 506

8.26507

 

163

168.23

5.70506

5.58506

5.34506

4.97506

4.49506

3.90506

3.23506

2.48506

1 .67506

8.17507

 

164

169.27

5.64506

5.52506

5.28506

4.92506

4.44506

3.86506

3.19506

2.45506

1 .65506

8.08507

 

165

170.30

5.58506

5.46506

5.22506

4.87506

4.39506

3.82506

3.16506

2.42506

1 .63506

7.99507

 

166

171.33

5.52506

5.40506

5.17506

4.81506

4.35506

3.78506

3.12506

2.40506

1.61506

7.90507

 

167

172.36

5.46506

5.35506

5.11506

4.76506

4.30506

3.74 506

3.09506

2.37 506

1 .59506

7.82507

 

168

173.39

5.40506

5.29506

5.06506

4.71506

4.25506

3.70506

3.06506

2.35506

1 .58506

7.73507

 

169

174.43

5.34506

5.23506

5.01506

4.66506

4.21506

3.66506

3.02506

2.32506

1 .56506

7.65507

 

170

175.46

5.29506

5.18506

4.95506

4.61506

4.16506

3.62506

2.99506

2.29506

1 .54506

7.57507

 

171

176.49

5.23506

5.12506

4.90506

4.56506

4.12506

3.58506

2.96506

2.27506

1 .53506

7.48507

 

172

177.52

5.17506

5.07506

4.85506

4.51506

4.08506

3.54506

2.93506

2.25506

1.51506

7.40507

 

173

178.55

5.12506

5.02506

4.80506

4.47506

4.03506

3.51506

2.90506

2.22506

1 .49506

7.32507

 

174

179.59

5.07506

4.96506

4.74 506

4.42506

3.99506

3.47506

2.87506

2.20506

1 .48506

7.24507

 

175

180.62

5.01506

4.91506

4.69506

4.37506

3.95506

3.43506

2.84506

2.17506

1 .46506

7.16507

 

176

181.65

4.96506

4.86506

4.64506

4.32506

3.90506

3.39506

2.81506

2.15506

1 .45506

7.09507

 

177

182.68

4.91506

4.81506

4.60506

4.28506

3.86506

3.36506

2.78506

2.13506

1 .43506

7.01507

 

178

183.71

4.85506

4.76506

4.55506

4.23506

3.82 506

3.32506

2.75506

2.11506

1 .42506

6.93507

 

179

184.75

4.80506

4.70506

4.50506

4.19506

3.78506

3.29506

2.72506

2.08506

1 .40506

6.86507

 

180

185.78

4.75506

4.65506

4.45506

4.14506

3.74506

3.25506

2.69506

2.06506

1 .39506

6.78507

 

181

186.81

4.70506

4.61506

4.40506

4.10506

3.70506

3.22506

2.66506

2.04506

1 .37506

6.71507

 

182

187.84

4.65506

4.56506

4.36506

4.06506

3.66506

3.18506

2.63506

2.02506

1 .36506

6.64507

 

183

188.88

4.60506

4.51506

4.31506

4.01506

3.62506

3.15506

2.60506

1 .99506

1 .34506

6.56507

 

184

189.91

4.55506

4.46506

4.26506

3.97506

3.58506

3.11506

2.57506

1 .97506

1 .33506

6.49507

 

185

190.94

4.51506

4.41506

4.22506

3.93506

3.55506

3.08506

2.55506

1 .95506

1.31506

6.42507

 

186

191.97

4.46506

4.37506

4.17506

3.89506

3.51506

3.05506

2.52506

1 .93506

1 .30506

6.35507

 

187

193.00

4.41506

4.32506

4.13506

3.84506

3.47506

3.02506

2.49506

1.91506

1 .28506

6.28507

 

188

194.04

4.37506

4.27506

4.09506

3.80506

3.43506

2.98506

2.47506

1 .89506

1 .27506

6.22507

 

189

195.07

4.32506

4.23506

4.04506

3.76506

3.40506

2.95506

2.44506

1 .87506

1 .26506

6.15507

 

190

196.10

4.27506

4.18506

4.00506

3.72506

3.36506

2.92506

2.41506

1 .85506

1 .24 506

6.08507

 

191

197.13

4.23506

4.14506

3.96506

3.68506

3.32506

2.89506

2.39506

1 .83506

1 .23506

6.02507

 

192

198.16

4.18506

4.10506

3.92506

3.64506

3.29506

2.86506

2.36506

1.81506

1 .22506

5.95507

 

193

199.20

4.14506

4.05506

3.87506

3.61506

3.25506

2.83506

2.34506

1 .79506

1 .20506

5.89507

 

194

200.23

4.10506

4.01506

3.83506

3.57506

3.22506

2.80506

2.31506

1 .77506

1.19506

5.82507

 

195

201.26

4.05506

3.97506

3.79506

3.53506

3.19506

2. 77506

2.29506

1 .75506

1.18506

5.76507

 

196

202.29

4.01506

3.93506

3.75506

3.49506

3.15506

2.74 506

2.26506

1 .73506

1.17506

5.70507

 

197

203.32

3.97506

3.88506

3.71506

3.45506

3.12506

2.71506

2.24 506

1 .72506

1.15506

5.64507

 

198

204.36

3.93506

3.84506

3.67506

3.42506

3.09506

2.68506

2.21506

1.70506

1.14506

5.58507

 

199

205.39

3.88506

3.80506

3.63506

3.38 506

3.05506

2.65506

2.19506

1.68506

1.13506

5.52507

 

200

206.42

3.84506

3.76506

3.60506

3.35506

3.02506

2.62506

2.17506

1.66506

1.12506

5.46507

 

201

207.45

3.80506

3.72506

3.56506

3.31506

2.99506

2.60506

2.14506

1 .64 506

1.10506

5.40507

 

 

202

 

208.49

 

3.76506

 

3.68506

 

3.52506

 

3.28506

 

2.96506

90

 

2.57506

 

2.12506

 

1.63506

 

1 .09506

 

5.34507

 

00000000000000000000000000000000000000000000000000000000

 

 

Table 7.9 Continued

 

203

209.52

3.72506

3.64506

3.48506

3.24506

2.92506

2.54506

2.10506

1.61506

1 .08506

5.28507

 

204

210.55

3.68506

3.61506

3.45506

3.21506

2.89506

2.51506

2.08506

1 .59506

1 .07506

5.23507

 

205

211.58

3.64506

3.57506

3.41506

3.17506

2.86506

2.49506

2.05506

1 .57506

1 .06506

5.17507

 

206

212.61

3.61506

3.53506

3.37506

3.14506

2.83506

2.46506

2.03506

1 .56506

1 .05506

5.11507

 

207

213.65

3.57506

3.49506

3.34506

3.11506

2.80506

2.44506

2.01506

1.54506

1 .04506

5.06507

 

208

214.68

3.53506

3.46506

3.30506

3.07506

2.77506

2.41506

1 .99506

1.53506

1 .02 506

5.01507

 

209

215.71

3.49506

3.42506

3.27506

3.04506

2.74506

2.38506

1 .97506

1.51506

1.01506

4.95507

 

210

216.74

3.46506

3.38506

3.23506

3.01506

2.71506

2.36506

1 .95506

1 .49506

1 .00506

4.90507

 

211

217.77

3.42506

3.35506

3.20506

2.98506

2.69506

2.33506

1 .93506

1 .48506

9.92507

4.85507

 

212

218.81

3.38506

3.31506

3.16506

2.94506

2.66506

2.31506

1.91506

1 .46506

9.82507

4.79507

 

213

219.84

3.35506

3.28506

3.13506

2.91506

2.63506

2.28506

1 .89506

1 .45506

9.71507

4.74507

 

214

220.87

3.31506

3.24 506

3.10506

2.88506

2.60506

2.26506

1 .87 506

1 .43506

9.61507

4.69507

 

215

221.90

3.28506

3.21506

3.07506

2.85506

2.57506

2.24506

1 .85506

1 .41506

9.51507

4.64507

 

216

222.93

3.24506

3.17506

3.03506

2.82506

2.55506

2.21506

1 .83506

1 .40506

9.40507

4.59507

 

217

223.97

3.21506

3.14506

3.00506

2.79506

2.52506

2.19506

1.81506

1 .38506

9.30507

4.54507

 

218

225.00

3.18506

3.11506

2.97506

2.76506

2.49506

2.17506

1 .79506

1 .37506

9.20507

4.49507

 

219

226.03

3.14506

3.07506

2 .94 506

2.73506

2.47506

2.14506

1 .77506

1 .36506

9.11507

4.45507

 

220

227.06

3.11506

3.04506

2.91506

2.70506

2.44506

2.12506

1 .75506

1 .34506

9.01507

4.40507

 

221

228.10

3.08506

3.01506

2.88506

2.68506

2.41506

2.10506

1.73506

1 .33506

8.91507

4.35507

 

222

229.13

3.04506

2.98506

2.85506

2.65506

2.39506

2.07506

1.71506

1.31506

8.82507

4.30507

 

223

230.16

3.01506

2.95506

2.82506

2.62506

2.36506

2.05506

1 .70506

1 .30506

8.72507

4.26507

 

224

231.19

2.98506

2.92506

2.79506

2.59506

2.34506

2.03506

1 .68506

1 .29506

8.63507

4.21507

 

225

232.22

2.95506

2.89506

2.76506

2.56506

2.31506

2.01506

1 .66506

1 .27506

8.54507

4.17507

 

226

233.26

2.92506

2.85506

2.73506

2.54506

2.29506

1 .99506

1.64506

1.26506

8.45507

4.12507

 

227

234.29

2.89506

2.82506

2.70506

2.51506

2.26506

1 .97506

1 .62506

1 .24506

8.36507

4.08507

 

228

235.32

2.86506

2.79506

2.67506

2.48506

2 .24506

1 .95506

1.61506

1 .23506

8.27507

4.04507

 

229

236.35

2.83506

2.77506

2.64506

2.46506

2.22506

1 .93506

1 .59506

1 .22506

8.18507

3.99507

 

230

237.38

2.80506

2.74506

2.61506

2.43506

2.19506

1.91506

1 .57506

1.21506

8.10507

3.95507

 

231

238.42

2.77506

2.71506

2.59506

2.41 506

2.17506

1 .89506

1 .56506

1 .19506

8.01507

3.91507

 

232

239.45

2.74506

2.68506

2.56506

2.38506

2.15506

1 .87506

1.54506

1.18506

7.92507

3.87507

 

233

240.48

2.71506

2.65506

2.53506

2.36506

2.12506

1 .85506

1 .52506

1.17506

7.84507

3.82507

 

234

241.51

2.68506

2.62506

2.51506

2.33506

2.10506

1 .83506

1.51506

1.16506

7.76507

3.78507

 

235

242.54

2.65506

2.59506

2.48506

2.31506

2.08506

1.81506

1 .49506

1.14506

7.67507

3.74507

 

236

243.58

2 .62506

2.57506

2.45506

2.28506

2.06506

1 .79506

1 .48506

1.13506

7.59507

3.70507

 

237

244.61

2.60506

2.54506

2.43506

2.26506

2.04506

1 .77506

1 .46506

1.12506

7.51507

3.66507

 

238

245.64

2.57506

2.51506

2.40506

2.23506

2.01506

1 .75506

1 .44506

1.11506

7.43507

3.62507

 

239

246.67

2.54506

2.49506

2.38506

2.21506

1 .99506

1 .73506

1.43506

1.10506

7.35507

3.59507

 

240

247.71

2.52506

2.46506

2.35506

2.19506

1 .97506

1.71506

1.41506

1 .08506

7.27507

3.55507

 

241

248.74

2.49506

2.44506

2.33506

2.16506

1 .95506

1 .69506

1 .40506

1 .07506

7.20507

3.51507

 

242

249.77

2.46506

2.41506

2.30506

2.14506

1 .93506

1 .68506

1 .38506

1 .06506

7.12507

3.47507

 

243

250.80

2.44506

2.38506

2.28506

2.12506

1.91506

1 .66506

1 .37506

1 .05506

7.05507

3.44507

 

244

251.83

2.41 506

2.36506

2.25506

2.10506

1 .89506

1 .64506

1 .36506

1 .04506

6.97507

3.40507

 

245

252.87

2 .39506

2.33506

2.23506

2 .07506

1 .87506

1 .62506

1.34506

1 .03506

6.90507

3.36507

 

246

253.90

2.36506

2.31506

2.21506

2.05506

1 .85506

1.61506

1 .33506

1 .02506

6.82507

3.33507

 

247

254.93

2.34506

2.29506

2.18506

2.03506

1 .83506

1 .59506

1.31506

1.01506

6.75507

3.29507

 

248

255.96

2.31506

2.26506

2.16506

2.01506

1.81506

1 .57506

1.30506

9.95507

6.68507

3.26507

 

249

256.99

2.29506

2.24506

2.14506

1 .99506

1 .79506

1 .56506

1 .29506

9.84507

6.61507

3.22507

 

250

258.03

2.26506

2.21506

2.11506

1 .97506

1 .77506

1 .54506

1 .27506

9.74507

6.54507

3.19507

 

251

259.06

2.24506

2.19506

2.09506

1 .95506

1 .75506

1 .52 506

1 .26506

9.64507

6.47507

3.15507

 

252

260.09

2.22506

2.17506

2.07506

1 .93506

1 .74 506

1.51506

1 .25506

9.53507

6.40507

3.12507

 

253

261.12

2.19506

2.14506

2.05506

1 .90506

1.72506

1.49506

1 .23506

9.43507

6.33507

3.09507

 

 

254

 

262.15

 

2.17506

 

2.12506

 

2.03506

 

1 .88506

 

1.70506

 

1 .48506

 

1 .22506

 

9.33507

 

6.27507

 

3.05507

 

0000000000000000000000000000000000000000000000000000

 

91

 

Appendix 7.11: Model Output for Salmonella Velocity

Excel formula used to calculate the Salmonella concentration at t = 1.03 s and x =

0.005 m. (i.e. cell D5):

D5 =B*(1/(MASTER!dsal/(2*'Effective Gap

Radius' ! D5))"O. 128)*((MASTER!Lsal/MASTER!dsal)"0. 128*‘Marinade

Velocity’!D5-SQRT(2*MASTER! g*(2*'Effective Gap

Radius'!D5)*(MASTER!rdsal-1)*(MASTER!Lsal/MASTER!dsal))*(1-

(MASTER!dsa1/(2*'Effective Gap Radius'!D5))"2))*((2*'Effective Gap

Radius'!D5)/dt)"n

Table 7.10: Predicted Salmonella velocity profile generated by Excel, given the input
variables listed in Table 4.1.

 

Salmonella Velocity

 

n

1

9

'4

4

‘3

R

7

R

0

1n

 

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

 

1 .5503

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0500

0.0500

0.0E+00

0.0E+00

0.0E+00

0.0E+00

 

7.4504

7.7504

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

 

2.75

7.4504

3.8504

3.9504

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

 

4.13

5.6504

5.6504

1 .9504

2.0504

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

 

5.50

5.6504

3.9504

3.8504

9.8505

9.9505

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

 

6.88

4.7504

4.7504

2.5504

2.4504

4.9505

4.9505

0.0E+00

0.0E+00

0.0E+00

0.0E+00

0.0E+00

 

8.26

4.7504

3.6504

3.5504

1 .5504

1 .4504

2.5505

2.5505

0.0E+00

0.0E+00

0.0E+00

0.0E+00

 

9.63

4.1504

4.1504

2.6504

2.5504

8.7505

8.5505

1.2505

1.2505

0.0E+00

0.0E+00

0.0E+00

 

ONGUIJSQN-ioh-

11.01

4.1504

3.4504

3.3504

1.8504

1 .7504

5.0505

4.9505

6.2506

6.2506

0.0E+00

0.0E+00

 

(D

12.39

3.7504

3.7504

2.6504

2.5504

1.1504

1.1504

2.8505

2.7505

3.1 506

3.1506

0.0E+00

 

.3
o

13.76

3.7504

3.2504

3.0504

1 .9504

1.8504

7.1505

6.7505

1 .6505

1.5505

1 .5506

0.0E+00

 

A
A

15.14

3.4504

3.4504

2.6504

2.4504

1.3504

1.2504

4.3505

4.1505

8.6506

7.6506

0.0E+00

 

.3
N

16.51

3.4504

3.0504

2.9504

1.9504

1.8504

8.7505

8.1 505

2.6505

2.4505

4.3506

0.0E+00

 

.3
O.)

17.89

3.2504

3.1504

2.5504

2.3504

1.4504

1.3504

5.7505

5.3505

1 .5505

1.2505

0.0500

 

.3
#

19.27

3.1504

2.9504

2.7504

2.0504

1 .8504

1.0504

9.1505

3.6505

3.3505

7.6506

0.0E+00

 

.3
01

20.64

3.0504

2.9504

2.4504

2.3504

1.5504

1.4504

6.8505

6.2505

2.2505

1.6505

0.0E+00

 

.3
a:

22.02

2.9504

2.7504

2.6504

2.0504

1.8504

1.1504

9.9505

4.5505

3.9505

1.1505

0.0500

 

.3
N

23.39

2.8504

2.8504

2.4504

2.2504

1.5504

1 .4504

7.7505

6.9505

2.8505

2.0505

0.0E+00

 

.3
m

24.77

2.8504

2.6504

2.5504

2.0504

1.8504

1.2504

1 .0504

5.3505

4.4505

1 .4505

0.0E+00

 

.3
(O

26.15

2.7504

2.6504

2.3504

2.1504

1.6504

1.4504

8.4505

7.4505

3.3505

2.2505

0.0E+00

 

N
0

27.52

2.6504

2.5504

2.4504

1.9504

1.8504

1.2504

1.1504

5.9505

4.8505

1.7505

0.0E+00

 

N
.3

28.90

2.6504

2.5504

2.2504

2.1504

1.6504

1.4504

9.0505

7.8505

3.8505

2.4505

0.0E+00

 

N
N

30.28

2.5504

2.4504

2.3504

1 .9504

1.7504

1.2504

1.1504

6.4505

5.1505

1.9505

0.0E+00

 

N
(A

31.65

2.4504

2.4504

2.2504

2.0504

1.6504

1 .4504

9.4505

8.0505

4.2505

2.5505

0.05400

 

N
#

33.03

2.4504

2.3504

2.2504

1.9504

1.7504

1.3504

1.1504

6.8505

5.3505

2.1505

0.0500

 

 

N
U!

 

34.40

 

2.3504

 

2.3504

 

2.1504

 

2.0504

 

1 .6504

92

 

1.4504

 

9.8505

 

8.2505

 

4.5505

 

2.6505

 

0.0E+00

 

Table 7.10 Continued

 

26

35.78

2.3504

2.2504

2.1504

1.8504

1.7504

1.3504

1.1504

7.1505

5.4505

2.2505

0.0E+00

 

27

37.16

2.3504

2.2504

2.0504

1.9504

1.6504

1 .4504

1.0504

8.2505

4.7505

2.7505

0.0E+00

 

28

38.53

2.2504

2.2504

2.1504

1.8504

1.6504

1.3504

1.1504

7.3505

5.5505

2.3505

0.0E+00

 

29

39.91

2.2504

2.1504

2.0504

1.9504

1.5504

1.4504

1.0504

8.2505

4.8505

2.7505

0.0E+00

 

30

41.28

2.2504

2.1504

2.0504

1.8504

1 .6504

1.3504

1.1504

7.5505

5.5505

2.4505

0.0E+00

 

31

42.66

2.1 504

2.1504

1.9504

1 .8504

1.5504

1.4504

1.0504

8.2505

5.0505

2.7505

0.0E+00

 

32

44.04

2.1504

2.0504

1.9504

1.7504

1.6504

1.3504

1.1504

7.6505

5.5505

2.5505

0.0E+00

 

33

45.41

2.1504

2.0504

1.9504

1 .8504

1.5504

1.3504

1.0504

8.2505

5.0505

2.7505

0.0E+00

 

34

46.79

2.0504

2.0504

1.9504

1.7504

1.5504

1.3504

1.1504

7.6505

5.4505

2.5505

0.0E+00

 

35

48.16

2.0504

2.0504

1.8504

1.7504

1.5504

1.3504

1.0504

8.1505

5.1505

2.7505

0.0E+00

 

36

49.54

2.0504

1.9504

1.8504

1.7504

1.5504

1.3504

1.1504

7.6505

5.4505

2.5505

0.0E+00

 

37

50.92

1 .9504

1.9504

1.8504

1.7504

1.5504

1.3504

1.0504

8.0505

5.1505

2.7505

0.0E+00

 

38

52.29

1.9504

1.9504

1.8504

1.6504

1.5504

1.2504

1.0504

7.6505

5.3505

2.5505

0.0E+00

 

39

53.67

1 .9504

1.9504

1.8504

1.6504

1.4504

1 .3504

1.0504

7.9505

5.1 505

2.7505

0.0E+00

 

40

55.05

1 .9504

1 .8504

1.7504

1.6504

1.4504

1.2504

1 .0504

7.6505

5.3505

2.5505

0.0E+00

 

41

56.42

1.8504

1.8504

1.7504

1.6504

1.4504

1.2504

9.9505

7.8505

5.0505

2.6505

0.0E+00

 

42

57.80

1.8504

1.8504

1.7504

1.6504

1 .4504

1.2504

1.0504

7.5505

5.2505

2.5505

0.0E+00

 

43

59.17

1.8504

1.8504

1.7504

1.6504

1.4504

1.2504

9.8505

7.6505

5.0505

2.6505

0.0E+00

 

44

60.55

1.8504

1.7504

1.7504

1.5504

1 .4 504

1 .2504

9.9505

7.4505

5.1 505

2.5505

0.0E+00

 

45

61.93

1.75-04

1.7504

1.6504

1.5504

1.4504

1.2504

9.6505

7.5505

4.9505

2.5505

0.0E+00

 

46

63.30

1.7504

1.7504

1.6504

1.5504

1.4504

1.2504

9.7505

7.3505

5.0505

2.5505

0.0E+00

 

47

64.68

1.7504

1.7504

1 .6504

1.5504

1.3504

1.2504

9.5505

7.4505

4.9505

2.5505

0.0E+00

 

48

66.05

1.7504

1.7504

1.6504

1 .5504

1.3504

1.1504

9.5505

7.2505

4.9505

2.4505

0.0E+00

 

49

67.43

1.7504

1.6504

1.6504

1.5504

1.3504

1.1504

9.3505

7.2505

4.8505

2.4505

0.0E+00

 

50

68.81

1.6504

1.6504

1.5504

1.4504

1.3504

1.1504

9.3505

7.1505

4.8505

2.4505

0.0E+00

 

51

70.18

1.6504

1.6504

1.5504

1.4504

1.3504

1.1504

9.1505

7.1505

4.7505

2.4505

0.0E+00

 

52

71.56

1.6504

1.6504

1.5504

1.4504

1.3504

1.1504

9.1505

6.9505

4.7505

2.3505

0.0E+00

 

53

72.94

1.6504

1.6504

1.5504

1 .4504

1.3504

1.1504

9.0505

6.9505

4.6505

2.3505

0.0E+00

 

54

74.31

1.6504

1.5504

1.5504

1.4504

1.2504

1.1504

8.9505

6.8505

4.6505

2.3505

0.0E+00

 

55

75.69

1.6504

1.5504

1.5504

1.4504

1.2504

1.1504

8.8505

6.8505

4.6505

2.3505

0.0E+00

 

56

77.06

1.5504

1.5504

1.4504

1.3504

1.2504

1.1504

8.7505

6.7505

4.5505

2.3505

0.0E+00

 

57

78.44

1.5504

1.5504

1.4504

1.3504

1.2504

1.0504

8.6505

6.6505

4.5505

2.2505

0.0E+00

 

58

79.82

1.5504

1 .5504

1.4504

1.3504

1.2504

1.0504

8.6505

6.5505

4.4505

2.2505

0.0E+00

 

59

81.19

1.5504

1.5504

1 .4504

1.3504

1.2504

1.0504

8.4505

6.5505

4.4505

2.2505

0.0E+00

 

60

82.57

1.5504

1.4504

1 .4504

1.3504

1.2504

1.0504

8.4505

6.4505

4.4505

2.2505

0.0E+00

 

61

83.94

1.5504

1.4504

1 .4504

1.3504

1.1504

1.0504

8.3505

6.4505

4.3505

2.2505

0.0E+00

 

62

85.32

1 .4504

1 .4504

1 .4504

1.3504

1.1504

9.9505

8.2505

6.3505

4.3505

2.1 505

0.0E+00

 

63

86.70

1.4504

1.4504

1.3504

1.2504

1.1504

9.8505

8.1505

6.2505

4.2505

2.1 505

0.0E+00

 

64

88.07

1 .4504

1.4504

1.3504

1.2504

1.1504

9.7505

8.0505

6.2505

4.2505

2.1505

0.0E+00

 

65

89.45

1.4504

1.4504

1.3504

1.2504

1.1504

9.6505

7.9505

6.1505

4.1505

2.1505

0.0E+00

 

66

90.83

1.4504

1.3504

1.3504

1.2504

1.1504

9.5505

7.8505

6.0505

4.1505

2.0505

0.0E+00

 

67

92.20

1 .4504

1.3504

1.3504

1.2504

1.1504

9.4505

7.8505

6.0505

4.0505

2.0505

0.0E+00

 

68

93.58

1.3504

1.3504

1.3504

1.2504

1.1504

9.3505

7.7505

5.9505

4.0505

2.0505

0.0E+00

 

69

94.95

1.3504

1.3504

1.2504

1.2504

1.1504

9.2505

7.6505

5.8505

3.9505

2.0505

0.0E+00

 

70

96.33

1 .3504

1.3504

1.2504

1.2504

1.0504

9.1505

7.5505

5.8505

3.9505

1 .9505

0.0E+00

 

71

97.71

1.3504

1.3504

1.2504

1.1504

1.0504

9.0505

7.4505

5.7505

3.9505

1.9505

0.0E+00

 

72

99.08

1.3504

1 .3504

1.2504

1.1504

1.0504

8.9505

7.3505

5.6505

3.8505

1 .9505

0.0E+00

 

73

100.46

1.3504

1.2504

1.2504

1.1504

1.0504

8.8505

7.3505

5.6505

3.8505

1 .9505

0.0E+00

 

74

101.83

1.3504

1.2504

1.2504

1.1504

1.0504

8.7505

7.2505

5.5505

3.7505

1 .9505

0.0E+00

 

75

103.21

1.2504

1.2504

1.2504

1.1504

9.9505

8.6505

7.1505

5.5505

3.7505

1.8505

0.0E+00

 

76

104.59

1.2504

1.2504

1.2504

1.1504

9.8505

8.5505

7.0505

5.4505

3.7505

1.8505

0.0E+00

 

77

105.96

1.2504

1.2504

1.1504

1.1504

9.6505

8.4505

7.0505

5.3505

3.6505

1.8505

0.0E+00

 

78

107.34

1.2504

1.2504

1.1504

1.1504

9.5505

8.3505

6.9505

5.3505

3.6505

1.8505

0.0E+00

 

79

108.72

1.2504

1.2504

1.1504

1.0504

9.4505

8.2505

6.8505

5.2505

3.5505

1.8505

0.0E+00

 

80

110.09

1.2504

1.2504

1.1504

1.0504

9.3505

8.1 505

6.7505

5.2505

3.5505

1 .7505

0.0E+00

 

81

111.47

1.2504

1.1504

1.1504

1.0504

9.2505

8.0505

6.7505

5.1505

3.5505

1 .7505

0.0E+00

 

82

112.84

1.2504

1.1504

1.1504

1.0504

9.1505

7.9505

6.6505

5.1505

3.4505

1.7505

0.0E+00

 

 

83

 

114.22

 

1.1504

 

1.1504

 

1.1504

 

1.0504

 

9.0505

93

 

7.9505

 

6.5505

 

5.0505

 

3.4505

 

1.7505

 

0.05-00

 

Table 7.10 Continued

 

84

115.60

1.1504

1.1504

1.1504

9.9505

8.9505

7.8505

6.4505

4.9505

3.3505

1.7505

0.0E+00

 

85

116.97

1.1504

1.1504

1.0504

9.8505

8.8505

7.7505

6.4505

4.9505

3.3505

1.6505

0.0E+00

 

86

118.35

1.1504

1.1504

1.0504

9.7505

8.7505

7.6505

6.3505

4.8505

3.3505

1 .6505

0.0E+00

 

87

119.72

1.1504

1.1504

1.0504

9.6505

8.6505

7.5505

6.2505

4.8505

3.2505

1.6505

0.05-+00

 

88

121.10

1.1504

1.1504

1.0504

9.5505

8.5505

7.4505

6.2505

4.7505

3.2505

1.6505

0.0E+00

 

89

122.48

1.1504

1 .0504

1.0504

9.3505

8.5505

7.4505

6.1 505

4.7505

3.2505

1 .6505

0.0E+00

 

90

123.85

1.1504

1.0504

9.9505

9.2505

8.4505

7.3505

6.0505

4.6505

3.1505

1.6505

0.0E+00

 

91

125.23

1.0504

1.0504

9.8505

9.1505

8.3505

7.2505

6.0505

4.6505

3.1505

1.5505

0.0E+00

 

92

126.60

1.0504

1.0504

9.7505

9.0505

8.2505

7.1505

5.9505

4.5505

3.1505

1.5505

0.0E+00

 

93

127.98

1.0504

1 .0504

9.6505

8.9505

8.1505

7.0505

5.8505

4.5505

3.0505

1.5505

0.0E+00

 

94

129.36

1 .0504

9.9505

9.5505

8.9505

8.0505

7.0505

5.8505

4.4505

3.0505

1.5505

0.0E+00

 

95

130.73

1.0504

9.8505

9.4505

8.8505

7.9505

6.9505

5.7505

4.4505

3.0505

1.5505

0.0E+00

 

96

132.11

9.9505

9.7505

9.3505

8.7505

7.8505

6.8505

5.6505

4.3505

2.9505

1 .4505

0.0E+00

 

97

133.49

9.8505

9.6505

9.2505

8.6505

7.7505

6.7505

5.6505

4.3505

2.9505

1.4505

0.0E+00

 

98

134.86

9.7505

9.5505

9.1505

8.5505

7.7505

6.7505

5.5505

4.2505

2.9505

1.4505

0.0E+00

 

99

136.24

9.6505

9.4505

9.0505

8.4505

7.6505

6.6505

5.5505

4.2505

2.8505

1 .4505

0.0E+00

 

100

137.61

9.5505

9.3505

8.9505

8.3505

7.5505

6.5505

5.4505

4.1505

2.8505

1 .4505

0.0E+00

 

101

138.99

9.4505

9.2505

8.8505

8.2505

7.4505

6.5505

5.3505

4.1505

2.8505

1 .4505

0.0E+00

 

102

140.37

9.3505

9.1505

8.7505

8.1 505

7.3505

6.4505

5.3505

4.1505

2.7505

1 .4 505

0.0500

 

103

141.74

9.2505

9.0505

8.6505

8.0505

7.3505

6.3505

5.2505

4.0505

2.7505

1 .3505

0.0E+00

 

104

143.12

9.1505

8.9505

8.5505

7.9505

7.2505

6.2505

5.2505

4.0505

2.7505

1.3505

0.0E+00

 

105

144.49

9.0505

8.8505

8.4505

7.9505

7.1505

6.2505

5.1505

3.9505

2.6505

1 .3505

0.0E+00

 

106

145.87

8.9505

8.7505

8.3505

7.8505

7.0505

6.1505

5.1505

3.9505

2.6505

1 .3505

0.0E+00

 

107

147.25

8.8505

8.6505

8.2505

7.7505

6.9505

6.0505

5.0505

3.8505

2.6505

1.3505

0.0E+00

 

108

148.62

8.7505

8.5505

8.2505

7.6505

6.9505

6.0505

4.9505

3.8505

2.6505

1.3505

0.0E+00

 

109

150.00

8.6505

8.4505

8.1505

7.5505

6.8505

5.9505

4.9505

3.8505

2.5505

1.3505

0.0E+00

 

110

151.38

8.5505

8.3505

8.0505

7.4505

6.7505

5.8505

4.8505

3.7505

2.5505

1.2505

0.0E+00

 

111

152.75

8.4505

8.2505

7.9505

7.4505

6.6505

5.8505

4.8505

3.7505

2.5505

1.2505

0.0E+00

 

112

154.13

8.3505

8.2505

7.8505

7.3505

6.6505

5.7505

4.7505

3.6505

2.5505

1.2505

0.0E+00

 

113

155.50

8.2505

8.1505

7.7505

7.2505

6.5505

5.7505

4.7505

3.6505

2.4505

1.2505

0.0E+00

 

114

156.88

8.1 505

8.0505

7.6505

7.1505

6.4505

5.6505

4.6505

3.6505

2.4505

1.2505

0.0E+00

 

115

158.26

8.1505

7.9505

7.6505

7.0505

6.4505

5.5505

4.6505

3.5505

2.4505

1.2505

0.0E+00

 

116

159.63

8.0505

7.8505

7.5505

7.0505

6.3505

5.5505

4.5505

3.5505

2.3505

1.2505

0.0E+00

 

117

161.01

7.9505

7.7505

7.4505

6.9505

6.2505

5.4505

4.5505

3.4505

2.3505

1.1505

0.0E+00

 

118

162.38

7.8505

7.6505

7.3505

6.8505

6.2505

5.4505

4.4505

3.4505

2.3505

1.1505

0.0E+00

 

119

163.76

7.7505

7.6505

7.2505

6.7505

6.1505

5.3505

4.4505

3.4505

2.3505

1.1505

0.0E+00

 

120

165.14

7.6505

7.5505

7.2505

6.7505

6.0505

5.2505

4.3505

3.3505

2.2505

1.1505

0.0E+00

 

121

166.51

7.6505

7.4505

7.1505

6.6505

6.0505

5.2505

4.3505

3.3505

2.2505

1.1505

0.0E+00

 

122

167.89

7.5505

7.3505

7.0505

6.5505

5.9505

5.1505

4.2505

3.3505

2.2505

1.1505

0.0E+00

 

123

169.27

7.4505

7.2505

6.9505

6.5505

5.8505

5.1505

4.2505

3.2505

2.2505

1.1505

0.0E+00

 

124

170.64

7.3505

7.2505

6.9505

6.4505

5.8505

5.0505

4.2505

3.2505

2.1 505

1.1505

0.0E+00

 

125

172.02

7.2505

7.1505

6.8505

6.3505

5.7505

5.0505

4.1505

3.2505

2.1 505

1 .0505

0.0E+00

 

126

173.39

7.2505

7.0505

6.7505

6.3505

5.6505

4.9505

4.1505

3.1505

2.1505

1 .0505

0.0E+00

 

127

174.77

7.1505

6.9505

6.6505

6.2505

5.6505

4.9505

4.0505

3.1505

2.1505

1 .0505

0.0E+00

 

128

176.15

7.0505

6.9505

6.6505

6.1505

5.5505

4.8505

4.0505

3.1 505

2.1505

1 .0505

0.0E+00

 

129

177.52

6.9505

6.8505

6.5505

6.1505

5.5505

4.8505

3.9505

3.0505

2.0505

1 .0505

0.0E+00

 

130

178.90

6.9505

6.7505

6.4505

6.0505

5.4505

4.7505

3.9505

3.0505

2.0505

9.9506

0.0E+00

 

131

180.27

6.8505

6.6505

6.4505

5.9505

5.4505

4.7505

3.8505

3.0505

2.0505

9.8506

0.0E+00

 

132

181.65

6.7505

6.6505

6.3505

5.9505

5.3505

4.6505

3.8505

2.9505

2.0505

9.7506

0.0E+00

 

133

183.03

6.6505

6.5505

6.2505

5.8505

5.2505

4.6505

3.8505

2.9505

1 .9505

9.6506

0.0E+00

 

134

184.40

6.6505

6.4505

6.2505

5.7505

5.2505

4.5505

3.7505

2.9505

1.9505

9.5506

0.0E+00

 

135

185.78

6.5505

6.4505

6.1505

5.7505

5.1 505

4.5505

3.7505

2.8505

1.9505

9.4506

0.0E+00

 

136

187.15

6.4505

6.3505

6.0505

5.6505

5.1 505

4.4505

3.6505

2.8505

1.9505

9.3506

0.0E+00

 

137

188.53

6.4505

6.2505

6.0505

5.6505

5.0505

4.4505

3.6505

2.8505

1.9505

9.2506

0.0E+00

 

138

189.91

6.3505

6.2505

5.9505

5.5505

5.0505

4.3505

3.6505

2.7505

1.8505

9.1 506

0.0E+00

 

139

191.28

6.2505

6.1505

5.8505

5.4505

4.9505

4.3505

3.5505

2.7505

1.8505

9.0506

0.0E+00

 

140

192.66

6.2505

6.0505

5.8505

5.4505

4.9505

4.2505

3.5505

2.7505

1.8505

8.9506

0.0E+00

 

 

141

 

194.04

 

6.1505

 

6.0505

 

5.7505

 

5.3505

 

4.8505

94

 

4.2505

 

3.5505

 

2.6505

 

1 .8505

 

8.8506

 

0.0E+00

 

Table 7.10 Continued

 

142

195.41

6.0505

5.9505

5.7505

5.3505

4.8505

4.1505

3.4505

2.6505

1.8505

8.7506

0.0E+00

 

143

196.79

6.0505

5.8505

5.6505

5.2505

4.7505

4.1505

3.4505

2.6505

1.7505

8.6506

0.0E+00

 

144

198.16

5.9505

5.8505

5.5505

5.1505

4.6505

4.0505

3.3505

2.6505

1 .7505

8.5506

0.0E+00

 

145

199.54

5.8505

5.7505

5.5505

5.1505

4.6505

4.0505

3.3505

2.5505

1 .7505

8.4506

0.0E+00

 

146

200.92

5.8505

5.7505

5.4505

5.0505

4.6505

4.0505

3.3505

2.5505

1 .7505

8.3506

0.0E+00

 

147

202.29

5.7505

5.6505

5.4505

5.0505

4.5505

3.9505

3.2505

2.5505

1 .7505

8.2506

0.0E+00

 

148

203.67

5.7505

5.5505

5.3505

4.9505

4.5505

3.9505

3.2505

2.5505

1.7505

8.1506

0.0E+00

 

149

205.04

5.6505

5.5505

5.2505

4.9505

4.4505

3.8505

3.2505

2.4505

1.6505

8.0506

0.0E+00

 

150

206.42

5.5505

5.4505

5.2505

4.8505

4.4505

3.8505

3.1505

2.4505

1.6505

7.9506

0.0E+00

 

151

207.80

5.5505

5.4505

5.1505

4.8505

4.3505

3.7505

3.1505

2.4505

1 .6505

7.9506

0.0E+00

 

152

209.17

5.4505

5.3505

5.1505

4.7505

4.3505

3.7505

3.1505

2.4505

1.6505

7.8506

0.0E+00

 

153

210.55

5.4505

5.3505

5.0505

4.7505

4.2505

3.7505

3.0505

2.3505

1.6505

7.7506

0.0E+00

 

154

211.93

5.3505

5.2505

5.0505

4.6505

4.2505

3.6505

3.0505

2.3505

1 .5505

7.6506

0.0E+00

 

155

213.30

5.2505

5.1505

4.9505

4.6505

4.1505

3.6505

3.0505

2.3505

1 .5505

7.5506

0.0E+00

 

156

214.68

5.2505

5.1505

4.9505

4.5505

4.1505

3.6505

2.9505

2.3505

1 .5505

7.4506

0.0E+00

 

157

216.05

5.1505

5.0505

4.8505

4.5505

4.0505

3.5505

2.9505

2.2505

1 .5505

7.4506

0.0E+00

 

158

217.43

5.1 505

5.0505

4.8505

4.4505

4.0505

3.5505

2.9505

2.2505

1.5505

7.3506

0.0E+00

 

159

218.81

5.0505

4.9505

4.7505

4.4505

4.0505

3.4505

2 .8505

2.2505

1.5505

7.2506

0.0E+00

 

160

220.18

5.0505

4.9505

4.7505

4.3505

3.9505

3.4505

2.8505

2.2505

1 .5505

7.1506

0.0E+00

 

161

221.56

4.9505

4.8505

4.6505

4.3505

3.9505

3.4505

2.8505

2.1505

1 .4505

7.0506

0.0E+00

 

162

222.93

4.9505

4.8505

4.6505

4.2505

3.8505

3.3505

2.8505

2.1505

1 .4505

7.0506

0.0E+00

 

163

224.31

4.8505

4.7505

4.5505

4.2505

3.8505

3.3505

2.7505

2.1505

1 .4505

6.9506

0.0E+00

 

164

225.69

4.8505

4.7505

4.5505

4.2505

3.7505

3.3505

2.7505

2.1505

1.4505

6.8506

0.0E+00

 

165

227.06

4.7505

4.6505

4.4505

4.1505

3.7505

3.2505

2.7505

2.0505

1 .4505

6.7506

0.0E+00

 

166

228.44

4.7505

4.6505

4.4505

4.1505

3.7505

3.2505

2.6505

2.0505

1.4505

6.7506

0.0E+00

 

167

229.82

4.6505

4.5505

4.3505

4.0505

3.6505

3.2505

2.6505

2.0505

1.3505

6.6506

0.0E+00

 

168

231.19

4.6505

4.5505

4.3505

4.0505

3.6505

3.1505

2.6505

2.0505

1.3505

6.5506

0.0E+00

 

169

232.57

4.5505

4.4505

4.2505

3.9505

3.6505

3.1505

2.6505

2.0505

1 .3505

6.4506

0.0E+00

 

170

233.94

4.5505

4.4505

4.2505

3.9505

3.5505

3.1505

2.5505

1 .9505

1.3505

6.4506

0.0E+00

 

171

235.32

4.4505

4.3505

4.1505

3.9505

3.5505

3.0505

2.5505

1 .9505

1.3505

6.3506

0.0E+00

 

172

236.70

4.4505

4.3505

4.1505

3.8505

3.4505

3.0505

2.5505

1 .9505

1 .3505

6.2506

0.0E+00

 

173

238.07

4.3505

4.2505

4.1505

3.8505

3.4505

3.0505

2.4505

1 .9505

1.3505

6.2506

0.0E+00

 

174

239.45

4.3505

4.2505

4.0505

3.7505

3.4505

2.9505

2.4505

1 .9505

1.2505

6.1506

0.0E+00

 

175

240.82

4.2505

4.2505

4.0505

3.7505

3.3505

2.9505

2.4505

1 .8505

1.2505

6.0506

0.0E+00

 

176

242.20

4.2505

4.1505

3.9505

3.7505

3.3505

2.9505

2.4505

1 .8505

1.2505

6.0506

0.0E+00

 

177

243.58

4.2505

4.1505

3.9505

3.6505

3.3505

2.8505

2.3505

1 .8505

1.2505

5.9506

0.0E+00

 

178

244.95

4.1505

4.0505

3.8505

3.6505

3.2505

2.8505

2.3505

1.8505

1 .2505

5.8506

0.0E+00

 

179

246.33

4.1505

4.0505

3.8505

3.5505

3.2505

2.8505

2.3505

1.8505

1.2505

5.8506

0.0E+00

 

180

247.71

4.0505

3.9505

3.8505

3.5505

3.2505

2.7505

2.3505

1.7505

1.2505

5.7506

0.0E+00

 

181

249.08

4.0505

3.9505

3.7505

3.5505

3.1505

2.7505

2.2505

1.7505

1 .2505

5.7506

0.0E+00

 

182

250.46

3.9505

3.9505

3.7505

3.4505

3.1505

2 .7505

2.2505

1.7505

1.1505

5.6506

0.0E+00

 

183

251.83

3.9505

3.8505

3.6505

3.4505

3.1505

2.7505

2.2505

1.7505

1.1505

5.5506

0.0E+00

 

184

253.21

3.9505

3.8505

3.6505

3.4505

3.0505

2.6505

2.2505

1.7505

1.1505

5.5506

0.0E+00

 

185

254.59

3.8505

3.7505

3.6505

3.3505

3.0505

2.6505

2.1505

1.6505

1.1505

5.4506

0.05-00

 

186

255.96

3.8505

3.7505

3.5505

3.3505

3.0505

2.6505

2.1505

1 .6505

1.1505

5.4 506

0.0E+00

 

187

257.34

3.7505

3.7505

3.5505

3.2505

2.9505

2.5505

2.1505

1.6505

1.1505

5.3506

0.0E+00

 

188

258.71

3.7505

3.6505

3.5505

3.2505

2.9505

2.5505

2.1505

1.6505

1.1505

5.2506

0.0E+00

 

189

260.09

3.7505

3.6505

3.4505

3.2505

2.9505

2.5505

2.1505

1 .6505

1.1505

5.2506

0.0E+00

 

190

261 .47

3.6505

3.5505

3.4505

3.1505

2.8505

2.5505

2.0505

1 .6505

1 .0505

5.1506

0.0E+00

 

191

262.84

3.6505

3.5505

3.3505

3.1505

2.8505

2.4505

2.0505

1 .5505

1.0505

5.1 506

0.0E+00

 

192

264.22

3.5505

3.5505

3.3505

3.1505

2.8505

2.4505

2.0505

1.5505

1 .0505

5.0506

0.0E+00

 

193

265.59

3.5505

3.4505

3.3505

3.0505

2.7505

2.4505

2.0505

1 .5505

1.0505

5.0506

0.0E+00

 

194

266.97

3.5505

3.4505

3.2505

3.0505

2.7505

2.4505

2.0505

1 .5505

1.0505

4.9506

0.0E+00

 

195

268.35

3.4505

3.4505

3.2505

3.0505

2.7505

2.3505

1.9505

1 .5505

9.9506

4.9506

0.0E+00

 

196

269.72

3.4505

3.3505

3.2505

3.0505

2.7505

2.3505

1 .9505

1 .5505

9.8506

4.8506

0.0E+00

 

197

271.10

3.4505

3.3505

3.1505

2.9505

2.6505

2.3505

1.9505

1 .4505

9.7506

4.8506

0.0E+00

 

198

272.48

3.3505

3.3505

3.1505

2.9505

2.6505

2.3505

1.9505

1 .4505

9.6506

4.7506

0.0E+00

 

 

199

 

273.85

 

3.3505

 

3.2505

 

3.1 505

 

2.9505

 

2.6505

95

 

2.2505

 

1.9505

 

1 .4505

 

9.5506

 

4.7506

 

0.0E+00

 

 

Table 7.10 Continued

 

200

275.23

3.3505

3.2505

3.0505

2.8505

2.6505

2.2505

1.8505

1 .4505

9.4506

4.6506

0.0E+00

 

201

276.60

3.2505

3.1505

3.0505

2.8505

2.5505

2.2505

1 .8505

1 .4505

9.3506

4.6506

0.0E+00

 

202

277.98

3.2505

3.1 505

3.0505

2.8505

2.5505

2.2505

1.8505

1 .4505

9.2506

4.5506

0.0E+00

 

203

279.36

3.2505

3.1505

2.9505

2.7505

2.5505

2.1505

1 .8505

1 .4505

9.1506

4.5506

0.0E+00

 

204

280.73

3.1505

3.1 505

2.9505

2.7505

2.4505

2.1505

1.8505

1 .3505

9.0506

4.4506

0.0E+00

 

205

282.11

3.1505

3.0505

2.9505

2.7505

2.4505

2.1505

1.7505

1 .3505

8.9506

4.4 506

0.0E+00

 

206

283.48

3.1505

3.0505

2.9505

2.7505

2.4505

2.1505

1.7505

1 .3505

8.8506

4.3506

0.0E+00

 

207

284.86

3.0505

3.0505

2.8505

2.6505

2.4505

2.1505

1.7505

1 .3505

8.7506

4.3506

0.0E+00

 

208

286.24

3.0505

2.9505

2.8505

2.6505

2.3505

2.0505

1.7505

1 .3505

8.7506

4.2506

0.0E+00

 

209

287.61

3.0505

2.9505

2.8505

2.6505

2.3505

2.0505

1 .7505

1 .3505

8.6506

4.2506

0.0E+00

 

210

288.99

2.9505

2.9505

2.7505

2.5505

2.3505

2.0505

1 .6505

1 .3505

8.5506

4.1506

0.0E+00

 

211

290.37

2.9505

2.8505

2.7505

2.5505

2.3505

2.0505

1.6505

1 .2505

8.4506

4.1 506

0.0E+00

 

212

291.74

2.9505

2.8505

2.7505

2.5505

2.2505

2.0505

1 .6505

1.2505

8.3506

4.0506

0.0E+00

 

213

293.12

2.8505

2.8505

2.6505

2.5505

2.2505

1 .9505

1 .6505

1.2505

8.2506

4.0506

0.0E+00

 

214

294.49

2.8505

2.7505

2.6505

2.4505

2.2505

1 .9505

1 .6505

1.2505

8.1506

4.0506

0.0E+00

 

215

295.87

2.8505

2.7505

2.6505

2.4505

2.2505

1 .9505

1 .6505

1 .2505

8.0506

3.9506

0.0E+00

 

216

297.25

2.7505

2.7505

2.6505

2.4505

2.2505

1 .9505

1 .5505

1 .2505

7.9506

3.9506

0.0E+00

 

217

298.62

2.7505

2.7505

2.5505

2.4505

2.1505

1 .8505

1 .5505

1.2505

7.9506

3.8506

0.0E+00

 

218

300.00

2.7505

2.6505

2.5505

2.3505

2.1505

1 .8505

1 .5505

1 .2505

7.8506

3.8506

0.0E+00

 

219

301.37

2.7505

2.6505

2.5505

2.3505

2.1505

1 .8505

1 .5505

1.1505

7.7506

3.8506

0.0E+00

 

220

302.75

2.6505

2.6505

2.5505

2.3505

2.1505

1 .8505

1 .5505

1.1505

7.6506

3.7506

0.0E+00

 

221

304.13

2.6505

2.5505

2.4505

2.3505

2.0505

1 .8505

1.5505

1.1505

7.5506

3.7506

0.0E+00

 

222

305.50

2.6505

2.5505

2.4505

2.2505

2.0505

1.8505

1 .4505

1.1505

7.4506

3.6506

0.0E+00

 

223

306.88

2.5505

2.5505

2.4505

2.2505

2.0505

1.7505

1 .4505

1.1505

7.4506

3.6506

0.0E+00

 

224

308.26

2.5505

2.5505

2.4505

2.2505

2.0505

1.7505

1 .4505

1.1505

7.3506

3.6506

0.0E+00

 

225

309.63

2.5505

2.4505

2.3505

2.2505

2.0505

1.7505

1 .4505

1.1505

7.2506

3.5506

0.0E+00

 

226

311.01

2.5505

2.4505

2.3505

2.1505

1 .9505

1 .7505

1 .4505

1.1505

7.1506

3.5506

0.0E+00

 

227

312.38

2.4505

2.4505

2.3505

2.1505

1.9505

1.7505

1 .4505

1.1505

7.1506

3.4506

0.0E+00

 

228

313.76

2.4505

2.4505

2.3505

2.1505

1 .9505

1.6505

1 .4 505

1.0505

7.0506

3.4506

0.0E+00

 

229

315.14

2.4505

2.3505

2.2505

2.1505

1 .9505

1 .6505

1 .3505

1 .0505

6.9506

3.4506

0.0E+00

 

230

316.51

2.4505

2.3505

2.2505

2.1505

1.9505

1.6505

1 .3505

1 .0505

6.8506

3.3506

0.0E+00

 

231

317.89

2.3505

2.3505

2.2505

2.0505

1.8505

1 .6505

1.3505

1.0505

6.8506

3.3506

0.0E+00

 

232

319.26

2.3505

2.3505

2.2505

2.0505

1 .8505

1.6505

1.3505

1.0505

6.7506

3.3506

0.0E+00

 

233

320.64

2.3505

2.2505

2.1505

2.0505

1 .8505

1.6505

1 .3505

9.9506

6.6506

3.2506

0.0E+00

 

234

322.02

2.3505

2.2505

2.1505

2.0505

1.8505

1.5505

1 .3505

9.8506

6.6506

3.2506

0.0E+00

 

235

323.39

2.2505

2.2505

2.1505

1 .9505

1.8505

1.5505

1 .3505

9.7506

6.5506

3.2506

0.0E+00

 

236

324.77

2.2505

2.2505

2.1505

1 .9505

1.7505

1.5505

1 .2505

9.6506

6.4 506

3.1 506

0.0E+00

 

237

326.15

2.2505

2.1505

2.1505

1 .9505

1 .7505

1 .5505

1 .2505

9.5506

6.3506

3.1 506

0.0E+00

 

238

327.52

2.2505

2.1505

2.0505

1 .9505

1 .7505

1.5505

1 .2505

9.4506

6.3506

3.1 506

0.0E+00

 

239

328.90

2.2505

2.1505

2.0505

1 .9505

1 .7505

1 .5505

1.2505

9.3506

6.2506

3.0506

0.0E+00

 

240

330.27

2.1505

2.1505

2.0505

1.8505

1 .7505

1 .4505

1.2505

9.2506

6.1506

3.0506

0.0E+00

 

241

331.65

2.1505

2.1505

2.0505

1 .8505

1.6505

1 .4505

1 .2505

9.1506

6.1506

3.0506

0.0E+00

 

242

333.03

2.1505

2.0505

1 .9505

1.8505

1.6505

1 .4505

1.2505

9.0506

6.0506

2.9506

0.0500

 

243

334.40

2.1505

2.0505

1 .9505

1.8505

1.6505

1 .4505

1 .2505

8.9506

6.0506

2.9506

0.0E+00

 

244

335.78

2.0505

2.0505

1 .9505

1 .8505

1.6505

1 .4505

1.1505

8.8506

5.9506

2.9506

0.0500

 

245

337.15

2.0505

2.0505

1.9505

1 .8505

1.6505

1 .4505

1.1505

8.7506

5.8506

2.8506

0.0E+00

 

246

338.53

2.0505

2.0505

1.9505

1 .7505

1.6505

1 .4505

1.1505

8.6506

5.8506

2.8506

0.0E+00

 

247

339.91

2.0505

1 .9505

1 .8505

1.7505

1.5505

1 .3505

1.1505

8.5506

5.7506

2.8506

0.0E+00

 

248

341.28

2.0505

1 .9505

1 .8505

1.7505

1.5505

1 .3505

1.1505

8.4506

5.6506

2.8506

0.0E+00

 

249

342.66

1 .9505

1 .9505

1.8505

1.7505

1.5505

1 .3505

1.1505

8.3506

5.6506

2.7506

0.0E+00

 

250

344.03

1 .9505

1.9505

1.8505

1.7505

1 .5505

1 .3505

1.1505

8.2506

5.5506

2.7506

0.0E+00

 

251

345.41

1 .9505

1 .9505

1 .8505

1.6505

1.5505

1 .3505

1.1505

8.1506

5.5506

2.7506

0.0E+00

 

252

346.79

1 .9505

1.8505

1 .8505

1.6505

1.5505

1 .3505

1.1505

8.1506

5.4506

2.6506

0.0E+00

 

253

348.16

1 .9505

1 .8505

1.7505

1 .6505

1.5505

1 .3505

1 .0505

8.0506

5.4506

2.6506

0.0E+00

 

 

254

 

349.54

 

1.8505

 

1.8505

 

1.7505

 

1.6505

 

1.4505

 

1 .2505

 

1.0505

 

7.9506

 

5.3506

 

2.6506

 

0.0E+00

 

96

 

Appendix 7.12: Model Output for Salmonella Concentration Profile

Excel formula used to calculate the Salmonella concentration at t = 1.03 s and x
= 0.005 m. (i.e. cell D5):
D5 =Concentration(D$3,dx,$A5,'Salmonella

Location'!$C$5:$IV$258,$C$5:$C$258)*(l/MASTER!$B$9)

Table 7.11: Predicted Salmonella concentration profile generated by Excel, given the
input variables listed in Table 4.1.

 

 

Salmonella Concentration Profile

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 => 0 1 2 3 4 5 6 7 8 9 10
j 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 .04 .04 0.05
0 0 # Entering cells/cm3
1 1.03 18139 16196 0 0 0 0 0 0 0 0 0
2 2.06 18139 32392 0 0 0 0 0 0 0 0 0
3 3.1 0 13775 44690 0 0 0 0 0 0 0 0 0
4 4.1 3 1 361 2 56844 0 0 0 0 0 0 0 0 0
5 5.16 1 1 524 671 34 0 0 0 0 0 0 0 0 0
6 6.19 11351 77268 0 0 0 0 0 0 0 0 0
7 7.22 10101 70092 16196 0 0 0 0 0 0 0 0
8 8.26 9938 78965 16196 0 0 0 0 0 0 0 0
9 9.29 9099 70893 32392 0 0 0 0 0 0 0 0
10 10.32 8949 78884 32392 0 0 0 0 0 0 0 0
1 1 1 1.35 8345 74036 44690 0 0 0 0 0 0 0 0
12 12.39 8208 81364 44690 0 0 0 0 0 0 0 0
13 13.42 7750 76131 56844 0 0 0 0 0 0 0 0
14 14.45 7626 72650 67134 0 0 0 0 0 0 0 0
15 15.48 7267 79138 67134 0 0 0 0 0 0 0 0
16 16.51 7152 75389 77268 0 0 0 0 0 0 0 0
17 17.55 6863 72498 86287 0 0 0 0 0 0 0 0
18 18.58 6758 78532 86287 0 0 0 0 0 0 0 0
19 19.61 6519 75479 95161 0 0 0 0 0 0 0 0
20 20.64 6423 73090 87089 16196 0 0 0 0 0 0 0
21 21 .67 6222 78645 87089 16196 0 0 0 0 0 0 0
22 22.71 6133 76131 95080 16196 0 0 0 0 0 0 0
23 23.74 5962 74003 86335 32392 0 0 0 0 0 0 0
24 24.77 5879 79253 86335 32392 0 0 0 0 0 0 0
25 25.80 5732 77042 81 364 44690 0 0 0 0 0 0 0
26 26.83 5654 75170 88284 44690 0 0 0 0 0 0 0
27 27.87 5525 73295 82939 56844 0 0 0 0 0 0 0
28 28.90 5453 78164 82939 56844 0 0 0 0 0 0 0
29 29.93 5339 76442 89427 56844 0 0 0 0 0 0 0
30 30.96 5270 74762 85524 671 34 0 0 0 0 0 0 0
31 32.00 5168 79376 85524 671 34 0 0 0 0 0 0 0
32 33.03 5104 77806 81517 77268 0 0 0 0 0 0 0
33 34.06 5012 76247 87551 77268 0 0 0 0 0 0 0

97

 

Table 7.11 Continued

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

34 35.09 4951 74847 84352 86287 0 0 0 0 0 0 0
35 36.12 4867 79192 84352 86287 0 0 0 0 0 0 0
36 37.16 4809 77751 81214 95161 0 0 0 0 0 0 0
37 38.19 4732 76421 78645 1 03285 0 0 0 0 0 0 0
38 39.22 4676 75120 84121 103285 0 0 0 0 0 0 0
39 40.25 4606 73909 81454 95080 16196 0 0 0 0 0 0
40 41 .28 4552 77973 81454 95080 16196 0 0 0 0 0 0
41 42.32 4486 76729 79253 102530 16196 0 0 0 0 0 0
42 43.35 4434 75570 84370 86335 32392 0 0 0 0 0 0
43 44.38 4373 74426 82090 93663 32392 0 0 0 0 0 0
44 45.41 4322 78285 82090 93663 32392 0 0 0 0 0 0
45 46.44 4265 77160 80103 88284 44690 0 0 0 0 0 0
46 47.48 4216 76056 84972 88284 44690 0 0 0 0 0 0
47 48.51 4162 75005 82930 95093 44690 0 0 0 0 0 0
48 49.54 41 15 78679 76442 89427 56844 0 0 0 0 0 0
49 50.57 4063 77601 81 148 89427 56844 0 0 0 0 0 0
50 51 .61 4017 76573 79376 95813 56844 0 0 0 0 0 0
51 52.64 3968 75559 83934 85524 67134 0 0 0 0 0 0
52 53.67 3923 74587 82281 91651 67134 0 0 0 0 0 0
53 54.70 3876 78048 82281 91651 67134 0 0 0 0 0 0
54 55.73 3833 77050 80667 87551 77268 0 0 0 0 0 0
55 56.77 3788 76086 79192 93372 77268 0 0 0 0 0 0
56 57.80 3746 751 37 83486 84352 86287 0 0 0 0 0 0
57 58.83 3702 78442 77751 90087 86287 0 0 0 0 0 0
58 59.86 3661 77486 81976 90087 86287 0 0 0 0 0 0
59 60.89 3619 76542 80596 86770 95161 0 0 0 0 0 0
60 61 .93 3579 75625 84708 86770 95161 0 0 0 0 0 0
61 62.96 3538 78784 79232 84121 1 03285 0 0 0 0 0 0
62 63.99 3499 77844 77973 89445 1 03285 0 0 0 0 0 0
63 65.02 3460 76927 81 978 89445 1 03285 0 0 0 0 0 0
64 66.05 3422 76023 80687 86704 1 1 1275 0 0 0 0 0 0
65 67.09 3383 75140 84592 86704 95080 161 96 0 0 0 0 0
66 68.12 3346 78127 79474 84370 102530 16196 0 0 0 0 0
67 69.15 3309 77222 78285 89419 102530 16196 0 0 0 0 0
68 70.18 3273 76336 82093 89419 102530 16196 0 0 0 0 0
69 71 .22 3236 75461 80925 87024 1 09859 161 96 0 0 0 0 0
70 72.25 3201 78320 80925 87024 93663 32392 0 0 0 0 0
71 73.28 3166 77430 79772 84972 1 00583 32392 0 0 0 0 0
72 74.31 31 31 76552 83446 84972 1 00583 32392 0 0 0 0 0
73 75.34 3097 75690 82307 82930 1 07392 32392 0 0 0 0 0
74 76.38 3063 78425 77601 87636 95093 44690 0 0 0 0 0
75 77.41 3030 77543 81188 87636 95093 44690 0 0 0 0 0
76 78.44 2997 76676 801 16 85763 1 01 581 44690 0 0 0 0 0
77 79.47 2964 75819 8361 9 85763 101581 44690 0 0 0 0 0
78 80.50 2932 78437 79062 83934 9581 3 56844 0 0 0 0 0
79 81 .54 2900 77565 82523 83934 9581 3 56844 0 0 0 0 0
80 82.57 2869 76704 81470 88409 9581 3 56844 0 0 0 0 0
81 83.60 2837 75856 80432 86701 101941 56844 0 0 0 0 0
82 84.63 2807 78362 80432 86701 91651 67134 0 0 0 0 0
83 85.66 2776 77497 79430 8501 3 97685 671 34 0 0 0 0 0
84 86.70 2746 76643 82736 8501 3 97685 671 34 0 0 0 0 0
85 87.73 2717 75800 8171 1 83486 1 03506 67134 0 0 0 0 0
86 88.76 2687 78200 81 71 1 83486 93372 77268 0 0 0 0 0
87 89.79 2658 77342 80717 8771 1 93372 77268 0 0 0 0 0
88 90.83 2630 76495 79737 86151 99106 77268 0 0 0 0 0
89 91 .86 2601 78817 79737 86151 99106 77268 0 0 0 0 0

 

 

 

 

 

 

98

 

 

 

 

 

 

 

Table 7.11 Continued

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

90 92.89 2573 77955 78784 84708 95643 86287 0 0 0 0 0
91 93.92 2545 77104 81908 84708 95643 86287 0 0 0 0 0
92 94.95 2518 76263 80933 88772 95643 86287 0 0 0 0 0
93 95.99 2491 78487 80933 83296 92246 95161 0 0 0 0 0
94 97.02 2464 77632 79982 87301 92246 951 61 0 0 0 0 0
95 98.05 2438 76788 83003 81 978 97569 95161 0 0 0 0 0
96 99.08 241 1 75953 82032 85937 97569 95161 0 0 0 0 0
97 100.1 1 2385 78083 82032 85937 89445 1 03285 0 0 0 0 0
98 101 .15 2360 77236 81082 84592 94694 103285 0 0 0 0 0
99 102.18 2334 76398 80144 88451 94694 103285 0 0 0 0 0
100 103.21 2309 78460 80144 83334 99812 103285 0 0 0 0 0
101 104.24 2285 77610 79226 87142 91821 1 1 1275 0 0 0 0 0
102 105.27 2260 76769 82084 87142 91821 1 1 1275 0 0 0 0 0
103 1 06.31 2236 78766 78320 85858 96870 1 1 1275 0 0 0 0 0
104 107.34 2212 77914 81146 85858 96870 111275 0 0 0 0 0
105 108.37 2188 77072 80226 84641 94352 1 18726 0 0 0 0 0
106 109.40 2165 76239 82991 84641 94352 1 18726 0 0 0 0 0
107 110.44 2141 78151 79317 88314 94352 118726 0 0 0 0 0
108 1 1 1 .47 21 18 77308 82052 83446 91892 126055 0 0 0 0 0
109 1 12.50 2096 76474 81 130 87074 91892 126055 0 0 0 0 0
1 10 1 13.53 2073 78325 81 130 82307 96659 109859 16196 0 0 0 0
1 1 1 1 14.56 2051 77481 80219 85894 96659 109859 16196 0 0 0 0
1 12 1 15.60 2029 76646 82865 85894 89739 1 16779 16196 0 0 0 0
1 13 1 16.63 2007 78438 79322 84731 94444 1 16779 16196 0 0 0 0
114 117.66 1986 77594 81940 84731 94444 116779 16196 0 0 0 0
1 15 1 18.69 1965 76759 81026 88234 94444 1 16779 16196 0 0 0 0
116 119.72 1944 78494 81026 83619 92251 123588 16196 0 0 0 0
1 17 120.76 1923 77650 80127 87080 92251 123588 16196 0 0 0 0
1 18 1 21 .79 1902 7681 5 82660 82523 96808 1 07392 32392 0 0 0 0
1 1 9 1 22.82 1 882 78495 79238 85945 96808 1 07392 32392 0 0 0 0
120 123.85 1862 77652 81744 85945 90320 1 13880 32392 0 0 0 0
121 124.88 1842 76818 80841 84852 94795 1 13880 32392 0 0 0 0
122 125.92 1822 78445 80841 84852 94795 1 13880 32392 0 0 0 0
123 126.95 1803 77603 79949 88196 94795 1 13880 32392 0 0 0 0
124 1 27.98 1784 76770 82374 83776 92829 1 07967 44690 0 0 0 0
125 129.01 1765 78346 79069 87081 92829 1 07967 44690 0 0 0 0
126 1 30.05 1746 77505 81468 87081 92829 1 07967 44690 0 0 0 0
127 131 .08 1727 76674 80573 86004 91047 1 14095 44690 0 0 0 0
128 1 32.1 1 1709 78200 80573 86004 91 047 1 14095 44690 0 0 0 0
1 29 1 33.14 1691 77361 79690 84942 95341 1 14095 44690 0 0 0 0
1 30 134.17 1673 78855 79690 84942 95341 1 14095 44690 0 0 0 0
131 135.21 1655 78010 82012 84942 95341 101941 56844 0 0 0 0
132 1 36.24 1637 77174 81 1 14 83912 93532 107975 56844 0 0 0 0
133 1 37.27 1620 78620 81 1 14 83912 93532 107975 56844 0 0 0 0
1 34 1 38.30 1603 77779 80228 87071 93532 107975 56844 0 0 0 0
1 35 1 39.33 1 585 76946 82476 82896 97707 107975 56844 0 0 0 0
1 36 140.37 1 569 78347 79352 86020 91886 1 13796 56844 0 0 0 0
1 37 141 .40 1 552 77508 81576 86020 91886 1 1 3796 56844 0 0 0 0
1 38 1 42 .43 1 535 78879 78487 84997 95998 103506 671 34 0 0 0 0
139 143.46 1519 78035 80687 84997 95998 103506 67134 0 0 0 0
140 144.49 1503 77201 82864 84997 90264 109241 67134 0 0 0 0
141 145.53 1487 78528 79809 83988 94328 109241 67134 0 0 0 0
142 146.56 1471 77689 81962 83988 94328 109241 67134 0 0 0 0
143 147.59 1455 76858 81071 87009 94328 109241 67134 0 0 0 0
144 148.62 1440 78144 81071 83003 92777 1 14796 67134 0 0 0 0
145 149.66 1425 77309 801 90 85991 92777 104662 77268 0 0 0 0

 

 

 

 

 

 

99

 

 

 

 

 

 

 

 

Table 7.11 Continued

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

146 1 50.69 1409 78568 801 90 85991 92777 104662 77268 0 0 0 0
147 1 51 .72 1 395 77729 82274 85991 92777 1 04662 77268 0 0 0 0
148 1 52.75 1 380 76899 81382 84986 96736 1 04662 77268 0 0 0 0
149 153.78 1365 78117 81382 84986 91260 110138 77268 0 0 0 0
150 154.82 1351 77283 80499 87908 91260 1 10138 77268 0 0 0 0
151 155.85 1336 78476 80499 84004 95165 1 10138 77268 0 0 0 0
152 156.88 1322 77639 82517 84004 95165 1 10138 77268 0 0 0 0
1 53 1 57.91 1308 78807 79628 86893 89841 1 06442 86287 0 0 0 0
1 54 1 58.94 1 294 77966 81624 83034 93701 106442 86287 0 0 0 0
1 55 159.98 1280 771 34 80740 85892 93701 1 06442 86287 0 0 0 0
1 56 161 .01 1267 78265 80740 85892 93701 106442 86287 0 0 0 0
1 57 162.04 1253 77431 82694 85892 93701 106442 86287 0 0 0 0
1 58 163.07 1240 78538 79867 8491 1 92259 1 1 1692 86287 0 0 0 0
159 164.10 1227 77701 81800 8491 1 92259 1 1 1692 86287 0 0 0 0
160 165.14 1214 78785 79004 87707 92259 102818 95161 0 0 0 0
161 166.17 1201 77945 80916 83942 96024 102818 95161 0 0 0 0
162 167.20 1 188 771 14 82808 83942 96024 102818 95161 0 0 0 0
163 168.23 1 176 78164 80043 86707 90906 107936 95161 0 0 0 0
164 169.27 1 163 77331 81914 86707 90906 107936 95161 0 0 0 0
165 170.30 1 151 78359 81914 82991 94622 107936 95161 0 0 0 0
166 171 .33 1 139 77524 81030 85726 94622 107936 95161 0 0 0 0
167 172.36 1 127 78530 81030 85726 94622 107936 95161 0 0 0 0
168 173.39 1 1 15 77694 80156 88431 89574 1 12985 95161 0 0 0 0
169 174.43 1 103 78679 80156 84758 93248 104860 103285 0 0 0 0
170 175.46 1091 77841 81968 84758 93248 104860 103285 0 0 0 0
1 71 1 76.49 1 080 78805 79292 87433 93248 1 04860 1 03285 0 0 0 0
172 177.52 1068 77967 81085 83805 96875 1 04860 103285 0 0 0 0
173 178.55 1057 78910 81085 83805 91942 1 09793 103285 0 0 0 0
174 179.59 1046 78071 8021 1 86452 91942 109793 103285 0 0 0 0
175 180.62 1035 77240 81966 86452 91942 1 09793 103285 0 0 0 0
176 181 .65 1024 78154 81966 82865 95529 101803 1 1 1275 0 0 0 0
177 182.68 1013 77323 81083 85483 95529 101803 1 1 1275 0 0 0 0
178 183.71 1002 78218 81083 85483 90660 106672 1 1 1275 0 0 0 0
1 79 184.75 992 77386 82800 85483 90660 1 06672 1 1 1275 0 0 0 0
180 185.78 981 78262 8021 1 84530 94203 1 06672 1 1 1275 0 0 0 0
181 186.81 971 77430 81910 84530 94203 106672 1 11275 0 0 0 0
182 187.84 960 78288 81910 84530 94203 106672 1 1 1275 0 0 0 0
183 188.88 950 77456 81029 87091 94203 106672 1 1 1275 0 0 0 0
184 189.91 940 78295 81029 87091 89437 1 1 1438 11 1275 0 0 0 0
1 85 190.94 930 77463 82691 83588 92940 1 03987 1 18726 0 0 0 0
186 191 .97 920 78285 80158 86121 92940 1 03987 1 18726 0 0 0 0
187 193.00 91 1 77453 81803 86121 92940 103987 1 18726 0 0 0 0
188 194.04 901 78258 81803 86121 92940 103987 1 18726 0 0 0 0
189 195.07 892 77427 80924 85166 96401 1 03987 1 18726 0 0 0 0
190 196.10 882 78214 80924 85166 91695 1 08693 1 18726 0 0 0 0
1 91 197.13 873 77384 82534 85166 91695 108693 1 18726 0 0 0 0
192 198.16 864 78155 80055 87645 91695 108693 1 18726 0 0 0 0
193 199.20 854 78918 80055 84223 951 17 108693 1 18726 0 0 0 0
194 200.23 845 78080 81647 84223 951 17 101365 126055 0 0 0 0
1 95 201 .26 836 78827 79195 86675 951 17 101365 126055 0 0 0 0
1 96 202.29 828 77990 80771 86675 90502 1 05979 1 26055 0 0 0 0
1 97 203.32 81 9 78721 80771 86675 90502 1 05979 1 26055 0 0 0 0
1 98 204.36 810 77886 82330 83293 93884 1 05979 126055 0 0 0 0
1 99 205.39 802 78601 79904 85719 93884 105979 1 26055 0 0 0 0
200 206.42 793 77767 81447 85719 93884 105979 126055 0 0 0 0
201 207.45 785 78468 81447 85719 93884 105979 126055 0 0 0 0

 

 

 

 

 

 

100

 

 

 

 

 

 

 

Table 7.11 Continued

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

202 208.49 777 77636 80573 881 18 93884 105979 126055 0 0 0 0
203 209.52 768 78322 80573 84774 92672 1 10536 126055 0 0 0 0
204 210.55 760 77491 82083 84774 92672 103616 1 32975 0 0 0 0
205 21 1 .58 752 78163 82083 84774 92672 103616 1 32975 0 0 0 0
206 212.61 744 78827 79709 87147 92672 103616 132975 0 0 0 0
207 213.65 736 77992 81203 87147 92672 103616 132975 0 0 0 0
208 214.68 729 78642 81203 83842 95977 103616 132975 0 0 0 0
209 215.71 721 77808 80332 86190 91502 108091 132975 0 0 0 0
210 216.74 713 78445 80332 86190 91502 108091 132975 0 0 0 0
211 217.77 706 77614 81794 86190 91502 108091 132975 0 0 0 0
212 218.81 698 78237 81794 86190 91502 108091 132975 0 0 0 0
21 3 219.84 691 78855 79472 85243 94770 1 08091 1 32975 0 0 0 0
214 220.87 684 78019 80918 85243 94770 108091 1 32975 0 0 0 0
215 221 .90 677 78623 80918 85243 94770 101283 1 39783 0 0 0 0
216 222.93 669 77790 82349 85243 94770 101283 1 39783 0 0 0 0
217 223.97 662 78381 80051 87541 90350 105703 1 39783 0 0 0 0
218 225.00 655 78966 80051 87541 90350 105703 1 39783 0 0 0 0
219 226.03 648 78130 81467 84310 93581 105703 139783 0 0 0 0
220 227.06 642 78703 81467 84310 93581 105703 1 39783 0 0 0 0
221 228.1 0 635 77869 80595 86582 93581 105703 1 39783 0 0 0 0
222 229. 1 3 628 78430 80595 86582 93581 105703 1 39783 0 0 0 0
223 230.16 622 77599 81980 86582 93581 105703 1 39783 0 0 0 0
224 231 .19 615 78149 81980 83387 96776 105703 139783 0 0 0 0
225 232.22 609 78692 79732 85635 92431 1 10049 139783 0 0 0 0
226 233.26 602 77859 81 103 85635 92431 103561 146271 0 0 0 0
227 234.29 596 78391 81 103 85635 92431 103561 1 30076 16196 0 0 0
228 235.32 590 78917 81 103 85635 92431 103561 130076 16196 0 0 0
229 236.35 583 78082 80235 87859 92431 103561 1 30076 161 96 0 0 0
230 237.38 577 78597 80235 84700 95590 103561 1 30076 16196 0 0 0
231 238.42 571 77765 81577 84700 95590 103561 1 30076 16196 0 0 0
232 239.45 565 78269 81577 84700 95590 103561 130076 16196 0 0 0
233 240.48 559 78769 81577 84700 91296 107854 130076 16196 0 0 0
234 241 .51 553 77935 80705 86900 91296 107854 130076 16196 0 0 0
235 242.54 547 78424 80705 86900 91296 107854 1 30076 16196 0 0 0
236 243.58 542 78907 80705 86900 91296 107854 1 30076 16196 0 0 0
237 244.61 536 78073 82018 83776 94420 1 07854 1 30076 16196 0 0 0
238 245.64 530 78546 79842 85952 94420 1 07854 1 30076 161 96 0 0 0
239 246.67 525 77715 81 141 85952 94420 101468 136462 16196 0 0 0
240 247.71 519 78179 81141 85952 94420 101468 136462 16196 0 0 0
241 248.74 514 78637 81 141 85952 94420 101468 1 36462 16196 0 0 0
242 249.77 508 77805 82427 85952 901 95 1 05693 1 36462 1 61 96 0 0 0
243 250.80 503 78255 80274 88105 90195 105693 136462 16196 0 0 0
244 251 .83 498 78699 80274 8501 7 93284 1 05693 1 36462 1 61 96 0 0 0
245 252.87 492 77867 81546 85017 93284 105693 136462 16196 0 0 0
246 253.90 487 78302 81546 85017 93284 105693 136462 16196 0 0 0
247 254.93 482 78732 81546 85017 93284 105693 136462 16196 0 0 0
248 255.96 477 77900 80675 87146 93284 105693 1 36462 16196 0 0 0
249 256.99 472 78321 80675 87146 93284 105693 1 36462 16196 0 0 0
250 258.03 467 78738 80675 87146 93284 1 05693 1 36462 1 61 96 0 0 0
251 259.06 462 77906 81920 84091 96339 105693 136462 16196 0 0 0
252 260.09 457 78314 81920 84091 92164 109868 136462 16196 0 0 0
253 261.12 452 78718 79813 86198 92164 103741 142589 16196 0 0 0

 

 

 

 

 

 

 

 

 

 

 

 

 

101

 

 

Appendix 7.13: Three-Dimensional Graphs of Model Output

The following three-dimensional graphs show the complete profile created by the
model for dry basis moisture concentration, effective gap radius, Darcy ’s velocity,
marinade velocity, Salmonella velocity, Salmonella location, and Salmonella

concentration at every time step.

 

 

 

r 2.66
~ 2.64 I
~ 2.62

2.60 Moisture
~2.58 Contentdb

 

 

 

 

--2.56
tlme(s) , .254
§ , .252

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Figure 7.1: Three-dimensional graph of the dry basis moisture content.

102

 

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b 8.E-05 veloc'ty

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Figure 7.3: Three-dimensional graph of Darcy’s velocity.

103

 

 

 

  
 
  

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Figure 7.5: Three-dimensional graph of Salmonella velocity.

104

 

 

 

 

 

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