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LIBRARY
Michigan State
University

 

 

 

This is to certify that the

dissertation entitled

Mathematical Modelling of Thermal Gelation of Myofibrillar
Beef Proteins and Their Interaction With Selected
Hydrocolloids.

presented by

Carlos Antonio Lever—Garcia

has been accepted towards fulfillment
of the requirements for

Ph.D. Food Science

degree in

 

 

    
   

{m

. ' W
Major professor

 

Date W ///l /?g

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

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MATHEMATICAL MODELLING OF THERMAL GELATION
OF MYOFIBRILLAR BEEF PROTEINS AND THEIR

INTERACTION WITH SELECTED HYDROCOLLOIDS.

BY

CARLOS ANTONIO LEVER-GARCIA

A DISSERTATION

Submitted to
Michigan State University
In partial fulfillment Of the requirements
for the decree
DOCTOR OF PHILOSOPHY

Deparment Of FOOd SCIODCC and Human Hutritlon

1988

507'3t/u5

ABSTRACT

MATHEMATICAL MODELLING OF THERMAL GELATION
OF MYOFIBRILLAR BEEF PROTEINS AND THEIR

INTERACTION WITH SELECTED HYDROCOLLOIDS.

BY

CARLOS ANTONIO LEVER-GARCIA

This study was designed to prov1de the experimental
base of a proposed mathematical model for predicting
thermally-induced gelation of myofibrillar beef proteins and
their interaction with selected hydrocolloids. Four
connercially available vegetable gums were evaluated. These
hydrocolloids were carrageenan. guar. locust bean and
xanthan gums. The mathematical model was designed to predict
the gelation of myofibrillar beef proteins as a function of
the time-temperature history of the process and.the protein
concentration on a dry basis. The thermally induced gelation
was measured as the Instron back-extrusion apparent

VISCOSItY. A model myofibrillar protein system '38 utilized.

CARLOS ANTONIO LEVER-GARCIA
The model system was a solution of extracted myofibrillar
beef proteins. Protein solutions were heated at 54. 64, 70.
80 and 85 °C. The time-temperature history of the process,
the back—extrusion apparent viscosity and the model
variables were calculated by a combination of commercial
software and computer programs developed specifically for

this experiment.

It was found that under the conditions of this study.
three terms of the model formerly reported as constants
(A'.a and a) were actually functions of the protein
concentration. These mathematical relationships were
developed and integrated into the basic model. The modified
mathematical model was found to predict (R3 from.0.88 to
0.97) the heat-induced gelation and the change in water
holding capacity of myofibrillar beef proteins as a function
of the time-temperature history of the process and the

protein concentration (dry basis).

It was hypothesized that vegetable gums contribute to
the framework of the thermally-induced protein gels.
Correlation between experimental and predicted gelation
values of gels from carrageenan. guar and locust bean gums

with.myofibrillar beef proteins (R2 from 0.85 to 0.91)

CARLOS ANTONIO LEVER-GARCIA
support this premise. As predicted by the mathematical model
these protein- hydrocolloid gels had a higher gel strength
and water holding capacity than the control. Under the
conditions of this experiment xanthan gum was found to
inhibit gelation. However this protein-gum solution had the
best water holding capacity value among all experimental

units.

To my Wife DELIA GUZHAN DE LEVER for her love.
support. 1‘1an WOPK and patience during my doctoral studies
and those long long winters.

To my son CARLOS RODRIGO and my daughter MARIANA
DELIA who have been the light of my eyes and the Joy of my
spirit.

 

 

ACKNOWLEDGMENTS

A special note of appreciation is given to Professor
James F. Price chairman of my doctoral committee for his
guidance. patience and support. His dedication as scientist
was a source of continuous inspiration during my doctoral

studies.

To Professor Charles Cress who taught me a new way of
thinking and open my eyes to the meaning of experimental
design. To Dr. Ron Morgan who taught me what mathematical
modelling is all about. To Drs. H.E. Zabik. A.H. Pearson and
A. Booren, who also served in my committee. thank you very

much for their assertive advice.

A1 DI“. Carlos Escandon D. una nota muy especial de
agradecimiento P01" 81 apoyo proporcionado a 10 largo de mis

estudios doctorales.

TO Fernando OSOPIO who helped me write the computer
programs and WithOUt whose help it would not have been
POSSIDIQ to find the solution for the thermal equations,

muchas [111161188 SPECIES I .

V1

For the financial assistance without which these
studies could not have been possible, I would like to thank
the Goverment of my country MEXICO through.the ConseJo
Nacional de Ciencia y Tecnologia (CONACYT). also the
Asociacion Hacional de Universidades e Institutos de
Educacion Superior (ANUIES) and the Universidad

Iberoamericana.

Vii

TABLE OF CONTENTS

Page
LIST OF TABLES . . . . . . . . . . . . . X
LIST OF FIGURES . . . . . . . . . . . . . . XI
INTRODUCTION . . . . . . . . . . . . . . . 1
REVIEW OF LITERATURE . . . . . . . . . . . . 5

Properties of Meat Proteins 5
Sarcoplasmic Proteins 6
Myofibrillar Proteins . 7
Tropomyosin Troponin Complex 9
Cytoskeletal Proteins . . . . . 10
Connective Tissue Proteins . . . . . 11

Role of Myofibrillar Proteins in Gelation . . . . 12

Gelation of Proteins . . . . . . . . i4

Mechanism of Protein Gelation . . . . i7

Mechanism of Gelation of Myofibrillar Proteins . . 20

Effect of Temperature in Gel Strength . . . . . 25

Theoretical Basis of the Mathematical Model . . . 27
Concentration Effects . . . . . . . 28
Rheological and Kinetics Background . . . . . 29

Characteristics of MydrocoIIOids . . . . . . . 36

MATERIALS AND METHODS . . . . . . . . . . . . 41

Experimental Design . . . . . . . . . . . 4i

Experiment I . . . . . . . . . . . . 41
Meat Sample Preparation . . . . . . . 4a
Isolation of Myofibrillar Proteins . . . . . 48
Heat Treatment . . . . . . . . . . . . 43
Gel Strength . . . . . . . . . 44
Evaluation of the Thermal Process . . . . . .45
Determination of Activation Energy . . . . . 51

Experiment 11 . . . . . . . . . . . . . 58

Methods of Analysis . . . . . . . .53
Determination of Water Molding Capacity . . . . 53
Buffer Preparation . . . . . . . . . . 54
Protein Determination . . . . . . . . . . 54
Statistical Analysis . . . . . . . . . . 54

Computer Programs Utilized . . . . . . . . 55
Program Rodrigo . . . . . . . . . . . . 55
Program Delia . . . . . . . . . . . . 56
Program Mariana . . . . . . . . . . . . 56

V111

Page

RESULTS AND DISCUSSION . . . . . . . . . . . 64
Determination of Activation Energy . . . . . . 64
Determination of (a) . . . . . . . . . . . 79
Determination of A' . . . . . . . . 88
Test and Verification of the Model . . . . . . 93

Test of the Model . . . . . . . . . . 93
Verification of the Model . . . . . . . . 96
Effect of Vegetable Gums . . . . . . . . . 102

SUMMARY AND CONCLUSIONS . . . . . . . . . . . 115

BIBLIWRAPHY I O 0 0 O 0 0 O O Q I 9 0 O 116

APPENDICES . . . . . . . . . . . . . . . 128
Appendix A. List of computer program Rodrigo . . . 128
Appendix B. List of computer program Delia . . . .135
Appendix C. List of computer program.Mariana . . . 140
Appendix D. Experimental values of Y’ and

TTH utilized in testing the
model (Figure 13). . . . . . . . 155

ix

LIST OF TABLES

Table Page

1. Thermally induced back-extrusion apparent
viscosity (Y’) for the time range of 30
to 350 sec and for five different
temperatures . . . . . . . . . . . . 66

8. Calculated constant values for
modelled experimental data cooked to five
different temperatures . . . . . . . . . 58

3. Time of cooking in seconds required
to produce selected thermally induced apparent
viscosity for five different temperatures . . 69

4. Estimated values for Ea (cal/mot)
for different Y's and temperature ranges . . . 71

5. Statistical analysis of the linear
regression of experimental Y’ values and
corresponding TTH values. Ea utilized was
29,500 cal/mol . . . . . ,. . . . . . 76

6. Statistical analysis of the linear
regression of experimental Y' values and
corresponding TTM values. Ea utilized was
20,000 cal/mol . . . . . . . . . . . 81

7. Calculation Of (a) and Kt values
using an average Of 20 TTH values
Ea used was 20,000 cal/mol . . . . . . . 84

8. Calculated values of equivalent
protein concentration. a and A' for
selected gum-protein combinations . . . . . 109

Figure

10.

11.

12.

13.

14.

LIST OF FIGURES

FlOW diagram Of computer program
Roarigo o o o o o o o 0

Flow diagram of computer program
Delia O 0 O 0 e O o O 0 0

Flow diagram of computer program
Mariana . . . . . . .

Estimation Of activation energy
range for five temperatures and three
thermal-induced apparent Viscosities

correlation Of Y' values Obtained

in the temperature range Of 54 to 84 OC and-

their estimated TTH . . . . . . .

Relationship between experimental
values of Y' and the Log of TTH . .

correlation Of Y' values Obtained

in the temperature range of 64 to 84°C and

their estimated TTH . . . . . . .
correlation between G and (a)

Plot of o versus R2 for
five protein concentrations . . .

correlation between G and
protein concentration (d.b.)

Correlation between protein
concentration and Y' . . . . . .

Correlation between protein
concentration and A' . . .

Experimental Y' values for
temperature range Of 54 to 54 0C Versus
values predicted DY the TTH model . .

Experimental Y' values for three

different protein concentrations versus
values predicted.by the TTH model . .

81

Page

57

58

60

7O

75

77

60

85

87

89

91

98

94

97

Figure

15.

16.

17.

18.

19.

200

21.

220

Dependency Of water release
for two protein

over time Of cooking.
concentrations . . . .

Experimental water release

values versus values predicted DY the

m mOde l 0 0 0 0 0

Experimental Y’ values for

carrageenan-protein solution and values

predicted by the TTH model

Experimental Y’ values for

guar-protein solution and values predicted

by the TTH model . . .

Experimental Y’ Values for

locust-bean-protein solution and values

Predicted by the TTH model

Experimental Y' values for

xanthan-protein solution and values

predicted by the TTH model

Effect Of protein gum combination

on a O 0 O O O 0

Water released by four gum- rotein

combinations (75 min at 70

xii

C)

Page

100

101

104

105

106

107

110

111

INTRODUCTION

The thermal gelation of myofibrillar beef proteins is
an important phenomenon which takes place in all heat
processed meats products. These series of chemical reactions
are one of the major factors affecting the final sensory

attributes Of meat products.

Men learned to process meat products by this mechanism
in ancient times. Egyptian hieroglyphics found in pyramids
are the first recorded use of manufactured meat products.
The great Greek philosopher and historian Homer describes in
“The Iliad" how the Greek soldiers. during the siege of
Troy. produced manufactured meat products by mixing meat
pieces with spices and salt and stuffing them in goat

stomachs before cooking.

In the centuries that followed. men learned to modify
the end-products of this thermal ly-induced gelation of meat
proteins by trial and error. However. these changes were not
accompanied by an understanding and a control of the
chemical reactions responsible for this gelation. It was not
until this century that knowledge about gelation reactions
began to appear in technical literature. with most of the

work concentrated in the last decade. It is now known that

protein-protein interaction is the functional event which is
related to the structural integrity of meat products through
orderly heat-induced protein aggregations. that myofibrillar
proteins play a major functional role. and that the complete
myosin molecule is necessary for attaining appreciable
continuity and strength in the protein matrix. All factors
that affect myofibrillar proteins will affect the final
texture and the quality of end products of the thermally
induced reactions. Some factors that are known to affect
this reaction are pH. type and concentration of salts. ionic
strength. concentration and type of protein and time and
temperature of cooking. The characteristic Viscosity of a
thermally produced protein gel is considered a measure of
the extent of gelation. and therefore. a function of the

same factors that affect protein-protein interactions.

During commercial meat processing operations several
factors remain constant e.g.. pH. concentration and type of
salts. However protein concentration and the time
temperature history of fine process is unique to each.batch.
and.therefore it was considered.that a mathematical model
was needed to predict the final product of this thermally
induced protein gelation as a function of the protein
concentration and the time temperature history of the

process.

If such a model could account for the overall time
temperature history of the process and the change in protein
concentration. it could greatly reduce research time and
costs. and therefore facilitate and decrease expense of food

product development.

A new type Of meat product that has been proposed for
development is a IOW fat product. However fats in meat
products have several functional properties. e.g.. they are
a reservoir for flavor compounds and they contribute to the
structure and texture Of the final product. Therefore.

reducing the fat content may alter the product quality.

To maintain the beneficial effect of fats in meat
products several compounds may be necessary. Among these are
the vegetable polysaccharides gums which seem to have
several advantages. Polysaccharides gums are a group of
chemical compounds that are extensively used in the food
industry to regulate viscosity. as texture modifiers. and to

form gels.

The objectives of this study were

1. To develop a mathematical model that predicts the
thermally induced gelation of myofibrillar beef proteins as
a function of the protein concentration and the time

temperature history Of the process.

2. To apply the model developed to the thermal gelation
of selected combinations of vegetable gums and myofibrillar

beef proteins and verify its validity.

L ITERATURE REV I EW

Gelation of proteins is an important phenomenon which
takes place in all fabricated meat products during thermal
processing. This tacky-sol transformation of meat proteins
to a gelled state results in formation of the ultimate
three-dimensional interlinked protein network. This
phenomenon involves both intramolecular (conformational) and
intermolecular changes in proteins. The protein network
physically (due to capillarity) and chemically (due to
noncovalent bonding) stabilizes water and physically or
structurally restrains dispersed fat (in comminuted.meats)
from rendering (Schmidt 9; l. 1981; Acton _e_t l. 1983;

Gossett _; _3. 1984). This review will focus on (1) the main
factors that affect gelation (meat proteins and chemistry
and mechanism of protein-protein interaction). (2) the basis
of a proposed mathematical model for predicting and (3)

general characteristics Of selected vegetable gums.

Progerties 2: Meat Proteins

 

The composition of lean meat can be approximated in a
broad sense as 75x water. aox protein. 3x fat and ex other
substances. The aoz protein. calculated from.6.25 times

percent nitrogen. includes 1 to 1.5% non-protein

nitrogeneous substances. such as amino acids. nucleotides.
creatine and traces Of Other nitrogeneous compounds (Schut.
1976; Forrest _g __1. 1975; Schmidt. 1957).

Muscle proteins have been traditionally classified
into various categories on the basis Of distribution.
organization. solubility and function in the living muscle.
Roughly they can be divided into three groups. 1. e.. the
sarcoplasmic proteins soluble in salt solutions Of lOW ionic
strength (< 0.1), the myofibrillar or structural proteins.
WhiCh are soluble in concentrated salt solutions (ionic
strength Of 0.5 to 0.5) and the connective tissue proteins.
WhiCh are insoluble in bOth. at least at low temperatures

(Szent-Gyorgyi. 1951; Perry. 1956).

Sarcoplasmic Proteins

The soluble proteins of the sarcoplasm located within
the sarcolemma are referred to as sarcoplasmic proteins.
Among them some albumins and.the so called myogens. to which
belong most of the enzymes of glycolitic pathway. are the
real water-soluble proteins (Schut. 1976). The other
fractions of sarcoplasmic proteins are soluble in low salt
concentration (ionic strength <=0.1). The recent trend is to
partition the sarcoplasmic proteins into four fractions by
sucrose gradient techniques involving ultracentrifugation.
They include a nuclear fraction. a mitochondrial fraction.

a microsomal fraction and a cytoplasmic fraction (Asghar and

Pearson. 1980; Bodwell and McClain. 1971). About 100
different proteins are known to be present in the
sarcoplasmic fraction (Scopes. 1970) which constitutes
about 30 to 35% of the total muscle protein or about 5% of
the weight of muscle. Despite their diversity sarcoplasmic
proteins have some common physicochemical characteristics.
They are globular or rod shaped in conformation. have low
viscosity. have isoelectric point between pH 6.0 and 7.0.
and.have molecular weights in the range 30.000 to 100.000

daltons (Bendall. 1964).

Mzofibrillar Proteins

The salt-soluble proteins which compose the
myofibrils within the muscle fibers are collectivelly
defined as the myofibrillar proteins. They constitute about
55 to cox of the total muscle protein or 10% of the weight
of the vertebrate skeletal muscle (Lawrie. 1974). The major
proteins in this category are the myosin-actin-actomyosin
group. the tropomyosin-troponin complex. the minor
myofibrillar components and the cytoskeletal proteins. Based
on their physiological functions in muscle. myofibrillar
proteins can be classified into two subgroups: (1) the

contractile proteins and (2) the regulatory proteins.

The myofibrillar proteins. myosin and actin. which.are
directly involved in the contraction-relaxation cycle Of

live muscle are termed "contractile". Myosin and actin are

the major components of thick and thin myofilaments.
respectively. Some of the salient characteristics of

contractile proteins were published by Bandman (1987).

Myosin is the major constituent of the thick
myofilaments of the sarcomeres. It accounts for
approximately 35% of muscle protein and is the most abundant
of all proteins found in muscle (Manson and Lowy. 1964). It
is a thread-like molecule with.a high length-to-diameter
ratio (40:1) having a molecular weight of 470.000 to 500.000

daltons (Frederiksen and Moltzer. 1968).

Actin is the major constituent of the thin myofilmment
and accounts for sex of the myofibrillar protein (Portter.
1974; Yates and Greaser. 1983). It exists in two forms:
globular (G-form) and fibrous (F-form) depending on
environmental conditions (Steiner _t _3. 1952). At
physiological concentrations of salt. globular G-actin
polymerizes to form filaments (F-actin)which can interact
with myosin filaments to produce mechanical energy for
biological movements (Bandman. 1987; Miyanishi and Tonomura.
1981; Pollard.gt a}. 1981). When actin is extracted with
water from.muscle tissue it is obtained in the globular
form. having a molecular weight of from 44.000 to 49.000
daltons (Sender. 1971; May gt 3;. 1973). Since the actin

filaments have no defined length F-actin does not have a

determined mo lecul ar weight.

In muscle. myosin and F-actin are present in a more or
less complexed form called actomyosin. Actomyosin is the
structural component which is responsible for contraction
and relaxation in muscle of living animals (Granicher and

Portzehl. i964).

Tropomyosin-groponin Compl ex
These myofibrillar proteins. which.are not directly

involved in cross-bridge formation but play a role
indirectly in the contraction-relaxion cycle. are called
regulatory proteins (Maruyama and Ebashi. 1970).
Tropomyosin and troponin together account for 9.5x and
12.05% of the muscle proteins and.have molecular weights of
36.000 and 70.000 daltons. respectively (Forrest g; 3;.
1975; Porzio and Pearson. 1977). Both.of these proteins are
located in the groove of the actin filament and play an
important role in the control of muscular contraction. in
particular as a regulating system.under the influence of

calcium ions (Murray and‘Weber. 1974).

During the last few years a number of new proteins
have been discovered in skeletal muscle. Guantitatively.
must of them are insignificant. but are believed to be
involved with.the regulation of the filamentous structure of
myofibrils. and hence they have been classified as minor

regulatory proteins. However. a precise function for many of

10

them is not yet clear. According to Asghar g; _l; (1985)
these proteins are distributed in different parts of the
ultrastructure of filaments such as the M-line (e.g..
M-protein or myomesin). A-band (e.g.. C-. F-. M-. I- and
x-protein). z-disk (e.g.. o-actinin. z-nin. and Eu-actinin).

and I-band (e.g.. B-actinin and y-actinin).

Cytoskeletal Proteins

Several researches have reported the existence of
longitudinal filaments other than those of actomyosin. Those
filaments look as if they are connecting the thin filaments
on either side of the z-line. Some researchers denoted them
as S-filament (Maruyama. 1980). Other filaments were also
seen as if they were connecting the edges of the thick
filaments to the z-line. and Sjostrand (1962) designated
them as gap filaments. Locker and Leet (1975) also supported
the existence of gap filament in the skeletal muscle. while
other workers used the term T-filaments in their description
(Maruyama. 1980). According to Wolosewick and Porter (1979)
a three-dimensional filamentous lattice connects most of
cytoplasmic structures. As the average diameter of these
filaments is about 10 nm. which is between that of actin (6
nm) and myosin (15 nm) filaments. they are referred to as
intermediate or 10 nm filaments (Bornstein and Sage. 1980;
Gard g; 3;. 1979; Lazarides. 1982; Price and Sanger. 1979;
Price and Sanger. 1980; Steiner gt 1. 1952; Wang and

Ramirez-Mitchell. i983). Chemically. intermediate filaments

11

differ from contractile and regulatory proteins in many
respects. and morphologically they resemble collagen fibers.
which in contrast. exist extracellularly. Since the
intermediate filaments are believed to strengthen the
architecture of myofibrillar system in muscle. they have
also been named cytoskeletal or backbone proteins (Granger
and Lazarides. 1978; Obinata _5 _l. 1981). These proteins
represent longitudinal intrafibrillar and transverse
interfibrillar bridges (Gracia-Nunzi and Franzini-Armstrong.
1980). The proteins which can tentatively be included in
this group are titin (or connectin). nebulin. desmin (or
skeletin). vimentin and symentin. Some important properties
of cytoskeletal proteins are published by Asghar t al.

(1985).

Connective Tissue Proteins
The interstitial space Of muscle cells (syncytia)

contains three proteins. namely. collagen. reticulin and
elastin. WhiCh are fibrillar in nature. Collectively they
are called connective tissue WhiCh, in fact. also contains
some globular mucoprotein and non—protein components such as
lipids and different mucopolysaccharides (hyaluronic acid.
chondroitin sulfate A. B and C. keratosulfate. heparitin
sulfate and.heparin in the form Of galactosamine or

glucosamine). However. the extracellular proteins around

12

individual muscle fiber consist mainly of fine reticular and
collagenous fibers which constitute the endomysium layer

(Bendall. 1964).

Collagen consists of a triple helix that contains a
higher hydroxyproline content than any other meat protein.
Collagen fibers shorten to about one-third.their original
length when heated to 70 00. At a temperature of about so 0c
or higher collagen is converted into gelatin. Reticulin.
resembling collagen in many respects. does not produce

ge l atin on heating.

Elastin is a rather unique protein because it contains
the uncommon amino acid residues desmosine and isodesmosine.
These amino acids are involved in the crosslinking of the
polypeptide chains and give elastin its characteristic
elastic properties. Unlike collagen. elastin is not
decomposed by heat. contains very little hydroxyproline. has
very little swelling ability and is extremely resistant to

acid and alkali (Bendall.1964).

 

Role 2: Myofibrillar Proteins i; Gelation

The presence of salt-extractable myofibrillar proteins
has been shown to be necessary for satisfactory binding in
both emulsion and restructured.meat products. Using model
emulsion systems. Acton and Saffle (1969). Miller 3; pl.

(1980) and Randall and Voisey (1977) showed that increasing

13

the proportion of salt-extractable myofibrillar proteins
produced a concurrent increase in binding quality. A similar
effect was observed by Acton (1972) and Siegel 53 al. (1978)

With sectioned and formed products.

In addition to the research done on the binding of
myofibrillar proteins as a group. work has been carried out
to determine the role of the individual myofibrillar
proteins in binding. Much of the initial work in this area
was carried out by Fukazawa t al. (1961a.b). SameJima

_t _l. (1969) and kakayama and Sato (1971a.b) using the
individual isolated myofibrillar proteins in model gelation
systems. It was generally concluded that myosin and
actomyosin were the proteins that produced.the greatest gel
strengths and. therefore, were the most important in

binding. In addition they found that in most 03.898

actomyosin was a more effective binding agent than myosin.

In contrast to these results. HacFarlane g; _l; (1977).
Ford 9; al_. (1978) and Turner gt a_l_. (1979) found that
myosin was superior to actomyosin in binding meat pieces
together in both.a model binding system and in a reformed
beef product. Although this difference seemed hard to
reconcile. an explanation is found in the work of Yasui
gt _1. (1960). Using a model gelation system they showed

that the addition 0f myosin t0 actomyosin produced a gel

that was much stronger than either myosin OP actomyosin when

14

used separately. Hence. the results obtained by HacFarlane
_; _g. (1977) is explained by the interaction of the added
myosin with the actomyosin present in the surface of the
meat to form a strong binding matrix and.the inability of

actomyosin to do 811111181417.

Gelation 21 Proteins
A discussion of gelation necessitates defining some
commonly used terms associated with.this phenomenon namely:

denaturation. aggregation. coagulation and gelation.

Denaturation has been defined as: (1) a process (or
sequence of processes) in which.the spatial arrangement of
the polypeptide chains within the molecule is changed from
that typical of the native protein to a more disordered
arrangement (Kauzmann. 1959). (a) as a process in which.a
protein or polypeptide is transformed from an ordered to a
disordered state without rupture of covalents bonds
(Scheraga. 1963). or (3) any process. except chemical
modification. not involving rupture of peptide bonds Which
causes a change in three-dimensional structure of a protein
from.its native in-vivo form.(Haschemeyer and Haschemeyer.
1973). These definitions suggest that denaturation is not an
'all-or-nothing" phenomenon but rather a continuous process
with.various areas of the protein molecule changing at

different rates (Paul and Palmer. 1972).

11
Dr

tc

15

Tanford (1988) qualified Kauzmann's definition by
requiring that there be no alteration in the protein's
primary structure. Denaturation can. therefore. be
restricted to the continuous process of native protein
structural changes involving the secondary. tertiary. or
quaternary structure in which.alterations of hydrogen
bonding. hydrophobic interactions. and ionic linkages occur
during the transition to the denatured state (Anglemier and

Montgomery. 1976).

Bond energies that contribute to native structure and
maintenance Of a protein's conformation were published by
Acton and DICK (1984). These bonds can 8180 be Viewed as
important in protein that 18 denatured. Once a new structure
18 formed. the same types 0f bonding can contribute to the

stability 0f the new structure (Acton and DICK. 1954).

Denaturation involves protein-solvent interactions and
leads to changes in physical properties. such as loss of
solubility of the protein. Sometimes unfolding of the
protein structure is considered part of denaturation
(Gossett gt 9;. 1984). Denaturation is usually irreversible
if the methods are drastic. the molecular weight of the
protein is large and aggregation occurs to prevent a return

to the native state.

16

Aggregation is a general term which.has been used to
describe many types of protein-protein interactions. with
formation of complexes of higher molecular weights
(Hermansson. 1979). Aggregation is usually governed by a
balance between attractive and repulsive forces. Attractive
forces can involve hydrogen bonds, covalent bonds such as
disulfide linkages. and hydrophobic association. whereas.
repulsive forces can involve coloumbic forces which.are
affected by the net charge of the protein molecule or the
ionic strength of the solution (Egelandsdal. 1980).
Aggregation causes the meat protein matrix to shrink which
limits the amount of water the matrix can bind and reduces
the strength of the forces immobilizing the water. The end
result of these changes is a decrease in water holding

capacity (Hermansson. 1988).

Coagulation is the random.protein-protein interaction
of denatured protein molecules. in which.polymer-polymer
interaction are favored over polymer-solvent reactions
(Schmidt gt 2;. 1981). The coagulum.is often turbid. and the
formation of the coagulum is usually thermally irreversible
(Shimada and.Hatsushita. 1981). A coagulumLmay settle out of
solution because randomness does not lead to an orderly

structural assembly 0f the final aggregate.

 

 

 

17

Gelation is the orderly interaction of proteins. which
may or may not be denatured and which leads to formation of
a three dimensional well-ordered structural matrix
(Hermansson. 1978). Polymer-polymer and polymer-solvent
interactions. as well as attractive and repulsive forces are
balanced such that a well-ordered matrix can be formed
(Schmidt 2; al. 1981). Since denaturation is involved in
this definition of gelation it is evident that native
protein structure is altered when the gelled protein matrix

18 formed.

Mechanism 2; Protein Gelation
The ClflSSlC explanation 0f protein gelation 18 the

two-step process proposed by Ferry (1948):

Native protein --> denatured protein --> aggregated protein

The first step is considered a denaturation process and the
second step an aggregation process. Comparison of the rate
of the denaturation step vs that of the aggregation step
helps determine gel characteristics. For example. Ferry
(1948) suggested that for a given rate of denaturation the
rate of aggregation will be slow if the attractive forces
betweenfthe denatured proteins chains are small. The
resulting gel will be a finer network of proteins chains.
will be less opaque and will ethibit less syneresis than one

with a faster rate Of aggregation. A coarser network Of

18

protein chains yields an opaque gel with large interstices
capable of holding solvent which is easily expressed from

the matrix.

Hermansson (1978) elaborated on Ferry's mechanism
stating that contrary to coagulation. where aggregation of
the protein molecules is random. gelation involves the
formation of a continuous network exhibiting a certain
degree of order. Furthermore. when aggregation is suppressed
prior to denaturation the resulting network can be expected
to eXhibit a higher degree of elasticity than if random
aggregation precedes denaturation. The slower the second
step relative to the first. the more orderly the denatured
chains will tend to orient themselves and therefore a more

finer gel network will be produced.

Ferry (1948) estimated that the number connecting
points between protein molecules was very small. and that
for gelation as few as 5—6 loci per chain were sufficient to
form a rigid network. The mechanism of gel formation differs
among proteins. predominantly in the type of interactions
which.stabilize the gel (Schmidt gt 3;. 1981). The type and
strength of these connections influence the response of the
gel to stresses such as mechanical agitation or temperature
change (Paul. 1972). Anglemier and.Hontgomery (1978) stated

that long proteinaceous fibers form a three-dimensional

19

network primarily through the establishment of interprotein
hydrogen bonds. and probably not through salt bridges which

would be highly solvated in aqueous gels.

Schmidt 5; _l. (1981) suggested that if aggregation
occurs simultaneously with denaturation an opaque.
less-elastic gel results. During the storage of a gel
syneresis or loss of fluid may occur as a result of the
formation of additional interprotein bonds which.decrease
the number of loci available for binding water and reduce

the amount Of intermolecular space available 1.0 imobilize

water through capillary forces.

Since the kinetics of the denaturation step relative to
the aggregation step appear to be important in determining
the type of gel produced. some kinetic terms that aid in
describing the gelation process will be reviewed. The first
is the reaction rate constant k (min’l). which.is obtained

from.the first order relationship (Lund. 1975):

_. .39... = kC (1)
dt
mere C 18 concentration aha ‘dC/dt 18 the rate at which

concentration decreases. Rearranging Equation (1) we obtain:

dC

C = kdt (a)

20

Integrating between limits C1 at time = o and C at time t

Equation 2 gives:

LnC = Ln C, _ kt (3)

A plot of Ln C vs t gives a line of slope -k.
The rate constant is usually temperature dependent and

can best be described by the Arrhenius equation:

k = S e-Eo/RT (4)

where S 8 frequency factor (min ")3

Ea 8 activation energy (cal/mole);
R a gas constant (1.987 cal/mole °K);
T - absolute temperature (°K).

A plot of Ln k vs i/T gives a straight line of slope

-Ea/R.

Hechanism g; Gelation g; myofibrillar Proteins

For muscle proteins during processing. thermal energy
is the single most important driving force in protein
transition from the native state to the denatured state. For
myosin and the actomyosin complex it is a continuous process
of native protein structural changes involving secondary.
tertiary and/or quaternary structure. Hydrogen bonding.
hydrophobic interactions and electrostatic linkages are
altered during transition.to the denatured state (Anglemier

and Montgomery. 1976). While electrostatic and hydrogen

21

bondings become weaker upon heating the potential for
hydrophobic bonding increases and with conformational
changes there 18 greater tendency for more interchain

hydrophobic interaction to occur (Acton and Dick. 1984).

Heat induced formation of a three-dimensional protein
matrix by myosin and actomyosin. termed gelation. can be
represented by two stages of reactions. Each.stage involves
distinct segments of the myosin molecules. More critically.
the stages occur in separate temperature regions during
heating. One stage of aggregation occurs between 30 and
50 °C and the second stage occurs at temperatures > 50 oC.
Thus. in the protein— protein interactions of myosin. each
stage by temperature region can be represented independently
using Perry’s two-step sequence of reactions. It is obvious
that in applying this to processed meats. heating to a final
internal temperature of between 65 and 71 °C involves both
stages as both temperature regions are traversed (Acton and

DICK. 1984).

The first stage involves aggregation Of the globular
head regions Of the molecule. It 18 an irreversible reaction
assuming heating will be continuous with continuous
temperature elevation On the system. Through studies with
the 3-1 fraction. m (heavy meromyosin) segment and myosin.
the aggregation 18 thought ‘10 be dependent On oxidation Of

-SH groups which are predominantly found in the globular

22

head region (SameJima gt 3;. 1981; Ishioroshi gt 3;. 1979).
While -SH group reduction (moles -SH/mole segment)
progressively increases from 20 to 70 0C. considerable
reduction of -SH content occurs in the early temperature
range of 20 to 50 °C for myosin or segments containing a

globular head position (Acton and Dick. 1984).

Studies from Ishioroshi g; 3;. (1980) support the role
of sulfhydryl group involvement in head-to-head aggregation
as one factor of the protein interactions. For sulfhydryl
group oxidation to occur it is necessary for head-to-head
aggregation to have been preceded by a rapid conformational
change in the head region. particularly if covalent
disulfide bonds are formed. Foegeding gt 3;. (1983) reported
that myosin gels heated to 70 °C were stabilized by
non-covalent and disulfide bonds. In addition. they reported
that gels heated to 50 °C were less difficult to
solubilize (by guanidine hydrochloride and urea) than gels

formed at 70 °C.

Liu 3; al. (1982) concluded on the basis of ease of
solubilization (by ex SDS) of actomyosin gels heated to
48 °C. that hydrophobic interactions were the predominant
force in actomyosin aggregation below so °C. Solubilization

of myosin gels heated to 50 °C by guanidine hydrochloride

23

and urea. as reported by Foegeding gt 3;. (1983) also
implies that hydrophobic and.hydrogen bonding are more

important than sulfhydryl group reduction.

The two-step sequence Of conformational change in the
head region and head-to-head aggregation satisfies Ferry’s

mechanism in the first stage between 30 and 50 0C (Acton and

Dick. 1984).

The second stage is associated with structural change
of the helical rod segment of myosin which culminates in

network formation through cross linking of these segments.

While the globular head interaction predominates in the
first stage there is also apparent early disruption of the
o-helix at the hinge region in moving from.the coiled-coil
(o-helix) to a random coil type structure in the same lower
temperature region. Further helical disruption of the tail
portions requires a higher energy input. thus these helical

alterations predominate in the second stage at temperatures

> 50 °C.

The coiled-coil to random coil conformational change in
the tail region is extremely important to aggregation
occurring in the second stage of events. The LHH (light

Ineromyosin) (Ishioroshi et 1. 1982) and.the myosin rod

24

(SameJima _3 _l. 1981) ultraviolet absorption difference
spectra at 285 nm. where the aromatic side chains of the
protein absorb. confirmed that absorption increases as both
segments are heated from 20 to 85 °C. In addition several
workers have reported an increase in binding and enhancement
of fluorescent intensity for actomyosin (Niwa. 1975) and
myosin (Lim and Botts. 1987) upon heating in the presence of
ANS (8-anilino-1-naphtalene sulfonate). ANS is a
fluorescence probe capable of binding with.hydrophobic
regions of proteins when.the conformational structure allows
reaction with.nonpolar residues. The binding of ANS is
initiated at temperatures beginning in the 35-45 °C

range. The fluorescent intensity increases with.further
temperature increase. The exposure of hydrophobic residues
facilitates hydrophobic interactions. and thus. increases

the potential for tail-to-tail cross-linking in establishing

the gel framework. (Acton and D1CK. 1954).

In the gelation type protein-protein interaction. it is
evident that there is a second reaction to aggregation
driven by thermal energy input where the product of the
first step becomes the reactor for aggregation in the second
step. This sequence of reactions or steps in the formation

Of three dimensional protein matrices has been proposed by

.Acton t l. (1983).

25

Effect gt Temperature gg _gt Strength
Yasui gt gt: (1980) investigated the effect of

temperature on rigidity of rabbit myosin and actomyosin
gels. They found that the gel strength started to increase
at 40 °C and reached a maximum at 60 °C. A similar result
was obtained by Grabowska and Sikorski (1976) using fish
myofibrils. with the difference being that the increase in
gel strength started at 30 °C and continued up to a
temperature of 80 °C. These results were confirmed in
principle by the work of Quinn gt gt. (1980) who showed.
using differential scanning calorimetry. that denaturation
of meat (beef) proteins begins at about 50 °C and continues
with increasing temperature up to 90 °C. This work and that
of Wright _t g (1977) shows that the temperature range of
denaturation of the different protein components is a
characteristic of the species of animal from.which.the
protein came. the pH and the ionic strength. This
information explains the varying results obtained by

different workers in this area. as each group used different

combinations Of animal species. PH and ionic strength.

Acton (1972) investigated the effect of temperature of
cooking on the binding ability of poultry loaves. The
results obtained on binding ability were basically the same
as those obtained by workers measuring gel strength of

isolated proteins. In essence. the binding strength.started

26

t0 increase at 40 oC and reached a maximum at 50 °C. after

which it decreased slightly with temperature up to 100 °C.

The effect of temperature on the binding ability of
crude myosin (beef) was investigated by Siegel and Schmidt
(1979). Their results showed that binding strength started
to increase at 55 °C. then increased linearly with
temperature to 80 °C but did not show the same decrease in
binding ability after 80 °C as reported by Acton (1972). The
difference in response of binding ability to temperature
obtained by the two different groups of workers may have an
explanation similar to that given for the variation in gels
of purified proteins. which was the difference in species of

animal. PH and ionic strength used by the two groups.

From all the gelation and meat binding studies. the
temperature region above 50 00 appears most critical. Gels
do not reach appreciable strength until the myosin tail
portion has undergone helix-coil transformation and
subsequent cross-linking. The myosin head region is
important since. from ultrastructure studies. it appears to
form the initial super-Junctions upon which.the super-thick
filament network interlinks (Siegel and.Schmidt. 1979:
SameJima _t _t: 1981). Similar studies of gelation of
natural actomyosin (Deng gt‘gt. 1978; Acton gt gt. 1981;
Liu t l. 1982; Ziegler and Acton. 1984) have shown that

the ultrastructure of actomyosin gels is one of thinner

27

filamentous strands with larger pore size distribution and a
different cross-linked appearance when compared to myosin
gels (Yasui gt gt. 1982). These ultrastructure studies
showing morphological differences between myosin and
actomyosin gels imply that in processed meats. differences
in textural attributes between prerigor and postrigor raw
materials may emerge in the finished product (Asghar t l.

1985).

From.the research that has been done on the effect of
temperature on gel-strength and binding ability of meat
proteins. it can be concluded that there is an interaction
between the temperature of heating and the presence and
concentration of different salts. The exact interaction has
not been clearly elucidated. but the implications is that
the temperature at which maximum binding occurs is dependent
on the presence of specific salts and.hence the ionic

strength and pH (Quinn gt gt. 1980).

Theoretical gggtg gt tgg Mathematical ggggt

To evaluate the effects of heat treatment on the
thermal gelation of myofibrillar beef proteins a universal
method is needed by which.all heating processes can be
quantified and compared to a common reference point. This

method would facilitate comparison between controlled

86

constant temperature processes and variable temperature-time
treatments and incorporate effects of variable heating

rates. variable protein concentration and added gums.

A mathematical model was developed by Morgan (1987)
and is based on fundamentals of protein kinetics and polymer
rheology. The model combines theories of how temperature and
protein concentration at constant shear rate affect
viscosity of non-Newtonian fluids and protein polymerization
(gelation) reaction kinetics and assumes that apparent

viscosity is a relative measure of gel strength.

Concentration Effects
A logarithmic relationship of moisture to dough
viscosity was proposed by Harper gt gt. (1971). Later the
same group used a semiempirical logarithmic mixing rule to
describe the effect of moisture content on viscosity of corn
flour dough.and proposed a formula that predicts
concentration effects on viscosity of suspensions or

solutions.

NC = NCO e 8 (concentration) (5)

*where C is concentration of total dry matter. 8 is the
parameter which quantifies relative effects of

concentration. NC 18 the viscosity index at a particular

29

concentration and Nco is the viscosity of the solvent.
This model assumes that C remains constant and that the

trapped water has a lubricant effect.

Rheological and Kinetics Background

A pseudo first-order reaction is used as starting point
to model the heat activated chemical changes responsible for
thermal gelation of myofibrillar beef proteins. The reaction
model is based.on the assumption that rheological and
kinetic theory presently used in studying plastic polymers
might be used as starting point for modelling gelation of
protein solutions. Considerable theory has been developed
for predicting polymerization phenomena of plastics and
similar materials. Mathematical relationships have been
developed using molecular and physical entanglement theories
to predict rheological properties of polymers (Ferry. 1970).
Sha and Darby (1976) used molecular weight data to
successfully predict apparent viscosity of polyethylene
melts which.had weight-average molecular weights ranging

from 57.700 to 139.000 daltons.

Several problems are encountered.when attempting to
draw analogies between plastic polymers and protein
reactions. Polymers generally undergo various reversible
melting and irreversible polymerization reactions during
thermoplastic extrusion. However. proteins undergo

irreversible denaturation with network entanglement and

30

possible cross-bonding. Generally. first or second order
reactions are assumed with reactive plastic polymer and
monomer species. whereas the thermal gelation of
myofibrillar beef proteins involves several higher order
reactions. However. perhaps the end effect of this
heat-mediated gelation of myofibrillar proteins could be
described via a first order reaction analogy (Blum. 1960;
Penny. 1967; Deng gt gt. 1976; Ziegler and Acton. 1984). and
then a simple reaction could be used to approximate the
average overall viscosity effect due to thermal denaturation

of the major myofibrillar proteins.

When a solution of myofibrillar beef proteins is
heated. the proteins undergo irreversible denaturation with
network entanglement and cross-linking that involve several
higher order reactions which occur simultaneously or in
cascade. These denaturation reactions will affect their
size. shape and molecular weight. Since apparent viscosity
is related to all of them. it seems logical to assume that
molecular changes caused by the gelation reaction will

significantly affect the material’s apparent V186081tY.

The pseudo first-order kinetic model used in this study
to model temperature-time history effects of myofibrillar

protein denaturation (gelation) On apparent viscosity

31

assumes that concentration of the reactive polymer species
will remain constant and predicts disappearance of the

monomer species.

The pseudo first-order reaction is described by the

formula:

Mc(t) .-_ Mge-kt (6)

where Hc is concentration of the monomer at time t. N1 is
the initial monomer concentration. and k is the first order
reaction rate constant. The molecular weight of a polymer

can be approximated by:
MWp = MWm DP (7)

where HWp is the polymer molecular weight. HWm is the
monomer molecular weight. and DP the degree of
polymerization (Williams. 1971). Williams. (1971) also
approximated for pseudo first order polymerization the

number—average degree of polymerization (DP) as

_ Mi — Mc )
DP - ( PC (3)

 

where PC 18 the reactive polymer species concentration and

Hi and Me as describe in Equation 6.

32

Ferry (1970) reported a power law relationship for
correlating the zero shear rate limiting Newtonian viscosity

(fl) Of polymers with their mblecular weight:
a
77 = k2(MWP) (9)

where k3 is a viscosity coefficient and a a

dimensionless constant. Theoretically. a is derived to be
1.0 for low molecular weight polymers and 3.5 for high
molecular weight polymers. Sha and Derby (1976) reported
that observed values of c found in the literature range from

3.4 to 8.0 for high molecular weight polymers.

Therefore. it was assumed that effects of changing
molecular weight during protein polymerization is described

(at constant shear rate) by:
a.
TIM») = k1(MWP) ( 10)

where k2 is assumed to be dependent on t and Hc according to
Equation 6 with a material-constant factor included in the

expression.

Equations 6. 7 and 8 were combined.with Equation 10
yielding n. as a function of reaction time (t). for constant

temperature (T).

33

um - u.( “if-"£3- )“(1-6-“ a (11)

Equation 11 describes the increase in protein gel
viscosity due to heat-induced denaturation and assumes that
total dry matter remains constat and shear rate is very be
low (near zero). This increase is a function of time for all
temperatures (T) greater than the threshold temperature

(Ta) .

The pseudo first-order reaction used in developing
Equation 11 assumes that temperature is constant and greater
than the reaction threshold temperature. The coefficient k1
is defined as the polymerization rate constant. Kinetic

theory implies that k1 is related to temperature by:

-Ed/RT
k1 = kte (12)

Where k1 is a specific reaction constant and Ed is the
activation energy of protein denaturation. According to
Eyring and Stearn (1939) k1 is related to absolute

temperature by:

'T kt ko (13)
h

 

k...

where kt is a transmission coefficient. kb is Boltzman's

constant and.h is Plank‘s constant.

34

Application of Equation 12 and 13 to Equation 11
requires that temperature remain constant with time.
However. temperature in most experimental conditions. as
well as in almost all commercial processes. will increase
from its initial temperature to process temperature in a
variable time. Therefore. each process will result in a
distinctive and variable temperature-time history within
meat products. To meet this need an integral
temperature-time history (TTH) developed by Horgan _t gt.

(1979) was incorporated into Equation 11. This function is

defined as:

00

TTH = f T(t) e-AEO/mdt W“

0

where Ea : activation energy. R = universal gas constant and
T(t) is the temperature-time profile above some minimum

threshold temperature (Td).

Morgan (1979) developed the TTH function based upon
work by Eyring and Stearn (1939). TTH is defined as zero for
all T(t) below the Td. Its assumed that the gelation
reaction is not initiated until the temperature is at or
exceeds Td and its effect is a multiplicative increase in
Viscosity due to the myofibrillar protein thermal activated

gelation.

35

Assuming constant moisture. Equations 11 and 14 could
be combined to give an expression for the incremental change
in apparent viscosity due to the thermal gelation of

myofibrillar beef proteins.

77(T—t) = 770(1.+A(1-_-€'°m)“) (15)

where n represents the viscosity of the myofibrillar

protein in solution before heating. ”A" is a parameter which
relates the ratio of maximumlheat-induced gel viscosity to
initial viscosity. "a" is related to the reaction rate
constant for protein denaturation and cross-linking. o is a
function.of the shear rate and a relative measure of
molecular entanglement during shear and TTH is defined by
Equation 14. Equation 15 is the basic model utilized in this
study for predicting the combined effects of temperature.
protein concentration. time-temperature history and selected
hydrocolloids gums (carrageenan. guar. locust bean and
xanthan) on apparent viscosity of myofibrillar beef proteins

solutions. where protein gelation was heat induced.

Similar models have been used to model TTH effects of
protein denaturation on apparent viscosity index (Horgan

gt 1. 1988: Harper t l. 1978; Smith.gt gt. 1988).

Harper gt 1. (1978) successfully applied this approach to

heat setting Of bovine plasma proteins suspensions. whereas

Smith e l. (1988) found good correlation when they appl1ed

36

such a simple reaction model to approximate the gel strength
in the thermal gelation of chicken myofibrillar protein
suspensions and drew the analogy between kinetic losses of
protein tertiary structure and pseudo first-order
polymerization. This analogy assumes that increases in
viscosity due to protein gelation is analogous to the
viscosity increase brought about by increased polymer
molecular weight during a classical polymerization process.
This evidence supported the idea that such an approach was
adequate for modelling the thermally-activated gelation of

myofibrillar beef proteins.

Characteristics gt szrocolloids

Macromolecular hydrocolloids or gums. as some are more
commonly known. are used by the food industry as texture
modifying agents in many different types of products. The
term gum.refers to a wide variety of compounds including
polysaccharides of plant and microbial origin. animal
proteins such as gelatin and some chemical derivatives of
cellulose (Andres. 1975). Many gums have the ability to form
gels at low concentration. physically binding water into a
three dimensional structure. The water held by these gels
eXhibits physical properties similar to those of free or
bulk water and is not easily removed from.the structure when

physically stressed.

37

Hydrocolloids have several properties that are valuable
to meat technologists and are used for several different
functions. such as stabilizers and structure forming
(gelling) agents. These functional properties are related in
part to the ability to imbibe and retain large amounts of
water. to interact with proteins and to bind fat

(Wallingford and Labuza. 1983).

Some of the gums that have been reported in meat
products are: xanthan. guar. locust bean and carrageenan.
Guar gum is a very effective water binder in comminuted
meat products. canned meats and pet foods. The anionic
xanthan gum and locust bean gum have been shown to prevent
fat separation in canned meat. whereas. carrageenan
stabilizes the texture of frankfurter emulsions against acid
deterioration (Pedersen. 1960; Abd El-Baki gt gt. 1981;

Fox. 1983).

Xanthan gum is a high molecular weight polysaccharide
gum produced by a pure culture fermentation of a
carbohydrate with.Xanthomonas cggpestris. and is purified by
recovery with isopropyl alcohol. dried and milled. The
linear portion of this colloid is composed of repeating
units of D-glucose and is chemically identical to cellulose.
The side chains which.accounts for xanthan's water
solubility are made up of D-mannose and D-glucuronic acid

subunits and also contain approximately 3% by weight

36

pyruvate (McNeely and Kang. 1973). It is a white to cream
colored powder that is readily soluble in hot and cold water
and is prepared as a mixture of potassium and sodium salts.
Because of its chemical structure it is able to form highly
viscous and stable solutions at low concentrations at room
temperature. These solutions eXhibit pseudoplastic
characteristics as a result of the rigid cellulosic backbone
that is stabilized by the side chains (Morris _t _t. 1977).
The principal properties exhibited by solutions of xanthan
gum which are important to the meat industry are: a high
degree of pseudoplasticity. a high tolerance to salts. very
high stability towards extremes of temperature. pH. ionic
strength and shearing force. an ability to suspend

particulate matter. the synergy with galactomannan gums and

the resistance to enzymatic degradation.

Guar gum is a seed gum.composed of linear chains of
D-mannose with numerous short side units composed of
D-galactose. It has a molecular weight of approximately
220.000 daltons and forms colloidal dispersions in cold
water (Goldstein and Alter. 1973). Guar gum is not highly
branched. with.the side chains consisting of single
galactose subunits. Therefore. guar dispersions. which.have
fairly high viscosities in comparison to more highly
branched molecules of equal molecular weights. should absorb
less water but still give a relatively high.water holding

capacity.

39

Locust bean gum consists of a linear chain of D-mannose
as does guar gum. However. it differs in the level of
substitution of D-galactose on the side chains. with one
substitution every fourth or fifth molecule of mannose
compared to every second molecule for guar gum (R01. 1973).
It has a molecular weight of the same order as guar gum.
approximately 310.000. Locust bean gumlhas a fairly low
water holding capacity (WHC) at 515g water per 100g dry gum
solids. The dispersability of locust bean gum at room
temperature appears to be the key to this gum’s low WHC.
Crystalline regions within the gum's structure fail to
solubilize at room temperature. only breaking up as the
solution is heated. Cold water dispersions of locust bean
gum have a significantly lower viscosity (about ten times)

than hot water dispersions (R01. 1973; Andres. 1975).

Carrageenan is not a well-defined substance. but rather
a designation (a family name) for a group of salts of
sulphated galactans. They have been defined as that group of
galactan polysaccharides extracted from.red algae
(Rhodophyceae) of the Gigartineceae. Solieracea. Hypneacae
and Phyllophoracea families. and.that have an ester content
of 202 or more and are alternately a 1-3. B 1-4
glycosidically linked. Various types or fractions of
carrageenan are defined according to idealized structures

and designated by greek letters lambda. kappa. iota. etc.

40

The different carrageenan fractions (kappa. iota.
lambda. etc.) occur in varying ratios in various red
seaweeds. By selection of seaweeds it is. therefore.
possible to obtain carrageenans which are predominantly of
one type. Blends with controlled intermediate properties are
produced by blending extracts or seaweeds before the

extraction Step.

Galactose is the most common repeating monomer. The
solubility of carrageenan depends on the hydrophilic
sulfate half-ester groups present and the galactopyranosyl
unit. and therefore. a range of solubility is found for the

various types of carrageenans (Stoloff. 1973).

Carrageenan is reported to have a molecular weight of
about 300,000 daltons (Marine Colloids. Inc.) which.1s much
lower than that of xanthan gum. Carrageenans are capable of
forming viscous solutions at low concentrations in cold
water with the viscosity dependent on temperature. pH.
concentration. type of carrageenan molecules and solutes

present.

METHODS AND MATERIALS

Experimental Design
This study was conducted in two parts. the first of

which.was a model system experiment designed as the .
experimental base for developing a mathematical model for
the thermal gelation of myofibrillar beef proteins. The
second part of this study was designed to test if the model
developed in part I could be applied to the thermal gelation
of myofibrillar beef proteins combined.with.selected

hydrocol lOidS.

Experiment t

This experiment was designed to be the experimental
base for testing a proposed.mathematical model developed by
Morgan _t gt. (1987) and to determine if it could be applied

to,a model meat system of myofibrillar beef proteins. The

mode 1 proposed was:

77(T-t) = no(1+A(1-e‘°"“)“)

where individual model components were described

previously.
This model describes the gelation of myofibrillar beef

41

42

proteins (measured as the back-extrusion apparent viscosity)
as a function of the time-temperature history of the process
and the protein concentration of the samples. The experiment

was carried out as described below.

gggt Sample Preparation. Beef was excised from the bottom
round (biceps femoris) of six young bulls slaughtered at
the meat laboratory facility (MSU) and allowed to age three
days. The exterior and seam fatty tissue and the epimysial
and perimysial connective tissue deposits were physically
removed prior to grinding the muscles. After passing the
tissue twice through a chilled grinder with plate orifices
of 4.8 mm diameter. the muscle mince was divided into lots
of approximately 600 g. vacuum packaged and stored at

-30 °C. until required for the experiment.

Isolation gt Mzofibrillar Proteins. Myofibrillar beef
proteins were extracted in a 2 °C cold room.following a
procedure describe by Eisele and Brekke (1981) and Smith

_t _t. (1988) with some modifications. The frozen ground
meat samples were allowed to thaw overnight in a 2 °C

cold room. A meat sample (about 600 g) was then blended for
30 sec in a Waring blender at maximum speed with 4 volumes
(2400 m1) of 0.1H sodium dhloride and 0.05M sodium phosphate
buffer. pH = 7.1. The suspension was stirred for 60 min at

1200 rpm using an electronically speed-control led stirrer

(Heller GT-Zl) equipped with a LH Jiffy Mixer stirrer shaft

43

(Thomas Scientific Apparatus. cat. 8634-820) to avoid air
incorporation into the protein suspension. The solution was
transferred to 250 ml centrifuge plastic bottles and
centrifuged at 9000 x G for 15 min at 0 °C in an automatic
refrigerated centrifuge (Sorval RC-2B). Any connective
tissue which accumulated on the propeller was discarded. The
supernatant containing fat and sarcoplasmic protein was
discarded and the pellet resuspended in 2400 ml of fresh
buffer. The extraction procedure of stirring. centrifuging.

and resuspension was repeated 3 times.

The final pellet was analyzed for nitrogen content and
adjusted to a selected protein concentration (1. 2. 3. 4. 5
or 6%) and to pH 6.5. Buffer salts concentration in the
water phase were 0.5z (w/w) sodium.chloride and 0.5% (w/w)

sodium pho sphate .

gggt Treatment. Protein suspensions were transferred to

16 x 100 mm disposable culture tubes (approximately 10 g per
tube) and thermally processed in a water bath (model FG-103.
Eberbach. Corp.. Ann Arbor. MI.) at six different
temperatures (54. 64. 70. 75. 80. and 85 00). Samples were
taken at variable intervals ranging from 30 to 13.000

sec. Heat-treated protein solutions were immediately
transferred to an ice-water bath. permitted to cool for 30

munutes and stored overnight at 2 °C.

44

During the thermal process the temperature at center of
each tube was monitored with a thermocouple thermometer
(model 450-TT. Omega Engineering. Inc. Stamford.CT.)
inserted in the center of the tube. Temperature was recorded
every 30 sec until the center of the tube reached the

temperature of water bath.

ggt Strength. Before measuring the gel strength. cooked
protein samples were transferred to a water bath.at 20 °C
and allowed to equilibrate for 2 h. Gel strength was
evaluated as the back extrusion apparent viscosity using an
Instron Universal Testing Machine (Model 4202. Canton. OH)
equipped with a 50 N load cell and coupled with a
microcomputer (Hewlett Packard 86B). The computer ran a
program specifically developed for this experiment
(program Rodrigo. appendix A). This program established
Instron operating variables: speed 20 mm/min. travel
distance 30 mm. load calibration cell 50 N. Distance (mm)
and Force (N) were read by the program every 300
milliseconds as a 7.33 mm.diameter plunger (flat tip)
penetrated the gel at constant speed. Distance-force data
were used by the computer to calculate the back-extrusion
apparent viscosity and.the apparent elasticity using the

procedure described by Hickson gt gt. (1982).

The peak plunger force was calculated as the equivalent

force of a linear force deformation triangle which resulted

45

in an area under the force deformation curve equivalent to
the observed curve. The deformation base of the triangle was
the same as that of the observed curve. The equation for

computing equivalent peak force (Fp) was:

2 AREA (16)

F =
p Lp

area under the force deformation curVe;
plunger travel distance..

where area
Lp

The viscosity index (n) was computed using Equation 17

(Hickson. gt gt. 1982).

l

77 = 5.7)('3)(1-K2)Ln(%)(1+fi <17)

where Vp = velocity of plunger (mm/min);
K a ratio of plunger diameter to tube diameter;
a = (1-K=)/(1+K=);
Lp = plunger travel distance (mm);
Fp = plunger force as describe before (N).

Evaluation gt tgg Thermal Process. The time-temperature
history (TTH) of the gels was calculated with.a
macrocomputer (Hewlett Packard 86B) using a program
specifically developed for this purpose (program Mariana.
Appendix C). Basically the program used the experimental

time-temperature data collected during the thermal gelation

Of the myofibrillar proteins and calculated the Fourier

46

number (Fo) and.the dimensionless temperature ratio 9

using Equations 18 and 19.

kt

(18)

F0:

 

Cp rzp

_ (TL—Too)

9 _
(Tc—Too)

 

(19)

where t = time in seconds;
r = internal radius of test tube;
Cp = caloric capacity calculated as 1.675
+ 0.025 Moisture;

p : density as weight/volume (1069 Kg/m3); -
T1 = initial temperature of sample (normally 20 00);
Tc = temperature at the center of test tube (°C);
T0° = temperature of the water bath (°C);

k : thermal resistant constant from Equation 20.

Thermal resistance values were not found in the
literature for this type of gel. Therefore a formula which
gave "k" as a function of temperature was developed. For
this purpose. data published for veal. lean beef. and pure
water were used (Heldman. 1985). Because the value of ”k"
changed with moisture and temperature. an average "k” for
902 moisture was calculated (i.e. the moisture expected in
this gel). Then using values for 40 and 80 °C. the

following linear relationship was created:

k = 0.274 + 1.4146 E(-—4) T (20)

47

where T: temperature in °C.

With the above data the Biot number was calculated for
each one of the experimental time-temperature points using
an approximation to the general equation. This approximation
was found to be good for a range of F0 values greater than
0.15. The approximation was calculated using a
simplification of the general formula for transient heat

transfer in an infinite cylinder (Equation 21).
2
. “lint-o
f
a = 2 Momfie (w

2 Bi
Idem) (In-sf ) ]

where A .. =

 

e = as defined in Equation 19:
Jo : Bessel function;
u = root 1 of Bessel function;
'r : radius of point for temperature 1;
R : radius of tube;
F0 2 as defined in Equation 18;
Bi : Biot number.

For the special case Of the center Of the tube (r/R -
i) and considering only the first root Of Ii Equation 21

could be written as:

2F

(9 == AuJo);L:}€3

or

46

Ln(6) = Ln(A1)+Ln(Jo§p1})—p.fFo (23)

Using values reported in the literature (Luikov. 1968).
it was found that for F0 > 0.15 Biot number was related to u

and A. (R3 = 0.99 and 0.989 respectively) by the

equations:
435842959
Ln(B.) = 0.08359... (24)
and
- 8£W8
Ln(Bi) = 0.060637 A: (25)

Combining Equations 34 and 25. 1t 18 found that for

this special case A1 is related tO u by:

Ln(A1) = 0.03974+O.5395 Ln(,u.) (26)

When n: 1. the argument of the Bessel function
approximates (0) and then Jotu): 1 and.therefore Ln(Jo[u))

- 0. When this value and Equation 26 are substituted in

Equation 23. it gives:

(27)
Ln(6)*uaFo-0.03974-0.5395 Ln(u)=o

49

A computer subroutine (program Mariana) using the secant
method was utilized to find n values that satisfied
Equation 27 and then utilized these values to calculate a

BiOt number for each experimental time-temperature point as:

4seeumo9

. 359
Bi':= 80 08 l1:

(28)
Followed by the average Biot for a user-selected range.

After defining the Bi number from experimental data.
the time-temperature profile was calculated based on a
transient heat transfer model for infinite cylinders
reported by Parker gt gt. (1970). This model predicts
temperature-time data for ten different radial positions
within a given test-tube cross-section based on the initial.
center and bath.temperatures for each treatment. The
cross-section was divided into ten concentric rings defined

by the preselected radial position.

The computer program utilized the general formula for
an infinite cylinder again (Equation 21). but in this case
r/R was a variable and u was calculated to its sixth.root.

Then the general equation could be represented as:

5 -IJ:Fb
6 = awash-.26

50

where u roots where calculated using Equation 30.

ano(;.Ln)-B.Jo(pn) = O (30)

The solution of the general equation gave a value of
e as function of r/R and time (as Fo). Using the calculated
value of 9. temperature Tc for any point r/R at any time
(F01) could be calculated by rearranging of Equation 19. as

shown in Equation 31.

T. = 6(T.—T-)+T. (31)

This generated an 11 x 60 time-temperature matrix
which was used to calculate the effect of the temperature at
each selected r/R point as an A' factor. The A' factor was

defined as:

-&VRT

A' = Te <32)

Activation energy:
Absolute temperature (°K):
Ideal gas constant.

where Ea
T
R

The TTH value was the accumulative effect of the
average A' of two points times the time interval between

these two points (30 sec) as represented by Equation 33.

51

 

TTH = §<A—(A‘2—1)> (33)

These procedures calculated the TTH for each ring

which.then were averaged over rings to obtain a final TTH

average.
Determination gt Activation Energy. The activation

energy of this pseudo first order reaction was calculated in
two steps. The first step estimated a range of values for
the activation energy. To do that. high. medium and low
values of apparent viscosity were selected in such a way
that values for time of cooking at different temperatures
were obtained for the same apparent viscosity (protein
concentration = 4.092). A plot of natural logarithm of time
versus the inverse of the absolute temperature of cooking
yielded a slope equal to -Ea/R where R = constant of ideal
gases (1.987 cal/°K-mol). These plots gave a range of
values for activation energy between 16.000 and 42.000

cal/mol.

The second step calculated the final Ea value by an
optimizing computer routine. TTH values for Ea = 16.000 and
42.000 cal/mol were plotted versus apparent viscosity and
the linear range selected. Selected values for the linear
region (all temperatures) were utilized as starting points.

The computer routine basically did the following: upon an

52

input of Ea value. calculated TTH values for each
temperature. plotted them versus n and calculated the
regression line. Then. Ea was adjusted and TTH values
recalculated until a maximum regression coefficient for the
regression line was obtained. A microcomputer (Zenith 148)
with the computer package Framework II (version 1.0) linked
to the graph-statistical package Plotit (version 1.1) were

used for these calculations.

A first estimated value for the material constants A
and a were calculated using the computer program.Mariana.
The final values of these constants were calculated using a
non-linear regression analysis (Marquardt. 1963). These
calculations were performed by a microcomputer (Zenith 148)
with the computer package Framework II and the graph

statistical package Plotit.

Experiment tt
This experiment was designed to test the hypothesis

that the mathematical model developed in experiment I could
be applied to the thermal gelation of myofibrillar beef
proteins combined with selected.hydrocolloids. The gums
utilized were Xanthan gum (Miles Laboratories. Inc.). Guar
gum (Colony Import & Export Corp.). Locust bean gum.(TIC
Gums. Inc.) and Carrageenan gums (Gelcarin XP 4039. FMC

Corp. ) .

53

The model meat system used was basically the same as
the one used in experiment I. The main differences were that
protein concentration was selected to be 2.5% and
temperature of cooking was 70 °C. Concentration of gums was
0.5% and they were added mixed with the salts (sodium

Chloride and phosphate salts).

Chemical and statistical analysis. as well as
mathematical calculations. were identical to those in
experiment 1, except that the Ea value was assumed tO be

equal to that calculated for the myofibrillar system.

Methods Qt Analysis

Determination gt ggtgp Holding Capacity

Water Holding Capacity (WHC) was determined on the
ruptured gel/protein suspensions after gel strength.testing.
The protein/gel in 16 x 100 ml culture tubes was centrifuged
at 2000 x G for 2 h.at 4 C in an automatic refrigerated
centrifuge (Sorval RC-3). The weight of the water released
was used to calculate the WHC as:

W.-W.

WHC = w, 100 (34)

 

where Wg 8 weight of the gel;
w. 8 weight of the water lost.

54

Buffer Preparation

Fresh buffer was prepared the day prior to the
experiment. The buffer batch was prepared as follows: 33.2 g
of monosodium phosphate monohydrate. 81.9 g of disodium
phosphate. 87.6 g of sodium chloride and 2797.3 g of glass
bidistilled deionized water were mixed with an magnetic
stirrer until all the salts dissolved (approximated 20
min.). The concentrated salt solution was tranferred to a
20 1 plastic bin and 12 1 of water added with agitation. The
pH of the solution was checked and found to be between 7.1
and 7.2. The buffer solution was allowed to cool overnight

in a cool room at 2 °C before use.

Protein Determination

Nitrogen was determined by the Micro-Kjeldhal procedure
(A.O.A.C.. 1985. 23.009) using a Buchii automated nitrogen
analyzer 322/342 equipped with.an Epson HX-20 minicomputer.
Protein samples weights were between 0.2 and 0.3 g and the
protein content determined as 6.25 times percent nitrogen.
Concentrations of chemicals were hydrocloric acid 0.05M.
boric acid 4% w/w and sodium hydroxide 30x w/w. Water was

glass bidistilled and deionized.

Statistical Analysis
Linear regression analysis was performed by the least

square method using the integrated graph-statistical program

55

Plotit (Eisensmith. 1985). Nonlinear regression analysis was
performed w1th.the algorithm for least squares estimation of
non-linear parameters developed by Marquard (1963) using the
integrated graph-statistical program Plotit (Eisensmith.

1961).

Computer Programs Utilized

This experiment required intensive use of computer time
to expedite the mathematical development of the TTH model
for myofibrillar beef proteins. The computer software
utilized were of two types: (a) availabe commercial programs
and (b) programs especially developed for this experiment.
The commercial programas utilized were: Framework II
(Ashton Tate. 1985). Plotit (Einsensmith. 1985) and

TK!Solver (Software Arts. Inc. 1963).

Specifically developed for this experiment were three
computer programs. They were named: program Rodrigo. program
Mariana and program Delia. These programs were developed in

Basic for a Hewlett Packard 86B computer.

Program.Rodrigo

This program was developed to: (1) calibrate and
control the working parameters of the texturometer INSTRON
(speed. travel distance. calibration of load cell.
calibration of plotter and calibration of load balance). (2)

to read every 300 milliseconds the distance-force data

56

generated as a 7.33 mm diameter plunger (flat tip)
penetrated the gel at predetermined and constant speed. (3)
to calculate the back extrusion apparent viscosity from
these data and (4) to save these values for future

reference. Figure 1 is a flow diagram of this program.

Program Delia

This program was developed to read raw data generated
and saved by the program Rodrigo. It calculated the back
extrusion apparent viscosity and the shear force at the tube
wall from.the distance-force data generated and saved during
the texture measurement. Figure 2 shows a flow diagram of

this program.

Program.Mariana

This program.was developed mainly to estimate the time
temperature history of the process. This program calculated
the following individual parameters of the thermal process:
the Fourier number. the e value. the Biot number. the time
temperature matrix for ten concentric rings. the TTH values
for calculated concentric rings and an all-over average TTH.
This program was also designed to calculate a first estimate
of some of the constants of the basic model (Equation 15)

namely: the (a). the d and ’1' values.

The thermal process was calculated using the

experimental time-temperature data collected during the

57

POWER
ON

 

 

LOAD PROGRAM

 

 

   
     

  

OUTPUT INSTROM
WORKING PARAMETER

 

 

CLEAR >- INPUT SAMPLE
MEMORY DESCRITPTION

 

 

 

 

 

 

l READ FROCE AND—T
DISTANCE DATA

 

 

 

COMPUTE AREA,
MAXIMUM FORCE AND
APPARENT VISCOSITY

 

 

 

 

 

SAVE RAw DATA
T0 DISK

 

 

 

 

 

yes

 

Figure 1. Flow diagram of computer program Rodrigo.

 

CLEAR
MEMORY

 

56

POWER
ON

 

INPUT

 

 

 

i

 

 

 

V’ FILENAME

READ DATA FROM
DISK

- 1

[COMPUTE AREA,MAXIMUM

 

 

 

 

FORCE AND APPARENT
VISCOSITY

 

     
 
    
  

. OUTPUT
RESULTS TO
PRINTE'

  

MORE
SAMPLES

 

     

?

Figure 2. Flow diagram of computer program Delia.

59

thermal treatment of the protein gels. These time
temperature values were utilized to calculate the Fourier
number and the 9 values. which in turn were utilized to
calculate the Biot number. The next step divided the cross
section of the test tube (used for cooking the gel sample)
into ten concentric rings with equal area and then

calculated the time-temperature profile for each ring.

The next step calculated the TTH value for each ring.
using time intervals of 30 sec. Final TTH values were
obtained by averaging individual TTH values for each ring at
every 30 sec. The program contains provision for a variable
Ea.

The second section of the program calculated values of
a and A using a regression analysis of selected values of n
and their corresponding protein concentration (C). whereas a
preliminary value of (a) was obtained by iterating several
values of Y' and their corresponding TTH values using

Equation 35.

Y1 8(1-e'm')“

Y2 8(1'e-flm)‘

(35)

 

Figure 3 shows a flow diagram of the steps followed by

this program.

60

Figure 3. Flow diagram of computer program Mariana.

61

POWER
ON

 

 

LOAD PROGRAM

 

 

 

 

 

 

INPUT SELECTION

CALCULATION OF:

(1) THERMAL PROCESS

(a) RING TIME-TEMPERATURE
‘P F - PROFILE

(3) TTH

 

 

(4) A' AND C
(5) AN ESTIMATED OF Ea
(6) AN ESTIMATED OF (a)

 

 

 

 

 

 

 

 

SELECTED
(1 ) no
CLEAR
MEMORY yes
.4 INPUT EXPERIMENTAL
TIME-TEMPERATURE DATA

 

 

 

COMPUTE F0. 0 AND Bi

 

 

 

 

 

 

COMPUTE RINGS RADII AND yes
TIME-TEMPERATURE PROFILE SELECT
2

 

 

 

   
 

  

OUTPUT
RESULTS TO
PRINTER

no

 
     

 

PROFILE TO DISK

i v

[SAVE TIME-TEMPERATURE I

 

 

 

62

L_. 1

LI NPUT WANT TO yes
CHANGE Ea VALUE

 

no INPUT NEW
VALUE OF 53,

 

 

 

COMPUTE A'. TTH PER~
RING AND ALL-OVER yes SELECT
AVERAGE TTH 3

 

   

no

 

4’33

yes

 

 

 

INPUT Y'ATCTIME yes
INFINITE AND PROTEIN ~ SELECT
CONCENTRATION DATA -

no

 

 

 

COMPUTE A'. d AND R2

 

 

     
     

OUTPUT
RESULTS TO
PRINTER

 

 

 

 

 

«yes

63

yes

 

INPUT SETS OF TIME-
TEMPERATURE DATA

 

 

COMPUTE AVERAGE Ea

 

 

 

      
    
  

OUTPUT
RESULTS TO
PRINTER

 
 
  

MORE

 

no

 

yes

CALCULATIONS

   

 

SELECT

no

 

 

 

INPUT 3 SET OF VALUES//’ yes
01“ 1" AND TTH -<

 

 

 

COMPUTE (a) FOR SELECTED
VALUES OF C

 

 

 

T

      
     
  

  

OUTPUT
RESULTS TO
PRINTER

  

MORE

 

CALCULATIONS
7

      

no

RESULTS AND DISCUSSION

Determination gt tgg Activation Energy thgg

The value of the Activation energy (Ea) of this
pseudo-first order reaction was calculated in two steps. The
first step estimated a range of values where the Ea value
was expected to be. whereas. the second step focused in that
range to calculate the final value. This first step was
based on kinetic theory. which predicts the reaction rate
constant for a given Ea. temperature and time period
(Equation 12). For a selected time condition Equation 12

could be written as:

-Eo/RT ;
k5 = kne (36)

If the natural logarithm of both terms is obtained and
this is followed by a rearrangement of the terms. then

Equation 36 could be written as:

Ln(t.) = Ln ._k_>+ E]. _ (37)
Equation 37 implies a linear relationship between Ln

time (sec) and.the inverse of the temperature (0!) of

cooking. The slope of the line is the Ea (cal/mol) divided

by the constant Of the ideal gases (R=1.987 cal/mol°x).

64

65

A basic assumption of these experiments was that the
heat-induced back-extrusion apparent Viscosity (Y') measured
the extent of the thermal induced gelation reaction of the
myofibrillar beef proteins. Therefore. by using the time to
obtain a predetermined reaction extent (Y': 1.0. 2.0. 2.5.
3.0 and 3.5) at different experimental temperatures it was

theoretically possible to calculate the Ea of this reaction.

This first step was carried out as follows: A protein
solution (4.22) was thermally processed at 54. 64. 70. 80
and 84 °C and the Y' calculated for different
time-temperature conditions (Table 1). The back-extrusion
apparent viscosity (BEAV) induced by the heat treatment was
found to be a function of time for the experimental range

and.follow the mathematical model described by Equation 38.

Y' = A(1-e"')c (38)
where Y'= thermally induced BEAV; .
A,B.C = constants related to the basic model (Equation
15);
t I time (sec).

Experimental Y' and related time values were utilized
to obtain the constant (A. B and C) values for each
temperature Of cooking. using a non-linear regression

algorithm (Marquard. 1963). The calculated constants A. B

Table i. Thermally induced back-extrusion apparent viscosity

66

(Y') for the time range of 30 to 350 sec and for five
different temperatures.

 

seconds.

30

90
120
150
180
210
240
270

330
360
390
420
450
480
510
540
570
600
630
660
690
720
750

 

T ERAT °C

g4 -g4 70 (go 84
.19 .06 .06 .11 .30
.30 .19 .32 .54 1.21
.39 .37 .75 1.15 2.15
.47 .56 1.25 1.81 2.83
.55 .77 1.75 2.40 3.28
.62 .99 2.20 2.91 3.54
.69 1.20 2.58 3.31 3.70
.75 1.42 2.90 3.63 3.79
.81 1.62 3.15 3.86 3.84
.87 1.81 3.35 4.04 3.87
.92 2.00 3.50 4.17 3.88
.98 2.18 3.62 4.27 3.89
1.03 2.34 3.71 4.34 3.90
1.08 2.49 3.78 4.39 3.90
1.13 2.64 3.84 4.43 3.90
1.17 2.77 3.88 4.46 3.90
1.22 2.89 3.91 4.48 3.90
1.26 3.01 3.93 4.49 3.90
1.31 3.11 3.95 4.50 3.90
1.35 3.21 3.96 4.51 3.90
1.39 3.30 3.97 4.52 3.90
1.43 3.38 3.98 4.52 3.90
1.47 3.45 3.98 4.52 3.90
1.51 3.52 3.98 4.53 3.90
1.55 3.58 3.99 4.53 3.90

 

67

and C values are shown in Table 2.

Calculated models were found to predict reasonably well
the experimental data (CD = 0.95 to 0.99). and therefore
were utilized to estimate the time required to achieve a
preselected Y' (1.0. 2.0. 2.5. 3.0 and 3.5) for each

temperature used. These values are shown in Table 3.

The first range of values for Ea was calculated by
plotting the Ln of time (sec) and the inverse of absolute
temperature (Figure 4). This method of calculating Ea value
assumes that the temperature of cooking is constant
throughout the thermal process and that the sample attains
the temperature of the process instantly. However.
experimental values showed that the equilibrium temperature
required a relatively long time. which is explained by the
relatively low heat transmission coefficient of this protein
solution. Experimental samples arrived at equilibrium
temperature (bath temperature) at about 10 min. which is the
time range where most of the thermal induced.changes in BEAV
occurred. This explains why different time-temperature
ranges produced different estimates of Ea. nevertheless
these values indicated the range where the Ea values were to

be found.

Values of Ea found by this method (Table 4) ranged from

14.147 to 33.989 cal. This range agrees with reports from

66

Table 2. Calculated constant values for modeled experimental
data cooked to five different temperatures.

 

 

TEMPERATURE CONSTANT”
92 6 e 9 R_.‘-’
54 4.603 0.2987e-3 0.679 0.964
64 3.995 0.3236e-2 1.774 0.991
70 3.997 0.9591e-2 3.046 0.952
60 3.950 0.15256-1 4.260 0.961
65 3.903 0.1940e-1 3.116 0.946

* A. B and C as defined by Equation 38;
R8 = coefficient of determination.

69

Table 3. Time of cooking in seconds required to produce a
selected thermal induced apparent viscosity for five
different temperatures.

 

APP T TEflEERATURE 2Q
VISQQSITY §4 64 70 £0 94

 

1 O 374 181 105 83 54
2 O 1123 329 167 129 83
2.5 1760 421 203 135 104
3 0 2500 538 260 188 130
3 5 3690 710 329 227 174

 

 

mommy 0E; C4

Ln Time (sec)

70

 

 

 

9-
1 VEa=14.147
8" DE8=20,647
. 0Ea=22,669
7-
5..
51
4-
0 17:13.5
‘ o 11:25
3 v n=1.0
.........I..rTrr.r.,........r,r..-.....]
0.0027 0.0028 0.0029 0.0030 0.0031

1/T (.K)
Figure 4. Estimation of activation energy range

for five temperatures and three thermal—induced
apparent viscosities (n).

71

Table 4. Estimated values for Ea (cal/moi) for different 1":
and temperature ranges.

 

TEMPERATURE HEAT-INDUCED APPARENT VISCOSITY
RANGE _OC 1. 0 2. 0 2. 5 3. 0 3. 5
54 to 84 14. 147 18. 828 20. 647 21, 664 22. 669
64 to 70 17. 513 25. 391 30. 264 30. 292 33. 989

80 to 84 28, 469 27. 629 26. 429 24. 434 17, 610

 

72

Ziegler and Acton (1984) who reported that the apparent heat
of activation for the heat mediated interaction of
actomyosin is in the range of 17.1 to 27.0 Kcal and with
Smith.gt al. (1988) who found that the activation energy for
the thermal gelation of myofibrillar poultry proteins is

20.000 cal.

The second step in calculating the Ea value was based
on a second assumption of this experiment namely: The
reaction extent (Y') is a function of the time-temperature
history of the process (TTH). other factors constant. This

is described by Equation 39.

t
Y'=F ( Ireflm dt ) (39)

Equation 39 implies that Y' values are a function of
the time—temperature history of the process. which.was one
of the basic assumptions of this experiment. Graphically
this relationship will produce (when selected Ea is correct)
one single curve when experimental Y’ is plotted versus the
time-temperature history of the process. independently of
the temperature of the water bath. When Ea is incorrect.
several curves (one for each temperature of processi will be

PI‘O duc e d.

73

The range of values for Ea calculated previously
(14,14? to 33.989 cal) where used as starting values. The
TTH value of the thermal process of the myofibrillar beef
proteins was calculated using a transient heat transfer
model for infinite cylinders (Parker 53 3;, 1970). It
predicted temperature-time data for ten different radial
positions within a given test-tube cross-section based on
the initial, center and bath temperatures for each treatment
as described before. The cross-section was divided into ten
concentric rings defined by the preselected radial position.
TTH was computed for each ring and then used to compute a
mass average TTH. Morgan 5; £9; (1987) demonstrated that
this ‘method significantly reduces error in estimating
Kinetic parameters. It more accurately accounts for the
variation in reaction rates within a sample due to
temperature-time profiles. They also concluded that using
the center temperature-time data. which is normally used in
lethality studies, results in errors commonly exceeding
100%. while using mass average temperature-time data
results in greater than 50x error, compared to the technique

0f using concentric rings to compute Kinetic parameters.

During the first stage of the gelation reaction. Y' is
a linear function of Ln (TTH). This property was used to
determine the Ea value. Those values of Y' that were in the
linear region for five different processing temperatures

(54, 54, 7°. 80 and 84 0C) were selected. These selected

74

experimental Y’ values were plotted against their
corresponding TTH values utilizing different values for Ea.
A linear regression analysis was performed each time a new
value of Ea was used until a maximum linear coefficient of
determination was obtained (CD: 0.96) when Ea had a value of
29,500 cal/mol. The relationship between experimental Y’
values and the values of the TTH model for the temperature
range of 54 to 84 °C and for the Y' range of O to 3.5 is
shown in Figure 5. The linear regression analysis is
presented in Table 5. During this thermal analysis it was
observed that the Y’ values for the thermal process at

54 °C lagged somehow behind values for higher temperatures
(Figure 5). This suggested that another reaction with.higher

Ea was exerting a significant influence at this temperature.

Even though the overall heat mediated aggregation
reaction has been reported to follow first order reaction.
the interaction of sarcoplasmic protein molecules apparently
proceeds through two steps. Acton 3; al. (1981) reported two
temperature reaction zones for the formation of natural
actomyosin aggregates in dilute solutions (0.5 mg/ml) and
for the formation of continuous structural aggregates in
more concentrated solutions (7.5 mg/ml). The second stage
was associated with structural changes of the helical rod

segment 0f 1111708111 W101! culminates in network formation

Y.

75

 

 

 

1
4.. R’= 0.92

J
3..
2-
1 - o

- / o
o I I I I I I I I I r I I I I I f I I I I I I
-15.5 -15.0 -l4.5 -14.0 -13.5

Log TTH
Figure 5. Correlation of Y‘ values obtained in the

temperature range of 64 to 84°C and their
estimated TTH value.

76

Table 5. Statistical analysis of the linear regression of
experimental Y’ values and corresponding TTH values. Ea
utilized was 29,500 cal/mol.

 

A N A L Y S I S O F V A R I A N C E

Source of Degrees of Sum of A Mean F
Variation Freedom Squares Square Value
Mean 1 137.5460 137.54600

Regression 1 18.4604 18.46035 250.2
Residual 23 1.6972 0.07379

Total 25 157.7036 $18. of F Value: .0000

R E G R E S S I 0 N S T A T I S T I C S

Regression Standard Student's T Confidence Limits

Coefficient Error Value Sig Lower .Upper
8(0) 31.70035 1.8567 17.07 .00 30.43 32.97
3(1) 2.05713 0.1301 15.82 .00 1.97 2.15

Coefficient of: Determination .916 Correlation .95

 

77

4
7 50:20.000 cal

34 °c
so °c
70 °C
54 °C
54 ’C
-7'.5

 

iliii

 

W

 

I I I I I I I I T
-9.0 —8.5 -8.0
Log TTH

Figure 6. Relationship between experimental
values of Y' and the Log of TTH.

78

through cross-linking of these segments. HonteJano gt 3;.
(1984) reported that a uniform and rapid increase in
rigidity of myofibrillar beef proteins started at 56 °C.
indicating the formation of stable. stiff and elastic
structured matrix. Other reports also indicate this type of
behavior (Ziegler and Acton. 1984; Liu gt a_l. 1982) and it
is consistent with the proposed reaction mechanism for the
formation of protein gels (Ferry. 1948).

step 1 step 2
xPn ---------- > de --------- > (Pd)x

where x is the number of protein molecules P. with n

denoting native state and d denatured state.

The helix to coil transformation starts at about 55 °C
and is the starting point of polymerization and gelation.
This will make the gelation process at 54 00 very slow
and even though the overall reaction remains as a first
order reaction. the predominant step (and.hence the
activation energy) that will predominate will be that of
some other reactions which.precede this step and start at
lower temperatures (dissociation of F-actin and the

conformational changes in the head of myosin).

The test of the model using temperature values from 64

to 84 °C notably increased the value of the coefficient of

79

determination of the regression line (CD: 0.98 versus 0.92)
and assigned to Ea a value 20.000 cal. This new linear
relationship of the experimental values of Y’ and the
values of TTH are shown in Figure 7. whereas the regression

analysis 18 shown in Table 6.

Experimental data suggest that the thermal gelation of
myofibrillar proteins does not occur at any significant rate
until the temperature of the process is above 54 °C and
probably is not significant before 60 °C. This agrees with
Yasui gt 3;. (1979) who reported that the gelation of myosin
reaches a maximum at 60-70 °C and with Ziegler and Acton
(1984) who concluded that the transition occurring at 55 °C
is possibly the most crucial. since gels do not attain
appreciable strength until this temperature is reached. This
also agrees with commercial practice for meat products where
the minimum temperature utilized to obtain a firm cooked
product is about 65 °C. For these reasons. it was decided to
consider only the temperature range of 64 to 84 °C. and

therefore, to consider a final Ea value of 20.000 cal/mol.

Determination 23 (a)
The value (a) represents the product of the

transmission coefficient (kt) and.the Boltzman's constant

Yl

80

4.

R2=0.98

 

 

 

Log' TTH

. Figure 7. Correlation of Y' values obtained
in the temperature range of 64 to 84 °C
and their estimated TTH.

81

Table 6. Statistical analysis of the linear regression of
experimental Y’ values and corresponding TTH values. Ea
utilized was 20.000 cal/mol.

 

A H A L Y S I S 0 F V A R I A N C E

Source of Degrees of Sum of Mean F
Variation Freedom Squares Square Value
Mean 1 109.27810 109.278

Regression 1 16.07738 16.0774 921.0
Residual 18 0.31420 0.0175

Total 20 125.6697 Sig. of F Value: .0000

R E G R E S S I 0 N S T A T I S T I C 8

Regression Standard Student's T Confidence Limits
Coefficient Error Value Sig Lower Upper
8(0) 23.34136 0.69271 33.70 .00 22.86 23.82
3(1) 2.55376 0.08415 30.35 .00 2.50 2.61

Coefficient of: Determination .981 Correlation .990

 

82

divided by Plank's constant or 2.83E+10 kt. Because (a) is
an implicit constant (can not be isolated) to calculate its

value an iteration process was required.

If the basic model (Equation 15) is considered at two
different reaction stages (stage 1 and 2) at a selected
protein concentration. they could be expressed by Equations

40 and 41.

)fi' = B(1-€'°"”')°' <40)

(41)

Y.’ = team-6“)“

The product 0f the division 0f Equations 40 and 41 gave

the following expression:

Y1 8(1_e-01TH.)¢

 

 

Y2 B(1-€’°"”‘)“ (42)
Equation 42 could also be written as:
Y1 B(1-€’°m')a
.___ _ :: “A (43)

Y2 B( 1 -e-°""')“

where H: 0 when the iterated value of (a) is correct.

83

Equation 43 indicates a relationship between (a) and a,
therefore. its solution for (a) requires the predetermination
of a. The value of a for myofibrillar beef proteins was not
found in the literature. but Horgan g; 3;; (1987) reported
that values of a for the gelation of proteins are in the
range of 1 to 3. Preliminary experiments yielded tentative
values of 0.52 and 0.80. Based on this. it was decided to
calculate the value of (a) with these five different values
of a (0.52. 0.8. 1.0. 2.0 and 3.0) and to obtain a graphic

relationship between these tWO material constants.

The value of (a) for each.a value. was calculated by an
iteration computer subroutine which.gave values to (a) until
the equality of Equation 43 was met (tolerance 1E-14). For
this purpose 20 values of Y' ranging in values from 0.54 to
3.54 and their corresponding TTH value were selected from
samples processed at 64. 70. 80 and 84 °C. These Y'-TTH
values were sorted and divided in two groups (ten lower and
ten higher) before being utilized in Equation 43. The
calculated average (a) value and (kt) value for each

preselected a 18 shown in Table 7.

The value of (a) was found to have a power relationship
with.a under the conditions of this experiment as shown
in Figure 8 (CD: 0. 99). This mathematical relationship is

expressed by Equation 44.

84

Table 7. Calculation of (a) and kt values using an average
of 20 TTH values. Ea used was 20,000 cal/mol.

 

A £§E§L is) 53

0.53 1.87e+7 8.98e-4
0.3 1.oae+a 4.91e-3
1.0 1.63e+8 7.84e-3
2.0 3.50e+8 1.68e-2
3.0 4.60e+8 2.21e-2

 

(a)

85

5E+08-

 

 

/O
/
4E+08— //
/
o/
/
3E+08- /
/
/
2E+08- /
f
1E+08- /!
/ .
J R _ 999
0 1 r I
o 1 2 3
a

Figure 8. Correlation between a and (a).

86

a = —0.114578 E(+10) + 0.13082 (E+10) alum“ ‘4‘“

The value of a and therefore of (a) required by the
mathematical model was calculated by assigning to a the
predetermined values of 0.5. 0.6. 0.7. 0.8. 0.9. 1.0. 2.0.
3.0. 5.0 and 8.0 followed by a calculation of their
corresponding (a) value. A non-linear regression algorithm
(Harquart. 1963) was utilized to fit the basic mathematical
model (Equation 15) to experimental data using these sets
of a and (a) values. The selected a-(a) combination was that

which.produced.the best coefficient of determination.

Experimental values were obtained from five different
experimental units processed at a similar temperature (70
0C) and.with protein concentrations 0f 85.3. 31.9. 38.4.

41.5 and 46.2% (dry basis).

The plot of the Ra’s determined for each.set of values
of a and (a) and for each.protein concentration is shown in
Figure 9. These results were not expected because they
suggest that the values of a are a function of the

protein concentration. when 1t W88 expected to be constant.

T0 investigate this possible correlation. the values

0f 0 which.produced the best coefficient 0f determination

 

 

 

87

 

1.0—
0.9-4
o.8~ ‘
0.7n
0.6- "
v—v 46.2% protein
.. H 38.4% protein
‘ H 31.9% protein
.. o—o 25.3% protein
0 5 o—o 41.5% protein
. . i 1 T l ' T ' f ' I

Figure 9. PLot of 0: versus R’ for five protein
concentrations. Protein concentration is in dry
basis.

88

were plotted against their corresponding protein
concentrations. It was found (Figure 10) that the values
of a decrease exponentially as the value of protein
concentration increases. This function is expressed by

Equation 45.
a = 0.734377 + 2074856 e'WW" (45)
Where C is protein concentration (dry basis).

Determination 93 A;

The value of A' was estimated using the special case
of the general model (Equation 15) for an infinite time of
cooking. Under this conditions the value of the second term
of the model tend to one and therefore the model could be

simplified to:
Yin = A' C (46)

where C 18 protein concentration on a dry 133818 and A' and G

are material constants.

To calculate this parameter. data were obtained from
five different experimental units thermally processed at
similar time-temperatures conditions but with protein

concentration on a dry basis varying between 25.3 and 44.62.

89

 

.1
_ R2=0.999
i
7 l
_ i
\
7 l
_ l
i
7 \
fl \
\
7 \
_ \
\
- \
_A \
\
- \
\
7 \
\
7 \
\‘—-__£__.o____
Ifi Ifi Ti T I I I I I I I I I I I I I I I I I I I I I

 

20 25 30 35 4O 45

Protein Concentration (0.0)

Figure 10. Correlation between a and protein
concentration (d.b.).

90

The thermally induced BEAV was calculated as mentioned
before. The experimental data was fitted to the basic model
(Equation 15) and Y’ at time infinite calculated using a
non-linear regression algorithm (Marquard. 1963). A plot of
the asymptotic values of Y' and their corresponding protein

concentration (dry b8818) 18 presented in Figure 11.

Equation 46 predicts the value of 7' at time infinite
for any concentration of protein where gelation occurs and

could be rearranged as:

A' = Y. (47)
Ce

 

Calculated values of A’ were found to increase as protein
concentration increased. A plot of these values (Figure 12)
exhibited a A' values increasing exponentially with.an
increase in protein concentration. Their calculated

mathematical relationship is described by Equation 48.

The dependency of A' on protein concentration as
described in Equation 48 set the calculated TTH model for

the thermal gelation of myofibrillar beef proteins as:

Y'=A'Cla ( 1’0”“ )a (49)

91

7.0 -,

6.0- /' .
5.0— /

 

 

4.0- . I /a
3.0- . ~ /

q /

/
2.0- . /
/
‘ /
/

1.0— i/

' R2: 0.998
0.0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I Ii

20 25 30 35 4o 45 50

Protein Concentration (d.b.)

Figure 11. Correlation between protein
concentration and Y'oo

Al

92

 

 

0.40-
? R’=O.997

0.30-4
.4 0

0.20-4

0.10-

0.00- e

—.10 IIrIIIIIITrITIIIIIITIIIIIIIIIII
20 25 30 35 40 45 50

Protein Concentration (d.b.)

Figure 12. Correlation between protein concentration
and A' (d.b.= dry basis).

93

Where 7': heat induced BEAV:
A': as describe by Equation 48:
a : as describe by Equation 45;
a : as describe by Equation 44;
TTH : the time temperature history of the process;
C : protein concentration in dry basis.

 

Maggy
The developed mathematical model (Equation 49)

theoretically described the thermally induced back extrusion
apparent viscosity of myofibrillar beef proteins. for
samples thermally processed on the temperature range of 64
to 84°C and.have a protein concentration on a dry basis
between 25.6 and 44.6%. According t0_this model. if the
protein concentration is kept constant. the model should
predict the value of the thermally induced BEAV as a
variable of the time-temperature history of the process.
This was tested as follows: Experimental samples from five
different experimental units with an approximate protein
concentration of 42.5% (dry basis) were cooked at 64. 69.
71. 80 and 84 °C and their Y' calculated as described
earlier. Experimental Y' values were plotted against values
predicted by the TTH model. The model was found to describe
reasonably well (CD:0.94) the experimental Y' values
obtained from.these experiments. Figure 13 shows
experimental 7’ values and values predicted by the model

against the time-temperature history of the process (TTH).

94

Figure 13. Experimental Y' values for temperature range of
64 to 84 0 versus values predicted by the TTH model.

i

95

okl

vmdnum

 

 

 

96

A second form of testing the model was by using the
protein concentration as a variable (at a constant
temperature). Under these conditions the model should
predict the value of Y’ as a variable of the protein
concentration. for any time of cooking. This was tested
using three different experimental units with a protein
concentrations on dry basis of 30.6. 38.4 and 46.2%. These
samples were thermally processed at 69 °C. Experimental Y’
values and predicted Y's were plotted against the
time-temperature history of the process and found to
correlate reasonable well (R3: 0.89. 0.91 and 0.94

respectively). These correlations are shown in Figure 14.

From.the information showed by Figures 13 and 14 it was
concluded that under the conditions of this experiment.
protein concentration on a dry basis in the range of 25.3 to
44.2% (dry basis) and a temperature in the range of 64 to
84 °C. the developed mathematical model (Equation 49)
predicts the change in Y’ (and presumedly the thermal
gelation of myofibrillar beef proteins) as a function of the
protein concentration and.the time-temperature history of

the process.

Verification u; the Hodel
A basic assumption of the model developed.was that the

heat induced BEAV (Y') is a measure of the thermal gelation

Y.

97

 

 

 

 

 

8‘ v R’=O.89
: o R’=0.91
7—i O R2=0.94 O
a 0
I
SJ 0 £—
.. e
e
0
5d
0
q o
4..
‘ : . e e
31 .
o
.4 V e
e v e
2- . V g 3—
. v
V
1 -(
i/gi v 3% PROTEIN
3 e 4% PROTEIN
0 ‘ e 5% PROTEIN
' l I I I I l ' l ' I ' l
0 0 2.0 4 O 6.0 8.0 10.0 12.0 14.0
1TH (1E-8)

Figure 14. Experimental Y' values for three different protein

concentrations versus values predicted by the TTH model.

98

of myofibrillar beef proteins. To verify this assumption a
parallel method of measuring the gelation reaction was
sought. The extent of the thermal gelation of proteins

can be measured in several ways namely: loss of solubility.
increase in turbidity. reduction of water holding capacity.
etc. (Acton and Dick. 1984). SameJima qt 3;. (1985) reported
that the water holding capacity and gelation properties are
the important factors that determine the quality of
comminuted meat products. They also reported that these two

properties are closely interrelated.

The water holding capacity is a measure of the water
released by the gel during the cooking process. When the
thermally induced protein matrix is formed. water is trapped
inside of the protein network. As the thermal process
continues. an increase in the points of interaction occurs
with
a reduction of the interstitial space occupied by the
trapped water. The net effect is a release of water which
then indirectly measures the extent of the polymerization

react 1011.

The water released was determined on the ruptured
gel/protein suspensions after gel strength testing. This
technique warranted that the experimental units had the

same Chemical composition and experimental treatment as

99

those for thermally induced BEAV. The experimental samples
were obtained from protein solutions thermally processed at
70 °C and with two different protein concentrations (30.6

and 38.42. dry basis).

The trend of experimental values of water released over
time of cooking (Figure 15) showed a tendency very similar
to the inverse of the basic TTH - Y' model described by
Equation 38. The next step was to test if the change in
water released during the thermal process could be described
by the model developed for Y’. Experimental data confirmed
the general knowledge that water holding capacity is
inversely related to protein concentration. Because Y' and
water holding capacity were inversely related. the
mathematical model (Equation 49) to be fitted to the water
released.had to be applied to the inverse of the protein

concentration.

Results showed (Figure 16) that the experimental values
of water released closely correlate with values predicted
by the basic model. Coefficients of determination were 0.99
and 0.97 for protein concentration on a dry basis of 30.6

and 38.4% respectively.

The correlation between water released and the TTH
model support the hypotesis that the thermally induced

gelation of myofibrillar beef proteins is measured by the

WATER RELEASED (g/lOg)

100

 

 

 

 

_______ :__________1
U D
a
D 4% PROTEIN
v 3% PROTEIN
I I I I I I
2000 3000 4000
TIME (sec)

Figure 15. Relationship of water released and time

of cooking for two protein concentrations.

WATER RELEASED (g/ 1 09)

101

 

o R’=0.993
o R’=0.982
4- —_—___3 ———————— o
. é/E D
3- l
l
J i __°.___.______°
l °
2— 1 /° -
/
1 lo
1 l o/
l /
1°!
1- l l
i I
l ,1)
III, a 39: PROTEIN
4% PROTEIN
0-4 I 1 I o I f I I I

 

T'l'l',l' l
0.0 2.0 4.0 6.0 8.0 10 12 14 16

TTH (1E-8)

Figure 16. Experimental water released values versus
values predicted by the TTH model.

102

Instron back-extrusion apparent Viscosity. This correlation
supports the hypothesis that the gelation process is a
function of the protein concentration and the
time-temperature history of the process. It also supports a
basic premise of this experiment. i.e.. The thermal gelation
of myofibrillar beef proteins could be predicted by the TTH

model described by Equation 49.

Effect 2; Vegetable Gums

It was theorized.that the addition of vegetable gums to
myofibrillar beef proteins solutions will have the effect of
increasing the equivalent protein concentration. and
therefore. alter the final three dimensional protein network
of the thermally-produced gel. Vegetable gums are
polysaccharides with different molecular weights. basic unit
composition and level of side-branching. but in general they
are more homogeneous and have more polar groups than
myofibrillar proteins. These polar groups are responsible
for the high.water holding capacity exhibited by these
compounds. They may also serve as linking points between
gums and myofibrillar beef proteins. therefore. helping in
the creation of the three dimensional network of the
thermally-induced gel produced by myofibrillar beef

proteins.

103

If vegetable gums interact with myofibrillar proteins
they should increase the apparent viscosity induced by the
thermal process. This will show up as an apparent increase
in the protein concentration (C) and the degree of
polymerization (A’). and produce a more orderly

entanglement (reduction of a).

To test these assumptions the following experiment was
carried out: An extracted myofibrillar beef protein solution
with a protein concentration of 29.9% (dry basis) was
divided into five lots. Each lot received 0.5% of a selected
vegetable gum (carrageenan. guar, locust bean or xanthan
gum) and was thermally processed at 70 °C. The TTH. Y’ and
water released values were then calculated as mentioned

before.

Experimental Y’ for all protein-gum combinations were
plotted against their corresponding TTH values and the
mathematical model developed for myofibrillar proteins
(Equation 49) fitted to each of them. Figures 17 to 20 show
plots of experimental Y' against TTH values for each gum

as well as values predicted by the TTH model.

The Y' at time infinite for each protein-gum
combination was calculated DY fitting Equation 48 t0 the
experimental data by means Of a non-linear regression

algorithm.(Harquard. 1963). The equivalent protein

YI

 

104

 

5.01
4.0- A
A
T__—————A' ———————
-l f A A
‘1,“ A 2
3.0- i
l
q l
l
l
20 I
‘ _ l
4 .
A
1.... ,1
l
_ I
l R’ = .912
l
0.0 I T T I fi . I I I I I I 1 I l
0 1 2 3 4 5 6
TTH (1E-7)

Figure 17. Experimental Y' values for carrageenan-protein

solution and values predicted by the TTH model.

YO

105

 

 

5.0-
1
4.0—
A
A
" A A
A ——————————————— -a
3.0- A a
1%: A
. A
q
4
2.0- :
j".
q
1.04 'l
l
. l
l 1?2 — .85
1
0'0 '1'1'1'1'1'1'
0 1 2 3 4 5
7714 (15-7)

Figure 18. Experimental Y' values for guar- protein

solution and values predicted by the TTH model.

YI

106

 

 

 

5.0—
4.0-1
4 A A
A A A
AA _______________
3.0— A if“.
/ A A A
A/ A
. I A
I A
P
2.0- l
i,
‘ ‘ 1.
1'
1.0-4
i
l
- l
i R2 = .904
0'0 l I F l I I i
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1TH (1E-7)

Figure 19. Experimental Y' values for locust bean -protein

solution and values predicted by the TTH model.

Y.

107

 

 

1.5-
1
1.0-1
‘ A
q A
A ”,..—-—
III l””’ A
. A ///
- A / A
, A /
05- 2A? A
- his”
g A
q P
l
q l
l R’ .. .77
0.0 I 4 I I I I I I I l T I I
O 1 2 3 4 5
TTH (1E—7)

Figure 20. Experimental Y' values for xanthan- protein

solution and values predicted by the TTH model.

108

concentration was calculated as the hypothetical protein
concentration which would be required to produce the same

value Of Y' after an infinite time Of cooking.

The calculated equivalent protein concentrations were
used to determine their corresponding A’ and a values using
Equations 45 and 48. Calculated equivalent protein
concentration. A' and a for the protein-gum solutions are
presented in Table 8. Relative change in a as affected by

the different protein-gum combinations is shown in Figure 21.

Carrageenan gum was found to contribute to the
protein-gum system by increasing the strength of the cooked
gel by 184.8% over that produced by the control. This meant
an increase in equivalent protein concentration of 121.2%

and a concomitant reduction of a of 19.4% (Table 8).

Carrageenan gum increased the water holding capacity by
a 130.7% over the control but had the lowest effect on water
holding capacity of any of the gums evaluated in this
experiment (Figure 22). This agrees in part with Wallingford
and Labuza (1983) who reported that carrageenan gum had the
second.highest water binding capacity (WBC) of nine gums in
low fat meat emulsions studied. Foegeding and Ramsey (1986)
reported.that carrageenan gum used at 1% with a 13% protein
meat batter increased the force to fracture (FF). the

hardness (H1) and the water holding capacity of

109

Table 8. Calculated values of equivalent protein

 

 

 

concentration. a and A' for selected gum-protein
combinations.

GUM Y' C+ a A'__
Control 1.82676 32.66 0.9186 0.0706
Carrageenan 3.37684 39.68 0.7403 0.2180
Guar 3.64221 40.11 0.7389 0.2306
Locust bean 3.02690 38.39 0.7461 0.1894
Xanthan 0.77900 7.76 43796 1.326E-17

 

+ Concentration

Of protein (dry basis)

 

 

 

 

. 7//////////////////A

 

 

”ZZZ? /

 

 

 

 

21. Effect of protein-gum combination on a.

 

 

d 1dl.—111-llq

- a
O O 0 0 O
8 6 4 2

100-

6 20 moz<Io m>_.—<...wm

Figure

111

 

cocx /
0

.. 7/
coo .

8.....3 Z

 

 

 

 

 

 

 

 

 

 

q _ 4 _ . _ _ .
O 0 O 0 0 0
0 8 6 4 2

ommfibum 15.35 u>_._.<._um

 

Figure 22. Water released by four gum-protein

combinations (75 min at 70°C).

118

frankfurters. They concluded.that among other gums studied.
carrageenan was the most beneficial in manufacturing low fat
frankfurters. When calculated model constants were
introduced in the TTH model it was found that the TTH model
predicted reasonably well (R3=0.912) the change of Y of

this protein-gum solution as a function of the time-

temperature hiStOI‘Y Of the process, as shown in Figure 17.

Guar gum increased the strength of the cooked gel by
193.9% over that produced by the control. The protein
samples containing guar gum had the highest Y’ among the
gum-protein combinations used in this experiment. The
increase in equivalent protein concentration was 122.8%

whereas a was decreased by 19.7% (Table 8).

Guar gum increased the water holding capacity by
138.94% over the control and was ranked.third. This seems to
disagree with.vallingford and Labuza (1983) who reported
that guar gum had approximately two-thirds of the WBC of
carrageenan. but because carrageenan gum is not a single
homogeneous compound but rather a.heterogeneous mixture of
several different polysaccharides. it is possible that their
sample was different than the one used in this experiment.
Foegeding and Ramsey (1986) reported that the addition of
guar gum to low fat frankfurters reduced force to fracture

(FF) and hardness DY almost 507.. While 0111" PQSUItS appear

113

not to support their findings, a direct comparison between
studies is difficult because variations in ingredients and
techniques. The calculated TTH model for the protein-guar
solution acceptably predicted (Ra: 0.86) the change of Y'

over time of cooking as shown in Figure 18.

Locust bean gum increased.the strength of the cooked
gel by 166.6%. This value is lower than that produced by
carrageenan and guar gums but this gum still produced an
increase in equivalent protein concentration of 117.6%.

with a concomitant reduction in a of 18.9% (Table 8).

Locust bean gum increased.the water holding capacity
the protein-gum solution by 140.19% (Figure 22). The
increase in water-retained during the cooking process
induced by this gum was slightly lower than that produced by
xanthan gum. This result supports the findings of Foegeding
and Dayton (1986) who reported that xanthan and locust bean
gums produced the lowest weight loss during cooking among
several meat-gums combinations. The calculated mathematical
model described reasonably well (R3: 0.904) the change in Y'
due to the time-temperature history of the process as

indicated in Figure 19.

114

Xanthan gum was found to behave as an inhibitor of the
gelling phenomenon. Gel strength of the xanthan-protein
solution was found to be only 42.6% of that produced by the
control. This value of Y’ is outside of the limits of the
mathematical model described by Equation 49 and therefore
the equivalent protein concentration (70.8%) and a (93.3%)

were calculated only as an exercise (Table 8).

The water holding capacity of the xanthan gum-protein
solution was the best of all experimental units (Figure 22)
showing an increase of 146.3% over the control. Similar
findings have been reported by Whiting (1984). who found
that xanthan gum added at 0.1 or 0.3% decreased cooking
losses and gel strength. It also agrees with.vallingford and
Labuza (1983) who described xanthan gum.as the best water

binder among several gums studied.

SUMMARY AND CONCLUSIONS

The objectives of this study were to develop a
mathematical model for predicting the effects of
time-temperature history and protein concentration in the
thermal gelation of myofibrillar beef proteins as well as
the interaction of these proteins with selected

hydrocolloids.

Thermal gelation of myofibrillar beef proteins was
measured as the thermally-induced Instron back-extrusion
apparent viscosity (Y’). water holding capacity was used as
parallel method of measuring gelation and to verify the

model developed for Y'.

The developed mathematical model has an Ea for the
thermal gelation of myofibrillar beef proteins of 20.000
cal/mol. Basic model constants (A'.a and a) were found
to be mathematically related to protein concentration. these
mathematical relationships were developed and integrated

into the basic model.

Experimental values Of Y' and WHC obtained under the

115

116

conditions of this experiment support the hypothesis that
the proposed mathematical model (Equation 49) can be used to
describe the thermally-induced gelation of myofibrillar beef
proteins as a function of the protein concentration (dry

basis) and the time-temperature history of the process.

It was also found that the TTH model could be used to
predict the water released during the thermal treatment as a
function of the protein concentration and the time-

temperature history of the process.

The TTH model was shown to describe reasonably well the
thermal gelation of solutions of myofibrillar beef proteins
and selected vegetable gums. Under the conditions of this
experiment xanthan gum was found to inhibit gelation. and
therefore. the TTH model could not be applied to this

protein-gum 80111121011.

More research is needed to study the effect of other
ingredients normally found in meat products (salts. lipids.
connective tissue. etc.) in the mathematical model. as well
as the relationship between back-extrusion apparent
viscosity and sensorial attributes of thermally gelled meat

products.

117

Fundamental knowledge of effects of temperature-time
history in meat gels should enhance the understanding of the
reaction kinetics involved in the thermally-induced gelation
process. This TTH model can be useful in studying and
predicting effects of process conditions on product quality

and significantly reduce experimental cost and time.

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H.. Ed. Academic Press. NY. p. 478.

Schmidt. G.R. 1987. Functional behavior of meat components
in processing. In: ”The Science of Heat and Heat Products”.
3rd ed. J.F. Price and B.S. Schweigert. Eds. Food and
Nutrition Press. Inc. Westport. CT. p. 413.

Schmidt. G.R.. Hawson. R.F. and Siegel. D.G. 1981.
Functionality of a protein matrix in comminuted meat
products. Food Technol. 6:236.

Schut. J. 1976. Heat emulsions. In: ”Food Emulsions."
Ferberg. 8.. Ed. Harcel Dekker. Inc. NY. p. 79.

Scopes. R.H. 1970. Characterization and study of
sarcosplasmic proteins. In: ”The Physiology and Biochemistry
of Huscle as Food. Vol.2.” Briskey. E.J.. Cassens. R.G. and
Harch. B.B.. Eds. University of Wisconsin Press. p. 471.

Sender. P.H. 1971. Huscle fibrils: Solubilization and gel
electrophoresis. FEBS Letters 17:106.

Shah. B.H. and Darby. R. 1976. Prediction of polyethylene
melt rheological properties from.molecular weight
distribution data obtained by gel permeation chromatography.
Poly. Eng. Sci. 16:8.

Shimada. I. and Hatsushita. S. 1981. Effects of salts and
denaturants on thermocoagulation of proteins. J. Agric. Food
Chem. 29: 16.

126

Siegel. D.G. and.Schmidt. G.R. 1979. Ionic. pH and
temperature effects on the binding ability of myosin. J.
Food Sci. 44:1686.

Siegel. D.G.. Theno. D.H. and Schmidt. G.R. 1978. Heat
massaging: The effect of salt. phosphate and massaging on
the presence of specific skeletal muscle proteins in the
exudate of a sectionated and formed ham. J. Food Sci.
43:327.

Sjostrand. F.S. 1962. The connections between A- and I-band
filaments in striated frog muscle. J. Ultrastruct. Res.
1:226.

Smith. D.H. 1987. Functional and biochemical changes in
deboned turkey due to frozen storage and lipid oxidation. J.
Food Sci. 62:22.

Smith. D.H.. Horgan. R.G. and Alvarez. V.B. 1988. A
generalized mathematical model for predicting heat-induced
chicken myofibrillar protein gel strength. J. Food Sci. Vol
No 63: In Press.

Smith. D.H.. Salih. A.H. and Horgan. R.G. 1987. Heat
treatments effects on warmed-over flavor in chicken breast
meat. J. Food Sci. 62:842.

Software Arts Inc. 1983. TKISolver. Scotts Valley. CA.

Steiner. R.F.. Laki. K. and Spicer. S. 1962. Light
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Stoloff. L. 1973. Carrageenan. In: " Industrial Gums.” R.L.
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Szent-Gyorgyi. A.G. 1961. Chemistry of Huscular Contraction.
2nd Ed. Academic Press. NY.

Tanford. C. 1968. Protein denaturation. Adv. Prot. Chem.
23:121.

Turner. R.H.. Jones. P.N. and HacFarlane. J.J. 1979. Binding
of meat pieces: An investigation of the use
myosin-containing extracts from pre- and post-rigor bovine
muscle as meat binding agents. J. Food Sci. 44:1443.

Wallingford. L. and Labuza. T.P. 1983. Evaluation of the
water binding properties of food.hydrocolloids by
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127

Wang. K. and Ramirez-Mitchell. R. 1983. A network of
transverse and longitudinal intermediate filaments is
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Whiting. R.C. 1984. Addition of phosphates. proteins and
gums to reduce-salt frankfurters batters. J. Food Sci. 48:1.

Wolosewick. J.J. and Porter. E.R. 1979. Hicrotrabecular
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Wright. D.J.. Leach. J.B. and.Wilding. P. 1977. Differential
scanning calorimetric studies of muscle and its constituent
proteins. J. Sci. Food Agr. 28:667.

Yasui. T.. Ishioroshi. H.. Nakano. H. and Samejima. K.
1979. Changes in shear modulus. ultrastructure and
spin-spin relaxation times of water associated with
heat—induced gelation of myosin. J. Food Sci. 44:1201.

Yasui. T.. Ishioroshi. H. and Samejima. K. 1980.
Heat-induced gelation of myosin in the presence of actin. J.
Food Biochem. 4:61.

Yasui. T.. Ishioroshi. H. and Samejima. x. 1982. Effect of
actomyosin on heat-induced gelation of myosin. Agric. Biol.
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Yates. L.D. and Greaser. R.L. 1983. Quantitative
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Ziegler. G.R. and Acton. J.C. 1984. Heat-induced transitions
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actomyosin. J. Food Biochem. 8:26.

 

APPEND I CES

APPENDIX A. List of the computer program Rodrigo.

100 REM "it: PROGRAM RODRIBO it!"

200 CLS e DISP " PROGRAM RODRIGO BY CARLOS A. LEVER"

300 REM "3 Program to Calculate Area, Apparent Viscosity,
Apparent Elasticity and Shear Rate at the TNS from
Back Extrusion Data 3”

400 DISP "TURN PONER ON" @ GOSUB 9900 8 CLEAR

500 REM tit! INSTROM CONTROL COMMANDS t!!!

600 DISP "Attach Plunger, set distance from envil= 3 cm. "
GOSUB 9900 @ CLEAR

700 DISP "Press 81 then 0 <ENTER>” e GOSUB 9900 @ CLEAR

800 DISP "Set Recorder Paper, Turn on line, CHart & servo."
G BOSUB 9900 G CLEAR

900 DISP "Press LOAD CAL <ENTER>" 8 GOSUB 9900 8 CLEAR

1000 DISP "Press LOAD BAL (ENTER) " e GOSUB 9900 e CLEAR

1100 DISP "Press IEEE (ENTER) " G BOSUB 9900 8 CLEAR

1200 PRINTER IS 701

1300 DIM SAMPIDSEBOJ,SAM$[15]

1400 DIM A(200),B(200),C(200)

1500 DIM AA(200).CC(200).S(8)

1600 DISP “CALIBRATION IS MANUAL, IF YOU WANT TO MAKE ANY

CHANGE"

1700 SET TIMEOUT 7310000

1800 ON TIMEOUT 7 GOTO 9200

1900 GOSUB 9900

2000 I=1

2100 CCP=0

2200 AO=704

2300 CLEAR

2400 GOSUB 25800

2500 OUTPUT AO a”K25,-30K32,3K24,2.5K31,3"

2600 OUTPUT AO ;"K21K26,0.0K27,-30.0"

2700 REM"!**PRINTER CONTROL COMMANDS #1!"

2800 DISP “Select % of Printer Scale Desire"

2900 DISP ”20 = 20%", “10 = 10%","5 = 5%"

3000 DISP "ENTER 3"

3100 INPUT NUMBER

3200 IF NUMBER=20 THEN GOTO 3600

3300 IF NUMBER=10 THEN GOTO 3800

3400 OUTPUT A0; “K13,20K15,6K19,5K20,2”

3500 SOTO 4000

3600 OUTPUT AO ; "K13,20K15,4Ki9,5K20,2"

3700 GOTO 4000

3800 OUTPUT AO; "K13,20K15,5K19,5K20,2"

3900 GOTO 4000

4000 OUTPUT AO :"K34,3"

4010 DISP "Enter Sample Identification Name (Up to 40

characters)"

4015 INPUT SAMPIDS

128

 

Audi-1.1.1

 

 

129

APPENDIX A. (continued)

4100 CLEAR

4105 DISP "Do you Want to Change Printer Scale“

4200 DISP "Yes=1","No=2"

4300 INPUT 88

4310 IF BB=1 OR 88=2 THEN 4400 ELSE BEEP 400,40

4320 DISP “ENTER 1 OR 2 ONLY” e GOTO 4105

4400 CLEAR

4500 IF BB=1 THEN 2700

4600 DISP G DISP "Ready to Start" a BEEP e DISP

4700 DISP ”Enter Sample Number and Time ( Up to 10
characters)"

4800 INPUT SAMS

5000 I=1

5100 AO=704

5200 OUTPUT AO 3"K25,-30K32,3K24,2.5K31,3“

5300 OUTPUT AO ; "K2"

5400 CLEAR

5500 OUTPUT AO ; "R2R27R3"

5600 ENTER AO ; A(I),B(I),C(I)

5700 I=I+1

5800 IF I>170 THEN GOTO 6100

5900 WAIT 397

6000 GOTO 5500

6100 OUTPUT AO ; ”K1"

6200 GOT 10300

6300 GOTO 6800

6400 DISP "YES=1", "NO=2"

6500 INPUT 88

6600 CLEAR

6700 IF 88=2 THEN GOTO 8000

6800 REM till! SAVING CONTROL PROGRAM tilt:

6900 DISP "Insert Disk to Store The Data. IthITIALIZEDtt"

7000 DISP "Enter File Name (Up to 8 characters)"

7100 INPUT FILN$

7200 CREATE FILN$,200,50

7300 ASSIGN# 1 TO FILNS

7400 PRINT# 1 3 SAMPIDS

7450 PRINT# 1 ; SAM6

7500 FORI=1 TO 170

7600 PRINT# 1 ; A(I),8(I),C(I)

7700 NEXT I

7800 GOTO 7900

7900 ASSIGN# 1 TO I

8000 DISP "Do you want to Repeat the Test?"

8100 DISP "YES=1","NO=2"

8200 INPUT 88

8300 CLEAR

8400 IF 88=2 THEN GOTO 9100

8500 DISP ”Please WAit"

8600 FOR NN=1 TO 170

APPEN

8700

8800

8900

9000

9100

9200

9300

9400

9500

9600

9700

‘9800

9850

9900

10000
10100
10200
10300
10400
10500
10600
10700
10800
10900
11000
11100
11200
11300
11400
11500
11600
11700
11800
11900
12000
12100
12200
12300
12400
12500
12600
12700
12800
12900
13000
13100
13200
13300
13400

DIX

A.

130

(continued)

A(NN),8(NN),C(NN)=0
AA(NN),CC(NN)=0
NEXT NN

GOTO 4100

END

FOR TT=O T0 6

STATUS 7,TT ;

S(TT)

PRINT "STATUS BYTE #";TT;" ="S(TT)
NEXT TT

PRINT “HP-18 Timeout”

SOTO 9100

! SUBRDUTINE: WAIT FOR <K1> TO BE PRESSED..
BEEP 4000,40 G DISP ”ENTER K1 KEY ONLY!"

DISP

"When ready,

Press <k1> to Continue."

ON KEY# 1 SOTO 10200
GOTO 10100
RETURN

U=0

PRINTER IS 701
FOR I=1 T0 170
AAII)=ABS (A(I) @ CC(I)=ABS (C(I)

NEXT

CLEAR

I

IF CCP=1 THEN 11500
IF CCP=2 THEN 12900

DISP
DISP

“Do you Want The Computer To EStimate Le and Lp"
IIYe5=1II ’ H No=2H

INPUT CCP

IF CCP=2 THEN 12900

REM tit Program to Calculate Le and Lp tittitttitt
FOR I=1 TO 170

IF A(I)<-.03 THEN GOTO 11900

NEXT I

R=I-2

IPV=CC(I-2)

FOR I=1 TO 170

IF A(I)<A(I+1) THEN 12400

NEXT I

8FV=CC(I)

PF=AA(I)

LPV=CC(165)

I=1

GOTO 18100

DISP "ALL DATA WILL BE DISPLAY AS:"

DISP

DISP "Force (N)”,"Distance (mm)","Sequential Number"
DISP

DISP ”Please take note of:”

DISP "-Sequential Value of Initial Distance Value
lam)”

 

 

 

131

APPENDIX A. (continued)

13500

13600
13700
13800
13900
14000
14100
14200
14300
14400
14500
14600
14700
14800

14900
15000
15100
15200
15300
15400
15500
15600
15700
15800
15900
16000
16100
16200
16300
16400
16500
16600
16700
16800
16900
17000
17100
17200
17300
17400
17500
17600
17700
17800
17900
18000
18100
18200

DISP "- Sequential Value of Break Points Distance and
Force"

DISP

DISP

DISP "To continue Press <k1>

ON KEY# 1 GOTO 14100

GOTO 14000

CLEAR

M=20

PAGESIZE 24

DISP "Force (N)","Distance (mm)","Sequential Number"
FOR I=1 TO M

DISP All),C(I),I

NEXT I

DISP "Do You Want To See More Values ?",”Yes=i";”
No=2"

INPUT 88

IF 88=2 THEN 16300
CLEAR

IF I>160 THEN 15900
M=M+20

DISP "Force (N)","Distance (mm),"Sequential Number"
FOR I=I TO M

DISP A(I),C(I),I

NEXT I

GOTO 14800

M=M+10

CLEAR

IF I=170 THEN 16300

GOTO 15400

CLEAR

M=M+10

PAGESIZE 16

DISP "DO YOU WANT TO SEE THE DATA AGAIN“

DISO "YES=1", "No=2"

INPUT 88

IF BB=1 THEN 14200

CLEAR

DISP "Sequential Value of Initial Distance Value=?"
INPUT I

IPV=CC(I)

R=I

CLEAR

DISP "Sequential Number of Break Distance ValueS?"
INPUT I

BFV=CC(I)

PF=AA(I)

LPV=CC(165)

DISP "Please Wait Area CAlculations in Progress"
U=U+1

APPENDIX

18300
18400
18500
18600
18700

18800
18900
19000
19100
19200
19300
19400
19500
19600
19700
19800
19900
20000
20100
20200
20300
20400
20500
20600
20700
20800
20900
21000
21100
21200
21300
21400
21500
21600
21700
21800
21900
22000
22100
22200
22300
23100
23200
23300
23400
23500
23600
23700
23800

A

SUM1=
FOR I
SUM1=
NEXT
FI=(A
CC(16
AREA
GOSUB
CLEAR
DISP
DISP
DISP
DISP
DISP
DISP
DISP
DISP
DISP
DISP
DISP
REM
DISP
DISP
INPUT
IF 88
CLEAR
DISP
DISP
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
CLEAR
REM
DISP
DISP
INPUT
IF 88
DISP

132

. (continued)

0

=R T0 165
SUM1+AA(I)¥(CC(I+1)-CC(I-1))*.5

I
A(R-1)*(CC(R)-CC(R-1))+AA(166)3(CC(166)-
5)))#.5

(U)=SUM1+FI

25000

SAMPID$;“ ":SAM6
"Area=";AREA (U);”mm“2"

"Plunger Velocity is="; 8(2);"mm/min"

"Apparent Viscosity is="; VI, poise"

"Apparent Elasticity is=";EA;“N/cm‘2"

"Shear Rate at the PWS is=";LW;"1/sec"

"Do you Want to Print These Values?"
ll Yes=1 H ’ H No=2"

BB
=2 THEN 23200

"PRINTIING IN PROGRESS, PLEASE WAIT" C DISP

"PRINTING IN PROGRESS, PLEASE WAIT“
"*ttltilitttt3*lititlttlilttlitttitttt"

SAMPID$; "35AMS

"Area=";AREA (U);"mm“2"

"Plunger velocity is =";8(2);"mm/min"
"Apparent Viscosity is="VI;"poise"
"Apparent Elasticity is=";EA;"N/cm“2"

"Shear Rate at the PWS is=";LW;"1/sec"

"Do You Want to Print the Raw Data?”
"YESgl u ’ 00N0=2u
BB
=2 THEN 24700
"PRINTING IN PROGRESS,PLEASE WAIT"

 

 

APPENDIX

23900
24000
24100
24200
24300
24400
24500
24600
24700
24800
24900
25000
25100
25200
25300
25400
25500
25600
25700
25800
25900
26000
26100
26200
26300
26400
26500
26600
26700
26800
26900
27000
27010
27020
27100
27200
27300
27400
27500
27600
27700
27800
27900
28000
28100
'28200
28300
28400
28500
28600

133

A. (continued)

PRINT"tittittttxtttitittxtttxttttttt"
PRINT
PRINT SAMPIDS
PRINT
PRINT “Force (N)","Distance (mm)","Sequential Number"
FOR I=1 TO 170
PRINT A(I),C(I),I
NEXT I
CLEAR
GOTO 6300
END
CLEAR
DISP "Please Wait Viscosity Calculations in Progress”
REM *tBASIC CALCULATIONStttt
LP=LPV—IPV
LE=BFV—IPV
F=2XAREA(U)/LP
VP=B(2)
GOTO 28400
REM IIVISCOSITY INDEX
VP=20
RI=4.255
RO=7.028
DISP "The Viscosity Constants Used Are:"
DISP
DISP
DISP
DISP
DISP
DISP "DO YOU WANT TO CHANGE THE VISCOSITY CONSTANTS ?"
DISP "YES=1","NO=2"

INPUT 88

IF BB=1 OR BB=2 THEN 27100 ELSE BEEP 400,40
DISP ”ENTER ONLY A 1 OR 2 " G GOTO 26800

IF BB=2 THEN GOTO 28300
CLEAR
DISP "VALUE OF PLUNGER VELOCITY IS=?“

INPUT VP
CLEAR
DISP “PLUNGER RADIUS,
INPUT RI
CLEAR
DISP ”INNER TEST TUBE RADIUS,
INPUT RO
CLEAR
RETURN
RETURN
DISP ”PLEASE WAIT ELASTICITY CALCULATIONS IN PROGRESS"
REM tiltApparent VIscosity Calculations xttttttt
K=RI/RO

CALCULATIONS****X*

"-Plunger Velocity = 20 mm/min"
"-Plunger Radius = 4.255 mm“
"-Inner Test Tube Radius 8 7.028mm"

IN mm, IS=?"

IN mm,IS=?"

134

APPENDIX A. (continued)

28700 ALFA=(1-K‘2)/(1+K“2)

28800 VI=1/(2!PI!VP)!(F/LP)!(1-K“2)!LOG(1/K)!(1+ALFA/LOG(K))
!6!10“8

28900 REM!!APPARENT ELASTICITY CALCULATIONS!!!

29000 SIGMA=SOR ((RO-RI)“2+LE‘2)

29100 ROA=PF/(PI!(RI“2+SIBHA!(RI+RO)I)

29200 EP=(RI+SIGMA-RO)/RO

29300 EA=ROA/EP!100

29400 REM !!!SHEAR RATE AT THE PWS!!!!

29500 LN=((’(ALFA/(LOG(K)+ALFA)I)!(VP/RI))/6O

29600 RETURN

29700 END

 

 

APPENDIX B. List of computer program Delia.

1000
1050
1100
1150

1200

1250
1300
1350
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
1950
2000
2050
2100
2150
2200
2250
2300
2350
2400
2450
2500
2550
2600
2650
2700
2750
2800
2850
2900
2950
3000
3050
3100
3150
3200
3250

DISP @ DISP e DISP
DISP " !!! PROGRAM DELIA BY CARLOS A. LEVER !!! " L
DISP

DISP “t! THIS PROGRAM USES DATA FROM INSTROM CONTROL
PROGRAM RODRIGO xx"

DISP ”TO CALCULATE APPARENT VISCOSITY AND APPARENT
ELASTICITY,"

DISP " FROM BACKEXTRUSION DATA "

REM

DISP e WAIT 5000 e CLEAR

u=o

CCP=0

PRINTER IS 701

DIM A(200),B(200),C(200),AA(200),CC(200)

DIM AREA(10)

DIM SAMPID$E853

DISP

GOSUB 10000

DISP "ENTER FILE NAME WITH INSTROM RAW DATA”

INPUT FILNAMEs

ASSIGN# 1 TO FILNAMES .
READ# 1 3 SAMPIDs ,
DISP SAMPIDs

DISP

FOR I=o TO 170

READ# 1 ; A(I),B(I),C(I)

AA(I)=ABS(A(I)) e CC(I)=AGS(C(I))

NEXT I

IF CCP=1 THEN 2600

IF CCP=2 THEN 3950

CLEAR

DISP "DO YOU WANT THE COMPUTER TO ESTIMATE Le AND Lp"
DISP ”YES=1","NO=2"

INPUT CCP

REM !! PROGRAM TO CALCULATE Le AND Lp DISTANCE xx

IF CCP=2 THEN 3950

FOR I=I TO 165

IF A(I)<-.03 THEN GOTO 2850

NEXT I

IPV=CC(I-1)

R=I-2

FOR I=I TO 165

IF A(I)<A(I+1) THEN 3100

NEXT I

BFV=CC(I)

PF=AA(I)

LPV=CC(165)

I=1

 

135

—-———

 

 

136

APPENDIX 8. (continued)

3300 GOTO 5800

3350 GOTO 3950

3400 DISP "ALL DATA WILL BE DISPLAY AS:"

3450 DISP "Force (N)“,"Distance (mm)","Sequential Number"

3500 DISP

3550 DISP "Please take note of:"

3600 DISP "-Sequential Value of Initial Distance Value (mm)"

3650 DISP "- Sequential Value of Break Points Distance and
Force"

3700 DISP

3750 DISP

3800 DISP "To continue Press <ki>

3850 ON KEY# 1 GOTO 3950

3900 GOTO 3900

3950 CLEAR

4000 M=20

4050 PAGESIZE 24

4100 DISP "Force (N)","Distance (mm)”,"Sequential Number"

4050 FOR I=1 TO M

4200 DISP A(I),C(I),I

4250 NEXT I

4300 DISP "Do You Want To See More Values 7",”Yes=1“;" No=2"

4350 INPUT BB '

4400 IF BB=2 THEN 5300

4450 CLEAR

4500 IF I>160 THEN 4850

4550 M=M+20

4600 DISP "Force (N)","Distance (mm),”Sequential Number“

4650 FOR I=I TO M

4700 DISP All),C(I),I

4750 NEXT I

4800 GOTO 4300

4850 CLEAR

4900 M=M+10

4950 IF I=170 THEN 5100

5000 GOTO 4600

5050 PAGESIZE 16

5100 DISP "DO YOU WANT TO SEE THE DATA AGAIN"

5150 DISO "YES=1", "No=2"

5200 INPUT BB

5250 IF BB=1 THEN 3350

5300 CLEAR

5350 PAGESIZE 16

5400 DISP ”INPUT INITIAL DISTANCE AND BREAK DISTANCE NUMBER"

5450 INPUT I,J

5500 IPV=CC(I)

5550 R=I

5600 BFV=CC(J)

5650 PF=AA(J)

5700 LPV=CC(165)

 

APPENDIX

5750
5800
5850
5900
5950
6000
6050
6100

6150
6200
6250
6300
6350
6400
6450
6500
6550
6600
6650
6700
6750
6800
6850
6900
6950
7000
7050
7100
7150
7200
7250
7300
7350
7400
7450
7500
7550
7600
7650
7700
7750
7800
7850
7900
7950
8300
8350
8400
8450

CLEAR

137

B. (continued)

DISP "Please Wait Area CAlculations in Progress”

U=U+1

SUM1=0

FOR I

=2 TO 165

SUM1=SUM1+AA(I)!(CC(I+1)-CC(I-1))!.5

NEXT
FI=(A
(CC(1

I
All)!(CC(2)-CC(1))+AA(166)!(CC(166)-
65)))!.5

AREA(U)=SUM1+FI

GOSUB
CLEAR
DISP
DISP
DISP
DISP
DISP
DISP

DISP

DISP
DISP
DISP
DISP
REM

GOTO

DISP
DISP

INPUT
IF BB
DISP

DISP

PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
PRINT
CLEAR
GOTO
DISP
DISP

9600

SAMPID$;" ":SAMs
"Area=";AREA (U);"mm“2"

"Plunger Velocity is="; B(2);"mm/min"

"Apparent Viscosity is="; VI;"poise"

"Apparent Elasticity is=";EA;"N/cm“2"

"Shear Rate at the PWS is=";LW;"i/sec"

7200

"Do you Want to Print These Values?“
IO Yes=1 ll ’ OI No=2"

BB

=2 THEN 8300

"PRINTIING IN PROGRESS, PLEASE WAIT"
“PRINTING IN PROGRESS, PLEASE WAIT"

“u

 

DISP

”!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"

SAMPIDS; " ":SAMs
"Area=";AREA (U);"mm“2"
"Plunger velocity is =”;8(2);"mm/min"

“Apparent Viscosity is="VI;"poise"

”Apparent Elasticity is=";EA;"N/cm“2"

"Shear Rate at the PWS is=”;LW;"1/sec"

9350
"Do You Want to Print the Raw Data?"
.0 YES=1 II ’ OI N082"

APPEN

8500
8550
8600
8650
8700
8750
8800
8850
8900
8950
9000
9050
9100
9150
9200
9250
9300
9350
9400
9450
9500
9550
9650
9700
9750
9800
9850
9900
9950
10000
10050
10100
10150
10200
10250
10300
10350
10400
10450
10500

10550
10600
10620
10730
10650
10700
10750
10780
10800

138

DIX 8. (continued)
INPUT 88
IF BB=2 THEN 9100

DISP "PRINTING IN PROGRESS,PLEASE WAIT”
PRINT"!!#!*!!!!!!!!it!!!t!!!!!!!!t!!!!!!!!!!!t"
PRINT
PRINT SAMPIDS
PRINT
PRINT "Force (N)","Distance (mm)","Sequential Number"
FOR I=1 TO 165
PRINT A(I),C(I),I
NEXT I
GOTO 9100
CLEAR
DISP "DO YOU WANT TO READ ANOTHER FILE ?"
DISP "1=YES","2=NO"
INPUT SINO
IF SINO=2 THEN 9550
FOR L=1 TO 165
A(L),B(L),C(L),AA(L),CC(L)=O
NEXT L
GOTO 1800
END
DISP "Please Wait Viscosity Calculations in Progress"
REM !!BASIC CALCULATIONS!!!! '
LP=LPV-IPV
LE=BFV-IPV
F=2¥AREA(U)/LP
VP=B(2)
GOTO 11150
REM ##VISCOSITY INDEX CALCULATIONS!!!!!!
VP=20
RI=4.255
RO=7.028
DISP "The Viscosity Constants Used Are:”
DISP
DISP "CONTANT (1)-Plunger Velocity = 20 mm/min"
DISP "CONTANT (2)-Plunger Radius = 4.255 mm"
DISP "CONSTANT(3)-Inner Test Tube Radius = 7.028mm"
DISP
DISP "IF YOU WANT TO CHANGE ANY OF THEM, INPUT
CONSTANT NUMBER "
DISP "IF NOT THEN INPUT 0"
INPUT BBGCLEAR'GIF BB=O THEN RETURN
IF BB=1 OR BB=2 OR B=3 THEN 10650 ELSE BEEP 400,40
DISP "ENTER ONLY A 1, 2 OR 3 " e GOTO 10500
IF BB=1 THEN GOTO 10800
IF BB=3 THEN 11000
IF BB=2 THEN 10900
CLEAR
DISP "ACTUAL PLUNGER VELOCITY IS=";VP;mm/min, INPUT

.7. 1.2%.“; lit... at,
.

 

139

APPENDIX 8. (continued)
NEW VALUE"
10850 INPUT VP G CLEAR 9 GOTO 1200
10900 DISP "ACTUAL PLUNGER RADIUS, IS";RI;"mm, INPUT NEW
VALUE“
10950 INPUT RI 0 CLEAR e GOTO 10200
11000 DISP “ACTUAL INNER TEST TUBE RADIUS IS";RO;"mm, INPUT
NEW VALUE" 4
11050 INPUT RO @ CLEAR @ GOTO 10200
11100 RETURN
11150 DISP .
11200 DISP "PLEASE WAIT ELASTICITY CALCULATIONS IN PROGRESS" A
11250 REM !!!!Apparent VIscosity Calculations !!!!!!!! .
11300 K=RI/RO '
11350 ALFA=(1-K“2)/(1+K“2)
11400 VI=1/(2!PI!VP)!(F/LP)!(1-K“2)!LOG(1/K)!(1+ALFA/LOG(K))
!6!10“8
11450 REM!!APPARENT ELASTICITY CALCULATIONS!!!
11500 SIGMA=SOR ((RO-RI)‘2+LE“2)
11550 ROA=PF/(PI!(RI“2+SIGMA!(RI+RO)))
11600 EP=(RI+SIGMA-RO)/RO
11650 EA=ROA/EP!100
11700 REM !!!SHEAR RATE AT THE PWS!!!!
11750 LW=((~(ALFA/(LOG(K)+ALFA)))!(VP/RI))/60
11800 RETURN '
11850 END

APPENDIX C. List of computer program Mariana.

1000
1050
1100
1150
1200

1250
1300
1350
1400

1450
1500
1550
1600
1650
1700
1750
1800
1850

1900
1950
2000
2050

2100
2150
2200
2250
2300
2350

2400
2450
2500
2550
2600
2650
2700
2750

2800
2850
2900
2950

3000

CLEAR @ DISP G DISP

DISP "!!! PROGRAM MARIANA BY CARLOS A. LEVER !!!!!"
REM !!!!

DISP

DISP ”!! PROGRAM TO CALCULATE THERMAL MODEL VARIABLES

!!!!!!!!!"

U=0

DIM tt(60), Ta(60), TT(15,65), Tt(15,65)

DIM t(100), Y(100), T(100), X(100)

DIM n(100), c(5,65), a(5,65), FO(80), TRA(60), RMU(30).
Fo(80)

DIM FLN$[8], SID$E80], TFLNSElO]

CCP=0

DIM PSIA (15,65), PSIAV(65), A_PRIME(15,65)

PRINTER IS 701

DISP

REM !!!!!! MODEL CALCULATION ROUTINES !!!!!!!!!!!!!!!!
REM

DISP

DISP " If you Want to Calculate the Thermal Process
(Bi, Fo,etc.), TYPE 1

DISP .

DISP " If you Want to Calculate A' and alpha, TYPE 2"
DISP

DISP "If you Want to Calculate First Raw Estimate of
Ea, TYPE 3“

DISP

DISP "If you Want to Calculate a(kt), TYPE 4"

DISP

DISP "If you Want to Calculate TTH, TYPE 5"

DISP

DISP “If you Want to Calculate Rings Temperature,
TYPE 6“

INPUT BB

CLEAR

IF
IF
IF
IF
IF
IF

BB=1
BB=2
BB=3
BB=4
BB=6
BB=5

THEN
THEN
THEN
THEN
THEN
THEN

2850
20000
21450
23100
10950
15300 ELSE BEEP 9 CLEAR 8 DISP "WRONG

CHOICE " G GOTO 1700
CLEAR
!!!!!!! Calculation of Thermal Process !!!!!!!!!!!
K=1 G D=2
DISP " Will Data Be Input From Keyboard (k) or from
Disk (D)"
DISP “Input k OR d" 0 INPUT BB

REM

140

APPENDIX

3050
3100
3150
3200
3250
3300
3350
3400
3450
3500
3550
3600
3650
3700
3750
3800
3850
3900
3950
4000
4050
4100

4150
4200
4250
4300
4350

4400
4450
4500
4550
4600
4650
4700
4750
4800

4850
4900
4950

5000
5050
5100
5150
5200
5250
5300

141

C. (continued)

IF BB=1 THEN 3700

IF BB=2 THEN 3150 ELSE CLEAR e GOTO 2950

CLEAR

DISP "Enter File Name"

INPUT FLNS

ASSIGN# 1 TO FLNs

READ # 1 ; SID$

DISP SIDs @ DISP

FOR x=1 TO 30

READ# 1 ; tt(x), T(x)

IF T(x)=0 then 3650

NEXT X

ASSIGN# 1 TO ! G GOTO 4750

DISP "INPUT TIME (sec) and Temperature (C)"
DISP "INPUT 0.0 when finish"

FOR x=1 TO 100

DISP "sample No." ; x

INPUT TT(x), t(x)

IF T(x)=0 THEN 4050

NEXT x

CLEAR e DISP " Storage of Data is Next" G DISP
DISP " Insert Disk to Store Data (!! Initialized
Disk !!)"

DISP "Enter File Name (up to eight Characters)"
INPUT FLN$ .

CREATE FLNS, 15,100

ASSIGN# 1 TO FLNs

DISP "Enter Sample Identification (up to 40
Characters)"
INPUT SIDS
PRINT” 1 ; SIDS
FOR PP=1 TO x
PRINT# 1 ; tt(PP),
NEXT PP

ASSIGN4 1 TO !
DISP "time", "Temperature"

Print "time”, "Temperature"

DISP 0 FOR PP=1 TO x-i C DISP tt(PP),
tt(PP), T(PP)

NEXT PP

REM !!!! Calculation of F0, [T], Bi, r/r !!!!!!!!!!!!
REM !!! Constants use in F0 and [TI are Defined
Next!!!!

Ti=20

Cp= 3.9216375

DEN= 1071.823

MOI= 90

Tin= 7O

REM !!!!! FO CALCULATION!!!!!!!!!!

CLEAR

T(PP)

T(PP) Q PRINT

APPENDIX

5350
5400
5450
5500
5550

5600
5650
5700
5750
5800
5850
5900
5950
6000
6050
6100
6150

6200
6250
6300
6350
6400
6450
6500
6550
6600
6650
6700
6750
6800
6850
6900
6950
7000
7050
7100
7150
7200
7250
7300
7350
7400

7450
7500
7550
7600
7650

142

C. (continued)

DISP “F0 AND [T] CONSTANTS ARE:"
DISP
DISP

DISP

Cp="; cp; "Kj/Kg-K"
Density (RO)="; DEN; "Kg/mc3"
Initial Temperature of Sample

"Constant (1) ;
"Constant (2) ;
DISP "Constant (3) ;
(To)="; Ti; “C"

DISP "Constant (4) Temp, of Water Bath=; Tin; "C"
DISP a
DISP "Do you Want to Change any of Them?" I
DISP "If YES then input Constant No." ‘
DISP "If NO then input 0"
INPUT 88

IF BB=0 THEN 7000

IF BB=2 THEN 6800

IF BB=3 THEN 6400

IF BB=4 THEN 6600

CLEAR

DISP “To Calculate New Cp,
Sample"

DISP "as PERCENTAGE
INPUT MOI
Cp=1.675+.025!MOI
GOTO 5300 .
CLEAR 2

DISP "Input New Value of To (C)"

INPUT Ti

GOTO 5300

CLEAR

DISP ”Input New Value of T (water bath- C)"
INPUT Tin

GOTO 5300

CLEAR

DISP "Input New Value for Density (RO).
INPUT DEN

GOTO 5300

REM !!!! FO FROMULA !!!!

RAR= 7.028/1000 ! r from mm to meters
RR=RAR!RAR ! r02
TOL=1!10“-5 ! For use in
MAXI=35 E IDEM than TOL
REM

DISP "Please Wait ,
FOR X=1 TO x-i

ThC= (.30776.775!10“-4!T(X)!.001730314961 !
Kj-m/sec-mAZ-C

THC=ThCl (DEN!Cp!RR)

FO(X)= tt (X)! THC

TRA (X)= (Tin-T(X)))/(Tin-Ti)

REM !!!! printing option for Checking values!!!!!!
REM !!! '

 

Input Moisture Content of

(O to 100)"

in Kg/m93"

Bessel and Bissect Subr.

F0 and [t] calculations are next"

to get

143

APPENDIX C. (continued)

7700
7750

7800
7850
7900
7950
8000
8150
8200
8250
8300
8350
8400
8450
8500
8550
8600
8650
8700

8750
8800
8850
8900
8950
9000
9050
9100

9150
9200
9250
9300
9350
9400
9450
9500
9550
9600
9650
9700
9750
9800
9850

9900
9950

DISP FO(X), TRA(X), tt (X)

PRINT USING "3A,ZD,2A,D.4D,X,5A,ZD,2A,D.4D, X,3D,X,4A";
"FO(”,X,")=”,FO(X),TETA (",X,")=”,TRA(X),tt (X),"sec."
NEXT X

CLEAR

REM !!!! CALCULATIONS FOR BI!!!!!!!!!!!

CLEAR L“
DISP "Calculations for Bi are next"

FOR X=1 TO x-1 I
FOU=FO(X) .
TETA=TRA(X)

“=1 l
XB=5

FA=FNFUN (XA)

FB=FNFUN (XB)

JJJ=1

GOSUB BISECT

RMU (X)=RD @ IF RMU (X)=5 THEN 8650 ELSE 8750

DISP “Value of Sample No (";X;") is out of Range"

DISP "Then a Very High Value of Bi (2!10“40) will be
assumed"

RD=0

NEXT X .

WAIT 5000 .

CLEAR

PAGESIZE 24

CLEAR

DISP "All values of Bi (Experimental) are Displayed”
DISP "Select Range to be Considered (First and Last
SEQUENTIAL Values)"

DISP

DISP

DISP "Bi “, ”Sequential Number"

PRINT G PRINT 9 PRINT

FOR X-i TO x-i

LBi= .08359!RMU(X)“4.35842959

Bi=EXP (LBi)

DISP Bi, X

PRINT "Bi (";X;")=”Bi

NEXT X

DISP G DISP G DISP "Press 1 to Continue"

INPUT BB

PAGESIZE 16

CLEAR ’

DISP "Input First and Last Value of Bi Selected
(Sequential Value)"

INPUT FIRST, LAST

RMUA=0

10000 FOR X=FIRST TO LAST
10050 RMUA=RMUA+RMU (X)

APPENDIX C.

10100
10150
10200
10250
10300
10350
10400
10450

10500
10550

10600

10650
10700
10750
10800
10850
10900
10950
11000
11050
11100
11150
11200
11250

11300
11350
11400
11450
11500
11550
11600
11650

11700
11750
11800
11850
11900
11950

12000

12050

12100
12150

144

(continued)
NEXT X
RMUAa=RMUAl(LAST+1-FIRST)
TEM
LBi = .08359!RMUAa“4.35843
CLEAR
CLEAR
DISP "average Bi="; Bi

DISP "Values Utilized Were From "; FIRST;" to "; LAST
@ PRINT @ PRINT

PRINT

PRINT "Average Bi="Bi,"Average MU=";RMUAa, “Average
M=”;1/8i

PRINT "Values Utilized Were From"; FIRST; " to "; LAST

8 PRINT @ PRINT

MUA=RMUAa

FOR X=1 TO x-i

RMU (X) =0

NEXT X

DISP

GOTO 11450

REM !!!! Calculations for r/R !!!!!!!!!!!!!!!

DISP ”Calculations for r/R and [T15 are Next"

DISP G DISP

Cp=3.9216375 G DEN+ 1071.823 0 MAXU=35

RAR= 7.028/1000 ! r from mm to meters

RR=RAR!RAR ! r92

Bi= 15.724 G Ti=20 e disp "Bi=";Bi, "Temp.
Initial=";Ti .

DISP "Input Temp of Water Bath “ 0 INPUT Tin
ThC=(.3077 + 6.775! 109-4 ! Tin)! .001730314961

THC= ThC/(DEN!Cp!RR)

AREA=155.1720074 @ TOL= 1!10“-5

IRADI= 7.028

NA=1O G TOLE= .01 G Ea= 20000

CLEAR 9 DISP e DISP "Value Of Ea ="; Ea

DISP "Number of Rings to be considered is";NA e DISP e
GOTO 12150

DISP "If you Want to change any of them TYPE Ea or NR"
DISP "To continue TYPE C”

INPUT 8860 IF 886=”Ea" THEN 12000

IF BB$= "NR" THEN 12050

IF BB$= “C" THEN 12150 ELSE 11950

BEEP 8 BEEP e DISP "Wrong Choice,try Again" 8 Wait
2000 e GOTO 11600

DISP "INPUT NEW VALUE OF Ea (Cal)“@INPUT EaG GOTO
11600

DISP "INPUT NEW VALUE OF NUMBER OF RINGS” e INPUT NA 0
GOTO 11600

Ea=2000

DISP

 

 

 

145

APPENDIX C. (continued)

12200 DISP "STORAGE OF t(I,J) Data Parameters are Next"

12300 DISP “ENTER FILE NAME ( UP TO 8 CHARACTERS)"

12350 INPUT TFLNs

12400 DISP " Please wait,intense calculations in progress"

12450 DISP G DISP " Calculation of MU roots (6) is next"

12500 XA= .1

12550 XB= 18

12600 JJJ=2 G X=1

12650 GOSUB BISECT2

12700 LL=1 G BEEP G DISP "The 6 MU's roots calculated"

12750 FOR OO=1 TO 6 G PRINT "R(";OO;")=";R(OO)@ NEXT 00

12800 RMU(LL)=R(LL)

12850 DISP @ DISP “Calculation of NODE temperatures are
next"

12900 FOR TIEMPO =0 TO 1800 STEP 30

12950 SUMl = 0

13000 X= TIEMPO/30 +1

13050 Fo(X)= THC!TIEMPO

13100 FOR Ii =0 TO NA

13150 ENE=SOR(Ii/NA)

13200 FOMU12=Fo(X)!RMU(LL)“2

13250 ROO=RMU(LL)!ENE

13300 VRMU=RMU(LL) -

13350 VFF=FNANJO(VRMU,ROO) ‘

13400 SUM1=SUM1+VFF

13450 DISP @ DISP "!! Calculation Round, Time"; TIEMPO;"SEC
RING";Ii;",ROOT";LL

13500 LL=LL+1 @ IF LL>6 THEN 13700

13550 RMU(LL)=R(LL)

13600 FOMU12=Fo(X)!RMU(LL)‘2

13650 GOTO 13250

13700 TETA=SUM1

13750 TT(Ii,X)=TETA!(Ti—Tin)+Tin

13800 BEEP @ DISP

13850 DISP " !! TEMP AT RING";Ii;"AND TIME";TIEMPO;"SEC,
=”;TT(Ii,X);"!!"

13900 IF TT(°,X)>=Tin-TOLE THEN 14150

13950 ll=1 G SUM1=0

14000 NEXT Ti

14050 NEXT TIEMPO

14100 CLEAR G GOTO 14450

14150 FOR RE=X TO 61

14200 FOR IE=O TO NA

14250 TT(IE,RE)=Tin

14300 NEXT IE

14350 NEXT RE

14400 PRINT SIDS

14450 PRINT USING "5A,11(X,5A)";"SEC","RO","R-i","R-2",”
R_3H ’ DIR-4H ’ CIR—.5" ’ DIR-6H ’ DIR-7” ’ IOR-ell ’ OUR-9M ’ OUR-1°"

14500 FOR J=1 TO 61

 

APPENDIX

14550

14600

14650
14700
14750
14800
14850
14900
14950
15000
15050
15100
15150
15200
15250
15300
15350
15400
15450
15500
15550
15600
15650
15700

15750
15800

15850
15900
15950
16000
16050
16100
16150
16200
16250
16300
16350
16400
16450
16500
16550
16600
16650
16700

146

C. (continued)

PRINT USING "5D,11(X,2D.20)”;J!30-30,TT(0,J),TT(1,J),
TT(2,J),TT(3,J),TT(4,J),TT(5,J),TT(6,J),TT(7,J),

TT(8,J), TT(9,J),TT(10,J)

DISP USING "50,11(X.2D.20)";J!30-30,TT(0,J),TT(1,J).
TT(2,J),TT(3,J),TT(4.J).TT(6.J).TT(6.J).TT(7.J).

TT(8,J), TT(9,J),TT(10.J)

NEXT J .4
CLEAR @ DISP "STORAGE OF T(i,j) DATA IS NEXT " e DISP

CREATE TFLNs, 11.500

ASSIGN# 1 TO TFLNs

PRINT# 1;SID$

FOR E=1 TO 61

FOR N=o TO NA

PRINTa 1;TT(N,E)

NEXT N

NEXT E

ASSIGN# 1 TO I

DISP "TTH CALCULATIONS ARE NEXT"
DISP

REM xx CALCULATIONS FOR TTH !!
DISP "TTH CALCULATIONS ARE NEXT"
Ea = 20000

DISP @ DISP " VALUE OF Ea Ea
DISP " IF YOU WANT TO CHANGE IT TYPE Ea"

DISP " TO CONTINUE TYPE C"

INPUT BBS e IF aas= "Ea" then 15750

IF 886 = "C” THEN 15850 ELSE 15700

BEEP @ BEEP G DISP " WRONG CHOICE, TRY AGAIN"e WAIT
2000 a GOTO 15450

DISP " INPUT NEW VALUE OF Ea " @ INPUT Ea e GOTO 15450
CLEAR @ DISP "DATA WILL BE INPUT FROM DISK (D) OR
KEYBOARD (K)"

DISP

INPUT ass
IF BBs="D"
IF BB$=”K"
CLEAR
DISP "ENTER FILE NAME"

INPUT FLNs

ASSIGN» 1 TO FLN$

READ» 1; SIDS

DISP SIDs e DISP

FOR E=1 TO 61

FOR NA=0 TO 10

READ» 1; TT(NA,E)

NEXT NA

NEXT E

ASSIGN# 1 TO 3 e GOTO 17850

DISP ”INPUT NUMBER OF RINGS CONSIDERED"
INPUT NA e CLEAR

@ DISP G GOTO 18150

THEN 16050
THEN 17850 ELSE BEEP Q GOTO 15300

APPENDIX

16750
16800
16850
16900
16950

17000
17050
17100
17150
17200

17250
17300
17350
17400

17450
17500
17550
17600
17650
17700
17750
17800
17850
17900
17950
18000

18050

18100
18150
18200
18250
18300
18350
18400
18450
18500
18550
18600
18650
18700
18750
18800
18850

-TT(8,J),

147

C. (continued)

DISP ”INPUT NUMBER OF SAMPLES PER RING"
INPUT J G CLEAR

FOR X=1 TO NA

FOR RR=1 TO J

DISP " INPUT TEMPERATURE (RING,TIME) OF SAMPLE
(II;X;H’H;RR;H)II

INPUT TT(X,RR)

NEXT RR

NEXT X

CLEAR G DISP " STORAGE OF DATA IS NEXT " G DISP

DISP "INSERT DISK TO STORE DATA (!! INITIALIZED
DISK!!)"

DISP " ENTER FILE NAME (UP TO EIGHT CHARACTERS)"
INPUT TFLN$

CREATE TFLN$,11,600

DISP "ENTER SAMPLE IDENTIFICACTION (UP TO 50
CHARACTERS)”

INPUT SID$

PRINT# 1 ; SIDs

FOR RR=1 TO J

FOR NA=0 TO X

NEXT NA

NEXT RR .

ASSIGN» 1 TP x .

CLEAR

BB=0 G T=0 G R=1.986 G I=0 G DT=30 G PSI=0 G PEPE=O
PRINT G PRINT G PRINT SIDs

FOR J=1 TO 61

DISP USING "5D,11(X,2D.2D)" ; J!30-30,TT(O,J),TT(1,J),
TT(2,J),TT(3,J),TT(4,J),TT(5,J),TT(6,J),TT(7,J),
TT(9,J),TT(10,J)

PRINT USING "5D.11(X,20.2D)" ;J!30-30,TT(O,J),TT(1,J).
TT(2,J),TT(3,J),TT(4,J),TT(5,J),TT(6,J),TT(7,J),
TT(8,J), TT(9,J),TT(10,J)

NEXT J

DISP " PLEASE WAIT, INTENSE CALCULATIONS IN PROGRESS"
BB=0 G T=0 G R=1.986 G I=O G DT=30 G PSI=0 G PEPE=0
PRINT G PRINT G PRINT SIDs

DISP G DISP "TRANSFORMATION FORM C TO K IN PROGRESS"
FOR J=1 TO 61

FOR I=0 TO 10

Tt=TT(I,J)+273.2

IF Tt<313.2 then 18550 else 18600

A_PRIME(I,J)=0 G GOTO 18650

A_PRIME(I,J)=Tt!EXP (-(Ea/R!Tt)))

NEXT I

NEXT J

DISP G DISP “CALCULATION OF A‘
FOR I=O TO 10

FOR J=2 TO 61

p1

 

IN PROGRESS"

148

APPENDIX C. (continued)

18900
18950
19000
19050
19100
19150
19200

19250
19300
19350
19400
19450
19500
19550
19600
19650
19700
19750
19800
19850
19900
19950
20000
20050
20100
20150
20200

20250
20300
20350
20400
20450
20500
20550
20600
20750
20800
20850
20900
20950
21000
21050
21100
21150
21200
21250
21300
21350

PSI=(A_PRIME(I.J)+A_PRIME(I,J))lszT
PEPE=PEPE+PSI

PSIA(I,J)=PEPE
NEXT J

PEPE=0 @ PSI=0
NEXT I

DISP G DISP ”TTH CLACULATIONS FOR EACH RING ARE IN
PROGRESS"

PSI=0

FOR J=2 TO 61

FOR I=0 TO 10
PSI=PSIA(I,J)+PSI
NEXT I
PSIAV(J)=PSI/10
PSI=0

NEXT J

DISP "TTH AVERAGE"

FOR J=2 TO 61

DISP "TTH AT " ;J!30-30; "SEC = ";PSIAV(J)
PRINT ”TTH AT " ;J!30-30; "SEC a ";PSIAV(J)
NEXT J

CLEAR

GOTO 1700 .

REM xxx ESTIMATION OF A' AND ALPHA xxxx

REM

SX,SY,SXY,SXX,SYY=0

FOR X=1 TO 20

DISP "INPUT Y' AT N (INFINITE) AND PROTEIN
CONCENTRATION (DRY BASIS) NO" ; x

DISP "INPUT 0,0 WHEN FINISH”

INPUT Y,C

IF Y=o THEN 20550

YY(X)=LGT (Y)

XX(X)=LGT (C)

GOTO 20800

GOSUB LINREG

DISP "A'=" ;10“a,”alpha=",b,";R“2=";r!r,"n=";x-1
GOTO 1700

REM xxxxx LINEAR REGRESSION SUBROUTINE xxxxx
SX(SX+XX(X)

SYY=SYY+YY(X)!YY(X)

SY=SY+YY(X)

SXY=SXY+YY(X)!XX(X)

SXX=SXX+XX(X)!XX(X)

CLEAR

NEXT X

LINREG:
b=(-((X-1)xSXY)+SxxSY)/(-((X—1)xSXX)+sxxSX)
a=(SX!SXY-SXX!SY)/(SX!SX-(X-I)!SXX)
r-((X-1)!SXY-SX!SY)/SOR (((X—1)xSXX-sxxSX)x((x-I)

 

149

APPENDIX C. (continued)

21400
21450
21500
21550
21600
21650
21700

21750
21800
21850
21900
21950
22000
22050
22100
22150
22200
22250
22300
22350
22400
22450
22500
22550
22600
22650
22700
22750
22800
22850
22900
22950
23000
23050
23100
23150
23200

23250
23300
23350
23400
23450
23500
23550
23600
23650
23700

!SYY -SY!SY))

RETURN

REM !!!!!! ESTIMATION OF Ea !!! !!!!!!!!!!!!!!!!!
T,t,Y=0

SX,SY,SXY,SXX,SYY=O

FOR x=1 TO 10

FOR X=1 TO 100

DISP " INPUT TEMPERATURE (IN C) AND TIME (SEC) FOR
EXP. NO. " ; X

DISP "ESTIMATION OF Ea No. ";x

DISP "WHEN FINISH INPUT 0,0"

INPUT T,t

IF T=0 THEN 22150

XX(X)=1/(t+273.2)

YY(X)=LGT (t)

CLEAR

GOTO 20850

GOSUB LINREG

r(x)=r

Ea(x)=b!1.986

DISP "Ea=";Ea(x),"R=;r(x),"VARIABLE No ”;x
PRINT "Ea=";Ea(x),“R=;r(x),"VARIABLE No ";x
DISP "DO YOU WANT TO CALCULATE ANOTHER Ea"
DISP "YES=1", "No=2"

INPUT 88

IF 88=2 THEN 22650

NEXT x

EA=O

FOR X=1 TO x

EA=EA+Ea(X)

NEXT X

Ed=EA/x

DISP "Ea AVERAGE OF“;x;"EXPERIMENTS IS =";Ed;”CAL/MOL"
PRINT " Ea AVERAGE IS =“;Ed;"CAL/MOL"

CLEAR

GOTO 1700

REM !! SUBROUTINE TO ESTIMATE (a) !!

DISP

DISP "CALCULATION OF (a) WILL BE THE AVERAGE OF 3 SETS

OF VALUES"

FOR m=1 TO 3

b=1

a=0

DISP "INPUT VALUES OF Y'(a),PSI(a), AND Y'(b),PSI(b)"
DISP “OF SAMPLE "' m

INPUT y(1). psia(1),Y(2),psia(2)

CLEAR

DISP "PLEASE WAIT INTENSE CALCULATIONS IN PROGRESS"
DISP "round";m

LHS=y(1)/y(2)

I'M.

 

150

fiat-1"“. - r—T

APPENDIX C. (continued)

23750 FOR X=1 TO 10

23800 FOR x=1 TO 100

23850 c=a+b!x

23900 RHS=(1-EXP (-(c!psia(1))))/(1-EXP (-(C!psia(2))))
23950 IF RHS>LHS THEN 24050

24000 NEXT x

24050 a=c+(x-1_!b

24100 b=b!.1

24150 NEXT X

24200 A(m)=a

24250 DISP ”a(";m;")=";A(m)

24300 NEXT m

24350 AV=(A(1)+A(2)+A(3))/3

24400 DISP "(a) AVERAGE IS ="; AV

24450 PRINT

24500 PRINT "(a) AVERAGE IS =”; AV

24550 PRINT

24600 PRINT "a(i)="; A(i),”a(2)=";A(2),"a(3)=";A(3)
24650 PRINT

24700 CLEAR

24750 GOTO 1700

24800 REM !!! SUBROUTINE DEFINITION OF BESSEL FUNCTION !!!!
24850 DEF FNBESJ (ROO)

24900

24950
25000
25050
25100
25150
25200
25250
25300
25350
25400
25450
25500
25550
25600
25650
25700
25750
25800
25850
25900
25950
26000
26050
26100
26150

TOL=1!10“-5 G DISP G DISP “CALCULATION OF BESSEL
FUNCTION, SAMPLE No.”;X
X2=ROO!ROO

FNBESJ=0

IF ROO>15 THEN 25600

SUM=1

TERM2=SUM

I=0

I=I+1

TERM+TERM2
TERM2=-(TERM!X2!.25/(I!I))
SUM=SUM+TERM2

IF ABS (TERM2)>ABSD (SUMxTOL) THEN 25250

FNBESJ=SUM

GOTO 25650

FNBESJ=SOR (2/(PIxROO))xCOS (ROO-PI /4)

FN END

END

REM xxxxxx DEFINITION OF FUNCTION FNFUN (RMU) xxxxxxx
DEF FNFUN (RMU)

RM=RMU

ATE=LOG (TETA)

FNFUN=RM!RM!FOU-.5395!LOG (RM)+ATE-.O3974

FN END

END

REM xxx SUBROUTINE BISECT xxxxxxxxxxxxxxxxx

REM

 

APPENDIX C. (continued)

26200
26250

26300
26350
26400
26450
26500
26550
26600
26650
26700
26750
26800
25850
26900
26950
27000
27100
27150
27200
27250
27300
27350
27400
27450
27500
27550
27600
27650
27700
27750
27800
27850
27900
27950
28000
28050
28100
28150
28200
28250
28300
28350
28400
28450
28500
28550
28600
28650

BISECT:
RM=O G RD=0 G DISP "CALCULATION IS IN SUBROUTINE
BISECT, SAMPLE No. ";X
X0=XA

FO=FA

X1=XB

F1=FB

XM+(XA+XB)/2

IF FO!F1>0 THEN 29250
REM

FOR J=1 TO MAXI

IF ABS (F1)>= ABS (PU) THEN 26800
GOTO 27100

X2=X0

XO=X1

X1=X2

F2=FO

FO=F1

REM
XX=X1-F1!(X1-X0)/(Fi-FO)
IF XB<XA THEN 27400

IF XX<XA THEN 27900

IF XX>XB THEN 27900
GOTO 27500

IF XX<XB THEN 27900

IF XX>XA THEN 27900

REM

DIFF= ABS (XX-X1)

IF DIFF<ABS (xxxTOL) THEN 29050
X0=X1

FO=F1

X1=XX

GOSUB SELECTI

NEXT J

REM xxxxxxxx

GOSUB SELECT2

IF FM=0 THEN 28850

IF FAxFM<o THEN 28450
XA=XM

FA=FM

FO=FA

X0=XA

F1=FB

X1=XB

GOTO 28750

XB=XM

X1=XB

FB=FM

F1=FB

X0=XA

_rlmr —

 

 

 

APPENDIX

28700
28750
28800
28850
28900
28950
29000
29050
29100
29150
29200
29250
29300
29350
29400
29450
29500
29550
29600
29650
29700
29750
29800
29850
29900
29950
30000
30050
30100
30150
30200
30250
30300
30350
30400
30450
30500
30550
30600
30650
30700
30750
30800
30850
30900
30950
31000
31050
31100
31150

162

C. (continued)

FO=FA

XM=(XA+XB)+2

IF ABS (XA-XB)>ABS
RM=XM

RD=XM

RETURN

END

RM=XX

RD=XX

RETURN

END
DISP
RD=5
RETURN

END

REM !!!!! FUNCTION FNJI

DEF FNJI (ROD)

X2=ROO!ROO

FNJ1=0

IF ROO >15 THEN 30200
SUM=ROO/2

TERM2=SUM

I=0

I=I+1

TERM= TERM2
TERM2=-(TERM!X2!.25/(I!(I+1)))
SUM=SUM+TERM2

IF ABS (TERM2)>ABS (SUM!TOL) THEN 29850

FNJ1=SUM

GOTO 30250

FNJ1=SOR (2/(PI!ROO))!COS (ROO-.75!PI)

FN END

END

REM !!!!!! SUBROUTINE SELECT1 !!!!!!!!!!!!!
SELECTI:

IF JJJ=1 THEN F1=FNFUN(X1)

IF JJJ+2 THEN Y=FNFX1(XX)

RETURN

END

REM !!!!!!!! SUBROUTINE SELECT2 !!!!!!!!!!!!!!!!!!
SELECT2:

IF JJJ=1 THEN FM=FNFUN(XM)

IF JJJ+2 THEN FM=FNFX1(XM)

RETURN

END

REM !!!!!! SUBROUTINE SELECT3 !!!!!!!!!

SELECT3:

IF LL=2 THEN XA=3.5

IF LL82 THEN XB=6

IF LL=3 THEN XA=6.5

(XM!TOL) THEN 26600

"ROOT NOT IN INTERVAL"

!!!!!!!!!!!!!

153

APPENDIX C. (continued)
31200 IF LL=3 THEN XB=9
31250 IF LL=4 THEN XA=9.5
31300 IF LL=4 THEN XB=11
31350 IF LL=5 THEN XA=13
31400 IF LL=5 THEN X8=15
31450 IF LL=6 THEN XA=16
31500 IF LL=6 THEN XB=17
31550 IF LL>6 THEN DISP "LL > 6" ELSE 31650
31600 DISP "VALUE OUT OF RANGE" 8 END
31650 RETURN
31700 END
31750 REM !!!! DEFINITION OF FNANJO(MU,ROO) !!!!!!!!!!!!!!!!
31800 DEF FNANJO(MU,ROO)
31850 AN=2!Bi/(FNBESJ(MU)!(MU!MU+Bi!Bi))
31900 FNANJO=AN!FNBESJ(ROO)!EXP (-FOMU12)
31950 FN END
32000 END
32050 REM !! SUBROUTINE VMUROO TO OBTAIN MU FROM ROOT EO. !!
32100 VMUROO:
32150 WW=O
32200 FA=FNFX1(XA)
32250 FB=FNFX1(XB)
. 32300 GOSUB BISECT
32350 RETURN
32400 END
32450 REM !!!! DEFINITION OF FUNCTION FNFXi !!!!!!!!!!!!!
32500 DEF FNFXi (MU1)
32550 FNFX1=MU1!FNJ1(MU1)-Bi!FNBESJ (MU1)
32600 WW=WW+1 G DISP
32650 DISP " MU1=“; MU1;" ROUND No."; WW
32700 FN END
32750 END
32800 REM !!!! SUBROUTINE BISECT2 !!!!!!!!!!!!!!!!
32850 BISECT2:
32900 N1+6 8 S=.2 @ DISP "CALCULATION IS IN SUBROUTINE
BISECT2"
32950 A1=XA G B=XB G WW=O
33000 A=A1
33050 N=0
33100 FOR I=1 T0 N1
33150 R(I)=INF 8 F(I)=INF G E(I)=INF
33200 NEXT I
33250 XX=A
33300 IF N>=N1 THEN 35650
33350 N=N+i
33400 GOSUB SELECTi
33450 F+Y
33500 A=A+S
33550 IF A>B THEN 35650
33600 XX=A

154

APPENDIX C. (continued)

33650
33700
33750
33800
33850
33900
33950
34000
34050
34100
34150
34200
34250
34300
34350
34400
34450
34500
34550
34600
34650
34700
34750
34800
34850
34900
34950
35000
35050
35100
35150
35200
35250
35300
35350
35400
35450
35500
35550
35600
35650
35700

GOSUB SELECT 1

P=F!Y

IF P>0 THEN 33450

IF P<0 THEN 34300

IF F<> 0 THEN 34000
XX=A-S

Y=F

R(N)=XX

F(N)=Y

A=A+S

Z=109-12

LET E(N)=Z

GOTO 33250

L=A-S

R=A

C=0

XX=(L+R)/2

GOSUB SELECTi

C=C+1

IF C>MAXI THEN 35300
IF ABS (Y)<TOL THEN 35050
P=F!Y

IF P<= 0 THEN 34900
L=XX

GOTO 34450

IF P=0 THEN 35050
R=XX

GOTO 34450

R(N)=Xx

F(N)=Y

Z=R-L

LET E(N)=Z

GOTO 33250

DISP "ROOT NOT FOUND IN"; MAXI; ”ITERATIONS”
DISP "FOR ROOT No.";N

Z=R-L
R(N)=(L+R)/2
F(N)=Y

LET E(N)=Z
GOTO 33250
RETURN

END

 

155

APPENDIX D. Experimental values of Y' and TTH utilized in
testing the model (Figure 13).

TEMPERATURE 92 PROCESSING

 

 

64 0C 70 oC
1: m I: '_r_'r_H
0.04 1.53E-10 0.55 5.08-10
0.08 1.53E-10 0.63 5.0E-10
0.10 1.53E-10 0.64 5.08-10
0.71 6.46E-10 0.97 2.0E-9
0.71 6.46E-10 1.18 2.08-9
0.71 6.46E-10 1.19 2.08-9
0.78 1.58E-9 2.35 5.228-9
0.88 1.58E-9 1.98 5.228-9
0.77 1.588-9 2.25 5.22E-9
2.21 4.5OE-9 3.00 1.21E-8
1.80 5.37E-9 3.23 1.21E-8
1.95 6.27E-9 3.21 1.21E-8
2.77 1.4OE-8 3.77 2.0E-8
3.42 1.40E-8 4.09 2.08-8
3.13 . 1.40E-8 4.58 2.08-8
4.05 2.40E-8 3.85 3.0E-8
3.79 2.40E-8 4.72 3.0E-8
3.58 2.4OE-8 4.86 3.0E-8
4.13 5.7OE-8 3.93 3.0E-8
4.43 5.708-8 3.44 3.03-8
4.87 5.70E-8 3.68 3.08-8
4.19 9.003-8 3.71 5.08-8
3.95 9.008-8 4.14 5.0E-8
4.17 9.008-8 3.56 5.0E-8
4.86 1.23E-7 4.13 7.0E-8
3.28 1.23E-7 4.42 7.08-8

4.20

10 a3E-7

APPENDIX D.

(continued)

 

80 °C
X.’ 21.11
0.26 3.08-10
0.35 3.08-10
0.66 1.14E-9
0.51 1.14E-9
0.76 1.14E-9
0.67 3.0E-9
0.83 3.0E-9
1.25 3.0E-9
2.06 4.53E-9
1.69 4.53E-9
1.80 4.53E-9
3.16 1.00E-8
3.43 1.008-8
2.56 1.00E-8
3.72 2.0E-8
3.72 2.0E-8
3.72 2.08-8
5.05 5.0E-8
3.89 5.0E-8
4.91 8.0E-8
4.10 8.08-8
4.55 8.0E-8
4.09 1.0E-7
3.97 1.0E-7
4.03 1.0E-7

156

TEMPERATURE g: PROC§§SING

 

x;
0.15
0.19
1.50
0.55
1.54
1.57
1.55
2.23
3.44
3.27
2.72
3.49
3.12
3.35
3.54
4.10
3.20
3.47
3.55
4.23
4.20
4.55
4.95
3.55
4.05

TTH

6.0E-10
6.0E-10
2.19E-9
2.19E-9
2.19E-9
4.72E-9
4.72E-9
4.72E-9
8.21E-9
8.2iE-9
8.21E-9
1.77E-8
1.77E-8
1.77E-8
3.0E-8
3.0E-8
3.0E-8
8.0E-8
8.0E—8
8.0E-8
.0E-7
.0E-7
.0E-7
.0E-7
.OE-7

NNNNN