
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/UF00089827/00001
Material Information
 Title:
 Numerical modeling of natural convection and conduction heat transfer in canned foods with application to online process control
 Series Title:
 Numerical modeling of natural convection and conduction heat transfer in canned foods with application to online process control
 Creator:
 Datta, Ashim Kumar
 Publisher:
 Ashim Kumar Datta
 Publication Date:
 1985
 Language:
 English
Subjects
 Subjects / Keywords:
 Boundary conditions ( jstor )
Cooling ( jstor ) Free convection ( jstor ) Heat transfer ( jstor ) Heating ( jstor ) Isotherms ( jstor ) Liquids ( jstor ) Velocity ( jstor ) Vorticity ( jstor ) Water temperature ( jstor )
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 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright Ashim Kumar Datta. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 000872393 ( alephbibnum )
14514671 ( oclc )

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Full Text 
NUMERICAL MODELING OF NATURAL CONVECTION
AND CONDUCTION HEAT TRANSFER IN CANNED FOODS
WITH APPLICATION TO ONLINE PROCESS CONTROL
By
ASHIM KUMAR DATTA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
Dedicated to the memory of my
grandfather and father
ACKNOWLEDGEMENTS
The author is indebted to Dr. Arthur A. Teixeira, his major
professor, for his encouragement, guidance and patience throughout
the course of the work and especially for his assistance in
preparing the manuscript.
The author is also highly grateful to Dr. Tom IP. Shih, Dr.
Chung K. Hsieh and Dr. Ruey J. Shyu for their time and extremely
valuable suggestions in the study of the natural convection heat
transfer. Further thanks to Dr. Khe V. Chau and Dr. Spyros Svoronos
for their interest in the study and helpful suggestions.
The importance of a good computing facility (number crunching
to be particular) in developing the natural convection model cannot
be overemphasized. The availability of the Cyber 170 computer
through Dr. John Gerber and the IFAS administration was vital for
that purpose. The author also greatly appreciates help from Dr.
Ellen Chen, who introduced him to the Cyber facility, and Victor,
Jimmy and Margaret for endless support on problems with the Cyber
system.
The mountain of data available from the transient natural
convection heating model would have been of little use without the
plotting capability provided by the excellent graphics software
(PLOTPAK) made available from Dr. William E. Dunn from the
Mechanical and Industrial Engineering Department of the University
of Illinois at UrbanaChampaign. Help by Subhasis Laha in physically
transporting the plotting software to Gainesville is also
appreciated. Availability of a hard copy plotter was the next
necessity and the author appreciates the equipment provided by Dr.
James W. Jones in the Agricultural Engineering Department.
Finally, the author deeply acknowledges the assistance from his
wife Anasua in typing this manuscript, and her encouragement and
perseverence during the study.
TABLE OF CONTENTS
ACKNOWLEDGMENTS .................... ...........................i
LIST OF SYMBOLS................................................ vi
LIST OF TABLES....................................................xi
LIST OF FIGURES............ ................................ xii
ABSTRACT ........................................ .......... xvi
INTRODUCTION...................... .............................
REVIEW OF LITERATURE.............................................5
Computer Control of Food Thermal Processing..................5
Overview of Food Thermal Processing........................5
Online Control of Thermal Processing....................10
Conduction Heating of Canned Foods.........................12
Studies Based on Analytical Solution.....................12
Numerical Studies .......................................13
Natural Convection Heating in Canned Foods..................14
Previous Works on Canned Liquid Foods..................15
Other Similiar Works on Natural Convection
Heating in Cylindrical Enclosures........................19
PROBLEM FORMULATION...........................................24
Natural Convection Heat Transfer in Canned Foods............24
Governing Equations and Boundary Conditions..............24
Boussinesq Approximation.................................26
Transformation of the PDEs.............................27
Nondimensionalization of the Variables.................28
Conduction Heat Transfer in Canned Foods...................30
Governing Equation..................................31
Boundary Conditions.............................. .....31
Computer Control of Conduction Heating Process..............32
METHODOLOGY .................................................... 35
Modeling of Natural Convection Heating in Cans..............35
Grid Generation and Grid Stretching......................36
Discretization of the Parabolic
Temperature and Vorticity Equation......................39
Time derivatives .............................. .....42
Linearization. ........................................44
Convection terms......................................45
Diffusion terms ...................................... 50
Discretization of the Elliptic
Stream Function Equation..................................61
Computational Boundary Conditions for the FDEs...........69
Convergence Criteria for Iteration .......................74
Algorithm for Iterative Solution.........................76
Coding of the Computer Program and Hardware..............80
Modeling of Conduction Heating in Cans......................80
Discretization and Solution of the Equation..............81
Coding of the Computer Program..........................82
Experimental Studies ....... ........ ....................83
Use of the Conduction Heating Model for Online Control.....86
RESULTS AND DISCUSSION.... ......................... ...... ......93
Natural Convection Heating in Canned Foods..................93
Transient Flow Patterns and Temperature Profiles.........94
Start of flow and conduction layer....................95
Radial temperature and velocity profiles.............106
Axial temperature profiles........................... 113
Slowest Heating Zone................ ..... ............. 115
Contrast with Conduction................................120
Assessment of the Numerical Method......................124
Convergence..........................................124
Selection of the grid size...........................125
Selection of the time step...........................130
Convergence of boundary vorticity...................134
A note on the half grid points.......................134
Conduction Heating in Cans.................................138
Analysis of Boundary Conditions........................139
Comparison of the Transient Temperature Values.......... 141
Effect of Sudden Pressure Drops in the Can.............. 141
Performance of the Online Control Logic...................141
CONCLUSIONS AND RECOMMENDATIONS..............................151
Conclusions................................................151
Recommendations............................................ 153
APPENDIX
A ALTERNATIVE FINITE DIFFERENCING OF
STREAM FUNCTION EQUATION................................154
B ALTERNATIVE FINITE DIFFERENCING OF BOUNDARY VORTICITY...155
C INPUT DATA FILE FOR NATURAL CONVECTION MODEL............156
D ADE FORMULATION FOR THE CONDUCTION EQUATION.............157
LIST OF REFERENCES.......... ..... .............. ................ 161
ADDITIONAL REFERENCES............................. ....... ....... 167
BIOGRAPHICAL SKETCH................................... ....... 172
LIST OF SYMBOLS
A = constant
a,b,c = coefficients in tridiagonal matrices
f = inverse slope of timetemperature graph
g = gravity
Fo = sterilization Fo value
H3
Gr = Grashof number = gB (Ti To) 7
h = outside heat transfer coefficient
H = height of cylinder
j = j value (a constant)
JO = Bessel function of order zero
k = thermal conductivity
n = normal direction
nz = number of grid points in z direction
nr = number of grid points in r direction
p = pressure
Pr = Prandtl number = pC /k = v/a
r = non dimensional distance in radial direction
r = distance in radial direction
R = radius of cylinder
= Rayleigh number = Gr*Pr
= time
= temperature at any point inside at time t > 0
= initial uniform temperature inside (at t = 0)
= boundary temperature at time t > 0
= retort temperature at time t
H 
= u = nondimensional velocity in vertical direction
= velocity in vertical direction
H
= v = nondimensional velocity in radial direction
= velocity in radial direction
 nondimensional distance in vertical direction
H
= distance in vertical direction
= Zvalue for organism
= thermal diffusivity = k/pCp
= coefficient of thermal expansion
= relaxation parameter for boundary vorticity
= designates a difference when used as a prefix
= convergence criterion for boundary vorticity
= convergence criterion for SOR
= vertical distance in computational plane = n(z)
TTi
= nondimensonal temperature
ToT
= first eigenvalue in z direction
= eigenvalues in z direction
= first eigenvalue in r direction
= eigenvalues in r direction
viii
= viscosity of liquid being heated
= kinematic viscosity of liquid being heated = p/p
= radial distance in computational plane = (r)
= density of liquid being heated
= deformation parameter in grid stretching
= t = nondimensional time
H2
= radial coefficients for kinematic consistency
= radial coefficients for kinematic consistency
= nondimensional stream function = 
Ha
= stream function defined by v = 
r ar r az
= nondimensional vorticity
av au H2 v U H2 
az ar a a
= vorticity
= optimum relaxation parameter for SOR
Subscripts
a
center
cool
heat
i
J
m
n
W
= ambient
= geometrical center of can
= cooling cycle
= heating cycle
= grid points in vertical direction
= grid points in radial direction
= grid point on vertical boundary (z = H)
= grid point on radial boundary (F = R)
= Wall
Superscripts
d = design value
f = normal to grid faces
k = iteration counts in SOR
m = iteration count within a time step
n = time step
t = tangential to grid faces
= "half" step in ADI method (same as n+l/2)
LIST OF TABLES
TABLE PAGE
1. Sequence of line types for identification of stream
function contours in figures 10ao........................96
2. Heating and cooling rates for various process
conditions........................... ..................... 140
3. Adjusted heating times and resulting lethality (Fo)
in response to process deviations using proposed on
line control logic and method of GiannoniSuccar and
Hayakawa ................................................. 148
LIST OF FIGURES
FIGURE PAGE
1. Isotherms and velocity profiles in a glass bottle at
various heating times (after Engelman and Sani, 1983).....18
2. Isotherms and velocity profiles in a can at various
heating times (after Sani, 1985).........................20
3. The grid system and the boundary conditions for the
cylindrical can...........................................40
4. Notations of various quantities defined on a non
uniform grid system in cylindrical geometry...............62
5. Algorithm for the iterative solution of the set of
equations in natural convection...........................77
6. Experimental setup for conduction heating..................84
7. Timings of various realtime computations for online
control of conductionheated food.........................88
8. Flow diagram for computer control of retort opera
tions with online correction of process deviations.......90
9. Isotherms near bottom wall predicted by the
convective model compared with the predictions
considering conduction (equation 83) only.................97
10a. Isotherm and streamlines in a cylindrical can after 1
second of heating .........................................99
10b. Isotherm and streamlines in a cylindrical can after 2
seconds of heating........ .......... ......................99
10c. Isotherm and streamlines in a cylindrical can after 3
seconds of heating.......................................100
lOd. Isotherm and streamlines in a cylindrical can after 4
seconds of heating......................... ......... ..... 100
10e. Isotherm and streamlines in a cylindrical can after 5
seconds of heating.................. ... .. ............ ..101
10f. Isotherm and streamlines in a cylindrical can after 6
seconds of heating....................................... 101
10g. Isotherm and streamlines in a cylindrical can after 7
seconds of heating......................................102
10h. Isotherm and streamlines in a cylindrical can after 8
seconds of heating...................................... 102
10i. Isotherm and streamlines in a cylindrical can after 9
seconds of heating.................... ... ....... ......... 103
10j. Isotherm and streamlines in a cylindrical can after
10 seconds of heating................. ........... ...... 103
10k. Isotherms and streamlines in a cylindrical can after
30 seconds of heating................... ................ 104
101. Isotherms and streamlines in a cylindrical can after
120 seconds of heating.......................... ....... 104
10m. Isotherm and streamlines in a cylindrical can after
300 seconds of heating.............. ..................... 105
1On. Isotherms and streamlines in a cylindrical can after
600 seconds of heating................................... 105
10o. Isotherms and velocity vectors in a cylindrical can
after 1800 seconds of heating...........................107
11. Predicted radial velocity profile at the sidewall at
mid height in a cylindrical can after 30 seconds of
heating ..................................................108
12. Predicted radial temperature profile at the sidewall
at mid height in a cylindrical can after 30 seconds
of heating ...............................................108
13. Isothermal vertical surface, showing boundary layer
with velocity profile u(x,y) and temperature profile
t(x,y) in water (Higgins and Gebhart, 1983)..............110
14. Predicted radial velocity profiles at the sidewall at
mid height in a cylindrical can after various heating
times .......................... ........................... 111
15. Observed radial velocity profiles at the sidewall at
mid height in water in a cylindrical can after
various heating times (from Hiddink, 1975)...............112
16. Comparison of numerically predicted axial temperature
profiles with observed (Hiddink, 1975) values at
various times during heating.............................114
xiii
17. Locations of the slowest heating points in water over
a heating period of 10 minutes in a cylindrical can......116
18. Migration of the slowest heating points in water over
a heating period of 10 minutes in a cylindrical can......116
19. Transient temperature at various locations during
heating of water in a cylindrical can....................119
20. Transient temperature at various locations during
initial times of heating of water in a cylindrical
can.......... ... ..... ..................... .......** 121
21. Temperature at the slowest heating point during
natural convection heating contrasted with that
during conduction heating in a cylindrical can...........122
22. Isotherms during natural convection heating contr
asted with those during conduction heating in a
cylindrical can..........................................123
23. Diagram of 39(0.9)x39(0.8) grid on the left and
58(0.85)x58(0.76) on the right...........................126
24. Isotherms after 90 seconds of heating computed using
39x39 grid (left half of the figure) compared with
computations using 58x58 grid (right half of the
figure) ..................................................127
25. Streamlines after 90 seconds of heating computed
using 39x39 grid (left half of the figure) compared
with computations using 58x58 grid (right half of the
figure) .................................................. 128
26. Transient slowest heating temperatures computed with
39x39 and 58x58 grid.................. .................... 129
27. Transient temperature at 1/3rd height on the center
line computed with 39x39 and 58x58 grid...................131
28. Time step history during a typical run of the
numerical heat transfer model for natural convection.....132
29. Isotherms in a cylindrical can calculated with two
different convergence criteria for boundary vorticity
(6 =0.0001 on the left and 6 =0.001 on the right)........135
30. Streamlines in a cylindrical can calculated with two
different convergence criteria for boundary vorticity
(6 =0.0001 on the left and 6 =0.001 on the right)........136
31. Transient center temperature during conduction
heating of Bentonite in a cylindrical can................142
32. Transient center temperature during conduction
cooling of Bentonite in a cylindrical can with and
without drop in retort pressure...........................143
33. Computer plot of reference process (no deviations)
showing 66.8 minutes of heating time and an
accomplished Fo of 6.24............ .............. ......144
34. Computer plot of a process that experienced a step
functional drop of 8.30C (150F) in temperature bet
ween 50th and 60th minute and still maintained an Fo
of 6.02.................................................. 146
35. Computer plot of a process that experienced a linear
drop in temperature between 40th and 60th minute
(from 00C to 11.1C (200F)) and still maintained an
Fo of 6.18..............................................147
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NUMERICAL MODELING OF NATURAL CONVECTION
AND CONDUCTION HEAT TRANSFER IN CANNED FOODS
WITH APPLICATION TO ONLINE PROCESS CONTROL
By
ASHIM KUMAR DATTA
December 1985
Chairman: Arthur A. Teixeira
Major Department: Agricultural Engineering
A numerical model was developed for predicting detailed flow
patterns and temperature profiles during natural convection heating
of liquids in a can (a cylindrical enclosure). The liquid (water)
was initially stagnant at a uniform temperature of 300C, and the
sidewall, top and bottom wall temperatures were suddenly raised to a
temperature of 1210C. A free (thermally insulated) top liquid
surface was also considered. Boussinesq approximation was used and
all other fluid properties were treated as constant.
Finite difference methods were used to solve the governing
equations in axisymmetric cylindrical coordinates. A vorticity
stream function formulation was used. Parabolic temperature and
vorticity equations were solved with the alternating direction
implicit method and upwinding the convective terms. The elliptic
stream function equation was solved using successive over relaxation
method. Grid stretching was used in both radial and axial
directions. Boundary wall vorticities were used to check for
convergence of the iteration process.
Plots of transient isotherms, streamlines and velocities were
provided. Calculated radial velocities and axial temperature
profiles at different times compared quite well with available
experimental data. Slowest heating points were located in the bottom
15% of the height of the container studied. These slowest heating
points migrated over time within this lower region of the can but no
particular pattern of migration was noted.
Transient can center temperature was also studied for the case
of conductionheated solid food. An alternating direction explicit
finite difference solution to the conduction equation compared very
well with analytical solutions and experimental results. The ability
of the finite difference model to perform under arbitrary variation
of boundary temperature allowed the model to be incorporated in an
algorithm for online control in batch thermal processing of
conductionheated food. The uniqueness of this control system in
maintaining the level of desired food sterilization for arbitrary
variation in process heating temperature was shown to offer
considerable advantages over other possible methods of online
control.
xvii
INTRODUCTION
Background and Justification
Thermal preservation of food involves application of heat
sufficient to destroy the microorganisms present in food that cause
spoilage. Commercial canning processes provide such heat treatment
to extend the shelflife of foods. In a typical canning process, the
container is filled with food and sealed. The sealed container is
heated in a closed vessel with steam or hot water long enough to
kill the microorganisms and is then cooled. In a production situa
tion, the heating medium (steam or hot water) temperature can
sometimes deviate significantly from design values during the
process. Heating the product more than the required amount would un
necessarily degrade its quality while wasting energy. However,
insufficient heating could lead to serious public health hazards and
cannot be allowed. Thus, food processors always have to meet this
minimum heat treatment (sterilization). Determination of the heating
time to meet the minimum sterilization demands that the actual
transient temperature history of the food during processing be known
predicted with a heat transfer model or actually measured).
ConductionHeated Foods
Traditionally, for conductionheated foods, the required
processing time is determined using the analytical solution to the
heat conduction equation. In practice, the constant medium temper
ature assumed by the analytical solution cannot always be achieved.
Thus, experimental correction factors have been proposed to adjust
the heating time, predicted by analytical solution, when the medium
temperature goes through certain (predefined) types of deviations.
However, deviations in practice are arbitrary, and it is not
conceivable to know the correction factors for all such situations.
Of course, the problem is trivial if the transient food temperature
is directly measured using a thermocouple; but this is both
inconvenient and impractical in a production situation. Better
models for predicting food temperature in response to arbitrary
fluctuations in medium temperatures are therefore needed.
A numerical finite difference model would be able to predict
the food temperature for truly arbitrary variations in the medium
temperature. However, the calculations required are relatively much
more complicated and time consuming. Using a microcomputer to
perform this would have two advantages. In addition to correctly
predicting food temperature for truly arbitrary heating conditions,
it can perform this automatically without any worker supervision.
Introduction of lowcost microcomputers for online process control
is becoming commonplace in the 1980s. Food industries are not an
exception to this situation. Use of microcomputers for online
control of conductionheated food is thus quite conceivable.
ConvectionHeated Foods
Unlike solid packed foods that are heated by conduction, thin
liquid foods heat through convection. In general such liquid foods
are mechanically agitated to improve the heat transfer. However,
there are situations where the food material cannot be agitated for
several reasons. The food then heats primarily by natural convec
tion. Inpackage pasteurization for liquids (e.g. beer) in bottles
or cans that cannot be agitated is an example. Broth, thin soups,
evaporated milk, most fruit and vegetable juices, fruits in syrup or
water, and pureed vegetables are also examples of products where
natural convection heating can occur when no agitation is used.
Flexible pouches that cannot be agitated to keep the package integ
rity are another important example. Fermentation is another process
where a natural convection heat transfer model (with heat generation
term included) can provide valuable insight.
Just as optimum heating times are required for conduction
heated foods, they are also needed for natural convectionheated
products. Thus heat transfer models are similarly needed. Since we
are interested in the minimum heat treatment the food receives, we
need to know the point or the region that heats the slowest. Because
convection sets up complex flow patterns, the location where the
temperature is minimum is by no means obvious. Only a study of
complete transient flow patterns and temperature profiles can give
such insight. Previous studies to achieve this objective have not
been successful. The use of more advanced numerical techniques and
better computing equipment would make it possible to study in detail
the natural convection heat transfer in canned liquid foods.
Such a natural convection heat transfer model could also have
several applications outside the food area. It may be used to study
the heating of buildings (without forced circulation), cooling of
gas turbine blades, storage of cryogenic fluids (e.g. liquid rocket
propellants), and the startup of chemical reactors. Since natural
convection heat transfer would involve a much higher degree of math
ematical complexity than a conduction heating model (and its appli
cation to online control), the primary objective of this study was
directed toward developing a heat transfer model for natural convec
tion; followed by improvements to existing conductionheat transfer
models for application to online process control as secondary
objectives.
Objectives
The objectives of this study were to
1. develop a mathematical model to predict temperature and
velocity profiles caused by natural convection heat
transfer in a closed cylinder,
2. investigate the need for possible improvements on an
existing conduction heat transfer model to improve its
suitability for use in online process control appli
cations, and
3. incorporate the conduction heating model in an online
process control algorithm for thermal processing of
canned food, and compare its performance against other
possible methods of online control.
REVIEW OF LITERATURE
This chapter begins with a brief introduction to thermal
processing as applied to the sterilization of canned foods. The need
for computer control of thermal processing is noted, along with the
importance of heat transfer models in such computer control systems.
Previously reported conduction heat transfer models used in studying
sterilization of conductionheated canned foods are described next.
The few studies attempting to model natural convection heat transfer
in the case of sterilizing canned liquid foods are noted. The few
other works on natural convection heat transfer in liquid inside a
vertical cylindrical enclosure are also reported.
Computer Control of Food Thermal Processing
An overview of thermal processing is presented in this section.
Problems with traditional approaches to processing of conduction
heated foods are then noted and the need for online control is
stressed. Some of the past approaches in thermal processing of con
ductionheated foods that could also be used online are discussed.
Overview of Thermal Processing
Thermal processing consists of heating foods, filled and
hermatically sealed in containers, in pressurized steam retorts at a
constant retort temperature for a prescribed length of time followed
by rapid cooling. These process times are calculated on the basis of
sufficient bacterial inactivation in each container to comply with
public health standards and reduce the probability of spoilage to a
specified minimum. Estimation of bacterial inactivation requires
understanding of thermal death kinetics of bacteria, along with the
temperature history the food (containing the bacteria) experiences
during the process. Following is a brief introduction to the thermal
death kinetics of bacteria (Ball & Olson, 1957; Stumbo, 1965).
Thermal death of bacteria follows first order kinetics. If C is
bacterial concentration (number of bacteria per unit volume of food
material) at time t,
dC 1 (1)
dt (0.4 34)OT
where (0.434)*D is the inverse of the rate constant. If Ci is the
initial concentration and C is concentration after time At
C
Ci At
log = (2)
C
The constant D (known as 0 value), which is the inverse of the rate
constant (multiplied by the factor 0.434), is the time required to
reduce the bacterial population by one log cycle at a specified
temperature. The temperature dependency of the D value is given by
T T
D = Do 10 Z (3)
where Do is the D value at some reference temperature To. The Z
value describes the temperature dependency of the lethal rate and is
related to the activation energy. The reference temperature To is
generally taken as 1210C (2500F).
From bacteriological considerations, a thermal process is
specified by noting the time (Fo) necessary to reduce the bacterial
population, from the initial (Ci) to some final safe value (C) at
temperature 1210C (2500F). Thus
C.
F0 = D log (4)
When the heating process is not at the constant temperature of
121C, introducing the temperature dependency of Do and equation (2)
for bacterial death,
T121
Z At
Fo = D 10 z
T121
= 10 Z At (5)
At is the time required at temperature T to achieve the designed
bacterial reduction. As the food material is heated, its temperature
T becomes a continuous function of time t and the expression for Fo
(equation 5) becomes
t T121
Z (6)
F = f 10 dt (6)
When the food being heated has a spatially varying temperature
profile, the chosen temperature T, for thermal process calculation,
is the temperature at the slowest heating point in the food
material. Thus, for conduction heating in a cylindrical can, the
chosen temperature T is at the geometrical center of the can.
The designed bacterial inactivation for a process is specified
by Fd which is the time required to achieve the desired
0
inactivation of bacteria when the same is heated at a constant
temperature (T) of 1210C. When temperature T is a function of time
(T(t)) instead, the objective of thermal processing is to select
process time t for the particular heating condition, such that
the equivalent Fo received (as given by eqn. 6) is greater than or
equal to the design value F. That is,
Fo(t) Fd (7)
Application to conductionheating. When applying thermal
processing to conductionheated food in a cylindrical container,
Ball and Olson (1957) used the solution to the heat conduction
equation in cylindrical coordinates.
TTR (X2+ 2)at
R E Ampe m P Jo(mr) Cos ( z) (8a)
TOTR m p
The solution was first approximated to
T TR (X2 + X2)cat
S= A e Jo(xr) Cos(xz) (8b)
T0 TR
when sufficient time has elapsed, so that the first term in the
analytical series solution (equation 8a) dominates. Equation (8b)
was then rearranged to give
j (TR TO)
t = f log TR T(9)
TR T
where
S 2.303 (10)
(f a + (10)
and
j (r,z) = A Jo(xr) Cos (xz) (11)
From the transient heat conduction equation (9) which gives
temperature T as a function of time t and the definition of Fo
(equation 6) which gives Fo as a function of temperature T, it is
possible to find Fo as a function of time t. However this cannot be
performed analytically. Upon solving the two equations numerically,
it is possible to know temperature Th achieved at the end of heating
for a given value of Fo. Ball and Olson (1957) tabulated ThTR
values as a function of F0 for particular values of f and j. Tradi
tionally, in thermal processing of conductionheated foods in cylin
drical containers, these tables are used to find ThTR for a given
Fo value. Using this (ThTR) value and equation (9), the process
time t is found.
Online Control of Thermal Processing
In a production situation, the heating time t specified
(assuming a constant value of TR) to achieve a given Fo value would
need to be adjusted for arbitrary variations in the heating medium
temperature TR from its specified constant value. Researchers in the
past have attempted this in one of the following three ways
Precalculated tabulations. Correction factors are tabulated to
be applied to th for certain possible types of deviations in
TR(t). Commercial retort control systems often use this technique
(LOGTEC system, 1984). Such tabulations cannot be exhaustive,
since the process temperature TR(t) could go through any random
deviation. Instead of tabulating such correction factors, Giannoni
Succar and Hayakawa (1982) developed expressions for correction
factors to be applied to th when TR(t) goes through a step drop. In
their work, other types of deviations had to be approximated to
closefitting step functions, and multiple drops could not be
considered. Getchell (1980) described a control system that would
attempt to maintain the design heating medium temperature and sound
an alarm when critical low or high limits for the temperature are
exceeded. The operator would then attempt to make necessary changes
in th.
Direct Measurement. Temperature at the can center T(t) is
measured online. This eliminates the need to assume constant
heating temperature. Measured T(t) is used with equation 1 to stop
the heating cycle at time th such that
F t Fd (12)
Fo(th) F
This approach was used by Mulvaney and Rizvy (1984). They used a
thermocouplefitted test can in every retort batch to obtain actual
temperature T(t). The system proposed by Navankasattusas and Lund
(1978) also planned to measure can center temperature online. In
commercial practice these methods are costprohibitive with regard
to production efficiency, and are considered impractical.
Numerical Models. Instead of measuring T(t) at the can center,
T(t) can be predicted for arbitrary variations in TR(t). In
general, analytical solutions for T(t) are of limited use for
arbitary variations of TR(t). Teixeira and Manson (1982) described
a numerical finite difference approximation to the heat conduction
equation for predicting T(t) from TR(t). This is conceptually quite
different from the approaches mentioned earlier. With a finite
difference model, T(t) can be predicted for truly arbitrary
variations in TR(t) and, of course, use of this model eliminates the
need for having a test can with an actual thermocouple inserted in
it. This model was first described and verified against published
procedures by Teixeira et al., 1969; and later verified against
experimental data and other published procedures for timevarying
boundary conditions by Teixeira et al. (1975 a,b).
However, in the proposed retort control algorithm of Teixeira
and Manson (1982), every time a deviation took place, a new value of
th was found through simulation assuming the deviated retort temper
ature continued for the rest of the heating process. This is unnec
essary because any further changes in the retort temperature would
make the estimated th value useless and fresh estimations of th
would require repeated timeconsuming computer simulations.
Using the general idea of Teixeira and Manson (1982), the
feasibility of several control algorithms that can work online with
available computer hardware needed to be investigated, and led to
the following studies as part of the work carried out in this
project:
1) Ways to obtain the transient temperature solution faster,
using other available methods of solving the conduction
equation.
2) The need to incorporate a convection (Robin's) boundary
condition to reflect the true situation, rather than the
temperature specified (Dirichlet) boundary condition used
by Teixeira et al. (1975a,b) in solving the conduction
equation.
Conduction Heating of Canned Foods
Since the pioneering work of Ball and Olson (1957) mentioned
earlier, conduction heating of canned foods has been studied
extensively. Consequently there is a vast amount of literature
existing on the subject.
Studies Based on Analytical Solution
The majority of the works on conductionheating of canned foods
were based on one of several different forms of the analytical solu
tion to the heat conduction equation. Also most of these solutions
were derived assuming, among other things, a step change in the
boundary (can wall) temperature from the initial to the heating
temperature and maintaining the boundary heating temperature
constant throughout the process. More general forms of boundary
conditions have also been formulated. Hayakawa (1964) developed the
general form for an analytical solution that could predict the
transient temperature distribution inside a cylindrical can of
conductionheated food when the boundary (can wall) temperature was
any Laplacetransformable function of time.
In practice, the heating medium temperature, which affects can
boundary temperature, is not under complete control and can go
through unexpected variations from its design value. For such
arbitrary variations in the boundary temperature, analytical
solutions are of limited use.
Numerical Studies
A finite difference analog of the heat conduction equation can
also be used to achieve the transient temperature of food at finite
but small increments of time. Such methods can utilize the actual
boundary temperature "read" for the incremental time and calculate
the interior temperature values based on this boundary temperature.
Teixeira et al. (1969) were the first to use a forward difference
explicit finite differencing method to study the transient
temperature distribution inside the cylindrical can. Use of the
explicit finite differencing severely limited the choice of grid
size and time increment, due to stability requirements for such a
method. In their later work (Teixeira et al., 1975a) an alternating
direction explicit (ADE) method (Saul'yev, 1957; Allada and Quon,
1966) was used that was unconditionally stable but still explicit.
This method was first applied to simulate thermal processing of
canned foods by Manson et al. (1974). Teixeira et al. (1975a,b)
compared outputs from the ADE method for using constant boundary
temperature and sinusoidal timevarying boundary temperature with
the results from corresponding analytical solutions. The analytical
results for constant boundary temperature agreed within 0.30C
(0.50F) with the results from the ADE method, except at interior
points near the surface after small initial time (less than 3
minutes). Similar agreement between the results of the analytical
and the ADE formulation was also obtained when a sinusoidally
varying boundary condition was used.
Teixeira (1971) also compared the ADE finite difference outputs
for conduction heating in a can with experimental data for constant
boundary temperature and for step increases in the boundary tempera
ture during heating. The agreement was good except at initial times.
In addition to this ADE method, other methods of solution to
the conduction equation needed to be investigated from the point of
view of their suitability of computation in realtime and ability to
accommodate arbitrary boundary temperature variations.
Natural Convection Heating of Canned Foods
Internal (in enclosures) natural convection flow problems have
received much less attention than the external ones. Also, internal
flow problems are generally more complex than the external ones.
Fewer works have been reported on internal flows in cylindrical
coordinates. Of the studies on natural convection heating in
cylindrical enclosures, only four studies could be located where
they were in reference to natural convection heating during the
thermal processing of liquid foods. Such thermal processing is
characterized by a rather large Grashof number, generally small
overall dimension of the container and water as the liquid. Perhaps
the food processing situation is unique to general heat transfer
studies, which would explain the fact that only one such study could
be located in the more common heat transfer literature.
Previous Works on Canned Liquid Foods
The rate of heat transfer in canned liquid foods was studied
using correlations, and slopes of timetemperature curves with
limited success (Blaisdell, 1963; BarreiroMendez, 1979). Possible
flow patterns were postulated by Blaisdell (1963) using dye and
aluminum powder tracers.
Stevens (1972) first tried to study detailed temperature and
flow patterns during natural convection heating of canned liquid
foods. He used an explicit finite difference formulation from the
work of Torrance and Rockett (1969). The can was treated as a ver
tical cylindrical enclosure. The fluid (ethylene glycol) inside the
can was initially at rest at a uniform temperature of 300C. The can
was then placed in hot water bath at about 1000C. The top surface of
the fluid was free (the can had headspace). The actual temperature
variation on the can boundary was noted and was used in the
numerical calculations. Temperature values were recorded at ten
different points inside the container. The recorded temperature
values agreed poorly with numerically calculated temperatures. The
numerical inaccuracy was attributed to insufficient grid points
which could not resolve the details of the flow. Several sources of
experimental error included inaccurate setting of boundary
temperature, nonzero initial velocity field, errors in thermocouple
measuring circuit and inaccurate placement of the thermocouples.
Some of these factors were thought to have caused three dimensional
movements in the experiment and could not be picked up by the two
dimensional numerical model.
Detailed temperature and velocity profiles during natural
convection heating of liquid foods in a cylindrical can were
extensively investigated by Hiddink (1975). The numerical solution
technique used was the explicit finite difference technique of
Barakat and Clark (1966) discussed later. The initial temperature
was considered uniform throughout. The top, bottom and sidewall were
suddenly raised to the heating temperature. The top surface of the
liquid was considered free (having headspace) and was thus treated
as thermally insulated. Several test fluids of low and high
viscosities (water, sucrose solution and silicone fluid respec
tively) were used. The numerical results included temperature and
streamline patterns. The overall flow patterns were visualized using
a "particle streak method." In this method small glittering
particles are suspended in the liquid. The particles are illuminated
by a flat narrow slit of light, exposed against a dark background
and photographed. Detailed study of flow in the boundary layer was
observed using a laserdopper velocimeter which is an improved
version of the "particlestreak method" mentioned above. Temperature
values were measured by inserting thermocouples at several points in
the liquid. The temperature variation on the top free surface was
observed using a Thermovision Infrared camera. Radial velocity
profiles were presented at different heights in the liquid and at
different times during heating. His experiment and other works (Chu
& Goldstein, 1973) had showed convective heat transfer from the
bottom, as expected, when the bottom was also heated. His numerical
results, however, could not pick up the convective eddies generated
by the bottom heating. Thus his numerical results had poor corre
spondence with his experimental results.
A finite element method was used by Engelman and Sani (1983) to
study inpackage pasteurization of fluids (beer) in bottles. In the
process studied, bottles entered the pasteurizer at 1.70C(350F) and
passed through several progressively hotter zones of hot water
spray, which raised the product temperature to 600C(1400F). This
temperature of 600C is maintained in the holding zone. The product
then passed through several progressively cooler zones, which low
ered the temperature to 70 800F. Since the product was fluid and
the package was not agitated, it was primarily heated through nat
ural convection. Experimental details for this study was provided by
Brandon et al. (1981). The bottle geometry was complex (figure 1)
and so were the boundary conditions. The bottom of the bottle was
20 SECONDS
42 SECONDS
Figure Isotherms and velocity profiles in a. glass
bottle at various heating times (after Engelman
and Sani, 1983)
treated as insulated and wall temperature specified as a function of
time. The numerical results were in good agreement with experimental
data.
Sanil (personal communication, 1985) also developed a similar
model for pasteurization of beer in cylindrical metal cans (aspect
ratio =4.2). Water was the test fluid (Pr=6.81). The initial
temperature was 350C and the sidewall boundary condition used was
approximately 600C. The top of the water level was free (the can had
headspace) and was treated as insulated. The temperature at the
bottom of the can was a function of radial distance. Temperature and
velocity profiles from this study are given in figure 2. Further
quantitative details on this study were not available. These results
were not compared with experiments and also not published.
Other Similar Works on Natural Convection Heating in Cylindrical
Enclosures
One of the earliest of the numerical studies on the subject of
natural convection heating in cylindrical enclosures was performed
by Barakat and Clark (1966), who were looking at thermal strati
fication and associated processes in liquid propellant tanks for
application to space flight. The vertical cylindrical enclosure,
they studied, was filled with water, and arbitrary temperature and
heat flux variation were imposed on its wall, while its bottom was
kept insulated. At the top free surface of the liquid, heat transfer
to the ambient was considered negligible and thus insulated. They
1R. L. Sani, CIRES, Campus Box 449, University of Colorado, Boulder,
Colorado 80309
asseasa as a I1.1
6 l 8 * * **  
Gsilla.
\ fIlimII s alllII1*
i11 a 6 ) as t i11
(LiI~LaaliIll)i
S\* * a a ) \\ .\
IS S S I s a
allalIilalasIla
Figure 2. Isotherms and velocity profiles in a can at
various heating times (after Sani, 1985)
o
i *:: : : ;illI;
I .. .. . ..
: .: : : : .: ;: : .:: : ;i : : : : : : :
. . . . . . ..
. . .. *... .. . . .. ... il**
. . * ** . . . . . . .
I1 i
S' . * **I( * * * . i
I I * **CII' i I .. t * ** *
* * * i .*....... ..** I
* * * * * e . * * * **
S*j * i* * , * * * * **
i * * s o il* i * * * * *I*
I S S **S * * 1*1 * * ****
*. * *** * * *I ***
I . . . . .. I* . S * ** *
Ie I IIil" ** * * *
S.* * * .. * *
Ir \ B r I I I .
F r ** * *
i* i ** CI is *
aa I I * * * l* I
.i gu ,2 con ,d.,*
Figure 2. contd.
had numerically calculated and plotted the isotherms and streamlines
inside the enclosure. In their experimental work, they recorded
temperature values at ten different points inside the cylindrical
enclosure. The observed temperature values deviated from the
numerical results by 10%. This discrepancy was attributed to the
small nonsymmetry in the imposed wall temperature which might have
produced three dimensional effects and, also, to heat losses from
the bottom and top of the container.
Torrance (1968) compared five separate finite difference
formulations for the problem of natural convection in cylindrical
enclosures. The five formulations were respectively by Barakat and
Clark (1966), Wilkes and Churchill (1966), Fromm (1964) and two of
his own. The physical problem chosen was the study of natural
convection flow induced in a vertical cylindrical enclosure by a
small hot spot centrally located on the bottom. One of his two
formulations was explicit and the other was implicit. The explicit
formulation was shown to be numerically superior in many ways to the
other four and it was also reported to have good. agreement with
experimental observations (Torrance and Rockett, 1969; Torrance et
al., 1969) for lower values of Grashof numbers. This was also the
formulation used by Stevens (1972) referred to earlier.
Kee et al. (1976) performed a numerical study of temperature
distribution and streamlines in a closed vertical cylinder
containing uniformly distributed heat generating tritium gas. The
walls were kept isothermal. An upwind method was used for the
convective terms. Numerical temperature values were compared with
experimentally determined temperatures at three axial and two radial
locations. Thermistor thermometers were used for temperature
measurement. The calculated temperature values were reported to be
within the limits of experimental error, over a range of Grashof
numbers.
Evans et al. (1968) measured the transient temperature field
during natural convection of fluid in a vertical cylinder due to
uniform heat flux. The top surface of the liquid was free and no
transfer of heat or mass was assumed at the liquidvapor interface
at the top. Dye tracers were used to qualitatively observe the flow
patterns. They reported symmetry about the vertical axis in the
temperature field. On the basis of the experimental observations,
the convective liquid flow was modeled in terms of three regions:
1. a mixing region at the top
2. a central core
3. a boundary layer rising at the heated wall
They developed three different equations to describe the three
regions. These equations were solved simultaneously to predict the
overall behavior of the system. They reported good agreement between
the model and their experimental data.
Shyu and Hsieh (1985) made a numerical study of unsteady
natural convection in thermally stratified water in a cylindrical
enclosure. They studied the effect of placement of insulation on
maintainence of the stratification. Transient flow and temperature
fields were provided for the study.
PROBLEM FORMULATION
The governing equations and boundary conditions for the natural
convection and conduction heating process together with the assump
tions are first described here. The online control problem for the
conduction heating process is then formulated.
Natural Convection Heat Transfer in Canned Foods
The equations governing natural convection heat transfer in
liquid foods together with the boundary conditions are first
presented. The Boussinesq approximation to the set of equations is
then discussed. To further facilitate the solution, new variables
are defined and the governing equations together with the boundary
conditions are transformed into a new set of equations.
Governing Equations and Boundary Conditions
In natural convection heat transfer, the driving force for the
liquid motion is the buoyancy caused by the density variations due
to change in temperature. The partial differential equations (PDEs)
governing such naturally convective motion of fluid in a cylindrical
space are the NavierStokes equations in axisymmetric cylindrical
coordinates (Bird et al., 1976) and are described below.
+ Tar
9r
+ u 3T
3z
 3
r +
DF
K l1 a 3
 ~r ( (r
p r ar
= + {
8z r
u")
av
ai )
0
aT ) 2T
r) + )
ar az2
3 3u
ar ar
+ 313
3r r 3r
1 (pV) + (p) =
r ar 8z
The boundary conditions are
sidewall
T=T1
32u
+ +
Dz2
(r )) + }
aZ2
u = 0
v = 0
centerline
O z < H
3T 0
3F
bottom wall
0 < < R
top wall
0 < < R
z= 0
z=H
T=T1
T=T1
=0 = 0
(17ad)
u =0
u = 0
v = 0
v = 0
Initially the fluid is at rest at uniform temperature
0 < < R
0 < < H
T=TO
u =0
v =0 (18)
Refer to list of symbols for definition of individual symbols.
As expected, the flow field (equations 14 & 15) is coupled with
the temperature field (eq. 13) through the density term. These
a
+ +
aF
av
p ( +
at
(13)
(14)
(15)
(16)
equations have been approximated for no viscous dissipation. Jaluria
(1980) discussed various situations under which viscous dissipation
can be neglected. All other fluid properties are treated as
constants except variation of density which is discussed below.
Boussinesq Approximation
The solution to the set of equations can be considerably
simplified using the Boussinesq approximation. In the Boussinesq
approximation; the density is treated as constant except in the body
force term. The density difference in the body force term is approx
imated as
p p = pB (T T ) (19)
and the density variation in the continuity equation is ignored.
This approximation requires both
B (T Ta) << 1 (20)
and ( )T g << 1 (21)
to be true. The second inequality is generally satisfied for
liquids. So is the situation with the first inequality. For
example, for natural convection in water, as in this problem
B = 0.0002 /C
T = wall temperature of the can = 1210C
Ta= initial temperature of liquid in can = 30C
27
Thus,
B (T Ta) = (0.0002)(12130) = 0.0182 << 1 (22)
Almost all the previous works on natural convection in similar
areas have used the Boussinesq approximation.
Transformation of the PDEs
To further facilitate numerical solution to the set of
equations, the primary variables are transformed to a new set of
variables. Vorticity (J) and stream function ( ) are defined in
terms of the primary variables u (axial velocity) and v (radial
velocity)
:a u ,a(23)
3z aF
u =1a v = (24a,b)
rF Fr a
Using vorticity so defined, the pressure (p) is eliminated from
the u and v momentum equations 14 and 15. These two equations gave
the parabolic transport equation for vorticity Next, using the
definition of (equation 23) and the definition of (equations
24a and 24b), the equation for is derived. This gave the vorticity
streamfunction formulation for the natural convection problem.
28
Nondimensionalization of the Variables
The variables in the equation are then nondimensionalized as
below.
z
H
H
T H2t
H2
r
r 
H
v H
a
(25)
T T1
1
eH
T0 T1
H2 
1
ir~F :
The resulting equations are the governing equations for natural
convection in the form used for the present study.
ae a(ue) 1 (rve) _a2 1 a aeo
T + z } = { TaiZ + r ) }
(26)
aw a(u) a3(vw) D a2W a 1 a(rw)
9T + { + } =T GrPrr + Pr{ yz+ F r ar (7)
(27)
 T + ( T )
1 a
u = 1 F
(29a,b)
1 a2p
r az
(28)
The boundary
variables c.
sidewall11
r=R/H
centerline
r=O
bottom wall
0
top wall
0
y conditions
an be written
0 < z < 1
0 z < 1
z=0
z=l
in the transformed and nondimensionalized
as
= 0
=a 0
@r= 0
o 0=O
0=0
W = 0
S= 0 u=0
^ 0 .0
S=0 u = 0
ar
S= 0 u=0
S= 0 u=0
(30)
with the initial conditions
R
0
W = 0
S= 0 u=0
The relative merits of solving the transformed variables (w,p)
as compared to the (u,v,p) system of primary variables were discuss
ed by Roache (1982). To solve the (u,v,p) system, another elliptic
Poisson equation would need to be formed for the pressure (p) vari
able, using the two equations for velocity, regardless of whether
transient pressure solutions are desired. Following are the relative
evaluations of the (u,v,p) system and the (w, ) system for two
1Explicit boundary conditions for vorticities on solid walls are
unavailable.
(31)
dimensional flows when no free surface is present, iterative methods
are used to solve the Poisson equation, and transient pressure solu
tions are not required (Roache, 1982).
1. The (w,( ) system needs to solve one parabolic equation for
w and one elliptic Poisson equation for i as compared to the
(u,v,p) system which requires the solution of two parabolic
equations for u and v and one elliptic equation for p.
2. With the (0, 4) system, two additional differential equations
(29a and 29b) are needed to get the velocities u and v, but
the u and v momentum transport equations are more complicated
than the w transport equation.
3. At noslip walls, the boundary conditions for u and v are
known explicitly walll = 0, wall = 0) as compared to wall
which is not known explicitly. This would be a great advantage
for using implicit methods for the (u, v, p) system but non
linear instabilities of the pressure term prevents the (u, v,
p) system from using implicit methods.
4. The elliptic equation that needs to be solved for the pressure
takes much longer to converge than the elliptic equation for
p (eq. 28).
Conduction Heat Transfer in Canned Foods
The PDE describing conduction heat transfer in a cylindrical
body of solidpacked food material is presented here. The boundary
conditions present during the thermal processing of canned foods are
described next.
Governing Equation
For strict conduction heating (no movement of the material) of
a homogeneous isotropic cylindricalshaped body of the food materi
al, the heat transfer is described by the classical heat conduction
equation
ae a2e 1 eo 9a2
TF ar + F F + z (32)
This is the energy equation (26) approximated for zero velocities
(diffusion term only). This equation is quite well behaved and easy
to solve. The resistance to heat transfer of the metallic walls of
the can was ignored.
Boundary Conditions
Canned foods are generally heated with hot water or steam and
cooled with water. Thus, the heat transfer coefficient at the out
side of the can wall could be comparatively small (for water) or
quite large (for condensing steam). In general, the conditions at
the boundaries can be written as
sidewall
r =R/H 0 z 1 3 + hH =0
+ ar o =O
centerline
r = 0 0 < z 1 = 0 (33ad)
bottom wall
3e hH
0 < r < R/H z = 0 az e= 0
top wall
0 < r < R/H z = 1 + hH = 0
z k
with the initial conditions
0 < r < R/H 0 < z < 1 9 = 1 (34)
Computer Control of ConductionHeating Processes
In controlling thermal processes, the objective is to meet the
designed level of bacterial sterilization (Fd) for the process,
irrespective of any retort temperature variation TR(t), and with a
minimum of overprocessing. The lethal effects of thermal processing
are achieved during the heating as well as cooling times (th and
tc). Thus, the objective function in thermal processing is to
S T121 + T121
minimize F(th +t) = O 10 dt + fth 10 dt (35a)
h
subject to the constraint: F (th+tc) > Fd (35b)
where the integral from equation 6 has been broken into separate
heating and cooling times. Fo(t) is the accumulated sterilization at
any time t. The first integral above is the contribution to the
sterilization Fo(t) from heating, and the second integral is the
contribution to the sterilization during cooling. Temperature T(t)
is taken to be the temperature of the slowest heating point in the
product. This is so that when the design F (F ) is satisfied at
this point, all other points in the product have also been satis
fied. For a cylindrical can, T(t) is the temperature at the can
geometric center at time t.
This transient temperature T(t) is a function of transient
retort temperature TR(t), the can wall outside heat transfer coeffi
cient (h), dimensions of the can (R and H), and thermal diffusivity
(a) of the product for a conductionheating food. Symbolically,
T(t) = f (TR(t), R, H, a) (36)
For conduction heating in a cylindrical can, the function f is the
heat conduction equation in cylindrical coordinates.
For a given can size and product, the two variables that can
possibly be controlled are the retort temperature TR(t) during
heating (saturated steam or hot water under pressure in the retort),
and the time period of heating th. Cooling is generally done using
available water at ambient temperature. This makes the contribution
to Fo from cooling (given by the second integral in equation 35a)
immune from further control once the cooling process is under way.
Also, both the heating and cooling temperatures can only be applied
to the boundary of the product making it a boundary control problem.
In the case of a process deviation, the heating medium tempera
ture may go through unexpected variations beyond the capacity of the
temperature controller. Thus in reality, only one variable can be
controlled. It is the time period of heating (th). The problem in
thermal processing is to specify th for arbitrary variations in
TR(t) during heating while the process is under way in realtime.
Instead of searching for the proper th through simulation every
time a deviation takes place as suggested by Teixeira and Manson
(1982), Fo(t) can be monitored in realtime using T(t) calculated
from the finite difference approximation to the heat conduction
equation (32) with measured boundary (retort) temperature TR(t). The
heating process could continue until time th such that
Fo(th) > F (12)
thus satisfying the required sterilization Fd for arbitrary varia
tions in retort temperature TR(t).
However, the contribution to Fo from cooling (the second
integral in equation 35a) should not be neglected for a conduction
heating product. Depending on the can center temperature at the
start of cooling, size of the can, and other factors in a conduction
heating situation, the contribution to Fo from cooling could be as
much as 40% of the total. This is'unlike convection heated products,
where the container is sometimes agitated and the product is
rapidly cooled, making a small contribution to Fo during cooling. To
avoid gross overprocessing of conduction heated product, therefore,
the cooling Fo must be considered. Also, as mentioned earlier, there
is no control over the possible contribution to Fo from cooling once
cooling is under way.
Thus, the contribution from cooling cannot be neglected, it is
not a constant, and it cannot be controlled. Therefore, it can only
be estimated before cooling is actually started. Such estimation of
cooling lethality can be done through simulation of the cooling
cycle assuming a constant cooling water temperature in the retort.
The objective here is to develop an algorithm to perform this simu
lation in realtime when other control actions are going on.
METHODOLOGY
Detailed development of the solution techniques for natural
convection and conduction heat transfer models are described in this
chapter. For the natural convection heating problem, formation of
finite difference equations, setting up of computational boundary
conditions, and development of the algorithm for the iterative
solution of the complete problem, are described first. Along the
same lines, solutions of the finite difference analog to the
conduction equation are described next. Experimental study of
conduction heating is then reported. With the thermal models
working, use of the conduction heating model for online control of
conductionheated food is then presented.
Modeling of Natural Convection Heating in Cans
Because of the inevitable coupling of the governing equations,
except for extremely simplified cases, most natural convection flows
of practical interest are too complex to be solved analytically.
Thus numerical computations are generally employed to obtain the
desired solution. The present problem of natural convection in a
cylindrical enclosure is no exception. Numerical solutions to this
and similar problems demand high speed digital computers with large
amounts of memory that were unavailable until the 1960s. Since then
numerical studies of natural convection heating in enclosures are
being reported constantly. Among the numerical methods, finite
difference methods are probably the most popular in computational
fluid dynamics. The same was chosen for the present study.
Several choices were available for the finitedifference
analogs, computational boundary conditions (treatment of boundary
data in the numerical solution, as compared to real physical
boundary values), and the technique and order of solving individual
PDEs in the overall computational cycle. The relative merits of
these choices are discussed, and appropriate selections are
described in this section.
Grid Generation and Grid Stretching
The discrete points at which finite difference equations are to
be solved constitute the grid system for the problem. For computa
tional efficiency, the number of grid points should be the minimum
that is required to resolve all the significant spatial variations
of the flow. Since this problem involves a rather large step
increase in the wall temperatures together with a high Rayleigh
number (Ra), strong boundary layers are expected to form near the
walls. This is true particularly for the side (vertical) wall. In
order to resolve these boundary layers adequately, a large concen
tration of grid points would be necessary near the walls. An alge
braic method was used to generate the nonuniform grid system. The
particular algebraic expression chosen was evaluated for the proper
ties listed below (Anderson et al., 1984; Ka'lney de Rivas, 1971).
Here (r,z) refers to the physical domain and (,n) refers to the
computational domain with AE and An as constants.
1. The mapping from (,n) to (r,z) must be one to one.
2. Grid lines should be smooth to provide continuous
transformation derivatives. This requires r(E) and z(n)
as well as r and d should be continuous over the closed
intervals 0
functions defined on a closed interval are bounded, this in
turn implies that dr and dz should be finite over the
whole interval. If dr becomes infinite at some point, the
d
mapping r(S) would give poor resolution near that point,
which cannot be improved by increasing the number of points,
since A r r () (AE) Similarly for dz/dn.
3. Close spacing of grids where large numerical errors are ex
dr 0 at the
pected. For this problem, that means = 0 at the
r=R/H
sidewall which ensures high resolution of the boundary layer
near r = R/H. Similarly at z = 0 and z = 1.
4. Absence of excessive grid skewness which can sometimes
exaggerate truncation errors (Raithby, 1976).
An algebraic transformation equation suggested by Kublbeck et
al. (1980) for natural convection in cavities was
1 (+ tan[T/2 (2z1) al) (37)
2 tani/2 a
where a is
stretching.
ultimately
mation z(n)
below.
a deformation parameter that varies the degree of
However, since the computer coding of the problem is
done in the physical domain, the inverse transfor
is more appropriate in this context which is given
z =1/2{1 +  tan ((2n 1) tan a)}
Tr a 2
(38)
This concentrates the grids on either end z = 0 and z = 1.
In the r direction, the above transformation is modified to
concentrate grids on one side (side wall, r = R/H) only. Thus
r = 2 tan1 ( tan o) 
r 2 H
(39)
To check if the transformations indeed satisfy requirements 1
through 4, we first see that both the transformations are one to
one. Next, looking into the transformation derivatives,
dr 2 R tanw/2 a
d 77 IFT 1 + 2 tan2f/2 a
(40a,b)
dz 1 2 2 tan n /2
n w a 1 + (2n 1)2 tan2Tr/2 a
which are continuous in 0 < 5 < 1 and 0 < n < 1 respectively.
Also
dz
Lt d I
a+1 dn z=0
S2 tanira/2
S 1 + tan2Lt
a+1 we 1 + tan2nr/2
=Lt 2tan (1 2) by L'Hospital's rule
a+1
= 0
Similarly
dz 2 tanWra/2
Lt = Lt = 0
a+ z=l a lr ao 1 + tanZna'/2
and
dr 2 tani/2 a
Lt = Lt 1+tan2/2= 0
a 2l ir=R/H = 1 7r (J 1 + tan27r/2 a
Thus the transformations provide closer spacing of grids near z = 0,
1 and r = R/H. Figure 3 shows the grid system with close spacing of
grids near the top, bottom and sidewall. As can be seen, there is no
excessive grid skewness.
Ka'lney de Rivas (1971) also showed that when conditions 2 and
3 are satisfied for any smooth function, "extra truncation errors"
due to the nonuniform grid system are improved from first order to
second order in An and AE Vinokur (1983) also discussed similar
types of stretching functions.
Discretization of the Parabolic Temperature and Vorticity Equations
Of the several finite difference formulations that are
available for any set of PDEs, careful choices needed to be made
depending on many factors including boundary conditions, geometry of
the problem, type of solution (steadystate or transient) desired,
range of parameters involved (particularly the Grashof number in
e .= 1
mj r2t Z
m, m1 ,j
m0, ?Z Zi2
or amj 0
ei ,0= Oi I
i,0o O1
iO
e~0= ei
= 0
SIr i 1 ~ I I I I I II
I 111111
j = 0 O0,j=
SO,j
O,j
I t I IIIIllllll lii
tI IIII IItlll I
i ,n=
e. =
i ,n=
i ,nf
0
1
24i ,nI
rn/2 r n r
n12n ni
j = n
1
29
S1,j
r~ Z
ii1
The grid system and the boundary conditions
for the cylindrical can
ii
I I I I I
Figure 3.
'"'
SI I I H IIIIIIIIIIIIlIII
.t  It  I I ill
4++++1
ilIFill
I I I I
natural convection problems), as well as realistic constraints on
available resources (Roache, 1982). For this study, the transient
temperature and velocity solutions are needed and the range of
Grashof number is 108 1010 Since the temperature and vorticity
equations are both parabolic and are very similar, they will be
discussed simultaneously in this section.
The performance of the various finite difference methods was
noted in terms of stability, truncation error, and conservation
property. Stability was defined as the controlled growth of the
roundoff errors (O'Brien et al., 1950). An uncontrolled growth of
the roundoff errors leads to instability in the finite difference
methods. The stability requirements often put restrictions on the
time step as a function of grid spacings. Empirical stability re
striction on grid size of the type Ax<8/jui,jl and Ay< 8/Ivi,jl
were also reported by Torrance (1968). Truncation error is the error
resulting from the discretization of the PDE. Conservation of any
physical entity (e.g. energy, vorticity) within the finite differ
ence grid system exists if the difference equations for the entity
are summed over all interior grid points and no spurious sources or
sinks of these quantities are found. Torrance (1968) found that
formulations by Barakat and Clark (1966) and by Wilkes and Churchill
(1966) did not conserve energy or vorticity and he did not recommend
the use of their equations. Another important property of finite
difference formulations is the transportive property. A difference
formulation is transportive if the effect of a perturbation in a
transport property is convected only in the direction of velocity.
Time derivatives. The treatment of the time derivative in
equations 26 and 27 can be broadly classified into implicit and
explicit methods. Explicit finite difference methods are generally
easier to understand and program. Explicit methods have been used by
Barakat and Clark (1966), Torrance (1968), Stevens (1972) and
Hiddink (1975) in studying natural convection heating inside a
cylindrical container. For most of the explicit methods, however,
there are stringent restrictions on the allowed time step, from the
stability considerations. In situations where large temperature
gradients, eddies and recirculations are present like in this study
grid points need to be fine enough to resolve the physical phen
omena. When using most of the explicit methods, such a fine grid
size could severely limit the allowed time step and thus require
prohibitive amounts of computer time.
Implicit finite difference methods involve more than one
advance timelevel value in the same equation and thus require
matrix inversion to calculate function values at a new time level.
These methods generally allow larger time steps due to better
stability properties. However, even though a larger time step is
allowed in the implicit method, it requires many more calculations
to solve that step. For an implicit scheme to save computer time in
the overall computation, it should allow several times larger a time
step than that allowed by an explicit method to achieve a given
accuracy critrion.a
aT. IP. Shih, Mech. Engg. Dept., Univ. of Florida. Personal Comm.
Instead of using a fully implicit set of equations, alternating
direction implicit (ADI) methods were introduced in companion papers
by Peaceman and Rachford (1955) and Douglas (1955). The advantage of
the ADI method over a fully implicit method is that the set of
equations, although implicit, is tridiagonal. Further the ADI method
as applied to linear equations has a formal order of accuracy of
O(At2,Az2,Ar2). The stability of this method is also unconditional
as determined by VonNeumann stability analysis. For irregularly
shaped regions, programming could get complicated (Roache, 1982) but
for simple rectangular regions, as is the present situation, that is
no particular problem.
The practical advantage of the ADI method over the explicit
method is, however, nothing like that indicated by VonNeumann
analysis (Roache, 1982). It is unstable both at large time steps
and/or insufficient convergence of boundary vorticity. In general,
many researchers have indicated that ADI methods do allow larger
time steps and faster overall computation, by a factor of two or
more, and furthermore allow second order accuracy in time. The ADI
method is a very popular method in computational fluid dynamics.
While solving the set of PDEs (2629) Barakat and Clark (1966)
compared explicit and implicit methods for their study and decided
against the use of the implicit method due to unavailability of the
vorticity boundary conditions at the wall (to be explained later).
They allowed vorticity boundary values to lag one time step, neces
sary for such explicit formulations. The lagging of boundary
vorticity is considered undesirable (Roache, 1982) and iterations
are required because of this situation. The iterations can also
serve the purpose of preserving second order accuracy in time of the
ADI method. The second order accuracy may otherwise deteriorate due
to the presence of nonlinear convective terms (discussed below).
Since the iterations are needed anyway, an implicit set of equations
can be solved, instead, utilising the iterations. As noted earlier,
Stevens (1972) and Hiddink (1975) also used the explicit
formulations of Barakat and Clark (1966) to study the present
problem and could not resolve the convective eddies present due to
the bottom heating. Since the ADI method may allow finer grid
spacings with less stricter restrictions on At, it may be able to
pick up the convective eddies present, without taking prohibitive
amounts of computer time. Thus the ADI method was chosen to solve
the temperature and vorticity equation in this study. The ADI
formulation is written below for the temperature equation 26. At the
"half" step
en+1/2 en a(uo) n+1/2 a(rve) n a2 n+ /2 1 n
AT/2 8z ar a z +r ar Ir r
and at the "full" step
n+1 en+ 1/2 (u) n+ /2 n(rvI+/2 1 n+1
e o +(ue) n+ a(rve) n al^e 1 n
At/2 + az I + r az= r arar
Linearization. The velocities in nonlinear convective terms
e.g a(ue)In+1/2
e.g. (u) should ideally be evaluated at the same time level
(n+ 1/2 in this case) as the (or ). However, since u and v
(n+ 1/2 in this case) as the 9 (or w). However, since u and v
depends on i which in turn depends on u this would couple all the
equations together and make it practically incomputable. To avoid
this, the nonlinear terms are linearized. Various ways of lineari
zing these terms have been discussed by researchers (Anderson et
al., 1984; Peyret and Taylor, 1983; Roache, 1982). A very simple and
common strategy in linearizing is to evaluate the velocity coeffi
cient at n time level. This is known as "lagging" the coefficients.
However in this case, the accuracy in time drops from second order
(for ADI) to first order. Roache (1982) noted that such "lagging" of
the velocities could still be second order accurate if the velo
cities u and v were slowly varying. An improvement over this
linearization procedure is to update the coefficients u and v as
iterations proceed. The procedure is repeated until the values u and
v converge. One way to achieve this would be to update un+1/2 and
vn+1/2 by solving the stream function equation at the "half" time
step of the ADI (Roache, 1982). Since solution of the Poisson stream
function equation is the most time consuming operation, it was
decided against this updating procedure.
Anderson et al. (1984) noted that "lagging" the coefficients
has been widely and satisfactorily used by previous investigators,
and, recommended the use of the same for its simplicity. Thus the
velocity coefficients u and v were "lagged" in this finite differ
ence study.
Convection terms. The presence of the Lonvection terms makes
the temperature and the vorticity equation qualitatively different
from other simple parabolic equations. Proper numerical treatment of
the convection term is important. Several different ways of finite
differencing these terms have been presented in the previous works.
These include spacecentered differencing and upwind or upstream
differencing. The upwind methods have several desired properties
that are not shared by the other finite difference formulations, as
applied to the convection term. Both the upwind methods are
transportive, meaning any perturbation is carried along only in the
direction of velocity. Both the methods are also conservative so
that they preserve the integral conservation relations in the
continuum equation. The first of the two upwind differencing methods
is as follows (Lilly, 1965; Forsythe and Wasow, 1960; Frankel, 1956)
(u) = uO. uO.
(ue) (u > 0)
az Az
(41)
uOi+1 ueO
u= i+1 i u(u < 0)
AZ
Accuracy of this method is 0(At,Az). This was used by Barakat and
Clark (1966) to solve equations 26 through 29. The second variation
of this upwind method is to treat the convection term as follows
(Gentry et al., 1966)
(ue) URR L UL
az AZ
where
OR= i (uR > 0)
= i+ (UR < 0)
(42)
eL = i (uL > 0)
= eO (uL < 0)
where uR and uL are velocities as explained below. This latter
upwind method also has a formal accuracy of 0(Az). However, it
behaves like it is secondorder accurate when the quantity
( 0 or w ) is slowly varying (Roache, 1982). Torrance (1968) demon
strated that the second upwind method is indeed superior to the
first upwind method (Roache, 1982). The second upwind method was
used in the present formulation.
The velocities uR and uL are at the right and left face
respectively of the grid volume as shown in figure 4. As will be
explained later, a kinematically consistent velocity formulation
(Parmentier and Torrance, 1975) was used. In this formulation,
transport velocities (uj and vij in figure 4) are defined normal
to the grid volumes. Accordingly,
f f
UR= ui,j UL= Uil,j
f f
R= i,j L ,jI
Following is a development of the finite differences for the
convection terms in the temperature equation. In the z direction,
a(ue) a(ue) an
3z an az
9(ue) 1
an az/an
R i,j L i1,j 1
An z z
Si+1 i1
2 An
z +2 z( UReij uLi,j
zi+1 zi_1
UR > O, UL > 0 (a)
Similarly
a(uo) 2
az zi+1 z1 (uR i+l,j uLi1,j)
z 2 z (URi+l,j uLei,j)
i+1 *i
z i+ z i l i, uL i,j)
^ i+1 1 1
UR < 0
UL > 0 (b)
UR < 0 UL < 0 (c)
UR > 0 UL < 0 (d)
To avoid the use of FORTRAN IF statements for checking the sign of
UR and uL in the equations (a) through (d), the equations are
combined as below
a(ue) (RUR ) i+l,j+(UR+uRlUL+ u)i,j(uL+ i1,j 2
t 2 z+ z i
1+1 
(43)
In the r direction,
3(rve) ag a(rve) 1 (rve)
ar ar 9 2r/la aE
1
r 1r
j+1 j1
2A9
vRrj+ I/i j vLrj 1/i ,j1
2
rj+l rj1Rrj+ /2 ,j
Simliarly
6(rve)
6r
2
=r+1 2 (vRr1Nj+ 1,j +l rJ. 1
rj+1 rj1 Rr+ 1/2 i + vLj ij
rj+12 rj1vR N / v i Lrj. If 1,]
VR < 0 vL > 0
VR < 0 vL < 0
VR > 0 VL < 0
Therefore, combining the equations
a(rvo)
ar
(VRIVR )r l 1/201 j+1+{VR+IVRI )r l/(LIvL )rr ji j(,L+ vLI )r J/i j1
rj+lrj1
(44)
VR>0
VL>0
VL
 v Lj 1i ,j1
Diffusion terms. Central differencing was used for the
diffusion terms in both the temperature and vorticity equation. The
diffusion terms in the temperature equation are finite difference
below.
a ao an a ae
3 (7) = Tz Tn az
0ae
1 TZ i+ 12,j
Saz/an An
a i
Tz 1/2,j
9i+1,j 1i,j i1,j
1 i+1 i zi zil
z z. i1 An
2i+1
2An
2 i__ Ei+l, ,j eil
z i+ zi1 i+ zi zi zi_
i+1 i +1 1 
(45)
1 a ae 1 ac a ae
ar a(r) F r ar (r
ae
rliJ^
A&
ao
ri/?
11
r ar
5
1 1
r rj+1 rj
j 2A
S ,j+1 ij ,j ,j1
rJI2 rj+ r 12 r. rj
4/ rJ+1 r j12 j 1
2. O e. .
r j(rj+1 r ) /2 j+1 rj/2 r 1
a20
az
(46)
The finite difference formulations for the time and space deriva
tives are then put together in the ADI formulation. For each of the
temperature and vorticity equations, the equations are written for
the "half" time step and for the "full" time step of the ADI method.
Algebraic simplifications are done to convert the equations at each
of the time steps to a tridiagonal systems of equations. A glimpse
of the tridiagonal matrices for the "half" and "full" step is then
included.
The tridiagonal systems of equations are solved using the back
substitution procedure (Smith, 1978). The time step At and the
mesh sizes Az, Ar were chosen in such a way that the coefficients
(ai, bi, ci in equation 48a) are all positive and the matrices
diagonally dominant (bi > ai + ci). Richtmyer and Morton (1967)
showed that these two conditions are sufficient to keep the round
off errors down, in the back substitution procedure. Introduction of
the second upwind differencing method satisfies the criterion that
coefficients ai, bi, and ci are all positive (see Kublbeck et al.,
1980). The second requirement of diagonal dominance was not investi
gated any further in this study. However, at all times whenever the
numerical values of the coefficients ai, bi, ci were visually
checked, the diagonal dominance was found to be satisfied.
ADI formulation for the temperature equation is now developed.
For convenience, the differential equation for temperature is
repeated. The finite differences corresponding to each of the
derivatives in the temperature equation are then assembled in the
same order.
a3 a(ue) 1 a(rve)
TF az +r ar
a2e 1 a a
S= z +TF (rr) }
At the "half" or intermediate step ("n+l/' written as "*" in short)
* n
9. .e. .
AT/2
{(eluR)Oi+1,jUn+lu uI +lul),_(j +lul)i *,j
Zi+lZi_1 I1uR R i U,
(V nlv n n n ,n+1+(v+i
'r(r j+lrj
*
n 0 n zn zn
r r j+ rjl1 rj+1 rj rj r j 1
r. r r r r r.r
3 j+1 j1 +1 j j j1
(47a)
At the "full" step
on+le .
0. J e
1,j 1,3
AT/2
1 n n + n n n n n 0*
z{( l i+1,j + u+ "I u"+ RuL ) i,j ul ( L i'1,j }
(V/20 +l I )r n+1 n r +1 n n n+1
S(v vj+ R Ij RL1 L r ij J 2vl )rj i 2ij1
+ rj (rj+lrj_1
z z
S2 il~ji
Zi+lzi L Zi+l zi
* *
. . .
1,3 i1,j
i ii
n+l n+1l
+ ( 2 r.0, .
2J j++1/ j+1 ij
r (rj+l r ) rj+ r.
3 j+1 31 j+1 3
,n+1 n+1
l j j1
'J'Jl
(47b)
Vn n ) r .? n ) n
(v L L /r iJ (vL L rl j 1
After algebraic manipulation, temperature equations at the "half"
time step
ai il ,j+bi i,jci i+,j = rhsi
(48a)
where
S zi+1 Zi+l i Zii1
1 + 1/ (n+U n n+1nj) + 1 1 +
S 1/ n 1 1 1
b = R RI) 2 i
i T z +1z. R R L + iz. zi zzi
C i : ,)
zi+1 11 '+'z +1 i
rhsi = dn +ee +fe
i1 ij1 1 i,j+l
1/2v n+ vnjl)r ' rj
d = r r I + 1FTr r rr 1
(r+l j+ 1 J 1 j J+ J 1
e =
vVc nl,_ rj _li( nl n r 1/
R R L v r i 1 rj + 2+.r
rj rj+, rj_1 rj ( j +I j ) I
5 1 r+rj rjrj
1 jVR )rj rj +/2
f andR
and
f
uR = ui,j
f
uL = ui
L 1,j
f
R= Vi,j
f
L ,j1
After algebraic manipulation, temperature equations at the "full"
time step
a n+l b l c +l = rhs
a i,j1 b1,j j i,j+l rhsj
(48b)
where
1/2(vn +n I )r r
j r.r. rr.r r. + r.
n Lin n
S(v +vnR )r (vL _V )rv j/ 1 r lr r 
S. r.(r + r ) jr. r. r.r.j+1
1I Rn n ]r r
j r. r. r. r. jl r. r .1rj)
j j3+1 1 J 3+11 j +1 3
rhsj = dl,j + eOj + fi+,j
u l n+ l 1
d = +
Zi+1zi1 Zi+lzii)(zZiz)
1 12 n n i n 1 1 1
(URn 1 RI ) i
f R +
zi+lzil (zi+lZi1)(zi+1zi)
and
f
uR = u j
R = ij
f
uL = ui.lj
L 1,j
f
VR = Vi j
f
L i ,j1
The tridiagonal matrices for the temperature equations at the
two ADI time steps are given below. At the "half" time step, the
equations are written for a column (going from the bottom wall to
the top wall). Specified boundary temperature is used both at the
top and bottom walls. *
b1 c1
a2 b2
a3
am2 bm_2 cm2
am1 bm1
rhsl+a1 0,j
rhs2
rhs3
rhsm2
rhsml+aml m,j
1 /I
(49a)
At the "full" step, equations are written for the rows (from
the centerline to the sidewall). At the centerline, the insulated
boundary condition (equation 75) is used due to symmetry. At the
sidewall, the specified boundary temperature is used.
blal cl i,1 rhs1
a2 b2 c2 i,2 rhs2
a3 b3 c3 ,i3 rhs3
an2 bn2 cn2 0i,n2 rhsn2
n+
an1 bn1 in1 rhsnl+cn1 i,n
n1i,n1 i,n
(49b)
Upwind differencing of the convective terms in the vorticity equation
a(uw) a(uo)
az) = similar to (
az az
(uRluRl)"i+l,j+ (uR+IURluL+IuLI)wi,j (uL+IuL)wil,j
zi+1 zi1
a(vw)
ar
(50)
(vR IVRI)wi,j+l + (VR+ IVRI VL+ IVLI)wi,j (VL+ IVLI)&i,j1
rj+1 rj1
j+1 j1
(51)
Upwind differencing of the diffusion terms in the vorticity equation
(52)
+2 i+ mij iij
zi+1 z1 i+1 zi zi z i1
a1 a(rM)3
as a a(r))
= 1' aTr ar
1
rj+1 rj1"
2A9
1 a(rw)
r+ 1/ ar
,j+ 1/?
AE
1 a(r O)
rj._ 1ar i J1/
2 rj+1i,j+l r mi
rj+ rjl rj+ 1/2(r+1 r.)
2 j+3+1 J
r.. j r. l )
 J I/2, j rj1
(53)
Central differencing of the buoyancy term
ar r+1 rj1
(54)
a2
az2
ADI for the vorticity equation
a + a(u) + (v+) a2 2 + a 1 a(rw)
S+ I ar } = GrPrD + Pr az ar+ r 3r()
At "half" or intermediate step,
* n
13i ,jWi ,j
AT/2
Zi+lZ 1Lj
1_ n nn l vn v nn nI
Iv v *v l ,
+(r 1rj 1I ,j+R1 R R L vn+ v ,j (vn j1
j+1 :1
n+1 n+1
1 e P
SGrPr2( i,j+1 i 2 Pr
j+1 J1 i+1 Zi1
n n
2 Pr rj+1 ij+l i,
rj+r ) r j+/2rj+ r )
'j+1 j+1 + /2 j+1 3
* *
mi+ljti ,j
i+lzi
i+1
* *
i ii
n n
rj i ,rj1 "ij1
rj. /2 rrj )
(55a)
At "full" step,
n+1 *
ST/2o.
AT/2
+ 1n n un+nln (un+unI
Z + (u I u I )uR i+l,j+(UR URI ULI i,_ UL il,j
i+1 1 l ,
1 n n n+1 n+lnln + n n+1 n lin+l
(rj +lrj1 )(vRv )wij+1 vR R L L ij L L L ij 1
n+1 n+1 *
GrPr2 ij 1 i ,j1 2 Pr Oi+1,j 'ij "Oi, i1,j
Sr +1 1 zi+zi1 zi+1zi z z i1
n+1 n+1
2 Pr rj+oi j+1 r ij
(rj+lrj1 r +1/2 (rj+lr) i
n+1 n+1
rjoij rj1 "i ,j1
r i/2 r j1r
(55b)
After algebraic manipulation
*
ai j+ bimj cii+,j = rhsi (56a)
where
1/2 (u +jul ) Pr
ai Z i+lZil zi+lziilzizi)
bi E z z (un+iuiun+tunI) + rz 1 +z 1
1 AT zi+ i1R R L i+ i1 +i+1 1i
1/2 (uR R ) Pr
c. +
1 z 
ci z i+1z1 ii+1z 1)(i+lzi
(n+l n+1
rhs. = D2. +Ewm .+Fwn 0.5GrPr2 ij+l j
S ,j1 1,i 3 i j+1 rj+1 rj
1/2 (v + vjI) Pr rj1
d = (v I+ L J
j+r 1 rj1/2 rj+1 rj_1 j r1
n n n nlr
1 i (VR+IVI IvI) Pr rj 1
1 R R L L Pr 1
e T= 1/2 r r r1 r r r rj _j/
S T/ j+1 j1 j+ rj1 2 rjlr] rj
1/2 (VRVR Pr rj+1
f=1 +
rj+1r1_ rj+1lrj1 Jrj /2 rj+lr )
After algebraic manipulation
aj n+l b mn(l n+l
a ,1 + b i, + c ,j+ rhs (56b)
where
1/2 (vn+ vj ) Pr rj_1
aj= (r + r. r r )Lr.r. )
j+1 j1 1/2 j+ j j1
(v n n n n
1 R+IVR vL+Iv L + Pr rj 1 +
bj= A + 2(j+lrjr (r rj )Irj (rj+rrj rj/2 1 rj1r )
j+1 j1 j+1 1
/2 (VR R Pr rj+
c.= r +
J r rj+l rjr.jlrjl/ rj rj
(en+1 n+1
rhs. dnw +en .+fn 0.5GrPr2 J+1 ,J
1,j 1, i+1,j r j+1 r
1/2 (u+I u~n I)
Zi+ +1_Zi 1 11)
z i+1z z. zi+1zi1)(zz
i+1 ii LI) 11
1/2 (n URI Pr
f =I +
Zi+lZ iI zi+lzi Zi+li)
The tridiagonal matrices for the vorticity equations at the two
ADI time steps are given below. At the "half" time step, equations
are written for the columns (from the bottom wall to the top wall).
Both the bottom and top wall vorticity values are used from previous
iteration.
b1 c1 01,j rhsl+alw ,j
a2 b2 C2 "2,j rhs2
 a3 b3 c3 m3,j rhs3
am2 bm2 cm2 "m2,j rhsm2
am_1 ,bm ml,j rhsml+aml om,j
(57a)
At the "full" time step, equations are written for the rows
(from centerline to sidewall). The centerline boundary vorticity is
specified (=0) and the sidewall boundary vorticity is used from
previous iteration.
n+1+l
b1 cl i,1 rhsl+al wi,O
a2 b2 C2 "i,2 rhs2
a3 b3 3 ,3 rhs3
an2 bn2 cn2 "i,n2 rhsn2
n+1
an1 bn i,n1 rhSnl+cnI "in
(57b)
Discretization of the Elliptic Stream Function Equation
The stream function equation (28), obtained from the definition
of a u is also a statement of the circulation theorem
f ~z ar
which is
f w dA = J ch (58)
area perimeter
element of area
Thus the difference formulation of the stream function equation has
to satisfy the circulation theorem in addition to conserving mass.
Difference approximations that conserve mass are more difficult to
achieve in a curvilinear coordinate system as compared to the
Cartesian coordinate system.
To satisfy both the circulation theorem and the conservation of
mass, Parmentier and Torrance (1975) introduced the concept of
tangential velocities parallel to the control volume faces and
transport velocities perpendicular to control volume faces (Figure
4). In what follows, their approach has been adapted for the non
uniform grid system.
The circulation theorem (equation 58) is written for a grid
volume around node (i,j) as shown in figure 4
ij ( 1/2z 1(rj rj
v t v t t u
S(vj i1,j)(j+/2r j 1/) (u juij1)(zi+1i/2z1/
t t t t
= 1i,j i ,j i,3 i',j1 (59)
ij zi+/2z 1/2 r.+ 2 r.
i + W zi 1/21/2
j+1/2
  Zi/2
I zjI
r]. l
Notations of various quantities defined on a
nonuniform grid system in cylindrical
geometry
Zi++I
iI
0
ii,'
I 1
r i./2 r] r
Figure 4.
iJ Z+Z/2
u 
ee
rj
The tangential velocities u ,j and v,j at the faces of the
grid volume are obtained by finite differencing equation 29
ut ij+ ij
ij rj+l/2 rj+ rj j
(60a,b)
v  ip
,t i+lj i ,j
i,j r (z i+ zi)
Substituting these definitions in equation (59)
1 i+l,jj 3J i,ji,j
I i+ Zi i
1 ri j+l ij 1iji,j1
rI r )I
rj+ 1/2r 1/2 rj+ 1/2rj+1r ) r.1/ rj 1
(61)
Note that the same finite difference equation (61) can be obtained
from central differencing the stream function equation (28) without
introducing the circulation theorem. This is shown in Appendix A.
After algebraic manipulation,
 "i = Ai,j+ B i,j + CJi + Di,j1 + Eij+1
(62)
where
B (z zi j z i ii rj { (rj+lrj) jr jr jl
+ 1 1/22+ 1 '/1
C+ r.zi.1
C [zi+ /2zi_ 1/2 )Zi+i )
D (rj+ 1/2rj. 1/2 Jrj. 1/2 [rrj1
E =1
(r+ 1/2r 1/2 rj+ 1/2 rj+1r
The grid point velocities are
interpolation of the transport veloc
then achieved
t
ities u. j and
J
th
r. r ? t 1 r1/21 r t
rj +/ j .r /2u 'i rj+ 1/2 rj 1/2 u 1
. i z l/? vt
vi 1j Zi+l/2 .i1/2 j,
+ i+z 1 zi t
zi+ 1/ zi 1/,
rough linear
t
i ,j
(63a)
(63b)
ui j
1 ,3
r i (z / J zi 1)
Second, transport velocities are defined normal to the grid volumes
*
in terms of a new set of stream function values i,
f i. ,j .,j1 ,f i, 1 i1,j (64a,b)
1, r ( 1/2j 1,j j z 1/2i zi1/2
The stream function values i,j are defined at the corner of the
grid volumes. These i,j values are related to the previous stream
function values ij through a consistency criterion which says
f
Lt u u = 0
Ar+0 i,
(65a,b)
Lt v v = 0
Ar+O
ti,j was defined as
,J 2 i,j + j) + X ( ,j+1+ i+1,j+1) (66)
then, using definitions of vj and v (equations 64b and
63b), it was shown that the consistency criterion (equation 65b)
require
j + xj = 1 (67)
J J
Similarly, using definitions of uj and u it was shown that
II >J ui j
the weakest condition that satisfies equation 65a was the recursion
relation
1 + (2j 1) xj (68)
j, 1(68)
Xj 2j + 1
Using boundary conditions on the axis, it was found that
X0 = 0.25 (69)
The transport velocities defined this way conserve volume flow
exactly and are consistent with the grid point velocity field as the
mesh is refined.
The transport velocities uj and v are the velocities
used in the energy and vorticity transport equations. The grid point
velocities u. i and vi, are the velocities plotted in the
velocity plots.
To solve for I from the vorticitystream function equation
61, we note that it is an elliptic Poisson equation as compared to
the parabolic equations for temperature and vorticity. Direct (non
iterative) methods to solve the equation are available that are
considerably (10 times) faster than an iterative method like the
successive overrelaxation technique (to be discussed later).
However, direct methods usually require large amounts of storage,
and available computer codes are often limited to rectangular or
simple domains or have restrictions on the type of boundary
conditions. Shih (in press) discussed suitability of direct methods
for "large" systems of equations. Direct methods also have round off
error propagation problem. With direct methods, roundoff errors can
be incurred at each mathematical operation, and simply accumulate
until final answers are obtained. When iterative techniques are
used, the presence of roundoff errors at the end of any given
iteration simply results in those unknowns being somewhat poorer
estimates for the next iteration. For practical purposes, roundoff
error in the final converged values of an iterative scheme is only
that accumulated in the final iteration (Hornbeck, 1975). A very
efficient direct method for solving Poisson's equation was described
by Schumann and Sweet (1976) and the code was available from the
National Council of Atmospheric Research (NCAR) software library.
However, on closer look the algorithm (GENBUN) was found to be
limited to coordinates transformed in one direction only. Since in
this study the boundary layers were expected to be present both
along sidewall as well as top and bottom wall, coordinate stretching
were considered necessary in either direction. This made the above
algorithm unsuitable for the present study.
A popular approach to solve such a steady state equation is to
formulate it as an unsteady equation and solve for the asymptotic
steady state solution. ADI methods have been used for this purpose.
These methods were known to be faster than the successive over
relaxation method (to be discussed later) for a rectangular region
(Birkoff et al., 1962). However, the optimum sequence of parameters
aR. Sweet. U. S. Dept. of Commerce. National Bureau of Standards.
325 Broadway, Boulder, Colorado. Personal Communication.
that makes the ADI method advantageous is not available in general.
Programming the ADI method for a nonrectangular region may get
complicated, and they are not known for sure to be faster (Roache,
1982).
Another popular method to solve the unsteady formulation of the
stream function Poisson equation is the successive overrelaxation
(SOR) method (Frankel, 1950). It is very simple and effective. In
SOR, the set of solutions are over relaxed by a factor 1 < n < 2
to speed up convergence. The exact value is found through simple
numerical experimentation. It is easier to program compared to ADI
methods. Most of the earlier researchers have used the SOR to solve
the stream function equation. In all the five formulations of
natural convection in a cylindrical geometry, studied by Torrance
(1968), SOR was used to solve the stream function equation. However,
it is time consuming and could end up taking 90% of the total time
spent to solve the problem. Nevertheless, SOR was chosen to solve
the stream function equation in the present study for its
simplicity. Following is the development of the SOR formulation.
Rewriting equation 62,
ij = C il ,j+Ci+,j+Dijl+Eij+l+ij
To get the SOR formulation,
k+1 1 k+1 k Dk+ + k (70)
i,j B 1,j i+l,j ij1 1i,j+l+ i,j
k+l k+ (71)
i,j ,j i,j
Using equation 71, stream function values ,j at all interior
points are updated. The entire sweep is repeated until the
convergence criterion
k+l k
Max ij 1 ,j (72)
i ,J
is satisfied. The value of e chosen for this study was 105. In a
*
transient calculation like this study, the optimum Q may change
*
from one time instant to another. However, the optimum Q found
through calculations during the very first time step was used
throughout the entire transient calculations.
Computational Boundary Conditions for the Finite Difference
Equations
A PDE such as equation 27 for w can describe a wide variety
flow situation. The flows (solutions) are distinguished only by the
boundary and initial conditions and flow parameters like the Grashof
number. Thus, specification of computational boundary conditions,
besides affecting numerical stability, greatly affects the accuracy
of the solution (Roache, 1982). Using the boundary conditions
already noted for the partial differential equations in the previous
chapter, the boundary conditions for the finite difference equations
are now derived.
The boundary condition for temperature is rather simple. When
the can wall temperature is specified, the computational wall
temperature value of the fluid next to the wall is made equal to the
specified value. On the adiabatic walls and at the centerline (which
is also considered adiabatic due to symmetry), a Taylor series
expansion gives
1 W + 0 An +1/2 j An2 + O(An3) (73)
ao
where stands for the derivative of temperature in the direction
normal to the wall and An being distance normal to the wall.
for = 0, (74)
for = +20W
OW+1 = W +1/2 820' An2
an2
= e + 0(An2) (75)
Thus if eW+1 = OW is set, wall temperature is evaluated from the
adjacent temperature and it is second order accurate even though
ao is only first order accurate (Roache, 1982).
In the ADI method, boundary conditions are needed at the
computational "half" step. Since wall temperature 0W is not a
function of time, the boundary temperature value at this "half" time
step is simply equal to eW.
Unlike temperature and velocities, vorticity and stream
function are not primary variables. This makes specification of
their boundary values difficult, particularly for vorticity.
Improper specification of boundary vorticity is found to be
destabilizing in general and the same was found in the present
problem. Vorticity is produced at noslip (wall) boundaries. It is
then diffused and convected into interior points. No explicit
expression for wall vorticity is available. It is related to the
interior stream function values by the vorticity streamfunction
equation (equation 28). However, this would make the vorticity
equation implicit with the stream function equation and the number
of simultaneous equations increase drastically. To avoid this,
Barakat and Clark (1966) chose explicit methods over implicit and
used vorticity boundary values calculated at the previous time step.
Wilkes and Churchill (1966) used the ADI method but similarly used
boundary vorticity values from the previous time step. Thus, in
their formulations, wall vorticity values lagged by one time step at
all times.
Roache (1982) discussed the effects of such lagging of boundary
vorticity values. It forces the use of a smaller time step since
larger time steps could make the solution inaccurate and unstable.
Since larger time steps are a major motivation for ADI, updating of
the boundary values through iteration is preferred. From the work of
Torrance (1968), Roache also noted that the convergence of the
boundary vorticity values actually imposed a timestep restriction
of the form AT < a / Az2 where "a" was some number dependent on the
problem and convergence requirements. Thus, due to the implicitness
of the boundary vorticity values, ADI actually has a timestep
restriction like the simplest of the explicit methods. The
definition of vorticity o was used at the boundaries to obtain
expressions for boundary vorticities. At the bottom boundary,
t
SO,j v,j
O,j (z l zO)
t t
UO,j UO,j1
( rj+ 1/rj 12
 1,j
r.z z12
since
t
Vo,j
t
uj= 0
_ ( ,jOj)
rj(z1 z0)
S0 from eqn 60a
and uj=j 0 from eqn. 60a
At the top boundary,
t
SVmj Vmlj
m,j= (z z
t t
Smj um,j1
( rj/ r.j
_ ml,j
r. (z z z z 1
J m m1 m m2
since
v .= 0
m ,j z 
S m ,jm1,j
Vml,jjr rn(ZmI)
 im1,j
 r.(z z )
I m m1
utj and u t , 0 from eqn. 60a
mij mll=,
(76a)
Ip,j
r z1
^J
(76b)
At the centerline,
t t t
i,0 Vil,0 ui,0 Ui,
i,0o z. i+ z.1, r i/ 0
=0 (76c)
since
t O and v, from eqn. 60b
1,0 1a1,0
t i 4,I i,0 2_ 1
io rl O) rli
ui,0=0
At the sidewall,
t t t
S,n Vil,n i,n i,n
i,n= (z+l/2 z l/ (rn rnl/
= in1 (76d)
r ni(r r )(r r
n n n1 n n
since
Vi, and vi 0 from eqn. 60b
1i ,n 1i l,n=
t i,n i,n1 1,n1
1i,n1 r/rn r r r n rl)
n /2 n n1 n_/2 n n1
u. =0
Identical expressions can be derived using Taylor series expansions
of equation 28 at the walls. This is shown in the appendix B.
These expressions are first order accurate. But it is the only
form consistent with the kinematic velocity formulation (Paramentier
& Torrance, 1975). The second order form using Taylor series
(Jensen, 1959) has also been used. However, sometimes the second
order form destabilized the calculations (Wilkes & Churchill, 1966)
and sometimes it was less accurate than the first order formulation
(Beardsley, 1969).
The vorticity boundary value at the computational intermediate
or "half" step of the ADI method is more difficult to set than the
corresponding temperature value because the vorticity boundary value
is also a function of time. The following simple way to define this
vorticity value on the boundary is often used and gives sufficiently
accurate results (Bontoux et al., 1978)
n+1/2m+1 n n+1m+l
W = 1/2(W+W ) (77)
where
n = wall vorticity value at time step n
m+l
n+1 = most recent estimate of wall vorticity at n+1
step using equation.
m+1
nmW2 = intermediate vorticity value between n and
(n+l)th time level at the (m+l)th iteration.
Israeli (1972) also used this equation to define intermediate (half
step) boundary vorticity though he used a different equation to
nm+l
calculate boundary vorticity wW
Convergence Criteria for Iteration
As discussed earlier, the set of finite difference equations
were solved iteratively until convergence. The measure of conver
gence could have several possibilities. Ideally, at convergence, all
the variables should have converged. For practical reasons, it would
be sufficient to check on the variable that converged the slowest.
This is so that when this variable has converged, all others would
have converged. However there are no general guidelines to choose
apriori the slowest converging variable. Interior velocity
components were suggested by Roache (1982) as a measure of
convergence. Frequently, researchers in the past have decided on the
vorticity boundary values as the variables to watch for the
iteration convergence of the entire problem. Roache (1978) referred
to tests on boundary vorticities as the most sensitive test for
convergence. The same variables were used to check for iteration
convergence in this study even though they may not have been the
variables that converged the slowest.
For the chosen variable, there are no definitive criteria for
the measure of convergence. A relative error criteria was used for
this study, which is
n+m+1 ~n+lm
< 6 (78)
n+1
at all solid boundaries. The boundary vorticities n+1 were found
from the interior stream function value using equation 76.
As an example of other measures of convergence, while studying
natural convection in a rectangular cavity, Kimura and Bejan (1984)
used the following convergence criteria on interior points
E E n+lm+1 n+1i
i j i ij < 6 (79)
1I1m+11 6
n+1
i j
together with a convergence criterion on the calculated Nusselt
number.
The choice of 6 is also problem dependent and no general
guidelines exist for it either. Even for very small values of 6,
the iteration of a solution may be stopped prematurely for certain
types of solutions which are not uncommon (Roache, 1982). Values
ranging from 103 to 104 were used in their study, and the
chosen value was the largest of these values that did not cause any
perceptible changes in the final solution.
Algorithm for the Iterative Solution
The algorithm for the iterative solution of the complete system
of finite difference equations (equations 47,55,61,64) is now
described. The grid system is generated first. The vorticity, stream
function and velocity values are set to zero which is their initial
condition (t = 0) since initially the liquid was stationary with
uniform initial temperature throughout the body of the liquid. The
nondimensional temperature values are set to one (initial temper
ature). The temperature boundary condition for t>0 is then enforced.
The set of equations are then solved for finite increments of time
(At) with iterations performed on boundary vorticity at each time
increment. This is described in figure 5 and explained as below. The
selection of time step the At is discussed under the RESULTS
section.
REPEAT OVER n
*n n"
lm
2=1.0
r
ne m 1.0
ITERATE OVER m
n
I r
=.
P +
r e1
n/2m= f, n, un+1m vn+lm, /2m)
Q 1 Q I
n+1m 2 n12m
0 Q f 2( .
n+1m n+1m
u Q v
"a a
n+lm
W )
n+lm
en + 1
n /2mo
wD = 0
2n1/2' f r n n4
n +3
w = f,(w, u
n+lm I
U) =O
r
n+1" f n 1/2
^ n+lm
) = Wr
ITERATE
m
n+lm

n+lm
, Q
,n/2m
, p 2
n+1m n+l1
uS vS
n1/2m n+1m
4 w 'a
n+1m
"r
f5 (n+1m
fo 0
(,n+lm
S 7 j,
n+m+l
= n+
=yw
m
S,, n+
+ (1y)w
r
UNTIL 
u(vn+1m
(u,v) l
r: solid boundary
Q: interior points
: centerline
I ^ *
m,w,w: temporary
storage
( n+1m
f r~l!
UNTIL DESIRED TIME REACHED
Figure 5. Algorithm for the iterative solution
of the set of equations
UNTIL 
n+m+1
n+1
"r
n+lm+l
Wr
n+1m n+lm
W( 5 O
1. The temperature equations (47a and 47b) were solved using
the ADI technique. This gave new temperature values after
incremental time At.
2. The vorticity equations (55a and 55b) were then solved
(also using the ADI technique). The interior vorticity
values calculated were provisional since old values were
used for boundary vorticity.
3. The stream function equations (61) were then solved for
interior stream function values using the point SOR
technique. This required only interior vorticity values.
Previous stream function values were used as an initial
guess.
4. Boundary vorticity values ( ab) at solid walls (top,
bottom and side) are first calculatedusing equation 76
based on the newly calculated stream function values.
However, before updating, the boundary vorticity values
(b) are underrelaxed as
m+1 m+1 m
n+1 n+l1 n+ (80)
"b = b + ( y) Wb
m+l
where n+ = boundary vorticity estimated using
equation and 76, with Oy
5. Steps 2 through 4 are repeated until the estimated wb
values (using equation 76) in two consecutive iterations
meets the convergence criterion
~n+lm+1 ~n+lm
ab %b
< 6 (78)
n+l
Wb
for top, bottom and sidewall. When the convergence
criterion is satisfied, temperature, vorticity and stream
function values are considered the transient solution at
this time instant.
6. The velocities are calculated from the stream function
values using equation 63 and 64
7. Steps 1 through 6 are repeated for every time increment
until the desired time is reached.
The underrelaxation of boundary values at step 4 has often
been considered necessary in the literature (Peyret and Taylor,
1983; Roache, 1978) for convergence of the solution. Under
relaxation is usually employed to make a nonconvergent iterative
process converge (Hornbeck, 1975). This was found quite true for the
present study. Without underrelaxation, the time step allowed for a
stable solution got severely reduced to the point that solution to
the set of equations became impractical because of the amount of CPU
time required. As the computations proceeded, there were two sets of
values in storage. These were the estimates of the temperature,
vorticity and stream function values at the present time step and
converged temperature, vorticity and velocity values at the previous
time step.
Coding of the Program and the Hardware
The algorithm presented in figure 5 for solving the vorticity
stream function formulation of the natural convection problem was
coded in FORTRAN 77 with individual modules for solving temperature,
vorticity, stream function, velocity and boundary vorticity values.
Structured programming practices were used. Use of GO TO statements
were avoided except in two situations to simulate a WHILE statement.
The WHILE statement was unavailable in the particular version of
FORTRAN 77. Single precision of the variables was known to carry 14
significant digits for the particular computer and the same was
taken to be sufficient for this problem.
The input data file for a typical run is in Appendix C.
The computer used was a CDC Cyber 170/730, with two 60 bit CPUs
crunching 60 bit words at 400 MHz, 20 parallel processors, and 1091K
of 60 bit word central memory.
The plotting was done using a plotting package from the
University of Illinois1. The convenience offered by the plotting
package cannot be overemphasized. The plots were viewed on a
Tektronix 4006 terminal and hardcopies were made on Tektronix 4662
plotters.
Modeling of Conduction Heating in Cans
The partial differential equation (PDE) governing conduction
heating of a cylindricalshaped solid was discretized and finite
1W. E. Dunn, Mechanical and Industrial Engineering Department,
University of Illinois at UrbanaChampaign. Personal Communication.
difference using an alternating direction explicit (ADE)
differencing method, which is explicit but still unconditionally
stable. The transient temperature values predicted by performance of
this ADE finite difference method was compared with analytical
solutions and experimental measurements.
Discretization and Solution of the Equation
Keeping in mind that the equation was needed to be solved in
realtime for online control purposes and the ADE method was
already tested (Teixeira et al., 1975; Teixeira, 1971), the same was
selected. The ADE formulation was first reported by Saul'yev (1957).
This was later translated and described by Allada and Quon (1966),
and was first introduced for applications to thermal process
simulation by Manson et al. (1974). The method as applied to the
diffusion equation is unconditionally stable (Saul'yev, 1964) for
temperaturespecified boundary condition. It has a formal order of
accuracy of O(At2, Ar2, AZ2 ). Being explicit, it does not require
inversion of matrices as would be required by an implicit method. In
a fluid flow problem, the ADE formulation is generally not preferred
since the presence of the convection term (in any difference form)
makes the complete formulation unstable or returns it to simple
explicit stability limits ( a / for a one
dimensional problem; Roache, 1982). Another shortcoming of this ADE
method is the fact that even though it is an explicit method, it is
actually implicit in boundary conditions. To handle this implicit
ness, either explicit expressions for the temperature values need to
be derived algebraically for each of the boundaries or the computer
has to solve a simultaneous set of equations at the boundaries (see
Appendix D). For Robin's boundary condition, the application of the
ADE method required separate algebraic expressions for the top, side
and bottom wall boundaries. Separate expressions were also required
for the top and bottom corners and the centerline. This is in sharp
contrast with the situation if the ADI method were used. Adjustments
would be necessary at the boundaries for the ADI method, but for use
of the ADE method it became clumsy (see appendix D).
For temperature specified at the boundary, this is trivial and
no additional manipulations as noted above are needed. Experiments
conducted later showed that because of the low thermal conductivity
of food (which is mostly water), the Biot number (= hR/k) is quite
large and it is the internal conduction resistance that is limiting.
Thus a convection boundary (Robin's) condition can be approximated
by a temperature specified (Dirichlet) boundary condition for this
purpose. The two shortcomings of the ADE formulation are, therefore,
of little consequence in the case of conductionheated food. The
equations developed for the convection boundary condition are given
in appendix D.
Coding of the Computer Program
The ADE formulation for transient temperature calculation using
temperature specified boundary condition was coded in PASCAL and
used as a subroutine in a real time control algorithm (to be
discussed later).
Experimental Studies
Experiments were performed to record the transient temperature
values inside cylindrical cans during conduction heating. The effect
of different surface heat transfer coefficients, that were obtained
by using different media (steam and water), on the transient
temperature response was also noted.
The experimental work consisted of series of heat penetration
tests on No. 303 cans filled with a 10% bentonite suspension as a
conductionheating food model system (Niekamp et al., 1984). Eight
cans were fitted with an Ecklundtype copperconstantan thermocouple
mounted midway up the can wall, so as to have the junctions located
at the geometric can center. In order to achieve the two different
levels of headspace vacuum, four of the cans were filled at 900C
and the other four cans were filled at 670C. The cans were
identified according to fill temperature and then processed with an
extended steamcook to stabilize the bentonite suspension. An
experimental retort was used that had been designed and built at the
University of Florida with the capability of using pressurized
water, pure steam, or steam/air mixtures under controlled rates of
recirculation. The retort was fitted with a special rack designed to
hold the eight cans suspended upright in the mainstream of the
retort chamber, as shown schematically in Figure 6.
The rack consisted of a stainless steel plate with eight
cicular holes through which the cans could be inserted. Special
fixtures were welded at the edge of each hole so that the cans could

Full Text 
124
natural convection heating, even though most of the core exhibits
uniform radial temperature due to stratification, the symmetry about
the mid height is lost. This is due to the convective loops from the
bottom heating creating isotherms completely different from the con
duction heating. Also formation of a boundary layer at the sidewall
in natural convection heating clearly separates it from the
conduction heating situation.
Assesment of the Numerical Method
Convergence. The solution to the set of finite difference
equations should approach the true solution to the partial differ
ential equations having the same initial and boundary conditions as
the mesh is refined. This is referred to as convergence of the
finite difference solutions. For linear PDEs, Lax's equivalence
theorem states that for a consistent finite difference approximation
to a properly posed initial value problem, stability is the
necessary and sufficient condition for convergence. The same theorem
has not been proven for nonlinear PDEs (Anderson et al., 1984).
Thus convergence cannot mathematically be proven for the present
problem.
However, as shown in the previous sections of this chapter, the
axial temperature profiles and the radial velocity profiles compared
very well quantitatively with available experimental data. Several
phenomena revealed by the solution including formation of the tem
perature and velocity boundary layer, stratification, and, Benard
type convection were expected, and, have all been observed by
89
It is unconditionally stable, thus allowing large time steps. It is
also explicit, thus taking small computation time per step as
compared to many implicit methods.
The time increments At, At^, At^ and the finite difference
calculations were adjusted so that the inequality (72) was
satisfied. On a DEC PDP 11/23, the following computation times were
found to be feasible
At 12 seconds
At. =1.5 seconds
Atp = 9 seconds.
The control logic flow diagram for the retort thermal process
is given in figure 8. The input data are first checked against
specifications for the product. If they agree, steam is turned on
and the computer completes the venting cycle. The retort temper
ature comes up and the computer through a controller interface,
attempts to maintain the design retort temperature T^ As the
heating cycle continues, retort temperature TR(t) is read at
intervals of At T(t) is then calculated from TR(t) using the
finite difference approximation of equation 32. Using equation 35a,
F0(t) is then calculated. The input data includes a specified
pheat ^ which is the F0 value normally achieved at a time when
heating would be terminated for a design total F of F ^ When
oo
F0(t) exceeds the value p^eat t the computer also simulates the
cooling cycle in addition to calculating T(t) and F0(t). If the
F0(t) accumulated so far, together with the simulated contribution
from cooling, exceeds the design total FQ value for the process
(Fq) i.e. when
23
experimentally determined temperatures at three axial and two radial
locations. Thermistor thermometers were used for temperature
measurement. The calculated temperature values were reported to be
within the limits of experimental error, over a range of Gras'nof
numbers.
Evans et al. (1968) measured the transient temperature field
during natural convection of fluid in a vertical cylinder due to
uniform heat flux. The top surface of the liquid was free and no
transfer of heat or mass was assumed at the liquidvapor interface
at the top. Dye tracers were used to qualitatively observe the flow
patterns. They reported symmetry about the vertical axis in the
temperature field. On the basis of the experimental observations,
the convective liquid flow was modeled in terms of three regions:
1. a mixing region at the top
2. a central core
3. a boundary layer rising at the heated wall
They developed three different equations to describe the three
regions. These equations were solved simultaneously to predict the
overall behavior of the system. They reported good agreement between
the model and their experimental data.
Shyu and Hsieh (1985) made a numerical study of unsteady
natural convection in thermally stratified water in a cylindrical
enclosure. They studied the effect of placement of insulation on
maintainence of the stratification. Transient flow and temperature
fields were provided for the study.
76
Z Z
i j
n+1
ij
m+1
0)
z z
i J
n+1
di j
n+1
dij
m+Ti
mi
(79)
together with a convergence criterion on the calculated Nusselt
number.
The choice of 6 is also problem dependent and no general
guidelines exist for it either. Even for very small values of <5,
the iteration of a solution may be stopped prematurely for certain
types of solutions which are not uncommon (Roache, 1982). Values
3 4
ranging from 10 to 10 were used in their study, and the
chosen value was the largest of these values that did not cause any
perceptible changes in the final solution.
Algorithm for the Iterative Solution
The algorithm for the iterative solution of the complete system
of finite difference equations (equations 47,55,61,64) is now
described. The grid system is generated first. The vorticity, stream
function and velocity values are set to zero which is their initial
condition (t = 0) since initially the liquid was stationary with
uniform initial temperature throughout the body of the liquid. The
nondimensional temperature values are set to one (initial temper
ature). The temperature boundary condition for t>0 is then enforced.
The set of equations are then solved for finite increments of time
(At) with iterations performed on boundary vorticity at each time
increment. This is described in figure 5 and explained as below. The
selection of time step the At is discussed under the RESULTS
section.
Figure 17. Locations of the slowest heating points in
water over a heating period of 10 minutes in a
cylindrical can
N
FKMCP
D I L A
Figure 18.
Migration of the slowest heating points in
water over a heating period of 10 minutes in a
cylindrical can
25
3T
3T
+ v
3T
+ U
3r
3T
3z
_ K ,13 3T w 32T >
= TT" (t~ (r ) + )
p p r 3r 3r 3z2
(13)
p
(. +
' 3t
P (
V
3t
+
 3 3 \ 3p J 1 3 32 1
V+u ) + p T (rr) + T" I + P9
3r 3z 3z F 3r 3r 3z2
v + G J ) = iE. + (Fv)) + }
3F 3z 3F 3F F 3F 3z2
(14)
(15)
I_i (rpv) + i(pG) =0
F 3F 3z
(16)
The boundary conditions are
sidewal1
F = R
0 < z < H
T=Ti
O
11
1 3
< 1
II
o
centerline
F = 0
0 < z < H
iI=o
l=.
tl
o
o
It
1 >
3F
3r
bottom wall
(17ad)
0 < F < R
z = 0
T=Tl
= 0
O
II
1 >
top wall
0 < F < R
z = H
T=Tl
G = 0
o
II
1 >
Initially the
fluid is at rest
at uniform temperature
0 < r < R
0 < z < H
o
i
ii
i
= 0
v = 0 (18)
Refer to list of symbols for definition of individual symbols.
As expected, the flow field (equations 14 & 15) is coupled with
the temperature field (eq. 13) through the density term. These
velocity (mm/s)
Figure 15. Observed radial velocity profiles at the
sidewall at mid height in water in a
cylindrical can after various heating times
(from Hiddink, 1975)
112
79
n+1
m+1
0).
~n+l
%
m
~n+lm
%
< 6
(78)
for top, bottom and sidewall. When the convergence
criterion is satisfied, temperature, vorticity and stream
function values are considered the transient solution at
this time instant.
6. The velocities are calculated from the stream function
values using equation 63 and 64
7. Steps 1 through 6 are repeated for every time increment
until the desired time is reached.
The underrelaxation of boundary values at step 4 has often
been considered necessary in the literature (Peyret and Taylor,
1983; Roache, 1978) for convergence of the solution. Under
relaxation is usually employed to make a nonconvergent iterative
process converge (Hornbeck, 1975). This was found quite true for the
present study. Without underrelaxation, the time step allowed for a
stable solution got severely reduced to the point that solution to
the set of equations became impractical because of the amount of CPU
time required. As the computations proceeded, there were two sets of
values in storage. These were the estimates of the temperature,
vorticity and stream function values at the present time step and
converged temperature, vorticity and velocity values at the previous
time step.
3
ConvectionHeated Foods
Unlike solid packed foods that are heated by conduction, thin
liquid foods heat through convection. In general such liquid foods
are mechanically agitated to improve the heat transfer. However,
there are situations where the food material cannot be agitated for
several reasons. The food then heats primarily by natural convec
tion. Inpackage pasteurization for liquids (e.g. beer) in bottles
or cans that cannot be agitated is an example. Broth, thin soups,
evaporated milk, most fruit and vegetable juices, fruits in syrup or
water, and pureed vegetables are also examples of products where
natural convection heating can occur when no agitation is used.
Flexible pouches that cannot be agitated to keep the package integ
rity are another important example. Fermentation is another process
where a natural convection heat transfer model (with heat generation
term included) can provide valuable insight.
Just as optimum heating times are required for conduction
heated foods, they are also needed for natural convectionheated
products. Thus heat transfer models are similarly needed. Since we
are interested in the minimum heat treatment the food receives, we
need to know the point or the region that heats the slowest. Because
convection sets up complex flow patterns, the location where the
temperature is minimum is by no means obvious. Only a study of
complete transient flow patterns and temperature profiles can give
such insight. Previous studies to achieve this objective have not
been successful. The use of more advanced numerical techniques and
better computing equipment would make it possible to study in detail
the natural convection heat transfer in canned liquid foods.
Water
T>30*C
!op121*C
sldo=121*C
bo!121*C
R4.19cm
H=10.67om
R57,0.76
Z=57,0.85
80.0001
300
260
220
180
140
100
60
20
20
60
Figure lOg. Isotherms and streamlines in a
cylindrical can after 7 seconds of
heating
After 8 sec
Water
r,=30*C
top121*C
slde=121*C
bot121*C
R4.19cm
H=10.67om
R57,0.76
Z=57,0.85
60.0001
300
260
220
180
140
100
60
20
20
60
Figure lOh. Isotherms and streamlines in a
cylindrical can after 8 seconds of
heating
9
The solution was first approximated to
V tr
U2 + X2) at
A e J0(xr) Cos(xz)
(3b)
when sufficient time has elapsed, so that the first term in the
analytical series solution (equation 8a) dominates. Equation (8b)
was then rearranged to give
t = f log
J
V T
(9)
where
and
_ 2.303
a (Xz + X2)
(10)
j (r,z) = A J0(xr) Cos (xz) (11)
From the transient heat conduction equation (9) which gives
temperature T as a function of time t and the definition of FQ
(equation 6) which gives F0 as a function of temperature T, it is
possible to find F0 as a function of time t. However this cannot be
performed analytically. Upon solving the two equations numerically,
it is possible to know temperature achieved at the end of heating
for a given value of F0. Ball and Olson (1957) tabulated T^Tr
values as a function of FQ for particular values of f and j. Tradi
tionally, in thermal processing of conductionheated foods in cylin
drical containers, these tables are used to find T^Tr for a given
F0 value. Using this (ThTR) value and equation (9), the process
time t is found.
125
110
O
o
go
ZJ
i
o
L
CD
Ct
70
50
*
*
t
30
* 39(0.9)X39(0.B) grid
y 58(0.76)X58(0.58) grid
i i
0 120 240 360 480
Time of heating (seconds)
Figure 26. Transient slowest heating temperatures computed
with 39x39 and 58x58 grid
600
no
<43
Superscripts
d = design value
f = normal to grid faces
k = iteration counts in SOR
m = iteration count within a time step
n = time step
t = tangential to grid faces
* = "half" step in ADI method (same as n+l/2)
x
169
Hunter, J. H. 1982. Field testing of a microcomputer potato storage
control system. ASAE paper no. 824031. Presented at the 1982
summer meeting of the ASAE at Madison, Wisconsin.
Kunhardt, E. E., and P. F. Williams. 1985. Direct solution of
Poisson's equation in cylindrically symmetric geometry: A fast
algorithm. 0. Computational Physics 57:403414.
Larkin, B. K. 1964. Some stable explicit difference approximations
to the diffusion equation. Mathematics of Computation 18(86):
196202.
Lenz, M. K., and D. B. Lund. 1977. The lethalityFourier number
method: experimental verification of a model for calculating
temperature profiles and lethality in conductionheating canned
foods. J. Food Sci. 42(4):989996.
Lenz, M. K., and D. B. Lund. 1977. The lethal ityFourier number
method: experimental verification of a model for calculating
average quality factor retention in conduction heating canned
foods. J. Food Sci. 42(4):9971001.
Lenz, M. K., and D. B. Lund. 1977. The lethal ityFourier number
method: confidence interval for calculated lethality and mass
average retention of conductionheating canned foods. J. Food
Sci. 42(4):10021007.
Levy, C., and R. G. Hlavacek. 1980. Computer control optimizes
retort cook for yield, quality, safety(canned foods). Food
Processing 41(11):7477.
Merson, R. L., R. P. Singh, and P. A. Carroad. 1978. An evaluation
of Ball's formula method of thermal process calculations. Food
Tech. 3:6672.
Miller, C. W. 1977. Effect of a conducting wall on a stratified
fluid in a cylinder. Paper no.77292. Presented at the AIAA 12th
Thermophysics Conference, Albuquerque, New Mexico.
Mitchell, A. R., and D. F. Griffiths. 1980. The finite difference
method in partial differential equations. John Wiley & Sons.
Olson, F. C. W., and J. M. Jackson. 1942. Heating curves theory and
practical application. Industrial Engg. Chemistry 34:337.
Ostrach, S. 1972. Natural convection in enclosures. Advances in Heat
Transfer. 8:161227.
Ozoe, H., K. Yamamoto, S. W. Churchill, and H. Sayama. 1976. Three
dimensional, numerical analysis of laminar natural convection in
a confined fluid heated from below. J. Heat Transfer, Trans of
the ASME 98:202207.
TEMPERATURE (C)
TIME (minutes)
Figure 33. Computer plot of reference process (no deviations) showing
66.8 minutes of heating time and an accomplished FQ of 6.24
ACCOMPLISHED F0 (minutes)
TemperatureCc) n Axial velocity (mm/sec)
igure 11
Predicted radial velocity profile at the
sidewall at mid height in a cylindrical can
after 30 seconds of heating
108
Distance from sidewail(mm)
Figure 12. Predicted radial temperature profile at the
sidewall at mid height in a cylindrical can
after 30 seconds of heating
lOf. Isotherm and streamlines in a cylindrical can after 6
seconds of heating 101
lOg. Isotherm and streamlines in a cylindrical can after 7
seconds of heating 102
lOh. Isotherm and streamlines in a cylindrical can after 8
seconds of heating 102
lOi. Isotherm and streamlines in a cylindrical can after 9
seconds of heating 103
lOj. Isotherm and streamlines in a cylindrical can after
10 seconds of heating 103
10k. Isotherms and streamlines in a cylindrical can after
30 seconds of heating 104
101. Isotherms and streamlines in a cylindrical can after
120 seconds of heating 104
10m. Isotherm and streamlines in a cylindrical can after
300 seconds of heating 105
lOn. Isotherms and streamlines in a cylindrical can after
600 seconds of heating 105
10o. Isotherms and velocity vectors in a cylindrical can
after 1800 seconds of heating 107
11. Predicted radial velocity profile at the sidewall at
mid height in a cylindrical can after 30 seconds of
heating 108
12. Predicted radial temperature profile at the sidewall
at mid height in a cylindrical can after 30 seconds
of heating 108
13. Isothermal vertical surface, showing boundary layer
with velocity profile u(x,y) and temperature profile
t(x,y) in water (Higgins and Gebhart, 1983) 110
14. Predicted radial velocity profiles at the sidewall at
mid height in a cylindrical can after various heating
times Ill
15. Observed radial velocity profiles at the sidewall at
mid height in water in a cylindrical can after
various heating times (from Hiddink, 1975) 112
16. Comparison of numerically predicted axial temperature
profiles with observed (Hiddink, 1975) values at
various times during heating 114
xi i i
106
counterclockwise loop extended over most of the can and represented
the primary phenomena of upward movement of hot liquid at the side
wall and its downflow at the core. Several small clockwise and
counterclockwise secondary loops resulting from bottom heating
appeared and died down over time. A smaller clockwise loop was per
sistently present near the bottom centerline. This clockwise loop
could also be seen in the numerical results of Sani (figure 2). At a
still later time of t=1800 seconds (calculated with a
20(0.85)x20(0.80) grid), the clockwise loop near the bottom center
line persists as can be seen in figure 10o.
Radial temperature and velocity profiles. The transient
behaviour of the liquid velocities near the wall can now be
explained by referring to figure 11. The basic concepts involved in
describing the boundary layer formation are the same as for forced
convective flow situations. The effect of viscosity and temperature
are confined to thin region near the wall. However, as mentioned
earlier, the flow field is coupled with the temperature field for
natural convection flow. It is the difference in temperature that
creates the buoyancy force which moves the fluid upward.
Liquid next to the wall experiences the largest temperature
rise compared to the core temperature. Thus the local Rayleigh
number is largest next to the wall. The buoyancy force resulting
from this large temperature difference contributes to largest velo
cities near the wall (liquid on the solid wall is at rest due to the
no slip boundary condition). The liquid rising due to the buoyancy
effect produces viscous drag on adjacent fluid which also moves up.
r
Figure 19. Transient temperature at various locations
during heating of water in a cylindrical can
153
Recommendations
Using the available hardware, the algorithm described for
controlling conductionheating process needs to be implemented.
Further work on the natural convection heat transfer model can be
pursued in two major areas. These are to improve the model to make
it less CPU time intensive and to extract more information relevant
to food thermal processing. Specifically some of the ideas that can
be tried include
1. further optimization of grid stretching by trying out no
stretching at the top and/or different stretching
parameters
2. updating of velocities every iteration (with possible need
for relaxation of the values)
3. study of single point and integrated lethality
4. change in the thermal properties corresponding to different
types of foods
5. possibility of incorporating variable thermal properties
(e.g. viscosity)
6. effect of different can sizes and/or aspect ratios.
8
T121
Fo = L
10
dt
(6)
When the food being heated has a spatially varying temperature
profile, the chosen temperature T, for thermal process calculation,
is the temperature at the slowest heating point in the food
material. Thus, for conduction heating in a cylindrical can, the
chosen temperature T is at the geometrical center of the can.
The designed bacterial inactivation for a process is specified
by which is the time required to achieve the desired
inactivation of bacteria when the same is heated at a constant
temperature (T) of 121C. When temperature T is a function of time
(T(t)) instead, the objective of thermal processing is to select
process time t for the particular heating condition, such that
the equivalent FQ received (as given by eqn. 6) is greater than or
equal to the design value F^ That is,
Fo Fo
(7)
Application to conductionheating. When applying thermal
processing to conductionheated food in a cylindrical container,
Ball and Olson (1957) used the solution to the heat conduction
equation in cylindrical coordinates.
TT0 (\2+ X2)at
= E S A e m p J0(xr) Cos (x_z) (8a)
T mp uv m p
T0TR P
136
Water
T,=30*C
top121*C
Slde121*C
b0T121*C
R=*4.19cm
M~10.G7cm
5U0.0001
5r=0.001
Contours
105
85
65
46
26
7
13
32
52
72
Streamlines in a cylindrical can calculated
with two different convergence criteria for
boundary vorticity (6=0.0001 on the left
and 6=0.001 on the right)
Figure 30.
TABLE OF CONTENTS
ACKNOWLEDGMENTS iii
LIST OF SYMBOLS vii
LIST OF TABLES xi
LIST OF FIGURES xii
ABSTRACT xvi
INTRODUCTION 1
REVIEW OF LITERATURE 5
Computer Control of Food Thermal Processing 5
Overview of Food Thermal Processing 5
Online Control of Thermal Processing 10
Conduction Heating of Canned Foods 12
Studies Based on Analytical Solution 12
Numerical Studies 13
Natural Convection Heating in Canned Foods 14
Previous Works on Canned Liquid Foods 15
Other Similiar Works on Natural Convection
Heating in Cylindrical Enclosures 19
PROBLEM FORMULATION 24
Natural Convection Heat Transfer in Canned Foods 24
Governing Equations and Boundary Conditions 24
Boussinesq Approximation 26
Transformation of the PDEs 27
Nondimensionalization of the Variables 28
Conduction Heat Transfer in Canned Foods 30
Governing Equation 31
Boundary Conditions 31
Computer Control of Conduction Heating Process 32
METHODOLOGY 35
Modeling of Natural Convection Heating in Cans 35
Grid Generation and Grid Stretching 36
Discretization of the Parabolic
Temperature and Vorticity Equation 39
Time derivatives 42
Linearization 44
v
ADDITIONAL REFERENCES
Ames, W. F. 1977. Numerical methods for partial differential equa
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Barakat, H. Z., and J. A. Clark. 1966b. On the solution of the dif
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Bhowmik, S. R., and K. Hayakawa. 1979. A new method for determining
the apparent thermal diffusivity of thermally conductive food.
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Blake, K. R., D. Poulikakos, and A. Bejan. 1984. Natural convection
near 4C in a horizontal water layer heated from below. Physics
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Briggs, D. G. 1975. Numerical solution of high Raleigh number two
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Bryan, K. 1966. A scheme for numerical integration of the equations
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Energy performance of a HTST citrus evaporator under digital
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conditions on twodimensional, laminar, natural convection in
rectangular channels. J. Heat Transfer, Trans, of the ASME
98:194201.
167
7
0 = 0
o
(3)
where D0 is the 0 value at some reference temperature T0. The Z
value describes the temperature dependency of the lethal rate and is
related to the activation energy. The reference temperature T0 is
generally taken as 121C (250F).
From bacteriological considerations, a thermal process is
specified by noting the time (F0) necessary to reduce the bacterial
population, from the initial (C^) to some final safe value (C) at
temperature 121C (250F). Thus
F
o
(4)
When the heating process is not at the constant temperature of
121C, introducing the temperature dependency of 0Q and equation (2)
for bacterial death,
F
o
D 10
T121
Z At
D
T121
= 10 1 At
(5)
At is the time required at temperature T to achieve the designed
bacterial reduction. As the food material is heated, its temperature
T becomes a continous function of time t and the expression for F0
(equation 5) becomes
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Arthur A. Teixeira, Chairman
Associate Professor of
Agricultural Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
L \.V f'( .
Khe V. Chau
Associate Professor of
Agricultural Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Mechanical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Tom IP. Sjrih
Assistant/Professor of
Mechanical Engineering
31
Governing Equation
For strict conduction heating (no movement of the material) of
a homogeneous isotropic cylindricalshaped body of the food materi
al, the heat transfer is described by the classical heat conduction
equation
30 320 1 30 320
3t Ir7 r 3r II7 K
This is the energy equation (26) approximated for zero velocities
(diffusion term only). This equation is quite well behaved and easy
to solve. The resistance to heat transfer of the metallic walls of
the can was ignored.
Boundary Conditions
Canned foods are generally heated with hot water or steam and
cooled with water. Thus, the heat transfer coefficient at the out
side of the can wall could be comparatively small (for water) or
quite large (for condensing steam). In general, the conditions at
the boundaries can be written as
sidewall
r = R/H 0 < z < 1 Â£+hH0 = o
center!ine
r = 0
0 < z < 1
_30
3r
= o
(33ad)
bottom wall
0 < r < R/H z = 0
_30
3Z
hH
0 = 0
top wall
Meo
0 < r < R/H
z = 1
171
Steele, R. J., and P. W. Board. 1979. Amendments to Ball's formula
method for calculating the lethal value of thermal processes. J.
Food Sci. 44(1):292293.
Steensen, F. 1980. Why use a microprocessor for process regulation ?
Food Engg. International 5(12):3538.
Teixeira, A. A. 1978. Conductionheating considerations in thermal
processing of canned foods. ASME paper no. 78WA/HT55.
Presented at the winter annual meeting of the ASME at San
Francisco, California.
Uno, J., and K. Hayakawa. 1980. A method for estimating thermal
diffusivity of heat conduction food in a cylindrical can. J.
Food Sci. 45(3):692695.
Uno, J., and K. Hayakawa. 1980. Correction factor of comeup heating
based on critical point in a cylindrical can of heat conduction
food. J. Food Sci. 45(4):353859.
Wilkes, J. 0. 1963. The finite difference computation of natural
convection in an enclosed rectangular cavity. Ph.D. Thesis.
University of Michigan, Ann Arbor.
Wolfshtein, M., R. S. Hirsh, and B. H. Pitts. 1975. Secondorder
noniterative ADI solution of nonlinear partial differential
equations. Advances in computer methods for partial differen
tial equations. Edited by R. Vichnevetsky. AICA, New Brunswick,
New Jersey.
Yang, S., and T. IP. Shih. 1985. An algebraic grid generation
technique for timevarying twodimensional spacial domains.
Computational Fluid Dynamics Laboratory Report no. 8502.
Department of Mechanical Engineering, University of Florida,
Gainesvil1e.
41
natural convection problems), as well as realistic constraints on
available resources (Roache, 1982). For this study, the transient
temperature and velocity solutions are needed and the range of
Grashof number is 108 1010 Since the temperature and vorticity
equations are both parabolic and are very similar, they will be
discussed simultaneously in this section.
The performance of the various finite difference methods was
noted in terms of stability, truncation error, and conservation
property. Stability was defined as the controlled growth of the
roundoff errors (O'Brien et al., 1950). An uncontrolled growth of
the roundoff errors leads to instability in the finite difference
methods. The stability requirements often put restrictions on the
time step as a function of grid spacings. Empirical stability re
striction on grid size of the type Ax<8/u^ j and Ay< 8/jvÂ¡j
were also reported by Torrance (1968). Truncation error is the error
resulting from the discretization of the PDE. Conservation of any
physical entity (e.g. energy, vorticity) within the finite differ
ence grid system exists if the difference equations for the entity
are summed over all interior grid points and no spurious sources or
sinks of these quantities are found. Torrance (1968) found that
formulations by Barakat and Clark (1966) and by Wilkes and Churchill
(1966) did not conserve energy or vorticity and he did not recommend
the use of their equations. Another important property of finite
difference formulations is the transportive property. A difference
formulation is transportive if the effect of a perturbation in a
transport property is convected only in the direction of velocity.
PROBLEM FORMULATION
The governing equations and boundary conditions for the natural
convection and conduction heating process together with the assump
tions are first described here. The online control problem for the
conduction heating process is then formulated.
Natural Convection Heat Transfer in Canned Foods
The equations governing natural convection heat transfer in
liquid foods together with the boundary conditions are first
presented. The Boussinesq approximation to the set of equations is
then discussed. To further facilitate the solution, new variables
are defined and the governing equations together with the boundary
conditions are transformed into a new set of equations.
Governing Equations and Boundary Conditions
In natural convection heat transfer, the driving force for the
liquid motion is the buoyancy caused by the density variations due
to change in temperature. The partial differential equations (PDEs)
governing such naturally convective motion of fluid in a cylindrical
space are the NavierStokes equations in axisymmetric cylindrical
coordinates (Bird et al., 1976) and are described below.
24
77
REPEAT OVER n
*n n
0) = 0)
m
l.(
M11 m
oj 1 = 1.0
ITERATE OVER m
n *n
o
in n+1
0) = 0)r
r1" *rira
G+1" Ui+1". vT1. em)
m
0)
)
1 /El
n/2 =
T
w* na+1)
n
%
II
f r n n+lm
vn+1
a
, ,m
n+1
IE
>r
0)
n+lm_
'a
f ( n+2" ,,n+l
US1
m
n^/2m_
2= 0
n+1
m
(0=0).
ITERATE
m
,n+l"_ f r n+1 'i
n fsCa J
m
UNTIL
,n+l

n+1
m
fl,old
n+lm
^Q,oId
 i m+1 ^.m
n+1 c ,n+l >
r f7l^ J
< e
n+1
m+1
0).
= yo)T
~n+l
m+1
+ (Iy)oj
n+1
UNTIL
"n+1
m+1
 0)
0)
< 6
. << m m
<< =feO
UNTIL DESIRED TIME REACHED
T: solid boundary
ft: interior points
: centerline
in ~ .
o),oj,o): temporary
storage
Figure 5. Algorithm for the iterative solution
of the set of equations
150
In this way, products which experience process deviations can
be released for shipment on schedule and supported with all
necessary documentation in compliance with FDA Canned Food
Regulation. Use of these controlled systems in industry should lead
to improved production efficiency without compromising product
quality and safety in canned food operations.
69
,k+l n
*1,j
Â¡*+*
+ (1fi } *1,j
(71)
Using equation 71, stream function values i>. at all interior
points are updated. The entire sweep is repeated until the
convergence criterion
Max
i ,j
.k+1 k
*i,j ^ij
< e
(72)
is satisfied. The value of e chosen for this study was 10^. In a
*
transient calculation like this study, the optimum ft may change
from one time instant to another. However, the optimum ft found
through calculations during the very first time step was used
throughout the entire transient calculations.
Computational Boundary Conditions for the Finite Difference
Equations
A PDE such as equation 27 for to can describe a wide variety
flow situation. The flows (solutions) are distinguished only by the
boundary and initial conditions and flow parameters like the Grashof
number. Thus, specification of computational boundary conditions,
besides affecting numerical stability, greatly affects the accuracy
of the solution (Roache, 1982). Using the boundary conditions
already noted for the partial differential equations in the previous
chapter, the boundary conditions for the finite difference equations
are now derived.
The boundary condition for temperature is rather simple. When
the can wall temperature is specified, the computational wall
49
In the r direction,
a(rv0) a? a(rv0) 1___ a(rv0)
ar ar a? ar/as a?
i vRrj+ V?0i ,j ~ vLrjV?9i,jl
rj+l~ rJl
AS
2AS
Vd>0
V_>0
j+i
 vLrjV2ei.ji]
Simliarly
6(rv0)
6r
VR < 0
VR < 0
vR > 0
Therefore, combining the equations
vL > 0
< 0
VL < 0
3(rv0)
ar =
(VlVl)ri4/,91,.i+l"
r...r. ,
J+l Jl
(44)
71
Improper specification of boundary vorticity is found to be
destabilizing in general and the same was found in the present
problem. Vorticity is produced at noslip (wall) boundaries. It is
then diffused and convected into interior points. No explicit
expression for wall vorticity is available. It is related to the
interior stream function values by the vorticity streamfunction
equation (equation 28). However, this would make the vorticity
equation implicit with the stream function equation and the number
of simultaneous equations increase drastically. To avoid this,
Barakat and Clark (1966) chose explicit methods over implicit and
used vorticity boundary values calculated at the previous time step.
Wilkes and Churchill (1966) used the ADI method but similarly used
boundary vorticity values from the previous time step. Thus, in
their formulations, wall vorticity values lagged by one time step at
all times.
Roache (1982) discussed the effects of such lagging of boundary
vorticity values. It forces the use of a smaller time step since
larger time steps could make the solution inaccurate and unstable.
Since larger time steps are a major motivation for ADI, updating of
the boundary values through iteration is preferred. From the work of
Torrance (1968), Roache also noted that the convergence of the
boundary vorticity values actually imposed a timestep restriction
of the form At < a / Az2 where "a" was some number dependent on the
problem and convergence requirements. Thus, due to the implicitness
of the boundary vorticity values, ADI actually has a timestep
restriction like the simplest of the explicit methods. The
Water
T,=30*C
loplEfC
slde121*C
bot121*C
R4.1.9cm
Hs*10.67om
R57,0.76
Z57,0.85
ao.oooi
300
260
220
180
140
100
60
eo
20
60
Figure lOe. Isotherms and streamlines in a
cylindrical can after 5 seconds of
heating
Figure lOf. Isotherms and streamlines in a
cylindrical can after 6 seconds of
heating
LIST OF TABLES
TABLE PAGE
1. Sequence of line types for identification of stream
function contours in figures lOao 96
2. Heating and cooling rates for various process
conditions 140
3. Adjusted heating times and resulting lethality (F0)
in response to process deviations using proposed on
line control logic and method of GiannoniSuccar and
Hayakawa 148
xi
51
The finite difference formulations for the time and space deriva
tives are then put together in the ADI formulation. For each of the
temperature and vorticity equations, the equations are written for
the "half" time step and for the "full" time step of the ADI method.
Algebraic simplifications are done to convert the equations at each
of the time steps to a tridiagonal systems of equations. A glimpse
of the tridiagonal matrices for the "half" and "full" step is then
included.
The tridiagonal systems of equations are solved using the back
substitution procedure (Smith, 1978). The time step At and the
mesh sizes Az, Ar were chosen in such a way that the coefficients
(a7, b.Â¡, c.Â¡ in equation 48a) are all positive and the matrices
diagonally dominant (b1 > ai + c^). Richtmyer and Morton (1967)
showed that these two conditions are sufficient to keep the round
off errors down, in the back substitution procedure. Introduction of
the second upwind differencing method satisfies the criterion that
coefficients aÂ¡, bj, and C] are all positive (see Kublbeck et al.,
1980). The second requirement of diagonal dominance was not investi
gated any further in this study. However, at all times whenever the
numerical values of the coefficients an, b.Â¡, Cj were visually
checked, the diagonal dominance was found to be satisfied.
ADI formulation for the temperature equation is now developed.
For convenience, the differential equation for temperature is
repeated. The finite differences corresponding to each of the
derivatives in the temperature equation are then assembled in the
same order.
83
Experimental Studies
Experiments were performed to record the transient temperature
values inside cylindrical cans during conduction heating. The effect
of different surface heat transfer coefficients, that were obtained
by using different media (steam and water), on the transient
temperature response was also noted.
The experimental work consisted of series of heat penetration
tests on No. 303 cans filled with a 10% bentonite suspension as a
conductionheating food model system (Niekamp et al., 1984). Eight
cans were fitted with an Ecklundtype copperconstantan thermocouple
mounted midway up the can wall, so as to have the junctions located
at the geometric can center. In order to achieve the two different
levels of headspace vacuum, four of the cans were filled at 90C
and the other four cans were filled at 67C. The cans were
identified according to fill temperature and then processed with an
extended steamcook to stabilize the bentonite suspension. An
experimental retort was used that had been designed and built at the
University of Florida with the capability of using pressurized
water, pure steam, or steam/air mixtures under controlled rates of
recirculation. The retort was fitted with a special rack designed to
hold the eight cans suspended upright in the mainstream of the
retort chamber, as shown schematically in Figure 6.
The rack consisted of a stainless steel plate with eight
cicular holes through which the cans could be inserted. Special
fixtures were welded at the edge of each hole so that the cans could
165
Schumann, U., and R. A. Sweet. 1976. A direct method for the
solution of Poisson's equation with Neumann boundary conditions
on a staggered grid of arbitrary size. J. of Computational
Physics 20:171182.
Shih, T. IP. In Press. Finitedifference methods in computational
fluid dynamics. Prentice Hall, Englewood Cliffs, New Jersey.
Shyu, R. J. 1985. Numerical study of unsteady natural convection in
cylindrical enclosures. Ph.D. Dissertation. University of
Florida.
Shyu, R. J., and C. K. Hsieh. 1985. Unsteady natural convection in
enclosures with stratified medium. ASME paper no. 85HT33.
Presented at the National Heat Transfer Conference at Denver,
Colorado. August 47, 1985.
Smith, G. D. 1978. Numerical solution of partial differential
equations. Oxford University Press, Oxford, England.
Stevens, P. M. 1972. Lethality calculations, including effects of
productmovement, for convection heating and brokenheating
foods in stillcook retorts. Ph.D. Thesis. University of
Massachusetts, Amherst.
Stumbo, C. R. 1965. Thermobacteriology in food processing. Academic
Press, New York and London.
Teixeira, A. A. 1971. Thermal process optimization through computer
simulation of variable boundary control and container geometry.
Ph.D. Thesis. University of Massachusetts, Amherst.
Teixeira, A. A., A. K. Datta, J. P. Adams, and W. R. Peterson. 1984.
Surface heat transfer considerations during immersion water
cooling of retorted foods. Proceedings of ICEF3, Dublin,
Ireland. I:(3), pp.23 Elsevier Applied Science Publishers, Ltd.
Barking, Essex, England.
Teixeira, A. A., J. R. Dixon, J. W. Zahradnik, and G. E.
Zinsmeister. 1969. Computer optimization of nutrient retention
in the thermal processing of conductionheated foods. Food Tech.
23(6):137142
Teixeira, A. A., K. Dolan, A. K. Datta, and J. P. Adams. 1985.
Retort depressurization effect on cooling rates in conduction
heating canned foods. Trans of the ASAE 28(2):645648, 656
Teixeira, A. A., and J. E. Manson. 1982. Computer control of batch
retort operations with online correction of process deviations.
Food Tech. 36(4):8590.
81
differenced using an alternating direction explicit (ADE)
differencing method, which is explicit but still unconditionally
stable. The transient temperature values predicted by performance of
this ADE finite difference method was compared with analytical
solutions and experimental measurements.
Discretization and Solution of the Equation
Keeping in mind that the equation was needed to be solved in
realtime for online control purposes and the ADE method was
already tested (Teixeira et al., 1975; Teixeira, 1971), the same was
selected. The ADE formulation was first reported by Saul'yev (1957).
This was later translated and described by Aliada and Quon (1966),
and was first introduced for applications to thermal process
simulation by Manson et al. (1974). The method as applied to the
diffusion equation is unconditionally stable (Saul'yev, 1964) for
temperaturespecified boundary condition. It has a formal order of
accuracy of 0(At2, Ar2, Az2 ). Being explicit, it does not require
inversion of matrices as would be required by an implicit method. In
a fluid flow problem, the ADE formulation is generally not preferred
since the presence of the convection term (in any difference form)
makes the complete formulation unstable or returns it to simple
explicit stability limits ( <1/2 > u ^ < 1 for a one
dimensional problem; Roache, 1982). Another shortcoming of this ADE
method is the fact that even though it is an explicit method, it is
actually implicit in boundary conditions. To handle this implicit
ness, either explicit expressions for the temperature values need to
92
Corrected process times for each of these three processing
situations are first noted for the online computer control
method. These are the times t^ when
rolh
cool^
'ho 'simulated
> F
d
o
is satisfied. The corrected processes as predicted by the CF method
were found by multiplying the normal process time with the
correction factors predicted by that method.
The accomplished FQ values for the processes corrected by the
online control system and processes corrected by the CF method were
calculated using equation 35a (the general method). In either case,
the can center temperatures required for use with equation 35a were
predicted using the ADE finite difference analog of equation 32.
94
Throughout this section, reference will be frequently made to
the experimental study of Hiddink (1975) mentioned in the REVIEW OF
LITERATURE section. Since his study involved more than one wall
temperature and other conditions, the particular conditions appli
cable to any situation are noted alongside. Reference will also be
made to the numerical finite element study of Sani on temperature
and velocity profiles during pasteurization of canned beer (also
mentioned in the REVIEW OF LITERATURE section).
Transient Flow Patterns and Temperature Profiles
The liquid adjacent to the sidewall, the top and the bottom
wall recieves heat from the hot walls. As the liquid is heated, it
expands and thus gets lighter. Liquid further from the sidewall is
still at a much lower temperature. The buoyancy force created by the
liquids at different temperatures forces the hot liquid, next to the
wall, upwards. A sharp drop in temperature occurs from the wall to
the core, creating a boundary layer. The hot liquid going up is
interrupted by the top wall and then travels radially towards the
core. The core liquid, being heavier, moves downwards and towards
the wall. Thus a recirculating flow is created with a boundary layer
at the sidewall, core flow around the centerline and a mixing region
at the top. The same qualitative observations were reported in the
experiments of Evans et al. (1968), Brandon et al. (1982), and
Hiddink (1975).
The arrow plots of velocity vectors create a picture that is
very busy and it is often hard to differentiate between the flow
56
Upwind differencing of the convective terms in the vorticity equation
3(uo))
3z
= similiar to
3(u0)
3z
. ^ur~Iur 1 ^Mj+i,j+ tylRlyluLl)Mi,j (yKPii.j
z.z. .
i+l il
(50)
3( Vw)
3r
(vR IvRpMjjn + (VR+ vr V KlK'.j ~ 1 vl1 ^i, jl
(51)
r .l1 r. .
J+l Jl
Upwind differencing of the diffusion terms in the vorticity equation
32cj
3Z2
zi+r zti
(1).,, .O). 0). tJ. .
' i+l.j i,J i,.i il.J
zi+r zi
zi zti
(52)
3 c1 3(ru)
3r 3r '
35 3 /l 3( raj)
IFF IT^r" 3r '
3( roi)
r.r. t '
J+l Jl
lk.
3r
3(ru)
3r
iJ+V? rjV? r i J V?
2AC
2
rj+r rji
A?
r.,,0). r.a). r.o). r. ,u. ,
f J+l 1,J+1 J 1,J J 1 ,J Jl 1Jll
1 r i Tp pi p] ip p i J
rj+V2(rj+r V
jV^rj" rji^
(53)
Central differencing of the buoyancy term
30
3r
9i,j+r 9iJl
rj+r rji
(54)
APPENDIX B
ALTERNATIVE FINITE DIFFERENCING OF BOUNDARY VORTICITY
Boundary conditions for vorticity at bottom wall
Let Az = zj z0= z1
Using Taylor's series expansion at (OJ)
3^1
+ l a2l>
*1J ^0,j+ 9z0jjAz T
OJ
AZ2 + 0(AZ3)
Now, using no slip conditions
u0,j 0 => fIfL
U 9 J
vj j = o > 4j? =o
0,J r or0J
and
i^O j= 0 by choice
=>
1 3 2ip
K1J 2 9zz
From equation
CO
= > (0,
oj
Az2 + 0(az3)
1 H2
OJ
r 9zz
1J
" 2 (uf
OJ
2^1
 IiJ,
rAz
oj 3r'r3pl,o.J r9z'lO,j
=> h.j = \ (o,jr)Az2 + (Az3)
Pi i
, similarly at other boundaries
rz'zV2
155
22
had numerically calculated and plotted the isotherms and streamlines
inside the enclosure. In their experimental work, they recorded
temperature values at ten different points inside the cylindrical
enclosure. The observed temperature values deviated from the
numerical results by 10%. This discrepancy was attributed to the
small nonsymmetry in the imposed wall temperature which might have
produced three dimensional effects and, also, to heat losses from
the bottom and top of the container.
Torrance (1968) compared five separate finite difference
formulations for the problem of natural convection in cylindrical
enclosures. The five formulations were respectively by Barakat and
Clark (1966), Wilkes and Churchill (1966), Fromm (1964) and two of
his own. The physical problem chosen was the study of natural
convection flow induced in a vertical cylindrical enclosure by a
small hot spot centrally located on the bottom. One of his two
formulations was explicit and the other was implicit. The explicit
formulation was shown to be numerically superior in many ways to the
other four and it was also reported to have good, agreement with
experimental observations (Torrance and Rockett, 1969; Torrance et
al., 1969) for lower values of Grashof numbers. This was also the
formulation used by Stevens (1972) referred to earlier.
Kee et al. (1976) performed a numerical study of temperature
distribution and streamlines in a closed vertical cylinder
containing uniformly distributed heat generating tritium gas. The
walls were kept isothermal. An upwind method was used for the
convective terms. Numerical temperature values were compared with
72
definition of vorticity to was used at the boundaries to obtain
expressions for boundary vorticities. At the bottom boundary,
to,
 voj"
.J Ul/2 z0)
t t
^rj+1/2~rj ^2

1J
r.z z\.
J 1 72
(76a)
since
'0,j
v0,j=0
rj(zr zoT
r.z.
J 1
Uq j= 0 and Ug ._j = 0 from eqn. 60a
At the top boundary,
t t t
Vm,j~ vml,j um,j" um,jl
r l2m W lV/2 rJW
_ m1 ,j
r (z z )(z z ij
j m m1 m m/2
since
(76b)
Vm 1= 0
m 9 j
(.Â¡tf , _)
m 9j m19j
vml,j r.(z z 7)
j m m1
V n
m1 ,j
r.(z z7)
j m m17
um,j a"d 41,j= 0
from eqn. 60a
98
T(z,t) Tx
jj
'0
1
erf (
/4oit
(83)
where erf is the error function.
Isotherms and streamlines during the first ten seconds of
heating are given in figures lOaj. These figures (lOaj) represent
results of computations using a 58(0.85)x58(0.76)* grid. Instability
resulted due to the presence of heated bottom layers with colder
liquid above them. This instability in the bottom layer gave rise to
"bursts" of convective cells in a very regular manner. Such
phenomena are referred to as Benard convection. However, because of
the very large initial temperature difference, this convection
occurred very rapidly in the present study (figures lOej) as
compared to its much slower occurrence reported by Shyu and Hsieh
(in press). The formation of these convective cells were responsible
for the convective heat transfer from the bottom. In the study of
Hiddink (1975), these were precisely the convective flow patterns
that could not be picked up by his numerical model. Along the side
wall, hot liquid rises and gets deposited at the top. This explains
the shape of isotherms at the top core region.
Transient isotherms and streamlines at later time instants of
30, 120, 300, and 600 seconds of heating are presented in figure
lOkn. These results (figures lOkn) were computed using
39(0.9)x39(0.8) grid. Two main flow loops could be seen. The larger
*The values 0.85 and 0.76 in parenthesis denote the stretching
parameter sigma (see equations 38 and 39) in z and r direction
respectively.
ACKNOWLEDGEMENTS
The author is indebted to Dr. Arthur A. Teixeira, his major
professor, for his encouragement, guidance and patience throughout
the course of the work and especially for his assistance in
preparing the manuscript.
The author is also highly grateful to Dr. Tom IP. Shih, Dr.
Chung K. Hsieh and Dr. Ruey J. Shyu for their time and extremely
valuable suggestions in the study of the natural convection heat
transfer. Further thanks to Dr. Khe V. Chau and Dr. Spyros Svoronos
for their interest in the study and helpful suggestions.
The importance of a good computing facility (number crunching
to be particular) in developing the natural convection model cannot
be overemphasized. The availability of the Cyber 170 computer
through Dr. John Gerber and the IFAS adminstration was vital for
that purpose. The author also greatly appreciates help from Dr.
Ellen Chen, who introduced him to the Cyber facility, and Victor,
Jimmy and Margaret for endless support on problems with the Cyber
system.
The mountain of data available from the transient natural
convection heating model would have been of little use without the
plotting capability provided by the excellent graphics software
(PLOTPAK) made available from Dr. William E. Dunn from the
149
The step function and the gradual drop in temperature are just
two examples of process deviations where the online computer
control system is able to maintain the desired sterilization. This
control system predicts the can center temperature using a finite
difference analog of equation 32. It then uses this temperature
value with a numerical (trapezoidal) method of integrating equation
6 (the general method). Both these steps place no restriction what
soever on the retort temperature variation. This enables the online
control system to maintain the desired sterilization for arbitrary
variations in retort temperature. This is not true for the correc
tion factor approach of GiannoniSuccar and Hayakawa (1982) which
requires the retort temperature drop to be a step function from an
otherwise constant value. Also, it is not apparent how their method
would correct a process with multiple drops, even if the individual
drops were step functional. The numerical method of calculating the
accumulated process F0 is a more direct approach than Ball's formula
method of calculating the process F0 and subsequent correction
factors.
Other benefit areas include realtime operational information
and complete documentation. Figures 33, 34, and 35 show a complete
history of the process including retort temperature, calculated can
center temperature and calculated F0. In addition to providing all
the required documentation for regulatory and quality control com
pliance, the plots will provide visual descriptions of the complete
process history. The graphs can clearly show if the process went
through any deviation and how the required F0 was still maintained.
r%
3
Figure 2. isotherms and velocity profiles in a can at
various heating times (after Sani, 1985) ^
o
yrrr
50
Diffusion terms. Central differencing was used for the
diffusion terms in both the temperature and vorticity equation. The
diffusion terms in the temperature equation are finite differenced
be!ow.
320 3 .30. 3n 3 .30x
TT ST'S?* = Iz j
oZ
30, _30i
_ i h+VpJ 9z* iVftj
3z/3n An
1
0.
1+1,j
zi+l
z
j+r z
2 An
i1
0. 0. .
_UJ \A,a
zi zil
An
zi+l zil
0. 0. .
 i+l,J i,J
zi+l
z.
1
9i ,j ~ 9il,j
zi zil
(45)
Ii_ (p!,
r 3r 3r;
r 3r 35 *^3F
dQ _30
 i 8r'i,j1/?
~ r 3r A5
85
0. 0. 0. 0. 1
1 ,J+1 1 ,J 1 ,J 1,J1
W
TJ ri+l rJl
rj+l ~ rj
A5
W
9i,j+i ~ Qi,j p
J^/2 rJ+1 r
0. 0. .
T,J
jJ/2 rj rjl
(46)
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
The following conclusions were drawn from this study.
Natural Convection Heating in Cans
1. A numerical model was developed that is capable of predicting
transient flow patterns and temperature profiles during
natural convection heating of liquids in a cylindrical metal
can. Plots of computed transient isotherms, streamlines, and
velocity vectors were provided.
2. Calculated radial velocity profiles and axial temperature
profiles compared well with available experimental data.
3. Qualitatively, the flow inside the can consisted of liquid
rising at the boundary wall, radial flow and mixing near the
top, and uniform core flow downwards at the center.
4. Thermal as well as velocity boundary layers formed at the side
wall.
5. During very small initial times, the heating from the bottom
of the can was restricted to conduction heating.
6. As time passed, magnitudes of the velocities reduced due to
decreasing temperature difference between the fluid and the
boundary walls.
151
Water
Figure 10a. Isotherms and streamlines in a
cylindrical can after 1 second of
heating
Water
T,=30*C
top121*C
slda=121*C
bot121*C
R4.19cm
h=10.67cm
R57,0.76
Z=57,0.85
fi0.0001
300
260
220
1B0
140
100
60
20
20
60
Figure 10b. Isotherms and streamlines in a
cylindrical can after 2 seconds of
heating
31. Transient center temperature during conduction
heating of Bentonite in a cylindrical can 142
32. Transient center temperature during conduction
cooling of Bentonite in a cylindrical can with and
without drop in retort pressure 143
33. Computer plot of reference process (no deviations)
showing 66.8 minutes of heating time and an
accomplished F0 of 6.24 144
34. Computer plot of a process that experienced a step
functional drop of 8.3C (15F) in temperature bet
ween 50th and 60th minute and still maintained an F_
of 6.02 146
35. Computer plot of a process that experienced a linear
drop in temperature between 40th and 60th minute
(from 0C to 11.1C (20F)) and still maintained an
F0 of 6.18 147
xv
44
are required because of this situation. The iterations can also
serve the purpose of preserving second order accuracy in time of the
ADI method. The second order accuracy may otherwise deteriorate due
to the presence of nonlinear convective terms (discussed below).
Since the iterations are needed anyway, an implicit set of equations
can be solved, instead, utilising the iterations. As noted earlier,
Stevens (1972) and Hiddink (1975) also used the explicit
formulations of Barakat and Clark (1966) to study the present
problem and could not resolve the convective eddies present due to
the bottom heating. Since the ADI method may allow finer grid
spacings with less stricter restrictions on At, it may be able to
pick up the convective eddies present, without taking prohibitive
amounts of computer time. Thus the ADI method was chosen to solve
the temperature and vorticity equation in this study. The ADI
formulation is written below for the temperature equation 26. At the
"half" step
n+1/2
0 
3(u9)
At/2
3z
n+ V2
3(rv0)
3r
320
n+V2
+ 1
r
30,
3r^ r3r)
and at the "full" step
3n+1_ 0n+1/2
Ax/2
3(U0)
3Z
n+1/2
3(rv0)
3r
n+1
320
Iz2
n+V2
r 3rv 3r;
n+1
Linearization. The velocities in nonlinear convective terms
e.g.
3(u0)
3z
n+ V2
should ideally be evaluated at the same time level
(n+ 1/2 in this case) as the 0 (or w). However, since u and v
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
S
Spyros Svoronos
Assistant Professor of
Chemical Engineering
This dissertation was submitted to the Graduate Faculty of the
College of Engineering and to the Graduate School and was accepted
as partial fulfillment of the requirement for the degree of Doctor
of Philosophy.
Dean, Graduate School
Water
T,30*C
!op121*C
slde = l21*C
bo121*C
R4.L9cm
H=*10.67om
R38,0.80
Z38.0.90
ao.oooi
105
83
65
45
23
30
50
Figure 10k. Isotherms and streamlines in a
cylindrical can after 30 seconds
of heating
After 30 sec
Water
T,= 30*C
lop121*C
slda=lÂ£jl C
bot121*C
R4.19cm
h=l O.G7om
R38.0.80
Z=38,0.90
80.0001
40
25
20
15
10
15
20
Figure 101. Isotherms and streamlines in a
cylindrical can after 120 seconds
of heating
109
Further away from the wall, near the core, the effect of positive
velocities upward is not felt. A thermal as well as a hydrodynamic
boundary layer is thus created. Due to recirculation of the water in
the enclosure, negative velocities appear in the core region. Since
the liquid is going up only through a small cross sectional area
(the thickness of the boundary layer) while its downward movement is
over a much larger area at the core, the downward velocity is much
smaller. This gave rise to the velocity profile as in figure 11.
Such velocity profiles were numerically computed by Shyu (1985),
Sani (personal communication, 1985) and experimentally observed by
Hiddink (1975) and Engelman and Sani (1983).
The thermal boundary layer formed is shown in figure 12. It can
be seen that the temperature did not drop continuously with distance
from the sidewall as would be the situation for natural convection
from a heated vertical wall in an infinite medium (see figure 13).
The temperature first dropped and rose somewhat to become the core
temperature. The rise in temperature from the minimum value in the
profile (closer to wall) to a higher value in the core is caused by
the downward flow of hot fluid in the core from the top.
The magnitude of the velocities computed at the boundary layer
(figure 14) compares excellent with the velocities experimentally
observed by Hiddink (1975) as shown in figure 15. For these two
figures the input data used were radius=8.9cm, height of liquid
=18.00cm, top liquid surface free (thermally insulated) and sidewall
and bottom wall temperatures of 107C. The maximum positive veloci
ties in the boundary layer computed in this study at three different
87
thus satisfying the desired sterilization value for arbitrary
variations in retort temperature T^(t). As noted earlier, the
computer needs to simulate cooling and perform other calculations in
realtime.
Figure 7 shows the timings of various calculations that need to
go on in realtime. For the simulation to be feasible in realtime,
the requirement that
Atj + At 2 < At (81)
has to be satisfied. There is some choice over selection of all
three time intervals At^, At2> At However, At cannot be increased
too much. This could make finite difference calculations of T(t)
from T^(t) inaccurate and unstable. Explicit finite difference
methods generally take less computation time as compared to implicit
methods and may be useful in reducing At^, while At2 can be reduced
by selecting finite difference methods that allow larger time steps
without going unstable. This makes it possible to simulate the
complete cooling cycle in a short time. Implicit finite difference
methods generally allow large time steps as compared to explicit
methods. Thus we see that the requirements for reducing At^ and
At2 are conflicting. It was necessary to look for finite differ
ence methods that could take large time increments while using as
little computation time as possible to calculate each step. The
alternating direction explicit (ADE) finite difference method used
by Teixeira and Manson (1982) does offer both of these advantages.
TEMPERATURE (C)
25
TIME (minutes)
Figure 35. Computer plot of a process that experienced a linear drop in
temperature between 40th and 60th minute (from 0C to 11.1C
(20F)) and still maintained an F 0f 6.18
ACCOMPLISHED Fn (minutes)
39
= Lt
a+l
1
2 tan (ttct/2)
by L1Hospital's rule
Smil arly
= 0
Lt
ff+l
dz
cTn
z=l
Lt
a>l
2
ir a
tanirg/2
1 + tanzira/2
= 0
and
Lt
a+l
dr
r=R/H
Lt
cr+l
2
TT O
tanu/2 g
1 + tanir/2 a
0
Thus the transformations provide closer spacing of grids near z = 0,
1 and r = R/H. Figure 3 shows the grid system with close spacing of
grids near the top, bottom and sidewall. As can be seen, there is no
excessive grid skewness.
Ka'lney de Rivas (1971) also showed that when conditions 2 and
3 are satisfied for any smooth function, "extra truncation errors"
due to the nonuniform grid system are improved from first order to
second order in An and AÂ£ Vinokur (1983) also discussed similar
types of stretching functions.
Discretization of the Parabolic Temperature and Vorticity Equations
Of the several finite difference formulations that are
available for any set of PDEs, careful choices needed to be made
depending on many factors including boundary conditions, geometry of
the problem, type of solution (steadystate or transient) desired,
range of parameters involved (particularly the Grashof number in
64
"i,j = A*1l .j+ B*1,j + C*1+l,j + ''"i.jl + E*T,j+l
(62)
where
A =
rjlZlJ/2'ZiVlZlZil)
B =
rj(zf4ziVlzHrz1 + Vzii!' rj4/2rjJ/2 rjJ/2lrj+rrj>+rji/2(rrrji>1
C =
WViV^i+r^
D =
lrj+V2rjV2Jrj1/2lrj'rji)
E =
tVV2rJV2Jr>V2tl>r'V
The grid point velocities are then achieved through linear
interpolation of the transport velocities u^f and v!f .
jJ J
u.. rj rjfe. ut.+ rjt.h.: ,rj_ ut.
J rj+V2 rjV2 1,0 rj+V2 rjv2 J1
(63a)
v. = J...~ \r.lA. vt + +V2..1.,. vt .
J Zi+V2" ZiV21,J Zi+V2 z1V *J
(63b)
53
After algebraic manipulation, temperature equations at the "half"
time step
ai0ii,j+bi0i,jci0i+l,j = rhsi
(48a)
where
l/2(uL+uL) ^ x
a zi+rzii zi+rz,i^zrzii)
bi
ci
. i + & (uMuSiuMum + i r L_ + i
Ax z_. .,z. ^ R I Rl L I LW z. .,z_. ^ z. ,,z. z.z_. ;
i+1 i1
i n
R'IUR
i+1 i1 i+1 i i i1
 Vu"u")
zi+rzii
tzi+rziiJtzi+rzf)
rhs, de^ J.J 3 fej j+1
' VVi'ji5 rjlrj+r'ViJirjrji)
I_ i.(vRtlVRlr.1^(,L,Ll)rJJfr 1 ,ri^
AT 2 rjirj+rrji) rjlrj+rrji^rj+i"rj rrrji
n i n i
f.. fryKil^ + /Jjft
pjrj+irjiJ rjtrj+rrjiJlrj+rrj)
and
UR =
i ,j
<
70
II
<
s*
C_i.
f
f
ilj
L" i,jl
120
more viscous, there may be significant differences between the
slowest heating temperature and the temperature at other locations.
The transient temperatures at different points in the container
experience rapid fluctuations due to changing flow patterns. This
is illustrated in figure 20 by enlarging the time scale.
Contrast with Conduction
For a thin liquid like water, natural convection speeds up the
heat transfer considerably as compared to a solid (having same
thermal diffusivity value) which can heat only by conduction. Figure
21 shows temperature profile obtained in water at the slowest
heating points using the natural convection heating model developed
in this study, and the temperature profile obtained using the ana
lytical heat conduction equation (equation 8a) at the geometrical
center in a solid product having the same thermal diffusivity as
that of water, heated under identical conditions. As can be seen,
the slowest point in the natural convectionheated can reached al
most the heating temperature (121C) in about 10 minutes. However,
the slowest point in the conduction heated food with the same
thermal properties reached only 101C after 1 hour! For liquids more
viscous, the difference in heat transfer may not be so pronounced.
A contrast of the uniformity of temperature distribution in the
conduction and natural convection heating situations is presented in
figure 22. The essential difference is that for a uniform specified
boundary temperature, conduction heating creates completely uniform
isotherms as expected from the conduction equation, whereas, for
166
Teixeira, A. A., C. R. Stumbo, and J. W. Zahradnik. 1975a.
Experimental evaluation of mathematical and computer models for
thermal process evaluation. J. Food Sci. 40:653.
Teixeira, A. A., G. E. Zinsmeister, and J. W. Zahradnik. 1975b.
Computer simulation of variable retort control and container
geometry as a possible means of improving thiamine retention of
thermally processed foods. J. Food Sci. 40:656.
Torrance, K. E. 1968. Comparison of finitedifference computations
of natural convection. Journal of the National Bureau of
Standards 72B(4).281301.
Torrance, K. E., L. Orloff, and J. A. Rockett. 1969. Experiments on
natural convection in enclosures with localized heating from
below. J. Fluid Mech. 36(1):2131.
Torrance, K. E., and J. A. Rockett. 1969. Numerical study of natural
convection in an enclosure with localised heating from below
creeping flow to the onset of laminar instability. J. Fluid
Mechanics 36(1):3354.
Vinokur, M. 1983. On onedimensional stretching functions for
finitedifference calculations. J. Computational Physics. 50:
215234.
Wilkes, J. 0., and S. W. Churchil. 1966. The finitedifference
computation of natural convection in a rectangular enclosure.
AIChE Journal 12(1):161166.
125
previous researchers. These indicate that the solution is reasonably
accurate.
Selection of the grid size. Many grid sizes ranging from 20x20
to 58x58 were tried. Increase in the number of the grid points
resulted in more iterations necessary to reach convergence per time
step. This was in addition to the increase in number of equations to
be solved. Thus computation time increased considerably with smaller
grid size. On the other hand, large grid sizes cannot resolve many
significant flow features. A grid system was chosen that resolved
all the significant flow features related to natural convection
heating of liquid foods and still consumed "reasonable" computer
time.
The isotherms and streamlines computed at 90 seconds for
39(0.9)x39(0.8) grid and 58(0.85)x58(0.76) grid (figure 23) are
shown in figures 24 and 25. The number in parenthesis are the
stretching parameter a in the grid stretching equations 38 and 39.
The streamlines are almost identical and temperatures are predicted
with less than 2% error for most of the enclosure. Thus, refining
grids beyond 39(0.9)x39(0.8) did not bring significant additional
information and the same grid was selected for the present study.
To further emphasize on the fact that use of 39(0.9)x39(0.8)
did not result in loss of any significant information in reference
to natural convection heating of liquid foods, the transient slowest
heating temperatures computed with the two grid sizes of
39(0.9)x39(0.8) and 58(0.85)x58(0.76) are shown in figure 26. The
figure shows that the two grid systems predict almost identical
170
Packer, G. J. K., and J. L. B. Gamlen. 1974. Calculation of
temperature measurement errors in thermocouples in convection
heating cans. J. Food Sci. 39:739743.
Patankar, S. V. 1980. Numerical heat transfer and fluid flow. Hemi
sphere publishing Corporation, Washington.
Pepper, 0. W., and S. 0. Harris. 1976. Numerical simulation of
natural convection in closed containers by a fully implicit
method. Numerical/Laboratory computer methods in fluid
mechanics. Presented at the winter annual meeting of the ASME,
New York, New York. Dec 510.
Pflug, I. J., J. L. Blaisdell, and R. C. Nicholas. 1965. Rate of
heating and location of the slowest heating zone in sweet fresh
cucumber pickles. Food Tech. 19(6):121126.
Phillips, T. N. 1984. Natural convection in an enclosed cavity. J.
Computational Physics. 54:365381.
Ramaswamy, H. S., M. A. Tung, and R. Stark. 1983. A method to
measure surface heat transfer from steam/air mixtures in batch
retorts. J. Food Sci. 48:900904.
Schneider, G. E., A. B. Strong, and M. M. Yovanovich. 1975. Finite
difference modelling of the heat conduction equation in general
orthogonal curvilinear coordinates using Taylor series ex
pansion. Advances in computer methods for partial differential
equations. Edited by R. Vichnevetsky. AICA, New Brunswick, New
Jersey.
Schwind, R. G., and G. C. Vliet. 1964. Observations and inter
pretations of natural convection and stratification in vessels.
Proceedings of the 1964 Heat Transfer and Fluid Mechanics
Institute, Berkeley, California. June 1012. Edited by W. H.
Giedt and S. Levy. Stanford University Press.
Seki, N., S. Fukusako, and M. Suguwara. 1977. A criterion of onset
free convection in a horizontal melted water layer with free
surface. J. Heat Transfer, Trans, of the ASME 99:9298.
Shyu, R. J., and C. K. Hsieh. In Press. Numerical simulation of
transient natural convection in enclosures heated from below. To
be presented at the ASME national meeting.
Shyy, W. 1985. A study of finite difference approximations to
steadystate, convectiondominated flow problems. J. Computa
tional Physics 57:415438.
Steele, D. J. 1980. Microprocessors and their application to the
control of a horizontal batch retort. IFST Proceedings
13(3):183193.
70
temperature value of the fluid next to the wall is made equal to the
specified value. On the adiabatic walls and at the centerline (which
is also considered adiabatic due to symmetry), a Taylor series
expansion gives
An +V2ISI An2 + 0(An3) (73)
W 9n lw
where stands for the derivative of temperature in the direction
normal to the wall and An being distance normal to the wall.
for f u 0 (74)
An2
W
= 0W + 0(An2) (75)
JW+1
+1/2
a2e
9n2
Vl = 0W + In
Thus if 0w+j = 0^ is set, wall temperature is evaluated from the
adjacent temperature and it is second order accurate even though
is only first order accurate (Roache, 1982).
In the ADI method, boundary conditions are needed at the
computational "half" step. Since wall temperature 0,^ is not a
function of time, the boundary temperature value at this "half" time
step is simply equal to 0^.
Unlike temperature and velocities, vorticity and stream
function are not primary variables. This makes specification of
their boundary values difficult, particularly for vorticity.
36
amounts of memory that were unavailable until the 1960s. Since then
numerical studies of natural convection heating in enclosures are
being reported constantly. Among the numerical methods, finite
difference methods are probably the most popular in computational
fluid dynamics. The same was chosen for the present study.
Several choices were available for the finitedifference
analogs, computational boundary conditions (treatment of boundary
data in the numerical solution, as compared to real physical
boundary values), and the technique and order of solving individual
PDEs in the overall computational cycle. The relative merits of
these choices are discussed, and appropriate selections are
described in this section.
Grid Generation and Grid Stretching
The discrete points at which finite difference equations are to
be solved constitute the grid system for the problem. For computa
tional efficiency, the number of grid points should be the minimum
that is required to resolve all the significant spatial variations
of the flow. Since this problem involves a rather large step
increase in the wall temperatures together with a high Rayleigh
number (Ra), strong boundary layers are expected to form near the
walls. This is true particularly for the side (vertical) wall. In
order to resolve these boundary layers adequately, a large concen
tration of grid points would be necessary near the walls. An alge
braic method was used to generate the nonuniform grid system. The
particular algebraic expression chosen was evaluated for the proper
17
by a flat narrow slit of light, exposed against a dark background
and photographed. Detailed study of flow in the boundary layer was
observed using a laserdopper velocimeter which is an improved
version of the "particlestreak method" mentioned above. Temperature
values were measured by inserting thermocouples at several points in
the liquid. The temperature variation on the top free surface was
observed using a Thermovision Infrared camera. Radial velocity
profiles were presented at different heights in the liquid and at
different times during heating. His experiment and other works (Chu
& Goldstein, 1973) had showed convective heat transfer from the
bottom, as expected, when the bottom was also heated. His numerical
results, however, could not pick up the convective eddies generated
by the bottom heating. Thus his numerical results had poor corre
spondence with his experimental results.
A finite element method was used by Engelman and Sani (1983) to
study inpackage pasteurization of fluids (beer) in bottles. In the
process studied, bottles entered the pasteurizer at 1.7C(35F) and
passed through several progressively hotter zones of hot water
spray, which raised the product temperature to 60C(140F). This
temperature of 60C is maintained in the holding zone. The product
then passed through several progressively cooler zones, which low
ered the temperature to 70 80F. Since the product was fluid and
the package was not agitated, it was primarily heated through nat
ural convection. Experimental details for this study was provided by
Brandon et al. (1981). The bottle geometry was complex (figure 1)
and so were the boundary conditions. The bottom of the bottle was
LIST OF REFERENCES
Aliada, S. R., and D. Quon. 1966. A stable, explicit numerical
solution of the conduction equation for multidimensional non
homogeneous media. Chemical Engineering Progress Symposium
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Anderson, D. A., J. C. Tannehill, and R. H. Pletcher. 1984.
Computational fluid mechanics and heat transfer. Hemisphere
Publishing Corporation, Washington, D.C.
Ball, C. 0., and F. C. W. Olson. 1957. Sterilization in food techno
logy. McGrawHill Book Company, Inc., New York.
Barakat, H. Z. 1965. Transient natural convection flows in closed
containers. Ph.D. Thesis. University of Michigan.
Barakat, H. Z., and J. A. Clark. 1966. Analytical and experimental
study of transient natural convection flows in partially filled
liquid containers. Proceedings of the Third International Heat
Transfer Conference. The Science Press, Ephrata 2:152162.
BarreiroMendez, J. A. 1979. Prediction of timetemperature rela
tionships and thiamine retention during thermal processing of
canned model systems heated by natural convection. Ph.D. Thesis.
The Louisiana State University and Agricultural and Mechanical
College.
Beardsley, R. C. 1969. Alternative finite difference approximations
for viscous generation of vorticity at a stationary fluid
boundary. Report GRD/692, Massachusetts Inst, of Tech.
Bird, R. B., W. E. Stewart, and E. N. Lightfoot. 1976. Transport
Phenomena. John Wiley and Sons, New York.
Birkoff, G., R. S. Varga, and D. Young. 1962. Alternating direction
implicit methods. Advances in Computers. Vol 3. Academic Press,
New York.
Blaisdell, J. L. 1963. Natural convection heating of liquids in
unagitated food containers. Ph.D. Thesis. Michigan State
University.
Bontoux, P., B. Gilly, and B. Roux. 1980. Analysis of the effect of
boundary conditions on numerical stability of solutions of
NavierStokes equations. J. of Computational Physics 36:417427.
161
96
Table 1.
Sequence of line types for identification of stream
function contours in figures lOao.
82
be derived algebraically for each of the boundaries or the computer
has to solve a simultaneous set of equations at the boundaries (see
Appendix 0). For Robin's boundary condition, the application of the
ADE method required separate algebraic expressions for the top, side
and bottom wall boundaries. Separate expressions were also required
for the top and bottom corners and the centerline. This is in sharp
contrast with the situation if the ADI method were used. Adjustments
would be necessary at the boundaries for the ADI method, but for use
of the ADE method it became clumsy (see appendix D).
For temperature specified at the boundary, this is trivial and
no additional manipulations as noted above are needed. Experiments
conducted later showed that because of the low thermal conductivity
of food (which is mostly water), the Biot number (= hR/k) is quite
large and it is the internal conduction resistance that is limiting.
Thus a convection boundary (Robin's) condition can be approximated
by a temperature specified (Dirichlet) boundary condition for this
purpose. The two shortcomings of the ADE formulation are, therefore,
of little consequence in the case of conductionheated food. The
equations developed for the convection boundary condition are given
in appendix D.
Coding of the Computer Program
The ADE formulation for transient temperature calculation using
temperature specified boundary condition was coded in PASCAL and
used as a subroutine in a real time control algorithm (to be
discussed later).
65
Second, transport velocities are defined normal to the grid volumes
*
in terms of a new set of stream function values xp .
' iJ
* * *
uf .. vf ... Vr ,
J YWT? rj+vJzi+i/2 iv^
ilJ
(64a,b)
The stream function values ik. are defined at the corner of the
grid volumes. These tp. values are related to the previous stream
T 5 J
function values ip. through a consistency criterion which says
Lt u. u.
Ar+0
i ,J i ,J
= 0
(65a,b)
Lt vi i vi i
ArK)
i). was defined as
. .
H ,3
3>.
2
2
^ <*1,J + +1+1,j> (66)
then, using definitions of v^ and v. (equations 64b and
63b), it was shown that the consistency criterion (equation 65b)
require
+ xj = 1
(67)
Similarly, using definitions of u and u.Â¡ it was shown that
*1 jJ >J
125
Figure 21. Temperature at the slowest heating point
during natural convection heating contrasted
with that during conduction heating in a
cylindrical can
47
0R
0.
i
Vi
0
L
Vi
(uR > 0)
(Ur < 0)
(42)
(uL > 0)
(uL < 0)
where Ur and ul are velocities as explained below. This latter
upwind method also has a formal accuracy of 0(az). However, it
behaves like it is secondorder accurate when the quantity
( 0 or in ) is slowly varying (Roache, 1982). Torrance (1968) demon
strated that the second upwind method is indeed superior to the
first upwind method (Roache, 1982). The second upwind method was
used in the present formulation.
The velocities uR and u^ are at the right and left face
respectively of the grid volume as shown in figure 4. As will be
explained later, a kinematically consistent velocity formulation
(Parmentier and Torrance, 1975) was used. In this formulation,
transport velocities (u[ _ and v( in figure 4) are defined normal
to the grid volumes. Accordingly,
UR
UL=
u
f
11J
v A.}
L vi,jl
Following is a development of the finite differences for the
convection terms in the temperature equation. In the z direction,
127
After 90 sec
Figure 24. Isotherms after 90 seconds of heating computed
using 39x39 grid (left half of the figure)
compared with computations using 58x58 grid
(right half of the figure)
6
constant retort temperature for a prescribed length of time followed
by rapid cooling. These process times are calculated on the basis of
sufficient bacterial inactivation in each container to comply with
public health standards and reduce the probability of spoilage to a
specified minimum. Estimation of bacterial inactivation requires
understanding of thermal death kinetics of bacteria, along with the
temperature history the food (containing the bacteria) experiences
during the process. Following is a brief introduction to the thermal
death kinetics of bacteria (Ball & Olson, 1957; Stumbo, 1965).
Thermal death of bacteria follows first order kinetics. If C is
bacterial concentration (number of bacteria per unit volume of food
material) at time t,
dC 1
dt (0.434)Dl
(1)
where (0.434)*D is the inverse of the rate constant. If Cn is the
initial concentration and C is concentration after time At
log
At
D
(2)
The constant D (known as 0 value), which is the inverse of the rate
constant (multiplied by the factor 0.434), is the time required to
reduce the bacterial population by one log cycle at a specified
temperature. The temperature dependency of the D value is given by
61
Discretization of the Elliptic Stream Function Equation
The stream function equation (28), obtained from the definition
of u = 4^. J is also a statement of the circulation theorem
3z 3r
which is
/ ai dA = / v Jl
(58)
area perimeter
element of area
Thus the difference formulation of the stream function equation has
to satisfy the circulation theorem in addition to conserving mass.
Difference approximations that conserve mass are more difficult to
achieve in a curvilinear coordinate system as compared to the
Cartesian coordinate system.
To satisfy both the circulation theorem and the conservation of
mass, Parmentier and Torrance (1975) introduced the concept of
tangential velocities parallel to the control volume faces and
transport velocities perpendicular to control volume faces (Figure
4). In what follows, their approach has been adapted for the non
uniform grid system.
The circulation theorem (equation 58) is written for a grid
volume around node (i,j) as shown in figure 4
(59)
Z i + V2 Z i V2 r j + V2 r j V2
11
This approach was used by Mulvaney and Rizvy (1984). They used a
thermocouplefitted test can in every retort batch to obtain actual
temperature T(t). The system proposed by Navankasattusas and Lund
(1978) also planned to measure can center temperature online. In
commercial practice these methods are costprohibitive with regard
to production efficiency, and are considered impractical.
Numerical Models. Instead of measuring T(t) at the can center,
T(t) can be predicted for arbitrary variations in TR(t). In
general, analytical solutions for T(t) are of limited use for
arbitary variations of TR(t). Teixeira and Manson (1982) described
a numerical finite difference approximation to the heat conduction
equation for predicting T(t) from TR(t). This is conceptually quite
different from the approaches mentioned earlier. With a finite
difference model, T(t) can be predicted for truly arbitrary
variations in T^(t) and, of course, use of this model eliminates the
need for having a test can with an actual thermocouple inserted in
it. This model was first described and verified against published
procedures by Teixeira et al., 1969; and later verified against
experimental data and other published procedures for timevarying
boundary conditions by Teixeira et al. (1975 a,b).
However, in the proposed retort control algorithm of Teixeira
and Manson (1982), every time a deviation took place, a new value of
tft was found through simulation assuming the deviated retort temper
ature continued for the rest of the heating process. This is unnec
essary because any further changes in the retort temperature would
make the estimated t^ value useless and fresh estimations of t^
y
V
5
P
a
T
4
X
u
)
*
viscosity of liquid being heated
kinematic viscosity of liquid being heated = y/p
radial distance in computational plane = 5(r)
density of liquid being heated
deformation parameter in grid stretching
t = nondimensional time
H2
radial coefficients for kinematic consistency
radial coefficients for kinematic consistency
nondimensional stream function = tt
na
I
r 3r
stream function defined by =
nondimensional vorticity
9v 9u H2 9v 9u> _
9Z 3r = a '
3z 3r
vorticity
optimum relaxation parameter for SOR
_ 1 s'f
r 9z
H2
a
Subscripts
a
center
cool
heat
i
j
m
n
W
ambient
geometrical center of can
cooling cycle
heating cycle
grid points in vertical direction
grid points in radial direction
grid point on vertical boundary (z = H)
grid point on radial boundary (r = R)
Wall
ix
148
Table 3. Adjusted heating times and resulting lethality (F0) in
response to process deviations using proposed online
control logic and method of GiannoniSuccar and Hayakawa
(1982).
Reference
Process
Step Drop (figure 33)
Drop duration 10 min.
Drop amount 15F
Gradual drop (figure 34
Drop duration 20 min.
Maximum drop 20F
No
deviation
Online
control
System
GiannoniS.
&
Hayakawa
0nline
Control
System
GiannoniS.*
&
Hayakawa
Time of
Heating
(minutes)
66.8
72.2
72.0
74.6
77.8
Total F0
(minutes)
6.24
6.11
6.02
6.18
7.49
* Treated as step drop of 20F for 20 minutes duration.
14
method. In their later work (Teixeira et al., 1975a) an alternating
direction explicit (ADE) method (Saul'yev, 1957; Aliada and Quon,
1966) was used that was unconditionally stable but still explicit.
This method was first applied to simulate thermal processing of
canned foods by Manson et al. (1974). Teixeira et al. (1975a,b)
compared outputs from the ADE method for using constant boundary
temperature and sinusoidal timevarying boundary temperature with
the results from corresponding analytical solutions. The analytical
results for constant boundary temperature agreed within 0.3C
(0.5F) with the results from the ADE method, except at interior
points near the surface after small initial time (less than 3
minutes). Similar agreement between the results of the analytical
and the ADE formulation was also obtained when a sinusoidally
varying boundary condition was used.
Teixeira (1971) also compared the ADE finite difference outputs
for conduction heating in a can with experimental data for constant
boundary temperature and for step increases in the boundary tempera
ture during heating. The agreement was good except at initial times.
In addition to this ADE method, other methods of solution to
the conduction equation needed to be investigated from the point of
view of their suitability of computation in realtime and ability to
accomodate arbitrary boundary temperature variations.
Natural Convection Heating of Canned Foods
Internal (in enclosures) natural convection flow problems have
received much less attention than the external ones. Also, internal
Water
T,= 30*C
top121'C
slda=121 *C
bot121*C
R4.19cm
h=10.07om
R3B.0.B0
Z=38,0.90
ao.ooot
25
20
15
to
7
5
10
After 300 sec
Figure 10m. Isotherms and streamlines in a
cylindrical can after 300 seconds
of heating
Water
T,= 30*C
top121*C
slda = l'21 *C
bot121*C
R4.19cm
H=10.07om
R3B.0.B0
Z=3B,0.90
ao.oooi
to
8
6
4
2
4
6
Figure lOn. Isotherms and streamlines in a
cylindrical can after 600 seconds
of heating
Analysis of Boundary Conditions
The inverse slopes of the heat penetration curve obtained
experimentally for various heating and cooling processes are given
in table 2.
The data presented are averages for several similar processes.
The standard deviations for these values among replicate cans and
replicate processes were in the order of 1.5 minutes. The results
show that fill temperature (a measure of head space vacuum) did not
have a significant effect on the heating or cooling rates. This was
expected since even though there was considerable difference in
vacuum produced in the two fill temperatures, the physical dimen
sions of the head spaces were not much different.
The difference in heat transfer rates between heating and
cooling were seen to be more pronounced than the difference between
two rates of heating by steam and hot water respectively. The dif
ference between heat transfer rates during heating and cooling was
explained primarily by changes in the thermal diffusivity of the
product because of considerable difference between the average tem
peratures experienced during heating and cooling (Datta et al.,
1984). The small difference between the rates of heating with steam
as opposed to using hot water was attributed to the much larger
resistance inside the can (large Biot number) due to very low
thermal conductivity of foods, which limits the rate of heat
transfer. This means that the convection boundary condition that
exists in an actual heating or cooling situation can be approximated
by a specified temperature boundary condition for calculation
purposes.
13
were derived assuming, among other things, a step change in the
boundary (can wail) temperature from the initial to the heating
temperature and maintaining the boundary heating temperature
constant throughout the process. More general forms of boundary
conditions have also been formulated. Hayakawa (1964) developed the
general form for an analytical solution that could predict the
transient temperature distribution inside a cylindrical can of
conductionheated food when the boundary (can wall) temperature was
any Laplacetransformable function of time.
In practice, the heating medium temperature, which affects can
boundary temperature, is not under complete control and can go
through unexpected variations from its design value. For such
arbitrary variations in the boundary temperature, analytical
solutions are of limited use.
Numerical Studies
A finite difference analog of the heat conduction equation can
also be used to achieve the transient temperature of food at finite
but small increments of time. Such methods can utilize the actual
boundary temperature "read" for the incremental time and calculate
the interior temperature values based on this boundary temperature.
Teixeira et al. (1969) were the first to use a forward difference
explicit finite differencing method to study the transient
temperature distribution inside the cylindrical can. Use of the
explicit finite differencing severely limited the choice of grid
size and time increment, due to stability requirements for such a
80
70
60
50
40
30
T
MID POINT ON AXIS
1/3RD POINT ON AXIS
SLOWEST POINT
20 40
Tima of hooting (seconds)
60
Figure 20. Transient temperature at various locations
during initial times of heating of water in a
cylindrical can
130
100
70
40
10
25
RETORT
AN CENTER
 15
TTTTTTT rTT TTTTJT I I I  I I I l' j T I I I
25 50 75 100 125 150
20
 10
 5
0
TIME (minutes)
34. Computer plot of a process that experienced a step functional
drop of 8.3C (15F) in temperature between 50th and 60th
minute and still maintained an Fn of 6.02
cr>
ACCOMPLISHED F0 (minutes)
63
The tangential velocities ik and .= at the faces of the
jJ 1 jJ
grid volume are obtained by finite differencing equation 29
u* .=
i
^i J+l'^iJ
'j+V2 rj+i* rj)
(60a,b)
ViJ:
N+U ~ ^ ,j
rj(zi+r zi J
Substituting these definitions in equation (59)
). .=
V zi+V2"zii/2
i+l,j i j
zi+r
z.
1
,J 11, J
z.z. .
1 11
1 r i J+r^i ,j ^i.j^i.jl 1
rj>Vfrj Vz Wj+ipjr rji/2irjrji)
(61)
Note that the same finite difference equation (61) can be obtained
from central differencing the stream function equation (28) without
introducing the circulation theorem. This is shown in Appendix A.
After algebraic manipulation,
zi+l
Figure 4.
Notations of various quantities defined on a
nonuniform grid system in cylindrical
geometry
125
110
O
O
(D 90
L_
O
L_
CD
CL
70
Â£
CD
y 50
*
*
*
* *
* 39(0.93X39(0.8) grid
v 58(0.763X58(0.58) grid
_i i i i
120 240 360 480
Time of heating (seconds)
600
Figure 27. Transient temperature at l/3rd height at the
centerline computed with 39x39 and 58x58 grid
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TITLE: Numerical modeling of natural convection and conduction heat transfer in
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872393)
PUBLICATION DATE: 1985
I,
Aj> k/V K
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30
dimensional flows when no free surface is present, iterative methods
are used to solve the Poisson equation, and transient pressure solu
tions are not required (Roache, 1982).
1. The (u,tji ) system needs to solve one parabolic equation for
a) and one elliptic Poisson equation for ip as compared to the
(u,v,p) system which requires the solution of two parabolic
equations for u and v and one elliptic equation for p.
2. With the (oi, ) system, two additional differential equations
(29a and 29b) are needed to get the velocities u and v, but
the u and v momentum transport equations are more complicated
than the u transport equation.
3. At noslip walls, the boundary conditions for u and v are
known explicitly (uwan = 0, vwan = 0) as compared to u ^
which is not known explicitly. This would be a great advantage
for using implicit methods for the (u, v, p) system but non
linear instabilities of the pressure term prevents the (u, v,
p) system from using implicit methods.
4. The elliptic equation that needs to be solved for the pressure
takes much longer to converge than the elliptic equation for
4> (eq. 28).
Conduction Heat Transfer in Canned Foods
The POE describing conduction heat transfer in a cylindrical
body of solidpacked food material is presented here. The boundary
conditions present during the thermal processing of canned foods are
described next.
85
be clamped tightly in place once inserted to the desired height. The
holes were located so that the cans stood two abreast in four
staggered rows. This configuration helped to assure that all the
cans received nearly equal exposure to the heating and cooling
media, and there could be no possible movement of the cans during
sudden pressure changes that could occur during initial cooling.
The same eight cans were used repeatedly in four different
processes with two replicates of each. The same retort temperature
(121C) and process time (75 minutes) was used in all four
processes. They differed in the type of heating medium used (pure
steam or pressurized water), and in the retort pressure during
cooling (either held constant at the level required during heating
or allowed to drop suddenly to near atmosphere with the introduction
of cooling water). The flow rate of recirculating cooling water was
adjusted so as to achieve a low mainstream velocity of 0.05 meters
per second between the cans. Transient temperatures and pressures
were recorded with a data logger.
While recording the transient temperature response at the can
center, it was observed that sudden depressurization of the retort,
during start of the cooling cycle, resulted in a very different
centertemperature profile during the early period of cooling. The
can center temperature was found to drop much more rapidly in the
case of sudden depressurization as compared to when the retort
pressure was maintained constant. It was postulated that such rapid
fall in the retort pressure caused cans to expand, thus creating
sudden loss of internal pressure causing the product to "boil"
159
At the top boundary, finite differencing the boundary condition
Qn+1 Qn+1
nz,j nzlj hHQn+l
Az k nz,j
(95)
and at the sidewall
in+1 en+1
i,nr i ,nrl hHQn+l
Ar ki ,nr
(96)
The finite differences for the boundary conditions and the governing
equations are now combined. At the upper corner, combining equations
93, 95 and 96
0n+1 = cf 0n .+ S(0n .+ 0n 9 .)
nz,nr AU+P+S)1 nzl,nrl 1 nzl,nrl nz2,nrl;
+ pf0n + 0n 1 + Hf_Qn 'll
nzl,nrl unzl,nr2J Rl nzl,nr2"
. h (P+Q/R)(1+yAz) S(l+YAr)
where A = (1+ PAr) (1+ rAz)
K K 1+P+S 1+P+S
Combining equations 93,95 at the top wall,
(97)
,n+l
nz,J A(1+P+S)
 {0n .+ S(0n .+ 0n 9 .)
Dj.c\ 1 nz1 ,j nz1 ,j nz2,j ;
+ P((l+ Az)0n+1 0n .+ 0n i i )
u k nz,j+l nzl,j nzl,jl'
+ %0n ,+ (1+ r Az)0n+1 )}
r^ nzl,jl v k 1 nz,j+l;j
(98)
where A = 1+ jrAz
1+P+S
APPENDIX A
ALTERNATIVE FINITE DIFFERENCING OF STREAM FUNCTION EQUATION
The stream function equation
_1 32^ j), 1_
' 7 ~Bz7 + 3r^ r 3r J
Using central differencing
B2ip
Bz7
_3 > 3i{>> 3p 3 > 3^\
3z '3z 3z 3n (Bz'
1
3z/3n
3^
3Z
i+V?J i V? J
An
*i+lJ~*i J ^iJ ~ ^ilj
z.z.
1+1 1
V zii
zi+r zii
2 An
2
zi+r zii
An
j^i+lj" *itj *i,j *il,j
zi+r zi
zt zii
)
3 /1 3iK
3rV 3r'
35 1_ (I Mi
3r 35 'r Br1
_3t{> 2. 3^1
i W jt.i+ v? riv,arli,.ii/,
r.i+r rii 45
2A5
f 1 > J y I ^ ^ 3 T > J
mv 1. ( r r l~
rj+r rji v4(rjn rjr rj4(rr rji>'
Combining the terms, we get equation 61.
154
162
Brandon, H., P. Pelton, and G. Staack. 1981. Stateoftheart
methodology for evaluation of pasteurizer heating and cooling
processes. MBAA Technical Quarterly 19(1):3440.
Briley, W. R. 1971. A numerical study of laminar seperation bubbles
using the NavierStokes equations. J. Fluid Mech. 47(4):713736.
Chu, T. Y., and R. J. Goldstein. 1973. Turbulent natural convection
in a horizontal layer of water. J. Fluid Mech. 60:141159.
Datta, A. K., A. A. Teixeira, and J. E. Manson. In Press. Computer
based retort control logic for precise online correction of
process deviations. J. Food Sci.
Douglas, J. Jr. 1955. On the numerical integration of
S2u/5x2 + 62u/6y2 = <5u/6t by implicit methods. J. Society of
Industrial Applied Mathematics 3(1):4265.
Engelman, M. S., and R. L. Sani. 1983. Finiteelement simulation of
an inpackage pasteurization process. Numerical Heat Transfer
6:4154.
Evans, L. B., R. C. Reid, and E. M. Drake. 1968. Transient natural
convection in a vertical cylinder. AIChE Journal 14(2):251259.
Forsythe, G. E., and W. R. Wasow. 1960. Finite difference methods
for partial differential equations. John Wiley and Sons, New
York.
Frankel, S. P. 1956. Some qualitative comments on stability
considerations in partial differential equations. Proceedings of
the Sixth Symposia in Applied Mathematics, AMS, 6:7375.
Fromm, J. 1964. The time dependent flow of an incompressible viscous
fluid. Methods of Computational Physics 3:345382.
Gentry, R. A., R. E. Martin, and B. J. Daly. 1966. An Eulerian
differencing method for unsteady compressible flow problems. J.
Computational Physics 1:87118.
Getchell, J. R. 1980. Computer process control of retorts. ASAE
paper no. 806511. Presented at the winter meeting of the ASAE
at Chicago, Illinois.
GiannoniSuccar, E. B., and K. Hayakawa. 1982. Correction factor for
deviant thermal processes applied to packaged heat conduction
food. J. Food Sci. 47(2):642646.
Hayakawa, K. 1964. Development of formulae for calculating the
theoretical temperature history and sterilizing value in a
cylindrical can of thermally conductive food during heat
processing. Ph.D. Thesis. Rutgers, The State University.
Water
T,=30*C
op121*C
Slde121*C
boI121*C
R4.19cm
H=10.67om
R57.0.76
2=57,0.85
8=0.0001
300
260
220
180
140
100
60
20
20
60
Figure lOi. Isotherms and streamlines in a
cylindrical can after 9 seconds of
heating
Water
T,= 30'C
top121'C
slde=121*C
bot=121'C
R=4.19cm
h=10.67cm
R57,0.76
Z=57,0.85
6=0.0001
300
260
220
180
140
100
60
20
20
60
A ter 10 sec
Figure lOj. Isotherms and streamlines in a
cylindrical can after 10 seconds
of heating
78
The temperature equations (47a and 47b) were solved using
the ADI technique. This gave new temperature values after
incremental time At.
The vorticity equations (55a and 55b) were then solved
(also using the ADI technique). The interior vorticity
values calculated were provisional since old values were
used for boundary vorticity.
The stream function equations (61) were then solved for
interior stream function values using the point SOR
technique. This required only interior vorticity values.
Previous stream function values were used as an initial
guess.
Boundary vorticity values ( Z^) at solid walls (top,
bottom and side) are first calculatedusing equation 76
based on the newly calculated stream function values.
However, before updating, the boundary vorticity values
(Zb) are underrelaxed as
n+1
m+1
= Y a).
n+1
m+1
+ (1 y)
n+1
m
(BO)
n+1
m+1
where = boundary vorticity estimated using
equation and 76, with 0
Steps 2 through 4 are repeated until the estimated ^
values (using equation 76) in two consecutive iterations
meets the convergence criterion
TIME INCREMEMT (seconds)
0.25
T
T
T
T
0.20
TIME OF SIMULATION (seconds)
Figure 28. Time step history during a typical run of the
numerical heat transfer model for natural
convection
GJ
ro
37
ties listed below (Anderson et al., 1984; Ka'lney de Rivas, 1971).
Here (r,z) refers to the physical domain and (5,n) refers to the
computational domain with AÂ£ and An as constants.
1. The mapping from (C,n) to (r,z) must be one to one.
2. Grid lines should be smooth to provide continuous
transformation derivatives. This requires r(s) and z(n)
as well as and ^ should be continuous over the closed
intervals 0
functions defined on a closed interval are bounded, this in
turn implies that
dr
cff
dr
and
dz
should be finite over the
whole interval. If becomes infinite at some point, the
mapping r(c) would give poor resolution near that point,
which cannot be improved by increasing the number of points,
d p
since a r (^) (as) Similarly for dz/dn.
3. Close spacing of grids where large numerical errors are ex
d p i
pected. For this problem, that means rr\ =0 at the
a5'r=R/H
sidewall which ensures high resolution of the boundary layer
near r = R/H. Similarly at z = 0 and z = 1.
4. Absence of excessive grid skewness which can sometimes
exaggerate truncation errors (Raithby, 1976).
An algebraic transformation equation suggested by Kublbeck et
al. (1980) for natural convection in cavities was
n = (1+ tan[ir/2 (2zl) p]j
2l
tann/2 a
(37)
COOLING
WATER
Figure 6.
Experimental setup for conduction heating
00
141
This is important for realtime calculation since the ADE method
becomes considerably more complicated when required to account for
convection boundary condition as compared to the specified temper
ature boundary condition.
Comparison of the Transient Temperature Values
The transient temperature values recorded at the can center as
heating progressed are reported in figure 31. The observed
temperature values are compared with the predicted values using the
analytical solution to (equation 8a) and the ADE formulation to the
conduction equation. As can be seen, there was excellent agreement
between the predicted and observed values.
Effect of Sudden Pressure Drops in the Can
Figure 32 illustrates the effect of sudden depressurization of
the retort on transient temperature values at the can center. This
had been shown to be due to internal product movement from sudden
loss of pressure inside the can (Teixeira et al., 1985) and empha
sizes the importance of maintaining the retort pressure during
cooling.
Performance of the Online Control Logic
The estimated process times and the accumulated FQ values for
the three process conditions (figures 33, 34 and 35) are presented
in table 3. For the reference process without any deviation (figure
33) the CF method predicted a correction factor close to 1 or no
73
At the centerline,
vi,0 vil ,0 ui ,0" ui,0
ui ,0" z. i. z.
i, z. i, ri. 0
i + 72 1 72 72
=0
(76c)
since
vlj g and v^ Q from eqn. 60b
t i.r *1,0 2*i,i
i,0 tl/2(rr 0) r\jV 1
ui,0=0
At the sidewall,
t t
v. v. .
i ,n il,n
U). = 1
u u ,
19 n in1
1>n" Ui+V2 ZiV^ (v vv)
 Ill,
i ,nl
r 1 ,(r r ,) (r ~r 1.)
nl/2 n n1 n n72
(76d)
since
v? n and v^ . = 0 from eqn. 60b
1 ) n i
. ip. ~ip. ip. ,
ut = i,n i ,nl i ,nl
i,nl r i.(r r TJ r i.(r r 7}
nV2 n n1 n72 n n1
u. = 0
i ,n
Identical expressions can be derived using Taylor series expansions
of equation 28 at the walls. This is shown in the appendix B.
These expressions are first order accurate. But it is the only
form consistent with the kinematic velocity formulation (Paramentier
& Torrance, 1975). The second order form using Taylor series
(Jensen, 1959) has also been used. However, sometimes the second
133
problem. For purposes of comparison, the explicit formulation of
Barakat (1965) for the set of equations (2629) was used. In their
formulation, they "lagged" the velocities u and v in the convection
terms and used a backward finite difference for these terms. Their
stability analysis showed the following restrictions on At
At (
i ,J
Az
2
(AZ)2
(Ar):
(86)
Non dimensionalization of the variables used in the present study is
different from that used by Barakat (1965). Accordingly equations
(8486) have been adjusted to conform to the present study. For Pr >
1, as is the present situation, inequality (85) is more restrictive
than (84).
For a nonuniform coordinate system, Ar and Az to be used are
their smallest values in the grid systems (Ka'lney de Rivas, 1971).
However, the allowed time step At here is also a function of grid
point velocities. Thus, using equation 85, At (and therefore At )
is calculated at all the grid points. For a 39(0.9)x39(0.8) grid
Min At. =
i, j ,J
.07seconds
Ra = Rayleigh number = Gr*Pr
t = time
T = temperature at any point inside at time t > 0
Tg = initial uniform temperature inside (at t = 0)
Tj = boundary temperature at time t > 0
Tr = retort temperature at time t
LI
u = = nondimensional velocity in vertical direction
a J
= velocity in vertical direction
H 
v = v = nondimensional velocity in radial direction
v = velocity in radial direction
z = q = nondimensional distance in vertical direction
z = distance in vertical direction
Z = Zvalue for organism
a
B
Y
A
6
e
n
0
A
A
P
A
A
m
thermal diffusivity = k/pCp
coefficient of thermal expansion
relaxation parameter for boundary vorticity
designates a difference when used as a prefix
convergence criterion for boundary vorticity
convergence criterion for SOR
vertical distance in computational plane = n(z)
TT l
= nondimensonal temperature
VT,
first eigenvalue in z direction
eigenvalues in z direction
first eigenvalue in r direction
eigenvalues in r direction
vi i i
43
Instead of using a fully implicit set of equations, alternating
direction implicit (ADI) methods were introduced in companion papers
by Peaceman and Rachford (1955) and Douglas (1955). The advantage of
the ADI method over a fully implicit method is that the set of
equations, although implicit, is tridiagonal. Further the ADI method
as applied to linear equations has a formal order of accuracy of
0(At2,Az2,Ar2). The stability of this method is also unconditional
as determined by VonNeumann stability analysis. For irregularly
shaped regions, programming could get complicated (Roache, 1982) but
for simple rectangular regions, as is the present situation, that is
no particular problem.
The practical advantage of the ADI method over the explicit
method is, however, nothing like that indicated by VonNeumann
analysis (Roache, 1982). It is unstable both at large time steps
and/or insufficient convergence of boundary vorticity. In general,
many researchers have indicated that ADI methods do allow larger
time steps and faster overall computation, by a factor of two or
more, and furthermore allow second order accuracy in time. The ADI
method is a very popular method in computational fluid dynamics.
While solving the set of PDEs (2629) Barakat and Clark (1966)
compared explicit and implicit methods for their study and decided
against the use of the implicit method due to unavailability of the
vorticity boundary conditions at the wall (to be explained later).
They allowed vorticity boundary values to lag one time step, neces
sary for such explicit formulations. The lagging of boundary
vorticity is considered undesirable (Roache, 1982) and iterations
91
f0W + (Fo001 ^simulated Fo <82>
is satisfied, the computer turns off the steam and lets in cooling
water. The computer still keeps reading the retort temperature
Tr(t) and calculating T(t) and F0(t). When T(t), the calculated
temperature at the can center, is below a certain specified value,
cooling is ended by stopping the cooling water flow and the water is
drained prior to unloading the retort.
At the end of the process a complete documentation of measured
retort temperature history Tp(t), calculated can center temperature
history T(t) and the accomplished FQ(t) is kept on file and can be
documented in both tabular and graphical form.
The control algorithm presented in figure 8, implemented in
realtime with a computer, can ensure designed sterilization F^ in a
product for arbitrary variations in retort temperature. The
performance of this system for correcting deviations was compared
with process corrections predicted by the correction factor (CF)
method of GiannoniSuccar and Hayakawa (1982). Comparison of the
two methods was based on
1) estimated process times, and
2) estimated accomplished F0 values.
Three different processes were considered:
1) reference process with no deviation,
2) step drop and later step rise in process temperature, and
3) gradual drop and later step rise in temperature.
PAGE 1
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PAGE 192
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140
Table 2. Heating and cooling rates for various process conditions.
f values (minutes) from eqn. 9
(Slopes of semi log timetemp graph)
Steam Water Water
Fill temperature Cook Cook Cool
44.8 44.8 51.2
43.8 44.7 50.9
67C (153F)
90C (194F)
160
Combining equations 94 and 95, at the top wall on the centerline
0
n+1
nz,0
1
A(1+2P+S)
{0
n
nz1,0
+
S(0
n
nz1,0
+
0
n
nz2,0
)
+ 2P((1+ y Az)0n+1
, 0 A+
nz,l nz1,0
zl.l
here A = U Â£az 
Combining equations 93 and 96 at the sidewall
(99)
i ,nr
1 {" i+ S((l+ Ar)0?*! Qn. ,+
1 i,nrl ^ k i+l,nr i ,nrl
A(l+P+S)
+ P(0Â¡ 1+ 0^ p __ <
^ i ,nrl i,nr2' R1 i,nr
n
il,nrl^
)} (100)
where A = 1+ {V 
Equations 93 and 94 are then used for the interior points. Moving
outwards, a completely different (algebraically) set of equations are
used. For the interior points, equations 91 and 92 are used. At the
boundary, equation 96 is used for the sidewall
0n+1
0n+l + ,l,nrl
,*nr TTfTr
At the top wall, equation 95 is used
0n+1
0n+i nzl,j
nzJ
and finally, using equation 96 for the upper corner
An+1 An+1 / h ,
0 =0 / 1+ t Ar
nz,nr nz1,nr k
where a is a deformation parameter that varies the degree of
stretching. However, since the computer coding of the problem is
ultimately done in the physical domain, the inverse transfor
mation z(n) is more appropriate in this context which is given
below.
z =1/2 {1 + tan^^n 1) tan a)}
(38)
This concentrates the grids on either end z = 0 and z = 1.
In the r direction, the above transformation is modified to
concentrate grids on one side (side wall, r = R/H) only. Thus
R
H
(39)
To check if the transformations indeed satisfy requirements 1
through 4, we first see that both the transformations are one to
one. Next, looking into the transformation derivatives,
dr 2 R
dÂ£ it a IT
tamr/2 a
1 + 52 tan2ir/2 a
(40a,b)
dz 1 2
2 tan tt a/2
dn ? it a
1 + (2n l)2 tan2ir/2 a
which are continuous in 0 < Â£ < 1 and 0 < n < 1 respectively.
95
patterns accurately. Streamlines, on the other hand, can provide all
the essential details of the flow patterns without the clumsiness.
This is particularly true in the initial times when velocities are
many times larger near the boundary layer than at the core. This
created a scaling problem for vector plots. Also, in the boundary
layer, grid points are very close together. Thus, for most of the
figures, streamlines are presented instead of velocity vectors. At
much later times (t=1800 seconds) when the flow approached steady
state, and the velocity magnitudes became comparable in the core and
in the boundary layer, velocity plots would be visually more
meaningful (see figure 10o).
The magnitudes of the streamlines in each figure are noted on
the right side of each figure under "5 =0.0001". To read the value
that corresponds to a particular stream function contour, the line
type of the contour is noted. Next, the order of that line type in
table 1 is noted counting from the top. The value of the contour is
the number in the same order from the top in the list accompanying
each figure.
Start of flow and the conduction layer. For very small times
after heating starts, the fluid nearest to the bottom wall is
restricted to being conductionheated. Figure 9 shows the area near
the bottom wall of the can. Further from the sidewall, the isotherms
in this figure compares very well with temperatures predicted at the
same height using the conduction equation in a semiinfinite
(0 < z < ) region (Ozisik, 1980).
26
equations have been approximated for no viscous dissipation. Jaluria
(1980) discussed various situations under which viscous dissipation
can be neglected. All other fluid properties are treated as
constants except variation of density which is discussed below.
Boussinesq Approximation
The solution to the set of equations can be considerably
simplified using the Boussinesq approximation. In the Boussinesq
approximation, the density is treated as constant except in the body
force term. The density difference in the body force term is approx
imated as
p p = p8 (T T ) (19)
a a
and the density variation in the continuity equation is ignored.
This approximation requires both
8 (T T ) 1 (20)
"'! ( )T g 5 1 <21>
to be true. The second inequality is generally satisfied for
liquids. So is the situation with the first inequality. For
example, for natural convection in water, as in this problem
8 = 0.0002 /C
T = wall temperature of the can = 121C
Ta= initial temperature of liquid in can = 30C
Figure 13. Isothermal vertical surface, showing boundary
layer with velocity profile u(x,y). and
temperature profile t(x,y) in water (Higgins
and Gebhart, 1983).
135
Wafer
T,=30*C
top121*C
s!da=121*C
boT121*C
R4.19cm
H=10.67cm
LR=*3S,0.80
LZ=38,0.9Q
50.0001
RR38,0.80
RZ38,0.90
5=0.001
After 30 sec
Isotherms in a cylindrical can calculated with
two different convergence criteria for
boundary vorticity (<5=0.0001 on the left
and <5=0.001 on the right)
Figure 29.
METHODOLOGY
Detailed development of the solution techniques for natural
convection and conduction heat transfer models are described in this
chapter. For the natural convection heating problem, formation of
finite difference equations, setting up of computational boundary
conditions, and development of the algorithm for the iterative
solution of the complete problem, are described first. Along the
same lines, solutions of the finite difference analog to the
conduction equation are described next. Experimental study of
conduction heating is then reported. With the thermal models
working, use of the conduction heating model for online control of
conductionheated food is then presented.
Modeling of Natural Convection Heating in Cans
Because of the inevitable coupling of the governing equations,
except for extremely simplified cases, most natural convection flows
of practical interest are too complex to be solved analytically.
Thus numerical computations are generally employed to obtain the
desired solution. The present problem of natural convection in a
cylindrical enclosure is no exception. Numerical solutions to this
and similar problems demand high speed digital computers with large
35
80
Coding of the Program and the Hardware
The algorithm presented in figure 5 for solving the vorticity
stream function formulation of the natural convection problem was
coded in FORTRAN 77 with individual modules for solving temperature,
vorticity, stream function, velocity and boundary vorticity values.
Structured programming practices were used. Use of GO TO statements
were avoided except in two situations to simulate a WHILE statement.
The WHILE statement was unavailable in the particular version of
FORTRAN 77. Single precision of the variables was known to carry 14
significant digits for the particular computer and the same was
taken to be sufficient for this problem.
The input data file for a typical run is in Appendix C.
The computer used was a CDC Cyber 170/730, with two 60 bit CPUs
crunching 60 bit words at 400 MHz, 20 parallel processors, and 1091K
of 60 bit word central memory.
The plotting was done using a plotting package from the
University of Illinois*. The convenience offered by the plotting
package cannot be overemphasized. The plots were viewed on a
Tektronix 4006 terminal and hardcopies were made on Tektronix 4662
plotters.
Modeling of Conduction Heating in Cans
The partial differential equation (POE) governing conduction
heating of a cylindricalshaped solid was discretized and finite
*W. E. Dunn, Mechanical and Industrial Engineering Department,
University of Illinois at UrbanaChampaign. Personal Communication.
66
the weakest condition that satisfies equation 65a was the recursion
relation
1 + (2j 1) Xj.L
ZjFT
(68)
Using boundary conditions on the axis, it was found that
Xo = 0.25 (69)
The transport velocities defined this way conserve volume flow
exactly and are consistent with the grid point velocity field as the
mesh is refined.
The transport velocities ut and vt are the velocities
jJ '
used in the energy and vorticity transport equations. The grid point
velocities u. and v. are the velocities plotted in the
i jj i
velocity plots.
To solve for ^ from the vorticitystream function equation
61, we note that it is an elliptic Poisson equation as compared to
the parabolic equations for temperature and vorticity. Direct (non
iterative) methods to solve the equation are available that are
considerably (10 times) faster than an iterative method like the
successive overrelaxation technique (to be discussed later).
However, direct methods usually require large amounts of storage,
and available computer codes are often limited to rectangular or
simple domains or have restrictions on the type of boundary
conditions. Shih (in press) discussed suitability of direct methods
Figure 2. contd.
method. Grid stretching was used in both radial and axial
directions. Boundary wall vorticities were used to check for
convergence of the iteration process.
Plots of transient isotherms, streamlines and velocities were
provided. Calculated radial velocities and axial temperature
profiles at different times compared quite well with available
experimental data. Slowest heating points were located in the bottom
15% of the height of the container studied. These slowest heating
points migrated over time within this lower region of the can but no
particular pattern of migration was noted.
Transient can center temperature was also studied for the case
of conductionheated solid food. An alternating direction explicit
finite difference solution to the conduction equation compared very
well with analytical solutions and experimental results. The ability
of the finite difference model to perform under arbitrary variation
of boundary temperature allowed the model to be incorporated in an
algorithm for online control in batch thermal processing of
conductionheated food. The uniqueness of this control system in
maintaining the level of desired food sterilization for arbitrary
variation in process heating temperature was shown to offer
considerable advantages over other possible methods of online
xvi i
control.
130
slowest heating temperatures at all times. The grid sizes also
predict identical transient temperatures at l/3rd height on the cen
terline as shown in figure 27.
Selection of the time step. A comprehensive stability ana
lysis, that included all the finite difference equations and the
boundary conditions, was not practicable. It was found that larger
time increments required more iterations per time step that could
actually increase the computation time for the overall problem. A
strategy was taken whereby the computer automatically adjusted the
time increment to restrict the number of iterations to about 15.
This was done in the following manner
1. If the number of iterations during a particular time
step is less than 15 and decreased from the number of
iterations required in the previous time step, the time
step is increased by 0.05 seconds.
2. Otherwise the time step (At) is decreased by 0.05
second.
The time step history for a typical run is given in figure 28. A
similar algorithm for adjusting the time step to limit the number of
vorticity iterations was used by Kee (1974) from the work of Briley
(1971).
Since implicit methods generally consume more CPU time per time
increment, it was decided to investigate how the time steps used in
the ADI method compare with time steps that could have been achieved
using a possible explicit finite difference formulation for the
APPENDIX D
ADE FORMULATION FOR THE CONDUCTION EQUATION
Alternating direction explicit formulation for the conduction
equation is noted below. For points other than center and with i,j
increasing ( moving away from center)
_30 320 320 _1_30
3x = 'dzT + Jr7 + r 9r
e"*}. e? .
1,J 1 ,3
At
01?
i+yu
 e? en+1+ e>?+1
1 ,J
il,j
Az
f: 0? 0?+^+ 0?+* .
1tJ+l ItJ 1.1 TtJ1
2
+ I
r
0n 0n+1
ij+l itjl
2Ar
0,?+l
i ,3
~{0? .+S(01?.i  0? + 0?+} )
DiC V 1+l.J 1,3 1l.j'
1+P+S
+p(ei,j+r i,j+ eiji)+r(8i,j+r sji)!
Q,n
ji+1
(91)
where P=Ax/Ar2, Q=Ax/2rAr, S=Ax/Az2
For points at center (j=0), for use while i increasing
80 a20 a20
3x ~dZ7 3r2
nn+l nn Qn nn Qn+1, nn+l 9frn n nn+l, n+l >
9i.o ei,o _9i+i.o' 9i,o ei.o+ Vi.o 2(0i,r ei,o si,o+ ViJ
! 2 + at2
Ax
0n+1
i,0
1+2P+S
bfeuo+s<0Hi,o 6m+ eii,o)+2P(si,r e?,o+ li>)
(92)
where o!?+^= 0? ^ used due to centerline symmetry
157
27
Thus,
6 (T T ) = (0.0002)(12130) = 0.0182 1 (22)
cl
Almost all the previous works on natural convection in similiar
areas have used the Boussinesq approximation.
Transformation of the PDEs
To further facilitate numerical solution to the set of
equations, the primary variables are transformed to a new set of
variables. Vorticity (to) and stream function ( ijj ) are defined in
terms of the primary variables u (axial velocity) and v (radial
velocity)
 3v 3
3z 3r
(23)
u
1
 r *
r 3r
1_
r 3z
(24a,b)
Using vorticity so defined, the pressure (p) is eliminated from
the u and v momentum equations 14 and 15. These two equations gave
the parabolic transport equation for vorticity Next, using the
definition of m (equation 23) and the definition of ij/ (equations
24a and 24b), the equation for ip is derived. This gave the vorticity
streamfunction formulation for the natural convection problem.
145
correction. The online computer method also maintained the process
time and the accumulated F0 close to their design values. Next, a
process with an idealized stepfunctional drop was considered. This
is shown in figure 34. The retort temperature underwent an idealized
step drop of 8.3C (15F) for a period of 10 minutes from the 50th
minute to the 60th minute. The CF method predicted an extended
heating time of 72 minutes, resulting in a total accumulated FQ
value of 6.02. The online control logic extended the heating time
to 72.2 minutes, resulting in a total accumulated F0 value of
6.11. Thus, both methods predicted very similar extensions to the
heating time and were able to maintain the required total FQ of 6.0.
A process that experienced a gradual drop (linear ramp) after
about 40 minutes into the cook cycle, was considered next (figure
35). The linear decrease in temperature continued for 20 minutes
with a rapid return to set point. The online computer method of
control extended the heating time to 74.6 minutes in order to still
maintain an F0 of 6.18. Since the CF method can only be applied to
step function deviations, the gradual drop in this case was treated
as a step drop of 20F for the 20minute duration. This represented
a more severe deviation than the actual ramp function, and the pre
dicted extension in process time would be expected to overcompensate
for the actual loss in lethality but would be sure to predict a safe
process. The results in Table 3 show how the CF method predicted a
process time of nearly 78 minutes (three minutes longer than neces
sary) resulting in a final F0 of 7.5 minutes (nearly 20% overproces
sing compared to the target FQ of 6.0 minutes).
18
20 SECONDS
42 SECONDS
Figure 1
Isotherms and velocity profiles in a. glass
bottle at various heating times (after Engelman
and Sani, 1983)
113
times of 30, 240, and 450 seconds are 35, 16 and 12 mm/second re
spectively. The observed maximum positive velocities at the same
time instants by Hiddink (1975) were 32, 18 and 12 mm/sec respec
tively. Thus the computed and observed values have good agreement.
As time passes, the enclosed liquid gets warmer so that the local
Grashof number at wall drops, which means that the buoyancy force
drops. Thus, as the figures show, the maximum velocities at the
boundary layer reduced with time in both computed and observed
values. The computed and observed profiles also agree in predicting
the thickening of the boundary layer as time passes due to further
penetration of heat. The thickness of the boundary layer was within
510% of the radius and agrees with observations of Hiddink (1975).
Shyu (1985) also reported boundary layer thickness of 510% of
radius but his computed velocities were much smaller and could be
attributed to a much smaller temperature difference imposed on the
wall in his case.
Axial temperature profiles. Temperature values along the axis
of the cylindrical can at different times are presented in figure
15. The axial temperature profiles were also available from the
experimental study of Hiddink (1975). These are plotted alongside.
The input data for figure 16 are radius=8.9cm, height of liquid
=18.0cm, top surface free and sidewall and bottom wall temperatures
of 120C. It can be seen that the numerical model predicts the ob
served values with about 5% discrepancy except at small times after
heating. Qualitatively, too, the predicted and observed shapes of
the temperature profiles at different times are very similar.
67
for "large" systems of equations. Direct methods also have round off
error propagation problem. With direct methods, roundoff errors can
be incurred at each mathematical operation, and simply accumulate
until final answers are obtained. When iterative techniques are
used, the presence of roundoff errors at the end of any given
iteration simply results in those unknowns being somewhat poorer
estimates for the next iteration. For practical purposes, roundoff
error in the final converged values of an iterative scheme is only
that accumulated in the final iteration (Hornbeck, 1975). A very
efficient direct method for solving Poisson's equation was described
by Schumann and Sweet (1976) and the code was available from the
National Council of Atmospheric Research (NCAR) software library.
However, on closer look the algorithm (GENBUN) was found to be
limited to coordinates transformed in one direction only3. Since in
this study the boundary layers were expected to be present both
along sidewall as well as top and bottom wall, coordinate stretching
were considered necessary in either direction. This made the above
algorithm unsuitable for the present study.
A popular approach to solve such a steady state equation is to
formulate it as an unsteady equation and solve for the asymptotic
steady state solution. ADI methods have been used for this purpose.
These methods were known to be faster than the successive over
relaxation method (to be discussed later) for a rectangular region
(Birkoff et al., 1962). However, the optimum sequence of parameters
aR. Sweet. U. S. Dept, of Commerce. National Bureau of Standards.
325 Broadway, Boulder, Colorado. Personal Communication.
APPENDIX C
INPUT DATA FILE FOR NATURAL CONVECTION HEAT TRANSFER MODEL
4.19
10.67
981
0.0002
0.0014612
0.0089
1
30.0
121.00
38
38
10000
1.82
0.00001
0.0001
2
1
1
1
19
38
1
1
1
0
0.5
40
120200
2
2
0.80
0.90
RADIUS OF CYLINDER IN CM(=1.65 INCHES)
HEIGHT OF CYLINDER IN CM(=4.20 INCHES)
G(ACCN. DUE TO GRAVITY) IN CM/SEC SQ.
VOLUMETRIC COEFF. OF WATER IN /DEG C AT 27 DEG C
THERMAL DIFFUSIVITY OF WATER IN CM SQ/SEC AT 25 DEG C
VISCOSITY OF WATER IN GM/CM/SEC AT 25 DEG SEC
DENSITY OF WATER IN GM/CC
INITIAL UNIFORM TEMPERATURE INSIDE CAN IN DEG C
WALL TEMPERATURES AT TIME GREATER THAN ZERO IN DEG C
NUM8ER OF GRID ELEMENTS ALONG HEIGHT
NUMBER OF GRID ELEMENTS ALONG RADIUS
INTERVAL OF PRINT IN MILLISECONDS
OPTIMUM RELAXATION PARAMETER FOR SOR
CONVERGENCE CRITERION FOR STREAM FUNCTION
CONVERGENCE CRITERION FOR VORTICITY
CENTERLINE BOUNDARY IS INSULATED
SIDEWALL BOUNDARY IS AT HEATING TEMPERATURE
TOP BOUNDARY IS AT HEATING TEMPERATURE
BOTTOM BOUNDARY IS AT HEATING TEMPERATURE
NUMBER OF RADIAL GRID POINTS TO PRINT
NUMBER OF Z GRID POINTS TO PRINT
STRETCHING IN Z DIRECTION
STRETCHING IN R DIRECTION
PRINT GRID COORDINATES
DO NOT FIND OPTIMUM OMEGA
VALUE OF RELAXATION PARAMETER FOR BOUNDARY VORTICITY
MAXIMUM ITERATIONS PERMITTED IN SOR
FINAL TIME IN MILLISECONDS
ONESIDED STRETCHING IN R DIRECTION
BOTH SIDED STRETCHING IN Z DIRECTION
VALUE OF SIGMA (EQN. 39) FOR STRETCHING R DIRECTION
VALUE OF SIGMA (EQN. 38) FOR STRETCHING Z DIRECTION
156
75
gence could have several possibilities. Ideally, at convergence, all
the variables should have converged. For practical reasons, it would
be sufficient to check on the variable that converged the slowest.
This is so that when this variable has converged, all others would
have converged. However there are no general guidelines to choose
apriori the slowest converging variable. Interior velocity
components were suggested by Roache (1982) as a measure of
convergence. Frequently, researchers in the past have decided on the
vorticity boundary values as the variables to watch for the
iteration convergence of the entire problem. Roache (1978) referred
to tests on boundary vorticities as the most sensitive test for
convergence. The same variables were used to check for iteration
convergence in this study even though they may not have been the
variables that converged the slowest.
For the chosen variable, there are no definitive criteria for
the measure of convergence. A relative error criteria was used for
this study, which is
~n+lm+^ ~n+lm
0) 0)
n+1
m
0)
< 6
(78)
~n+1
at all solid boundaries. The boundary vorticities o> 1 were found
from the interior stream function value using equation 76.
As an example of other measures of convergence, while studying
natural convection in a rectangular cavity, Kimura and Bejan (1984)
used the following convergence criteria on interior points
Can center temperature(c)
Figure 31. Transient center temperature during conduction
heating of Bentonite in a cylindrical can Â£
ro
12
would require repeated timeconsuming computer simulations.
Using the general idea of Teixeira and Hanson (1982), the
feasibility of several control algorithms that can work online with
available computer hardware needed to be investigated, and led to
the following studies as part of the work carried out in this
project:
1) Ways to obtain the transient temperature solution faster,
using other available methods of solving the conduction
equation.
2) The need to incorporate a convection (Robin's) boundary
condition to reflect the true situation, rather than the
temperature specified (Dirichlet) boundary condition used
by Teixeira et al. (1975a,b) in solving the conduction
equation.
Conduction Heating of Canned Foods
Since the pioneering work of Ball and Olson (1957) mentioned
earlier, conduction heating of canned foods has been studied
extensively. Consequently there is a vast amount of literature
existing on the subject.
Studies Based on Analytical Solution
The majority of the works on conductionheating of canned foods
were based on one of several different forms of the analytical solu
tion to the heat conduction equation. Also most of these solutions
RESULTS AND DISCUSSION
Results for the numerical natural convection heat transfer
model are presented and discussed first. Next, numerical and experi
mental results are presented for conduction heat transfer in canned
foods. Finally, performance of the online control logic (using
finite difference conduction heat transfer model) is demonstrated
for conductionheated foods.
Natural Convection Heating in Canned Foods
The results of the numerical model for natural convection heat
transfer are described here with water as a model liquid food.
Unless otherwise noted, the input data file for the results is given
in Appendix C. Development of the flow patterns and temperature
profiles are first discussed and diagrams of the same are given at
various times during heating. The diagrams are qualitatively and
quantitatively compared with previous numerical studies and
available experimental data. Location and migration of the slowest
heating points inside the can are then discussed. Also, variation of
the slowest heating temperature with time was compared with that of
transient temperature at other locations. Finally, comments are
included on the numerical aspects of the model.
93
* L 1 * v ~~ ^
30 50 70 90 110 120
Temperaiure(c)
Figure 16.
Comparison of numerically predicted axial
temperature profiles with observed (Hiddink,
1975) values at various times during heating
117
geometric center the coldest at all times (see figure 22). Tradi
tionally, in natural convectionheating products, the point at l/3rd
the height of the container on its axis was considered to be the
slowest heating point at all times. This study shows that, not only
is there no single point that is coldest at all times, the cluster
of points (or the slowest heating zone) is located away from the
centerline. The clockwise loop formed at the bottom (see figure 10)
forces liquid upwards near the centerline. This clockwise loop was
also clearly seen in the calculations of Sani (figure 2). Because of
the hot liquid going up near the centerline at the bottom, the
temperature is much higher in this region (see the isotherms in
figures 2 and 10 near bottom) and it is unlikely that the lowest
temperature would occur near the bottom centerline. Also, there is
no obvious reason why one particular point would stay the slowest at
all times. Hiddink (1975) also noted the same from his experimental
observations. He did not observe any well defined slowest heating
point. Instead, he referred to a "slowest heating region" near the
bottom of the container and reported no stagnant fluid that stays
consistently at one slowest heating point. The coldest region found
in this study, therefore, was a donutshaped region near the bottom
of the container, away from the sidewall and centerline, as shown in
figure 17.
Migration of the slowest heating points. Figure 18 shows how
the coldest point in the can moved as heating proceeded. The order
of movement is shown in figure 18 by following the letters in alpha
betical order. Between two consecutive letters of the alphabet, the
90
Figure 8, Flow diagram for computer control of retort
operations with online correction of process
deviations
134
The actual time step used was 0.15 seconds which is twice the
possible time step for this explicit method.
Convergence of boundary vorticity. Insufficient convergence of
boundary vorticity can lead to inaccuracy and instability in the
solution. Convergence criteria (S) of 10^ and 10"^ were used for
the boundary vorticities in two sets of computation with all other
input values unchanged. The temperature and stream function values
obtained from these two computations are presented in figures 29 and
30. As can be seen, there is very little difference between using
the two different criteria for convergence. To be on the
conservative side, <5 =10"^ was used in all the calculations.
A note on the half grid points. Confusion exists over
evaluation of half grid points at the control volume faces for non
uniform grid system. If we use the half points in the computational
(n 5) domain, as is the case if we derive in the following way:
9(rve) 9n 9(rve) 9(rve) 9ji
9r 9r 9n 9n 9r
(87)
r.j, r.
>1 rjl
Then rj+l/2 should be defined by
r
(38)
54
After algebraic manipulation, temperature equations at the "full"
time step
nN+1 j. k nfl+1 An+1 _
a . i+ b .0  c.0 ,i = rhs
J ijJ1 j i,j J i,j+l J
(48b)
where
aj
1,, n n.
VyvLl)ri.i/;, r.i^
rj(rj+rrjiJ(rjrji)
,= 1_+i ^ + rj=b
3 iT 2rj(rj+rrji) rjtrj+rrji) r+irj rrrji
V^lvgl)^ rlHl/
Wv V>rrjin'>rrj)
rhsj d9*l,j + ee1,J + f6i+l,j
VKH\I) 1
zi+rzii +( zi+rziiiUrzii)
= i lJz(uS+iuSiuMu[1n l(l +* )
At z.inz_. 1 R I RI L I LN z. ,z. ^ z..,z. z.z. ;
i+1 "i1
i+1 "i1 "i+1 "i "i "i1
V2(Urur1)
zi+rzii
f =
+ izi+rzi.iKzi+rzV)
and
u
VR =
f
i J
u
L
u
f
11J
V
f
i Jl
CAN CENTER TEMPERATURE (c)
143
Figure 32.
Transient center temperature during conduction
cooling of Bentonite in a cylindrical can with
and without drop in retort pressure
28
Nondimensionalization of the Variables
The variables in the equation are then nondimensionalized as
be!ow.
z =
u =
5
a
t =
t
H2
0 =
T T1
To T1
H2 
O) = (i)
a
* = TG *
a
(25)
The resulting equations are the governing equations for natural
convection in the form used for the present study.
ae r a(u0l 1 a(rv0) , a20
T? + i + 7 Sr Â¡ = (
, _1 ^ r 9 'i i
+ r ar^rar J J
(26)
3o)
Tr
+ {
a(uc) a(vt) j. nr,r a r l a(ru>) ^ ,
3z + Tr~ } GrPr% + Pr{ gj? + gjrC 7 5p ) J
 )
l a^jj ^
7Sr J
(27)
(28)
1 8i>
r lz
_ 1 a
" TW
v
(29a,b)
17. Locations of the slowest heating points in water over
a heating period of 10 minutes in a cylindrical can 116
13. Migration of the slowest heating points in water over
a heating period of 10 minutes in a cylindrical can 116
19. Transient temperature at various locations during
heating of water in a cylindrical can 119
20. Transient temperature at various locations during
initial times of heating of water in a cylindrical
can 121
21. Temperature at the slowest heating point during
natural convection heating contrasted with that
during conduction heating in a cylindrical can 122
22. Isotherms during natural convection heating contr
asted with those during conduction heating in a
cylindrical can 123
23. Diagram of 39(0.9)x39(0.8) grid on the left and
58(0.85)x58(0.76) on the right 126
24. Isotherms after 90 seconds of heating computed using
39x39 grid (left half of the figure) compared with
computations using 58x58 grid (right half of the
figure) 127
25. Streamlines after 90 seconds of heating computed
using 39x39 grid (left half of the figure) compared
with computations using 58x58 grid (right half of the
figure) 128
26. Transient slowest heating temperatures computed with
39x39 and 58x58 grid .....129
27. Transient temperature at l/3rd height on the center
line computed with 39x39 and 58x58 grid 131
28. Time step history during a typical run of the
numerical heat transfer model for natural convection 132
29. Isotherms in a cylindrical can calculated with two
different convergence criteria for boundary vorticity
(6 =0.0001 on the left and 6 =0.001 on the right) 135
30. Streamlines in a cylindrical can calculated with two
different convergence criteria for boundary vorticity
(6 =0.0001 on the left and 6 =0.001 on the right) 136
xi v
164
Mulvaney, S. J., and S. S. H. Rizvi. 1984. A microcomputer
controller for retorts. Trans, of the ASAE 27(6):19641969.
Navankasattusas, S., and 0. 8. Lund. 1978. Monitoring and
controlling thermal processes by online measurement of
accomplished lethality. Food Tech. 32(3):7983.
Niekamp, A., K. Unklesbay, N. Unklesbay, and M. Ellersieck. 1984.
Thermal properties of bentonitewater dispersions used for
modelling foods. J. Food Sci. 49:2831.
O'Brien, G. G., M. A. Hyman, and S. Kaplan. 1950. A study of
numerical solution of partial differential equations. J.
Mathematics and Physics 29:223251.
Ozisik, M. N. 1980. Heat conduction. John Wiley and Sons, New York.
Parmentier, E. M., and K. E. Torrance. 1975. Kinematically consis
tent velocity fields for hydrodynamic calculations in curvi
linear coordinates. Journal of Computational Physics 19:404417.
Peaceman, D. W., and H. H. Rachford, Jr. 1955. The numerical
solution of parabolic and elliptic differential equations. J.
Society Industrial Applied Mathematics 3(1):2841.
Peyret, R., and T. D. Taylor. 1983. Computational methods for fluid
flow. SpringerVerlag, New York.
Raithby, G. 0. 1976. Skew upstream differencing schemes for problems
involving fluid flow. Comput. Methods Appl. Mech. Engg. 9:153
164.
Richtmyer, R. D., and K. W. Morton. 1967. Difference methods for
initialvalue problems. Interscience Publishers, Inc., New York.
Roache, P. 1978. Semidirect calculation of steady two and three
dimensional flows. Numerical methods in laminar and turbulent
flow. Proceedings of the 1st international conference at
University College Station, Swansea, U.K. Edited by C. Taylor,
K. Morgan and C. A. Brebbia. John Wiley and Sons, New York.
Roache, P. 1982. Computational fluid dynamics. Hermosa Publishers,
Albuquerque, New Mexico.
Saul'yev, V. K. 1957. A method of numerical solution for the
diffusion equation. Dokl. Akad. Nauk SSSR. 115(6):10771079 and
117(1):3639.
Saul'yev, V. K. 1964. Integration of equations of the parabolic type
by the method of nets. Translated from Russian by G. J. Tee.
Pergamon Press, New York.
15
flow problems are generally more complex than the external ones.
Fewer works have been reported on internal flows in cylindrical
coordinates. Of the studies on natural convection heating in
cylindrical enclosures, only four studies could be located where
they were in reference to natural convection heating during the
thermal processing of liquid foods. Such thermal processing is
characterized by a rather large Grashof number, generally small
overall dimension of the container and water as the liquid. Perhaps
the food processing situation is unique to general heat transfer
studies, which would explain the fact that only one such study could
be located in the more common heat transfer literature.
Previous Works on Canned Liquid Foods
The rate of heat transfer in canned liquid foods was studied
using correlations, and slopes of timetemperature curves with
limited success (Blaisdell, 1963; BarreiroMendez, 1979). Possible
flow patterns were postulated by Blaisdell (1963) using dye and
aluminum powder tracers.
Stevens (1972) first tried to study detailed temperature and
flow patterns during natural convection heating of canned liquid
foods. He used an explicit finite difference formulation from the
work of Torrance and Rockett (1969). The can was treated as a ver
tical cylindrical enclosure. The fluid (ethylene glycol) inside the
can was initially at rest at a uniform temperature of 30C. The can
was then placed in hot water bath at about 100C. The top surface of
the fluid was free (the can had headspace). The actual temperature
19
treated as insulated and wall temperature specified as a function of
time. The numerical results were in good agreement with experimental
data.
Sani* (personal communication, 1985) also developed a similar
model for pasteurization of beer in cylindrical metal cans (aspect
ratio =4.2). Water was the test fluid (Pr=6.81), The initial
temperature was 35C and the sidewall boundary condition used was
approximately 60C. The top of the water level was free (the can had
headspace) and was treated as insulated. The temperature at the
bottom of the can was a function of radial distance. Temperature and
velocity profiles from this study are given in figure 2. Further
quantitative details on this study were not available. These results
were not compared with experiments and also not published.
Other Similar Works on Natural Convection Heating in Cylindrical
Enclosures
One of the earliest of the numerical studies on the subject of
natural convection heating in cylindrical enclosures was performed
by Barakat and Clark (1966), who were looking at thermal strati
fication and associated processes in liquid propellant tanks for
application to space flight. The vertical cylindrical enclosure,
they studied, was filled with water, and arbitrary temperature and
heat flux variation were imposed on its wall, while its bottom was
kept insulated. At the top free surface of the liquid, heat transfer
to the ambient was considered negligible and thus insulated. They
*R. L. Sani, CIRES, Campus Box 449, University of Colorado, Boulder,
Colorado 80309
After algebraic manipulation
, n+1 k n+1 n+1 ^
a to. + D.). + c.,1 = rhs.
J i>Jl j i ,j j i,j+l j
(56b)
where
lh (vvl)
Y_tVHr WVhW
Pr rjl
r n, i ni n, i ni >,
1 (VIvrK+IvlN
bj' 4? +
* vR~VivlI J Pr rj r 1 1
2trj+rrjiJ lrj+i'rii) rjty> trj+i'rj) Wi
Cj
WKl) ....
rj+i"rji trj+rrji>jJ/2 lrj+rrj)
Pr rj+l
rhs = do)1? .+e)? .+fajr?,1 .
J il,J i,j l+l,j
0.5GrPr2
(Gn+l+10n+l .)
^ iJ+l
rj+rrji
V2(uMuP) Pr
zi+rzii C zi+rziiH*izii)
e
1_
At
il
Pr
zi+rzii
zi+rzi
zizii
f
zi+rzii
115
Slowest Heating Zone
In the absence of agitation, there is very little mixing of the
liquid during natural convection heating. Consequently, the liquid
does not experience the same timetemperature history at all points
inside. Points at the top stay at a consistently higher temperature
than at the bottom. However, at the very bottom, since the bottom
wall is heated, the temperature is higher again. Somewhere in
between the liquid stays consistently at a lower temperature than
the rest of it. From a food processing point of view, these slower
heating points (or regions) are of utmost importance. The liquid
(and therefore the bacteria carried with it) at these points experi
ences much less exposure to heat (less sterilization) than the rest
of the product. Several studies in the past (Blaisdell, 1963;
Hiddink, 1975: Stevens, 1971) attempted to locate these slowest
heating points under various situations. For this study, these
slowest points were investigated with respect to locations,
migration, and temperature values at different times during heating.
Location of the slowest heating points. To find the locations
of the transient slowest heating points, the lowest temperature at
any time instant can be located. When more than one location was
found with the same lowest temperature, the point closest to the
bottom, and closest to the centerline was chosen. In figure 17,
these points are plotted for every 30 seconds during the heating.
The figure points out that the same physical location in the
container does not stay the coldest at all times. This is in
contrast to conductionheating where symmetrical heating makes the
107
TT3TTT223L
V * ***^*
V L Ju r.
120.99C
t
V !* ic i Ufi
\ vt i ///^
I i l
il
llllUF
/
120^96^0 _
Water
T,=30*C
ToplSlC
Sld8=121*C
3or121*C
R4.19cm
H=10.67om
N=19
M = 19
5=0.0001
After 1600 sec
Figure 10o. Isotherms and velocity vectors in a
cylindrical can after 1800 seconds of heating
Axial velocity Axial velocity Axial velocity(mm/sec)
40
111
Figure 14. Predicted radial velocity profiles at the
sidewall at mid height in a cylindrical can
after various heating times
68
that makes the ADI method advantageous is not available in general.
Programming the ADI method for a nonrectangular region may get
complicated, and they are not known for sure to be faster (Roache,
1982).
Another popular method to solve the unsteady formulation of the
stream function Poisson equation is the successive overrelaxation
(SOR) method (Frankel, 1950). It is very simple and effective. In
SOR, the set of solutions are over relaxed by a factor 1 < n < 2
to speed up convergence. The exact value is found through simple
numerical experimentation. It is easier to program compared to ADI
methods. Most of the earlier researchers have used the SOR to solve
the stream function equation. In all the five formulations of
natural convection in a cylindrical geometry, studied by Torrance
(1968), SOR was used to solve the stream function equation. However,
it is time consuming and could end up taking 90% of the total time
spent to solve the problem. Nevertheless, SOR was chosen to solve
the stream function equation in the present study for its
simplicity. Following is the development of the SOR formulation.
Rewriting equation 62,
To get the SOR formulation,
'.0
F (A*1,j+c*i+l,j+0*i Jl+E*1,j+l+l,j ) (70)
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NUMERICAL MOOELING OF NATURAL CONVECTION
AND CONDUCTION HEAT TRANSFER IN CANNED FOODS
WITH APPLICATION TO ONLINE PROCESS CONTROL
By
ASHIM KUMAR DATTA
December 1985
Chairman: Arthur A. Teixeira
Major Department: Agricultural Engineering
A numerical model was developed for predicting detailed flow
patterns and temperature profiles during natural convection heating
of liquids in a can (a cylindrical enclosure). The liquid (water)
was initially stagnant at a uniform temperature of 30C, and the
sidewall, top and bottom wall temperatures were suddenly raised to a
temperature of 121C. A free (thermally insulated) top liquid
surface was also considered. Boussinesq approximation was used and
all other fluid properties were treated as constant.
Finite difference methods were used to solve the governing
equations in axisymmetric cylindrical coordinates. A vorticity
stream function formulation was used. Parabolic temperature and
vorticity equations were solved with the alternating direction
implicit method and upwinding the convective terms. The elliptic
stream function equation was solved using successive over relaxation
xvi
60
The tridiagonal matrices for the vorticity equations at the two
ADI time steps are given below. At the "half" time step, equations
are written for the columns (from the bottom wall to the top wall).
Both the bottom and top wall vorticity values are used from previous
iteration.
bl ci
32 b2 C2
33 b3 C3
am2 bm2
aml
()*i
l.J
rhsl+alw0,j
rj
CM
3
rhs2
3,j
rhs3
cm2
V2,j
rhsm2
bml
rbsml+aml m,j
(57a)
At the "full" time step, equations are written for the rows
(from centerline to sidewall). The centerline boundary vorticity is
specified (=0) and the sidewall boundary vorticity is used from
previous iteration.
Yl+l
bl "C1
i.l
~a2 b2 c2
1,2
a3 b3 c3
wi ,3
=
I an2 bn2 _cn2
i ,n2
l anl bnl
l
. ,, n+1
rhsl+al
rhs
rhs:
rhsn_2
(57b)
Water
T,=30'C
top121*C
slde=121*C
bot121*C
R4.19om
H=10.67om
R57.0.76
Z=57,0.85
ao.oooi
300
260
220
180
140
100
60
20
20
6
Figure 10c. Isotherms and streamlines in a
cylindrical can after 3 seconds of
heating
Water
T,= 30*C
topiei*C
slde=121*C
boi121'C
R4.19cm
h=10.07cm
R57.0.76
2=57,0.85
ao.oooi
300
260
220
180
140
100
60
20
20
60
After 4 sec
Figure lOd. Isotherms and streamlines in a
cylindrical can after 4 seconds of
heating o
O
32
with the initial conditions
0 < r < R/H 0 < z < 1 0=1 (34)
Computer Control of ConductionHeating Processes
In controlling thermal processes, the objective is to meet the
designed level of bacterial sterilization (F^) for the process,
irrespective of any retort temperature variation TR(t), and with a
minimum of overprocessing. The lethal effects of thermal processing
are achieved during the heating as well as cooling times (t^ and
tc). Thus, the objective function in thermal processing is to
t Jill1 t +t Id21
minimize F0(th+tc^ = ^ 10 ^ dt + c 10 Z dt (35a)
uh
subject to the constraint: F0(th+tc) > ^o (35b)
where the integral from equation 6 has been broken into separate
heating and cooling times. FQ(t) is the accumulated sterilization at
any time t. The first integral above is the contribution to the
sterilization F0(t) from heating, and the second integral is the
contribution to the sterilization during cooling. Temperature T(t)
is taken to be the temperature of the slowest heating point in the
product. This is so that when the design Fq(Fq) is satisfied at
this point, all other points in the product have also been satis
fied. For a cylindrical can, T(t) is the temperature at the can
geometric center at time t.
29
The boundary conditions in the transformed and nondimensional ized
variables can be written as
sidewall 1
r=R/H
0 < z < 1
0=0
o
11
u=0
v=0
centerline
r=0
bottom wall
0 < z < 1
iÂ£ = o
9r u
O) = 0
e
n
o
U 0
3r
v=0
0
z=0
0=0
o
ii
u=0
v=0
top wall
0
Z=1
0=0
4> = 0
u=0
v=0
(30)
with the initial conditions
0
0 < z < 1
0=1
(o = 0
i) = 0
u=0
v=0
(31)
The relative merits of solving the transformed variables
as compared to the (u,v,p) system of primary variables were discuss
ed by Roache (1982). To solve the (u,v,p) system, another elliptic
Poisson equation would need to be formed for the pressure (p) vari
able, using the two equations for velocity, regardless of whether
transient pressure solutions are desired. Following are the relative
evaluations of the (u,v,p) system and the ) system for two
^Explicit boundary conditions for vorticities on solid walls are
unavailable.
45
depends on ij> which in turn depends on to this would couple all the
equations together and make it practically incomputable. To avoid
this, the nonlinear terms are linearized. Various ways of lineari
zing these terms have been discussed by researchers (Anderson et
al., 1984; Peyret and Taylor, 1983; Roache, 1982). A very simple and
common strategy in linearizing is to evaluate the velocity coeffi
cient at n time level. This is known as "lagging" the coefficients.
However in this case, the accuracy in time drops from second order
(for ADI) to first order. Roache (1982) noted that such "lagging" of
the velocities could still be second order accurate if the velo
cities u and v were slowly varying. An improvement over this
linearization procedure is to update the coefficients u and v as
iterations proceed. The procedure is repeated until the values u and
v converge. One way to achieve this would be to update un+^2 and
vn+V2 by solving the stream function equation at the "half" time
step of the ADI (Roache, 1982). Since solution of the Poisson stream
function equation is the most time consuming operation, it was
decided against this updating procedure.
Anderson et al. (1984) noted that "lagging" the coefficients
has been widely and satisfactorily used by previous investigators,
and, recommended the use of the same for its simplicity. Thus the
velocity coefficients u and v were "lagged" in this finite differ
ence study.
Convection terms. The presence of the convection terms makes
the temperature and the vorticity equation qualitatively different
from other simple parabolic equations. Proper numerical treatment of
16
variation on the can boundary was noted and was used in the
numerical calculations. Temperature values were recorded at ten
different points inside the container. The recorded temperature
values agreed poorly with numerically calculated temperatures. The
numerical inaccuracy was attributed to insufficient grid points
which could not resolve the details of the flow. Several sources of
experimental error included inaccurate setting of boundary
temperature, nonzero initial velocity field, errors in thermocouple
measuring circuit and inaccurate placement of the thermocouples.
Some of these factors were thought to have caused three dimensional
movements in the experiment and could not be picked up by the two
dimensional numerical model.
Detailed temperature and velocity profiles during natural
convection heating of liquid foods in a cylindrical can were
extensively investigated by Hiddink (1975). The numerical solution
technique used was the explicit finite difference technique of
Barakat and Clark (1966) discussed later. The initial temperature
was considered uniform throughout. The top, bottom and sidewall were
suddenly raised to the heating temperature. The top surface of the
liquid was considered free (having headspace) and was thus treated
as thermally insulated. Several test fluids of low and high
viscosities (water, sucrose solution and silicone fluid respec
tively) were used. The numerical results included temperature and
streamline patterns. The overall flow patterns were visualized using
a "particle streak method." In this method small glittering
particles are suspended in the liquid. The particles are illuminated
REVIEW OF LITERATURE
This chapter begins with a brief introduction to thermal
processing as applied to the sterilization of canned foods. The need
for computer control of thermal processing is noted, along with the
importance of heat transfer models in such computer control systems.
Previously reported conduction heat transfer models used in studying
sterilization of conductionheated canned foods are described next.
The few studies attempting to model natural convection heat transfer
in the case of sterilizing canned liquid foods are noted. The few
other works on natural convection heat transfer in liquid inside a
vertical cylindrical enclosure are also reported.
Computer Control of Food Thermal Processing
An overview of thermal processing is presented in this section.
Problems with traditional approaches to processing of conduction
heated foods are then noted and the need for online control is
stressed. Some of the past approaches in thermal processing of con
ductionheated foods that could also be used online are discussed.
Overview of Thermal Processing
Thermal processing consists of heating foods, filled and
hermatically sealed in containers, in pressurized steam retorts at a
5
58
After algebraic manipulation
* *
a w. i + bco* eCo.
i ll,j i i,j
. u).,, = rhs
i i+l,j i
where
(56a)
V2 (u"+u") Pr
31 = zi+rzii +12i+rziiJizrzii)
bi
i_ + ljl (uMuJIuMuPIl + ( 1 + k 1
At z.,,z. ^ R I R! L I !_N z..,z ^ z.,,z. z.z. J
i+l i1
i+l "i1 "i+l "i "i "i1
ci
V2 (ur'1urI) + Pr
zi+rzii izi+rzii^zi+rzi^
rhs. =
fe^1 en.+l. )
Do)9 ,+Ew; .+Fc!Â¡ .0.5GrPr2
i,Jl i,J i,J+l rj+l"rjl
d =
V2(vJ+v")
Pr r
j1
(rj+rpjij + rj^irJ+irjiKrrrjiJ
1 ,.K+KK+Kn
fcVk
r.Llr. ,
J+l J1
lrj+rrji)tV/2trj+rrj) +
^2 (VR~1VR I)
rj+rrji irjtrrji>jJ/21'j+r'V
Prjj+l
Water
T,= 30*C
1oplSl*C
slde=121*C
bot121*C
R4.19cm
H**10.67om
r,
toriti*c
boriei*C
R4.10offl
Hal0.07*m
After 300 sec
Figure 22. Isotherms during natural convection heating
contrasted with those during conduction
heating in a cylindrical can
LIST OF FIGURES
FIGURE PAGE
1. Isotherms and velocity profiles in a glass bottle at
various heating times (after Engelman and Sani, 1983) 18
2. Isotherms and velocity profiles in a can at various
heating times (after Sani, 1985) 20
3. The grid system and the boundary conditions for the
cylindrical can 40
4. Notations of various quantities defined on a non
uniform grid system in cylindrical geometry 62
5. Algorithm for the iterative solution of the set of
equations in natural convection 77
6. Experimental setup for conduction heating 84
7. Timings of various realtime computations for online
control of conductionheated food 88
8. Flow diagram for computer control of retort opera
tions with online correction of process deviations 90
9. Isotherms near bottom wall predicted by the
convective model compared with the predictions
considering conduction (equation 83) only 97
10a. Isotherm and streamlines in a cylindrical can after 1
second of heating 99
10b. Isotherm and streamlines in a cylindrical can after 2
seconds of heating 99
10c. Isotherm and streamlines in a cylindrical can after 3
seconds of heating 100
lOd. Isotherm and streamlines in a cylindrical can after 4
seconds of heating 100
lOe. Isotherm and streamlines in a cylindrical can after 5
seconds of heating 101
xi i
46
the convection term is important. Several different ways of finite
differencing these terms have been presented in the previous works.
These include spacecentered differencing and upwind or upstream
differencing. The upwind methods have several desired properties
that are not shared by the other finite difference formulations, as
applied to the convection term. Both the upwind methods are
transport!'ve, meaning any perturbation is carried along only in the
direction of velocity. Both the methods are also conservative so
that they preserve the integral conservation relations in the
continuum equation. The first of the two upwind differencing methods
is as follows (Lilly, 1965; Forsythe and Wasow, 1960; Frankel, 1956)
a(ue) u6i u9il
9z Az
u9i+r u6i
AZ
(u > 0)
(u < 0)
(41)
Accuracy of this method is 0(At,Az). This was used by Barakat and
Clark (1966) to solve equations 26 through 29. The second variation
of this upwind method is to treat the convection term as follows
(Gentry et al., 1966)
9(u0)
3Z
Vr
Vl
AZ
where
BIOGRAPHICAL SKETCH
Ashim Kumar Datta was born in the city of Hooghly and raised in
the city of Krishnagar in the state of West Bengal in India. He
attended the Collegiate school in Krishnagar where he received the
Higher Secondary degree in the Fall of 1974. He completed his
Bachelor of Technology in agricultural engineering from the Indian
Institute of Technology at Kharagpur in the Fall of 1979. He then
attended one year of postgraduate studies in statistical quality
control and operations research from the Indian Statistical
Institute, Calcutta. Later, he proceeded to receive his Master of
Science degree in agricultural engineering from the University of
Illinois at UrbanaChampaign in the Summer of 1982. He hopes to
receive his Doctor of Philosophy degree in the Fall of 1985, majoring
in agricultural engineering, from the University of Florida at
Gainesville. He is married to Anasua Datta.
172
138
would change in case of a nonuniform coordinate system. Parmentier3
recommended that equation 89 be used even in a nonuniform grid
system. This means grid faces would still be half way between grid
points in a nonuniform coordinate system. The recursion relation of
X (equation 68) would probably change with a nonuniform
J
coordinate system. However, in this study, the same x (equation
J
68) was used as in the case of a uniform coordinate system. Another
interesting point noted by Parmentier*5 was that the need for using
kinematic consistency criterion was, less critical for fine grid
system and vice versa.
Conduction Heating in Cans
The importance of surface heat transfer coefficients during the
actual processing of conductionheated food is described first in
this section. The temperature values actually measured in the can
center are then compared with the corresponding values predicted by
the ADE finite difference method and the analytical solution. After
thus confirming the accuracy of the finite difference method of
solution, a discussion is presented on how drastically such
prediction using finite differences would be wrong in case there is
sudden pressure drop during the process.
a^E. M. Parmentier, Box no. 1846, 324 Brook Street, Brown Univ.,
Providence, RI 02912. Personal Communications.
55
The tridiagonal matrices for the temperature equations at the
two ADI time steps are given below. At the "half" time step, the
equations are written for a column (going from the bottom wall to
the top wall). Specified boundary temperature is used both at the
top and bottom walls. #
bl Cl
32 t>2 "c2
93 b3 "c3
rhsi+ai0>j
rhS2
rhs3
am2 bm2 cm2
"aml bmll
m2,j
m1 ,j
rhsm_2
rhsml+aml0m,j/
(49a)
At the "full" step, equations are written for the rows (from
the centerline to the sidewall). At the centerline, the insulated
boundary condition (equation 75) is used due to symmetry. At the
sidewall, the specified boundary temperature is used.
bl"al C1
a2 b2 c2
 83 b3
'c3
V
rhs^
i ,2
rhs?
0i ,3
rhs3
bn2 "cn2
Gi ,n2
rhsn_2
_anl bnl
p,nl
irhsnl+cnlQ^n
(49b)
52
30 A r 3(U0), 1 3(rv0) r 320 ,13 r_30 ^ ,
3t i ~W~ + p 3r i = 1 Tz2 + r Sr''V J >
At the "half" or intermediate step ("n+V' written as in short)
* n
0. .0. .
i ,3 i ,3
At/2
. (vKi v^,, Wi
rjlrj+rrji)
* * *
0 0.,. .0. 0. .0. n .
2 r 1+1, J 1,3 1,J 11,J i
zi+rzii zi+r2i zrzii '
Jir:+rrji)l
(\ t _n _n \
rj+v? ^9j j+r0T j ri yj6i ,j~9j ,ji^
rj+r rj
r.r. ,
0 Jl
(47a)
At the "full" step
o"+*0* .
i ,J i ,J
Ax/2
"i +1
^ V KIK+1.J ^ur+ Kl <+ C^NlKu
, n i nM n+l n. nM ji+1 n i nM ji+1 n. nn n+l
< vKi^i,3+i+(vivRi>jVi.r(vIvlI,i cvl+i*Li)pji/ii,ji
rjtrj+rrji1
*
0.l1 .0. 0. .0. .
1+1,J 1 ,J 1,3 11,Jj
zi+rzii zi+rzi
zizil
,n+l ji+1
n+1 jn+1
+ r. (r.,ir. V)^
J 3+1 31
j+
/ JITi JTi N / JTi .11 i I \
V? ^ ,j+l"T ,J rj V? ^1 3 ~ 1 j!
rj+l'rj
r.r. ,
J Jl
(47b)
48
3(u0) a(u0) 3n
3z 3ri 3z
_ 3(u0) 1
3r) 3z/3n
Un0. U. 0. 4 .
R i,] L il,j
An
zi+r zii
2 An
ziU' z
uR01,j uL01l,j) R>0 UL>0
Si miliarly
3(u0)
3z
z'Vx zi'J1 Kei+l,j Vi1,j) UR < 0 UL > 0
O
 fun0,. u, 0 1 Uo < 0 Ui < 0
zi+r zii 1R 1 1,3 L K L
z2T ^R6i,j uLei,j) Ur > 0 UL<0
(a)
(b)
(c)
(d)
To avoid the use of FORTRAN IF statements for checking the sign of
ur and ul in the equations (a) through (d), the equations are
combined as below
3(u0) (UR~IUR
5F" "
i+i j+(VIur1~VK1)0 j~(V1ul1)9tij
r
Zi+1 zil
(43)
Dedicated to the memory of my
grandfather and father
158
For points other than center and with i, j decreasing (moving towards
center)
de 320 320 _1_30
3t ~ 3z2 + 3r2 + r 3r
eO+t
0?
hi 
At
0? 01? 0?+^+ Q"+] .
11,J 1 ,J 1 ,J 1+1, J
2
+
f. 01? Q^1. 0^1 ,
1>J1 1.J 1,J 1,J+1
AZ2
en. ..
i Jl
0
n+1
i J+l
2Ar
e?H =
',J l+p+S
b{s?.j+s(9?i.r e"l Cj>
* p<0Ur 9u+ 1
(93)
For points at center (j=0), for use while i ,j decreasing
30 320 320
3t = JzT + 1FI
,n+l
,n
,n
,n
0': &: n o'.1 n 0. i+ 0^i n 2 (0? o'? n 0^+i+ ? i)
1,0 1,0 11,0 1,0 l ,0 1+1,0 1 i,l i,0 1,0 i,1J
At Az2 aF2
0n+1
i,0
le?,o+s(0ii,o sm+ 9i+i,o)+2P(0i,r lo+ 0>
n+1
1+2P+S
(94)
88
Calculate T(t)
8 update f^(t)
Simulate cooling
to estimate F0C0*
At
2
H
t 1
measure
T(t)
t+At
T
measure
TR(t+At)
where
At. = Time taken to calculate T(t) from measured T(t)
1 using finite difference approximation to equation 4,
At2 3 Time taken to simulate cooling assuming it starts at
time instant t, and
At = Time interval between reading retort temperatures
TR(t).
Figure 7.
Timings of various real time computations for
online control of conductionheated food
74
order form destabilized the calculations (Wilkes & Churchill, 1966)
and sometimes it was less accurate than the first order formulation
(Beardsley, 1969).
The vorticity boundary value at the computational intermediate
or "half" step of the AOI method is more difficult to.set than the
corresponding temperature value because the vorticity boundary value
is also a function of time. The following simple way to define this
vorticity value on the boundary is often used and gives sufficiently
accurate results (Bontoux et al., 1978)
n+vr1
w
1/2 (
n+1
w
m+1
(77)
where
n+1
m+1
wall vorticity value at time step n
most recent estimate of wall vorticity at n+1
step using equation.
n/2"
W
m+1
intermediate vorticity value between n and
(n+l)th time level at the (m+l)th iteration.
Israeli (1972) also used this equation to define intermediate (half
step) boundary vorticity though he used a different equation to
+,m+l
calculate boundary vorticity ^ 1
Convergence Criteria for Iteration
As discussed earlier, the set of finite difference equations
were solved iteratively until convergence. The measure of conver
163
Hiddink, J. 1975. Natural convection heating of liquids, with
reference to sterilization of canned foods. Agricultural
Research Reports #839. Center for Agricultural Publishing and
Documentation, Wageningen, Netherlands.
Higgins, J. M. and 8. Gebhart. 1983. The stability of vertical
buoyancyinduced flow in cold water. J. Heat Transfer, Trans, of
the ASME 105:767773.
Hornbeck, R. W. 1975. Numerical methods. Quantum Publishers, New
York.
Israeli, M. 1972. On the evaluation of iteration parameters for the
boundary vorticity. Studies in Applied Mathematics LI(1):6771
Jaluria, Y. 1980. Natural convection heat and mass transfer.
Pergamon Press, Oxford, England.
Jensen, V. G. 1959. Viscous flow round a sphere at low Reynolds
numbers (<40). Proceedings of the Royal Society of London,
Series A, 249:346366.
Ka'lney de Rivas, E. 1971. On the use of nonuniform grids in finite
difference equations. J. of Computational Physics 10:202210.
Kee, R. J. 1974. A numerical study of natural convection inside a
horizontal cylinder with asymmetric boundary conditions. Ph.D.
Thesis. University of California, Davis.
Kee, R. J., C. S. Landram, and J. C. Miles. 1976. Natural convection
of a heatgenerating fluid within closed vertical cylinders and
spheres. J. Heat Transfer, Trans, of the ASME 98:5561.
Kimura, S., and A. Bejan. 1984. The boundary layer natural
convection regime in a rectangular cavity with uniform heat flux
from the side. J. Heat Transfer, Trans, of the ASME 106:98103.
Kublbeck K., G. P. Merker, and J. Straub. 1980. Advanced numerical
computation of twodimensional timedependent free convection in
cavities. International Journal of Heat and Mass Transfer
23:203216.
Lilly, D. K. 1965. On the computational stability of numerical
solutions of timedependent nonlinear geophysical fluid dyna
mics problems. Monthly Weather Review 93(1):1126.
LOGTEC CCS8 Retort Management System. 1984. Central Analytical
Laboratories, Inc., 2600 Marietta Street, Kenner, Louisiana.
Manson, J. E., C. R. Stumbo, and J. W. Zahradink. 1974. Evaluation
of thermal processes for conduction heating foods in pearshaped
containers. J. Food Sci. 39:276281.
ADI for the vorticity equation
Â£ f + 1W ( GrPr^ + Pr{ & ^ I I'M ) (
At "half" or intermediate step,
a). to. .
1 ,J 1 ,J
At/2
, 1 rr n i nM ,r n.i ni n.i nn ( n.i nn i
ZHrzil f(Vl'JRlh+l,j+(VlURuL+luLlh,jK+l'JLlhl,j !
1 rr n i ni >, n lI n^i n< n^i ,ni >, n r n.i.ni > n
{(vRvRl)i,j+l+iVlvRvL+lvLl)i,j CVIvlD1,M
*
0).
* *
0). C
fln+l _n+l *
= _GrPr2 (Isitlliill) + 2 pr { ii+lj~hii ,j i J'^il.j j
r *< z < Z > < z 4 ** z z z 4
i li
rj+rrji
zi+rziil zi+rzi
n
2 Pr
lrj+rrjiJ
At "full" step,
n+1 *
(J .
1 ,J 1 ,J
A/2
rj+i mi,j+rrj Mitj
j+
r. . r.)
1/2lrj+l
n n
r. to. .r. i o). 1
J T ,J Jl 1J1
rjV2^rjiJ
(55a)
1 rr n i ni, L( nLi ni n^i ni ^ r n.i nn i
+ I^ji((uRuRm+ijj+{vuRuL+uL)ijj(vluLDil,jl
1 ,, n i nM n+1 ,r n,i nt n L ( n i ^ n+1 ( n ni, n+1 ,
+(rjtrrjT>f(Vr'1 VrI )i .j+l (VlvrIvlvlI )1,3 (vlVJ H ,jl)
* *
0). 0).
fln+l n+1 *
= _GrPr2 r9i ,j+l~9i ,jl 3 + 2_Pr rM1+lfj;.MitJ JZlzUi
rj+rrji zi+rzii zi+rzi zizii
n+1 n+1 n+1 n+1
2 Pr rrj+lui ,j+lrjMi ,j rji,j "rjl ui,jl i
(rj+rrji) 1 rj+ v2 lrj+ipjj" rjv2l rj'rjiJ
(55b)
168
Clever, R., and I. Catton. 1972. Steady natural convection in a
vertical cylinder at large Prandtl number. Proceedings of the
1972 Heat Transfer and Fluid Mechanics Institute, edited by R.
8. Landis and G. J. Hordemann.
Cunsolo, D., and P. Orlandi. 1978. Accuracy in non orthogonal grid
reference systems. Numerical methods in laminar and turbulent
flow. Edited by C. Taylor, K. Morgan, and C. A. Brebbia. John
Wiley and Sons, New York.
Dickerson, R. W. Jr. 1969. Simplified equations for calculating
lethality of the heating and cooling phases of thermal
inactivation determinations. Food Tech. 23(3):108111.
Evans, L. B., and N. E. Stefany. 1966. An experimental study of
transient heat transfer to liquids in cylindrical enclosures.
Chemical Engg. Progress Symposium Series 62(64):209215.
Fairweather, G., and A. R. Mitchell. 1967. A new computational
procedure for ADI methods. SIAM J. of Numerical Analysis
4(2):163170.
Fitzpatrick, J. A. 1980. The application of computers and
electronics to process control in Thames (sugar) refinery. The
International Sugar Journal 82(980):231236.
Harrell, R. C., G. A. Kranzler, and D. C. Davis. 1981. Micro
processor control of a solar fruit juice pasteurizer. ASAE paper
no. 815030. Paper presented at the 1981 summer meeting of the
ASAE at Orlando, Florida.
Hayakawa, K. 1974. Response charts for estimating temperatures in
cylindrical cans of solid food subjected to time variable
processing temperatures. J. Food Sci. 39:10901098.
Hayakawa, K. 1978. A critical review of mathematical procedures for
determining proper heat sterilization processes. Food Tech.
32(3):5965.
Heliums, J. 0. 1961. Finite difference computation of natural
convection heat transfer. Ph.D. Thesis. University of Michigan.
Ann Arbor.
Hess, C. F., and C. W. Miller. 1979. Natural convection in a
vertical cylinder subject to constant heat flux. Int. J. Heat
Mass Transfer. 22:421430.
Hogue, R. L., and V. A. Jones. 1982. Hardware recommendations for
computerized steam injection UHT processing. Paper no. 82
6017. Presented at the 1982 summer meeting of the ASAE at
Madison, Wisconsin.
42
Time derivatives. The treatment of the time derivative in
equations 26 and 27 can be broadly classified into implicit and
explicit methods. Explicit finite difference methods are generally
easier to understand and program. Explicit methods have been used by
Barakat and Clark (1966), Torrance (1968), Stevens (1972) and
Hiddink (1975) in studying natural convection heating inside a
cylindrical container. For most of the explicit methods, however,
there are stringent restrictions on the allowed time step, from the
stability considerations. In situations where large temperature
gradients, eddies and recirculations are present like in this study
grid points need to be fine enough to resolve the physical phen
omena. When using most of the explicit methods, such a fine grid
size could severely limit the allowed time step and thus require
prohibitive amounts of computer time.
Implicit finite difference methods involve more than one
advance timelevel value in the same equation and thus require
matrix inversion to calculate function values at a new time level.
These methods generally allow larger time steps due to better
stability properties. However, even though a larger time step is
allowed in the implicit method, it requires many more calculations
to solve that step. For an implicit scheme to save computer time in
the overall computation, it should allow several times larger a time
step than that allowed by an explicit method to achieve a given
accuracy critrion.3
aT. IP. Shih, Mech. Engg. Dept., Univ. of Florida. Personal Comm.
137
However, in the work on kinematically consistent velocities by
Parmentier and Torrance (1975) the half points were mid points
between grids in physical domain, i.e. they are defined by
>v2
V2(V rj+l>
(89)
but the equations developed in their study were meant for an uniform
grid system. For an uniform grid system, a is zero and the two
definitions of rj+l/2 mer9e into one* It is interesting to see that
apparently the same equation (87) can be derived with this
definition (89) of rj+l/,, by writing
3(rvQ) VRrj+ V?i ,j~ VLrj V^i Jl +
0. v, r.
3r
r.
j+ Vi
rjV2
0( Ar)
(90)
2^vRr.j+y?6j ,.j vLriv,3i..iiJ
r.,. r. n
J+l Jl
using equation 89
The catch is that equation 90 now turns out to be first order accu
rate due to "skewed" central differencing instead of being second
order accurate for regular central differencing as in equation 87.
Thus, use of rj+1/2 deflned by equation 89 together with equation
87 would drop the order of accuracy. However, it is expected that
with such a definition, transport velocities could be a better rep
resentative for the fluxes at the interface. It is unclear, from
their 1975 work how some of the equations for calculating velocities
2
ConductionHeated Foods
Traditionally, for conductionheated foods, the required
processing time is determined using the analytical solution to the
heat conduction equation. In practice, the constant medium temper
ature assumed by the analytical solution cannot always be achieved.
Thus, experimental correction factors have been proposed to adjust
the heating time, predicted by analytical solution, when the medium
temperature goes through certain (predefined) types of deviations.
However, deviations in practice are arbitrary, and it is not
conceivable to know the correction factors for all such situations.
Of course, the problem is trivial if the transient food temperature
is directly measured using a thermocouple; but this is both
inconvenient and impractical in a production situation. Better
models for predicting food temperature in response to arbitrary
fluctuations in medium temperatures are therefore needed.
A numerical finite difference model would be able to predict
the food temperature for truly arbitrary variations in the medium
temperature. However, the calculations required are relatively much
more complicated and time consuming. Using a microcomputer to
perform this would have two advantages. In addition to correctly
predicting food temperature for truly arbitrary heating conditions,
it can perform this automatically without any worker supervision.
Introduction of lowcost microcomputers for online process control
is becoming commonplace in the 1980s. Food industries are not an
exception to this situation. Use of microcomputers for online
control of conductionheated food is thus quite conceivable.
33
This transient temperature T(t) is a function of transient
retort temperature TR(t), the can wall outside heat transfer coeffi
cient (h), dimensions of the can (R and H), and thermal diffusivity
(a) of the product for a conductionheating food. Symbolically,
T(t) = f (TR(t), R, H, a) (36)
For conduction heating in a cylindrical can, the function f is the
heat conduction equation in cylindrical coordinates.
For a given can size and product, the two variables that can
possibly be controlled are the retort temperature TR(t) during
heating (saturated steam or hot water under pressure in the retort),
and the time period of heating t^. Cooling is generally done using
available water at ambient temperature. This makes the contribution
to F0 from cooling (given by the second integral in equation 35a)
immune from further control once the cooling process is under way.
Also, both the heating and cooling temperatures can only be applied
to the boundary of the product making it a boundary control problem.
In the case of a process deviation, the heating medium tempera
ture may go through unexpected variations beyond the capacity of the
temperature controller. Thus in reality, only one variable can be
controlled. It is the time period of heating (t^). The problem in
thermal processing is to specify t^ for arbitrary variations in
TR(t) during heating while the process is under way in realtime.
Instead of searching for the proper t^ through simulation every
time a deviation takes place as suggested by Teixeira and Manson
(1982), F0(t) can be monitored in realtime using T(t) calculated
SIDEWALL
97
BOTTOM
Figure 9. Isotherms near bottom wall predicted by the
convective model compared with the predictions
considering conduction (equationy<83j only
/
NUMERICAL MODELING OF NATURAL CONVECTION
AND CONDUCTION HEAT TRANSFER IN CANNED FOODS
WITH APPLICATION TO ONLINE PROCESS CONTROL
By
ASHIM KUMAR DATTA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
10
Online Control of Thermal Processing
In a production situation, the heating time t specified
(assuming a constant value of TR) to achieve a given F0 value would
need to be adjusted for arbitrary variations in the heating medium
temperature TR from its specified constant value. Researchers in the
past have attempted this in one of the following three ways
Precalculated tabulations. Correction factors are tabulated to
be applied to t^ for certain possible types of deviations in
TR(t). Commercial retort control systems often use this technique
(LOGTEC system, 1984). Such tabulations cannot be exhaustive,
since the process temperature TR(t) could go through any random
deviation. Instead of tabulating such correction factors, Giannoni
Succar and Hayakawa (1982) developed expressions for correction
factors to be applied to th when TR(t) goes through a step drop. In
their work, other types of deviations had to be approximated to
closefitting step functions, and multiple drops could not be
considered. Getchell (1980) described a control system that would
attempt to maintain the design heating medium temperature and sound
an alarm when critical low or high limits for the temperature are
exceeded. The operator would then attempt to make necessary changes
in th.
Direct Measurement. Temperature at the can center T(t) is
measured online. This eliminates the need to assume constant
heating temperature. Measured T(t) is used with equation 1 to stop
the heating cycle at time t^ such that
W > Fo
(12)
Mechanical and Industrial Engineering Department of the University
of Illinois at UrbanaChampaign. Help by Subhasis Laha in physically
transporting the plotting software to Gainesville is also
appreciated. Availability of a hard copy plotter was the next
necessity and the author appreciates the equipment provided by Dr.
James W. Jones in the Agricultural Engineering Department.
Finally, the author deeply acknowledges the assistance from his
wife Anasua in typing this manuscript, and her encouragement and
perseverence during the study.
LIST OF SYMBOLS
A = constant
a,b,c = coefficients in tridiagonal matrices
f = inverse slope of timetemperature graph
g = gravity
F0 = sterilization F0 value
H 3
Gr = Grashof number = ge (Ti T0)
h = outside heat transfer coefficient
H = height of cylinder
j = j value (a constant)
Jg = Bessel function of order zero
k = thermal conductivity
n = normal direction
nz = number of grid points in z direction
nr = number of grid points in r direction
p = pressure
Pr = Prandtl number = yCp/k = v/a
r = = non dimensonal distance in radial direction
r = distance in radial direction
R = radius of cylinder
vii
86
resulting in product movement and much faster cooling rates. Details
of the work that showed direct evidence of internal product movement
during sudden depressurization of conductionheated foods are
presented elsewhere (Teixeira et al., 1985).
The transient temperature response recorded at the can center
during conduction heating was compared with the temperature at the
can center predicted by the ADE analog to the heat conduction
equation (equation 32). The same was also compared with the
analytical solution to the equation 32 (heat conduction equation).
To study the effect of surface heat transfer coefficients, the
transient temperature data were plotted on semilog paper with
( T TR) /( T0 TR) on the log scale and time on the linear scale.
From equation 9, the inverse slope of this graph is the fvalue. The
different inverse slopes (fvalues) thus calculated, for heating
with steam and heating and cooling with water, were compared. A
higher surface heat transfer coefficient was expected to result in
lower fvalues (steeper slopes) and vice versa.
Use of Conduction Heating Model for Online Control
The finite difference model developed for predicting temper
ature T(t) was incorporated in the software for the online control.
Using the temperature at the can center T(t), F0(t) was monitored in
realtime with measured can boundary (retort) temperature TR(t). The
process would continue until time tR such that
Fo Fod
(7)
128
Water
T,=30*C
top12i#C
s!de=121*C
bot=121*C
R4.19cm
H=10.07cm
6l0.0001
6r=0.00 Of
Contours
45
35
30
22
25
32
40
45
200
200
Streamlines after 90 seconds of heating
computed using 39x39 grid (left half of the
figure) compared with computations using 58x53
grid (right half of the figure)
Figure 25.
4
Such a natural convection heat transfer model could also have
several applications outside the food area. It may be used to study
the heating of buildings (without forced circulation), cooling of
gas turbine blades, storage of cryogenic fluids (e.g. liquid rocket
propellants), and the startup of chemical reactors. Since natural
convection heat transfer would involve a much higher degree of math
ematical complexity than a conduction heating model (and its appli
cation to online control), the primary objective of this study was
directed toward developing a heat transfer model for natural convec
tion; followed by improvements to existing conductionheat transfer
models for application to online process control as secondary
objectives.
Objectives
The objectives of this study were to
1. develop a mathematical model to predict temperature and
velocity profiles caused by natural convection heat
transfer in a closed cylinder,
2. investigate the need for possible improvements on an
existing conduction heat transfer model to improve its
suitability for use in online process control appli
cations, and
3. incorporate the conduction heating model in an online
process control algorithm for thermal processing of
canned food, and compare its performance against other
possible methods of online control.
152
7. A slowest heating region was located away from the centerline
in a donutshaped region near the bottom 15% of the height of
the container studied.
8. The slowest heating points also migrated within this region
but with no particular pattern of migration.
9. The transient temperature values at the slowest heating points
were close to, but different from the transient temperature at
l/3rd height on the axis of the container.
Conduction Heating Model and Online Control
1. A realtime retort control algorithm was developed. The
algorithm is capable of continious calculations of
accomplished sterilization, online correction of arbitrary
deviations in retort temperature and printed documentation.
2. A numerical ADE conduction heat transfer model was adapted for
use with the retort control algorithm. The large Biot number
associated with low thermal diffusivity of food permitted the
use of temperature specified boundary condition at the can
wall.
3. A numerical ADE conduction heat transfer model was also
written for use with finite heat transfer coefficient at the
can wal1.
4. For the numerical heat transfer models to predict temperatures
accurately, the pressure in the retort should be carefully
controlled and sudden drops during cooling avoided.
INTRODUCTION
Background and Justification
Thermal preservation of food involves application of heat
sufficient to destroy the microorganisms present in food that cause
spoilage. Commercial canning processes provide such heat treatment
to extend the shelflife of foods. In a typical canning process, the
container is filled with food and sealed. The sealed container is
heated in a closed vessel with steam or hot water long enough to
kill the microorganisms and is then cooled. In a production situa
tion, the heating medium (steam or hot water) temperature can
sometimes deviate significantly from design values during the
process. Heating the product more than the required amount would un
necessarily degrade its quality while wasting energy. However,
insufficient heating could lead to serious public health hazards and
cannot be allowed. Thus, food processors always have to meet this
minimum heat treatment (sterilization). Determination of the heating
time to meet the minimum sterilization demands that the actual
transient temperature history of the food during processing be known
predicted with a heat transfer model or actually measured).
1
126
I
Figure 23. Diagram of 39(0.9)x39(0.8) grid on the left
and 58(0.85)x58(0.76) on the right
Figure 3. The grid system and the boundary conditions
for the cylindrical can
34
from the finite difference approximation to the heat conduction
equation (32) with measured boundary (retort) temperature T^(t). The
heating process could continue until time t^ such that
Fo Fo <12>
thus satisfying the required sterilization for arbitrary varia
tions in retort temperature Tr(t).
However, the contribution to F0 from cooling (the second
integral in equation 35a) should not be neglected for a conduction
heating product. Depending on the can center temperature at the
start of cooling, size of the can, and other factors in a conduction
heating situation, the contribution to FQ from cooling could be as
much as 40% of the total. This is unlike convection heated products,
where the container is sometimes agitated and the product is
rapidly cooled, making a small contribution to F0 during cooling. To
avoid gross overprocessing of conduction heated product, therefore,
the cooling F0 must be considered. Also, as mentioned earlier, there
is no control over the possible contribution to FQ from cooling once
cooling is under way.
Thus, the contribution from cooling cannot be neglected, it is
not a constant, and it cannot be controlled. Therefore, it can only
be estimated before cooling is actually started. Such estimation of
cooling lethality can be done through simulation of the cooling
cycle assuming a constant cooling water temperature in the retort.
The objective here is to develop an algorithm to perform this simu
lation in realtime when other control actions are going on.
Convection terms 45
Diffusion terms 50
Discretization of the Elliptic
Stream Function Equation 61
Computational Boundary Conditions for the FDEs 69
Convergence Criteria for Iteration 74
Algorithm for Iterative Solution 76
Coding of the Computer Program and Hardware...... 80
Modeling of Conduction Heating in Cans 80
Discretization and Solution of the Equation 81
Coding of the Computer Program 82
Experimental Studies 83
Use of the Conduction Heating Model for Online Control 86
RESULTS AND DISCUSSION 93
Natural Convection Heating in Canned Foods 93
Transient Flow Patterns and Temperature Profiles 94
Start of flow and conduction layer ....95
Radial temperature and velocity profiles 106
Axial temperature profiles ...113
Slowest Heating Zone 115
Contrast with Conduction 120
Assessment of the Numerical Method 124
Convergence 124
Selection of the grid size 125
Selection of the time step 130
Convergence of boundary vorticity 134
A note on the half grid points 134
Conduction Heating in Cans 138
Analysis of Boundary Conditions 139
Comparison of the Transient Temperature Values 141
Effect of Sudden Pressure Drops in the Can. 141
Performance of the Online Control Logic 141
CONCLUSIONS AND RECOMMENDATIONS 151
Conclusions 151
Recommendations 153
APPENDIX
A ALTERNATIVE FINITE DIFFERENCING OF
STREAM FUNCTION EQUATION 154
B ALTERNATIVE FINITE DIFFERENCING OF BOUNDARY VORTICITY...155
C INPUT DATA FILE FOR NATURAL CONVECTION MODEL 156
D ADE FORMULATION FOR THE CONDUCTION EQUATION 157
LIST OF REFERENCES 161
ADDITIONAL REFERENCES... 167
BIOGRAPHICAL SKETCH 172
vi
118
elapsed time is 30 seconds. For clarity, two or more cold points at
the same location are drawn slightly shifted. As can be seen, the
movement is quite random and no particular order is apparent; but
remains well within the lowest 15% of the can height.
Transient temperature at the slowest heating points. The tran
sient values of the temperature at the slowest heating points were
then investigated. An average of the temperatures at three slowest
heating points (nearest to the bottom) was considered for this
purpose. This average temperature, therefore, did not correspond to
any one particular physical location. The average slowest heating
temperature value was plotted with time (figure 19). For comparison,
transient temperature at the mid height and l/3rd (from the bottom)
height on the axis is also plotted. The l/3rd height on the axis has
traditionally been taken to be the slowest heating point.
Qualitatively, the transient temperature curves were as
expected. The slowest heating temperature, by definition, stayed
consistently lower than the other two temperatures. Also, the
slowest heating temperature was rising consistently since heat was
constantly entering the can. The transient temperature at the l/3rd
height stayed closer than the mid point to the slowest heating tem
perature. Although, there are definite differences between the tem
peratures at all times, the magnitude of these differences is
probably not sufficiently great to warrant any change from the tra
ditional practice of using the l/3rd height location. However water,
due to its low viscosity, heats rather quickly and this explains the
little variation of temperature within the container. For a liquid

