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MODELING THE RHEOLOGICAL BEHAVIOR OF
GELATINIZING STARCH SOLUTIONS

By
Kirk David Dolan

A DIS SERTATION

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

DOCTOR OF PHILOSOPHY
in
Agricultural Engineering

Department of Agricultural Engineering

1989

(.3. at! g N,

5Q73742

ABSTRACT

MODELING THE RHEOLOGICAL BEHAVIOR OF
GELATINIZING STARCH SOLUTIONS

By
Kirk David Dolan

"Viscosity" of gelatinizing starch dispersions is used in
quality control and process design around the world. A
comprehensive model to predict viscosity for an arbitrary
process has not been proposed in the published literature. The
purpose of this study was to present a generalized mathematical
scheme to identify and measure factors influencing viscosity
development during gelatinization in food processing: shear
rate, temperature, moisture content, temperature-time hiStory.
and strain history.. Back extrusion and mixer viscometry
techniques were chosen to illustrate a general experimental
technique, because variables were known and highly controlled.
The results (within 10 % standard deviation of errors) suggested
that, for larger processes, the limiting factor will be
measurement accuracy rather than fit of experimental data in
determining parameters.

A back extrusion technique was used to estimate effect of
gelatinization only. An index of apparent viscosity of a 13.7%
gelatinized cornstarch solution at conStant shear rate and strain
hiStory was measured at 20 C. Activation energy of
gelatinization was estimated as 210. kJ/mol over the range 81-

95 C, and decreased in the range 95-105 C.

A Brookfield RVTD mixer viscometer was used to gelatinize
5.5, 6.4, and 7.3% (d.b.) native corn starch dispersions.
Torque response, the dependent variable, was used to estimate
parameters in a generalized model. The independent variables
were impeller speed,temperature, (50-95 C), moisture content,
temperature-time history, and strain history. Predicted and
experimental pasting curves were compared. A simplified and
rapid procedure for estimating parameters from an arbitrary
pasting curve was proposed, and applied to a bean starch
dispersion.

Dispersions thickened with decreasing temperature and had an
Arrhenius activation energy between 6.4 and 11.5 kJ/mol.

There was no evidence that retrogradation caused this effect.
Maximum viscosity depended on cook temperature, ranging from
a 150- to a 220-fold increase at 85 and 95 C, respectively.

First order reaction kinetics was accurate (9.8% standard
deviation) in describing viscosity increase during gelatinization.
Torque decayed exponentially after initial gelatinization.
Comparison of relative effects showed a pasting curve could be
predicted by knowing only the rate of gelatinization during
heating rise, the rate and extent of breakdown during shear

decay, and the torque response to temperature during cooling.

To my father and mother,
who introduced me to my
Eternal Father,
who has taught me,

"Let not the wise man boast of his wisdom
or the strong man boast of his strength
or the rich man boast of his riches,

but let him who boasts boast about this:
that he understands and knows me,
that I am the Lord,
who exercises kindness, justice,
and righteousness on earth,
for in these I delight."

Jeremiah 9:23, 24

iv

ACKNOWLEDGEMENTS

I thank my closest friend, encourager, and leader-Jesus of
Nazareth--for giving me health and wise counselors to complete
this work. I also thank Him for all the friends, both Americans
and internationals, who have helped make my life full.

I am grateful to Dr. Steffe for his encouragement,
availability, and guidance. I especially thank him for
suggesting a major change in our initial proposal, which was
too ambitious. His moral support was invaluable.

I appreciate Dr. Ofoli's insight about the many mathematical
details of my projects. I thank him for correcting this work so
promptly several times, as a favor to me.

I thank Dr. Morgan for challenging me to apply theory and
take risks, necessary qualities for work in China especially.
His influence on me cannot be underestimated.

I thank Dr. Jayaraman for his valuable suggestions,
particularly about parameter estimation, and application of
polymer to starch rheology.

I appreciate Dr. Markakis' insight about starch chemistry,
and his challenge to combine the physical with the theoretical.

I thank Dr. Beck, who, even when not on my committee,
helped interpret my data and suggested a transformation to

improve parameter estimation for the model.

I thank Dr. Gilliland for his suggestions, given on such short
time, concerning experimental design and analysis.

I am deeply indebted to Kevin Rose for his indefatigable
work on my experimental apparatus. He was always available,
and did far more than I ever asked. His willingness to analyze
my daily equipment problems late into the night, even while he
was working on his thesis, testifies to his servant's heart. I
hope to learn much more from him in China.

I am grateful to Kevin Mackey for his ideas about data
analysis and his help in learning software.

I thank Denny Welch for his shop work when time was short.

I appreciate so much the moral support of my housemate,
Andy Granskog, and the editing and encouragement of Cindy
Phelps. I also thank Mary Sokalski and Kay Cook, and their
prayer groups, for their concern.

I am very grateful to Songyos Ruengsakulrach and Naruemon
Srisuma for their tireless work with slides, the Apple computer,
and Ecklund thermocouples.

Thanks extends to Marta Olsen of Apple Computer and Walt
MacClay of Strawberry Tree, for their donations of a Mac SE

personal computer and data acquisition system, respectively.

vi

TABLE OF CONTENTS

List of Tables ............................................................................... x

List of Figures .............................................................................. xi

Nomenclature ................................................................................ xiv
=/ 1. Introduction .......................................................................... .3:

2. Back extrusion used in modeling rheological behavior

of starch solutions throughout gelatinization ...................... 3
2.1. Abstract ....................................................................... 3
\/2.2. Introduction ............................................................... {3"

2.2.1 Rheological Models for Starch “
Solutions ....................................................... 4
2.2.2 Other Rheological Models Applicable to
Starch ............................................................. 6
2.3. Modeling Approach ..................................................... 8
2.3.1. General Mathematical Model ........................... 7
2.3.1.1. Shear rate effect ................................ 10
2.3.1.2. Temperature effeCt ............................. 11
2.3.1.3. Moisture content effect ..................... 11
2.3.1.4. Temperature- time history effect ....... 12
2.3.1.5. Strain history effect .......................... 15
2. 3. 2. Advantages of the Model ................................. 16
2.3.2.1. Simplicity of form ............................. 16
2. 3. 2. 2. Predictive nature ................................ 16
2.3.2.3. Generality ........................................... 17
2. 3. 2. 4. Ease of measurement ......................... 17
2.3.3 Simplified Model for Starch
Gelatinization .................................................. 18
2.4. Materials and Methods ............................................... l9
2.4.1.Choice of Variables ......................................... 19
2.4.2.Experiments ...................................................... 20
2.4.2.1.Apparent viscosity ...................... 21
2.4.2.1.l.Temperature-time history ....... 25
2.5 Results ........................................................................ 28
2.6 Discussion ................................................................... 32
2.7 Problem for Calculating the Cook Time of a
Steam Infusion Process .............................................. 34
2.8 Summary and Conclusions ......................................... 34

vii

3. Mixer viscometry used in modeling rheological

behavior of gelatinizing starch solutions. ........................... 37
3.1. Abstract ....................................................................... 37
-/3.2. Introduction ............................................................... g}?
3.3. Theoretical Considerations ........................................
3.4. Practical Considerations ............................................ 50
3.5. Materials and Methods ............................................... 53
3.5.1. Experimental plan ......................................... 53
3.5.2. Apparatus ....................................................... 59
3.5.3. Procedure ....................................................... 63
3.5 4. Analysis ......................................................... 65
3.5.4.1. Analysis Involving All
Parameters .......................................... 65
3.5.4.2. Simplified Analysis ........................... 71
3.6. Results ........................................................................ 72
3.7. Discussion ................................................................... 91
3.7.1. Comparison of parameters with
literature ........................................................ 91
3.7.2. Gelatinization effects ................................... 91
3.7.3. Moisture content effects ............................... 92
3.7.4. Temperature Effects ...................................... 92
3.7.5. Strain history effeCts .................................... 93
3.7.6. Predicted pasting curves .............................. 94
3.7.6.1. Heating rise ........................................ 96
3.7.6.2. Shear decay ........................................ 96
3.7.6.3. Cooling ............................................... 97
3.7.6.4. Relative influences of
independent variables ........................ 97
3.7.6.5. Simplified Analysis ........................... 98
3.8. Conclusions ................................................................ 101
4. Industrial applications of rheological modeling .................. 103
4.1. Prediction of velocity profile .................................... 103
4.2. Rapid parameter estimation ........................................ 105
5. Overall Summary and Conclusions ....................................... 108
6. Suggestions for future research ........................................... 110
6.1. Limitations of the model ............................................ 110
6.2. Other applications ...................................................... 111
7. Bibliography .......................................................................... 114
8. Appendices ............................................................................. 119
8.1 Appendix A Parameter Estimation Analysis ............... 119

8.2 Appendix B Observed equilibrium torque of
gelatinizing native corn starch to determine
effects of moisture content and temperature-time

hiStory (corresponding to Table 3.2) .......................... 124

8.3.Appendix C Time and temperature data ..................... 126
8.3.1.Appendix C.l Time and temperature data

for each sample in Appendix B. ..................... 126
8.3.2.Appendix C.2 Time and temperature data

for each sample used in back extrusion ......... 145

viii

.Appendix D Computer programs (written by K.

Dolan unless otherwise indicated) to estimate k,

a, and DEg, from normalized torque versus
temperature-time history in Appendices B and C ....... 157
.Appendix E. Calculated viscous activation

energy (AEV) of gelatinized native corn starch to
determine effects of temperature (corresponding

to Table 3.3) ................................................................. 173
.Appendix F Observed equilibrium torque of

gelatinized native corn starch to determine effects

of shear rate (corresponding to Table 3.4) ................. 174
.Appendix G Calculated strain history parameters

of gelatinized native corn starch (corresponding

to Table 3.5) ................................................................. 175
.Appendix H Example of mathematical procedure

to correct and normalize raw torque during a

pasting curve to a reference temperature (60 C)

and reference temperature-time history (zero) at

10 5 intervals. ............................................................... 176

ix

Table
Table

Table

Table

Table

Table

Table

Table
Table

Table
Table

Table

LIST OF TABLES

Physical meaning of terms in Eq. 2.2. ................. 9

Summary of studies reporting viscosities of
gelatinized starch dispersions. ............................. 33

Experimental design for determining effeCts
of moisture content and temperature-time
history .................................................................... 35

Experimental design for determining effects
of temperature ........................................................ 35

Experimental design for determining effects
of shear rate ........................................................... 37

Experimental design for determining effects

of strain history ..................................................... 37
Experimental design to produce pasting

curves ..................................................................... 39
Procedure to estimate the parameters ................... 47
Analytical design and step-wise correction

for Table 3.2. ......................................................... 47
Calculation steps .................................................... 47

Comparison of parameter estimates to those
in literature for native corn starch ....................... 54

Procedure for rapid analysis of parameters
using a tube viscometer. ....................................... 87

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.1a

.1b

.1c

LIST OF FIGURES

Mechanism of starch gelatinization (from Remsen
and Clark, 1978)... ............................................... 13

Typical force-penetration curve obtained from
back extrusion testing at a constant plunger
velocity.. ................................................................ 22

Schematic of test tube in plastic holder
during back extrusion.... ...................................... 24

Use of temperature-time history to collapse a
series of curves, each at constant
temperature, to one curve... ................................. 26

Average normalized viscosity versus time for
13.7% (d.b.) starch solutions at 20 C and
constant shear rate after heating at 81, 84,

85, 91. 95101. and 105 C. .................................. 29

Figure 2.5 with the abscissa replaced by
temperature-time history (Regression does
not include values at 101 and 105 C.) ................. 30

Residual (measured value-predicted value)
versus temperature-time history for
regression line in Figure 2.6. ............................... 31

Experimental apparatus of mixer viscometer
and bath system. .................................................... 60

Brookfield small sample adapter with flag

impeller and copper-constantan

thermocouple. ......................................................... 61
Flag impeller. ......................................................... 62
Equilibrium torque at 60 C versus time of

heating at 85, 88, 92, and 95 C for 6.4% (d.
b.) starch dispersions. .......................................... 74

xi

Figure .3
Figure .4
Figure .5
Figure .6
Figure .7
Figure .8
Figure .9
Figure .10
Figure .11
Figure .12
Figure .13
Figure .14
Figure .15
Figure Aul

Equilibrium torque at 60 C versus time of
heating at 85, 92, and 95 C for 7.3% (d. b.)
starch dispersions. ......... ' ....................................... 7 5

Superposition of Figures 3.2 and 3.3, with
torque normalized and time transformed to
temperature-time history. Regression line
corresponds to Eq.3.3 (42 of 100 points). .......... 76

Experimental and predicted torque and

temperature versus time for 6.4% (d.b.)
cornstarch dispersion, using constant

parameters for prediction .................... 79

Pasting curve of Figure 3.5, using varying
strain history and temperature parameters for

prediction. .............................................................. 80
Replicate of Figure 3.6, except final

temperature is 60 C. .............................................. 81
Pasting curve with all variables as in Figure

3.6, except heating time equal to 8 min. ............. 82
Replicate of Figure 3.8 ......................................... 83

Pasting curve with all variables as in Figure
3.8, except cooling rate was increased. ............... 84

Pasting curve with all variables as in Figure
3.8, except starch concentration equals 6.4%
(d.b.). ..................................................................... 85

Pasting curve with all variables as in Figure
3.6, except rpm equals 50. ................................... 86

Pasting curve with all variables as in Figure
3.6, except maximum temperature equals 85
C. ............................................................................ 87

Pasting curve of Figure 3.6, with four

predicted curves: Curve 1, prediction

without temperature and strain history terms;
Curve 2, prediction without the temperature

term; Curve 3, prediction without the Strain
history term; Curve 4, prediCtion with all

terms ....................................................................... 89

Pasting curve for 6% (d.b.) bean starch
Phaseolus vulgaris var. seafarer. ......................... 90

Sensitivity coefficients for Eq. 3.3. .................... 120

xii

Figure A.2 Ratio of X2/X3 versus k‘I’ to determine extent
of linear dependence. ............................................ 122

xiii

A3

Nomenclature

relative amount of viscosity increase due to
gelatinization (dimensmnless)

material constant related to the effective molecular
weight of the protein (dimensionless)

power law coefficient relating shear-rate effects on the
molecular weight of denatured protein (dimensionless)

relative amount of viscosity reduction due to
irreversible mechanical degradation (dimensionless) .

exponent measuring the lubricating effects of moisture
on dough viscosity (dimensionless)

dry weight of starch per unit weight of dispersion
(decimal)

protein concentration, dry weight basis
arbitrary constant (units depend on text definition)
arbitrary constant (units depend on text definition)

parameter related to the rate of mechanical degradation
(dimensionless)

maximum force for ideal back-extrusion, Figure 3.3 (N)

arbitrary constant

consistency coefficient (Pa 5")

pseudo-consistency coefficient at reference temperature
T, and reference moisture content MC, (N m min")

reaction transmission coefficient, affecting rate of
gelatinization ([K s]'1)

length of pipe over which pressure drop occurs (m)
distance traveled by plunger (m)
torque (N m)

torque corrected to reference values of moisture content
and maximum temperature (N m)

ungelatinized torque (at ‘I’=0) (N m)

xiv

Moo

M'...
MC
MC,

0to

[3.

fully gelatinized torque (as ‘P—900) (N m)
torque before shearing (at <D=0) (N m)

torque after shearing (as <b—>°°) (N m)
moisture content (dry basis, decimal)
reference moisture content (dry basis, decimal)
impeller speed (revolutions per minute)

power law flow index (dimensionless)

weight of swollen starch per unit weight of dry starch
(decimal)

universal gas constant (8.314 J/mol K or 1.986 cal/mol
K), or radius of pipe (m) (Section 5.2)

absolute temperature (K)

threshold temperature for gelatinization (K)
initial temperature (K)

arbitrary reference temperature (K)

maximum temperature reached by a sample (K)
time (s)

total time of experiment (5)

normalized torque (dimensionless)

Greek

exponent describing molecular weight effect on
viscosity (dimensionless)

intercept coefficient relating effect of denatured protein
molecular structure on viscosity at low shear rates (5“)

material constant related to the maximum viscosity
increase due to protein gelation (dimensionless)

viscometric shear (strain) rate (5'1)

XV

Y average viscometric shear rate (5'1)

AEg "activation energy" of gelatinization (J/mol)

AEV viscous activation energy of temperature effects (J/mol)

AP magnitude of pressure drop (Pa)

At elapsed time (s)

e exponent relating effects of moiSture level to the extent
of protein denaturation (dimensionless)

n apparent viscosity index (Pa 5)

no apparent viscosity at zero time (Pa 5)

nus ungelatinized apparent viscosity, equal to no(Pa s)

110., apparent viscosity as ‘P—)°°, or cb—mo (Pa 3)

niTmm'MCapparent viscosity corrected to reference values of
shear rate, temperature, maximum temperature, and
moisture content.(Pa s)

1.1, Heinz-Casson high-shear limiting viscosity at reference
temperature T, and reference moisture content MC, (Pa
5)

o, Heinz-Casson yield stress at reference temperature T,
and reference moisture content MC, (Pa)

‘1’ integral temperature-time history (K s)

‘P' k‘I’ (dimensionless)

‘1", reference value of ‘P' (dimensionless)

‘1’" defined by Eq. A.5 (s)

(D integral strain. history (dimensionless)

i thermal diffusivity (m2/s)

xvi

1. Introduction

Viscosity is an objective and convenient measure of the
extent of gelatinization in starch dispersions. Numerous
excellent studies have explained the effects of shear rate and
starch concentration, but none of those reviewed suggested a
comprehensive procedure for predicting viscosity. Furthermore,
few studies have differentiated among the faCtors affeCting
gelatinization most. The purpose of this dissertation was to
identify and quantify variables influencing starch viscosity
development most, and determine relationships among the
variables. The goal of this work was to test a comprehensive
model in benchtop experiments, then in piIOt plant conditions,
and finally in full-scale processing. This dissertation presents
results for benchtop experiments, and proposed a procedure for
pilot-plant and industrial trials.

This work is composed, primarily, of two journal articles.
The comprehensive model considered in b0th was based on the
work of Morgan et al. (1989), where the rheological behavior of
protein dough was modeled during extrusion cooking. The
proposition was that starch viscosity could be described as a
function of shear rate, temperature, moisture content,
temperature-time history, and strain history. The variables had
to be measurable during or before the process to use them for
prediction. The first article investigated only the temperature-
time history effect, because that term had not previously been

applied to starch. The work involved a back extrusion

technique to measure a "viscosity index" for 13.7% corn starch
dispersions at 20 C. .

The second article investigated all the terms and their
interactions, but emphasis was placed on temperature-time
history and strain history effects. Published literature indicated
that strain history effects had not been previously quantified for
starch solutions. A mixer viscometer was used to measure
torque for 5.5-7.3% (d.b.) corn and 6% (d.b.) bean Starch
dispersions. The model is equipment-independent, so bOth
devices were chosen to demonstrate how to apply the model to
any system. The model was shown to accurately: predict a
typical industrial paSting curve.

The final section of this dissertation addresses an additional
objective: to use the results of both articles to present a

simple, generalized model and experimental design to evaluate

starch gelatinization behavior for industrial applications.

2.1 Abstract

A generalized a priori theoretical model relating apparent
viscosity of protein dough to several independent variables was used to
model gelatinized starch dispersions. Independent variables in the
original model were shear rate, temperature, moisture content,
temperature-time history and strain history. The model is applied here
to corn starch dispersions gelatinized using various temperature-time
treatments. Apparent viscosity of a 13.7% gelatinized cornstarch
solution at constant shear rate and constant strain history was
measured at 20 C using la back extrusion technique. Activation
energy of gelatinization was estimated as 210. kJ/mol (50. kcal/mol)

over the range 81-95 C, and decreased in the range 95-105 C.

2.2 Introduction

The largest single food group in the human diet is cereal grains.
Starch, the primary constituent of cereal grains. is used in different
ways by the food industry (Lund, 1984). Foods containing starch
are processed over a range of temperatures and concentrations.
Among these products, in order of increasing concentration, are
soups and gravies, puddings, custards and doughs. Starch is used as
a thickening agent and as a processing aid, such as corn starch used
to dust work surfaces or in-process material to prevent sticking
(Whistler et al., 1984).

"Gelatinization" is typically. defined as the physicochemical
phenomenon of swelling of starch granules as they imbibe water at

temperatures sufficient to destroy the birefringence of the granules.

a

The process occurs as the starch/liquid system is heated above a
characteristic "gelatinization temperature." Below this temperature,
birefringence of the starch granules is preserved. For a population
sample of granules, gelatinization temperature usually varies over a
10°C temperature range, indicating distribution of different
gelatinization temperatures. Viscosity of dilute starch suspensions in
the early heating stages will increase mainly because amylase is
released while, in later stages, viscosity increases further due to
interaction of extragranular material and swelling of the granules
(Lund, 1984).

Apparent viscosity can be used to quantify the thickening
effect of starch. One application is the prediction of minimum
pressure or minimum wall shear stress for flow of a processed
fluid. This information can aid in preventing plugging of pipes, a
costly problem in industrial processing. Knowledge of the
viscosity of starch-thickened foods is needed to design process
systems with optimum operating performance as well as superior texture
and product quality. Other applications include mixing systems,

aseptic processing, and steam infusion.

2.2.1 Rheological Medels for Starch Solutions

The Visco-Amylo-Graph (C.W. Brabender Instruments. Inc., 50 E.
Wesley St., S. Hackensack, NJ 07606) is an empirical instrument
used industrially to simulate effects of processing conditions on
the rheological behavior of starch solutions. Thus. the ability of
the instrument to predict flow properties depends upon the knowledge

base of rheological behavior of solutions. This knowledge is usually

S
unavailable when new products are being developed which involve changes
in formulation or process conditions.

Several authors have presented models of the apparent viscosity of
starch solutions. Christianson and Bagley (1983) and Bagley and
Christianson (1982) found for dilute (less than 26%, g starch/g soln)
corn starch and wheat starch dispersions, apparent viscosity/(C*Q)
exponentially increased with C*Q, where C*Q equals the grams of
swollen starch per gram of dispersion. Q, the grams swollen starch per
grams initial dry starch, increased non-linearly with increasing
temperature, showing the dependence of viscosity on temperature.

Bagley, Christianson and Beckwith (1983) proposed an
exponential dependence of intrinsic viscosity on the volume
fraction of swollen corn and wheat starch granules for volume
fraction between 0.6 and 1.0. Evans and Haisman (1979) suggested
the viscosity was a function of volume fraction for gelatinized
corn, potato, and tapioca starch solutions up to 10% starch.

A parameter related to apparent viscosity is yield stress.
Bagley and Christianson (1983) found a yield stress existed for 11-
13% gelatinized wheat starch dispersions measured at 23°C, and found no
yield stress for 10-14% dispersions measured at 600C. Christianson and
Bagley (1984) reported yield stresses existed in 11 and 12% cornstarch
solutions, and did not exist in 8 and 10% dispersions. They found that
yield stress depended on temperature-time (T—t) history. Wong and
Lelievre (1982) described yield stress of 1.6-8.2% wheat starch
solutions as a function of starch concentration, swelling
capacity, and the number fraction of large granules in the starch.

The dependence of viscosity and yield stress of starch

6
solutions on T-t history can be inferred from the data of Bagley and
Christianson (1982) and Christianson and Bagley (1983, 1984).
However, in none of the above-mentioned models was the T-t history
explicitly included. Other models for food doughs, discussed in the
next section, clearly separate the opposing effects of
temperature and temperature-time history.

All of the previously mentioned studies modeled viscosity of starch
based on variables measured at the end of the test. such as volume
fraction and swelling capacity. The researchers did not correlate
these variables with in-process conditions, such as temperature and
time. The first-order modeling of Suzuki et a1. (1976), Bakshi and
Singh (1980) and Kubota et al. (1979) is different because it
correlates end measurements to temperature and time during the test.
Suzuki et al. (1976) reported AEg for cooked rice as 80 and 37 kJ/mol
for temperature ranges of 75-110 and 110-1500C respectively. Kubota et
al. (1979) found ABS equal to 59 kJ/mol between 70 and 85°C for rice
starch. Bakshi and Singh (1980) gave AEg values of 78 and 44 kJ/mol
for rough rice in the ranges 50-85 C and 85-1200C respectively, and A88
equal to 100 and 40 kJ/mol for brown rice in the ranges 50-85 and 85-

120°C respectively.

2.2.2 Other Rheological Models Applicable to Starch

The molecular mechanisms acting in starch systems with excess water
and in those with limited water are different. In the first case
dispersed starch undergoes gelatinization, swells. and forms a thicker
dispersion. In the latter case, starch undergoes melting and granules

seldom swell; the latter material resembles more a glass. whereas the

7
first system is a dispersion of deformable particles. Therefore the
purpose of presenting dough viscosity models is not to suggest the
phenomena are similar; rather, it is to propose that in both cases,
temperature-time history and temperature may be treated as two separate
independent process variables with opposite effects on viscosity.

Cuevas and Puche (1986) applied dimensional analysis to describe
the apparent viscosity index (a relative indicator) and consistency of
corn dough. They varied the speed and measuring temperature of a
Brookfield viscometer, and the concentration of the corn dough (for 35
and 40% corn flour).

Harper et al. (1971) and Cervone and Harper (1978) predicted

viscosity of cereal doughs and pregelatinized corn flour as a

power law function of shear rate, an exponential function of l/T and
an exponential function of moisture content. Bloksma (1980) found
that unless heating was "extremely slow," (less than .01 K/s) the

viscosity of wheat flour doughs was a function of the actual

temperature and thermal history.

In comparison to the effect of other variahles, protein
denaturation and starch gelatinization drastically increase
solution viscosity. Both phenomena occur at temperatures above a

certain level, and continue toward completion as long as that

threshold is exceeded. Thus, there are two effects above the
threshold: the ”thinning effect" of higher temperatures which
decreases viscosity, and the integral T-t kinetic history effect of

. gelatinization (or denaturation) which increases the viscosity.
Some researchers have separated these two opposite effects in

their models. The development of Roller (1975) was used by Remsen and

8
Clark (1978), who tested 22—35% MC (wet basis) defatted soy flour
doughs. Janssen (1984) expressed apparent viscosity of an
extruded food containing starch or protein as a function of shear
rate, temperature and T-t history. Harper et a1. (1978) presented
a model for apparent viscosity as a function of T-t history and

moisture content of bovine plasma protein suspensions.

2.3 Modeling Approach
2.3.1 General Mathematical Model

Morgan et al. (1989) proposed a mathematical model describing
apparent viscosity of denaturing protein doughs as a function of
shear rate, temperature, moisture content, T-t history, and strain

history

n 1/n (AEv/RT)(l/T-l/Tr) + b<Mc-Mcr>l

n(i.T.MC.W.¢) - [('0/1)n+(”r) 1 e[

{1+A“(1-e'kw)°} {1 — 3(1-e'd¢): (2.1)

Eq. 2.1 was developed for this study by translating the approach of
Morgan et a1. (1989), from a protein denaturation-based phenomenon
where water content is limiting (doughs) to a starch gelatinization
model where excess water is available. Table 2.1 describes the
analogy/physical meaning of each term in Eq. 2.1. Further explanation
is reported in Morgan et al. (1989).

A number of assumptions were made in developing Eq. 2.1:

-No elastic effects,

-No compositional effects from materials other than starch and

Table 2.1. Physical meaning of terms in Eq. 2.1.
1m mm

nll/n

[(ro/1)n+(pr) shear rate effect

e[(AEv/RT)(1/T'1/Tr)i

temperature effect

e[b(Mc-MCr)] moisture content effect

(1+Aa(l-e-kw)a) gelatinization (T-t history) effect

(1-B(l-e'd°)l strain history effect

10

moisture content,

-No dependence upon maximum shear rate

-No explicit volume fraction dependence,

-The effects of gelatinization on viscosity may be approximated by

first-order reaction kinetics,

-Homogeneous, isotropic medium.
The similarities and differences in the model with respect to starch
versus protein are discussed in the following sections. The use of a
dough viscosity model for dilute systems has been justified by a review
of literature for excess-water systems, not for water-starved systems.
1,- .. ; .- . . ~ ; -~ — . -; ‘11 a; t e :as_bi it o :-9 1-

1- --°-‘i'--r--°'v°‘:i‘~ :. '39 toth-w. t -‘ '1

2.3.1.1 Shear rate effect.

Christianson and Bagley (1983, 1984), Bagley and Christianson (1982),
Wong and Lelievre (1982), Doublier (1981), and Evans and Haisman (1979)
in separate studies investigated the effect of shear rate on the
apparent viscosity of dilute wheat starch, corn starch. and tapioca
starch solutions. All results showed shear-thinning behavior for
the gelatinized solutions, but no attempt was made to quantify and/or
correlate thermal history effects.

Christianson and Bagley (1984), Bagley and Christianson (1982), Lang
and Rha (1981), and Evans and Haisman (1979) measured yield stresses
of gelatinized corn and wheat starch dispersions in separate
studies. Their findings indicate a model for starch dispersion

viscosity must account for non-Newtonian behavior including the

11
presence of a yield stress. The Herschel-Bulkley model describes
both phenomena. However, at high shear rates this model approaches
infinite or zero viscosity depending on the power law index This
behavior creates a problem if gelatinized starch solutions have
a finite limiting viscosity , such as those experienced in high-shear
processes. A simple model accounting for non-Newtonian behavior,
yield stress, and finite limiting viscosity is the Heinz-Casson
model, (Table 2.1, first term) used by Christianson and Bagley (1984),

and selected for this study.

2.3.1.2 Temperature effect.

Within the starch literature reviewed, the work of Doublier
(1981) with wheat starch pastes was the only one in which temperature
effects were measured separately from thermal history effects, i.e.
gelatinization was complete before measuring the change in viscosity
with temperature. His plot of data shows adequate agreement (no
measure of variance was given) with the viscosity model suggested by
the Eyring kinetic theory (Eyring and Stern, 1939). The Fyring model is

the second term in Table 2.1.

2.3.1.3 Moisture content effect.

Doublier (1981) used a power-law model to describe the dependence of
apparent viscosity of 0.1 to 2.5% wheat starch solutions on moisture
content. Harper et a1. (1971) used an exponential model. Bagley and
Christianson (1982,. 1983) and Christianson and Bagley (1983)
presented plots showing the effects of moisture content on the

viscosity of wheat starch and corn starch dispersions. ' The

12
data suggest an exponential decrease of viscosity with increasing
moisture content. Morgan et a1. (1989) used the same assumption.

The last three studies, excluding Morgan et al. (1989),
investigated dilute solutions, where water did not limit
gelatinization. Thus, the moisture term in Eq. 2.1 (Table 2.1, third
term) describes the lubricating effect of water between the starch
granules. If water were limiting (which may be the case with dough),
moisture content would have a reverse effect upon viscosity, because

gelatinization would depend on moisture content.

2.3.1.4 Temperature-time history effect.

Thermal (T-t) history is distinguished from temperature in
that the former depends upon the path. Thus, if the starch
solution viscosity was dependent on only temperature, its value would
be the same at 80 C, whether a sample had been brought to 80 C
in 30 seconds or in 5 minutes. In fact, the viscosity differs
according to the temperature exposure over time. Furthermore, T-t
history and temperature have opposing effects on fluid viscosity.
Greater T-t histories increase viscosity to a limit. whereas greater
temperatures decrease viscosity.

Remsen and Clark (1978) presented a simplified model of the
gelatinization process (Figure 2 1). The fact that gelatinization
begins with separate granules and ends with a network of linked
particles suggests a "pseudo" polymerization. After exceeding a
threshold temperature, the viscosity of starch solutions increases to a
maximum over time as gelatinization occurs. Conversely. as shown by

the second term in Table 2.1, an increase in temperature causes a

 

13

Raw starch .ranules made up of
amylose (he IX) and amylopectm
(branched).

Addition of water breaks up amylose
crystallinity ranules and disrupts
helices. Granu es swell.

Addition of heat and more water
causes more sw lling. Amylose begins
to diffuse outo granule.

Granules“ now containin mostly
amylopecttn, have collapse and are
held in a matrix of amylase forming a
ge .

Figure 2.1 Mechanism of starch gelatinization (from Remsen

and Clark, 1978).

14

decrease in viscosity, as expected for most fluids.

Remsen and Clark (1978), Harper et al. (1978), Janssen (1984), and
Morgan et al. (1989) have modeled the T-t history effect on the
viscosity of cereal dough, bovine plasma, starch and protein foods,
and defatted soy flour, respectively. Harper et al. (1978) and Morgan
et al. (1988) assumed protein denaturation could be approximated by
a "pseudo" first-order reaction, and made the process analogous to
polymerization. Janssen (1984) assumed both protein denaturation and
starch gelatinization could be approximated by first-order
kinetics, and Suzuki et a1. (1976) proposed a first-order model for
the gelatinization of rice starch.

There is a major difference between protein denaturation and starch
gelatinization. Denaturation is a chemical reaction, where a three-
dimensianal structure is lost as hydrogen bonds are broken.
Gelatinization is both a physical and chemical process. The physical
process. is hydration and swelling of granules, with leaching of
amylase and amylopectin molecules into the solution. The physica-
chemical process is water breaking intermolecular hydrogen bonds and
replacing them with water-palysaccharide hydrogen bonds. However, the
overall effect of both processes is similar. Full denaturation results
in unravelled aggregated protein , and gelatinization results in a
loose matrix of granules and long chain molecules. In both cases the
net effect is an increase in viscosity.

Therefore, the gross effect of starch gelatinization on
viscosity was made analogous to that of a first-order
condensation polymerization. The same assumption was made by Morgan et

al. (1989) for protein denaturation (Table 2.1, fourth term).

15

2.3.1.5 Strain history effect.

Starch granules were assumed to undergo irreversible damage due to

mechanical degradation, solubilization of granules, and
depolymerizatian of starch. Any reversible degradation caused by
breakdown of starch flocculates was not considered. The shear (also

called strain ) rate was used as a measure of the degradation and
subsequent decrease in viscosity (thixotrapy). The strain rate-time
effect (strain history term in Table 2.1)is prominent in high shear
processes.

Shear rate and strain history are two separate effects, similar
to temperature and T-t history. A given strain rate produces an
instantaneous stress response, as shown by a rheogram (stress versus
shear rate). However, while undergoing shear, the Viscosity of a
fluid may decrease asymptotically to a limit (thixotrapy). showing the
effect of strain history. As an illustration, Wong and Lelievre
(1982) made measurements as quickly as possible because the viscosity
of their starch solutions at higher shear rates drifted down with
time. There are few models of starch solution Viscosity as a
function of strain history. Diosady et al. (1985) proposed a
model describing intrinsic viscosity of raw starch solutions as a
function of the fraction of starch fully cooked and the product of
stress and time.

Since both protein and starch solutions show an asymptotic decrease
in apparent viscosity as strain is applied, the strain history term
(Table 2.1) proposed by Morgan et a1. (1989) is also appropriate for

starch.

16

2.3.2 Advantages of the Model
2.3.2.1 Simplicity of form.

Albert Einstein once said, "Everything should be made as simple as
possible, and no simpler." For example, a linear function to describe
the temperature profile in unsteady-state heat transfer is

simple but unacceptably inaccurate. A higher-order polynomial may be

accurate but unwieldy. A compromise can be made by using the
simplest form still retaining acceptable accuracy; in this case,
perhaps a parabola or exponential. The model used in this work was

developed under the same concept, that the best model is simple yet

accurate .

2.3.2.2 Predictive nature.

A general type of model is yn - f(xn), where y11 are dependent
variables and xn are independent variables. However. some models are
of the form y1 - f(y2). The difference in words is that y1 in the
second model cannot be predicted; that is, yl is known only after the
test when y2 is measured. The current work distinguishes the two
models by referring to the first as "predictive" and to the second as
"dependent." An example of the difference between the two models is
the directions for cooking a cake. Typical instructions are "bake the
cake for 40 min at 3500?, or until a knife placed into the center comes
out clean." The time and temperature suggested are independent
variables predicting the cleanliness of the knife (y2)’ which in turn
predicts that the cake is done (yl).In this research. any independent

variable which could not be measured before or during a process was

l7
discounted.

An example of a dependent rheological model is that of Bagley and
Christianson (1982) and Christianson and Bagley (1983). They proposed
apparent viscosity (yl) as a function of the amount of swollen
starch per dry starch (yz) of the final product. Another example is
the intrinsic viscosity model of Diosady et al. (1985), where he uses

the fraction of starch fully cooked as an independent variable.

2.3.2.3'Generality.

Independent variables with universally-recognized, objective
definitions were used, rather than those created specifically for
this work. For example, shear rate, temperature, time, and

moisture content are used for any fluid and are strictly defined,

whereas "degree of gelatinization" or "fraction fully cooked"
(Diosady et al., 1985) are substance-specific and have various
definitions.

2.3.2.4 Ease of measurement.

"Intensive" or "specific" properties (those independent of
mass) were preferred over "extensive" properties (those dependent on
mass), because measurement techniques were easier and more accurate
for the former. The model also became more general because it
was less substance-specific.

Shear rate, temperature, and time are intensive properties.
Shear rate is calculated using the velocity profile of the substance.
Temperature is measured by a thermocouple, and time by a clock.

Although moisture content is an extensive property, it can be

18
measured when formulating the sample. These measurement
techniques of intensive variables are contrasted to those for
extensive properties. The techniques for measuring volume
fraction and swollen weight of starch seem to be more difficult

(Bagley, Christianson and Beckwith, 1983).

2.3.3 Simplified Model for Starch Gelatinization
For this study all variables in Eq. 2.1 were constant except the T-t

history. Therefore, the simplified form of Eq. 2.1 is

n '"ug[1 + Aa(l-e-kw)a] (2.2.0)

resulting in apparent viscosity as an exponential function of W,where

“f
e -f[ T(t)exp(-AEg/RT(t) ]dt if r zrg
o
and W-O if T<Tg (2.2.1)

and

"us _ [(fa/71)n+(”r)n]1/n e[(AEv/RT1)(1/T1-l/Tr) + b(MC1-MCr)]

*(1-B(1-e‘d°1)i (2.2.2)

from Eq. 2.1, where 11,T1, MCl’ and @1 are arbitrary. It is proposed
that Eq. 2.2 be used to predict the relative increase in viscosity of
a dilute starch solution at any point in the gelatinization

process. There are three distinct features of Eq. 2.2:

l9
1) The gelatinization effect is additive.

2) The value of the constant A is the relative increase in
viscosity caused by gelatinization. For example, if A?- l, the
final viscosity (as W goes to infinity) increases 100%, relative

to the ungelatinized viscosity.

3) The gelatinization effect has an exponential dependence upon T-t
history, which in turn has an exponential dependence upon
inverse absolute temperature. This "double"-exponential
dependence, similar to first-order Arrhenius kinetics, describes

the drastic increase of viscosity during gelatinization.

The value of ungelatinized viscosity and A give a range of the
physically possible viscosity values. The integral T-t function and
its coefficient (k) characterize the extent of reaction. In this
research, the viscosity of the cornstarch solution increased 80 times

(A? equal to 80 in Eq. 2.2) after complete gelatinization.

2.4 Materials and Methods
2.4.1 Choice of Variables

A parameter in a multi-variable model can be found by
experimentally holding all variables constant except the one
independent variable associated with the parameter. The dependent
variable then varies and the parameter is fit statistically.

Eq.2.2 shows that k, a, and AEg are the parameters and apparent

20
viscosity is the dependent variable. However, the associated
independent variable is not obvious because A28 is within an integral
(Eq. 2.2.1). Since an integral must be used for processes where
temperature is a function of time, AEg cannot be solved for
explicitly, but is approximated by an iterative procedure. The

parameter k is lumped with AEg'

2.4.2 Experiments

Corn starch solutions varying from 1322 to 13.7% starch were
hand-mixed, to minimize strain history, using distilled water at 23
C. Moisture content was determined from sets of three 12 g samples

oven-dried 24 hr at 105 C.

The gelatinization threshold temperature was approximated by
heating a sample at 3 C per min. between parallel plates in a
Rheometrics Fluid Spectrometer (Model 8400). The gap was 0.8 mm and
the steady shear rate was 100 1/s. A 30-gauge thermocouple (Omega
Engineering, One Omega Drive, P.O. Box 4047, Stamford. CT 06907) lead

was attached to the top plate with tape (Scotch) to press the tip

against the plate. Silicone grease was applied to the tip to assure
contact to the top plate. Another thermocouple monitored bath
temperature at the bottom plate. The gelatinization temperature

was approximated as a weighted average of the top plate
temperature and bath temperature at the time when the viscosity began
to increase continuously, as measured by the spectrometer. Based on
an estimate of thermal resistance, the bath temperature was weighted
twice as much as the top plate temperature.

The starch solution was subjected to different T-t histories.

21
Based on prior observations of the gelatinization and gelling
phenomena, 125 x 13.5-mm (inner diameter) screw-top glass test tubes,
one-third full of the solution were heated in an ethylene
glycol/water bath (Aminco top-loading heating/cooling bath, Silver
Springs, MD) at constant temperature. Seven temperatures were
used: 81, 84, 85, 91, 95, 101, and 105 C . For each temperature,
sets of two to three tubes were heated for various times ranging
from 20 s to 20 min, depending upon the glycol bath temperature. The
tubes were shaken in a container at at 250 rpm, a speed determined
visually to prevent settling of starch granules. The temperature
at the radial center of the solution and 12 mm below the solution
surface was measured every 10 s by a needle-nose T-type thermocouple
(Omega Engineering, one Omega Drive, P.O. Box 4047. Stamford, CT
06907) fixed in each of one to four representative tubes for each
bath temperature. The average measured temperature at each time was
used for T-t history calculations. At the prescribed cook time,
the tubes were removed from the bath, and immediately placed in an ice

bath at O C to reduce the temperature below Tg within 30 seconds.

2.4.2.1 Apparent viscosity.

The solutions were allowed to gel at room temperature. from 18 to 22
C. The apparent viscosity of each tube was characterized using a back
extrusion device (Figure 2.2), a technique described by Harper et al.
(1978). A stainless steel plunger was forced through the gel at a
constant rate of 50 mm/min over a distance of 25 mm. measured from
the original surface of the gel. The force required to move the

plunger was recorded graphically by an Instron (Model 4202)

22

 

 

 

 

 

1% L 4|

Time (or Distance)

Figure 2.2 Typical force-penetration curve obtained from
back extrusion tcSting at a constant plunger
velocity

23
Universal Testing Machine (Instron Corp., Canton, MA).

The area under each curve was used to convert it to a right
triangle of equivalent area with the same travel distance Lp (Figure
2.3), thereby yielding a height equal to Fp, the maximum force.
This procedure is a modification of that of Harper et al. (1978). The
ratio of Fp to Lp times a constant was used as a relative value of

viscosity (Hickson et al., 1982).

The dimensionless form of Eq. 2.2 is

 

0 -kW a
n/nus -1 - A (l-e ) (2 3)
As W goes to infinity,
o
"m/"ug -1 - A (2.4)
Dividing Eq. 2.3 by Eq. 2.4 yields
(fig/nus) -1 ng-nug
Eq. 2.5 was used for plotting the results. For each bath

temperature, a curve of normalized viscosity (n-no)/(nm-no) versus time
was fit. The multiplicative form of Eq. 2.1 allows the effects of a
varying temperature of measurement and moisture content to be

cancelled by using normalized viscosity.

24

 

 

125mm

 

Figure 2.3

 

    

 

Instron smel plunger
8.5 mm
1 4- 11 mm
test tube (inside diameter 13.5 mm)
:3: m1. gel
4 5333233533} ‘plastic container

 

///.

mem -——>l

 

 

Schematic of test tube in plastic holder
during back extrusmn

25
2.4.2.2 Temperature-time history.
According to Eq. 2.5, all y versus time curves at different
temperatures can be converted to one master curve, y versus W. (Figure
2.4), once the proper AF.g is determined. Because W was an integral,

the following iterative procedure was developed:
1) An estimate of A88 was made, based on values reported in literature.

2) By using the representative measured temperature-time history (T-t)
and the estimate of AEg, W was calculated for each group of replicates
taken out of the hot bath at a specific time and temperature.

First the ”mass-average W" value was calculated. The gelatinized
starch inside the tube was treated as an infinite solid cylinder at
uniform initial temperature To, subjected to two step changes in
environment temperature: first, an increase to the hot bath
temperature, and second, a decrease to the ice-bath temperature at 0°C.
Heisler's chart (Holman 1976, Figure 4-13) was used to calculate
temperature at 10 different radii as a function of the measured center
temperature. The combined thermal resistance of the glass tube wall
and the heat transfer coefficient between the tube wall and bath fluid
was estimated as at least 20 times less than the internal gelatinized
starch
resistance. Thus, at each time the mass average W value

1

mass av. W- 2f T(r)e
0

(-AEg/(RT(I))(r/R)d(r/R) (2.6)

o>..=o ago o. .333an2
Efmcoo 3 some .3336 he mar—om
a omgszoa o. for»... 92:73:33.3.3 Lo an: v.~ 3:»;

we:

 

26

 

. L
l .1...“
am

.
~
as «Staten; \V xx

 

Ban:

0

q

my MB «or
xv

 

 

IIIIIIIIIIIIII I.. 8: IIM‘\ hWillis-3559... 8: .

27

was calculated by Simpson's rule (Hornbeck, 1975). Eq. 2.6 is the

integration over volume. These values were integrated over time using
the trapezoidal rule (Hornbeck, l975),to obtain the integral
temperature-time history W (Eq. 2.2.1) for each replicate group. y

versus. V were then plotted as points for each replicate group for all

bath temperatures, using one value of AEg, estimated in 1.

3) Using a computer routine, AEg was varied and the procedure outlined
in step 2 was repeated for each change. The coefficient k was lumped
with AES because k was not varied independently. but forced to change
with the fit of the equation every time AEg was changed. The agreement
between y and' W was measured by a Marquardt nonlinear regression
(Draper and Smith, 1981) of all points using the statistical computer
routine Plot-it (Eisensmith, 1987). The AEg yielding the greatest
coefficient of determination (R2) was used, as was the corresponding k
for the fit to Eq. 2.5. A linear transformation on Eq. 2.5 was
rejected because the resulting variance of residuals increased
dramatically with W (discussed in Neter et al., 1985. p.467-469)

The effect of different heating bath temperatures was shown by
regression on successively smaller sets of data: 1) all points, 2) all
points except those at 105 C, and 3) all points except those at 101 and

105 C.

28
2.5 Results

Figure 2.5 shows the seven curves of average normalized viscosity
versus time, and Figure 2.6 shows the same curves transformed to one
curve (normalized viscosity versus integral T-t history). with the
regression line and 95% confidence band. The coefficient of
determination was 0.843 for the regression of all values except those
for 101 and 105 C (Figure 2.6).The activation energy yielding the
greatest coefficient of determination was AEg equal to 210.kJ/mol (50.0
kcal/mol). The coefficient of W in Eq. 2.2.0. k. was equal to
0.846x 1026 (K s) 1. The estimate of the exponent a was 0.494. Each
point in Figure 2.6 represents an average of two or three viscosity
values at the same e value. The regressions on all points and on
all points except those at 105 C gave lack-of-fit significances less
than 0.0001. The regression on all points gave AEg equal to 170 kJ/mol
(40. kcal/mol) and a coefficient of determination of 0.755.

Figure 2.7 is a plat of residuals (measured value - predicted value)
versus 0*1026 for the regression line in Figure 2.6. There were two
to three replicates in each of 31 sets, giving 87 total observations.
The standard deviation and absolute mean of residuals was 0.09 (9.0%
of full-scale) and 0.076, respectively. In addition. the F value for
lack-of-fit was 0.76 at a significance of 78.2%.

The weighted average "threshold" gelatinization temperature was 65.
C. The average moisture content of the raw starch varied from 8.0
to 12%. The average viscosity indexes at 99% gelatinization were

estimated as 4300, 7300, 9600, and 7800 cp at 81. 85, 91, and 95 C,

respectively.

29

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32
2.6 Discussion
The regression on all point shows that Eq. 2.5 is inadequate to
include the full range from 81 C to 105 C. A better fit for all
temperatures may be found by expressing AEg as a. function of
temperature, or as a different constant within certain temperature
ranges (Suzuki et al. 1976, Kubota et al. 1979. Bakshi and Singh,
1980). This method is a concession for using first-order kinetics to
describe what is probably a mixed-order reaction (Lund. 1984). The
three separate regressions revealed AEg decreased at higher
temperatures, in agreement with Suzuki et al. (1976) and Bakshi and
Singh (1980). Most likely a different level of gelatinization, having
a lower activation energy, is triggered at higher temperatures.
The 0.843 coefficient of determination and 78.2% significance of
F lack-of-fit (Figure 2.6) indicate the usefulness of the model form
y- (l-exp(-kW))° for the temperature range 81-95 C. However, the 0.98
rather than 1.0 coefficient in the regression line (Figure 2.6)
indicates the difficulty of deciding what value of viscosity is
the limiting value.
Data for 101 and 105 C (Figure 2.6) were not included in the

regression, but were plotted to show how far they lay from the rest of

the data. This excessive deviation may have been caused by the
increased error at high temperatures, specifically in the
calculation of T-t history. Errors in temperature measurement are

magnified in W as temperatures increase, because of the exponential
dependence of W on inverse temperature (Eq. 2.2.1).
At a given T-t history, the viscosity is lower when heated at

higher temperatures (Figure 2 6). This result suggests there is a

33
limit to how quickly water can enter and hydrate the starch granules.
Although the calculated T-t history may be greater, the

diffusion of water in and the diffusion of amylose out cannot proceed

any faster. Another limiting factor is the increased amount of water
vapor produced as temperatures approach 100 C. Not only is heat lost
to vaporize liquid water, but the vaporized water cannot enter and

swell the granule as liquid water can.

In addition, the values at 101 and 105°C may have been affected by
the more complete disintegration of granules and the increased
solubility of the starch. Christianson and Bagley (1983) also mention
the increased amount of solubles at temperatures greater than 94°C.

Figure 2.7 shows the residual for every observation in the Figure
2.6 regression. The standard assumptions (Beck and Arnold, 1977) are
that errors are additive, have zero mean with constant variance, are
uncorrelated, and have a normal probability distribution. Randomness
and lack of trends (Figure 2.7) suggest that the errors are additive.
An absolute mean of .068 does not seriously violate the assumption of
zero mean, and the number of positive and negative residuals is 47 and
40, respectively. The "band" of residuals is approximately horizontal
and of constant width, showing a constant variance with W, unlike the
results for the linear form of Eq. 2.5. The number of changes in sign
(48) is more than half the total observations (87). According to Beck
and Arnold (1977, p.409) the errors are uncorrelated. In summary, the

standard assumptions for residuals are valid.

34
2.7 Problem for Calculating the Cook Time of s Steam Infusion Process.
The model developed in this study can be used to solve process
engineering problems. One example is provided in this section. Assume
that a steam-infusion process requires a final apparent viscosity
(0) is 700 cp, and the material has an ungelatinized viscosity (nu ) of

8
20 cp. The following fundamental information is provided from prior

rheological measurements: k - 2.3 x 1019 (K min) 1, AEg - 150. kJ/mol,
a-l.0, and A - 49. Calculate the required cook time considering a
sequence of steps:
1. Make sure desired n (700cp) is less than maximum nco - nug(l+A) -
20(1+a9)-1000 cp
2. Calculate the normalized viscosity (Eq. 2.5): y— (700-
20)/(1000-20)-0.69
3. Rearranging Eq. 2.5,
w(.69)- -ln(l-y)/k - -ln(l-.69)/(2.3 x 1019)- 5.1 x 10'20
4. Calculate cook time for a constant cooking temperature 85 C -
358 K. Rearranging Eq. 2.2.1

At_ t a(ass/RT)

/T-

-20 a
(5.1 x 10 )K min exp[(lS x 10 J/mol)/(8.3la J/mol K *358 K)]
/358 K -l.l min- 66 sec- cook time

This type of analysis would be appropriate for steam infusion problems

involving starch-thickened fluids, such as many baby food products.

2.8 Summary and Conclusions
A general model was used to simulate starch viscosity development
during gelatinization. Only the thermal history effect of

gelatinization was experimentally verified. However, the authors

3S

propose a comprehensive model, including independent variables of shear
rate, temperature, moisture content, T-t history. and strain
history, for describing viscosity of dilute starch solutions. The
shear rate dependence is described by the Heinz-Casson model, including
a yield stress and finite limiting viscosity at high shear. The
temperature effect is modeled using the Arrhenius relationship,
and the influence of moisture content is assumed to be exponential.
The gross effect of gelatinization on viscosity is modeled using
first-order kinetics, leading to a temperature-time history term. The
effect of strain history is approximated as exponential.

Evaluation of the T-t history effect indicates the relative
influence of gelatinization upon apparent viscosity can be modeled
as a function of one parameter only, temperature-time history.
Expression of "activation energy" as a functions of temperature
will improve the model.

Unlike starch viscosity models in the literature, this model does
not require measurement of the end product. The model is general
in that it can be used for any system geometry, process. or equipment,
and can predict apparent viscosity at high shear rates. The form
of the model (Eq. 2.1) allows each term to represent the complete
effect of one variable (Table 2.1). Therefore, any one term may be
easily replaced by another suggested form. For example, the
Heinz-Casson model can be directly replaced by the power-law form for
apparent viscosity.

The model can be tested and modified by estimating the
parameters for several types of starch. Future research should be

directed towards a verification of each term to assess which are most

36
influential and which may be neglected. The assumption of independence
of variables should also be investigated. The model can then aid in
full-scale simulations of the processing of any starch-thickened fluid,

if gelatinization is the overriding cause of viscosity increase.

37

3. Mixer viscometry used in modeling rheological

behavior of gelatinizing starch solutions.

3. 1. Abstract

"Viscosity" of gelatinizing starch dispersions is used in
quality control and process design around the world. A
comprehensive model to predict viscosity for an arbitrary
process has not been proposed in the published literature. The
purpose of this study was to to present a generalized scheme to
identify and measure factors influencing viscosity development
during gelatinization in food processing. A mixer viscometer
was selected to illustrate a general experimental technique,
because variables were known and highly controlled. The
results (within 10 % standard deviation) suggested that, for
larger processes, the limiting factor will be measurement
accuracy rather than fit of the model.

A Brookfield RVTD mixer viscometer was used to gelatinize
5.5, 6.4, and 7.3% (d. b.) native corn starch dispersions.
Torque response, the dependent variable, was used to eStimute
parameters in a generalized model. The independent variables
were impeller speed,temperature (SO-95 C), moisture content.
temperature-time history, and strain history. PrediCted and
experimental paSting curves were compared. A simplified and

rapid procedure for estimating parameters from an arbitrary

38

pasting curve was proposed, and applied to a bean starch
dispersion.

Dispersions thickened with decreasing temperature and had an
Arrhenius aetivation energy between 6.4 and 11.5 kJ/mol.
There was no evidence that retrogradation caused this effeCt.
Maximum viscosity depended on cook temperature, ranging from
a 150- to a 220-fold increase at 85 and 95 C, respectively.
First order reaction kinetics was accurate (9.8% standard
deviation) in describing viscosity increase during gelatinization.
Torque decayed exponentially after initial gelatinization.
Comparison of relative effects showed a pasting curve could be
predicted by knowing only the rate of gelatinization during
heating rise, the rate and extent of breakdown during shear

decay, and the torque response to temperature during cooling.

39

3.2. Introduction

Starch provides the major source of energy in the diet.
Across the world, some cereal grain, usually rice, wheat, or
corn; constitutes the major source of food, with starch
comprising about 75% of the grain (Hodge and Osman, 1976).
Whistler (1984) suggests why starch will continue to be a
dominant indu5tria1 raw material: the birth of enzyme
engineering allowing low cost conversion of starch to D-
glucose; use of starch as a feedstock for alcohol and as an
additive of tertiary oil recovery systems; and the growing world
population .

Starch is consumed as part of the grain or is isolated as
refined starch for use in foods, papers, adhesives, and textiles.
For the refined starch applications, "gelatinized" starch is used.
Atwell et al. (1988) presented definitions of "gelatinization."

"pasting," and "retrogradation,’ based on a survey of 67

attendees of the Starch Science and Technology Conferences.

They agreed upon the following definition of gelatinization:
Starch gelatinization is the collapse (disruption) of
molecular orders within the starch granule manifested in
irreversible changes in properties such as granular swelling.
native crystallite [sicl melting, loss of birefringence, and
Starch solubilization. The point of initial gelatinization and
the range over which it occurs is governed by Starch
concentration, method of observation, granule type, and
heterogeneities within the granule population under
observation.

There are many ways to measure extent of gelatinization of

starch. Qualitative methods include phatography (scanning

4O

electron microscopy) and measuring loss of birefringence, while
quantitative methods include measurement of volume fraction
and solubility. A macromolecular view can consider viscosity
or gel formation, while a micromolecular view may consider
glycosidic bonds or diffraction patterns. In spite of many
analysis methods, the gelatinization process is Still poorly
understood (Lund, 1984)

For industrial use, one prefers to characterize Starch
gelatinization with an efficient and inexpensive method
simulating process conditions. A tube viscometer identical in
size to process equipment would be inexpensive, and would
avoid the problems of scale-up. However, if all process
conditions were tested, the method would be inefficient.
”Obviously, starch products are pasted and used under a wide
variety of conditions, and no standard cycle of cooking and
cooling can be devised which will be generally representative of
their diverse applications" (Mazurs et al., 1957) A more
efficient procedure is to model all effects important in industrial
processing, because the model can be used to interpolate more
accurately.

None ofethe literature reviewed investigated a comprehensive
model, but the tools to build one are found in the collective
results. First, a quantitative, as opposed to qualitative
approach, is more accurate and allows mathematical modeling.
Secondly, a macro- as opposed to micromolecular method is
usually simpler and is already used by industry in the

Visco/amylo/Graph, which measures torque (viscosity) increase.

41

Published literature also provides enough data on starch
viscosity to suggest a comprehensive model. Furthermore, fluid
dynamics has already developed equations relating viscosity to
temperature and velocity,two design criteria for food
processing. In short, we already know how to apply viscosity;
what is needed is a way to predict it.

A comprehensive model for protein doughs was proposed by
Morgan et al. (1989). Dolan et al. (1989) applied only one term
of the model to starch dispersions. The model may be direCtly
applicable to a food process, but has not yet been thoroughly
investigated. Processing conditions are difficult to control.
Furthermore, some processes may mask the effects of less
significant variables. Therefore, the present Study required a
device having highly controlled conditions, and showing the
effects of all five variables: Shear rate, temperature, moisture
content, temperature-time history, and strain history. The
Brookfield RVTD mixer viscometer was chosen because of its
small sample size, constant Shear rate, Speed, simplicity, and
reproducibility. Since the model is equipment-independent, it
will work for any instrument, as long as the variables can be
measured. Therefore, the current Study used the Brookfield
device as an example of how to estimate model parameters.

Other instruments were considered. In starch viscometry, the
Brabender Visco/amylo/Graph (C. W. Brabender Instruments,
South Hackensack, N.J.) is most frequently used. The
Brabender instrument records, in arbiti'ary units, the torque

required to balance the developing Starch viscosity during a

42

programmed heating and cooling cycle (Zobel, 1984). Shearing
occurs between pins during cup retation at constant speed. The
Visco/amylo/Graph is accepted worldwide as the Standard
instrument to measure and record the gelatinizing properties of
starches and starch-containing products as a function of time,
temperature, and rate of shear (Shuey and Tipples, 1980).

An advantage of the Visco/amylo/Graph is that the effective
shear rate is close to that in the mouth, which implies that
rankings of viscosity derived from Brabender readings will
agree with sensory consistency judgments (Wood and
Goff, 1973). Another advantage is the large data base already
existing for starches. A third advantage would be the conStant
rate of temperature change claimed by the manufacturer, but
Osorio and Steffe (1988) showed the rate is n0t constant. The
disadvantages of the instrument are a) multiple and varying
shear rates around the pins; b) intermittent shearing action;
c)measurement and control of starch temperature is at an
arbitrary point and not at the location where the maximum
temperature occurs; d) long testing times (between 45 and 120
min); e) large (500 ml) sample size (Voisey et al. 1976), f) lack
of sensitivity., and g) lack of instrument-to-instrument
reproducibility (Steffe et al., 1989).

Despite disadvantages a through d, the Visco/amylo/graph can
be used as a quality control device. For example, incoming
Starch is accepted or rejected based on instrument standards.
However, the fan that temperature and Shear rate are unknown

makes using the instrument for prediction of processing

43

conditions complicated and difficult. Many studies Show
correlation between the shearing aetion and the shear rate. By
calibration with absolute devices, Wood and Goff (1973)
determined effective shear rates of the Brabender Viscograph,
an earlier model of the Visco/amylo/graph. Goodrich and Porter
(1967), and Blyler and Daane (1967) estimated rheological
parameters from a Brabender torque rheometer, an inStrument
having a complex shearing motion, as does the
Visco/amylo/Graph. Lee and Purdon (1969) converted
Brabender plastograph (an instrument giving relative readings)
curves to Instron flow curves. In these Studies, the shear rate
was an "overall" shear rate, and the temperature was measured
at only one location in the sample.

Thus, to estimate processing parameters, a device is needed
that measures sample temperature throughout (or makes
temperature gradient negligible) and keeps a conStant shear rate.
Lancaster (1964), Voisey et al. (1976) and Paton and Voisey
(1977), Steffe et al. (1988), and Walker et al. (1988) presented
instruments which meet these requirements, namely the Cooking
Viscometer, the Ottawa Starch Viscometer, the Brookfield RVTD
mixer viscometer, and the Rapid Visco-Analyzer. In addition.
all these devices gave peak viscosity in 1-5 min, compared to 45
min for the Visco/amylo/Graph. Freeman and Verr (1972) also
presented a rapid procedure to measure paste development with
the Brookfield Syncro-Lectric viscometer. Bhattacharya and
Sowbhagya (1981) presented a rapid Brabender viscograph test

with a 50% saving in time and 20% saving in flour weight.

44

The present study also used a rapid mixer viscometry
technique to measure torque, but results were given as estimated
parameters (empirical coefficients) in a comprehensive model.
Parameters are then used to predict a pasting curve as a function
of temperature, temperature-time history, and strain history.

There is a need to remove equipment-dependence from
rheological readings for starch; there is also a need for a
comprehensive viscosity model. Janas and Tomasik (1986)
concluded "more general description of properties of pastes
...can be achieved ...by means of a scope of parameters
independent of measurement conditions [italics added]." Lund
(1984) stated "studies on the kinetics of starch gelatinization
are very limited," and "currently there is no definitive kinetic
model for Starch gelatinization." In the same work, Lund

remarks ...it is highly questionable to develop a kinetic model
for gelatinization,..." because there are multiple-order reaCtions
occurring (Lund, 1987). However, after the initial stage of
gelatinization, a first-order model became more accurate (Lund.
1987). Janas and Tomasik (l986)wrote, "Due to lack of a
precise theory and suitable devices as well as the nature of the
material, results of rheological studies of starch pasting are
deprived of any general meaning."

In light of the difficulties and yet the need for a partial

solution. the-9hisstiu-of-this-sv.stk-uas-tc-identifrandmsasute
interim{litessins-xisscsitx-dexelenmsnt-dutiag-gslatiat;ation..

45

3.3. Theoretical Considerations
Morgan et al. (1989) proposed a model for viscosity of

protein doughs:
.1.) + b (MC - MCr)]
. or n Tr
n(y,T,MC.‘P.d>)=[(7)+(ur)] e

*[1+ [3, [A3 (MC)8 Cpl“)? (1 - {NYC}? [1 - B (1 - 6‘” )]

(3.1.0)
where
-AEg
if RT(t)
‘I’ = T(t)e dt T.>.Tg
0
=0 T<T8
. (3.1 1)
it-
(b = 'ydt
0 (3.1.2)

Simplifications were made in the gelatinization term of the
Morgan et al. (1989) model. Only 100% corn and bean Starch
were used in the current Study, so dry basis Starch
concentration (CD) was 1.0. In an excess-water syStem, the
molecular weight of gelatinized Starch did not depend on
moisture, so 8=O. Although shear rate was varied, the
dependence of a on shear rate was n0t inveStigated; therefore. [3

and A3 were combined into one constant, A.

46

The Morgan et al. (1989) generalized a priori model then

1 [ii

n(?, T, MC, ‘I’, (D) = K?) + (11,)":l n e

*[1+ A°‘(1 - {WW [1 - B (1 - e‘d¢)]

becomes

 

i1?) + b (MC — MCQ]

v-llv-

(3.1.4)

The application of the above model to starch dispersions is
discussed in Chapter 2. Eq. 3.1 is assumed to be separable in
all its variables; that is, each independent variable appears in a
separate term, and the parameters are constant. The
experimental design in this study was also set up to ensure this
condition. There are at least two cases for which this
assumption is inappropriate. The first is when parameters are
functions of any independent variable. The other, a more
serious Situation, is for a term to vary in form as an
independent variable changes.

Occurrence of the first kind does not invalidate the model;
rather it requires that experiments are run over the range of the
variable causing the violation. Then the functional form of the
parameter can be substituted in Eq. 3.1. Both types of
violations were investigated in the current study. The first kind
occurred with temperature and strain history parameters; the
second kind did not occur.

In this study, the yield stress was negligible. Therefore, the
shear rate term (Heinz-Casson model) in Eq. 3.1.4 was replaced

by the power law model, 1] = K('y)“’l Eq. 3.1.4 was transformed

47

by the power law model, 1] = KW)":l Eq. 3.1.4 was transformed
by replacing 11 on the left side with torque (M) following two

basic assumptions of mixer viscometry: Shear Stress is directly
preportional to torque, and Shear rate is directly proportional to

impeller speed. Multiplying by “rand substituting N for 7 yields

[111516; _ .TL) + b (MC-MCJ]
n r

M( N, T, MC, ‘1", <I>)=K,N e

*{1+A°‘(1-e"*" )a} {1-B(1-e‘d¢)}

(3.2.0)
{—AEV 1 k)
. lf R'I‘(t)+n
‘P=k‘P=J T(t)e dt
0 (3.2.1)

where K, is a pseudo-consistency coefficient at reference
temperature T, and reference moisture content Mcr. ‘I"
incorporates k with ‘I’, to avoid large scaling factors caused by
the exponential. Following the substitution of N for '7, Eq.
3.1.2 now becomes

tf
(I) =J' th
0 (3.2.2)

The proportionality constants from mixer viscometry were
absorbed into the parameters Kr and (1. Eq. 3.2 is a more
convenient form for use in the current work than Eq. 3.1.4.
The two equations are, however, equivalent.

To predict a pasting curve, all parameters (a total of ten) of

Eq. 3.1.1 and 3.2 must be known: For the N term, the

parameters are Kr and n; for the T term, AEV; for the MC term.

b; for the ‘I’ term, A, AEg, k and 0t; and for the (D term, B and

48

(1. As explained in the experimental plan, each parameter or
group of parameters was estimated by holding all but one
independent variable constant during data collection. The
difficulty with this approach iS that interaction of variables
forces one to take data outside the desired range, and then
extrapolate back into the range. For example, one cannOt
measure the temperature effect above 75 C because
gelatinization obscures it. Therefore, temperatureeffect was
found in two ways : 1) by eStimating the parameter below 75 C
and then assuming it was accurate above 75 C; and 2) by
estimating the parameter above 75 C after gelatinization was
complete. Effects of temperature-time and strain hiStories had
to be divorced from each other in the same way.

Thus, for parameter estimation, there were five simplified
forms of Eq. 3.2, corresponding to the five independent
variables. When only '7, T, or MC was varied, the form of Eq.
3.2 was straightforward (Morgan et al., 1988). When only ‘1’

was varied, Eq. 3.2 collapses to an analog of Eq. 2.5

a

M=(l- e-W.) -

-yv
M—‘Mo (3.3)

where Mo and M... are the ungelatinized and fully gelatinized

torques, respeCtively.

Similarly, if only (I) is varied, Eq. 3.2 becomes

49
M=M§[1-B(1-e"d¢)] (3.4.0)

Letting ¢—>°° gives

M..=Mo[1-B] . (3.4.1)
Eq. 3.4.1 provides a definition of B, so that Eq. 3.4.0 can be

expressed in the form of Eq. 3.3

M-
A=(l- e‘d¢)=y¢
M~"M0 (3.4.2)
where Mo and M'... are the torques before and after shearing. "d"
indicates the rate at which torque decays. "B" is the relative

amount of viscosity decrease caused by degradation. Both d and
B may be functions of “i, T, and ‘I’, as already discussed. The
experimental plan Shows how (1 and B were estimated at

different values of the three independent variables.

50

3.4. Practical Considerations

Dispersion concentrations of 5.5, 6.4, and 7.3% (g dry
Starch/g soln) were selected to correspond to the Amylograph
standard concentration, 7% (g bone-dry starch/g soln), (Shuey
and Tipples, 1980). The viscosity of these solutions was below
the minimum necessary to register accurately during back
extrusion with a SO-N load cell on the Instron Testing Machine,
used by Dolan et al.(1988). Furthermore, the thickest
dispersions, cooled to 25 C after 17 min at 95 C, were net
solids. Using a flag impeller, the mixer viscometer (Brookfield
RVTD) gave different torque readings at different axial
distances along a culture tube. Therefore, 6-8% dispersions had
to be mixed continuously to avoid concentration gradients.

Table 3.1 summarizes the literature reviewed reporting
apparent viscosity of starch dispersions as a function of the
independent variables in Eq. 3.1. Nine studies fit shear Stress
(t) to Shear rate (:Y) with a shear-thinning model, and four of
the Studies included yield Stress. The same nine Studies
presented viscosity varying with moisture content MC, but only
two (5 and 7) fit the data to a model where MC was the only
independent variable. In b0th studies, the model used was the
power-law, n=C1(MC)8. Although there was no model for MC
in the other studies (1-4 and 13-14), the viscosities presented
suggested an exponential or power-law increase with starch
concentration. Although there were five studies varying

temperature , only Doublier (1981) used more than two

51

temperatures. He used the Arrhenius relationship (temperature
term in Table 2.1), valid for most fluids. Nine studies varied
the thermal history, but only studies 8-12 reported and
quantified it, with a first-order kinetic model different from this
study. The first four studies (1-4) did not report temperature as
a function of time, so the absolute thermal history was
unknown. Studies 8-12 used a first-order kinetic model,
reported temperature over time, and estimated aetivation
energies (AEg). Only Doublier (1981, 1987) investigated time-
dependent behavior, though without a specific strain history
function

Based on Table 3.1, the Shear rate- and moisture content-
dependence of starch dispersions has been well investigated and
has shown consistent behavior. The temperature-dependence is
less established, and may have to be assumed conStant above
gelatinization temperatures, when gelatinization interferes. The
temperature-time history-dependence has been investigated five
different ways, with five models based on first-order kinetics.
The strain history-dependence is unknown other than that starch
dispersions are thixotropic. Therefore, the experimental plan
placed emphasis on varying thermal and strain history, while
still varying shear rate, temperature, and moisture content to

correct the data for use in the comprehensive model (Eq. 3 2).

5 2
Table 3.1. Summary of Studies reporting viscosities of gelatinized starch

 

dispersions.
Independenta
Starch Conc. __ya‘mb_]e__ Dependent
type % db Method ‘1 T MC ‘I’ (D variable Reference
corn 5—26 concentric x x x xb vol. frac. , 1.Christianson&
cylinder viscosity Bagley (1983)
wheat 7-25 concentric x x x xb vol. frac., 2.Bagley & Chris-
cylinder viscosity tianson (1982)
wheat 8-15 concentric x x x xb yield 3.Bagley & Chris-
cylinder stress tianson( 1983)
corn 8-14 concentric x x x xb yield 4.Christianson &
cylinder stress Bagley (1984)
corn up to concentric x x vol. frac., S.Evans & Hais-
potato 10 cylinder viscosity, man (1979)
tapioca yield stress
wheat 1.6- cone and x x x viscosity 6.Wong &
8.2 plate yield stress Lelievre(1982)
wheat 0.3-8 concentric x x x xc viscosity 7.Doublier (1981)
cylinder
rice excess plastometet x strain 8.5uzuki et al.
water (1976)
rice 6 & 30 capill. . x viscosity 9.Kubota et al.
potato rheom (1979)
rice excess iodine blue x enthalpy 10.Bakshi &
water Singh (1980)
rice excessDSC x enthalpy 11.Lund & Wirakar-
water takusumah(1984)
pctato l8 DSC x enthalpy 12.Pravisani et al.
(1985)
maize 5-10 viscograph x x xc viscosity 13.Doublier(l987)
wheat conc. cyl.
corn 3.3 conc. cyl x x viscosity l4. Colas (1986)

 

 

 

 

 

 

 

3?. shear rate; T, temperature; MC, moisture content; ‘I’, temperature-time
history; (D, Strain history .
bReported temperature and time separately; did not use a temperaturetime

function

 

cReported shear rate and time separately; did not use a strain history function

 

 

53

3.5. Materials and Methods
3.5.1. Experimental plan

In all experiments, the dependent variable, torque was
directly proportional to viscosity at constant shear.

Independent variables were impeller speed (directly proportional
to shear rate), moisture content, temperature, temperature-time
history and strain history. For any given run, impeller speed
and moisture content were constant. The assumption of a
separable model (Eq. 3.2) was teSted by conducting trials over a
range of conditions. In the experimental design, there were five
divisions--four for estimating the parameters, and one for
testing the results obtained in the first four divisions.

One cannot always independently estimate all parameters
(constants) of a given model. Clearly, parameters appearing in
groups cannot be estimated, because any combination resulting
in the same group value will work. Sometimes n0t even all
these groups may be found (Beck and Arnold, 1977). Hence,
before designing experiments, one Should determine which
parameters can be estimated (identifiability ), and what ranges
of independent variables give the most accurate eStimate.
Appendix A describes how to make bOth determinations for Eq.
3.2.

Table 3.2 Shows the experimental design for determining
effects of moisture content and thermal history. The 10 C
temperature range was chosen to approximate the range of the

Visco/amylo/graph where raw corn starch thickens most rapidly

54
Table 3.2.Experimenta1 design for determining effects of
moisture content and temperature-time historya

 

         

  
 
 

 

 

 

 

mpeller Temp g Starch/ Fluid JacTet Heating
seed r m C solution ‘
10,15
100 60 5.5 92 20,25
85 25.30
95 2,4,6,8,10,12
100 60 6.4 92 3,5,7,9,11,15
88 3,3.5,4,5,8,12,15
85 3.3.5.é.§.2.12.l§
95 2,2.25,2.5,4,7,l()
100 60 7.3 92 2.5,3,7,20,25
85 3,35,45,8512

 

aExperiments conducted in duLlicate

Table 3.3.Experimenta1 design for determining effects of

 

     

 

temperaturea
Impeller Temp g Starch/ Fluid Jacket Heating
s-dec .r_ R- C m - SWto . .Tm- ' '
3b
100 50-70 6.4 95 4b
5b
6b
100 60-95 6.4 95 8g

 

 

a‘AE‘ZV was eStimated from each run
bExperiments conduCted in duplicate
CThe coolingphase of Set 2.1 Table 3.5 (triplicate)

 

 

 

55

Approximately equivalent ranges of ‘I’ were achieved by
Shortening heating times as glycol jacket temperatures were
increased. Only equilibrium torque measurement (M¢_,,,,) was
used, and each trial was duplicated. The purpose of this design
was to use the 6.4 and 7.3% results to estimate gelatinization
parameters and the 5.5, 6.4, and 7.3% results to eStimate the
moisture content parameter.

Table 3.3 describes the experimental design for determining
effeCts of temperature. AEV was estimated for each test by
varying temperature only, a procedure made possible by the
small sample size. Different heating times were used to check
the assumption that AEV was independent of ‘I’. With smaller.
heating times (Table 3.3), when the sample was incompletely
gelatinized, the temperature range had to be less than Tg to
avoid influence of gelatinization. For greater heating times,
when gelatinization was complete, the torque was recorded as
the sample was cooled, because temperature-time history and
strain history effects had already reached a plateau.

In addition, the purpose of the experimental design for
determining effects of shear rate (Table 3.4) was to estimate
the shear index, n. The range of rpm was the operating range of
the viscometer.

The experimental- design for determining effects of strain
history is given in Table 3.5. There were two divisions, Sets
1.1-1.3 (division 1) and Set 2.1 (division 2). In division 1,
only strain history was varied in thixotropic (shear breakdown)

studies. Torque over time was recorded at the specified

56
Table 3.4.Experimenta1 design for determining effects of shear

rate8
Impeller Temp 3 Starch/ Fluid Jacket Heating
C , a 1 8.110110 e _m__ min)

 

        

5 60 6.4 95 3a

 

5 50 6.4 95 12b

 

aExperiments conducted in duplicate
bFrom experimental pastingcurve (Table 3.6)

 

 

 

Table 3.5.Experimenta1 design for determining effeCtS of Strain
historya

Test Impeller Temp g Starch/ Fluid Jacket Heating

et___se__d. __ C -_ _um ____Cme min)

 

     

 

 

 

1.1 50 60 6.4 95 12
J!)
1.2 100 60 6.4 95 8
4
1.3 100 50 6.4 95 12
95 8
2.1 100 82-95 6.4 95 3-8b

 

aExperiments conducted in duplicate
bExperiments conducted in triplicate

 

 

 

57

conditions. N, T, and ‘P were varied separately to check the
assumption that d and B were constant over those ranges. All
but one set of tests (Table 3.5) were at temperatures less than
Tg (65 C) to avoid interference from gelatinization or thermal
degradation. Each trial was duplicated. In division 2, strain
history was allowed to vary simultaneously with T and ‘1’. The
purpose was to determine if strain history interacted with T or
‘1'.

The experimental design to produce pasting curves (Table
3.6) to test the predictive ability of the model (Eq. 3.2). N, T,
MC, ‘1’, and Tm were varied separately to produce eight pasting
curves. Torque and temperature were measured over all time.
The curve of Set 1 was predicted first using a separable model,
and then the curves of all sets were predicted using an
inseparable model to find the improvement in fit. For the
separable model, all parameters from the results of Tables 1-5
were held constant, and Eq.3.2 was solved for M. For the
inseparable model, shear rate, MC, and ‘1’ parameters were held
constant, while the T and (D parameters were allowed to vary
according to the results of Table 3.3 and Set 2.1, Table 3.5.
respectively. With the substitution of these functions for the T
and (D parameters, Eq. 3.2 was again used to predict M. The
model was further tested by using bean Starch and decreasing

the final temperature to 5 C (Set 6, Table 3.6).

58

 

 

Table 3.6. Experimental design to produce pasting
curves
gTe st Impeller Finalc :I‘e emp g Starch/ Fluid Jacket Heating

7 Se s__-eed 7 _ _ solutionem time (min)

        
     

 

 

 

 

 

 

 

 

 

 

 

 

1:

2 100 60 6 4 95 12

8a

_3.___LQQ 6Q 7.3 ii 8
4 SQ SQ fiaé 9i 1_2_b

j 199 SQ 6A Ji 12.

6 199 5 6 95 210

 

 

aExperiments conducted in triplicate
bUsed to estimate shear index (Table 3.4)
cPurified bean starch Phaseolus vulgaris var. seafarer

 

 

 

59

3.5.2. Apparatus

Figure 3.1 is a schematic of the experimental apparatus. The
equipment and procedure in this work were a modification that
used by Steffe et al. (1988). The difference was that in this
study, three rather than two fluid baths were used, to gain more
rapid temperature change. There were also three sets of two
valves in this study, rather than multiple ports on two valves.
The apparatus mixes continuously, makes continual temperature
measurements, and produces peak viscosity for corn starch
within five minutes. T-typc (30-gauge) thermocouples were
calibrated in boiling distilled water at known elevation and
barometric pressure. Maximum error is 1.0 C (Omega
Engineering, Stamford, CT). Two automatic timers (GraLab

, models 171 Timer and 625 Timer/Intervalometer, Dimco-Gray
Co., Centerville, OH) were used to measure elapsed time and
control the sequencing of the ethylene glycol to the jacket.
Maximum error is 0.1 5. Torque was measured on an arbitrary
lOO-unit scale by a viscometer (Brookfield RVTD, Brookfield
Engineering Laboratories, Inc., Stoughton, MA). Published
accuracy and reproducibility are 1% of range in use and 0.2%.
respectively. A visual record of torque over time was kept on
an analog chart recorder (OmniScribe recorder, model 85217-5.
Houston Instrument, Austin TX). In the strain history
experiments only (Table 3.5), torque over time was recorded
using a hand calculator (HP 41CX, Hewlett Packard, Corvallis.
OR) to read (through an HP 3468A Multimeter) and print (to an

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and copper-constantan thermocouple.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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63

HP 82162A Thermal Printer) the viscometer torque voltage. The
torque and temperature over time for the paSting curves (Table
3.6) and experiments to determine temperature effects (Table
3.3) were recorded on a personalcomputer (Apple Macintosh SE.
Apple Computer, Inc.,Cupertino, CA) with a 16-bit data
acquisition board and accompanying software (ACSE-l6-8 board
and Analog Connection WorkBench software, Strawberry Tree
Computers, Inc., Sunnyvale, CA).
3.5.3. Procedure

Native corn starch, Melojcl (National Starch and Chemical
Corporation, Bridgewater, NJ), was used for all experiments.
The material was received as a white powder with an equilibrium
moisture content of 8.4%, measured by drying at 104 C for 48
hr. Chronological order and time duration of experiments were
first the 6.4% dispersion, three weeks; followed by the 7.3%.
one week; and finally the 5.5%, two days. Distilled water at 26
i 2 C was added to the corn starch to give a 50-g dispersion.
with pH between 5.8 and 6.2. Each dispersion was mixed at
approximately 800 rpm (Corning PC-351 Hot-Plate Stirrer) for
two minutes. A pipette (Gilson pipetman, Rainin Instrument
Co., Inc., Woburn, MA) was used to transfer 12 ml in 4 m1
increments, to the Brookfield small sample adapter chamber.
The chamber was immediately placed inside the water jacket and
the impeller was turned on at 100 rpm to keep the starch solids
from settling. No settling was observed. At the completion of
each test, the base of the chamber was checked for settling of

solids. Heating of the sample began within 15 5. Heating time

 

64

for any one sample varied between 2 and 25 min, depending on
heating temperature, which varied between 85 and 95 C. Time
to reach maximum temperature was about 8 min. Temperature
was measured every 10 s by a voltmeter (HP 3497A Data
Acquisition/Control Unit) and computer program (HP-85) Except
for Sets 1-3 of the strain history tests, the impeller r0tated
continuously at 100 rpm. Excluding Set 2.1 of the strain history
tests (Table 3.5), the impeller rotation speed was varied as
follows: 100 rpm until torque was 10 (Brookfield units, full
scale equals 100 or 7187.dyne cm), 50 rpm between torque
equal to 10 and 15, 20 rpm between 15 and 20, and 0 rpm until
heating and cooling were completed and measuring temperature
was reached. This procedure gave minimum breakdown while
still maintaining homogeneous mixtures. Then the viscometer
was restarted at the appropriate rpm (20, 50, or 100)
simultaneously with the calculator. Mechanical degradation was
recorded as decreaSing torque over time.

Temperature of the Starch solution was calibrated with the
chamber thermocouple temperature, to account for conduction
along the sheath and thermocouple, and the lag time. Heating
and cooling cycles were duplicated for each of the four target
temperatures. The impeller was replaced by a 1.5 mm-outer
diameter, 175 mm-long aluminum sheath rotated by hand
between 50 and 70 rpm. Four T-type insulated thermocouples
(30-gauge) were wound around the sheath. The soldered tip of
each thermocouple was placed where the flags would sweep in

the sample: two tips at 4 mm above the sample chamber base,

65

one at 19 mm, and one at 34 mm. Thus, temperatures of these
four points in the starch and the chamber thermocouple were

measured every 10 3 during hand-mixed runs. Although heat
was added at the chamber wall, the radial temperature gradient
was insignificant due to thorough mixing.

3.5.4. Analysis

3.5.4.1. Analysis Involving All Parameters

In the gelatinization experiments (Table 3.2), the analytical
procedure of Dolan et a1. (1988) was used with modification.

At each target temperature and moisture content, temperatures at
three axial positions in the Starch were correlated linearly to
chamber thermocouple temperature. Two to three replications
were used.

The procedure for estimating parameters is illustrated in
Table 3.7. Table 3.7a shows the analytical design and Step-
wise correction for Table 3.2, while Table 3.7b shows the
calculation steps for Tables 3.2-3.5. The third column of Table
3.7a is measured torque, M. The right margin shows which
parameters were eStimated from particular groups of data. Table
3.7a is interpreted as follows. There were five horizontal
blocks. The MC parameter, b, was estimated from each of the
first three blocks (yielding b1, b2, and b3), because only MC
was varied at three constant maximum temperature (Tm) values.
The average b was used to correct all M to a common MC
(MMc,the second column). Now the only variable in the first
three blocks was Tm. Therefore, ACl was estimated from MMC in

the firm three blocks (see the right margin). ACl was used to

66
Table 3.7. Procedure to estimate the parameters

Table 3.7a Analytical design and step-wise correction for Table
3.2.

 

Dependent

Jamaal; .

29:1:ch __JndemdenLXAnam:_
\{I' Mm M N T MC ‘P' (D Tm

Nr Tr MC] Wit (bf
MCz T1111

1
MC 2 Tmz

l
MCz

 

 

 

Nr=100 rpm
Tr =60 C

‘I”,—>°O
(Dr—>00

Table 3.7b. Calculation steps

Using data for gelatinization effeCt (Table 3.2)

 

for each Tm, fit In M = lnC +b(MC) -> estimate};
use average b=(b1+b2+b3)/3

b(MC,-MC)
correct all M to MC; —>MMC=MC

 

 

 

 

 

from a plot of Mm... versus Tm-) estimate_Aa = f(Tm)
normalize MMC(Eq. 3.3) -)y\y' = (MMC-Mo)/(MMC,oo—M0)

 

 

 

 

 

fit yw' =(1-e"*")°l -> Qilim§§§-kL-QL-Afii

7

67

Table 3.7b (Cont' d)

Using data for temperature effect (Table 3.3)

 

for each experiment, use M (N,MC,‘I" ,¢,)
fit In M = lnC + AEv/(RT) aestjmatLAEv
l

Using data for shear rate effect (Table 3.4)

 

 

fit In M = 1nC.+ n ln(N) #:511133th

Use previously determined parameters to estimate the final one

for each experiment, use M (N,T,MC,,‘I" ,<D,) l

 

 

.9 KT=MO/(N,)n (3.6) I

 

 

Using data for strain history effect (Table 3.5 Sets 1.1-1.3)

 

 

for each experiment, use M(N,T,‘I", Tm)
fit ln(M-Mm) = ln(Mo-M”) - d*N At
figstjmatgfldhfi for each data set
where B= (MO-M...)/Mo

 

 

Using data for strain history effect (Table 3.5 Set 2.1), find d
and B while T and ‘I’ vary simultaneously

 

 

for each experiment, use M(N,,T,MC,,‘I" , Tm,)
fit ln[(M/Mo)-(M.,/Mo)] = lnB-d*NAt (8.3)
—>:stimate-'.'oxctalll- .-B.

 

 

68

correct all MMC to a common Tm by normalizing to yxy' (first
column). Since yxp' had now been corrected for MC and Tm, the
only remaining variable in all five blocks was ‘I”. Therefore,
k, a, and AEg were estimated from all yxy' (right margin).
The method for estimating b, M”, and A“ is straightforward.
As ‘I”—-)°°, A“ = (Mn-MOVMO . The gelatinization parameters
AEg, k, and a (Eqs 3.1.1 and 3.3) were eStimated by
minimization of the sum of squares of residuals ([observed
yxp']-[predicted ytp']) as follows: The Optimal value of A15g
was found using a minimization routine (quadratic interpolation
method, routine name UVMIF, Math Library, IMSL, Inc.); the
initial guess of AEg was based on reported values; for each
iteration in this routine, AEg was held constant, and k and a
were estimated simultaneously by sequential nonlinear
regression (Box-Kanemasu method, Beck and Arnold, 1977).
The reported k and a are those corresponding to the optimal
AEg, to which the routine converged. 3011'] the routine and the
nonlinear regression package were written in Fortran 77, and
were run on a VAX-11/750 VMS 4.7 computer. Average CPU
execution time was 42 s. In summary, gelatinization tests
(Table 3.2) provided five parameters: b, A, AEg, k, and 0t
Table 3.7 shows how the temperature parameter AEv and
shear index n were estimated. The tenth parameter Kr was
estimated at N,, Tr, MCr, ‘I"=0, and <D=0. Eq. 2 then reduces

to

69
IVIO=Krmr>n (3.5)

from which K, can be obtained

K,=MO/(N,)n . (3.6)
All terms on the right side of Eq. 3.6 were known constants,
yielding K,.

In the strain hiStory experiments assuming a separable model
(Sets 1-3, Table 3.5), the only independent variable was (D.

Eq. 3.4.3 was rewritten as

M-M;=(M,,-M;,)e‘d¢ (3.7)
The thixotropic parameter "d" (Equation 5) was estimated by
linear regression of ln(M-M'm) versus (NAt) during the first 15
min (Table 3.7). B (Eq. 3.4.1) was solved for in the same
manner as was A, yielding B = (M'o-M”)/M'o. d and B were
correlated to N, ‘I” , and T, in that order. Taking d as an
example, (1 at ‘1", , T, was correlated firm to N, and then all d' s
were corrected to N, . The procedure was repeated for
correlation to ‘1" and finally to T.

With Set 2.1 in Table 3.5 (for testing whether (1 and B
interacted with the independent variables), d and B were found
by factoring out, from the torque data, all effects except strain

history. Then Eq. 3.2 was solved for the strain history term:

70

[3:101]: - %) + b (MC,- MO]
{1-B(1—"“")}=Me =1";
e K,N“{1+A“(1-c“‘")a} M“

Mo was calculated after all other parameters were known.

 

(3.8.1)

Rearranging Eq. 3.6.1 yields

i={l ”B(1‘C—d¢)}=(1-B)+Be'd¢=&+Be‘d¢

M0 M0 (3.8.2)

Thus, the data for Set 2.1 were fit according to

1n[-M—--Nl-‘f]=1nB-d¢
M0 M0 (3.8.3)

This method is an "extraction" of parameters from a pasting
curve, once all other parameters are already known. In

summary, strain history tests provided two parameters, d and B.

71

3.5.4.2. Simplified Analysis

To test the fit of the model with a different material and a
minimum of experiments, parameters for bean starch were
estimated from one curve only. Since N and MC were constant
for any one curve, only the T, ‘P', and <I> parameters were
estimated. The procedure involved three step-wise corrections
of the data, corresponding to the three variables. A paSting
curve was divided into three sections: 1) heating rise--from the
beginning of temperature increase until peak torque; 2) shear
decay--from peak torque until beginning of cooling; and 3)
cooling--from beginning of temperature decrease until final
torque. Results (from Table 3.6) show that cooling was the
only region where a single variable, T, had a significant effeCt.
The same results show that ‘I” and T were the only significant
variables during heating rise, and that <15 parameters depended
on T and ‘1”. Therefore, parameters were estimated for T (using
cooling data only), ‘1" (using cooling and heating rise data
only), and (D (using all data), in that order, following the same

type of sequential correCtion shown in Table 3.7a.

72
3. 6. Results

The estimated parameters are listed in Table 3.8 and are
compared to literature values when possible.

In gelatinization experiments (Table 3.2), coefficients of
determination (R2) for the chamber thermocouple calibrations
ranged from 0.94 to 0.99. During the most rapid heating
(Tm=95 C ), chamber thermocouple readings lagged average
starch temperature by as much as 4 C. Average moisture content
parameter b was equal to - 0.511 with an 15.% coefficient of

.0367Tm 1

variance, and the equation used for A was Aa=6.80e
with R2 = 0.90. Average ungelatinized torque (Mo) was 0.2 /
(7.19 x 103) N m. Because published literature used a power-
law rather than an exponential for the moisture content effect,
the parameter (g)for the power law was estimated as -3.22 with
a 15% coefficient of variance (Table 3.8). Both moisture
content parameters (b and g) were greater in magnitude at lower
temperatures.

Figures 3.2 and 3.3 show the increase of torque with time
and temperature at constant rpm for the 6.4 and 7.3%
dispersions. Figure 3.4 shows 42 of the same data, corrected to
6.4% starch concentration, with torque normalized (Eq. 3.3) and
time transformed to ‘1" (Eq. 3.1.1). The 42 data chosen were
those with y less than 0.95, because in this region the torque
was most sensitive to the parameters (Appendix A). Parameter
estimates were AEg = 740. kJ/mol, k = 2.36 (K s)'1 (scaled to
‘1’), and a = 0.310. Although the regression converged, AEg

and k were correlated (a change in one caused a compensating

73

 

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77

change in the other). Standard deviation of the residuals was
9.8% .

In fitting the Arrhenius relationship for temperature effects, the
lowest correlation for any sample was R2 = 0.92. A15v appeared
to increase rapidly with ‘P' and then approach a limit, behavior

similar to viscosity. The trend of AEv was fit to two lines:

92.139" + 600 ‘P'< 0.3216
R2 = 0.77 (3.9.1)

AEV/R

AEv/R = 0.3.770‘1" + 765 ‘P'z 0.3216
R2 = 0.58 (3.9.2)
where 0 g‘P’s 300

For shear effects, the average shear index was 0.204 with a
Standard deviation of 0.004. The lowest correlation for any
sample was R2 = 0.94. Parameter K, was equal to 0.11 dyne cm
(min)n x 7187. (Eq. 3.6).

For strain history effects with all other variables held
constant (division 1, Table 3.5), the parameters d and B were

correlated to N, ‘1", and T as follows:

78

d(N,‘P',,Tr) = (58.3/N - 0.1385)/60,000 R2 = 1.00 (3.10.
d(N,,‘P',T,) = (0.34 exp(-0.0049\P') + 0.4)/60,000 R2 = 0.86 (3.10.
d(Nr,‘I",,T) = (0.043(T-273) - 2. 1)/60,000 R2 = 0.94 (3.10.
B(N,‘P',,T,) =0.001N +0.19 R2=0.70 (3.11.
B(N,,‘I-",T,) = 0.012exp( 0.00519") + 0.10 R2 = 0.93 (3. 11.
B(N,,‘I",,T) = 0.0088(T-273) - 0.23 R2 = 0.78 (3.11.

For curve fitting, the reference values used were Nr = 100, Tr =
60 C, and a finite scaled value of ‘1", = 1250. The torque
decrease within the first minute was a more rapid decay than
that of the remainder of the curve.

For the pasting curves of division 2, Table 3.5, after all
effects except strain history had been factored out (Eq. 3.6),
torque decay was noticeable only after peak torque. Average d
and B for the three trials were 6.38 and 0.39, with coefficients
of variance 20 and 5%, respectively. The average ‘1" when
maximum torque occurred was 6.84. These results were used
for predicting the final pasting curves (Table 3.6)

Figure 3.5 shows the predicted and experimental paSting
curves (Table 3.6, Set 1), assuming the model is separable.

The constant parameter values were b= - 0.511, d = 0.425,
B=0.280, and AEv/R = 2200 K. The remaining parameters were
as listed in Table 3.8.

Figures 3.6-3.13 show the predicted curves (in the order of
Table 3.6) using d and B extracted from real pasting curves (Set
2.1, Table 3.5). Shear rate, MC, and ‘1" parameters were set to
the estimates from this study, and ABV was varied according to

Eq. 3.11. (1) parameters were varied as follows: d and B were

79

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88
set to zero until ‘1" = 6.84, then held constant at 6.4*(100/N)
(inverse proportionality suggested by Eq. 10.1) and 0.39,
respectively, until beginning of cooling (Set 2.1, Table 3.5
results). During cooling, B was varied linearly with
temperature from 0.39 at 95 C to 0.29 at 60 C, while d was
varied according to Eq. 3.9.

The experimental curve of Figure 3.6, with three different
predictions, each with different terms of Eq. 3.2 left out, is
presented in Figure 3.14. Figure 3.15 shows the experimental
and predicted torque for bean starch (Set 6,Table 3.6). The
estimated parameters were, in order: for T, AEV=7.8 kJ/mol;
for ‘P, AEg = 430 kJ/mol, k = 0.48 (scaled to ‘P), a = 0.42; and
for (b, d=2. 1, 8:0. 16.

89

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91

3.7. Discussion
3.7.1. Comparison of parameters with literature

K, (Table 3.8) was the only equipment-dependent parameter,
because it is a scaling constant for the given system. In this
work, there was no advantage to give it absolute significance by
estimating § according to standard mixer viscometry techniques
(Rao, 1975).

Parameters n, k, and AEg (Table 3.8) did not compare well
with those published in literature. The value of n in this Study
was for corn starch paste gelatinized at 95 C, whereas the
maximum temperature in other studies was 90 C. Further study
is needed to show how shear-thinning increases with maximum
temperature. k and A8; are discussed in the following section.
All other parameters (Table 3.8) have absolute physical
significance, but AEV and a were not found for corn starch.
3.7.2. Gelatinization effects

Although the A83 estimated in this study was more than six
times greater than those reported for rice starch (References 8-
12, Table 3.1) and p0tato starch (Reference 9, Table 3.1), the
values cannot be compared because different mathematical
models were considered. The primary reason is that unlike the
Standard equation for first-order kinetics, Eq. 3.3 has the
exponent 0. Another reason is that the current study used a
different zero heating time reference viscosity. Other studies
(References 8-12, Table 3.1) used a arbitrary point during early

stages of gelatinization (Lund, 1984). This study used

92

ungelatinized viscosity, which was about equal to the viscosity
of water. Furthermore, the cited researchers estimated AEg
from isothermal experiments; the temperature was varied in, the
current study. In addition, there were different combinations
of AEg and k with identical sum of squares (correlation).
Therefore, the model (Eq. 3.2) prediCted y accurately (9.8%
Standard deviation), but could not estimate a unique activation
energy of gelatinization. Thus, the value of A138 is n0t to be
taken as an absolute physical parameter, nor was thepurpose of
this research to estimate the value; rather, the purpose was to
eStimate torque response, which was done with the combination
of k and AES' To estimate A83, (1 must be set equal to 1.0 (the
model of References 8-12, Table 3.1). If the values are Still
unusually high, additional is0thermal experiments muSt be
conducted.
3.7.3. Moisture content effects

There was no difference (15.% coefficient of variance) in the
exponential model used in this study and the power-law model
for moisture content. However, The moisture content range was
not intended to differentiate between the two models. There
was correlation between b and Tm (stronger MC effect at lower
Tm), but the range of Tm (10 C) was small enough to accept an
average b.
3.7.4. Temperature Effects

AEV increased with the extent of gelatinization (represented
by ‘1", Eq. 3.9). Physical chemistry prediCts that the response

of fluids to temperature depends on the molecular Structure

93

(Bird et al., 1960). Since gelatinizing starch is changing its
"macromoelecular structure", a dependence of AISv on ‘P' is
expected. This dependence could be further investigated by
comparing temperature response of starches with different ratios
of amylose, a linear molecule, and amylopectin, a branched
molecule. The ratio for native corn starch is approximately 1
amylose to 3 amylopeCtin.

3.7.5. Strain history effects

The rate and extent of breakdown should also depend on the
molecular structure (size and shape). The network of
macromolecules in gelatinizing Starch changes as b0th T and ‘1"
increase. Therefore, the dependence of B and d on T and ‘1"
(Eq. 3.10.2,3.10.3, 3.11.2,3.11.3) confirms predicrions from
physical chemistry.

There were two divisions of Strain history results. The first
(Sets 1-3, Table 3.5) showed that (D effects depend on ‘I” and
T, in contrast to the assumption that Eq. 3.2 parameters were
constant. Therefore, observations from division 1 are true only
when all variables other than (D are constant. The second (Set
2.1, Table 3.5) was an attempt to bypass this problem by
estimating <l> parameters while ‘I” and T were increasing (during
a pasting curve). This interference was unavoidable, because
‘1" could not be held constant at T greater than Tg. These two
divisions will be discussed in order.

In division 1, the product d*N varied within 25% (Eq.
3.10.1). Therefore, the estimate of a constant d*N was used in

predicting pasting curves (Table 3.6). The rate of breakdown

94

was greater at lower thermal histories (Eq. 3.10.2) and at
higher temperatures (Eq. 3.10.3), results consistent with
intuition about thinner fluids. (1 was more dependent on T than
on ‘P' (40% decrease compared to 400% increase, respeCtively).
B, the relative amount of breakdown, was strongly influenced
by ‘I” and T, but not by N (Eq. 3.11). Over the range of
increasing ‘P'and T, B varied from 0.10 to 0.28 (Eq. 3.11.1),
and from 0.30 to 0.60 (Eq. 3.11.2), respectively. This increase
in susceptibility to breakdown was caused by the greater
pliability of the swollen granules (Christianson and Bagley,
1983).

The rate of breakdown (d) experienced during pasting
(d=0.0064, division 2) was two to three times greater than that
(d=.00200 at 95 C) when ‘P'and T were held constant (division
1, Eq. 3.10.3). During paSting, B reached its maximum value
(at 95 C) more quickly than predicted (Eq. 3.11.2), implying B
is less dependent on ‘P'at higher temperatures. Therefore,
results of division 1 were inaccurate for predicting pasting
curves, as discussed in the following section.

3.7.6. Predicted pasting curves

The predictive ability of the model (Eq. 3.2) was tested by
using the parameters estimated at constant conditions (Tables
3.2-3.5) to predict torque when T, ‘1", and (D varied
Simultaneously (Table 3.6). Figure 3.5 shows the prediction
assuming all parameters are constant. The lack of a predicted
shear decay indicated (1 was undereStimated (division 1). The

peak torque was underpredicted and the final torque at 50 C was

95
overpredicted because the temperature parameter (AEV, Table
3.3) estimated at long cook times was too large. Therefore, the
predictions of Figure 3.5 imply Eq. 3.2 would fit better if d, B,
and A8,, were allowed to vary with ‘1",in contraSt to the the
assumption that parameters were constant. Eq. 3.9 confirms the
dependence of Arrhenius aetivation energy (AEV) on extent of
gelatinization. From beginning to end of the pasting process,
AEV doubled (Eq. 3.9). Therefore, errors in AEv were
magnified in predictions at longer cook times. At the
beginning of gelatinization, the granules are rigid particles
suspended in water. Viscosity is controlled by the water phase,
whose viscosity change with temperature is negligible compared
to that of the paste. As gelatinization progresses, the granules
swell and form a network whose viscosity responds to
temperature.

Figures 3.6-3. 13 show the robustness of the model (Eq. 3.2)
over the range of conditions in Table 3.6. For all parts except
the cooling section of Figure 13, the predicted follows the trend
of the experimental curve, with approximately a constant error
magnitude. In Figures 3.6-3.13, T, ‘P', and (b are changing
simultaneously within each figure.

The ability of the model to predict for different cooling
temperature, heating time, cooling rate, moisture content, shear
rate, and maximum temperature is shown separately in Figures
3.6 and 3.7, 3.8 and 3.9 (replicates), 3.10, 3.11, 3.12, and

3.13, respectively.

96

The qualitative and quantitative characteristics of each region
of a pasting curve (heating rise, shear decay, and cooling) will
be discussed in order.
3.7.6.1. Heating rise

The heating rise was overpredicted for all curves except
Figure 3.11. The first-order kinetic model was least accurate
during the beginning of gelatinization, where multiple-order
reactions occur (Lund, 1984; Biliaderis, 1986). Although the
rate of torque rise (governed by k and AEg) was eStimated from
fully gelatinized torque measurements (Table 3.2), the results
were also valid at the beginning of gelatinization. Therefore,
there was negligible effeCt of strain hiStory on gelatinization,
because the time of initial torque rise did not shift by more than
5%.
3.7.6.2. Shear decay

At the beginning of shear decay, since d and B are "turned

on at a specific ‘1" value, there is some overprediction of the
peak torque in Figs. 6-13. Using a step-change for d and B was
chosen because the exponential decay model (Eq. 3.4) fit only
the data after the peak. In the region near the peak, there is
interaction of gelatinization and strain history. In Figures. 3.6-
3.13, after the peak, the error magnitude was approximately
constant (Figures. 3.8, 3.9, and 3.11), showing the rate
parameter d was insensitive to the conditions varied. Figure
3.11 was the only prediction where shear decay began late,
suggesting the onset of shear decay is dependent not only on ‘1"

but also on the torque magnitude (Shuey and Tipples, 1980, p.

97

3). The assumption of a "time constant" = d*N (Figure 3.12)
was reasonable, as shown by the prediction (since N had been
halved, d was doubled). The error of Figure 3.12 resembled
that of the other figures, showing the estimate of conStant shear
index n was valid over all temperatures considered.
3.7.6.3. Cooling

Cooling in Figures 3.6-3. 12 began at different values of ‘1",
yet the cooling rise was predicted with a consistent trend. At
lower ‘F'values (Figure 3.13, where maximum temperature was
85 C), the prediction did not follow the experimental trend. At
this incomplete stage of gelatinization, change in Starch
viscosity caused by cooling from 85 to 50 C was undetectable.
The ‘P'value at this point (700 s. at 85 C) was approximately
equal to ‘P'after 180 s.at 95 C (Figures 3.6—3.12), suggesting
temperature effects were negligible during heating rise. The
granules at 85 C, unlike those at 95 C, had n0t swollen
sufficiently to reveal any effect of temperature.
3.7.6.4. Relative influences of independent variables

Figure 3.14 summarizes all the results by showing the
relative influence of each term in Eq. 3.2 throughout the
representative pasting process of Figure 3.6. The reference
temperature was 95 C. There was less than 5% difference
between Curve 1 and Curve 2 during heating rise (time less than
220 5), showing the overriding influence was gelatinization,
with minimal contribution from temperature. Between peak
torque (time equal to 220 s) and beginning of cooling (time

equal to 730 5), temperature varied less than 5 C and

98

gelatinization approached a limit, so strain history was the
primary influence. In this shear decay region, Curve 2,
prediction without temperature correction, was virtually
identical to prediction with temperature correction (Figure 3.6),
because the experimental temperature was within 2 C of 95 C.
After cooling began, gelatinization ended and the rate of torque
decay was negligible, so temperature was the only influence.
Curve 3 shows the rise caused by cooling. Therefore, for any
one pasting curve, there are at most two variables
simultaneously causing torque to vary. In summary, Figure
3.14 shows that a corn starch pasting curve can be predicted by
knowing only the rate of gelatinization during heating rise, the
rate and extent of breakdown during shear decay, and the torque
response to temperature during cooling.

The usual interpretation of set-back is that retrogradation,
the association of swollen granules, causes the viscosity
increase (Mazurs et al., 1957; Freeman and Verr, 1972; Shuey
and Tipples, 1980). In this study, setback occurred in 20 5, too
short a time for retrogradation to occur. Furthermore, for fully
gelatinized and sheared pastes,a small as a 2 C decrease resulted
in a reversible viscosity increase over 30 5. Cooling in the
Brabender device occurs in 30 min., when retrogradation is jusr
beginning. Therefore, what has been attributed to
retrogradation is really a cooling effect.
3.7.6.5. Simplified Analysis

Figure 3.15 shows that Eq. 3.2 can be applied with a

minimum of data. Although the shear decay region was n0t

99

visible (Figure 3.15), it was revealed when T and ‘P'effects
were removed. The shear decay was one-third (dbean/dcom) as
fast and four-tenths (Bbean/Bcorn) as great as that for
cornstarch, and was therefore obscured by gelatinization. AEV
showed a distinct increase in the range 30-5 C, compared to 95-
30 C, resulting in the undereStimation at the end of cooling.
Although activation energy for temperature effects depended on
‘1’ (Eq. 3.9), A1?" can be set to a constant as long as all the data
are corrected to a reference temperature. The estimated
parameters for ‘I"and (D changed to compensate for the
inaccuracy of AIS.v at small ‘I” Therefore, all the parameters,
not just those for ‘1” were correlated. This method can be used
only if M is desired. For example, one cannot use the constant
AISv to calculate the peak torque if T was equal to 70 C.

For corn and bean starch dispersions (Figures 3 6-3.15), the
Brookfield RVTD viscometer gave more accurate data in less
time (8 min. compared to 45 min.) than the Visco/amylo/Graph.
The greater accuracy in measuring shear rate and temperature
made the Brookfield a convenient tool for modeling. The
results of the RVTD viscometer could be recorded as families of
curves or as model parameters, which help explain the
gelatinization process. Because of these advantages, the
Brookfield device should be considered as a replacement or
companion to the Brabender instrument.

This type of modeling (Eq. 3.2) is instructive, because it
brings together present knowledge and takes a risk by assuming

independent effects. The advantage is that if the assumption is

100

correct, the experimental time will be decreased to a fraction of
that required for a full faCtorial design. For example, even if
the assumption is wrong for three variables, but correct for one,
the number of experiments will be reduced from (2x3)4=1296 to
(2x3)3=216, where 2x3 represents duplicates at three values of
each independent variable. Therefore, this kind of modeling can

benefit all kinds of experimental research.

101

3.8. Conclusions

1.

Each effect of shear rate, temperature, moisture content,
temperature-time history, and Strain history, as presented in
the Morgan et al. (1988) model, was necessary to account for
the torque response of pasted corn and bean starch
dispersions. Having fit highly controlled experiments (5.5-
7.3% d.b. corn starch dispersions at 50-95 C, and 6% bean
starch at 5-95 C), the model can be applied next to piIOt plant
conditions.

Although each of the five variables affected viscosity, there
was only one variable controlling torque response during
each period of corn starch pasting: gelatinization during
heating rise; strain history during shear degradation; and
temperature during cooling.

The most significant interactions were between the
temperature parameter and temperature-time history, and
between Strain history parameters and temperature-time
history. However, there was no change in the form of the
terms, only in the values of the regression parameters.
Therefore, the model was not changed, but the parameters
were estimated as functions of temperature-time history.
Strain history parameters should always be estimated last,
because they are the most sensitive.

Dispersions thickened with decreasing temperature, behaving
as typicalfluids. Arrhenius activation energy was between

6:4 and 11.5 kJ/mol. There was no evidence that

102

retrogradation caused this effect, because the cooling took
place in 20 s.

. The Brookfield RVTD viscometer gave more accurate data in
less time (8 min. compared to 45 min.) than the
Visco/amylo/Graph. The greater accuracy in measuring shear
rate and temperature made the Brookfield a convenient tool
for modeling. Because of these advantages, the Brookfield
device should be considered as a replacement or companion to

the Brabender instrument.

103

4. Industrial applications of rheological modeling

4.1. Prediction of velocity profile

Since there are already analytical solutions for velocity
profiles of various non-Newtonian models, we wish to write Eq.
3.1 in the same form as these models. Replace the Heinz-
Casson model in Eq. 3.1 with the Herschel-Bulkley model. All
terms except the shear rate term are combined into the

consistency coefficient K:

M y, T, MC. ‘1'. 0) = K (10*1

(4.1.0)
where

[LEV 1-; +b(MC-MC)]
K=e R (T T’) r [1+A“(1—e“‘“’)a][1-B(1—e‘d¢)]

(4.1.1)
A starch-thickened fluid food gelatinizing at temperature T1 is
entering a holding tube at the same temperature. Assume radial
velocity v, and viscous dissipation are negligible. The velocity

profile for a power-law fluid in laminar flow is

Al, .1. n 21:}. 0+;
v,(r)=(fi)“(m)(a " -r n ) (4.2)

Given a constant n, Eq. 4.2 indicates the velocity profile
uniformly decreases as K increases during gelatinization. Since
the flow rate is not varying, this behavior is not possible.

Therefore, the pressure drop (AP/L) must not be constant as the

velocity decreases. The flow rate is

104
1.

AP 3n+1 n
Q=n(fi)nR n (3n+l) (4.3)

There are at least two ways to approach the problem. The first

 

 

way is to assume n constant, divide the pipe into incremental
lengths, solve for the pressure drop over a length using Eq.
4.3, and substitute it into Eq. 4.2 to solve for the velocity
profile over that length. This method should give an "order of
magnitude" estimate.

'The second way is to assume the pressure drop is constant,
and determine how n changes with K, through experiment. This
method is more practical for a process engineer.

For non-ismhermal flow, such as when a starch-thickened
fluid begins to gelatinize in a heat exchanger, one may couple

Eq. 4.2 with the energy equation:

 

g t (3% laT]
arz r (4.4)
At each time , a numerical scheme can be used to calculate the
temperature profile, which is then substituted to find the
velocity profile. For many industrial problems, the most
important criterion is the maximum velocity to calculate the
"worst-case" hold time of the fluid after it has reached a
constant temperature. The assumption of negligible yield stress

and the use of Eq. 4.2 and 3.14 should be sufficient for most

processes.

105

4.2. Rapid parameter estimation

Based on the results of this work, the following experimental
design and estimation procedure is suggested to characterize
viscosity of a gelatinizing fluid (Table 4.1). Given the time
restrictions of industry, we followed the principle of "getting
the most from the least." See Dail et al. (1989) are designing a
tube viscometer to find rheological properties under aseptic
processing conditions.

The interpretation of Table 4.1 is similar to that of Table 3.7a.
The analytical design is in the upper part of Table 4.1, and the
calculation steps are the equations in the lower part. The fifth
column, 11, is measured viscosity. The first correction is made
on this column and is placed in the column to the immediate
left. The step-wise correction proceeds in this manner, with
subsequent corrections moving towards the left. The correCtion
equation for each column is given directly underneath the
column.

The overall procedure is to hold all variables constant except
one, and estimate the parameter for that variable. Then use that
parameter to correct all 11 to a reference value of that variable,
and repeat the entire process again. Therefore, the independent
variables had to be selected such that each varied separately,
with all others constant. Temperature must be varied below the
gelatinization temperature, to keep gelatinization from obscuring
the temperature effect. Therefore, T1, T2, and T3 (all greater
than T8) are used to estimate maximum gelatinized n as a

function of T, but T4 and T5 (below gelatinization) are

106

Table 4.1. Procedure for rapid analysis of parameters using a

tube viscometer.

 

 

 

 

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lliw-Jlo Vim-76 T4 (< T8) T1 ‘1’1a (1)1a
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T2 (> T21 T2
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AEV.

 

encloses the data used to estimate parameters at each Step.
For example, only the enclosed 11., data are used to estimate

aIn tube viscometry, One can expect ‘I’l->°° (fully gelatinized)
and 01 50 (negligible mechanical degradation).

 

 

107

necessary to estimate the Arrhenius temperature effeCt, i.e.
thinning at higher temperatures. The experiments at T1, T4, and
T5 are all gelatinized at T1 (Table 4.1, Tm column), and then
the last twoare cooled in a second heat exchanger to T4 and T5.
The order of analysis is as follows: estimate K and n from
the enclosed 1] data, and correct all n to '71-; estimate AF.v from
enclosed 11?, and correct all 11-7 to T,; eStimate dependence of
maximum viscosity on Tm from enclosed “7,1" and correct 311714”

to Tmr; estimate b from enclosed nirrm’ and correct all n‘r'rTm to

MC,. If b fits poorly , check if it varies with Tm, and fit

accordingly.

108

5. Overall Summary and Conclusions

This work brought together proven knowledge with
hypothesis, and categorized objective, measurable variables
affecting starch solution viscosity. The effects leaSt
investigated (temperature, temperature-time history, and strain
history) in published literature were emphasized, and then
combined with those most investigated (shear rate and moisture
content).

The Morgan et al. (1988) model was applied to 5.5-7.3% and
13.7% native corn starch, and 6% bean starch dispersions,
using two different methods: back extrusion and mixer
viscometry. The independent variables influencing the
dependent variable, viscosity, were shear rate, temperature,
moisture content, temperature-time history, and strain history.
Each of these five was varied individually with all others held
consrant to eStimate the model parameters. The parameters were
used to predict paSting curves in the mixer viscometer, where
temperature, temperature-time history, and Strain history
changed simultaneously. Results showed that, except for
temperature and Strain history parameters, this experimental
procedure yielded accurate results. Therefore, the temperature
parameter was estimated at various temperature-time histories.
Then, the Strain history parameters were reestimated more
accurately by allowing temperature and temperature-time hiStory
to change simultaneously. The final results indicated the model

could apply to systems larger than the Brookfield viscometer

109

and Instron Testing machine, if the independent variables are
ineasurable.

This type of modeling (Eq. 3.2) is instruCtive, because it
brings together present knowledge and takes a risk by assuming
independent effects. The advantage is that when the assumption
is correct, even for only one of several variables, the
experimental time will be decreased to a fraction of that
required for a full factorial design. Therefore, this kind of
modeling can benefit many different kinds of experimental

research.

110

6. Suggestions for future research

6.1. Limitations of the model

The model (Eq. 3.2) is not invalidated if a different form for
any one of the terms is found. Any of the terms can be replaced
with the new form. The model is invalidated when enough
interaCtionS are shown to require factorial experiments. The
value of the model is that it requires a minimum of experiments,
while still explaining the physical process. In this work, the
Strain hiStory parameters were estimated last, because they were
highly dependent on ‘IJ'and T, and at Short times, ‘P'could nOt
be held constant at high T. If any Other parameters were as
dependent as these, the difficulty of estimation would probably
make the model (from the practical standpoint) unacceptable.
The complexity of the model increases if two or more
independent variables change Simultaneously. Future research
on starches should estimate strain history parameters last, when
other variables are changing.

The model does n0t account for ingredients, requiring new
parameters for a new substance. The correlation of the MC
parameter to Tm may cause problems over a large temperature
range, but may also be insignificant at higher temperatures.

The power-law relationship for MC (M=C[MC]b)was valid in
this work and preferred in published literature over the
exponential (Eq. 3.2) form. The correlation of k and A58
prevents finding their absolute value, if one wishes to compare

activation energies in literature. Based on the sensitivity

111

coefficients, the following subStitution to remove correlation
was proposed by J.V. Beck (Mich. St. Univ, Mech. Eng. Dept.,
E. Lansing MI): define a new parameter BI=AEg(k)1/1°0. Solve
for A88 in terms of k and B1, and substitute it into Eq. 3.3. y
Should become virtually insensitive to k over a 100-fold change,
and [31 can be estimated without correlation.

Another drawback of the model is that non-linear regression
is required to estimate the three gelatinization parameters.
Although a poorer fit will result, a can be set to one, and Eq.
3.3 can be manipulated to allow linear regression. This method

can give a "first estimate."

6.2. Other applications

The present study intentionally used a device in which all
five variables-~shear rate, temperature, moisture content,
temperature-time history, and strain history--affected response.
For other situations, one should firSt judge which, if any, of
the variables may be insignificant. For example, b0th history
functions will probably be constant in aseptic food processing.

A more accurate method to estimate gelatinization effects is
the DSC method of Lund and Wirakartakusumah (1984).
However, this method may give parameters useless for process
conditions, because there is no conStant Shearing.

Future research of starches should always include a complete
history of temperature, rather than saying only that a "sample
temperature was within 2 C of the target within 12-20 min."

(Christianson and Bagley, 1983). Then the explanation of

112

viscosity as a function of volume fraction (Christianson and
Bagley, 1983) may be extended to viscosity as a function of
temperature-time history.

Strain history has barely been inveStigated. For pipe flow,
especially in aseptic conditions with long holding tubes, it is
unknown whether strain history effect follows exponential decay
as a function of average or wall shear rate. The exponential
decay may be invalid at the high Shear rates in pipe flow. For
low shear rates, viscosity may be more influenced by the
maximum shear rate than by Strain history. A "Spike" increase
in shear rate could have a greater overall effect than the
constant shearing. Extrusion provides this wide range of shear
rates for doughs. During constant shearing, thermal
degradation effects are difficult to separate from Strain history
effects. Back extrusion is one method to avoid shearing.

The elastic components of Starch solutions may become
important for smaller diameter pipes. Effects of or have n0t
been measured. Theoretically, 0t represents the molecular
weight effect on viscosity, and is a function of shear rate.

In addition to applying the model to a tube viscometer
simulating process conditions (Sec. 4.2), one can apply it to a
steam infusion or steam injecrion process. For any pipe flow,
the minimum residence time can be estimated (Seetion 3.2.1),
and possibly verified using phosphorescent particles. If the
model fits small ranges of independent variables, the ranges
should be extended to include several decades. Because of the

model's generality, it can be applied to all kinds of gums.

113

doughs, and starches, including modified starches, and mixtures
containing starches which gelatinize only at higher
temperatures. In this last case, the activation energy 'of
gelatinization may have to be modeled as a function of
temperature (Dolan et al., 1989). The Strain history term has
already been applied to pumping of time-dependent materials,
such as mayonnaise. If the model is valid for homogeneous
foods, one may attempt to apply it to foods with particulates.
Finally, the procedure of collecting into one expression known
terms, and assuming separable variables, can be applied to many

materials and many variables other than viscosity.

114

7. Binography

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Bakshi, A.S., and Singh, R.P. 1980. Kinetics of water
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Beck, J.V., and Arnold, K.J. 1977. Parameter Estimation in
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Bird, R.B., Stewart, W.E., and Lightfoot, EN. 1960.
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Cervone, N.W., and Harper, J.M. 1978. Viscosity of an
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115

Christianson, D.D., and Bagley, E.B. 1983 Apparent
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Christianson, D.D., and Bagley, E.B. 1984. Yield stresses in
dispersions of swollen, deformable cornstarch granules.
Cereal Chem. 61:500-503.

Colas, B. 1986. Flow behaviour of crosslinked corn starches.
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Collins, E.A., and Bauer, W.H. 1965. Analysis of flow
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Cuevas, R., and Puche, C. 1986. Study of the rheological
behavior of corn dough using the farinograph. Cereal
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Dail, R., and Steffe, J.F. 1989. Tube viscometry for aseptic
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Diosady, L.L., Paton, D., Rosen, N., Rubin, L.J., and
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single-screw extruder: mechano-kinetic breakdown of
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Dolan, K.D.,Steffe, J.F., and Morgan, R.G. 1989. Back
extrusion and Simulation of viscosity development during
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Doublier, J.L. 1981. Rheological studies on starch--f1ow
behavior of wheat starch pastes. Starch/Staerke 33:415-
420.

Doublier, J.L. 1987. A rheological comparison of wheat,
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Draper, N.R., and Smith, H. 1981. Applied Regression
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Eisensmith, Scott P. 1987. Plot-it. Statistics and plotting
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Evans, I.D., and Haisman, D.R. 1979. Rheology of gelatinised
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116

Freeman, J.E., and Verr, W.J. 1972. A rapid procedure for
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Goodrich, J.E., and Porter, R.S. 1967. A rheological
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Harper, J.P., Rhodes, T.T., and Wanninger, L.A. Jr. 1971.
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Harper, J.P., Suter, D.A., Dill, C.W., and Jones, ER, 1978.
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Hickson, D.W., Dill, C.W., Morgan, R.G., Sweat, V.E., Suter,
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117

Lang, E.R., and Rha, C. 1981. Determination of the yield
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Mazurs, E.G., Schoch, J., and Kite, F.E. Graphical analysis of
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119

8. Appendices

8.1. Appendix A. Parameter Estimation Analysis

The model for gelatinization effects on torque (Eqs. 3 and
1.1) Show that two parameters, k and A89, appear as a group.
although AEg is in an exponential and an integral. Therefore,
these parameters may be linearly dependent. The sensitivity

coefficients for Eq. 3.3 are

 

_ 3y _ -k‘l’ ‘1 —k‘l‘
Xi-fi - a(l—e )1n(1—e ) (A.1)
1 iii“)
a— RT(l)
X2 = gy- = ore-W (l-e‘w) r T(t ) e dt
1‘ o (A.2)
[-AEBJ
.. 3y __ 40v 401' “'11)!“ RN)
X3— 3(AES) - kae (l—e ) R 0 e dt (A.3)

Figure A.1 is a plot of these sensitivity coefficients, using
parameters values from Dolan et al. (1988). X1 is obviously
nOt linearly dependent, but X2 and X3 appear to have
proportional magnitudes over the entire range.

To find whether X2 and X3 are linearly dependent, divide Eq.
A.2 by Eq. A.3

120

6.0 .3. .0.. 3:30....000 3.3.3000

_.< 2:30

 

 

 

 

 

 

 

 

 

 

.33 00......

 

 

AC.
0F 0 .0 n N _. 0
1111111111 .....Illl/ o.o
/r/t/.IHIJ/
//,., / //« 3
a 38 . 9.. ....— u 0 //./ r//
Tum-IV ./.a~ //

 

 

05" i

.111... ..lllrslllll- 8. ~H

 

 

 

 

 

 

 

 

 

 

 

 

 

310913111900 Ail/queues

 

 

_ _ R ‘1’
X3 - k -AE - k ‘1’"
‘fl—‘J ‘
RT(t)
j .
0 (A.4)
where
{-1358}
If RT(t)
‘1’": I e dt
0 (A.5)

Since R and k are constants, Eq. A.4 shows that for constant T,
X2 and X3 are linearly dependent, because X2/X3=-RT/k. For
time-varying temperature, k and AEg will be more difficult to
estimate unambiguously as the ratio ‘I’I‘P"approaches a
constant.

Figure A.2 shows ‘F/‘P" versus k‘l’ for a typical thermal
history in this work. Between k‘I’ equal to 0.1 and 2.0, ‘P/‘P"
varied more at the higher AEg (12% decrease) than at the lower
(approximately 5%). This small (5%) change explains why we
cannOt visually detect the linear independence in Figure A. 1.
According to Figure A.2, k and 1588 can be estimated
independently at least between k‘I‘ equal to 0 and l, and we
expect more accuracy with greater values of AEg. According to
Figure A.1, y (portion gelatinized) is mOSt sensitive to all
parameters in the range k‘I’ equal to 0 to 1 (0-37% gelatinized).
At k‘l’ greater than 2 (86% gelatinized), values of y have less
and less influence on the estimates (Figure 3.17). Therefore,
the majority of the data used to estimate gelatinization

parameters were at y less than 80% gelatinized. The values near

122

0000000000 .000: ..0
.0200 058.300 0. 0.0. m:m.0> vanx 00 0.3.. ~.< 00:»;

.00.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0 ... n N _ 0
on.
1.111.111.1011-.. 1 l1
11F 0.010 1/ 11
21-8. 96.». 1 1
1.1.1.1. , 305201.04
_ A r CON
m ..\/ .
_r 1.-. .\ (id... -1111
1-1 . 911...: ...: .11.Lrll 11¢“- -. 1-1.411111311LY1111 .11 // M
2310 _ /\1/
Fill 111111.11- - 4 Ag 8.NHM .1. .. CNN
1 M 3.38.1.2 / .
A
. ..... ...
.1-.. 1.... .
_ 0 .
.
1 SN

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.a/s

123

100% gelatinization were used to calculate the maximum torque

(M... Eq.3.3).

8.2.Appendix B

1 2 4
APPENDIX B

Observed equilibrium torque of
gelatinizing native corn starch to
determine effects of moisture content
and temperature- -time history
(corresponding to Table 3. 2)

 

 

 

 

 

 

 

Impeller Fluid Heating Equil.
speed Tempg starch/ Jacket time Sample torque
(rpm) (C) 3 soln Temp(C) (min) number(s)(Brkf1d units)

95 10 95 30.1
15 96 29.6
100 60 5.5 92 20 97 26.3
25 98 25.3
85 25 99 19.1
30 100 18-3
2 1 6.9
4 2,3 35,368
6 4,5,6 42.7,38.3,4l.1
100 60 6.4 95 8 7,8 41.1,41.3
10 9,10 425,420
12 11,12 44.0,42.0
14 13,14 43. 0,442
16 15-16-.----17460410391
3 18 21. 6
5 22.23,24 35.5,35. 3, 35. 0
7 25,26,27 32.0.38 2,38. 5
100 60 6.4 92 9 28.29 39. 5,38. 5
11 30.31 39. 4,39. 5
15 32,33 40. 4,42. 3
20 34 42.8
25 35 42-3
3 36,37 83,99
3.5 38,39 l7.0,17.5
4 40, 41 27. 4,306
5 42, 43, 44 27. 6,3 .7,32 3
100 60 6.4 88 8 45, 46 33.8,3l.7
12 47,48 34.1,33.3
15 50.51 32.8,34.0
25 49 34-4
3 52,53 2.1,3.2
3.5 54,55 82,69
4 56,57 12.4,11.5
100 60 6.4 85 6 58,59 224,249
9 60,61 293,302
12 62,63 33.3,31.9
15 64 3 .8
25 65,66 34.4,33.6

 

 

Appendix B (Cont' d)

125

 

 

 

 

427 67 137. " "7
2.2 68.69 27.7.28.6
2.5 70,71 36.8,39.3
100 60 7.3 95 4 72,73 52450.7
7 74,75 63.9.65.6
10 76 61.6
15 11 J0-6
2.5 78.79 255,173
3 80.81 42.5,42.0
100 60 7.3 92 7 82,83 58.0,61.3
20 84 64.6
25 85 63-6
3 86 13.3
3.5 87,88 15.2,21.1
100 60 7.3 85 4.5 89.90 34.1,37.9
8 91,92 492,522
12 93 52.5
25 94 51.6

 

 

8.3.Appendix C

126
APPENDIX C

Time and temperature data

8.3.1.Appendix C.l Time and temperature data for each

sample in Appendix B.

Format (for entering computer program in Appendix D):

7

92
76
34.2

T1 t1

Starch concentration ("6" means 5.5%, "7"
means 6-4%. "8" means 7.3%)

indicates new sample
fluid jacket temperature
sample number

torque (Brookfield units (full scale=100),
corresponding to Appendix B)

Temperature (C) time (5)

last data for this sample

start with new sample

2.1
66
66.2
70.3
71.6
72.6
7303
74.6
76.4
76.2
76.6
77.4
77.9
77.6
72.7
66

66
63
3.2
67.3
69.3
71.4
72.6
73.9

76.1
76.9
77.7
76.4
76.9
79.6
79.4
76.7
70.3
66.6

6.2
66.2
66.2
69.6
71.6
72.6
73.7
74.9
76.2
77.3
76.1
76.6
79.3
79.6
79.9
60.6
61.2
61.1
77.3
70.3
64.6

240

70

90
100
110
120
130
140
160
160
170
160
190
200
210

110
120
130
140
160
160
170
160
190

210
220

128

”PHD
0000

00000000
00””
.6

o
00900NQU~6

470

490
610

66.1
66.1
66.2
66.2
64.4
76.3
70.2
63.7

32.6
67.6
69.9
71.3
72.3
73.3
74.7
76.9
76.6
77.6
76.3
76.9
79.4
79.9
60.6
.102
61.6
62.3
62.7

63.3
63.6
63.6

64.1
64.2
64.4
64.6
64.6
64.7
64.7
64.6
64.9

66.1
66.1
66.1
66.2
66.2
66.2
66.3
66.3
66.4
66.4
66.4
66.6
66.6
66.6
66.6

60.1
72.2
66.6

620
690
730

760
760

129

33.6
66.

70.6
72

74.1
76.3
76.2

77.7
76.3
76.6
79.4

60.6
61.2

210

61.3
66.6
66.3
70.7
72.3
73.5
74.7
75.9
76.9
77.7
76.3
76.9
79.6
60.4
60.6
76.3
66.6
63.4
61.1

66
9.9

61.6
66.6

60
60

69.1
71.6
73.2
74.4
76.7
76.9
77.7
76.4

79.6
60.3
61.2

62.7
83.3
63.3
79.9

130

71.6
63.6

27.4
60.2

66.1
70.6
72.6

240
260

50
60

60

66.9
71.3
72.7
73.6
76.3
76.6
77.6
76.3
79.1
79.9
60.7
61.6
62.6
63.1
63.7
3‘02
64.6

66.3
66.6
66.6

66.2
66.3
66.9
61.7
73.6

33.6
66.6
66.9
71.2
72.

74.1
76.3
76.6
77.4
76.1
76.7
79.4
60.2

61.9
62.6
83.3
63.6
64.3
64.7

.303
66.6
66.7
66.9
66.1
66.2
86.3
66.6
66.6
66.6
66.6
66.6
66.9

67

760

60
70
60
90
100
110
120
130
140
160

132

64.4
64.6
66.1
66.3
66.6
66.6
66.9
66.1
66.2
66.3
66.6
66.6
66.7
66.6
66.9
66.9

67

67.1.

67.1
67.2
67.2
67.3
67.3
67.4
67.4
67.4
67.6
67.6
67.6
67.6
67.9
67.9
67.9

62.7
76.6
66.6

60
32.6
67.6
70.1
71.6
72.9
74.2
76.6
76.6
77.3
77.9
76.7
76.6
60.3
61.2
61.9
62.6
63.2
63.7
64.2
64.6
64.9
66.2
66.6
66.7
66.9
66.1

66.2
66.4
66.6
66.6
66.7
66.9
66.9

67.1
67.1
67.2
67.3
67.3
67.4
67.4
67.6
67.6
67.6
67.6
67.7
67.7
67.6
67.6
71.9
66.9

21.6
62.1
66.6
69.6
72.1
73.6
76.2
76.6
77.6
76.3
79.2
60.2
61.4
.206
63.4
63.6
60.6

69.7
66.4
64.7
63.6
63.2
62.7
62.2

92

26.1
6‘01
66.1

71
72.6
74.3
76.9
77.6

76.9

140
160

170
160
190
200
210
220
230

63.2
67.6
71.1
72.6
74.2
76.9

77.9
76.9
79.9
61.1
62.6
63.6
64.3

66.7
66.4
66.9
67.4
67.6
66.1
66.4
66.6
66.6

134

66.1
66.4
66.7
8.09

69.3
69.6
69.6
69.7
69.9
69.9
90.1
90.2
90.2
90.3
90.4
90.4
90.6
90.6
.’07
66.1
76.6
67.4

92

39.6
64.6
66.6
71.6
73.6
76.3
76.7
77.7
76.7
79.6
60.9
62.3
63.6
64.2
64.9
66.7
66.4
66.9
67.4
67.6
66.2
66.6
66.7
66.9
69.1
69.2
69.4
69.6
69.6
69.6
69.9
69.9

90.1
90.2
90.2
90.3
90.4
90.4
9005

90.4
90.6
90.6
90.6
90.7
90.7
90.1
66.7
76.1
66.1

36.6
62.2
67.1
70.3
72.6

76.7
77.1
77.9
76.9
79.9

62.3
63.4
64.1
64.6
66.6
66.1
66.6
67.1
67.6
67.9
66.3
66.6
66.6

69.1
69.3
69.6
69.6
69.7
69.6
69.9

90.1
90.1
90.2
90.2
90.3
90.3
90.4
90.4
90.6
90.6
90.1
66.1

67.3
92

39.4
66

60

135

76.7
77.4
76.2
7902
60.3
61.6
62.7
.30‘

64.7
8502
66.6
66.3
66.6

67.6

66.2

120

67
67.6

66.3
66.6
66.6
69.1
69.3
69.4
69.6
69.6
69.9

9001
90.1
90.3
90.3
90.4
90.4
9005
90.6
90.6
90.6
90.7
90.6
90.6
90.7
90.6
90.7
90.7

360
390

410
420
430
470
600
610
630

490

600
620
630

670

77.7
76.9

61.4
93.7
63.6
64.4
66.1
66.9
66.6
67.1
67.7

66.2

66.7

69.1

69.6

90

137

90.4
90.6
91.2
91.6
91.6
92.1
92.3
92.6

290

310
320
330
340
360
360

60
70
90
100

94.4

96
14
44.2
66
71
73
76
76
77
76

360

600
720

60
90
100
110
120

74.2
76.3

76.4
77.3
76.4
79.3

60.6
61.2
61.4
76.3
70.7

34.1

139

33 310

”4 33° 71:23 133
32.2 330 77 170
7"“ 3‘° 77.9 130
69.5 330 73.3 133
° ° 73.3 200
33 3a 210
’1 30.3 220
‘9': . 31.1 230
‘5 5° 31.3 233
33.1 73 .2 250
7°'2 3° 32.3 230
71" 9° 32.7 270
72.7 100 33 23°
7‘ 11° 33.2 230
75 13° 33.3 300
73.3 133 33.3 31°
7"5 1‘° 33.3 320
77.3 130 ‘3‘, 33°
73.3 130 .‘.1 3.0
79.3 170 ..., 330
3° 1‘° 33.3 330
30.3 130 3‘.‘ 37°
31.2 200 33.3 33°
31.3 210 34., 39°
32.3 220 ,‘.5 ‘00
32.7 230 33.7 41°
'3 "° 33.7 323
33.3 230 33.3 ‘30
33.7 270 33.3 ‘50
33.3 233 3, 330
3‘ 3’° 33 370
"'3 3°° 33.1 330
33.3 310 33.3 ‘90
33.3 320 30“ 50°
33.3 330 73.2 510
"°‘ 3‘° 33.7 320
33.7 330 o o

33.3 330 33

33.3 373 ,3

33.9 330 52.5
3’ 3’° 37.2 70
‘3 ‘°° 33.3 30
33.1 310 70.9 90
33.1 320 71., 10°
33.2 333 73.1 110
"°’ “° 73.2 123
33.3 370 ,5 130
"33 ::: 73.3 130
73.3 130
33“ 5°° 77.3 130
73.2 313 73.3 17°
33.2 323 79.3 130
° ° 30 190
" 33.7 200
9’ 31.2 210
32.2 31.3 220
33.3 70 32.2 230
‘7'7 3° 32.3 230
59" 9° 32.9 230
70.3 100 33.2 230
71.3 110 33" 270
72.9 120 33., 28°
73.9 130 33.3 29°

73.3 130

63.9 300

490

1606
1616
1626
1636

60
70

140

90.6
90.9
91.1
91.1
91.3
91.4
91.4
91.2
66.6
76.6
67.9

92
61.3

66.3
70.4

100
110
120
130

160
160
170
160
190

210
220

60
70

71.7
73
74.3
76.6
76.7
77.6
79.3
60.7
61.6
62.6
63.7
64.6
66.2
66.6
66.2
66.7
67.2
67.6

66.4
66.7
69.1
69.6
69.7
90.3
90.3
90.4
90.6
90.6
90.9

91.1
91.1
91.2
91.2
91.7
64.7
74.1
64.6

92

64.6
67.6
70.3
71.6
73
74.3
76.4
76.6
77.6
79.2
60.4
61.4
62.4
63.1
63.9
64.6
66.1
66.6
66.1
66.6
66.9
67.3
67.7
8801

76.6
60.6
79.6
74.4
66.2

96

26.6
67.2

71.6
72.9
74.4
76.6
7702
76.7
60.6
60.6
76.4
66.7

96

36.6
67.6

460
470

600
610
660
670

1600
1610
1620
1630
1640

70
60

100
110
120
130
140
160

170
160

60

70
71.6
73.1
74.6
76.6
77.2
76.6
60.4
61.6
62.3
79.3
73.6
66.2

96

71
39.3
66.4
70.9
720‘
73.6
76.2
76.6

79.7
61.4
62.6
63.1
60.1
73.7
67.4

62.4
66.7
71.1
72.6
73.9
76.4
76.6
76.2
79.9
61.6
62.6
8309
64.6
66.7
66.3
66.9
67.6
66.1
66.7
69.2

64.6
76.6
69.2

96

60.7
66.3
70.6
72.2
73.6

142

96

66.6
66.3
70.7

72
73.4
74.6
7601
77.4
76.9
60.7

63.1
64.1
64.9
66.6
66.2
66.6
67.4
67.9

66.9
69.3
69.6
90.2
90.6

91.3
91.6
91.9
92.2
92.6
92.7
92.9

93.1
93.2
93.3
93.4
92.7
66.6
76.6
72.1
66.4

61.6
“03
71.1
72.9
74.7
76.3
77.9
60.1
61.6
63.1
64.3
66.4
66.4
67.2

67.6

66.6
69.1
69.7

93.4
93.6
93.7
93.6
93.6
93.9
93.9
93.6
93.6
93.9
93.9
94
94
94.1
94.1
94.2
94.2
94.4
92.9
66.2
79.4

70.3

66
66.4
650‘

19.1
67.4
6’03
70.7
71.6
72.6
73.7
74.7
76.6
76.2
77.6
76.1
7.07
79.1
7’06
79.9
60.3
60.7
61.1
61.4
61.6
62.2
62.6
62.6

63.3
63.6
63.7
63.6

64.1
64.3
64.4
64.6
64.6
8‘06
64.7

143

440
460
460
470
490

610
620

660
690
600

26.3
69.7
71.9
73.6
76.1
7605
77.6
76.6
79.6
60.3
61.1

62.9
63.7
64.4

66.6
66.1
66.6

67.4
.708
66.2
66.6
66.6

69.2
69.4
69.6

83?.

00 0
000030
3033 330
“006”“

000.000
0000~l~30

li‘O‘D‘O‘O‘O‘O|O‘0 1010101030303010
Unoauoa3arar-r-r-

0 0 0 0 0 0 0 0 0 0 0 0
slflhdidtdlfitihihfi

74.4

1200
1210
1220
1230
1240

630

1602
1612
1622
1632
1642

144

66
66.6
67.1
67.6

66.4
66.9
69.4
90.2
90.6

91
9103
91.6
91.9
92.1
92.3
92.4
92.6
92.7
92.6
92.9

93.4

66.6

145

8.3.2.Appendix C.2 Time and temperature data for each
sample used in back extrusion

Order of data is from least to greatest bath temperature: 81,

84,85,91,95,101, and 105 C.
Format:

heating temp TIA. 113,...11135;
number of samples

sample letter time Temperature

new sample number

81

.99 .962

3'6'3'6'0'3'3'0'3'6'6'3'6'6’6'6’6’6’6’6’3’3'6'6'6'6'6'6'6'6'6’6'6’6'3’6’6’6'6'6'6’6'6’6'6’3’0'6'0'0'6'6'6'6'0'0’6'6’3’3'0'0'FHF

100
110
120
130
140
160
160
170
160
190
200
210
220
230
240
260
260
270
260
290
300
310
320
330

146

.906 .73 .62 .429 .164 0 0 0 0 0
116
66.6
67.9
66.9
69.6
71.1
72.3
73.6
74.6
76.4
76.2
76.9
77.4
77.6
76.1
76.6
76.7
79.0
79.2
79.4
79.6
79.6
79.6
79.6
79.6
79.6
79.7
79.7
79.7
79.7
79.7
79.7
79.7
79.7
79.6
79.6
79.6
79.6
79.6
79.6
79.6
79.6
79.9
79.9
79.9
79.9
79.9
79.9
79.9
79.9
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.1
60.1
60.1
60.1
60.1
60.1

DIIUDUIIIIIIIU D’UDDUFUV”0",F’FVFVVV’0’.”””'79D’7”DVVVV”'7’,

1110

147

76.6
76.7
79.0
79.2
79.4
79.6
79.6
79.6
79.6
79.6
79.6
79.7
79.7
79.7
79.7
79.7
79.7
79.7
79.7
79.6
79.6
79.6
79.6
79.6
79.6
79.6
79.6
79.9
79.9
79.9
79.9
79.9
79.9
79.9
79.9
60.0
800°
600°
60.0
60.0
60.0
60.0
60.0
800°
60.1
60.1
60.1
60.1
60.1
60.1
60.1
60.1
60.2
60.2
60.2
60.2
8002
60.2
60.2
60.2
60.3
60.3
60.3
60.3
60.3
60.3

0000 000-
00000000O000000000000000000000000OOOOOOOOOOOOOOOOOOOO

000

630
640
660
660

100
110
120

60.3
79.0
73.6
66.2

66.6
67.9
66.9
69.6
71.1
72.3
73.6
74.6
76.4
76.2
76.9
77.4
77.6
76.1
76.6
76.7
79.0
79.2
79.4
79.6
79.6
779.6
79.6
79.6
79.6
79.7
79.7
79.7
79.7
79.7
79.7
79.7
79.7
79.6
79.6
79.6
79.6
79.6
79.6
79.6
79.6
79.9
79.9
79.9
79.9
79.9
79.9
79.9
79.9
60.0
60.0
60.0
60.0
60.0
76.6
73.6
66.0
39
66.6
67.9
66.9

'6

69.6
71.1
72.3
73.6
74.6
76.4
76.2
76.9
77.4
77.6
76.1
76.6
76.7
79.0
79.2
79.4
79.3
79.6
79.6
79.6
79.6
79.6
79.7
79.7
79.7
79.7
79.7
79.7
79.7
79.7
79.6
79.6
79.6
76.6
73.4
66.6

66.6
67.9
6.09
69.6
71.1
72.3
73.6
74.6
76.4
76.2
76.9
77.4
77.6
76.1
76.6
76.7
79.0
79.2
79.4
79.6
79.6
79.6
79.6
79.6
76.2
73.0
66.4
11

66.6

148

00000

67.9
66.9
69.6
71.1
7203
730’
74.6
76.4
73.9
66.7

66.6
67.9
66.9
67.4

149

89 .99 .975

’7’
3,’fi”"fi’F’F’VFv,’fi"?”""UUF’FUU"F’V’VF’F””"’FUFF,’

680

.934 .837 .741 .591 .466 0 0 0 0 0
62
71.0
71.9
72.9
74.2
75.5
76.7
77.6
78.4
79.1
79.7
80.3
80.7
81.1
81.5
81.8
82.2
82.4
82.6
82.9
83.0
83.2
83.3
83.5
83.6
83.7
83.8
83.9
83.9
84.0
84.1
84.1
84.2
84.2
84.3
84.3
84.3
84.3
84.3
84.3
84.3
84.3
84.3
84.4
84.4
84.4
84.4
84.4
84.4
84.4
84.4
84.4
84.4
84.4
84.4
8‘05
84.5
84.5
84.5
84.5
84.5
83.6
78.0
50
71.0

71.9
72.9
74.2
75.5
76.7
77.6
78.4
79.1
79.7
80.3
80.7
81.1
81.5
81.8
82.2
82.4
82.6
82.9
83.0
83.2
83.3
83.5
83.6
83.7
83.8
83.9
83.9
84.0
84.1
84.1
84.2
84.2
84.3
84.3
84.3
84.3
84.3
84.3
84.3
84.3
84.3
84.4
84.4
84.4
84.4
84.4
84.4
83.6
78.0

71.C
71.9
72.5
74.:
75.!
76.1
77.!
78.4
79.:
79.’
80.;
80..
81.
81.
81.
.82.

uuuuuuuunuuuuuuu GOODUUUDOOUDUUUUDDDUUDOODD OOOOOOOOOOOOOGOOOOOOOO

150

230

100
110
120
130
140
150
160
170
180
190
200
210
220

82.4
82.6
82.9
83.0
83.2
83.

83.5
83.6
83.7
83.8
83.9
83.9
84.0
84.1
84.1
84.2
84.2
84.3
84.3
84.3
83.4
77.8

71.0
71.9
72.9
74.2
75.5
78.7
77.6
78.4
79.1
79.7
80.3
80.7
81.1
81.5
81.8
82.2
82.4
82.6
82.9
83.0
83.2
83.3
83.5
33.6
82.7
77.1

71.0
71.9
72.9
7‘02
75.5
76.7
77.6
78.4
79.1
79.7
80.3
80.7
81.1
81.5
81.8
80.9

0000000 Q‘Vflflflflflflflfl I

120

75.3

71.0
71.9
72.9
74.2
75.5
76.7
77.6
78.4
79.1
79.6
77.5

710°
71.9
72.9
7‘02
74.4
72.3

35'.972 .974 .919 .929

’3’???V,F’VF’FD”’”3’9’7’U',”’FD,UDU,”VV”B’V’FDV’VVF’,’

640

60
70
80
90

59
71.8
72.5
73.2
74.5
7508
77.0
78.1
79.0
79.7
80.3
80.8
81.2
81.6
82.0
820‘
82.7
83.0
83.2
83.4
83.6
83.8
83.9
84.0
84.1
84.3
84.3
84.4
84.4
84.4
84.4
84.5
84.5
84.5
84.6
84.6
84.6
84.7
84.7
84.7
84.7
84.8
84.8
84.8
84.9
84.9
84.9
85.0
85.0
85.0
85.0
85.1
85.1
85.1
85.2
8502
84.2
79.8
72.5
64.3
52
71.8
72.5
7302
74.5

151

OOOOOOOOOOOOOOOOO ...-...IIIII-......UUUUIIIII-I-UIUIIUIIIUIIIUIUI

100
110
120

140
150
160
170
180
190
200
210
220
230

250
260
270
280
290
300
310
320
330

350
360
370
380
390
400
410
420
430

450
460
470
480
490
500
510
520
530
540
550
560
570

60
70
80

100
110
120
130
140
150
160
170
180
190
200
210
220

75.8
77.0
78.1
79.0
79.7
80.3
80.8
8102
81.6
82.0
82.4
82.7
83.0
83.2
830‘
83.6
3308
83.9
84.0
84.1
84.3
84.3
84.4
84.4
84.4
84.4
84.5
84.5
84.5
84.6
84.6
84.6
84.7
84.7
84.7
84.7
84.8
84.8
84.8
8‘09
84.9
84.9
85.0
85.0
85.0
84.2
79.8
72.5

71.8
72.5
7302
74.5
75.8
77.0
78.1
79.0
79.7
80.3
80.8
81.2
8106
82.0
82.4
82.7
83.0

00000000000000000

OUUUOUUOUUOUUDOUODOODUU

230

250
260
270
280
290
300
310
320
330
340
350
360
370
380
390

83.2
830‘
83.6
83.8
83.9
84.0
84.1
84.3
8‘03
84.4
84.4
84.4
84.4
84.5
84.2
79.8
72.5

71.8
72.5
7302
74.5
75.8
77.0
78.1
79.0
79.7
8003
80.8
81.2
81.6
82.0
82.4
82.7
83.0
83.2
83.4
83.0
78.0
70.0

152

9| ,99 .954 .77 .575 .482 .3 - 110 65.7
21
50 59.1
60 70.2
70 70.8
80 71.9
90 74.0
100 75.4
110 77.0
120 78.7
130 80.2
140 81.4
150 82.6
160 83.6
170 84.6
180 85.4
190 86.0
200 86.7
210 87.3
220 85.5
230 84.5
240 78.0
250 68.9
15
50 59.1
60 70.2
70 70.8
80 71.9
90 74.0
100 75.4
110 77.0
78.7
130 80.2
140 81.4
150 82.6
160 80.8
170 79.9
180 73.3
190 64.2
10
50 59.1
60 70.2
70 70.8
80 71.9
90 74.0
100 75.4
110 77.0
120 76.5
130 73.1
140 66.9
8
50 59.1
60 70.2
70 70.8
80 71.9
90 74.0
100 74.5
110 72.6
120 67.8
7
50 59.1
60 70.2
70 70.8
80 71.9
90 72.5
100 68.8

50 59.1
70.2
70 70.8
80 68.8
90 64.0

00000 0
O
0

000000 00000000 0000000000 000000000000000 ’0’?"””””"””'
...
N
0

99 .9 .96 .87 .73 .7 .6 .44 .27

00000000000000000 000000000000000000000 0’000000’3000’533”’577’

50

60

70

80

90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270

210

24
68.6
7306
74.1
75.1
76.6
78.1
79.5
80.8
81.9
82.9
83.8
84.5
85.2
85.8
86.3
86.8
87.2
87.6
87.9
88.2
87.7
83.9
76.4
67.6
21
6.06
730‘
74.1
75.1
76.6
78.1
79.5

80.8'
8109

82.9
83.8
84.5
85.2
85.8
8603
86.8
87.2
85.7
81.9
74.4
65.6
20

68.6
73.6
74.1
75.1
76.6
7801
79.5
80.8
81.9
82.9
83.8
84.5
85.2
85.8
86.3
86.8
86.3

153

000

000000000 00000000000 00000000000000

000000 0000000

OHHHH

220
230
240

82.5
75.0
65.2

68.6
73.6
74.1
75.1
76.6
78.1
79.5
80.8
81.9
82.9
83.8
83.3
79.4
72.0

68.6
73.6
74.1
75.1
76.6
78.1
79.5
80.8
8003
76.5
69.0

6.06
7306
74.1
75.1
76.6
78.1
78.3
74.5
67.0

68.6
73.6
74.1
75.1
76.6
76.1
72.3

68.6
73.6
74.1
75.1
74.6
70.8

68.6
73.6
73.1
69.3

154

35 .99 .819 .818 .705 0 0 0 0 0 0 0 0

15

50 71.0

60 72.9

70 74.5

80 76.8

90 79.1

100 81.1

110 82.8

120 84.3

130 85.6

140 86.8

150 87.7

160 88.6

170 87.0

81.2

190 72.7
13

50 71.0

60 72.9

70 74.5

80 76.8

90 79.1

100 81.1

110 82.8

120 84.3

130 85.6

140- 86.8

150 85.5

160 79.4

170 70.9
10

50 71.0

60 72.9

70 74.5

80 76.8

90 79.1

100 81.1

110 82.8

81.5

130 75.4

140 66.9
5

50 71.0

60 72.9

70 74.5

80 76.8

90 69.1

0000000000000 0,00,00,0000000
..o
0
0

000000 0000000000
*5
N
O

155

10' 99 .796 .602 .36 .16 0 0 0 0 0 0 0

11

40 58.8

50 70.8

60 73.3

70 75.7

80 78.6

81.5

100 83.8

110 85.9

120 87.8

130 80.5

140 67.1
8

40 58.8

50 70.8

60 73.3

70 75.7

' 80 78.6

90 81.5

77.0

110 68.4
5

40 58.8

50 70.8

60 73.3

70 75.7

80 71.2
4

40 58.8

70.8

60 73.3

70 68.8
3

70.8

60 66.3

070,000,???
0
0

00000 00000000
...
0
0

156

:05 965 .882 .655 .535 0 0 0 0 0 0 0 0
8
50 72.0
60 75.5
70 79.0
80 82.5
85.5
100 86.7
110 79.1
120 71.0
7
50 72.0
60 75.5
70 79.0
80 82.5
90 83.6
100 78.0
110 67.9
5
50 72.0
75.5
70 79.0
80 80.0
90 72.5
4
50 72.0
60 75.5
70 76.7
80 69.1

vvrryvvv
3

00000 00000 0000000
0
0

157
APPENDIX D

8.4.Appendix D. Computer programs (written by K.
Dolan unless otherwise indicated) to

estimate k, a, and AEg, from normalized
torque versus temperature-time history
in Appendices B and C

Figure D.1 Flow diagram of computer programs

Must compile all programs with REAL*8 using "fortran
filename.for/g_floating" to allow numbers greater than 1028

On VAX, use LINK
MINUVB,INITB,SETUPB,FPSIUV,NLINB,MODSENSUV,IMSL/
LIB

 

mm

Call INITB
Call UVMIF

Write minimum sum of squares and corresponding value of AEg
END

 

 

 

 

INITE

Read time and Temperature

Calculate calibrated Temperature (TT,TM,TB) along small
sample adapter axis

END

 

 

UVMIF IMSL generic minimization routine, given f(x) over a
range

Repeat

Use RNLINA(AE8)
Until a tolerance (x<tol) is met
END

 

 

 

RNLINAmEg)

Call SETUPUV(AEg)

Estimate k and a for a given AEg , by using MODSENSUV(AEg)
END

 

 

 

158

 

SETUPUV (AE8) sets up ‘1’
‘1‘: FPSIZ(AE8)

Scale ‘1’ between 0 and 11.0
END

 

 

MODSENSUV(AEg)

Calculate sensitivity coefficients
END

 

 

'F'fiSIUV

Calculate ‘1’
END

 

 

 

159

EMMMXE“

INTEGER MAXFNpNOUT

REAL BOUNDIFIFX,STEP,X,XACC.XGUESS,E
EXTERNAL RNLINA,UMACH,UVMIF,INIT

CALL INIT
OPEN(70,FILE-'E',STATU5-'UNKNOWN')
READ(70,*)XGUESS,STEP,XACC

CLOSE (70)

BOUND-20.0

MAXFN-SO

CALL UVMIF(RNLINA,XGUESS,STEP,BOUND,XACC,MAXFN,E)
WRITE(7,*)'IER-',IER
Fx-RNLINA(E)
CALL UMACH(2,NOUT)
WRITE(7,99999)E*83130.,FX
99999 FORMAT(' THE minimum is at ',7x,f14.2,//,' The function '
& , 'value is ', £14.11)
end

16()

"INITb" for UVMIN AND NLIBO with 2 parameters AND ALL DATA(6.7,8%)

BATCH JOB!

THIS PROGRAM READS TEMPERATURES AND TIMES FROM ONE FILE,

THEN CALCULATES AND STORES TT(),TM(), AND TB(), AND TIME()

FOR PSI SCALING. MUST SET UP Tt DATA FILE WITH LAST VALUE

HAVING GREATEST PSI

INPUT: TEMP, TIME, Y(SAMPLE)

OUTPUT: TEMP SPLIT INTO TT, TM, TB; TIME. AND Y(MAX SAMPLE TEMP),

SENT TO FUNCTION FPSI

SUBROUTINE INIT

DIMENSION TEMP(lOO, 60)

DIMENSION PS(S, 5), T(loo, 5), y(100), SIG2(100),B(S), Z(5), A(S),
S P(S, 5), 33(5), VINV(S, 5)

DOUBLE PRECISION P,PS,T

PARAMETER (UR-83130.)

COMMON T,Z,BS,I,ETA,PS,P,B,A,y,SIG2,MODL,VINV,NP

COMMON/TITEM/ TT(100,60),TM(100,60),TB(100,60),TIME(100,60)

COMMON/NUM/ NPT(100),ISAM

COMMON/SS/SYP,SLEAST

COMMON/FILE/DFILE

COMMON/CORR/TORQ(100),CTORQ(100),NUMSAM(100),TMAX(100)

CHARACTER *14 FNMR.FNMW.DFILE,OUTFIL

INTEGER TARGT(100),CONC(100)

REAL ETAINF.M

BO(TMP)-.511

Ol')(30(4)(10tl(-)

C INITIALIZE SUM.OF SQUARES. RUNNING VALUE OF LEAST SS
SYP--1
SLEAST-IOO

00

WRITE(*,*)’ENTER FILENAME FOR READING Tt DATA'
OPEN(45,FILE-'NAME',STATU3-'UNXNOWN')
READ(4S,10)FNMR

C ***OPEN INPUT AND OUTPUT FOR BECK

C WRITE(*,*)‘ENTER THE NAME OF THE INPUT BECK DATA FILE'
READ(45,'(A40)') DFILE

WRITE(*,*)'ENTER THE NAME OF THE OUTPUT FILE'
READ(45,'(A40)') OUTFIL
OPEN(7,FILE'OUTFIL,STATU5-'NEW')

t")

CLOSE(4S)
lO FORMAT(A)
OPEN(lO,FILE-FNMR.STATU5-'UNKNOWN')

0

***L IS SAMPLE #, N Is 3 OF TIME INTERVALS WITHIN A SAMPLE

L-l
N-1
NUMSAM(1)-O
C
READ(10.*)IFLAG
READ(10,*)TEMP(L,N),TIME(L,N)
C WHILE TEMP IS > 0 DO READ TEMP AND TIME
30 IF(TEMP(L.N).GE.O)THEN
C ***NEW CONCENTRATION? STORE IT UNTIL CALCULATIONS PERFORMED

IF(INT(TEMP(L,N)).EQ. 0.AND.TIME(L,N).GT.O)THEN
IFLAG'INT(TIME(L,N))

ENDIF

***NEW SAMPLE?
IF(INT(TEMP(L,N)).EQ. O)THEN

()

()

O

00

000

O O()OUO ('10
t
O

161

***SAMPLE# > 1?
IF(NUMSAM(L).GE.1)THEN
***MAX # 0F PTS FOR LAST SAMPLE
NPT(L)-N-l

***MAX TEMP. FOR THIS SAMPLE?
J-NPT(L)
IF(TEMP(L,J).GE.TEMP(L,J-l))THEN

TMAX(L)-TEMP(L,J)

***SLOPE FOR RAPID COOLING CURVE
MP.0771*TMAX(L)-5.7
TEMP(L,J+1)'TMAX(L)-M*10
TEMP(L,J+2)-TMAX(L)-M*20
TIME(L,J+l)-TIME(L,J)+10
TIME(L,J+2)'TIME(L,J)+20
NPT(L)-J+2

ELSE
J-J-l
GO TO 60
ENDIF

***INCREASE SAMPLE #
L-L+1
ENDIP
***END 'SAMPLE >1' IF LOOP
READ(10,*)TARGT(L)
READ(10.*)NUMSAM(L)
READ(10.*)TORO(L)
CONC(L)-IPLAG
N-o
ENDIP
***END 'EOUAL TO ZERO' IF LOOP
N-N+1 .
READ(10.*)TEMP(L,N),TIME(L,N)
GO TO 30
END IF
***END READING ALL SAMPLES LOOP
*rrqu # OP PTS POR LAST SAMPLE
do 150 i-1,L-1 -
WRITE(*,*)'I-'.I,' NPT-',NPT(I)
CONTINUE

***CALCULATE TT,TM.TB ACCORDING TO TARGET TEMP.
ISAM IS NUMBER OP SAMPLES
ISAM-L-l
WRITE“, *) 'ISAM" , ISAM
DO 50 I-1.ISAM
IHEAT-NPT(I)-2
***COOLING TEMPS NOT CALIBRATED
DO 65 x-IREAT+1,IREAT+2
TB(I,K)-TEMP(I,K)+273.15
TM(I,K)-TEMP(I,K)+273.15
TT(I,K)-TEMP(I,K)+273.15
CONTINUE
IF(TARGT(I).EQ.9S.AND.CONC(I).EQ.7)THEN
DO 70 R-1,IREAT
TB(I,K)-.819*TEMP(I,K)+18.66+273.15-1.45
TM(I,K)-.981*TEMP(I,K)+2.83+ 273.15-1.45
TT(I,K)-.990*TEMP(I,K)+l.44+ 273.15—1.4s

80

90

110

t'F)O ()0
L.)

J

162

CONTINUE
TMAX(I)-(TB(I,IHEAT)+2*TM(I,IHEAT)+TT(I,IHEAT))/4-273.lS
ELSEIP<TARGT(I).Eo.95)THEN
DO 80 x-1,IHEAT
TB(I,K)-.834*(TEMP(I,K)-l.45)+16.72+273.lS
TMtI,K)-.818r(TEMP(I,x)-1.45)+I9.9 +273.15
TT(I,K)-.725*(TEMP(I,K)-1.4S)+27.86+273.15
CONTINUE
TMAX(I)-(TE(I,IHEAT)+2*TM(I,IHEAT)+TT(I,TREAT))/4-273.15
ELSEIF<TARGT(I).EQ.92)THEN
DO 90 K-1.IHEAT
TB(I,K)-.836*(TEMP(I,K)-l.45)+16.47+273.15
TM(I,K)-.823*(TEMP(I,K)-1.4S)+19.1 +273.15
TT(I,K)-.852*(TEMP(I,K)-1.4S)+l6.36+273.l$
CONTINUE
TMAX(I)-(TB(I,IHEAT)+2*TM(I,IHEAT)+TT(I,IHEAT))/4-273.15
ELSEIP(TARGT(I).EQ.88)THEN
DO 100 K-1.IHEAT
TB(I,K)-.807*(TEMP(I,K)-1.4S)+18.43+273.15
TM(I,K)-.865*(TEMP(I,K)-1.4S)+14.7S+273.15
TT(I,K)-.886*(TEMP(I,K)-1.45)+12.83+273.15
CONTINUE
TMAX(I)'(TB(I.IHEAT)+2*TM(I,IHEAT)+TT(I,IHEAT))/4-273.lS
ELSEIF(TARGT(I).EQ.85)THEN
DO 110 x-1.IHEAT
TB(I,K)-.871*(TEMP(I,K)-1.45)+12.01+273.15
TM(I,K)-.784*(TEMP(I,x)-1.45)+19.96+273.15
TT<I,x)-.738*(TEMP(I,K)—1.4S)+23.47+273.15
CONTINUE ,
TMAX(I)-(TB(I,IHEAT)+2*TM(I,IHEAT)+TT(I,IHEAT))/4-273.15

WRITE(7,*)'YOU BOOFED ON ME! NO MATCH FOR TARGET TEMPS!’
STOP
ENDIF

CONTINUE

DO 137 L'1,ISAM
TC'TMAX(L)
***MAX VISCOSITY FOR THIS TEMP. AND CONC.
IF(CONC(L).EQ.6)THEN
CTORQ(L)-TORQ(L)*EXP(BQ(TC)*.9)
ELSEIF(CONC(L).EQ.7)THEN
CTORQ(L)-TORQ(L)
ELSEIF(CONC(L).EQ.8)THEN
CTORQ(L)-TORQ(L)*EXP(BQ(TC)*-.9)
ENDIF
ETAINF-1.3S9*EXP(.036716*TC)
WRITE(*,*)'ETA INFINITY-',ETAINF,'TMAX",TC
***SCALE Y ACCORDING TO MAX VISC.
Y(L)-(CTORQ(L)-.2)/(ETAINF-.2)
WRITE(*,*)'SAMPLE #-',NUMSAM(L),' TOTAL-',l,' Y-',Y(L)

CONTINUE
CLOSE(lO)

RETURN
END

1633

"SETUPUV" SUEROUTINE for UVMIN WITHIN NLIBO.FOR FOR 2 parameters

THIS PROGRAM KEEPS TRACK OF SUM OF SQUARES AND E. CALCULATES

AND SCALES PSI AFTER NEW E INPUT

FOR PSI SCALING. MUST SET UP Tt DATA FILE WITH LAST VALUE

HAVING GREATEST PSI

INPUT: TT,TM.TB.E.TIME

OUTPUT: SCALED PSI

SUBROUTINE SETUP(E)

REAL*8 FPSI.T.Q.SCLB.J

DIMENSION TEMP(100.60)

DIMENSION PS(S.S).T(100.S).y(lOO).SIG2(lOO),B(S),Z(S),A(S),
S P(5,5).BS(S),VINV(5,S)

DOUBLE PRECISION P.PS

PARAMETER (UR-83130.)

COMMON T,Z,BS.I.ETA,PS.P,B.A.y,SIG2.MODL.VINV,NP

COMMON/TITEM/ TT(100,60),TM(100,60),TB(100,60),TIME(100.60)

COMMON/NUM/ NPT(100),ISAM

COMMON/SS/SYP.SLEAST

OO()(')()()0

C ***WRITE THE SUM OF SQUARES OF RESIDUALS/SIGMA
WRITE(7.*)'OLD SYP-',OLDSYP.' NEW SYP-',SYP
WRITE(7,*)'OLD/NEW5'.OLDSYP/SYP.'LEAST/NEW".SLEAST/SYP
OLDSYP-SYP

GO

***KEEP RUNNING VALUE OF LEAST SS
IF(SYP.LT.SLEAST.AND.SYP.GT.0)THEN
SLEAST-SYP
END IF

***WRITE WHAT E IS
WRITE(7,'(/" E-",Fll.2." J/mol"," E/RlE4-",F8.3)')E*UR.E

***CALCULATE NEW PSI
DO 50 I-l.ISAM

T(I.1)-FPSI(E)
SO CONTINUE

0000

c ***SCALE ALL PSI'S
1'0
q-T(ISAM.1)
IP(q.GT.1.1 .AND.q .LE. 11) THEN
RETURN

ELSEIF(q.LE.1.1)THEN
300 IF(q.LE.1.1)THEN
q'q*10
j-int(j+l.l)
GO TO 300
ENDIF
ELSEIF(q.GT.11)THEN
400 IF(q.GT.11)THEN
q'q/ 10
j-int(j-1.l)
GO TO 400
ENDIF
ENDIF
sclb-10.0**J
DO 500 I-l.ISAM
T(I,l)-T(I,1)*sclb
500 CONTINUE

164

WRITE(7, *) 'SCALING FACTOR- 10 TO THE' . 27+J. ' POWER’
RETURN
END

165

c "FPSI2.FOR” for NLIBO.FOR with 2 parameters
c calculates psi by trapezoidal rule integration over Space
c and then time for small sample adapter
REAL*8 FUNCTION FPSI(E)
REAL*8 dpsi(60),ar(60),art.arm.arb.ul.u2,u3,t
REAL top.mid.bot
DIMENSION ps(5,5),t(100,5),Y(lOO),Si92(lOO),b(S),
5 2(5).a(5). P(5.5).b8(5).VinV(5.5)
DOUBLE PRECISION p.ps
PARAMETER (R-8.313.scl-le27,sc12-le4)
COMMON t.z,bs.i,eta,ps.p,b,a,y,sigz.modl,vinv,np
COMMON/TITEM/ tt(lOO,60),tm(lOO,60),tb(100.60), time(100,60)
COMMON/NUM/ NPT(100),ISAM
c
c initialize
FPSI-O
n-l
nmax-npt(I)
c ***calculate psi for ETA
c
c ***WHILE 'this sample' DO 'integration'
DO 100 n-1.nmax
top-tt(i,n)
mid-tm(i,n)
bot-tb(i.n)
C ***don't calculate exponents less than -88
ul-E*sc12/top
u2-E*scl2/mid
u3-E*sc12/bot
art-O
armPO
arb-O
c
c ***integrate over DISTANCE (axially)
c ***individual integrands

art-top*exp(-ul)*scl
armrmid*exp(-u2)*scl
arb-bot*exp(-u3)*scl
c ***weight each integrand
ar(n)-(art+2*arm+arb)/4.0
write(*,*)'ar-',ar(n)/scl .
write(*,*)'top-',top-273.lS.'mid-',mid-273.ls,'bot-',bot-273.15

0000

***integrate over TIME
IF(n.gt.l)THEN
dpsi(n)-(time(i,n)-time(i,n-l))*(ar(n)+ar(n-l))*0.5
FPSI-FPSI+dpsi(n)
ENDIF
100 CONTINUE

RETURN '
END

1636

FUNCTION RNLINA(E)
CCCCCCCCC PROGRAM DESCRIPTION CCCCC
C
C WRITTEN BY JAMES V. BECK
C NOVEMBER 1985

Ciit********i*************ii*******tit!*****t***********tt*******tc

000

‘0

L C
CVCCCCCCCC VARIABLE IDENTIFICATION CCCCCCCC
c C
c C
CtttIttt*fiti'i********t*******fi***********t*************rtttttkirtc
c C
L

CDCCCCCCCC DIMENSION BLOCK BLOCK 0000
C C
c C

REAL Y(100).SIG2(IOO),B(5).Z(S),A(5),BS(5),VINV(5,5)

REAL BSS(5),CG(5),BSV(S).R(5.5).TORQ(100).CTORQ(IOO),TMAX(IOO)
REAL*8 PS(5,5),P(5.S).PSV(S.5).SUMP.PSQl.PSQZ.PSQ.T(lOO.5)
INTEGER NUMSAM(100)

CHARACTER*40 DFILE.OUTFIL.GFILE

C C
Ct*1tit*tttttittitttttttttttttitttitit1112**fitttitttfitttrttitirtttc
C C
COCCCCCCCC COMMON BLOCK BLOCK 0100
C C

COMMON T.Z.BS.I.ETA.PS.P.B.A.Y.SIG2.MODL,VINV,NP

COMMON/SS/SYP.SLEAST

COMMON/FILE/DFILE

COMMON/CORR/TORQ.CTORO.NUMSAM.TMAX
C C
C C
Cttwttwtitwttitttiit**t****************Itt****t**t*t**t*witttiiiiic
C C
CACCCCCCCC DATA BLOCK BLOCK 0200
C C

DATA EPS.EPSS.IIN.IOUT/l.0E-30.0.0001.S.7/
C C
Ctittittiitifititittt***************itittitttitttittttttttwtwtttttRC
C C
CICCCCCCCC INITIALIZATION BLOCK BLOCK 0400
C CHANGE DOLAN 88/9/12 C

OPEN(S.FILE-DFILE.STATUS-'OLD') C

C .
Ctttitwtittitttt*ttttt***ttt****twwritttttt*ttttirrrttttwtwtrrtttwc
C C
CPCCCCCCCC PROCESS BLOCK BLOCK 050g
C
C --- START INPUT

C 88/8/2 addition Dolan
CALL SETUP(E)
200 READ(S,*) N.NP.NT.ITMAX.MODL.IOPT.IPRINT

60 FORMAT(IX.'NO DATA POINTS IN FILE 222')
IP(N.LE.0) THEN
WRITE(7,60)
STOP
ENDIP
3 WRITE(*.'(/.9x."N",8x.''MP'',8x,"NT'',Sx,"ITMAx",Sx,"MODE
C +L".6x."IOPT",4x."IPRINT")')
C WRITE(*,’(7IIO.4F10.4)') N,NP,NT,ITMAX,MODL,IOPT,IPRINT

WRITE(7.'(/.9x,"N",8x,"MP",3x,"NT",sx,"ITMAx".5x,"MODE

C --—
C ---
C --—

ISO

10

313

0000 0m

19

99
C ---

167

+L".6X."IOPT",4X,"IPRINT")')

WRITE(7.'(7I10.4FIO.4)') N.NP,NT.ITMAX.MODL.IOPT.IPRINT

IF IOPT-O THEN ON THE ZED AND SUCEEDING STACKED CASES. THE DATA IS
NOT REPRINTED.

IF IPRINT-l. EXTRA PRINT OUT OF ETA. RESIDUALS 8(1),... ARE GIVEN.
READ(5.*)(B(I).I-1.NP)

WRITE(7,'(10X."B(".Il.") - ",F16.5)’) (I.B(I),I-1,NP)

DO 150 Jl-ZIS
ES(J1) ' 0
CONTINUE

IF(IOPT.LE.0) THEN
WRITE(7.'(/.6X."J".5X."SAMP 3''.6X"Y(J)".3X.''TORQUE(J)H
+.3X."CORR.TORQ(J)".5X."T(J.l)".4X."TMAX")')
DO 10 IZ'l.N
READ(5.*)J.SIG2(J)
WRITE(7.'(2IB.6F12.5)') J.NUMSAM(J).Y(J).TORQ(J).CTORQ(J)
+ .(T(J.KT).KT‘1.NT).TMAX(J)
SIG2(J) ' SIG2(J)*SIG2(J)
CONTINUE
END IF

DO 2 IP-lINP
DO 2 KP'1.NP
PS(KP.IP) - 0
P(KP.IP) ' 0
CONTINUE
WRITE(7.'(/,5X."P(1.KP)".9X."P(2,KP)".9X."P(3,KP)'..9X.
+"P(4.KP)".9X."P(5.KP)")')
DO 6 IP-IINP
READ(5.*)(PS(IP.KP).KP'1.NP)
WRITE(7.'(5D16.5)') (PS(IP.KP).KP‘1.NP)
CONTINUE
ADDITION DOLAN 38/9/8
CLOSE(5)

ADD BLANK CARD AFTER LAST INPUT CARD

r-‘END INPUT

DO 18 IL'1.NP
BS(IL)'B(IL)
CG(IL) ' 0
CONTINUE
DO 19 IP-erP
DO 19 KP'1.NP
P(KP.IP) - PS(KP.IP)
CONTINUE
I ' 0
MAX ' 0

MAX - MAX + l
START BASIC LOOP GIVES B(I) AND SY

SY - 0

DO 100 I3-1.N
I ' I3
CALL SENS(E)
CALL MODEL(E)
RISD - Y(I)-ETA

168

SY ' SY + RISD*RISD/SIG2(I)
SUM ' 0
DO 20 K-1.NP
DO 20 L‘1.NP
SUM.' SUM.+ Z(L)*P(K,L)*Z(K)
20 CONTINUE
DELTA ' SIGZ(I) + SUM
DO 29 JJ-1.NP
A(JJ) - 0
29 CONTINUE
DO 30 JA-1.NP
DO 30 KA-IINP
A(JA) - A(JA) + Z(KA)*P(JA.KA)
30 CONTINUE
CS - 0
DO 40 JC'1.NP
CS - CS + Z(JC)*(B(JC)-BS(JC))
CG(JC) ' CG(JC) + Z(JC)*RISD/SIG2(I)
40 CONTINUE
C - Y(I) ' CS ' ETA
DO 50 IB'1.NP
B(IB) ' B(IB) + (A(IB)*C)/DELTA
50 CONTINUE
DO 41 ISV51.NP
DO 41 JSV51.NP
PSVIJSV.ISV) ' P(JSV.ISV)
41 CONTINUE
DO 52 IV51.NP
DO 52 IU-IV.NP
SUM? ' 0
DO 51 KP'1.NP
DO 51 JP-1.NP
IP(KP'IV.EQ.0.0R.JP‘IU.EQ.0) GOTO 51
P501 ‘ PSV!KP.JP)*PSVIIU.IV)
P502 ' PSVKIU.KP)*PSVTIV.JP)
PSQ ’ PSQI ‘ PSQZ
IF(DABS(PSQI)+DABS(PSQZ).LT.l.E-15) THEN
RP - PSQ * l.E15
ELSE
RP - PSQ / (DABS(PSQI)+DABS(PSQZ))
END IF
RP - ABS(RP)
RPP ' RP ‘ 1.0E‘12
IF(RPP.LE.0.0) THEN
PSQ ' 0
END IF
SUMP ' SUMP + Z(JP)*Z(KP)*PSQ
51 CONTINUE
P(IU.IV) - (PSVKIU.IV)*SIG2(I)+SUMP)/DELTA
52 CONTINUE
DO 53 IV‘2.NP
IVM.' IV - I
DO 53 IU - 1.IVM.
P(IU.IV)' P(IV.IU)
53 CONTINUE
IF(IPRINT.GT.0) THEN
IF(I.EQ.1) THEN
WRITE(7,'(//,8X,"I",9X,"ETA",8X,"RESIDUALS".6X."B(
gl)".llX."B(2)".llX."B(3)",llX."B(4)",llX."3(5)")')
END IF

169

WRITE(7,'(I10,7E15.7)')I,ETA,RISD,(B(JC),JC'1,NP)

END IF
100 CONTINUE
C “- END BASIC LOOP, GIVES B(I) AND SY
C “' START BOX-KANEMASU MODIFICATION
C
C START BOX-KANEMASU MODIFICATION

103 SS‘SY/2.0
105 D0 210 IBS'1,NP
B(IBS)- BSV(IBS)
210 CONTINUE
WRITE(IOUT,212)
212 FORMAT(7X,'USE BSV(IBS)')
GOTO 211
104 CONTINUE
DO 102 IBS‘1,NP
BSS(IBS)- BS(IBS)
102 CONTINUE
ALPHA- 2.0
AA- 1.1
110 ALPHA- ALPHA/2.0
DO 116 IBs-1,NP
BS(IBS)' BSS(IBS) + ALPHA*( B(IBS)-BSS(IBS) )
BSV(IBS)- BS(IBS)
116 CONTINUE
INDEx-O
G- 0.0
DO 115 IP-leP
DELB- BS(IP)‘BSS(IP)
G- G + DELB*CG(IP)
RATIO- DELB/( BSS(IP)+EPS )
RATIO- ABS(RATIO)
IF(RATIO-EPSS)113,113,114

113 INDEX- INDEX+1
WRITE(IOUT,314)
314 FORMAT(7X,'MAX',8X,'NP',5X,'INDEX',8X.'IP')
WRITE(7,'(7I10,4F10.4)') MAX,NP,INDEX,IP
114 CONTINUE
C WRITE(7,122) I,Y(I),ETA.RISD,Z(IP),XYP,DELB,SIG2(I)
115 CONTINUE
SYP- 0.0
DO 117 I3-le
I-I3

CALL MODEL(E)
RISD” Y(I)'ETA
SYP- SYP + RISD’RISD/SIG2(I)
117 CONTINUE
IF(NP‘INDEX)106,106,107
105 H'1.0
GOTO 132
107 CONTINUE
SYN- SYP*0.999
1C9 WRITE(7,108) ALPHA,SYP,SY
108 FORMAT(3X,'ALPHA TOO SMALL,ALPHA",F12.6r2X,'SYP",515.6,2Xr
)f'SY',E15.5)
WRITE(7,1001)

170

MPITMAX

IF(MAx-M)99,99,101
101 CONTINUE

IF(IPRINT)133,133,134
133 IPRINT-IPRINT+1

GOTO 99
134 CONTINUE

RNLINA-SYP

RETURN
1000 CONTINUE
C

CLOSE(IIN)

CLOSE(IOUT)
C C
CtttxI*******tt***********t*t**t*****tttttttttttrttttt*twtiwrttrtic
C C
CECCCCCCCC ERROR MESSAGES BLOCK 0900
C C
C ' C
Ctitwrttitiittitttttfli*tittttttttttiiittttttttrttittttttttttiwttitc
C C
CFCCCCCCCC FORMAT STATEMENTS BLOCK 9000
C C
C C

Citrt*ti*********tttittrttttttttt*ttttttttttrtttttt*tttritt*ttttttc
C

STOP

END

171

1001 FORMAT(le.'Z(l)',12x,'Z(2)',12x,'Z(3)',12x,'z(4)',12x,'Z(S)')
1002 FORMAT(6E16.5)
DO 1003 J'le
CALL SENS(E)
WRITE(7,1002) (Z(I),I'1,NP)
1003 CONTINUE
C CHANGE DOLAN 88/9/12
RNLINA'SYP
RETURN
112 CONTINUE
SKSUM‘ SY ' ALPHA*G*( 2.0-1.0/AA )
IF(SYP-SKSUM)131I1311130
130 H- ALPHA * ALPHA*G/( SYP’SY+2.0*ALPHA*G )
GOTO 132
131 CONTINUE
H- ALPHA*AA
132 CONTINUE
DO 113 IBN- leP
B(IBN)- BSS(IBN) + H * ( B(IBN)‘BSS(IBN) )
118 CONTINUE
211 CONTINUE
WRITE(IOUT,121)
121 FORMAT(7X,'MAX'II2X,'H',14X,'G',13X,
f'SY',12X,'SYP')
WRITE(7,122) MAXIHIGISYISYP
122 FORMAT(I10,1F15.8,4E15.7)
WRITE(7,°(10X,"B(",I1,") - H,E16.5)') (I,E(I),I'1,NP)
C END BOX'KANEMASU MODIFICATION
WRITE(7,'(/,5X,"P(1,KP)",9X,"P(2,KP)",9X,"P(3,KP)"I9XI
@"P(4,KP)",9X,"P(S,KP)")')
WRITE(7,207) (P(IP,KP),KP'1,NP)
206 CONTINUE
207 FORMAT(5D15.7)
WRITE(7,135)
135 FORMAT(5X,'CORRELATION MATRIX.)
DO 136 IR'1,NP
DO 136 IR2'11IR
AR- P(IR,IR) * P(IR2:IR2)
R(IR,IR2)- P(IR,IR2)/SQRT(AR)
136 CONTINUE
DO 137 IR'1pNP
WRITE(7,'(7E15.7)') (R(IR,III),III‘1,IR)
137 CONTINUE
PS(IPS'IPS)' (1.0E+7) * P(IPS,IPS)
126 CONTINUE
127 FORMAT(3X,'IPS",I4,3X,'PS(IPS,IPS)",D15.8)
DO 119 IP‘lINP
P(IP,KP)‘ PS(IP,KP)
119 CONTINUE
DO 120 IP-lINP
BS(IP)' 8(IP)
CG(IP)- 0.0
120 CONTINUE
WRITE(7;314)
WRITE(7, ' (7I10;4F10.4) ') MAXH'P, INDEX, IP
IF(NP'INDEX)101,101,123
1:3- CONTINUE

00

(‘J

1'72

"MODSENSUV.FOR" for NLIBO.FOR with 3 parameters
subroutine model(E)
dimension ps(5,5),t(100,5), y(100), sigZ(lOO),b(S),z(S),a(S),
p(5,5),bs(5),vinv(5,5)
double precision p,ps,psi,t
common t,z,bs,i,eta,ps,p,b,a,y,sigz,modl,vinv,np
psi-t(i,l)
IF (bs(l).LT.O)THEN
bs(l)--bs(1)
ENDIF
IF (bs(2).LT.O)THEN
bs(2)--bs(2)
ENDIF
eta-(l-exp(-bs(l)*psi))**bs(2)
return
end

subroutine sens(E)

dimension ps(5,5),t(100,5),y(100),sigz(100),b(5),z(5),a(5),
P(SIS)Ib3(S)IVinV(SIS)

common t,z,bs,i,eta,ps,p,b,a,y,sig2,modl,vinv,np

double precision p,ps,psi,t

parameter(R-8.313)

iflag-l means psi/T(t), -0 means psi

psi-t(i,l)

IF (bs(l).LT.0)THEN
bs(1)--bs(l)

ENDIF

IF (bs(2).LT.O)THEN
bs(2)-bs(21

ENDIF

d-exp(-bs(1)*psi)

g-l-d

f-g**(bs(2)-l.0)

sensitivity coefficients

z(l)-bs(2)*d*psi*f
2(2)-f*g*aloq(9)
2(3)-g**bs(2)

return
end

173
.APPENDIX E

8.5.Appendix E. Calculated viscous activation energy
(AEV) of gelatinized native corn starch
to determine effects of temperature
(correspondin to Table 3.3)

 

 

 

 

ImpellerTemp Fluid eating Carculated Calculated
speed Rangeg starch/ Jacket time ‘1’, scaled AEV
(rpm) (C) g_soln Temp(C) (min) (K s) _(K)

3 0836,0322 787,614
4 O.188,0.154 909.1178
100 50-70 6.4 95 5 3.5,3.8,4 707,641,584
6 60.2150 10881926
567 1176
60-95 8 452 998

446 1408

 

 

174
APPENDIX F

8.6.Appendix F. Observed equilibrium torque of
gelatinized native corn starch to

determine effects of shear rate

 

 

(corres ondin to Table 3.4)
Elma Heating Equil.

 

 

 

 

Impeller Temp
speed Range g starch/ Jacket time torque
(rpm) (C) j soln Temp(C) (min) (Brkfld units)
0.5 14.8
1 15.2
2.5 16.8
5 60 6.4 95 8 18.8
10 . 21.8
20 26.2
50 34.7
400 44-1
0.5 15.3
1 15.7
2.5 17.1
5 60 6.4 95 8 19.1
10 22.0
20 26.3
50 34.7
400 43.9
0.5 18.6
1 19.1
2.5 21.5
5 50 6.4 95 12 23.9
10 27.5
20 32.7
50 42.7
100 53.7

 

 

 

11 I... 1 1.11 II - Illillln‘lIuIllllll ..lllli

 

175
APPENDIX G

8.7.Appendix G. Calculated strain history parameters of
gelatinized native corn starch

(correspondirgfito Table 3.5)

 

 

 

 

 

Impeller Fluid Heating Cafculated
speed Temp g starch/ Jacket time ‘1’, scaled
(rpm) (C) gsoln Temp(C) (min) (Ks) -d"‘1000 B
100 fi40 0.498 0.2176
936 0.352 0.312
50 60 6.4 95 12 303 0.992 0.230
407 1.115 0.203
20 335 2.775 0.235
449 2.165 0-203
8 117 0.644 0.107
100 0.555 0.111
100 60 6.4 95 104 0.669 0.121
4 2.9 0.757 0.149
4-3 0.712 0.121
50 12 556 0.407 0.268
100 60 6.4 518 0.331 0.283
95 8 195 2.2 0.330
329 1.19 0.32

 

 

176
APPENDIX H

Appendix H. Example of mathematical procedure to correct and normalize raw
torque during a pastingcurve to a reference temperature (60 C) and
reference temperature-time history (zero) at 10 5 intervals.

(3%) (r't'ii

Temp. correction factor ac

A5.
T- 217’v+600

1w -1+ A'(1- a""")

 

I
(217"’°°°)'( (00.273.131-17'373 '31)

temp correction f actor-o
w correction f actor-fl
normalizing factor(1/K,)-(l+219.8)‘( 1-0.2864)/44.16

 

 

 

torque
torque corrected
temp. Experi torque corrected to 60 C,
starch correction mental counted to 60 C, v-O,
Temp w factor to torque to 60 C w-O normalized
C K-s N-m N-m N-m N-m
58.7 0.000 1.007 1.041 0.17 0.169 0.162 0.579
65 0.000 0.974 1.234 0.15 0.154 0.12 0.445
70.2 0.000 0.948 1.984 0.15 0.158 0.080 0.285
73.9 0.000 0.93 3.804 0.17 0.183 0.048 0.171
7 5.9 0.000 0.921 6.35 0.24 0.261 0.041 0.146
78.1 0.000 0.911 10.715 0.35 0.384 0.036 0.128
7 8.7 0.001 0.909 14.756 0.55 0.605 0.041 0.146
80.6 0.003 0.9 21.636 0.8 0.889 0.041 0.147
81.5 0.008 0.896 29.576 1.32 1.473 0.050 0.178
82.4 0.017 0.892 38.791 2.57 2.881 0.074 0.265
82.8 0.029 0.89 47.722 5.01 5.630 0.1 18 0.421
83.9 0.052 0.885 60.175 8.05 9.102 0.151 0.540
84.4 0.086 0.881 72.304 1 1.15 12.656 0.175 0.625
84.8 0.13 0.878 83.869 14.81 16.877 0.201 0.718
86 0.208 0.869 101.179 18.34 21.100 0.209 0.744
85.7 0.305 0.866 1 13.315 22.05 25.445 0.225 0.801
86.5 0.424 0.858 125.993 25.79 30.055 0.239 0.851
87 0.598 0.848 139.28 29.22 34.435 0.247 0.882
86.8 0.788 0.842 147.466 32 37.902 0.25 7 0.917
87.6‘ 1.02 0.83 157.146 34.7 41.298 0.263 0.938
87.9” 1.34 0.828 164.625 36.96 44.070 0.268 0.955
88.2. 1.72 0.827 170.346 38.63 46.149 0.271 0.967
89.1 2.29 0.822 178.746 39.72 47.712 0.267 0.952
88.9 2.99 0.823 180.173 40.4 48.484 0.269 0.960
89.7 3.84 0.819 184.345 40.75 49.149 0.267 0.951
89.9 4.86 0.817 186.203 40.91 49.419 0.265 0.947
90 6.13 0.817 186.757 40.79 49.326 0.264 0.942
90.3 7.56 0.815 188.721 40.54 49.136 0.260 0.929
91 9.52 0.811 193.014 40.2 48.960 0.25-1 0.905
91.3 12.1 0.81 195.037 39.76 48.557 0.249 0.888
91.2 14.9 0.809 195.037 39.26 47.970 0.246 0.878
91.9 18.4 0.805 199.529 38.66 47.495 0.238 0.849

Appendix H. (Cont’d).

91.9
92.2
92.6
92.9
93.2
93.1
93.1
93.3
94.1
93.3

94
94.1
93.7
94.2

94.4'

94.5
94.1
94.4
94.1
94.6
94.1
94.6
94.1
94.5
94.3
94.3
94.6
93.9
94.2

91
87.3
83.3

79
76.5
73.7
71.9
70.1
68.2
67.2
65.6
64.8
63.8

m
U
—

MMMMQQOOO‘G
b‘iu'osboiuoei—hhia'

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22.6
27.3
33.1
40.1
48.5
57.6
66.6
76.1
89.1
102
115
130
145
160
178
198
217
235
252
272
292
311
330
348
368
386
407
426
441
451
452
452
452
452
452
452
452
452
452

0.805
0.802
0.8
0.797
0.794
0.793
0.792
0.789
0.783
0.785
0.78
0.777
0.777
0.772
0.768
0.764
0.763
0.759
0.758
0.752
0.751
0.746
0.746

199.529
201.675
204.124
206.038
208.254

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999999999999Pppppppppwpppwpppppppp

177

36. 43

40. 63
41.61

42. 4
43.03
43.54
43.98
44.32
44.62
44.92
45.18
45.41
45. 64

45.8
45.95

COO-—
COWG—

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tJ|JN————u————-——-——-—-———————_——o——e——-u——-—NNNNNN
bbom&———&O\\IUw—I~J—OO\OONW\OO‘OO—W&M

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