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Dynamic heat transfer in composite miniature structures

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Title:
Dynamic heat transfer in composite miniature structures
Creator:
Ariet Antiga, Mario, 1939-
Publication Date:
Language:
English
Physical Description:
xii, 208 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Analog computers ( jstor )
Asbestos ( jstor )
Heat ( jstor )
Heat sinks ( jstor )
Heat transfer ( jstor )
Insulation ( jstor )
Mathematical models ( jstor )
Modeling ( jstor )
Simulations ( jstor )
Thermal batteries ( jstor )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis--University of Florida, 1965.
Bibliography:
Bibliography: leaves 127-129.
Additional Physical Form:
Also available online.
General Note:
Manuscript copy.
General Note:
Vita.

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Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Mario Ariet Antiga. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit 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:
13409735 ( OCLC )
0021888922 ( ALEPH )

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DYNAMIC HEAT TRANSFER IN

COMPOSITE MINIATURE STRUCTURES



















By

MARIO ARIET ANTIGA










A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
April, 1965













ACKNOWLEDGMENTS


The author wishes to express his sincere appreciation to

Professor Robert D. Walker, Jr., whose interest, advice, and criticism

stimulated and guided this research program; to Dr. Herbert E. Schweyer

for his guidance and advice throughout his career.

He wishes to thank Mr. Henry R. Wengrow without whose assistance

this work would not have been possible, Dr. Mack Tyner and Mr.-Mario

Padron for their helpful discussions and suggestions, and the members of

his Supervisory Committee Dr. T. M. Reed, Dr. R. G. Blake and Dr. R. W.

Kluge. A special appreciation is due Mr. Bruce T. Fairchild, Mr. H. R.

Wengrow and Dr. F. P. May for the use of their AMOS program, and

Mr. Roberto Vich for his assistance on the drawings.

The author also wishes to acknowledge the financial assistance

of the Harry Diamond Laboratories, Army Materiel Command, and its

technical representatives Messrs. R. H. Comyn and Nathan Kaplan, who

by their encouragement and support made it possible to conduct this

investigation.






/













TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS ....................... ......................... ii

LIST OF TABLES................................. ... ... .......... vi

LIST OF FIGURES................................................. vii

ABSTRACT........................................................ x

CHAPTER

I. INTRODUCTION ................................... ......... 1

II. THEORY OF HEAT TRANSMISSION............................. 8

Geometrical Considerations in Battery Life
Optimization................................... 16

III. ANALOG COMPUTER MODEL I................................. 22

A. Description of the Model........................... 22

B. Heat Transfer Coefficients......................... 24

C. Development of the Mathematical Model.............. 27

D. Analog Computer Solution........................... 31

E. Discussion of Results............................... 38

1. Temperature Histories in a Standard Simulated
Battery.................................. 42

2. Effect of Heat Sink on Cell Temperatures....... 42

3. Effect of Insulation Parameters on Cell
Temperatures............................. 44

4. Variation of Heat Generator Parameters........ 44

5. Variation of Cell Parameters.................. 49

6. Effect of Rate and Level of Heat Generation by
Chemical Reactions in Cells.............. 56





iii













TABLE OF CONTENTS (Continued)


Page
CHAPTER

IV. ANALOG COMPUTER MODEL II ............................... 62

A. Description of the Model........................... 62

B. Development of the Mathematical Model.............. 65

C. Analog Computer Solution........................... 67

D. Discussion of Results.............................. 67

1. Geometrical Shape of Insulation Elements...... 76

2. Simulation of Standard Thermal Battery......... 78

3. Effect of a Metal Layer Next to the Core on
Core Temperature......................... 80

4. Effect of Varying Insulation Arrangements on
Core Temperature. ....................... 80

V. DIGITAL COMPUTER MODEL III.............................. 88

A. Description of the Model........................... 88

B. Development of the Mathematical Model.............. 90

C. Finite Differences Approximation................... 94

D. Development of the Computer Program................ 94

E. The AMOS Program ................................... 97

F. Discussion of Results.............................. 99

1. Simulation of Standard Thermal Battery........ 101

2. Effect of Varying Insulation Arrangements on
Core Temperature......................... 104

3. Effect of Idealized Insulating Materials on
Core Temperature........................ 108

4. Effect of Core Radius on Core Temperature..... 109

5. Effect of Intra-Cell Heat Generation.......... 114



iv













TABLE OF CONTENTS (Continued)


Page

CHAPTER

6. Effect of Heat Sink Temperature on Core
Temperature .............................. 114

7. Effect of Changes in the Heat Capacity of the
Heat Generators on the Core Temperature.. 114

8. Core Temperature of Improved Thermal Battery.. 118

9. Effect on Core Temperature of Delayed Heat
Generation Within the Insulation......... 118

10. Effect of Change in the Volume of the
Battery on Core Temperature.............. 121

VI. CONCLUSIONS AND RECOMMENDATIONS......................... 123

LIST OF SYMBOLS................................................. 125

LITERATURE CITED................................................ 127

APPENDICES..................................................... 130

A. Details of Analog Model I............................... 131

B. Details of Analog Model II.............................. 138

C. Details of Digital Model III .......................... 148

D. Details of the Computer Program........................ 156

BIOGRAPHICAL SKETCH... ............ ................. ............ 209


















V













LIST OF TABLES


Table Page

1 Differential-Difference Equatiohs for Mathematical
Model I.............................................. 28

2 Coefficients for Programmed Differential-Difference
Equations: Model I.................................. 32

3 Average Physical Properties and Dimensions of Thermal
Battery Components: Model I........................ 36

4 Effect of Parameters and Changes on the Life to 4000C of
a Simulated Thermal Battery.......................... 40

5 Differential-Difference Equations for Mathematical
Model II............................................. 69

6 Coefficients for Programmed Differential-Difference
Equations: Model II................................. 71

7 Average Physical Properties and Dimensions of Thermal
Battery Components: Model II........................ 75

8 Effect of Changes in the Insulation Structure on Life to
4000C of Simulated Thermal Batteries................. 77

9 Effect of Parameters and Changes on Life to 400 C of a
Simulated Thermal Battery............................ 105

A-1 Summary of Runs........................................... 132

A-2 Summary of Potentiometer Settings......................... 134

B-1 Summary of Runs........................................... 143

B-2 Summary of Potentiometer Settings......................... 144

C-l Data Used in Simulation of Standard Thermal Battery....... 150

C-2 Summary of Runs........................................... 151










vi-












LIST OF FIGURES


Figure Page

1 Schematic Diagram of Thermal Battery.................. 4

2 Temperature Histories in Homogeneous Right Circular
Cylinders ........................................ 21

3 Schematic Diagram of Battery Described by Model I..... 23

4 Analog Computer Diagram of Model I.................... 30

5 Experimental Versus Computed Results.................. 39

6 Temperature Histories of Elements in Standard Simula-
ted Thermal Battery.............................. 43

7 Effect on Cell Temperature of Changing the Temperature
of the Heat Sink ............................. 45

8 Effect on Cell Temperature of Reducing the Thermal
Conductivity of the Insulation................... 46

9 Effect on Cell Temperature of Increasing the Thermal
Conductivity of the Insulation... .............. 47

10 Effect on Cell Temperature of Changing the Thickness
of the Insulation................................ 48

11 Effect on Cell Temperature of Changing Thickness of
Heat Generators.................................. 50

12 Effect on Cell Temperature of Compressing Heat Genera-
tors............................................. 51

13 Effect on Cell Temperature of Changing the Thickness of
the Cell......................................... 53

14 Effect on Cell Temperature of Changing the Enthalpy
of the Cell...................................... 55

15 Effect on Cell Temperature of Changing the Magnitude
of the Intra-Cell Chemical Heat Generation...... 57

16 Change of Initial Rate of Intra-Cell Chemical Heat
Generation Term................................. 59

17 Change of Decay Rate of Intra-Cell Chemical Heat
Generation Term.................................. 61



vii













LI:ST CT C2S (C*i7 n iaed)





18Schemaic ~iAram of Battery Described by .odel II...

S Anag Co ar Diagram of Model II.................. 68

"V20 T araure Histories in Stand&ard Simulated Thermal


S T ,..p .,Ge 3istories in Simulated Thermal Battery
arka Mtal Next to Co rere................. .. 81

-2 1CaSe-atuc~H a Eistories in Simulatea The-eal Eattery
-wi~th vayin Insulation C z,"nI; n ....... .

3 Yristories in Simualated he la Lterzey with.
V ying insulation Arranngeme t. ................ 30

Im Ter-ertura Histories in Simulated Thermal Battery
t. th Vryln; Insulation Arrangememnt ............ 8

25 Te,:perature Histories on Simulated Thermal Battery
with Varying Insulation Arrangement............. 87

'h chem~ tic Liagram of Battery Described by Model III.. 93

:7 Temperature Profiles in Standard Simulated Thern.al
Battcry............................ ................103

S E~ect on Cell Temperature of Various Insulation
Arrangements ................. ...... .......... ... 107
29 Effect on Cell Te s perature of Idealized Changes in
the Thermal Properties of the Inulaetion........ 110

30 effect on Insulation Teimperature of Idealized Changes
in iThe ':l Proyer:ies of the Insulation ......... 111

31 Effect on Cell Temperature of Different Idealizad
Insula rs ........................................ 112

32 Effect on Cell -Teperature of Changes in Core Radius. 113

3 Effe c on Call Temperature of Changes in the Intra-
h_ KErtl Generation .................... ......... 115

14 Effect on Cell Te ...,ature of the Heat Sink Tempera-
ture...................................... ........ 11



viii













LIST OF FIGURES (Continued)


Figure Page

35 Effect on Cell Temperature of Changes in the Heat
Capacity of the Heat Generators................. 117

36 Effect on Cell Temperature of Suggested Design
Improvements .................................... 119

37 Effect on Cell Temperature of Delayed Heat Generation
Within the Insulation........................... 120

38 Effect on Cell Temperature of a Change in the Volume
of the Battery.................................. 122

A-1 Analog Circuit of Heat of Reaction Term.............. 137

B-l Details of the Shape of Insulation Elements.......... 139

C-l Analytical Versus Computed Results for the Cooling of
a Homogeneous Sphere............................ 149
































ix













Abstract of Dissertation Presented to the Graduate Council in
Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy



DYNAMIC HEAT TRANSFER IN COMPOSITE
MINIATURE STRUCTURES



By



Mario Ariet Antiga



April, 1965



Chairman: Prof. Robert D. Walker, Jr.

Major Department: Chemical Engineering


The inability to measure temperatures in small objects accurately

and at all desirable locations during rapid temperature transients

points up the need for valid alternate methods of solving such prob-

lems. In this investigation, temperature histories were computed

for specific locations in a simulated thermal battery, and the effect

of changing the construction configuration and the physical properties

of the materials were studied in order to optimize battery life.

Two analog computer models were developed to simulate the

thermal battery. These models were based on the assumption of "well-

mixed" elements having the thermal resistance "lumped" at the inter-

faces between adjacent elements. They provided simulations of the

temperature histories in thermal batteries which were in satisfactory



x













agreement with experience. The analog studies provided considerable

insight into heat transfer processes in these units, and the effects of

changes in configuration and properties of materials. The results

of these models were also very helpful in the development of the more

accurate digital computer model.

The digital computer model consisted of a sphere having a

core made up of an inner well-mixed section and an outer section where

temperature gradients existed. The core was surrounded by six

concentric spherical layers of insulating materials which could be

assigned any value of physical properties. The partial differential

equations describing this model were approximated using finite-

difference techniques by a system of differential-difference equations

which in turn were solved by the Adams-Moulton-Shell numerical inte-

gration method. Among the more important findings of this work are:

1. Of the insulation materials and configuration studied

maximum battery life is achieved with all-Thermoflex insulation.

2. If it is necessary to use layers of more than one insula-

tion material, maximum battery life is achieved with as much Thermoflex

as possible adjacent to the core.

3. Insulation layers prepared by mixing a poor insulator with

a good one are less efficient than proper arrangements of layers consist-

ing of pure materials.

4. An increase in the volumetric heat capacity of the heat

generators leads to an increase in the battery life.

5. An increase in the intra-cell heat of reaction leads to



xi












an increase in battery life providing it is spread over a sufficiently

long period of time.

6. Placement of suitable heat'.generators in the insulation

leads to substantially increased battery life.

7. An increase in battery size generally results in an

increase in battery life because the capacity of the heat reservoir

increases more rapidly than the rate of heat losses. Conversely, the

smaller the battery the more serious the heat transfer problem.

8. The sphere is the most efficient shape because it has

the smallest area for a given volume. The greater the deviation of an

actual shape from a sphere the greater the heat transfer problem.

9. The high cut-off temperature imposed by a high-melting

electrolyte leads to an optimum core to insulation volume ratio. For

the system studied this optimum ratio turned out to be close to that cho-

sen for the standard.

10. The optimum insulator for a high cut-off temperature is

not the insulator with the lowest thermal diffusivity; it is, rather,

the one with the lowest thermal conductivity and a value of the volu-

metric heat capacity which is a function of the cut-off temperature.









xii






xii













CHAPTER I


INTRODUCTION


The construction and operation of thermal batteries was first

discussed by Goodrich (1), who defined them as electrochemical power

supplies based upon electrolytes of various inorganic salts which re-

main solid and nonconducting at all storage temperatures. He indicated

that for isolated performance, an integral heat generating source

(based on chemical reaction using gaseous, liquid or solid fuels) is

required to raise the temperature of the cell above its melting point.

Vinal (2), and Selis et al. (3) have presented descriptions of differ-

ent electrochemical systems, such as Mg/LiCl-KCl-K2Cr204/Ni and

Mg/LiCl-KCl/FeOx,Ni which perform satisfactorily in thermal batteries.

McKee (4) enumerated the following advantages of thermal

batteries:

1. Permit high voltages

2. Large currents may be drawn from them

3. Indefinite storage

4. Operation over a wide range of temperature extremes

5. No maintenance

6. Use in any position

7. Ruggedness

He stated that the chief limitation is that they are relatively short

lived, the implication being that this results from loss of heat, for

otherwise life could be increased essentially indefinitely by adding



1









2



more cell reactants. Johnson (5), and Hill (6) presented data on heat

generating systems which can achieve temperatures of about 7000C in

about 1 second. Some of these systems contain zinc metal while

others are based on KMnO4, CaF2, MgF2, Fe, etc. Numerous other

systems have been devised from study of the thermodynamic properties

of the reactants and products of reactions.

Possibly the classic case of utilizing a chemical reaction to

produce heat is the thermite process. Generally the reactants in

these reactions are in the form of very fine powder and are chosen so

as to be gasless, or nearly so. These reactions are essentially

instantaneous, and they give off a great deal of heat. Temperatures

of the order of 200000C are commonly attained.

The design of thermal batteries constitutes a challenging

complex engineering problem because of the stringent heat transfer

limitations required to construct a useful unit. The chemical compo-

sition of the materials which compose the thermal battery and the

details of its construction are subject to security classification, but

enough unclassified information is available to permit the heat transfer

problem to be defined meaningfully.

A thermal battery can be thought of as an assembly of three
/
main types of elements:

1. Heat generators, which constitute an essentially instan-

taneous heat source.

2. Electrolytic cells, which become activated when the









3



electrolyte, which is solid at ambient temperature, melts as it receives

heat given off by the heat generators.

3. Insulation, which serves to decrease the rate of heat

loss from the assembly to the surroundings.

In a thermal battery the heat generators and electrolytic cells,

which may be shaped in the form of flat circular cylinders, can be

arranged in a stack in which each cell is in contact with two generators

and vice versa. This cell-generator core is then surrounded by the

insulation. A schematic diagram of such a unit is shown in Figure 1.

The operation of the battery is initiated when the heat

generators are set off. The large quantity of heat given off almost

instantaneously by these generators is transferred rapidly to the

electrochemical cells with the result that the solid electrolyte melts

and the cell begins to generate electrical power. The assembly loses

heat to the surroundings owing to its high relative temperature, and

the electrochemical reaction within the cell proceeds until the electro-

lyte approaches its freezing point. In this work the elapsed time

between the reaching 400 C and cooling tract to 400 C is referred to

as the life of the battery.

In a heat transfer study one of the obvious objectives would be

to maximize the life of the battery for a given battery volume subject

to other constructional and operational constraints. Maximization of

battery life above 4000C was, therefore, made the primary objective of

this investigation. The life of a thermal battery can be estimated

reasonably well if temperature histories can be obtained at the required






























S/H-- eat Generator



---Electro-
SChemical Cell


S" Insulation











NOTE: Section removed
for clarity




,Figure 1 Schematic Diagram of Thermal.Battery








5



locations within the battery. Although it might appear that the

desired time-temperature relations could be obtained either by direct

measurement or by classical mathematical methods, further considera-

tion of the problem will reveal inherent limitations in both the

experimental and the formal computational approaches.

Although thermometers, thermocouples, and similar devices are

generally quite adequate temperature measurement instruments for many

physical phenomena, there exist other situations, such as that encoun-

tered here, where the instrumental response will not be adequate. For

example, large differences can arise between the indicated and actual

temperatures of an object as a result of response lags of the sensing

device during rapid temperature changes, or as a result of heat losses

through the measuring device if the object under study is small in size

relative to the sensing device. Such errors are in general difficult to

evaluate.

Alternatively the temperature histories at points of interest

may be obtained by solving the mathematical model which describes the

given physical situation. For the case of unsteady-state heat transfer,

the mathematical model will consist of one or more partial differential

equations. Even if the errors associated with a temperature measuring

device should not be serious, a valid calculation procedure would have

a tremendous advantage of producing temperature histories of numerous

points simultaneously. Moreover, the experimental approach would subject

the system to the disruptive influence of the numerous measuring devices

required for direct measurements. Where the object under study is









6



relatively small the errors resulting from heat losses through the

various measuring instruments might be the most significant mode of

heat loss thereby making the observations meaningless.

Another significant advantage of the mathematical model approach

to this problem as compared to the experimental approach is that the

experimental testing of the battery requires a statistical analysis

involving numerous replications because of the random variations of the

materials making up the battery. In the mathematical model, materials

having exact properties are assumed and hence valid conclusions can be

drawn from a much smaller number of trials.

There is another consideration which is very significant from

the design viewpoint, namely, the mathematical model allows for the

possibility of evaluating the performance of a battery constructed of

idealized materials. If the results indicate that a significant

improvement could be obtained with such materials, efforts could be

directed towards the development of these materials.

In many cases of practical interest, such as the one under

consideration, formal mathematical solutions of partial differential

equations are very difficult or impossible to evaluate because mater-

ials may not be homogeneous, thermal properties may vary, or the boundary

or initial conditions may be complex. However, the availability of

electronic computers makes possible the solution of complex mathematical

models by various approximation techniques.

In this study, mathematical models of the dynamic heat transfer

in thermal batteries were developed with simplifying assumptions which









7



made their solution on the computers feasible. In the initial stage

of the investigation the analog computer was utilized. With the insight

gained from the analog studies, a more/complete mathematical model was

developed and programmed for the IBM 709 digital computer.

The investigation had two chief objectives:

1. To provide temperature histories for specific locations

within a battery.

2. To study the effect of changing the construction configura-

tion and properties of the materials required for the construction of

the battery so that design specifications could be made to optimize its

performance.

The optimization criterion was defined as the maximum life of

the battery. Thus, the configuration yielding the maximum battery life

was considered optimum. As mentioned before, the/effect of idealized

materials was studied with the idea that if the inclusion of certain

idealized material increased the life of the assembly significantly,

the desirability of developing such a material would be indicated.












CHAPTER II


THEORY OF HEAT TRANSMISSION


The second law of thermodynamics states that heat energy

always flows in the direction of the negative temperature gradient,

i.e., from a hot body to a cooler one.

There are three distinct methods by which this migration of

heat takes place:

1. Conduction,in which the heat passes through the substance

of the body itself.

2. Convection,in which heat is transferred by relative

motion of portions of the heated body.

3. Radiation,in which heat is transferred directly between

portions of the body by electromagnetic radiation.

Although the three kinds of heat transmission generally occur

together, fortunately one or the other often prevails in practical

cases. Therefore, separate laws governing each kind of heat transfer

have been developed and may be used in such cases. Superposition of

these laws is also often possible and used (7).

The basic law of heat conduction is:

= -k AT (II-1)
A AL

In this and the next two equations, Q, denotes the time rate of heat

flow, i.e., the heat energy flowing through a constant area, A, in unit

time. The rate of heat flow, Q, may be considered constant for the time




8









9


being. Equation (II-1) relates to steady state transfer in a plane

plate of thickness AL with a perfectly insulated edge: the two free

surfaces being held at the temperature difference AT. The parameter,

k, which may be considered a constant for the time being, is called

thermal conductivity.

Equation (II-1) originates from Biot (8), but it is generally

called Fourier's equation because Fourier (9) used it as a fundamental

equation in his analytic theory of heat.

For heat convection the following equation was first recommended

by Newton (10):

Q = HAAT (11-2)

Equation (11-2) relates to the heat transfer between a surface and a

fluid in contact with it, the temperature difference being AT. The

factor H is called surface coefficient of heat transfer, film coeffi-

cient of heat transfer, or simply coefficient of heat transfer. This

expression is often referred to as Newton's cooling law, but it is

really a definition of H. This point will be discussed later in more

detail.

For the total radiation, equations of the form

Q = AT4 (11-3)

have been used since Stefan (11) found this relation and Boltzmann

(12) proved it theoretically for a perfectly black surface. Equation

(11-3) relates to the emission of radiation from a surface at the

absolute temperature T. The factor a is a natural constant known as

the Stefan-Boltzmann constant, or the constant of total black-body









10



radiation. For surfaces not absolutely black, 0 must be modified if

the Stefan-Boltzmann law is to be applicable.

The basic equation of heat accumulation for small linear

changes of temperature is

Q = pC VAT (11-4)


where Q is the heat accumulation in unit time, in the volume, V, of a

medium of density, p, and specific heat, C when the temperature

increases by AT in a time interval At.

From the above fundamental relations a great deal of knowledge

has been developed. The application of mathematics permits the evalua-

tion of heat transfer processes by different modes, in different geomet-

rical shapes, and subject to varied initial and boundary specifications.

Carslaw and Jaeger (13) have presented a very complete formal mathe-

matical treatment of heat conduction problems. Jakob (7) considered

all forms of heat transfer in his work, and provided theoretical or

empirical solutions to a great variety of heat transfer problems.

McAdams (14) presents a very complete treatment of the heat transfer

problem from the practical design engineering point of view.

Specifically in the field of heat transfer by conduction, many

physical situations can be described by relations which, are amenable

to solution by formal mathematical techniques. Other studies (15,16,17)

have treated composite bodies, mostly the laminated wall having no

interfacial resistance. However, Siede (18) considered a composite

system having resistance between layers.

In most of the formal mathematical solutions to heat conduction









11



problems the assumption of constant thermal and physical properties

is usually made. Friedman (19), and Yang (20) have studied the effect

of these assumptions and have shown that in some situations significant

errors may result from their use. The solutions to most realistic

problems involving conduction heat transfer usually involve infinite

series of terms which may or may not converge rapidly. Therefore, it

is sometimes quite difficult to obtain a numerical answer from the

general mathematical solution.

In order to make the results of formal mathematical treatment

more applicable to practical problems Gurney and Lurie (21), Groeber

(22), Olson and Schultz (23), Newman (24), and others (14,25,26,27)

have presented graphs or charts showing temperature versus time or

geometrical location for different parameters. Such parameters as

thermal diffusivity,a = k_, surface convection coefficients, and
PCp

geometric shapes are usually employed. The geometrical shapes consi-

dered are limited to homogeneous infinite plates, infinite cylinders,

spheres or objects of such shape that heat flow can be considered

unidirectional.

It is generally conceded that formal mathematical methods are

capable of solving only the simpler situations of geometry and boundary

conditions in heat conduction problems. Many practical situations yield

a mathematical model which can only be solved by approximation methods.

Numerical, graphical and analog techniques are the most common tools

for handling complex heat conduction problems. Although these methods

are approximate, they can, in principle, be extended to any degree of









12



closeness of approach to the exact solution given by formal mathematical

techniques. Their only limitation is the amount of effort (time and/or

money) involved. In addition, as mentioned before, the formal mathemat-

ical solution also requires considerable effort if a precise numerical

answer is desired owing to the usual infinite series form of the solution.

Later, an example will be given of a problem where an approximate

approach actually required less effort to yield an answer of a given

accuracy than the effort required to evaluate the formal mathematical

solution to the same degree of accuracy.

Graphical methods for solving heat conduction problems were

first developed by Binder (28), and Schmidt (29) based on the calculus of

finite differences. Many improvements and extensions of the basic

method have been made (30,31). The work of Longwell (32) is of parti-

cular significance to this investigation because it treats graphically

the motion of the freezing boundary in the heat transfer process involv-

ing the phase change from liquid to solid. This is probably the

mechanism by which the electrochemical cells become inoperative. In

general it can be stated that graphical methods are useful only when

low accuracy is sufficient in the solution of a problem. If a high

degree of accuracy in the solution is attempted, this procedure becomes
/
prohibitively cumbersome.

It has been known for many years that different physical pheno-

mena can be described by the same mathematical relations; in such cases

they are said to be analogous processes. Langmuir, Adams and Meikle

(33) seem to have been the first to make use of the analogy between









13



thermal and electrical conduction; they solved a problem based on the

similarity between a flow-temperature field and an electrical flow-

voltage field of the same geometrical configuration.

Beuken (34), and Paschkis (35) developed large-scale, permanent

analog devices whose principal elements were resistors and condensers,

and they were able to solve unsteady-state heat transfer problems. The

chief drawback of these analog devices is that they are expensive to

construct, and are usually capable of simulating only the type of

system for which they were specifically designed. Even relatively minor

modifications of the original system can be cumbersome and expensive.

The type of analog devices discussed above depend for their

operation upon the existence of a direct physical analogy between the

analog and the prototype system under study. Such an analogy is

recognized by comparing the characteristic equations describing the

dynamic or static behavior of the two systems. An analogy is said to

exist if these characteristic equations are identical in form,and the

initial and boundary conditions are the same. Such a similarity is

possible only if there is a one-to-one correspondence between elements

in the analog and in the prototype system. For every element in the

original system there must be present in the analog system an element

having similar properties, i.e., an element having a similar excita-

tion-response relationship; furthermore, the elements in the analog

must be interconnected in the same fashion as the elements in the

original system.

The other major class of analog system includes mathematical

rather than physical analogs. The behavior of the system under study,









14



or the problem to be solved is first expressed as a set of algebraic

or differential equations. An assemblage of computing units or elements,

each capable of performing some specific mathematical operation, such

as addition, multiplication or integration, is provided, and these

units are interconnected so as to generate the solution of the problem

(36).

The availability in recent years of high-speed digital computers

has augmented the interest in numerical methods based on the calculus

of finite differences as an efficient tool for the solution of complex

heat flow problems. Emmons (37) utilized the relaxation method developed

by Southwell (38) for the solution of two and three-dimensional steady

state heat transfer processes. Although the relaxation technique can

be used for unsteady-state problems (39), explicit time iteration

procedures, such as the one developed by Dusinberre (40), are generally

preferred to relaxation methods because they can be adapted more

readily to digital computation.

The explicit finite-differences technique has in general the

limitation that it is difficult to evaluate the accuracy of the solution.

If the criteria of "stability" and "convergence" are satisfied, the

accuracy is determined by the number of increments used, and it can be
/
improved at the expense of increased effort (41). The convergence

criterion is the requirement that the exact solution be approached

by the approximate solution as the number of increments approaches

infinity. The stability criterion means that the error introduced into

the computation, owing to the limited number of digits which a given









15



computer can carry, must not increase in magnitude as the computation

proceeds. These criteria have been studied by a number of investigators

(41,42,43). Therefore, for a numerical.method which is stable and

convergent when applied to a system of equations, the finite-difference

technique can yield any degree of accuracy desired. The only restric-

tion is the amount of effort required.

Brian (44),and Douglas (45) developed implicit difference

methods which are unconditionally stable, usually at the expense of

increased computational effort. Yavorsky, et al. (46) utilized the

explicit type finite-difference formulation, and solved on a digital

computer the problem of heating homogeneous cylindrical briquettes.

Dickert (47) used the explicit finite-difference approach for the

solution on an IBM 650 digital computer of the unsteady-state heat

transfer in a composite finite cylinder. Actually the physical model he

simulated was a simplified version of the thermal batteries which are

the subject of this study.

Horne and Richardson (48) developed a model to simulate the

performance of batteries at low ambient temperatures. It was programmed

on a digital computer and it was based on well-mixed sections with

lumped thermal resistance at the interface.

In this investigation, two different mathematical models were

developed and solved on an analog computer in order to benefit from the

advantageous features of the instrument, such as the essentially

instantaneous availability of the answer, the continuous display of

the results (usually in an oscilloscope or a plotter), and the immediate









16


response of the system to a change in one of the parameters. All of

these features made the analog computer the initial choice in this

study. Later, after sufficient insight had been gained from the analog

studies and when increased accuracy was desired, a model was developed

to be programmed on the digital computer. The results revealed very

interesting aspects 'of heat transfer phenomena, and provided a good

simulation of thermal batteries.

A. Geometrical Considerations in
Battery Life Optimization

Some conclusions can be drawn from purely geometrical consi-

derations with respect to the optimum shape of a thermal battery under

the criterion of maximum life. Since the rate of heat transfer is

directly proportional to the area and the rate of temperature change is

inversely proportional to the volume, it is clear that the smaller the

area of a body, the lower the rate at which it will loose heat, all

other things being equal. Therefore, a hot body of a given volume will

remain hot longer, the smaller its area.

If, for the moment, the geometrical shape of.the battery is

restricted to right circular cylinders, elementary mathematical consi-

derations show that for a cylinder having radius R and height h, the

total area is given by


A = 27R2 + 27Rh (11-5)

while the volume is given by

V = iR 2h (11-6)

If the volume is considered to be fixed, the area can be expressed by









17



A= 271R2 + 2nR V (11-7)


which can be differentiated with respect to R and equated to zero

to give

dA 2V
S= 47nR -- = 0 (11-8)
"dR 2
R

Equation (11-8) may be solved for the volume to obtain

V = 2nR3 (11-9)

This would be the value of the volume corresponding to a minimum area,

but

V 7R2h (II-10)

hence

h = 2R (II-11)

Equation (II-11) makes clear that the right circular cylinder having

the minimum area per unit volume is one having its height equal to its

diameter.

Similarily it can be shown that for the case of orthogonal

parallelepipeds the volume is given by

V = xyz (11-12)

while the area is given by

A = 2xy + 2xz + 2yz (11-13)

If the volume is considered to be fixed the area can be expressed as
2xV 2yV
A = 2xy + 2xV + 2V (11-14)
xy xy

which can be differentiated partially with respect to x and y to give

aA 2V
S= 2x -2 =0 (11-15)
x









18


aA 2V
= 2y 2 0 (11-16)
y

These equations may be solved for x and y,respectively, to give

x = V1/3 and y= Vl/3 (11-17)
1/3
which results in a value of z = V when substituted into equation

(11-12). Hence the orthogonal parallelepided whose outside area is a

minimum for a given total volume is the cube.

Forsyth (49) has shown by the calculus of variations that the

sphere is the solid generated by rotation which has the maximum volume

for a given area. The sphere is likewise the solid having the maximum

volume for a given area out of all possible solids, but this is more

difficult to demonstrate rigorously It can be shown specifically

that the sphere has a lower ratio of area to volume for a given volume

than the cylinder having equal height and diameter, which in turn has

a lower ratio than the cube., For a volume of V the radius of the sphere

is given by 1/3

R =3- V (11-18)

and the area is given by 1/3

A = 47T( (11-19)

For the cylinder, the radius is given by
( V\ 1/3
c- = IV1/ (11-20)

and the area 2/3

A = ,47r (1I-21)

Hence, for the same volume V, the ratio of the area of the sphere to the









19



area of the cylinder is

A 2/3
s-= = 0.825 (II-22)
A 4
c

For a cube the side 1 is given by

1 = VI/3 (11-23)

and the area

A = 6V2/3 (II-24)
cu

Therefore, the ratio of the area of the sphere to the area of the cube

is
A 2/3
cs = 47 3 = 0.804 (11-25)
A 6 41-
cu

These considerations indicate that, if it were feasible to

construct thermal batteries in a spherical shape, this would be the

optimum configuration from the heat transfer standpoint. There are

other restrictions which make this shape impractical, hence the next

most efficient shape is that of a right circular cylinder having its

diameter equal to its height.

Figure 2 illustrates the temperature histories of the center point

of different homogeneous right circular cylinders having equal volumes

and different height to diameter ratios, and having initial temperatures
0 /
of 500 C everywhere except at the surface where the temperature is

assumed to be constant at zero degrees. These curves were evaluated from

tables presented by Olson and Schultz (23). The parameter shown on the

curves is the height to diameter ratio. The volume of all cylinders

is that of the cylinder having a height of 3.0 cm (equal to its









20



diameter). A temperature history for the center point of a sphere

having the same volume is also shown. ,For right circular cylinders

of constant volume, Figure 2 indicates that the rate of cooling increases

drastically when the height to diameter ratio is made less than the

optimum. The rate of cooling also increases when the height to diameter

ratio is made greater than the optimum but the effect is less than in

the former case.




























/












500--



Sphere


450



0

400 R = 3.0 cm
44I


o R

S350




S=0.14 =0.38 = 4.3
300-





.250-
0 20 40 0
Time, Sec.
Figure 2 Temperature Histories in Homogeneous Right Circular Cylinders













CHAPTER III


ANALOG COMPUTER MODEL I


A. Description of the Model


The first model developed to simulate a thermal battery con-

sisted of a cylindrical stack of alternating heat generators and cells

surrounded by insulation. While the actual number of generators and

cells in a real thermal battery may vary, it was assumed that the core

was composed of three cylindrical generators and two cells surrounded by

top, bottom and lateral insulation. A schematic diagram of the model

is shown in Figure 3.

Consideration of the physical dimensions of the elements in the

battery led to some assumptions. The very small relative thickness of

the elements compared to other dimensions, such as diameter of the cells,

heat generators and top insulation, and height for the lateral insula-

tion, suggests that the major portion of the heat transfer is an axial,

rather than a radial, process. This suggests-that the temperature with-

in each element would be rather uniform, hence it was assumed that each

element was "well-stirred", i.e., that its temperature was uniform

throughout. This assumption is more valid for some elements than for

others. For example, the cells consist (during the operating life of

the battery) of molten electrolyte, and, since intra-cell chemical and

electrochemical reactions may be occurring simultaneously, it appears

that the mobility of the ions in the electrolyte provides a relatively





22










23









R
0

R

----------------

Top Insulation h
*r-4
Top Generator


Cell H

-- Middle Generator -





e-1




Fiure 3 Schematic Diaram of Battery Described by Model










Figure 3 Schematic Diagram of Battery Described by Model I









24



well-mixed element having an approximately constant temperature. There

also exists the possiblity that some convection currents might be

established, but this is doubtful owing to the small thickness of the

cell. The deviation of the heat generators and insulation from this

assumption would be of the same order of magnitude because their thermal

properties are comparable.

The cylindrical symmetry of the model makes necessary the

consideration of only the top half of the unit.

The heat generators achieve their maximum temperature of about

2200C (47) in a length of time which is negligible compared to the

rest of the heat transfer process. Therefore, it was assumed that they

reached their maximum temperature instantaneously, and this high tempera-

ture becomes the initial driving force of the heat transfer system.

There is a heat of reaction from intra-cell chemical reactions.

The experimental data describing this phenomenon are very uncertain. It

is known that the heat of reaction increases rapidly at the beginning

of the operation, reaches a maximum, and then decays. A triangular

shape was assumed for the heat of reaction-time relationship. Both the

shape and magnitude of this effect were based on educated guesses of

experienced investigators (50), and it is the only factor in this study

not based on experimental or computed physical data.

B. Heat Transfer Coefficients

The chief consequence of the assumption of well-mixed elements

in the battery is that the heat transfer process which occurs under these

conditions become one of convection rather than conduction. Because









25



there can be no temperature gradient through any single element, all

of the resistance to heat transfer appears "lumped" at the interfaces

between elements. Therefore, a pseudo-heat transfer coefficient must

be calculated by appropriately lumping the heat transfer resistances of

two adjacent elements at the interface between them (based, of course,

on their thermal conductivities and the mean path traveled by the

heat).

The pseudo-heat transfer coefficients were evaluated by

considering the two elements to constitute a series arrangement for the

resistance to heat flow. For the case of a cell and a heat generator,

the coefficient had the following form:

Total = Resistance + Resistance (III-1)
Resistance of Cell of Generator

h1 1 hc_ 1 ht (111-2)
2 2-
H k~ 2 k 2
g-c g


H = 1 (III-3)
g-c
1 h
k 2 k\ 2
c g

where H is the heat transfer coefficient between the heat generator
g-c
and the cell, and k and h are the thermal conductivities and heights,

respectively. The heat transfer coefficients involving the lateral

insulation were obtained by calculating the radius equivalent to one-half

the volume of the interior element, and considering the mean distance

the distance from this radius to the outer radius of the element. The

mean distance in the case of the lateral insulation is, of course, one-

half of the thickness. Therefore, for a value of the radius of the








26



element of 1.56, the following heat transfer coefficient was obtained:


H =l 1 (111-4)
g-li
1 k i i 1--(.46)
-i + kg
k.2 k
i\ g

All other heat transfer coefficients were computed in a manner similar

to those discussed above.

Because of the high initial temperature of the heat generators,

radiation rather than convection or conduction is the principal mechanism

of heat transfer while the generators are incandescent. A pseudo-

convection heat transfer coefficient was calculated for this period

based on the laws of radiation. It was arbitrarily decided that 7500C

was the temperature where the principal mechanism changed from radiation

to "convection". The heat transfer coefficient describing the radiation

transfer can be obtained as follows (51):

q = A F 2(T4 T24) (III-5)

where q is the rate of heat transfer, and A is the area of the heat

transfer surface. F is a dimensionless factor to allow for interchange

between gray surfaces; a is the Stefan-Boltzmann constant

(4.92 x 10- kgcal/m hr k ), and T is the temperature in degrees Kelvin.


F =1 1 (111-6)

__. + i 1) + 1( )
12

where F is a dimensionless geometrical factor to allow for net radiation

between black surfaces including the effect of refractory surfaces, and









27



E is the dimensionless emissivity.

The equivalent heat transfer coefficient is obtained from

4 4
H g F 12(T T2 H (111-7)
g-c-R AT + g-c

where T1 is evaluated as the arithmetic average of the fourth powers

of the extreme values of the heat generator temperature during the

radiation period and T2, the cell temperature, was evaluated similarily.

AT was taken as the geometric mean of the extreme values of the tempera-

ture differences (52). Detailed computations of all these coefficients

are shown in Appendix A.


C. Development of the Mathematical Model

A differential heat balance around each element (i.e. cell,

generator, etc.) gives the equations shown in Table 1 with notation

having the significance indicated below:

H = pseudo-convection heat transfer coefficient, cal
2 o
cm sec C

h height of element, cm(See Figure 3.)

R = radius of element, cm(See Figure 3)

R = outside radius of assembly, cm(See Figure 3)

T = temperature of element, oC

t = time, sec


Subscripts

a = refers to the ambient

c = refers to the cell

g = refers to the generators











TABLE 1


DIFFERENTIAL DIFFERENCE EQUATIONS FOR MATHEMATICAL MODEL I


Top Insulation

-RR2Hti-a(Tti Ta) 2RhtiHti-li(Tti li)- R2Hti-g(Tti Ttg) PiCpiR 2h ti (III-8)
dt

Initial Condition For Equation III-8 at t = 0 Tti = T (III-9)

Top Generator

2 2 2 dT 00
R2H ig(T 27Rh Htg g (litg Tl) RHg (T Tc) = pgC R2hdt (III-10)
ti-g tg ti tg g-li tg li g-c tg c g t-

Initial Condition For Equation III-10 at t 0 Tt = 22000C (III-11)

Cell
2 2 2 dT
-RH (T T ) 2Rh H li(T T i -7R H (T Tr) = pc PR2h dT (111-12)
g-c tg c -li c li g-c ng c c pc e

Initial Condition For Equation III-12 at t = 0 T = T (111-13)

Middle Generator

-R2H (T T) 2Rh H (Tmg Tli)= pC R2h dTm (III-14)
g-c mg c g g-li mg li gpg t dt
2 2

Initial Condition For Equation 111-14 at t = 0 T = 22000C (111-15)
mg











TABLE 1 (Continued)


Lateral Insulation


t27Rhi ti-li li Tti 2 tg g-li (Ti Ttg 27RhcHc-li Tli c 2R g-lili mg
2

-27Ro(hti + htg + h + h )H-a(Ti Ta) = Cp2(RoR)(ht + + h + h dTi ( -16)
Sti tg t li-a i a i i o ti tg c dt
2 2

Initial Condition For Equation III-16 at t = 0 Tli = T (III-17)
i a (1 7











49V \ 9--44









-100 0 I.C., T tg= 2200 0C








-li 49 9-


-T 42

45 12
S -T -T
6 43m c 7 44 mg
-T mg 25 43 \

"gmg g

T1 22







-T
111




-Y c-11 4- T -5




















Figure 4

Analog Computer Diagram of Model I
0
--00 (3-

Fiur
AnlgCmue iarmo oe


-au
11o









31



Subscripts continued

i = refers to the insulation

1 refers to lateral

m = refers to middle i.e. mg refers to middle generators

R = refers to radiation (See equation III-7)

t = refers to top i.e. ti refers to the top insulation

Example

Hti.li refers to the heat transfer coefficient between the

top insulation and the lateral insulation.


D. Analog Computer Solution

The preceding equations in Table 1 can each be solved for the

derivative of temperature with respect to time, and they constitute a

system of ordinary first-order linear differential equations which is

readily amenable to solution with the aid of an analog computer. The

EASE 1032 Analog Computer of the Chemical Engineering Department at

the University of Florida was utilized in this investigation.

The basic element of the analog computer is the electronic

amplifier which can serve as an integrator or a summer depending on

whether a capacitor or a resistor is connected across the amplifier.

Many references discuss in detail the theory and operation of analog

computers (36,53). The equations describing the model, were programmed

on the analog computer, and the analog computer circuit is shown in

Figure 4. Table 2 illustrates the algebraic form of the relations

between the parameters which contribute to each potentiometer setting.









32


TABLE 2


COEFFICIENTS FOR PROGRAMMED DIFFERENTIAL
DIFFERENCE EQUATIONS: MODEL I


Pot. No. Mathematical Expression Pot.Setting

1 1 H .t + hti li + Hti 0.663
p .C .h R. R 91
Spi ti
H
2 ti-g 0.123
pC h
g pg tg

3 1 H + 2ht H + H 0.744
p ti-g R g1 -c
g Cpghtg

4 Hti-g 0.286
PiCpihti

5 -c 0.143
pC h
c pc c

6 htg g-li 0.042
p.C .( o l)(ht + 3/2 h + h )
Spi- R ti tg c

7 ti Hti-li 0.0319

PiC pi( l)(hi + 3/2 htg + h)
SR t tg c

8 p1 H + cH l + H.. 0.2907
pcC Ch g- c-li -c

9 2 H +t H 0.126
gCpghtg LC R

AH h
11 c 0.0035
100
2H
12 2-c 0.123
PgCpg tg








33



TABLE 2 (Continued)


Pot. No. Mathematical Expression Pot. Setting

h H
14 c c-li 0.1395

R
Pp. ( /2)(h11 + 3/2 h + h )

15 (htg/2)Hg-li 0.021
p.C .( o l)(hti + 3/2 h + h )
R hi. tg c

17 H Ta 0.0037
17 ti-a a
Pipih ti

18 R/R(Hl-a T) 0.0063
18 o li-a a
lO1piC (O- 1)
R

19 c-li + 3/2 H -l tg + Hti-liti
.C (~o 1)(hti + 3/2 ht + h )
i tg c


Ro/R(ht + 3/2 htg + h)Hli-a 0.8644

PiC i(o l)(hti + 3/2 ht + h )


20 2ti-li 0.0107
Rp.C P
RPiCpi

21 2H-li 0.006
Rp C
g pg

2H 0.0046
22 c-li 0.0046
Rp cC
c pc









34



TABLE 2 (Continued)


Pot.No. Mathematical Expression Pot. Setting


23 H-li 0.003
Rp C
g pg

24 H -c 0.615
pC h
g pg tg

25 g-c 0.143
pC h
Pc pc c


31 Time Scale 0.050

32 Initial Rate of Heat Generation 0.050

34 Decay Rate of Heat Generation 0.0071

42 2 H + htg H 0.615-
g-c-R g-11
pC h R
g pg tg

43 2H-c-R 0.615
pC h
g pg tg

44 1 H + tg H l + Hg R 0.321"
44hI Htig g-11 g-c-R
pCh R
g pg g

45 g-c-R 0.715
p C h
c pc c

2h 1
49 1 H + c Hc + H 0.143



50 g-c-R 0.308
p C h
g pg tg

53 g-c-R 0.715
PcCpc h
c pc c









35



These factors are, of course, a direct consequence of the form of the

system of differential equations.

Table 3 lists the values of the physical parameters for all the

materials which constitute the "standard" battery being simulated. These

are the values used in obtaining the numbers shown in Table 2 for each

potentiometer setting in the simulation of the standard battery. In

other runs, the values of some of these parameters were changed

judiciously to investigate their effect on the performance of the

battery.

It has been pointed out earlier that a set of switches was

arranged to automatically change the value of the heat transfer

coefficient to account for radiation heat transfer at the heat generator-

cell interfaces when the heat generator temperature is above 7500C.

This temperature was chosen for the change from radiation to convection

heat transfer because it seemed reasonable and it gave good agreement

with the known activation times of certain thermal batteries. More-

over, it led to realistic cell peak temperatures.

The representation of the intra-cell heat generation by a

triangular heat of reaction term also required a number of switches to

provide an appropriate simulation on the analog computer. The details

of these switching arrangements are shown in Appendix A.

Also a manual switch was installed which permitted the inclu-

sion or exclusion of the heat of reaction due to the electrochemical

process in the cells.









36



TABLE 3


AVERAGE PHYSICAL PROPERTIES AND DIMENSIONS
OF THERMAL BATTERY COMPONENTS: MODEL I


Insulation

p, g/cc = 0.193

C, cal = 0.232
Sg.C

k, cal cm = 0.0002
g C cm.

h, cm. = 0.156

R, Cm. = 1.56

Ro, cm. = 1.72


Generator

p, g/cc = 1.25

C cal = 0.130
p 0
gC

k, cal cm = 0.0005
g C sq. cm.

h, cm. = 0.10

R, cm. = 1.56


Cell

p, g/cc = 3.48

C cal = 0.201
p
gC









37


TABLE 3 (Continued)


Cell continued

k, cal. cm. = 0.10
gC sq. cm.

h, cm. = 0.10

R, cm. 1.56









38


E. Discussion of Results


The great flexibility of the analog computer permitted the

investigation of a wide range of parameters and geometrical arrange-

ments which might affect the life of a thermal battery. A summary of

the more important results is given in Table 4 and they are discussed

below.

The model treated in this work has several limitations and some

caution should be exercised in attributing too much significance to

small effects. On the other hand, large effects are probably correct and

qualitative conclusions based on them should be sound. Probably the

most serious limitation of this model is that it is not based on conduc-

tion but on pseudo-convection heat transfer coefficients. Temperature

gradients within an element are thus precluded, and this is known to be

incorrect. However, the temperature gradients within an element do not

appear to be large (except perhaps for the first few seconds in the heat

generators), and the model appears to simulate the temperature histories

of elements in an actual battery rather well (see Figure 5).

It should be noted also that the complexity of the physical

models which can be studied on an analog computer is limited by the

capacity of the computer, and the time required to obtain meaningful

results is greatly increased when the model is made more complex

because there are more components and all of them (amplifiers, poten-

tiometers, capacitors, etc.) must perform satisfactorily for the

results to be valid. The problem which was here programmed on the

























400 -






300






. 200






100 i-i
0 20 40 60 80
t, sec. .

Figure 5 Experimental Versus Computed Results
*









40


TABLE 4


EFFECT OF PARAMETERS AND CHANGES ON THE LIFE
TO 4000C OF A SIMULATED,THERMAL BATTERY


Heat Sink Temperature

-650F 250C 160F 1600Life 25 Life
-650Life 25-Std.

Sec. Sec. Sec. Sec. Dimen-
sionless

A. Insulation

1. Thermal Conduc-
tivity

0.0001 64 87 97 1.52 1.61
0.0002 (Std.) 25 48 60 2.40 1.00
0.0003 18.5 32.5 41 2.21 0.68

2. Thickness

2X* 40 54 65 1.62 1.12

B. Heat Generators

1. Thickness

2X 48 1.00

2. Enthalpy

p = 2X, h = 1/2 X 75 1.57

C. Electrochemical Cells

1. Thickness

1.2X 54 1.12
1.5X 66 (4 sec. actv.) 1.38
2X 100 (17 Sec. actv.) 2.08


X = times standard value of parameter being varied









41



TABLE 4 (Continued)


Heat Sink Temperature

-650F 250 1600F 1600Life 250Life
-650Life 25uStd.

Sec. Sec. Sec. Sec. Dimen-
sionless

C. Electrochemical Cells
(Continued)

2. Enthalpy (pC )

1.2X* 38 0.79
0.8X 35 0.73

D. Intra-Cell Heat
Generation

1. Level

None 30 0.63
Std. 48 1.00
1.5X 60 1.25
3X 100 2.08

2. Rise Rate

0.5X 45 0.94
Std. 48 1.00
2X 48 1.00

3. Decay Rate

0.5X 52 1.08
Std, 48 1.00
1.3X 44 0.92



X = times standard value of parameter being varied









42



EASE 1032 Analog Computer represented essentially the limit of the

capabilities of the instrument.

1. Temperature Histories in a Standard Simulated Battery

The temperature histories of five elements in the standard

simulated thermal battery are shown in Figure 6. These elements are

(1) top heat generator, (2) center heat generator, (3) cell, (4) top

insulation, (5) lateral insulation.

From Figure 6 it can be seen that both of the heat generators

release their heat rapidly to the cell and the top insulation. Within

five seconds after activation the temperature of the heat generators is

below the cell temperature, but the center heat generator is only

slightly cooler than the cell. Thus, it can be said that the cell

is heating the generators after the first few seconds.

The cell is shown to reach a temperature of 400 C in less than

0.5 seconds, but little importance should be attached to this because

the response of the recorder was not particularly good for times of less

than one second. Figure 6 indicates that the cell reaches a peak

temperature of about 5500C in approximately five seconds which is in

good agreement with experience. All of the curves in Figure 6 are in

reasonably good agreement with what one would expect for a heat transfer

system of the type under consideration.

2. Effect of Heat Sink Temperature on Cell Temperatures

As one might expect, the temperature history of a cell is strong-

ly dependent on the heat sink temperature. The data summarized in

Table 4 indicate that the cell life above 4000C is approximately twice




43





1000-





875




Heat Sink Temperature 250C

750-



Top Generator

625


5500C
0


50 Cell
500


P /Middle Generator


375- ... 4000C
375-





250-
0Top Insulation



Lateral Insulation

125-

/




0
0 10 20 30 40 50 60
Time, sec.
Figure 6 Temperature Histories in Standard Simulated Thermal Battery









44



as great when the heat sink temperature is +1600F instead of -650F.

Room temperature results are intermediate. Figure 7 illustrates the

temperature histories for a standard cell at the three heat sink

temperatures.

3. Effect of'Insulation Parameters on Cell Temperatures

Thermoflex insulation has a thermal conductivity of about
-1 o -1 -2
0.0002 cal. cm. g C cm and this has been adopted as the stan-

dard insulation type. The thermal conductivity of asbestos is around

0.0003. In order to assess the effect of a large improvement in insula-

tion properties, one run was made with a hypothetical insulation having

a thermal conductivity of 0.0001. These results are tabulated in Table

4, and it may readily be seen that thermal conductivity of the insula-

tion is an important factor in the life of the cell. A comparison of

the life above 4000C at a 250C heat sink temperature shows that a 50%

variation in life might be expected with the sort of variation in thermal

conductivity studied. Figures 8 and 9 show the curves for cell tempera-

tures at these conditions.

Table 4 also shows that doubling the insulation thickness (for

Thermoflex) results in relatively little increase in life except at a

heat sink temperature of -650F. Where life is now minimal at low heat
/
sink temperatures, increasing the insulation thickness would appear to

be a quite promising means of increasing cell life. These curves are

shown in Figure 10.

4. Variation of Heat Generator Parameters

Two heat generator parameters were studied: (1) a variation in





45





750





625


__5500C


500
Heat Sink 1600F

Heat Sink 250C -
0

4000C
375 -
I- -----Heat Sink -65F


S Heat Sink -650F

250 No intra-cell chemical
heat generation





125






0-
0 -1 --------------------------------------------- --I ---
0 10 20 30 40 50 60
Time, sec.
Figure 7 Effect on Cell Temperature of Changing the Temperature of the Heat Sink





46




750






625
625 k= 0.0001


5500C


500


o0 o Heat Sink 1600F

4000C
375

Heat Sink 25 C



Heat Sink -650 0
250





125





0-
0 10 20 30 40 50 60

Time, sec.
Figure 8 Effect on Cell Temperature of Reducing the Thermal Conductivity of the Insulation





47






750



ki= 0.0003

625

5500C



500



0 4000C
__ ______ 400C_________4__

375 Heat Sink 1600F
S375

S/ Heat Sink 250C



250 Heat Sink -65





125





0
0 10 20 30 40 50 60
Time, sec.
Figure 9 Effect on Cell Temperature of Increasing the Thermal Conductivity of the Insulation





48






750



h ti 2X

625


S5500C


500
at Sink 1600F

0 Heat Sink 250C


-4- 400 c
375 Heat Sink

-Heat Sink -65F
Heat Sink -650F

250 No intra-cell
chemical heat generation





125






0 10 20 30 40 50 60

Time, sec.
Figure 10 Effect on Cell Temperature of Ghanging the Thickness of the Insulation









49



the thickness of the heat generators, and (2) a variation in their

enthalpy.

The effect of doubling the thickness of the heat generators is

illustrated in Figure 11. The net effect on cell life appears to be

quite small. Although doubling the thickness of the heat generator

increases the heat available for the cells,the consequent increase in

the size of the core requires the same cell to heat a larger volume

which more than offsets the gain in heat. It should be recalled that

the cells serve as the heat sources after the first few seconds (see

Figure 6). Thus there is a slightly higher peak temperature, arrived

at later than in the standard arrangement because of the longer heat

path in the generator, but the cell cools more rapidly than the

standard once cooling starts because the area of the core is larger

and thus heat losses are greater. The net effect of the change is

essentially zero as far as the cell life above 400C0 is concerned.

If the standard heat generator is compressed the heat genera-

tion per unit volume of generator increases. In Figure 12 the effect

of compressing the heat generator to one-half of its original thickness

is portrayed. It may readily be seen that the peak temperature and rise

time of the cell are not greatly affected. However, the rate of heat
/
loss of the cell is greatly reduced as compared to the standard arrange-

ment owing primarily to the reduction in core area and an increase

in cell life over 4000C of about 60% is observed.

5. Variation of Cell Parameters

Three cell parameters were varied: (1) cell thickness, (2) cell





50





750





625


550 C

htg= 2X
500 /--
00 / Standard

o, C



c 375





250





125





0
0 10 20 30 40 50 60
Time, sec.
Figure 11 Effect on Cell Temperature of Changing Thickness of Heat Generators






S51






750





625


5500C
O =1/2 X, = 2X
g g
500
500 -Standard



oaa _______--- 4000C

S 375


cu


250





125






0 -
0 10 20 30 40 50 60
Time, sec.
Figure 12 Effect on Cell Temperature of Compressing Heat Generators









52



enthalpy, and (3) intra-cell chemical heat generation. The third of

these parameters is treated in a separate subsection because several

factors involved were studied.

Assuming that the standard components of a cell are used regard-

less of thickness, it is obvious that the enthalpy of a cell is propor-

tional to its thickness. One may also observe that the heat transfer

paths are also lengthened for both heating and cooling, and one would

expect a thick cell to both heat and cool more slowly than a standard

one. The effect of varying cell thickness is illustrated in Figure 13,

and one does, indeed, observe these effects. Since the heat input is

constant (except for the intra-cell chemical heat generation built in),

the peak temperatures decrease as the cell thickness increases; however,

the intra-cell heat generation begins to contribute more heavily as cell

thickness increases and when the thickness is increased by 50% the peak

temperature is actually determined by the intra-cell heat generation.

Thus the peak temperatures are reached a fairly long time after activa-

tion in these cases.

The activation times of thick cells are also increased, and be-

come prohibitive for very thick cells. It appears that a thickness

increase of no more than 50% can be tolerated unless activation times

of more than 20 seconds are permissible, or unless other geometric

arrangements are used.

It is possible in principle to add to the cell materials which

can change its enthalpy. For example, a material having a transition in

the temperature range of interest might be added. It is obvious that






53






750






625


5500C

Standard
500 h = 1.2X

ic ------_
o0 hc= 1.5X
(U I

u 375
Ph= 2X





250






125







0 10 20 30 40 50 60
Time, sec.
Figure 13 Effect on Cell Temperature of Changing the Thickness of the Cell









54


the heat of fusion of the electrolyte in the cell cannot be of any

assistance for the performance of thermal batteries must suffer badly

when the temperature approaches the freezing point because of the change

in the electrolytic conductivity of the electrolyte. Therefore, this

factor represents a hypothetical change in parameters which would merit

serious investigation if it should appear to contribute strongly to cell

life.

In Figure 14 the effect of a 20% change in cell enthalpy is

demonstrated. The most obvious effect is that on peak temperature.

Substantial decreases in the cell enthalpy while maintaining the same

heat input from heat generators would result in overheating of the

cell. A substantial increase in cell enthalpy with no change in heat

input would result in the cell just barely becoming activated. Clearly

the only practical approach would be to adjust heat generator input

to the enthalpy requirements of the cell.

Since the cells act as the primary heat reservoir after the first

few seconds of operation, it is clear that increasing the cell enthalpy

should be beneficial everything else being the same. This, in fact, is

seen to be the case in Figure 14, where the slopes of the cooling por-

tions of the curves are in proportion to the cell enthalpy. Thus, a

combination of changes in cell enthalpy, by means of domposition or

thickness changes, and in the heat generator by similar means would

appear to offer possibilities in the way of meeting varying specifica-

tions of time of activation and life.

Another possibility appears here, namely construction of duplex

cells and heat generators, which have a portion of each made very thin





55




750









500 Stn .5500C


500 / Standard

pcC P 1.2X

-__ _-~--- ____ __ ____- 4000C
375





250





125





0
0 10 20 30 40 50 60

Time, sec.
Figure 14 Effect on Cell Temperature of Changing the Enthalpy of the Cell









56


for fast activation, and a larger portion which activates slowly but

serves as a heat reservoir to prolong life. Unfortunately the capa-

city of the analog computer did not permit a problem of this complexity

to be studied.

6. Effect of Rate and Level of Heat Generation by Chemical Reactions
in Cells

It has been noted earlier that the reactants in the electro-

chemical reaction can also react chemically to produce heat but no

electricity. While it might appear that any such reaction would be

wasteful, it turns out not to be so since the major limitation on cell

life appears to be heat losses rather than exhaustion of reactions, and

these intra-cell chemical reactions generate heat at a point where it

is most effective in keeping the electrolyte molten.

The effect on all temperature histories of these chemical

reactions is illustrated in Figure 15 for a heat sink temperature of

250C. It is seen that the peak temperature of 5500C is reached at about

4 seconds, and that the life of the cell above 4000C is about 48 sec.

when the normal heat generation is used.

A word about the heat generation is in order. The general shape

of this function (which is approximate, of course) is also shown in

Figure 15. The shape and the average rate of heat inppt to the system

are based on experiments performed earlier at the Energy Conversion

Laboratory of the University of Florida, and on educated guesses of

experienced investigators familiar with the design and operation of a

number of types of thermal batteries. In this study the normal rate of
3
heat generation was chosen to be 15 cal per cm per sec because of the





57






750





625


5500C


500

= 1. 5X
0o_----_ Standard

a3 ~~---^_ ~ ~- ~-- __ ~ ~ __ /n~
375 AH = 0.75X ---5
No heat of reaction




250-





125- Form of the intra-cell chemical heat generation







0 o o0 20 4 5'0 60
Time, sec.
Figure 15 Effect on Cell Temperature of Changing the Magnitude of the Intra-Cell Chemical Heat Generation









58



cell chemistry assumed. The heat generation rates of other cell

reactions is covered by the range of heat generation terms used.

From Figure 15 it may be seen that the peak temperature and the

time to reach it are not strongly dependent on the value of the cell

heat generation term unless very energetic and extensive chemical

reaction occurs. A change in the heat generation term of 50% appears

to change the life above 4000C by about 20% without exceeding allowable

peak temperature. Figure 15 also indicates that a heat generating

reaction producing heat at approximately three times the rate in the

normal situation would lead to a relatively small increase in the

peak temperature, but it would result in a delay to reach the peak

temperature of approximately 30 seconds and would result in approximately

twice the life above 4000C.

Figure 16 indicates the shape of the heat,generation functions

programmed in these experiments. The peak heat generation rate and

the decay rates were held constant and the rise rate varied from one-

half to twice the standard rate. The data in Table 4 indicate that

this change produced essentially no change in'the cell temperature

history. Activation times and peak temperatures turned out to be

essentially unaffected.

This kind of effect resulted, however, because the heat genera-

tion term chosen as a standard is such as to affect only the cooling

portion of the temperature history of the cell in any significant way.

If, for example, a system employing an intra-cell chemical heat genera-

tion rate more than twice as large as the standard should be studied, a




59






60





Slope 2X
50 // --





40
/Standard




r4 30
> / Slope = 1/2 X





20



S/ /
/
10 -




i/

0 10 20 30 40 50 60
Time, sec.

Figure 16 Change of Initial Rate of Intra-Cell Chemical Heat Generation Term









60


much more pronounced effect on the temperature history of the cell would

be noted.

Figure 17 illustrates the kinds of changes in decay rate studied.

Here the rise rate and peak generation rate were held constant and the

decay rate varied from one-half to 1.3 times that of the standard. It

is clear from examination of time-temperature curves of a cell (not

presented here) and of the data in Table 4 that only a small effect is

noted. There is no effect on activation time and peak temperature, but

the cooling rate of the cell is affected slightly.

In summary, it does not appear that alternation of either rise

or decay rates of the intra-cell chemical heat generation function

within broad limits would strongly affect the temperature history of a

cell. However, the level of this type of heat generation, i.e., a

change in the rate of chemical heat generation per unit volume could

have a significant effect.






61





60





50 -~

i Slope = 0.5X



40 /
Standard




30 / Slope = 1.3X
30 .





20





10
/'l







0 -
0 10 20 30 40 50 60

Time, sec.
Figure 17 Change of Decay Rate of Intra-Cell Chemical Heat Generation Term













CHAPTER IV


ANALOG COMPUTER MODEL II


A. Description of the Model


The next phase of the investigation concerning dynamic heat

transfer in thermal batteries called for a study of the effect of the

nature and structure of the insulation on the performance of thermal

batteries. It should be obvious that Model I, which was discussed in

the preceding chapter is inadequate to simulate an insulation region

composed of more than one insulating material.

This model was developed primarily to be able to study the

effect of different combinations of insulating materials on the battery

performance. In order to accomplish this simulation and at the same

time keep the problem within the practical capabilities of the avail-

able analog computer (EASE 1032 Analog Computer), a number of simpli-

fying assumptions were made.

The most significant simplifying assumption of this model as

compared to Model I is that the cell-generator stack is considered to

have a homogeneous core having the average physical properties of the

electrochemical cells and heat generators. Furthermor'e, this homogen-

eous core is assumed to be well-mixed with no temperature gradients.

Therefore, only one ordinary differential equation describes the heat

balance considerations in the core.

This model is more complicated than Model I in that the insula-

tion is considered to consist of four elements rather than one (see


62









63



Figure 18). This model makes it possible to consider a core surrounded

by four successive layers of different thermal characteristics, each,

of course, having no internal temperature gradient, and having the

thermal resistance lumped at the interfaces as in Model I.

Another important difference between the two models resides in

the manner in which the battery activation is achieved. In the case

of Model I it was assumed that the heat generators constitute an

essentially instantaneous heat source, and, in order to make the set

of assumptions consistent, it was postulated that the initial tempera-

ture of the heat generator was 22000C, which is the approximate tempera-

ture of the center of the heat generators. The temperature gradient

between the heat generator and the cell was the driving force which

promoted the heat transfer.

In the case of Model II, since only one element is assumed to

simulate the cell-generator stack, it is meaningless to consider core

temperatures of the order of 22000C, first because the enthalpy of the

generators does not permit such average core temperatures and secondly

because there exists a high-temperature limit of about 55000C above

which the electrochemical system in the cells undergoes undesirable

transformations (3). Therefore, after examination of the temperature

histories of cell and generators as shown in Figure 6,'where it was

observed that the temperature of cell and generators was essentially

the same only a few seconds after activation and that it remained so

for the rest of the operating period, it was decided to assume for the

core a heat generation term equivalent in calories to the enthalpy of the









64





















CORE 1 2 3 4

















igure 18 Schematic Diagram of Battery Described by Model II


/









65



hot generators of Model I. This heat generation term was assumed to be

an exponential decay function which was judged to be an adequate simula-

tion of the sudden energy burst of the'heat generators.

Another difference between the two models is that in Model I

the boundary between top and lateral insulation was considered to be

vertical and in this model, the boundary between the corresponding top

and lateral elements in the insulation zone was assumed to be an

imaginary surface cutting diagonally from the outer corner of the core

to the outer corner of the assembly. This arbitrary shape assumed for

the insulation elements has some rather interesting consequences in

the form of the coefficients of the differential-difference equations

which describe the system. This point will be discussed in more detail

later.

One final simplification of this model relative to Model I was

the elimination of the heat of reaction term due to the cell. This was

done mainly because the effect of the intra-cell heat of reaction had

been evaluated in Model I, and because its inclusion would have complica-

ted this model without contributing appreciably to the qualitative

differentiation between different insulation arrangements.


B. Development of the Mathematical Model

It is shown in Appendix B that when the boundary surface

between corresponding top and lateral insulation elements is assumed

to have the shape illustrated in Figures 18 and B-lb, the volume of the

top insulation element is given by









66


VTi = (R +RR + R) (III-1)
Ti 3 1i+1 + i+l i

where V is the volume of the element, h its height, and R is the radius.

The common area of contact between the top and lateral elements is

given by the expression

A = 7h(R + Ri ) (111-2)
TL1 i i+l

and the volume of the lateral insulation element is given by

V = h(2R2 + R R) + n7(R 2 R )S (111-3)
L1 3 i+l i i+1 R+1+ R

where S1 is the height of the element as shown in Figure B-lb.

The heat transfer coefficients were calculated in a manner

identical to the one utilized in Model I, that is, by assuming the

thermal resistance between two adjacent elements to be lumped at the

interface. This resistance was evaluated by consideration of the mean

heat path between two adjacent elements and their respective thermal

conductivities. The numerical values of the heat transfer coefficients

for Run 15 are shown in Appendix B.

The differential-difference equations were developed by heat

balance considerations of each of the elements making up the simulated

battery. If the following substitutions are utilized for the purpose

of simplification, the differential-difference equations which describe
/
the system take the form shown in Table 5.

F. (R2 + RR. + (111-4)
1 3- i+1 i 1+1 i

and

G C ih(2R RR R ) + 2Cp 2 (R (II-5)
i "3 i+1l i i+l -1









67



C. Analog Computer Solution

The equations shown in Table 5 were programmed on the analog

computer and the circuit is shown in Figure 19.

The potentiometer settings corresponding to Run 15 are listed

in Table 6 together with the corresponding mathematical expression

for each potentiometer. The values of the physical parameters used

in these simulations are shown in Table 7.

It should be noted that Amplifier 17 in the computer diagram

represents the exponential decay type of heat generation attributed

to the core. This exponential function was designed so that the inte-

gral over time of this relation, namely the total amount of heat

generated by unit volume of core, resulted in a core peak temperature

near the maximum chosen. The value which produced the most realistic

core peak temperature (5250C) corresponded to 353 ,cal./g of heat

generator in the core (see Appendix B).

Because of the high initial rate of change of temperature in the
-.4
core and the nature of the simplifying assumptions of this model, it was

realized that the model could not predict these temperatures accurately

during the first few seconds of operation. In any case, the primary con-

cern in this study was with the cooling period, and with the effect of

changes in the thermal properties of the materials and'the geometric

arrangement of the assembly on the cooling rate.

D. Discussion of Results

This model is a variation of Model I and it is based on

essentially the same assumptions as the latter; therefore, many of

the considerations discussed in the preceding chapter are applicable












- L3 1 T2 i
1-T

S4 2 5 -T
-T (

-L -L
-- 3---- 4_ 12







_T3 7 1 5 T -T 2 Q2 -T














---7i -C 11i 14 4---

-8 2 -A
1 -AC23








1- -29 19
-L2-L






















A 32 T
L-L



















-C 1 I.C. = 40.0 Volts













Figure 19
1Analog Comuter Diagram of Model II
-TA Cmt \ o \








69


TABLE 5

DIFFERENTIAL-DIFFERENCE EQUATIONS FOR
MATHEMATICAL MODEL II

Top Element 4

R5 4A R 243 (R + R )h dT4
S2---(T4 A) (T4 T3)- F (T4 L4)= d-
F4 4 ) 4 34 F4 T dt

Top Element 3

R4HT43 R3 HT32 (R + R3)h3 dT3
F (i 3 T 4) F (T3 T2) F HTL3(T3 L3) \ dt
3 3 3

Top Element 2

R3 HT32 R2 HT21 (R3 + R2)h dT2
F 2 F2 3)2 HTL2(T2 L2) dt

Top Element 1
2R2
R2 T21 R 1 HTIC (R2 + R1)h1 dT1
F1 (T1 T2) F (l c) F2 HTL1(T1 L )= dt

Core

TIC 2LIC AH dC
CpcS 1 T 1) Pcpc PcCpe
Lateral Element 4

2R5S5LAG 2R4S4HL43 (R5 + R )h4, dL4
S 4 (L A)- G (L4 L3 G L4(L T) = dt

Lateral Element 3

2R4 4143 2R33HL32 (R4 + R3 )h
G3 (L3. L)- (L3 L2)- G3 HTL3(L3 3)

= dL
dt








70


TABLE 5 (Continued)

Lateral Element 2

2R 3S3HL32 2R2S2 HL21 (R3 + R2)h2
(L2 L- G (L- ) 3 + (L-T
2 2 2 L3) 2 (L2 L G2 TL2(L2 2
dL2
dt
Lateral Element 1

2R2S2 21 2R1LIC (R2 + Rl)h
G1 (L1 L2 G (L1 C -- G L(L1 T1)
dL1
dt








71

TABLE 6

COEFFICIENTS FOR PROGRAMMED
DIFFERENTIAL-DIFFERENCE EQUATIONS: MODEL II

Pot. No. Mathematical Expression Pot. Setting

1 R52HT4A + R2HT43 + (R5 + R4)h HTL4 26.35
F4


F4
2


F4
5 RH43 1.824
F

6 (R5 + R4)h HTL 0.086
F4

2 R42 T43 + R2HT32 + (R + R3)h3HTL3 34.31




F3
2





12 (R4 + R3)hHTL3 0.0014
F3

3 R32HT32 + R22 HT21 + (R3 + R2)h2HTL2 0.2314
F2

15 R32 T32 0.1762
F.. .









72


TABLE 6 (Continued)


Pot. No. Mathematical Expression Pot. Setting

16 R22 H21 0.0544
F2

17 (R3 + R2)h2HL2 0.00075
F2

8R22HT21 + R2T1C + (R2 + Rl)hlHTL 1.155
F1

22 R22HT21 0.635
F1

23 R12HT1C 0.516
F

24 (R2 + Rl)hlHTL1 0.0040
F1

9 I1C 2H 0.0121
pC S R p C
Pc pcS1 1Rc pc

27 AH 0.1

25 H'IC 0.00357
pC S
Pc pc 1

28 2HL1C 0.0085

R1P c pc

29 AH 0.500
pc
c PC

19 (R5 + R4)h4HTL + 2R5S5HLA4 + 2R S HL43 26.31
G








73


TABLE 6 (Continued)

Pot. No. Mathematical Expression Pot. Setting

31 (R5 + R4)h 4HT 0.043
G4

32 2R5S HLA4 24.44
G4

33 2R4S4HL43 1.824
G4

20 (R4 + R3)h3HTL3 + 2R S4HL43 + 2R3S3HL32 34.31
G

35 (R4 + R3)h3HTL3 0.0007
G3

36 2R4S4HL43 32.94
G3

37 2R3S3HL32 1.371
G3

21 (R3 + R2)h2HTL2 + 2R3S3G + 2RSHL21 0.2310
G2
02
39 (R3 + R2)h2HTL2 0.00038
G2

40 2R3S3HL32 0.1762
G2
~2

41 2R2S2HL21 0.0544
G2









74


TABLE 6 (Continued)


Pot. No. Mathematical Expression Pot. Setting

26 (R2 + Rl)hlHTL1 + 2R2S2L21 + 2R1S1HLIC 1.153
GI

43 (R2 + RI)hlHTL1 0.0020
GI
1

44 2R2S2HL21 0.635
G1

46 2R1SIHlC 0.516
GI



















i


)I *









75


TABLE 7


AVERAGE PHYSICAL PROPERTIES AND DIMENSIONS
OF THERMAL BATTERY COMPONENTS: MODEL II


Core

p 2.36 gms/cc

C 0.183 cal/gmC
P

k 0.05 caL cm
cm C sec.

Mica

p 2.7

C 0.206
P

k 0.001

Asbestos

p 1.5

C 0.308
P

k 0.0004

Thermoflex

p 0.193

C 0.232

k 0.0002

Metal
Steel

p 7.9

C 0.12
P
k 0.11









76



here. The size of the available analog computer was again an

important limitation which restricted the complexity that could be

built into the model. In spite of this.limitation, it was decided

to continue with the analog computer study because it was believed that

it could lead to qualitative information which would be of value in

more precise studies utilizing a digital computer. Thus some valuable

qualitative information was obtained which considerably facilitated

the development of the digital computer model. The results of the

analog simulation of Model II are summarized in Table 8.

1. Geometrical Shape of Insulation Elements

The geometrical shape assumed for the insulation elements in

Model II is really no more arbitrary than.that assumed for Model I

but it has two advantages.

1. Compared to the shape assumed for the insulation elements

in Model I, the shape used for the insulation elements in Model II

reduced by one the number of elements -with which each insulation element

was in contact, thus considerably simplifying the differential-difference

equations of the system with the obvious consequence of a simplifica-

tion in the analog computer circuit.

2. The shape assumed for the insulation elements in Model II
/
yields expressions for the area and volume of the corresponding top

and lateral insulation elements which illustrate the similarity of the

radial and axial modes of heat transfer in a cylindrical structure

described by this approximate model (see Appendix B). The similarity

lies in the fact that the assumed shape yields values for the area and









77



TABLE 8


EFFECT OF CHANGES IN THE INSULATION STRUCTURE ON LIFE
TO 4000C OF SIMULATED THERMAL BATTERIES



Structure
Life at Heat Sink 25Life
Run No. Element No. Material Temp. of 25?sec. 250Std.

15 4 Metal (Standard) 52 1.00
3 Mica
2 Asbestos
1 Thermoflex

16 4 Thermoflex 25 0.50
3 Mica
2 Asbestos
1 Metal

17 4 Asbestos (New Stan- 95 1.00
3 Asbestos dard)
2 Thermoflex
1 Thermoflex

18 4 Thermoflex 90 0.95
3 Thermoflex
2 Asbestos
1 Asbestos

19 4 Asbestos 97 1.02
3 Thermoflex
2 Thermoflex
1 Thermoflex

20 4 Thermoflex 130 1.37
3 Thermoflex
2 Thermoflex
1 Thermoflex









78


volume of a lateral insulation element which are twice the values

obtained for these quantities in the corresponding top insulation ele-

ment. Thus these elements have the same ratio of area to volume and

the equations describing the differential heat balance in these elements

have very similar coefficients, which is equivalent to very similar

potentiometer settings for corresponding potentiometers as shown in

Table 6 for the case of potentiometers 1 and 19, 4 and 32, and many

others. Figure 23 illustrates the almost identical temperature

histories of the corresponding top and lateral insulation elements

in Model II as a result of the above discussed similarity in their

descriptive equations.

These results provided some justification for the assumption

of spherical symmetry utilized in the digital computer model.

2. Simulation of Standard Thermal Battery

The standard thermal battery which, in the framework of this

model, consisted of a cylindrical core surrounded in order by Thermo-

flex insulation, asbestos insulation, a mica layer and a metal can,

was simulated and the results are shown in Figure 20. A life above

4000 C of 52.5 seconds was indicated. This figure is in reasonable

agreement with experimental results.

One significant aspect of this run was that the mica and metal

layers did not rise appreciably above the temperature of the heat sink.

In view of these results, the mica and metal layers could be

considered extensions of the heat sink. This assumption liberated a

section of the analog computer which could be used to simulate in more






79






S750 Configuration
4 = Metal
ll3 = Mica
I 2 = Asbestos
1 = Thermoflex
625

5500C

Core

500




*4 --I------- ---- 4000C
1, 375




Thermoflex
250-------





125 -
SAsbestos


Mica and Metal

0-
) 10 26 30 40 50 60
Time, sec.
Figure 20 Temperature Histories in Standard Simulated Thermal Battery










80


detail the heat transfer process in other sections of the battery.

3. Effect of a Metal Layer Next to the Core on Core Temperature

Some considerations indicated-that it might be advantageous

from the standpoint of maximization of battery life, to place the metal

can next to the core instead of on the outside of the assembly. The

intuitive justification of this arrangement was based on the high heat

capacity of the metal. It appeared that the metal would absorb the

heat given off by the core and act as a buffer between the core and

the insulation. The results of a run testing this idea clearly

revealed the inefficiency of this arrangement which clearly yielded a

lower battery life than the standard as shown in Figure 21. The reason

for this result is that the metal layer is essentially an addition of

inert material to the core (inert in the sense that there is no heat

generation associated with it)which only serves to increase its heat

transfer area and hence the rate of heat loss from the core to the

insulation.

4. Effect of Varying Insulation Arrangements on Core Temperature

The negligible temperature rise of the outer mica and metal

layers discussed before for the case of the standard battery allowed a

more precise simulation of the effect of different insulation arrange-

ments on the life of the battery. The number of well-mixed elements

used to simulate the insulation space, which in the case of the standard

battery is composed of a layer of Thermoflex and a layer of asbestos,

was doubled. This allowed for the existence of a step-type temperature

gradient in the simulation of an actual layer of homogeneous material




81





750
Configuration
4 = Thermoflex
3 = Mica
2 = Asbestos
1 = Metal
625


55000


500



cu I4000C
0 Core0


0 375
I./





250
L2
L3


125 L4





0
0 10 20 30 40 50 60
Time, sec.
Figure 21 Temperature Histories in Simulated Thermal Battery with Metal Next to Core









82


such as Thermoflex. Figure 22 illustrates this effect on the simula-

tion of the standard thermal battery. The results shown in this figure

indicate that the mathematical model was very sensitive to this type

of change. The standard battery life was almost doubled as a result

of inaccuracies introduced by the new mode of simulation. It should

be noted that battery life is a very stringent test for agreement be-

tween two models because of the low rate of cooling of the battery which

makes small changes in the rate of cooling have a large effect on

battery life.

The significant discrepancy between the lives of the two

simulations of the standard battery is believed to be a consequence of

the assumptions of the model. The heat transfer coefficients

utilized in the model are calculated on the basis of the thermal resis-

tance of the two adjacent elements. The introduction of extra insula-

tion elements reduces the thickness of the element in contact with the

core and makes the value of the core-insulation coefficient more

dependent on the core. Therefore, in a sense it may be stated that

increasing the number of insulation elements decreases the relative

approximation of the core results. Since in this case, the core was

being represented by only one element already having by far the greatest

volume of all the elements present,the intended improvement in the

simulation of the standard battery, actually resulted in a poorer

approximation.

In spite of this disagreement between the two simulations of the

standard battery, a valid qualitative comparison between different




83





750-
Configuration
4 = Asbestos
3 = Asbestos
2 = Thermoflex
1 = Thermoflex
625


5500C


500

Core

S---4000 C

C 375

|I 37 ---_ ~~ ~L1



250 L2





125 -./ __ L3


L4


0 1I
0 20 40 60 80 100 120
Time, sec.
Figure 22 Temperature Histories in Simulated Thermal Battery with Varying Insulation Arrangement









84



insulation arrangements should exist using, of course, the new

standard as the basis for the comparison. Figure 23 shows the effect

of interchanging the asbestos and Thermoflex layers to be detrimental

to the life of the battery. Figure 24 shows that a slight increase in

battery life is obtained when three layers of Thermoflex and only one

outer layer of asbestos are utilized. This result indicates the

desirability of having as much Thermoflex as possible adjacent to the

core. Figure 25 illustrates the obvious arrangement which results as

a consequence of the data in Figure 24, this is, an all Thermoflex

insulation. In connection with this figure it should be noted that,

although it predicts the longest life of all the arrangements tested,

the magnitude of the predicted life is clearly in disagreement with the

trend established in the preceding figures. The reason for this is

probably experimental error in setting the numerous potentiometers

in the analog circuit, and the fact that this particular experiment was

carried out on a different occasion which means that all the potentiome-

ters had to be set, as opposed to the other three arrangements which

were run on the same day. For these runs only a few potentiometers had

to be changed to affect the simulation of a particular insulation

arrangement.




Il~i i!




85





750
Configuration
4 = Thermoflex
3 = Thermoflex
2 = Asbestos
1 = Asbestos
625


5500 C


500

Core
0
W) 4000C

S375 -
^ '------ _________^ Core


a)

L2


250
L3




125 -
T4 and L4





0 20 40 60 80 100 120

Time, sec.
Figure 23 Temperature Histories in Simulated Thermal Battery with Varying Insulation Arrangement




86





750 -
Configuration
4 = Asbestos
3 = Thermoflex
2 = Thermoflex
1 = Thermoflex
625


5500C



500
Core


o4000C








250



L3

125

L4



0 -- -I- I I
0 20 40 60 80 100 120
Time, sec.

Figure 24 Temperature Histories in Simulated Thermal Battery with Varying Insulation Arrangement




87






750
Configuration
4 Thermdflex
3 Thermcflex
2 Thermcflex
1 Therm flex
625

5500C



500 Core


o 4000C


J 375





250





125 -







0 20 40 60 80 100 120
Time, sec.

Figure 25 Temperature Histories in Simulated Thermal Battery with Varying Insulation Arrangement













CHAPTER V


DIGITAL COMPUTER MODEL III


A. Description of the Model


The digital computer model which was programmed on the IBM 709

computer was that of a multi-layered sphere consisting of a well-

mixed spherical core surrounded by a core film in which heat conduction

is occurring; this, in turn, is surrounded by spherical shells of insula-

tion, and the whole is immersed in an infinite heat sink. This particu-

lar physical model was arrived at after careful consideration of the

results of the two analog computer models and the requirements of the

problem.

One important factor in the use of the digital computer for

solution of problems of this type is the matter of the convergence of

the approximate (digital) solution to the exact one. If this factor is

considered together with the dimensionality of the problem, that is, the

number of dimensions involved in the partial differential equation, it

may be seen that the amount of effort (time and/or money) necessary to

achieve an accurate solution, i.e., assymptotic approach of the digital

computer solution to the exact one, becomes prohibitive. Therefore,

simplifying assumptions must ordinarily be made which will lead to a

solution on the computer, at reasonable expense, and at the same time

conserve enough characteristics of the physical model so that the

results may be meaningful.

The result of the analog computer study on Model II (a right



88




Full Text

PAGE 1

DYNAMIC HEAT TRANSFER IN COMPOSITE MINIATURE STRUCTURES By MARIO ARIET ANTIGA A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA April, 1965

PAGE 2

ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to Professor Robert D. Walker, Jr., whose interest, advice, and criticism stimulated and guided this research program; to Dr. Herbert E. Schweyer for his guidance and advice throughout his career. He wishes to thank Mr. Henry R. Wengrow without whose assistance this work would not have been possible, Dr. Mack Tyner and Mr. Mario Padron for their helpful discussions and suggestions, and the members of his Supervisory Committee Dr. T. M. Reed, Dr. R. G. Blake and Dr. R. W. Kluge. A special appreciation is due Mr. Bruce T. Fairchild, Mr. H. R. Wengrow and Dr. F. P. May for the use of their AM0S program, and Mr. Roberto Vich for his assistance on the drawings. The author also wishes to acknowledge the financial assistance of the Harry Diamond Laboratories, Army Materiel Command, and its technical representatives Messrs. R. H. Comyn and Nathan Kaplan, who by their encouragement and support made it possible to conduct this investigation.

PAGE 3

TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES vi LIST OF FIGURES vii ABSTRACT. x CHAPTER I INTRODUCTION 1 II. THEORY OF HEAT TRANSMISSION 8 A. Geometrical Considerations in Battery Life Optimization 16 III ANALOG COMPUTER MODEL 1 22 A. Description of the Model 22 B. Heat Transfer Coefficients 24 C. Development of the Mathematical Model 27 D. Analog Computer Solution 31 E. Discussion of Results 38 1. Temperature Histories in a Standard Simulated Battery 42 2. Effect of Heat Sink on Cell Temperatures 42 3. Effect of Insulation Parameters on Cell Temperatures 44 4. Variation of Heat Generator Parameters 44 5 Variation of Cell Parameters 49 6. Effect of Rate and Level of Heat Generation by Chemical Reactions in Cells 56 in

PAGE 4

TABLE OF CONTENTS (Continued) Page CHAPTER IV. ANALOG COMPUTER MODEL II 62 A. Description of the Model 62 B. Development of the Mathematical Model 65 C. Analog Computer Solution 67 D. Discussion of Results 67 1. Geometrical Shape of Insulation Elements 76 2. Simulation of Standard Thermal Battery 78 3. Effect of a Metal Layer Next to the Core on Core Temperature 80 4. Effect of Varying Insulation Arrangements on Core Temperature 80 V. DIGITAL COMPUTER MODEL III .' 88 A. Description of the Model 88 B. Development of the Mathematical Model 90 C. Finite Differences Approximation 94 D. Development of the Computer Program 94 E. The AM0S Program 97 F. Discussion of Results. 99 1. Simulation of Standard Thermal Battery 101 2. Effect of Varying Insulation Arrangements on Core Temperature 104 3. Effect of Idealized Insulating Materials on Core Temperature 108 4. Effect of Core Radius on Core Temperature..... 109 5. Effect of Intra-Cell Heat Generation..... 114 IV

PAGE 5

TABLE OF CONTENTS (Continued) Page CHAPTER 6. Effect of Heat Sink Temperature on Core Temperature 114 7. Effect of Changes in the Heat Capacity of the Heat Generators on the Core Temperature.. 114 8. Core Temperature of Improved Thermal Battery.. 118 9. Effect on Core Temperature of Delayed Heat Generation Within the Insulation 118 10. Effect of Change in the Volume of the Battery on Core Temperature 121 VI CONCLUSIONS AND RECOMMENDATIONS 123 LIST OF SYMBOLS 125 LITERATURE CITED 127 APPENDICES 130 A. Details of Analog Model 1 131 B. Details of Analog Model II 138 C. Details of Digital Model III & 148 D. Details of the Computer Program 156 BIOGRAPHICAL SKETCH 209 / v

PAGE 6

LIST OF TABLES Table Page 1 Differential-Difference Equations for Mathematical Model I... 28 2 Coefficients for Programmed Differential-Difference Equations: Model 1 32 3 Average Physical Properties and Dimensions of Thermal Battery Components: Model 1 36 4 Effect of Parameters and Changes on the Life to 400 C of a Simulated Thermal Battery 40 5 Differential-Difference Equations for Mathematical Model II 69 6 Coefficients for Programmed Differential-Difference Equations: Model II 71 7 Average Physical Properties and Dimensions of Thermal Battery Components: Model II 75 8 Effect of Changes in the Insulation Structure on Life to 400C of Simulated Thermal Batteries 77 9 Effect of Parameters and Changes on Life to 400 C of a Simulated Thermal Battery 105 A-l Summary of Runs 132 A-2 Summary of Potentiometer Settings ....". 134 B1 Summary of Runs 143 B-2 Summary of Potentiometer Settings 144 C-l Data Used in Simulation of Standard Thermal Ba'ttery 150 C-2 Summary of Runs 151 vi-

PAGE 7

LIST OF FIGURES Figure — Pa se 1 Schematic Diagram of Thermal" Battery 4 2 Temperature Histories in Homogeneous Right Circular Cylinders 21 3 Schematic Diagram of Battery Described by Model I 23 4 Analog Computer Diagram of Model 1 30 5 Experimental Versus Computed Results 39 6 Temperature Histories of Elements in Standard Simulated Thermal Battery 43 Effect on Cell Temperature of Changing the Temperature of the Heat Sink 45 8 Effect on Cell Temperature of Reducing the Thermal Conductivity of the Insulation 46 Effect on Cell Temperature of Increasing the Thermal Conductivity of the Insulation. ./' 47 10 16 Effect on Cell Temperature of Changing the Thickness of the Insulation 4g 11 Effect on Cell Temperature of Changing Thickness of Heat Generators 50 12 Effect on Cell Temperature of Compressing Heat Generators CT 13 Effect on Cell Temperature of Changing the Thickness of the Cell 53 14 Effect on Cell Temperature of Changing the Enthalpy of the Cell „ 55 15 Effect on Cell Temperature of Changing the Magnitude of the Intra-Cell Chemical Heat Generation 57 Change of Initial Rate of Intra-Cell Chemical Heat Generation Term 59 17 Change of Decay Rate of Intra-Cell Chemical Heat Generation Term. 6^ Vll

PAGE 8

JIGw 25 ^Continued) Sattsry 79 =rature Histories in Simulated Thermal Battery rrature Histories in Si-auiaaad "hernial lattery srature Histories in Sir^nluaed Thermal Lattery with Varying Insulation Arrangement C5 ilated Thermal Battery neura riistorias in Sim „th Varying Insulation Arrangement, GU v .-. mperature histories on Simulated Taarmai Battary with Varying Insulation Arrangement S7 han^sic Biagram of Battery Describee hy Model III.. 93 -/ Temperature trollies in Standard Simulated Thermal 23 Effect on Gall Temperature of Various Insulation Arrangements •:; 107 29 Effect en Cell Temperature of Idealized Chaages in Jy ^-j-"aC. en msulcit ion i iiTipsr&i-urt: oz idea i.12 sd Cn&ng&s jL.,1 iUciu-ui -.luJ^^-iCu w j_ Cue Xiiaua.SL.lOZl Ill iiirxsct on ^eii IS:inp6XS.cu2rG cz Dxf ££3T£iit v fir A, jv, w L. ij 'a„i iGtnparaturfi ot Changes in the Intra.. vi 1 1 A

PAGE 9

Figure LIST OF FIGURES (Continued) Page 35 Effect on Cell Temperature of Changes in the Heat Capacity of the Heat Generators 117 36 Effect on Cell Temperature of Suggested Design 1 1 q Improvements 1 J y 37 Effect on Cell Temperature of Delayed Heat Generation Within the Insulation I 20 38 Effect on Cell Temperature of a Change in the Volume of the Battery 122 A-l Analog Circuit of Heat of Reaction Term 137 B-l Details of the Shape of Insulation Elements 139 C-l Analytical Versus Computed Results for the Cooling of a Homogeneous Sphere *-* IX

PAGE 10

Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DYNAMIC HEAT TRANSFER IN COMPOSITE MINIATURE STRUCTURES By Mario Ariet Antiga April, 1965 Chairman: Prof. Robert D. Walker, Jr. Major Department: Chemical Engineering The inability to measure temperatures in small objects accurately and at all desirable locations during rapid temperature transients points up the need for valid alternate methods of solving such problems. In this investigation, temperature histories were computed for specific locations in a simulated thermal battery, and the effect of changing the construction configuration and the physical properties of the materials were studied in order to optimize battery life. Two analog computer models were developed to simulate the thermal battery. These models were based on the assumption of "wellmixed" elements having the thermal resistance "lumped" at the interfaces between adjacent elements. They provided simulations of the temperature histories in thermal batteries which were in satisfactory

PAGE 11

agreement with experience. The analog studies provided considerable insight into heat transfer processes in these units, and the effects of changes in configuration and properties of materials. The results of these models were also very helpful in the development of the more accurate digital computer model. The digital computer model consisted of a sphere having a core made up of an inner well-mixed section and an outer section where temperature gradients existed. The core was surrounded by six concentric spherical layers of insulating materials which could be assigned any value of physical properties. The partial differential equations describing this model were approximated using finitedifference techniques by a system of differential-difference equations which in turn were solved by the Adams-Moulton-Shell numerical integration method. Among the more important findings of this work are: 1. Of the insulation materials and configuration studied maximum battery life is achieved with all-Thermof lex insulation. 2. If it is necessary to use layers of more than one insulation material, maximum battery life is achieved with as much Thermoflex as possible adjacent to the core. 3. Insulation layers prepared by mixing a poor insulator with a good one are less efficient than proper arrangements of layers consisting of pure materials. 4. An increase in the volumetric heat capacity of the heat generators leads to an increase in the battery life. 5. An increase in the intra-cell heat of reaction leads to xx

PAGE 12

an increase in battery life providing it is spread over a sufficiently long period of time. 6. Placement of suitable heat' .generators in the insulation leads to substantially increased battery life. 7. An increase in battery size generally results in an increase in battery life because the capacity of the heat reservoir increases more rapidly than the rate of heat losses. Conversely, the smaller the battery the more serious the heat transfer problem. 8. The sphere is the most efficient shape because it has the smallest area for a given volume. The greater the deviation of an actual shape from a sphere the greater the heat transfer problem. 9. The high cut-off temperature imposed by a high-melting electrolyte leads to an optimum core to insulation volume ratio. For the system studied this optimum ratio turned out to be close to that chosen for the standard. 10. The optimum insulator for a high cut-off temperature is not the insulator with the lowest thermal diffusivity; it is, rather, the one with the lowest thermal conductivity and a value of the volumetric heat capacity which is a function of the cut-off temperature. xn

PAGE 13

CHAPTER I INTRODUCTION The construction and operation of thermal batteries was first discussed by Goodrich (1), who defined them as electrochemical power supplies based upon electrolytes of various inorganic salts which remain solid and nonconducting at all storage temperatures. He indicated that for isolated performance, an integral heat generating source (based on chemical reaction using gaseous, liquid or solid fuels) is required to raise the temperature of the cell above its melting point. Vinal (2), and Selis et al. (3) have presented descriptions of different electrochemical systems, such as Mg/LiCl-KCl-^Cr^/Ni and Mg/LiCl-KCl/FeO ,Ni which perform satisfactorily in thermal batteries. McKee (4) enumerated the following advantages of thermal batteries: 1. Permit high voltages 2. Large currents may be drawn from them 3. Indefinite storage 4. Operation over a wide range of temperature extremes 5. No maintenance / 6. Use in any position 7. Ruggedness He stated that the chief limitation is that they are relatively short lived, the implication being that this results from loss of heat, for otherwise life could be increased essentially indefinitely by adding

PAGE 14

more cell reactants. Johnson (5), and Hill (6) presented data on heat generating systems which can achieve temperatures of about 700 C in about 1 second. Some of these systems. contain zinc metal while others are based on KMnO, CaF MgF_ Fe, etc. Numerous other systems have been devised from study of the thermodynamic properties of the reactants and products of reactions. Possibly the classic case of utilizing a chemical reaction to produce heat is the thermite process. Generally the reactants in these reactions are in the form of very fine powder and are chosen so as to be gasless, or nearly so. These reactions are essentially instantaneous, and they give off a great deal of heat. Temperatures of the order of 2000 C are commonly attained. The design of thermal batteries constitutes a challenging complex engineering problem because of the stringent heat transfer limitations required to construct a useful unit. The chemical composition of the materials which compose the thermal battery and the details of its construction are subject to security classification, but enough unclassified Information is available to permit the heat transfer problem to be defined meaningfully. A thermal battery can be thought of as an assembly of three / main types of elements: 1. Heat generators which constitute an essentially instantaneous heat source. 2. Electrolytic cells which become activated when the

PAGE 15

electrolyte, which is solid at ambient temperature, melts as it receives heat given off by the heat generators. 3. Insulation which serves to decrease the rate of heat loss from the assembly to the surroundings. In a thermal battery the heat generators and electrolytic cells, which may be shaped in the form of flat circular cylinders, can be arranged in a stack in which each cell is in contact with two generators and vice versa This cell-generator core is then surrounded by the insulation. A schematic diagram of such a unit is shown in Figure 1. The operation of the battery is initiated when the heat generators are set off. The large quantity of heat given off almost instantaneously by these generators is transferred rapidly to the electrochemical cells with the result that the solid electrolyte melts and the cell begins to generate electrical power. The assembly loses heat to the surroundings owing to its high relative temperature, and the electrochemical reaction within the cell proceeds until the electrolyte approaches its freezing point. In this work the elapsed time between the reaching 400 C and cooling tract to 400 C is referred to as the life of the battery. In a heat transfer study one of the obvious objectives would be to maximize the life of the battery for a given battery volume subject to other constructional and operational constraints. Maximization of battery life above 400 C was, therefore, made the primary objective of this investigation. The life of a thermal battery can be estimated reasonably well if temperature histories can be obtained at the required

PAGE 16

Heat Generator ElectroChemical Cell Insulation NOTE: Section removed for clarityFigure 1 Schematic Diagram of Thermal Battery

PAGE 17

locations within the battery. Although it might appear that the desired timetemperature relations could be obtained either by direct measurement or by classical mathematical methods, further consideration of the problem will reveal inherent limitations in both the experimental and the formal computational approaches. Although thermometers, thermocouples, and similar devices are generally quite adequate temperature measurement instruments for many physical phenomena, there exist other situations, such as that encountered here, where the instrumental response will not be adequate. For example, large differences can arise between the indicated and actual temperatures of an object as a result of response lags of the sensing device during rapid temperature changes, or as a result of heat losses through the measuring device if the object under study is small in size relative to the sensing device. Such errors are in general difficult to evaluate. Alternatively the temperature histories at points of interest may be obtained by solving the mathematical model which describes the given physical situation. For the case of unsteady-state heat transfer, the mathematical model will consist of one or more partial differential equations. Even if the errors associated with a temperature measuring device should not be serious, a valid calculation procedure would have a tremendous advantage of producing temperature histories of numerous points simultaneously. Moreover, the experimental approach would subject the system to the disruptive influence of the numerous measuring devices required for direct measurements. Where the object under study is

PAGE 18

relatively small the errors resulting from heat losses through the various measuring instruments might be the most significant mode of heat loss thereby making the observations meaningless. Another significant advantage of the mathematical model approach to this problem as compared to the experimental approach is that the experimental testing of the battery requires a statistical analysis involving numerous replications because of the random variations of the materials making up the battery. In the mathematical model, materials having exact properties are assumed and hence valid conclusions can be drawn from a much smaller number of trials. There is another consideration which is very significant from the design viewpoint, namely, the mathematical model allows for the possibility of evaluating the performance of a battery constructed of idealized materials. If the results indicate that a significant improvement could be obtained with such materials, efforts could be directed towards the development of these materials. In many cases of practical interest, such as the one under consideration, formal mathematical solutions of partial differential equations are very difficult or impossible to evaluate because materials may not be homogeneous, thermal properties may vary, or the boundary or initial conditions may be complex. However, the availability of electronic computers makes possible the solution of complex mathematical models by various approximation techniques. In this study, mathematical models of the dynamic heat transfer in thermal batteries were developed with simplifying assumptions which

PAGE 19

made their solution on the computers feasible. In the initial stage of the investigation the analog computer was utilized. With the insight gained from the analog studies, a more complete mathematical model was developed and programmed for the IBM 709 digital computer. The investigation had two chief objectives: 1. To provide temperature histories for specific locations within a battery. 2. To study the effect of changing the construction configuration and properties of the materials required for the construction of the battery so that design specifications could be made to optimize its performance. The optimization criterion was defined as the maximum life of the battery. Thus, the configuration yielding the maximum battery life was considered optimum. As mentioned before, the -effect of idealized materials was studied with the idea that if the inclusion of certain idealized material increased the life of the assembly significantly, the desirability of developing such a material would be indicated.

PAGE 20

CHAPTER II THEORY OF HEAT TRANSMISSION The second law of thermodynamics states that heat energy always flows in the direction of the negative temperature gradient, i.e., from a hot body to a cooler one. There are three distinct methods by which this migration of heat takes place: 1. Conduction in which the heat passes through the substance of the body itself. 2. Convection in which heat is transferred by relative motion of portions of the heated body. 3. Radiation in which heat is transferred directly between portions of the body by electromagnetic radiation. Although the three kinds of heat transmission generally occur together, fortunately one or the other often prevails in practical cases. Therefore, separate laws governing each kind of heat transfer have been developed and may be used in such cases. Superposition of these laws is also often possible and used (7). The basic law of heat conduction is: / 2 = k 4I (ii-D A AL In this and the next two equations, Q, denotes the time rate of heat flow, i.e., the heat energy flowing through a constant area, A, in unit time. The rate of heat flow, Q, may be considered constant for the time

PAGE 21

being. Equation (II-l) relates to steady state transfer in a plane plate of thickness AL with a perfectly insulated edge: the two free surfaces being held at the temperature difference AT. The parameter, k, which may be considered a constant for the time being, is called thermal conductivity. Equation (II-l) originates from Biot (8), but it is generally called Fourier's equation because Fourier (9) used it as a fundamental equation in his analytic theory of heat. For heat convection the following equation was first recommended by Newton (10): Q HMT (H-2) Equation (II-2) relates to the heat transfer between a surface and a fluid in contact with it, the temperature difference being AT. The factor H is called surface coefficient of heat transfer, film coefficient of heat transfer, or simply coefficient of heat transfer. This expression is often referred to as Newton's cooling law, but it is really a definition of H. This point will be discussed later in more detail. For the total radiation, equations of the form Q = 0AT 4 (H-3) have been used since Stefan (11) found this relation and Boltzmann (12) proved it theoretically for a perfectly black surface. Equation (II-3) relates to the emission of radiation from a surface at the absolute temperature T. The factor a is a natural constant known as the StefanBoltzmann constant, or the constant of total black-body

PAGE 22

10 radiation. For surfaces not absolutely black, O must be modified if the Stefan-Boltzmann law is to be applicable. The basic equation of heat accumulation for small linear changes of temperature is Q = pC VAT (II-4) P At where Q is the heat accumulation in unit time, in the volume, V, of a medium of density, p, and specific heat, C when the temperature increases by AT in a time interval At. From the above fundamental relations a great deal of knowledge has been developed. The application of mathematics permits the evaluation of heat transfer processes by different modes, in different geometrical shapes, and subject to varied initial and boundary specifications. Carslaw and Jaeger (13) have presented a very complete formal mathematical treatment of heat conduction problems. Jakob (7) considered all forms of heat transfer in his work, and provided theoretical or empirical solutions to a great variety of heat transfer problems. McAdams (14) presents a very complete treatment of the heat transfer problem from the practical design engineering point of view. Specifically in the field of heat transfer by conduction, many physical situations can be described by relations which, are amenable to solution by formal mathematical techniques. Other studies (15,16,17) have treated composite bodies, mostly the laminated wall having no interfacial resistance. However, Siede (18) considered a composite system having resistance between layers. In most of the formal mathematical solutions to heat conduction

PAGE 23

11 problems the assumption of constant thermal and physical properties is usually made. Friedman (19), and Yang (20) have studied the effect of these assumptions and have shown that in some situations significant errors may result from their use. The solutions to most realistic problems involving conduction heat transfer usually involve infinite series of terms which may or may not converge rapidly. Therefore, it is sometimes quite difficult to obtain a numerical answer from the general mathematical solution. In order to make the results of formal mathematical treatment more applicable to practical problems Gurney and Lurie (21), Groeber (22), Olson and Schultz (23), Newman (24), and others (14,25,26,27) have presented graphs or charts showing temperature versus time or geometrical location for different parameters. Such parameters as thermal diffusivity,a = k__, surface convection coefficients, and ^ P geometric shapes are usually employed. The geometrical shapes considered are limited to homogeneous infinite plates, infinite cylinders, spheres or objects of such shape that heat flow can be considered unidirectional It is generally conceded that formal mathematical methods are capable of solving only the simpler situations of geometry and boundary conditions in heat conduction problems. Many practical situations yield a mathematical model which can only be solved by approximation methods. Numerical, graphical and analog techniques are the most common tools for handling complex heat conduction problems. Although these methods are approximate, they can, in principle, be extended to any degree of

PAGE 24

12 closeness of approach to the exact solution given by formal mathematical techniques. Their only limitation is the amount of effort (time and/or money) involved. In addition, as mentioned before, the formal mathematical solution also requires considerable effort if a precise numerical answer is desired owing to the usual infinite series form of the solution. Later, an example will be given of a problem where an approximate approach actually required less effort to yield an answer of a given accuracy than the effort required to evaluate the formal mathematical solution to the same degree of accuracy. Graphical methods for solving heat conduction problems were first developed by Binder (28), and Schmidt (29) based on the calculus of finite differences. Many improvements and extensions of the basic method have been made (30,31). The work of Longwell (32) is of particular significance to this investigation because it treats graphically the motion of the freezing boundary in the heat transfer process involving the phase change from liquid to solid. This is probably the mechanism by which the electrochemical cells become inoperative. In general it can be stated that graphical methods are useful only when low accuracy is sufficient in the solution of a problem. If a high degree of accuracy in the solution is attempted, this procedure becomes prohibitively cumbersome. It has been known for many years that different physical phenomena can be described by the same mathematical relations; in such cases they are said to be analogous processes. Langmuir, Adams and Meikle (33) seem to have been the first to make use of the analogy between

PAGE 25

13 thermal and electrical conduction; they solved a problem based on the similarity between a flow-temperature field and an electrical flowvoltage field of the same geometrical configuration. Beuken (34), and Paschkis (35) developed large-scale, permanent analog devices whose principal elements were resistors and condensers, and they were able to solve unsteady-state heat transfer problems. The chief drawback of these analog devices is that they are expensive to construct, and are usually capable of simulating only the type of system for which they were specifically designed. Even relatively minor modifications of the original system can be cumbersome and expensive. The type of analog devices discussed above depend for their operation upon the existence of a direct physical analogy between the analog and the prototype system under study. Such an analogy is recognized by comparing the characteristic equations describing the dynamic or static behavior of the two systems. An analogy is said to exist if these characteristic equations are identical in form, and the initial and boundary conditions are the same. Such a similarity is possible only if there is a one-to-one correspondence between elements in the analog and in the prototype system. For every element in the original system there must be present in the analog system an element / having similar properties, i.e., an element having a similar excitation-response relationship; furthermore, the elements in the analog must be interconnected in the same fashion as the elements in the original system. The other major class of analog system includes mathematical rather than physical analogs. The behavior of the system under study,

PAGE 26

14 or the problem to be solved is first expressed as a set of algebraic or differential equations. An assemblage of computing units or elements, each capable of performing some specific mathematical operation, such as addition, multiplication or integration, is provided, and these units are interconnected so as to generate the solution of the problem (36). The availability in recent years of high-speed digital computers has augmented the interest in numerical methods based on the calculus of finite differences as an efficient tool for the solution of complex heat flow problems. Emmons (37) utilized the relaxation method developed by Southwell (38) for the solution of two and three-dimensional steady state heat transfer processes. Although the relaxation technique can be used for unsteady-state problems (39), explicit time iteration procedures, such as the one developed by Dusinberre (40), are generally preferred to relaxation methods because they can be adapted more readily to digital computation. The explicit finite-differences technique has in general the limitation that it is difficult to evaluate the accuracy of the solution. If the criteria of "stability" and "convergence" are satisfied, the accuracy is determined by the number of increments used, and it can be improved at the expense of increased effort (41). The convergence criterion is the requirement that the exact solution be approached by the approximate solution as the number of increments approaches infinity. The stability criterion means that the error introduced into the computation, owing to the limited number of digits which a given

PAGE 27

15 computer can carry, must not increase in magnitude as the computation proceeds. These criteria have been studied by a number of investigators (41,42,43). Therefore, for a numerical. method which is stable and convergent when applied to a system of equations, the finite-difference technique can yield any degree of accuracy desired. The only restriction is the amount of effort required. Brian (44), and Douglas (45) developed implicit difference methods which are unconditionally stable, usually at the expense of increased computational effort. Yavorsky, et al. (46) utilized the explicit type finite-difference formulation, and solved on a digital computer the problem of heating homogeneous cylindrical briquettes. Dickert (47) used the explicit finite-difference approach for the solution on an IBM 650 digital computer of the unsteady-state heat transfer in a composite finite cylinder. Actually the physical model he simulated was a simplified version of the thermal batteries which are the subject of this study. Home and Richardson (48) developed a model to simulate the performance of batteries at low ambient temperatures. It was programmed on a digital computer and it was based on well-mixed sections with lumped thermal resistance at the interface. In this investigation, two different mathematical models were developed and solved on an analog computer in order to benefit from the advantageous features of the instrument, such as the essentially instantaneous availability of the answer, the continuous display of the results (usually in an oscilloscope or a plotter), and the immediate

PAGE 28

16 response of the system to a change in one of the parameters. All of these features made the analog computer the initial choice in this study. Later, after sufficient insight had been gained from the analog studies and when increased accuracy was desired, a model was developed to be programmed on the digital computer. The results revealed very interesting aspects 'of heat transfer phenomena, and provided a good simulation of thermal batteries. A. Geometrical Considerations in Battery Life Optimization Some conclusions can be drawn from purely geometrical considerations with respect to the optimum shape of a thermal battery under the criterion of maximum life. Since the rate of heat transfer is directly proportional to the area and the rate of temperature change is inversely proportional to the volume, it is clear that the smaller the area of a body, the lower the rate at which it will loose heat, all other things being equal. Therefore, a hot body of a given volume will remain hot longer, the smaller its area. If., for the moment, the geometrical shape of. the battery is restricted to right circular cylinders, elementary mathematical considerations show that for a cylinder having radius R and height h, the total area is given by ; A = 2-nR 2 + 2-nRh (II5) while the volume is given by V = 7lR 2 h (II-6) If the volume is considered to be fixed, the area can be expressed by

PAGE 29

17 A = 27lR 2 + 271R V (II-7) 7IR 7 which can be differentiated with respect to R and equated to zero to give •di = 47TR ^1 = (II_8) R Equation (II-8) may be solved for the volume to obtain V = 27lR 3 (H-9) This would be the value of the volume corresponding to a minimum area, but V = 7iR 2 h (11-10) hence h = 2R (II-ll) Equation (II-ll) makes clear that the right circular cylinder having the minimum area per unit volume is one having its height equal to its diameter Similarily it can be shown that for the case of orthogonal parallelepipeds the volume is given by V = xyz (11-12) while the area is given by A = 2xy + 2xz + 2yz (11-13) If the volume is considered to be fixed the area can be expressed as A = 2xy+2 ^+^ (11-14) xy xy which can be differentiated partially with respect to x and y to give ^ 2x *! (11-15) X

PAGE 30

18 U-fr.if.O (11-16) 1 y r These equations may be solved for x and y, respectively, to give x V l/3 and y = V 1/3 (11-17) 1/3 which results in a value of z = V when substituted into equation (11-12). Hence the orthogonal parallelepided whose outside area is a minimum for a given total volume is the cube. Forsyth (49) has shown by the calculus of variations that the sphere is the solid generated by rotation which has the maximum volume for a given area. The sphere is likewise the solid having the maximum volume for a given area out of all possible solids, but this is more difficult to demonstrate rigorously It can be shown specifically that the sphere has a lower ratio of area to volume for a given volume than the cylinder having equal height and diameter, which in turn has a lower ratio than the cube.. For a volume of V the radius of the sphere is given by ^ s ym and the area is given by ,„ / f 3v S I 47T R = — VI (11-18) A 47T -rH (11-19) For the cylinder, the radius is given by 1/3 C \ 7T and the area R = — V ) (11-20) A 4TT \zr\ (11-21) 2/3 C \ TX Hence, for the same volume V, the ratio of the area of the sphere to the

PAGE 31

19 area of the cylinder is 2/3 f = (! J = 0.825 (11-22) For a cube the side 1 is given by and the area 1 = V 1 3 (11-23) A = 6V 2/3 (11-24) cu Therefore, the ratio of the area of the sphere to the area of the cube is 2/3 s 4tt f 3 = 0.804 (11-25) cu A 6 V 47T These considerations indicate that, if it were feasible to construct thermal batteries in a spherical shape, this would be the optimum configuration from the heat transfer standpoint. There are other restrictions which make this shape impractical, hence the next most efficient shape is that of a right circular cylinder having its diameter equal to its height. Figure 2 illustrates the temperature histories of the center point of different homogeneous right circular cylinders having equal volumes and different height to diameter ratios, and having initial temperatures of 500 C everywhere except at the surface where the temperature is assumed to be constant at zero degrees. These curves were evaluated from tables presented by Olson and Schultz (23). The parameter shown on the curves is the height to diameter ratio. The volume of all cylinders is that of the cylinder having a height of 3.0 cm (equal to its

PAGE 32

20 diameter). A temperature history for the center point of a sphere having the same volume is also shown. ,For right circular cylinders of constant volume, Figure 2 indicates that the rate of cooling increases drastically when the height to diameter ratio is made less than the optimum. The rate of cooling also increases when the height to diameter ratio is made greater than the optimum but the effect is less than in the former case.

PAGE 33

21 co U QJ a •H rH o •H O 42 H (4 co o C o> 50 O g o CO 0) >H u o 4-) CO •H K 0) 4J n) M QJ 0 6
PAGE 34

CHAPTER III ANALOG COMPUTER MODEL I A. Description of the Model The first model developed to simulate a thermal battery consisted of a cylindrical stack of alternating heat generators and cells surrounded by insulation. While the actual number of generators and cells in a real thermal battery may vary, it was assumed that the core was composed of three cylindrical generators and two cells surrounded by top, bottom and lateral insulation. A schematic diagram of the model is shown in Figure 3. Consideration of the physical dimensions of the elements in the battery led to some assumptions. The very small relative thickness of the elements compared to other dimensions, such as diameter of the cells, heat generators and top insulation, and height for the lateral insulation, suggests that the major portion of the heat transfer is an axial, rather than a radial, process. This suggeststhat the temperature within each element would be rather uniform, hence it was assumed that each element was "well-stirred", i.e., that its temperature was uniform throughout. This assumption is more valid for some elements than for others. For example, the cells consist (during the operating life of the battery) of molten electrolyte, and, since intra-cell chemical and electrochemical reactions may be occurring simultaneously, it appears that the mobility of the ions in the electrolyte provides a relatively 22

PAGE 35

23 j F r^ o > i\. r Top Insulation a o 4J tO t-i 3 en (3 M 1 h t. Top Generator • Cell i >-l M • H Figure 3 Schematic Diagram of Battery Described by Model I

PAGE 36

24 well-mixed element having an approximately constant temperature. There also exists the possiblity that some convection currents might be established, but this is doubtful owing to the small thickness of the cell. The deviation of the heat generators and insulation from this assumption would be of the same order of magnitude because their thermal properties are comparable. The cylindrical symmetry of the model makes necessary the consideration of only the top half of the unit. The heat generators achieve their maximum temperature of about 2200 C (47) in a length of time which is negligible compared to the rest of the heat transfer process. Therefore, it was assumed that they reached their maximum temperature instantaneously, and this high temperature becomes the initial driving force of the heat transfer system. There is a heat of reaction from intra-cell chemical reactions. The experimental data describing this phenomenon are very uncertain. It is known that the heat of reaction increases rapidly at the beginning of the operation, reaches a maximum, and then decays. A triangular shape was assumed for the heat of reaction-time relationship. Both the shape and magnitude of this effect were based on educated guesses of experienced investigators (50), and it is the only factor in this study not based on experimental or computed physical data. B. Heat Transfer Coefficients The chief consequence of the assumption of well-mixed elements in the battery is that the heat transfer process which occurs under these conditions become one of convection rather than conduction. Because

PAGE 37

25 there can be no temperature gradient through any single element, all of the resistance to heat transfer appears "lumped" at the interfaces between elements. Therefore, a pseudo-heat transfer coefficient must be calculated by appropriately lumping the heat transfer resistances of two adjacent elements at the interface between them (based, of course, on their thermal conductivities and the mean path traveled by the heat). The pseudo-heat transfer coefficients were evaluated by considering the two elements to constitute a series arrangement for the resistance to heat flow. For the case of a cell and a heat generator, the coefficient had the following form: Total = Resistance + Resistance (III-l) Resistance of Cell of Generator H 1 (IH-3) where H is the heat transfer coefficient between the heat generator g-c and the cell, and k and h are the thermal conductivities and heights, respectively. The heat transfer coefficients involving the lateral insulation were obtained by calculating the radius equivalent to one-half the volume of the interior element, and considering the mean distance the distance from this radius to the outer radius of the element. The mean distance in the case of the lateral insulation is, of course, onehalf of the thickness. Therefore, for a value of the radius of the

PAGE 38

26 element of 1.56, the following heat transfer coefficient was obtained: H ., 1 (III-4) All other heat transfer coefficients were computed in a manner similar to those discussed above. Because of the high initial temperature of the heat generators, radiation rather than convection or conduction is the principal mechanism of heat transfer while the generators are incandescent. A pseudoconvection heat transfer coefficient was calculated for this period based on the laws of radiation. It was arbitrarily decided that 750 C was the temperature where the principal mechanism changed from radiation to "convection". The heat transfer coefficient describing the radiation transfer can be obtained as follows (51): q A 1 F 12 CT(T 1 4 T 2 4 ) (III-5) where q is the rate of heat transfer, and A is the area of the heat transfer surface. F is a dimensionless factor to allow for interchange between gray surfaces; is the Stefan-Boltzmann constant (4.92 x 10 kgcal/m hr k ), and T is the temperature in degrees Kelvin. / F 12 = 1 (III-6) iT + \~l' I + T 2 \~2~ l l where F is a dimensionless geometrical factor to allow for net radiation between black surfaces including the effect of refractory surfaces, and

PAGE 39

27 C is the dimensionless emissivity. The equivalent heat transfer coefficient is obtained from 4 4* H F 12 g(T l T 2 } H (IH-7) g-c-R ^ + g-c where T.. is evaluated as the arithmetic average of the fourth powers of the extreme values of the heat generator temperature during the radiation period and T„, the cell temperature, was evaluated similarily. AT was taken as the geometric mean of the extreme values of the temperature differences (52). Detailed computations of all these coefficients are shown in Appendix A. C. Development of the Mathematical Model A differential heat balance around each element (i.e. cell, generator, etc.) gives the equations shown in Table 1 with notation having the significance indicated below: H = pseudoconvection heat transfer coefficient, cal 2 o cm sec C h height of element, cm(See Figure 3.) R = radius of element, cm(See Figure 3 ) R = outside radius of assembly, cm(See Figure 3 ) o o T = temperature of element, C t = time, sec Subscripts a = refers to the ambient c = refers to the cell g = refers to the generators

PAGE 40

28 oo i Oi CM I I m W Q § < CJ M H eg o En 22 O H H 3 cy w w u W 1-1 < H H as w Pi w § H T3 •t4 CM CJ oo H i •H 4J 60 •r-4 J-l EC H i i •H •U SB c o -H 3 CO c 1-1 a o H i CO I CM H II O II CO 3 O cfl 3 cr W M o 3 O •H T3 C o CJ !-i O 4-J CO u 0) c cu e> D. O H Ml H "3 b0 Ml a. o oo Q. II H t M O oo H I oo i 00 00 -3 CM 00 60 O O CN CM 00 o II 4-1 o 7-1 I c o •rl 4-1 CO 3 W W o 3 •H 4J •H 3 O U CO •H 3 Ol 4-1 H h ro T3 II o ,3 CM H f o a. o o a H i oo 1 o I 00 CM PS C j=! I CM s 00 rc CJ I 00 o II 4-J M H M 3 O cd 3 a" M H O 3 13 3 O •H 3 M O 4-J CO M 01 3 QJ o ID 13 •H S oc E |C CM 00 3u 60 oo e o I oo u o o o CN CN op o II 4-1 CO i
PAGE 41

vO 29 M f I H 1 •H 1-1 •H H I-H 4-1 v^ H "O •H *o i-l /-n • 60 JCM 03 ^ x: a w + J2 1 CM H + II 60 /— \ u r O £1 H H i + •H H 4-1 i-l J3 H v-' V-' e"> -H ftf i-H 1 1 y P3 •-s W N^ 13 o £ O 0) ^3 CM 3 I •H II •H CM o 4-1 4J •1-1 B i Q. 4J o ca 60 II U /-v I-l H H* 3 •3 i -iH J
PAGE 42

30 60 I O O o o CM CM 60 o o o CM CM w H I H i H I H I H I 60 H I i o o H i 60 H I 0 EH i M 0) M-l O I u 60 efl •H Q H CD 3 a g o u 60 O .—I cti C < CD 3 60

PAGE 43

31 Subscripts continued i = refers to the insulation 1 refers to lateral '. m = refers to middle i.e. mg refers to middle generators R = refers to radiation (See equation III-7) t = refers to top i.e. ti refers to the top insulation Example H refers to the heat transfer coefficient between the ti-li top insulation and the lateral insulation. D. Analog Computer Solution The preceding equations in Table 1 can each be solved for the derivative of temperature with respect to time, and they constitute a system of ordinary first-order linear differential equations which is readily amenable to solution with the aid of an analog computer. The EASE 1032 Analog Computer of the Chemical Engineering Department at the University of Florida was utilized in this investigation. The basic element of the analog computer is the electronic amplifier which can serve as an integrator or a summer depending on whether a capacitor or a resistor is connected across the amplifier. Many references discuss in detail the theory and operation of analog computers (36,53). The equations describing the model, were programmed on the analog computer, and the analog computer circuit is shown in Figure 4. Table 2 illustrates the algebraic form of the relations between the parameters which contribute to each potentiometer setting.

PAGE 44

32 TABLE 2 COEFFICIENTS FOR PROGRAMMED DIFFERENTIAL DIFFERENCE EQUATIONS:. MODEL I Pot. No. 1 Mathematical Expression p.C .h *i pi ti H £i=g p C h. *g pg tg p C h *g Pg tg H ti-g p.C .h r i pi ti H g-cp C h c pc c H + 2h ti H + H ti-a — — ti-li ti-g K H + 2h tg H + H ti-g -£B g-li gR g-c h H tg g-li ,R p.C .(o l)(h + 3/2 h + h ) r x pi — ti tg c R ti ti-li ,R p.C .( o l)(h,.. + 3/2 h + h ) H i pi — ti tg cPot. Setting 0.663 0.123 0.744 0.286 0.143 0.042 0.0319 11 12 p C h c pc c p C h g Pg tg AH h c c 100 H + c H + H g-c — c-li g-c H + tg H g-c —^ g-li 2H g-c p C h g Pg tg 0.2907 0.126 0.0035 0.123

PAGE 45

33 TABLE 2 (Continued) Pot. No. Mathematical Expression Pot. Setting 14 h c H c-li 0.1395 15 ( V 2) Vli 0.021 P^pA" fr tl + 3/2 h + h c ) R 17 H ti-a T a 0037 Pi C pi h ti 18 R o /R(H li-a T a ) 0-0063 lOOp.C .( R o 1) 19 ViA + 3/2 Vii h t g + H ti-ii h ti Pi C piQl>Ch tl + 3/2h tg+ V R + R Q /R(h t + 3/2 h fcp +h c )H u a 0.8644 Pi c P i^ • 1)(h ti + 3/2 h tg + v : R 20 2H ti-li 00107 Rp.C r i pi Pi 21 2H g -li / 0.006 Rp C r g Pg 22 2 Vli 00046 Rp C r c PC

PAGE 46

34 TABLE 2 (Continued) Pot. No. 23 H g-li Rp C *g PS 24 H g-c p C h r g Pg tg 25 H K-C p C h c pc c 31 Time Sea 32 Initial 1 34 Decay Ra 42 2 P C h g Pg tg 43 2H g-c-R p C h r g Pg tg 44 1 p C h /g Pg g 45 H g-c-R p C h c pc c 49 1 p C h c pc c 50 H g-c-R p C h. g Pg tg 53 H g-c-R p C h c pc c Mathematical Expression H D +S& H -,g-c-R -jp g-li H 2h. + te H + H ti-g -jS* g-li g-c-R. 2h H + c H -, + H g-c-R -rc-li g-c-R Pot. Setting 0.003 0.615 0.143 0.050 0.050 0.0071 0.6150.615 0.321* 0.715 0.1430.308 0.715

PAGE 47

35 These factors are, of course, a direct consequence of the form of the system of differential equations. Table 3 lists the values of the physical parameters for all the materials which constitute the "standard" battery being simulated. These are the values used in obtaining the numbers shown in Table 2 for each potentiometer setting in the simulation of the standard battery. In other runs, the values of some of these parameters were changed judiciously to investigate their effect on the performance of the battery. It has been pointed out earlier that a set of switches was arranged to automatically change the value of the heat transfer coefficient to account for radiation heat transfer at the heat generatorcell interfaces when the heat generator temperature is above 750 C. This temperature was chosen for the change from radiation to convection heat transfer because it seemed reasonable and it gave good agreement with the known activation times of certain thermal batteries. Moreover, it led to realistic cell peak temperatures. The representation of the intra-cell heat generation by a triangular heat of reaction term also required a number of switches to provide an appropriate simulation on the analog computer. The details of these switching arrangements are shown in Appendix A. Also a manual switch was installed which permitted the inclusion or exclusion of the heat of reaction due to the electrochemical process in the cells.

PAGE 48

36 TABLE 3 AVERAGE PHYSICAL PROPERTIES AND DIMENSIONS OF THERMAL BATTERY COMPONENTS: MODEL I Insulation p, g/cc = 0.193 C cal = 0.232 p jyz k, cal cm = 0.0002 „ 2 g C cm. h, cm. = 0.156 R, Cm. = 1.56 R cm. = 1.72 o' Generator p, g/cc = 1.25 C cal = 0.130 P ^ k, cal cm -= 0.0005 o_ g C sq. cm. h, cm R, cm. 0.10 = 1.56 Cell p, g/cc = 3.48 C cal = 0.201 P g c

PAGE 49

Cell continued 37 TABLE 3 (Continued) k, cal. cm. = 0.10 gC sq. cm. h, cm. = 0.10 R, cm. 1.56

PAGE 50

38 E. Discussion of Results The great flexibility of the analog computer permitted the investigation of a wide range of parameters and geometrical arrangements which might affect the life of a thermal battery. A summary of the more important results is given in Table A and they are discussed below. The model treated in this work has several limitations and some caution should be exercised in attributing too much significance to small effects. On the other hand, large effects are probably correct and qualitative conclusions based on them should be sound. Probably the most serious limitation of this model is that it is not based on conduction but on pseudo-convection heat transfer coefficients. Temperature gradients within an element are thus precluded, and this is known to be incorrect. However, the temperature gradients within an element do not appear to be large (except perhaps for the first few seconds in the heat generators), and the model appears to simulate the temperature histories of elements in an actual battery rather well (see Figure 5). It should be noted also that the complexity of the physical models which can be studied on an analog computer is limited by the capacity of the computer, and the time required to obtain meaningful results is greatly increased when the model is made more complex because there are more components and all of them (amplifiers, potentiometers, capacitors, etc.) must perform satisfactorily for the results to be valid. The problem which was here programmed on the

PAGE 51

39 o 00 o \0 CO u i-H 3 CO cu D &, B o u w 3 CD • h o 0) o 01 > •fr. to H *, JJ_ C H JO o o o o o o CO o <1 o o CM o o

PAGE 52

40 TABLE 4 EFFECT OF PARAMETERS AND CHANGES ON THE LIFE TO 400C OF A SIMULATED THERMAL BATTERY Heat Sink Temperature •65F Sec. A. Insulation 2X* 40 Heat Generators 1. Thickness 2X 2. Enthalpy p = 2X, h • 1/2 X Electrochemical Cells 1. Thickness 1.2X 1.5X 2X 25C Sec, 54 48 75 160F Sec. 1. Thermal Conductivity 0.0001 64 87 97 0.0002 (Std.) 25 48 60 0.0003 18. 5 32.5 41 2. Thickness 65 160 Life Sec. 1.52 2.40 2.21 1.62 54 66 (4 sec. actv.) 100 (17 Sec. actv.) 25Life 65Life 25Std. Dimensionless 1.61 1.00 0.68 1.12 1.00 1.57 1.12 1.38 2.08 X times standard value of parameter being varied

PAGE 53

41 TABLE 4 (Continued) Heat Sink Temperature -65F 25C 160F Sec. Sec. Sec, 160 Life -65 u Life Sec. 25Life 25 u Std. Dimensionless C. Electrochemical Cells (Continued) 2. Enthalpy (pC ) 1.2X* 0.8X D. Intra-Cell Heat Generation 1. Level None Std. 1.5X 3X 2. Rise Rate 0.5X Std. 2X 3. Decay Rate 0o5X Std. 1.3X 38 35 30 48 60 100 45 48 48 52 48 44 0.79 0.73 0.63 1.00 1.25 2.08 0.94 1.00 1.00 1.08 1.00 0.92 X times standard value of parameter being varied

PAGE 54

42 EASE 1032 Analog Computer represented essentially the limit of the capabilities of the instrument. !• Temperature Histories in a Standard Simulated Battery The temperature histories of five elements in the standard simulated thermal battery are shown in Figure 6. These elements are (1) top heat generator, (2) center heat generator, (3) cell, (4) top insulation, (5) lateral insulation. From Figure 6 it can be seen that both of the heat generators release their heat rapidly to the cell and the top insulation. Within five seconds after activation the temperature of the heat generators is below the cell temperature, but the center heat generator is only slightly cooler than the cell. Thus, it can be said that the cell is heating the generators after the first few seconds. The cell is shown to reach a temperature of 400C in less than 0.5 seconds, but little importance should be attached to this because the response of the recorder was not particularly good for times of less than one second. Figure 6 indicates that the cell reaches a peak temperature of about 550 C in approximately five seconds which is in good agreement with experience. All of the curves in Figure 6 are in reasonably good agreement with what one would expect for a heat transfer / system of the type under consideration. 2 Effect of Heat Sink Temperature on Cell Temperatures As one might expect, the temperature history of a cell is strongly dependent on the heat sink temperature. The data summarized in Table 4 indicate that the cell life above 400C is approximately twice

PAGE 55

rn 1000 875 750 43 Heat Sink Temperature 25C Top Generator 625 500375 550C Middle Generator 250 125 10 Top Insulation Lateral Insulation 20 40 30 Time, sec. Figure 6 Temperature Histories in Standard Simulated Thermal Battery 50 60

PAGE 56

44 as great when the heat sink temperature is +160 F instead of -65 F. Room temperature results are intermediate. Figure 7 illustrates the temperature histories for a standard cell at the three heat sink temperatures. 3. Effect of Insulation Parameters on Cell Temperatures Thermoflex insulation has a thermal conductivity of about 0.0002 cal. cm. g" J C~ cm and this has been adopted as the standard insulation type. The thermal conductivity of asbestos is around 0.0003. In order to assess the effect of a large improvement in insulation properties, one run was made with a hypothetical insulation having a thermal conductivity of 0.0001. These results are tabulated in Table 4, and it may readily be seen that thermal conductivity of the insulation is an important factor in the life of the cell. A comparison of the life above 400 G at a 25 C heat sink temperature shows that a 50% variation in life might be expected with the sort of variation in thermal conductivity studied. Figures 8 and 9 show the curves for cell temperatures at these conditions. Table 4 also shows that doubling the insulation thickness (for Thermoflex) results in relatively little increase in life except at a heat sink temperature of -65 F. Where life is now minimal at low heat sink temperatures, increasing the insulation thickness would appear to be a quite promising means of increasing cell life. These curves are shown in Figure 10. 4. Variation of Heat Generator Parameters Two heat generator parameters were studied: (1) a variation in

PAGE 57

45 750 r 625 500 375 250 125 30 Time, sec. Figure 7 Effect on Ceil Temperature of Changing the Temperature of the Heat Sink

PAGE 58

46 750 -r 625 k i= 0.0001 550C /^ 500 f ~~ ~~~^-~-_^^ ""^-^ -^_Heat Sink 160F 400C 375 -~~-^^ Heat Sink 25C^~-~---^_ Heat Sink -65F "~~-~ -~_^^ 250 125 o -\ 10 20 "i 30 40 50 60 Figure 8 Time, sec. Effect on Cell Temperature of Reducing the Thermal Conductivity of the Insulation

PAGE 59

47 750 625 500 375 250 125 10 20 40 Figure 9 30 Time, sec. Effect on Cell Temperature of Increasing the Thermal Conductivity of the Insulation 50

PAGE 60

48 u u ctf u a g H 30 Time, sec. Figure 10 Effect on Cell Temperature of Ghanging the Thickness of the Insulation

PAGE 61

49 the thickness of the heat generators, and (2) a variation in their enthalpy. The effect of doubling the thickness of the heat generators is illustrated in Figure 11. The net effect on cell life appears to be quite small. Although doubling the thickness of the heat generator increases the heat available for the cells, the consequent increase in the size of the core requires the same cell to heat a larger volume which more than offsets the gain in heat. It should be recalled that the cells serve as the heat sources after the first few seconds (see Figure 6). Thus there is a slightly higher peak temperature, arrived at later than in the standard arrangement because of the longer heat path in the generator, but the cell cools more rapidly than the standard once cooling starts because the area of the core is larger and thus heat losses are greater. The net effect of the change is essentially zero as far as the cell life above 400 C is concerned. If the standard heat generator is compressed the heat generation per unit volume of generator increases. In Figure 12 the effect of compressing the heat generator to one-half of its original thickness is portrayed. It may readily be seen that the peak temperature and rise time of the cell are not greatly affected. However, the rate of heat loss of the cell is greatly reduced as compared to the standard arrangement owing primarily to the reduction in core area and an increase in cell life over 400 C of about 607o is observed. 5. Variation of Cell Parameters Three cell parameters were varied: (1) cell thickness, (2) cell

PAGE 62

50 750 625 550 C 500 375 250 125 10 20 40 Figure 11 30 Time, sec. Effect on Cell Temperature of Changing Thickness of Heat Generators 50 60

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51 750 625 550C 500 375 250 125 10 20 40 30 Time, sec. Figure 12 Effect on Cell Temperature of Compressing Heat Generators 50 60

PAGE 64

52 enthalpy, and (3) intra-cell chemical heat generation. The third of these parameters is treated in a separate subsection because several factors involved were studied. Assuming that the standard components of a cell are used regardless of thickness, it is obvious that the enthalpy of a cell is proportional to its thickness. One may also observe that the heat transfer paths are also lengthened for both heating and cooling, and one would expect a thick cell to both heat and cool more slowly than a standard one. The effect of varying cell thickness is illustrated in Figure 13, and one does, indeed, observe these effects. Since the heat input is constant (except for the intra-cell chemical heat generation built in), the peak temperatures decrease as the cell thickness increases; however, the intra-cell heat generation begins to contribute more heavily as cell thickness increases and when the thickness is increased by 507, the peak temperature is actually determined by the intra-cell heat generation. Thus the peak temperatures are reached a fairly long time after activation in these cases. The activation times of thick cells are also increased, and become prohibitive for very thick cells. It appears that a thickness increase of no more than 50% can be tolerated unless activation times of more than 20 seconds are permissible, or unless other geometric arrangements are used. It is possible in principle to add to the cell materials which can change its enthalpy. For example, a material having a transition in the temperature range of interest might be added. It is obvious that

PAGE 65

750 625 53 550C 500 375 250 125 ~£ 10 20 Figure 13 30 40 Time, sec. Effect on Cell Temperature of Changing the Thickness of the Cell 60

PAGE 66

54 the heat of fusion of the electrolyte in the cell cannot be of any assistance for the performance of thermal batteries must suffer badly when the temperature approaches the freezing point because of the change in the electrolytic conductivity of the electrolyte. Therefore, this factor represents a hypothetical change in parameters which would merit serious investigation if it should appear to contribute strongly to cell life. In Figure 14 the effect of a 20% change in cell enthalpy is demonstrated. The most obvious effect is that on peak temperature. Substantial decreases in the cell enthalpy while maintaining the same heat input from heat generators would result in overheating of the cell. A substantial increase in cell enthalpy with no change in heat input would result in the cell just barely becoming activated. Clearly the only practical approach would be to adjust heat generator input to the enthalpy requirements of the cell. Since the cells act as the primary heat reservoir after the first few seconds of operation, it is clear that increasing the cell enthalpy should be beneficial everything else being the same. This, in fact, is seen to be the case in Figure 14, where the slopes of the cooling portions of the curves are in proportion to the cell enthalpy. Thus, a combination of changes in cell enthalpy, by means of Composition or thickness changes, and in the heat generator by similar means would appear to offer possibilities in the way of meeting varying specifications of time of activation and life. Another possibility appears here, namely construction of duplex cells and heat generators, which have a portion of each made very thin

PAGE 67

55 375 250 125 10 20 40 Figure 14 30 Time, sec. Effect on Cell Temperature of Changing the Enthalpy of the Cell 50 60

PAGE 68

56 for fast activation, and a larger portion which activates slowly but serves as a heat reservoir to prolong life. Unfortunately the capacity of the analog computer did not permit a problem of this complexity to be studied. i 6. Effect of Rate and Level of Heat Generation by Chemical Reactions in Cells It has been noted earlier that the reactants in the electrochemical reaction can also react chemically to produce heat but no electricity. While it might appear that any such reaction would be wasteful, it turns out not to be so since the major limitation on cell life appears to be heat losses rather than exhaustion of reactions, and these intra-cell chemical reactions generate heat at a point where it is most effective in keeping the electrolyte molten. The effect on all temperature histories of these chemical reactions is illustrated in Figure 15 for a heat sink temperature of 25C. It is seen that the peak temperature of 550 C is reached at about 4 seconds, and that the life of the cell above 400 C is about 48 sec. when the normal heat generation is used. / A word about the heat generation is in order. The general shape of this function (which is approximate, of course) is also shown in Figure 15. The shape and the average rate of heat input to the system are based on experiments performed earlier at the Energy Conversion Laboratory of the University of Florida, and on educated guesses of experienced investigators familiar with the design and operation of a number of types of thermal batteries. In this study the normal rate of 3 heat generation was chosen to be 15 cal per cm per sec because of the

PAGE 69

750 -r 57 625 500 a) u % 375 250 125 AH = 3X No heat of reaction Form of the intra-cell chemical heat generation 10 20 30 Time, sec. Figure 15 Effect on Cell Temperature of Changing the Magnitude of the Intra-Cell Chemical Heat Generation 550C 60

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58 cell chemistry assumed. The heat generation rates of other cell reactions is covered by the range of heat generation terms used. From Figure 15 it may be seen that the peak temperature and the time to reach it are not strongly dependent on the value of the cell heat generation term unless very energetic and extensive chemical reaction occurs. A change in the heat generation term of 50% appears to change the life above 400C by about 20% without exceeding allowable peak temperature. Figure 15 also indicates that a heat generating reaction producing heat at approximately three times the rate in the normal situation would lead to a relatively small increase in the peak temperature, but it would result in a delay to reach the peak temperature of approximately 30 seconds and would result in approximately twice the life above 400 C. Figure 16 indicates the shape of the heat generation functions programmed in these experiments. The peak heat generation rate and the decay rates were held constant and the rise rate varied from onehalf to twice the standard rate. The data in Table 4 indicate that this change produced essentially no change in the cell temperature history. Activation times and peak temperatures turned out to be essentially unaffected. This kind of effect resulted, however, because the heat generation term chosen as a standard is such as to affect only the cooling portion of the temperature history of the cell in any significant way. If, for example, a system employing an intra-cell chemical heat generation rate more than twice as large as the standard should be studied, a

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60 -r 59 50 40 o > 30 20 10 30 Time, sec. Figure 16 Change of Initial Rate of Intra-Cell Chemical Heat Generation Term 50 60

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60 much more pronounced effect on the temperature history of the cell would be noted. Figure 17 illustrates the kinds, of changes in decay rate studied. Here the rise rate and peak generation rate were held constant and the decay rate varied from one-half to 1.3 times that of the standard. It is clear from examination of time-temperature curves of a cell (not presented here) and of the data in Table 4 that only a small effect is noted. There is no effect on activation time and peak temperature, but the cooling rate of the cell is affected slightly. In summary, it does not appear that alternation of either rise or decay rates of the intra-cell chemical heat generation function within broad limits would strongly affect the temperature history of a cell. However, the level of this type of heat generation, i.e., a change in the rate of chemical heat generation per unit volume could have a significant effect.

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60 -r 61 50 40 30 2010 Slope = 0.5X 40 30 Time, sec. Figure 17 Change of Decay Rate of Intra-Cell Chemical Heat Generation Term 50 60

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CHAPTER IV ANALOG COMPUTER MODEL II A. Description of the Model The next phase of the investigation concerning dynamic heat transfer in thermal batteries called for a study of the effect of the nature and structure of the insulation on the performance of thermal batteries. It should be obvious that Model I, which was discussed in the preceding chapter is inadequate to simulate an insulation region composed of more than one insulating material. This model was developed primarily to be able to study the effect of different combinations of insulating materials on the battery performance. In order to accomplish this simulation and at the same time keep the problem within the practical capabilities of the available analog computer (EASE 1032 Analog Computer), a number of simplifying assumptions were made. The most significant simplifying assumption of this model as compared to Model I is that the cell-generator stack is considered to have a homogeneous core having the average physical properties of the electrochemical cells and heat generators. Furthermore, this homogeneous core is assumed to be well-mixed with no temperature gradients. Therefore, only one ordinary differential equation describes the heat balance considerations in the core. This model is more complicated than Model I in that the insulation is considered to consist of four elements rather than one (see 62

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63 Figure 18). This model makes it possible to consider a core surrounded by four successive layers of different thermal characteristics, each, of course, having no internal temperature gradient, and having the thermal resistance lumped at the interfaces as in Model I. Another important difference between the two models resides in the manner in which the battery activation is achieved. In the case of Model I it was assumed that the heat generators constitute an essentially instantaneous heat source, and, in order to make the set of assumptions consistent, it was postulated that the initial temperature of the heat generator was 2200 C, which is the approximate temperature of the center of the heat generators. The temperature gradient between the heat generator and the cell was the driving force which promoted the heat transfer. In the case of Model II, since only one element is assumed to simulate the cell-generator stack, it is meaningless to consider core temperatures of the order of 2200 C, first because the enthalpy of the generators does not permit such average core temperatures and secondly because there exists a high-temperature limit 'of about 550 C above which the electrochemical system in the cells undergoes undesirable transformations (3). Therefore, after examination of the temperature histories of cell and generators as shown in Figure 6, 'where it was observed that the temperature of cell and generators was essentially the same only a few seconds after activation and that it remained so for the rest of the operating period, it was decided to assume for the core a heat generation term equivalent in calories to the enthalpy of the

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64 .Figure 18 Schematic Diagram of Battery Described by Model II /

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65 hot generators of Model I. This heat generation term was assumed to be an exponential decay function which was judged to be an adequate simulation of the sudden energy burst of the' heat generators. Another difference between the two models is that in Model I the boundary between top and lateral insulation was considered to be vertical and in this model, the boundary between the corresponding top and lateral elements in the insulation zone was assumed to be an imaginary surface cutting diagonally from the outer corner of the core to the outer corner of the assembly. This arbitrary shape assumed for the insulation elements has some rather interesting consequences in the form of the coefficients of the differential-difference equations which describe the system. This point will be discussed in more detail later. One final simplification of this model relative to Model I was the elimination of the heat of reaction term due to the cell. This was done mainly because the effect of the intra-cell heat of reaction had been evaluated in Model I, and because its inclusion would have complicated this model without contributing appreciably to the qualitative differentiation between different insulation arrangements. B. Development of the Mathematical Model It is shown in Appendix B that when the boundary surface between corresponding top and lateral insulation elements is assumed to have the shape illustrated in Figures 18 and B-lb, the volume of the top insulation element is given by

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66 V T1 ~A + 1 + Vi+l + J> (m 1> where V is the volume of the element, h its height, and R is the radius. The common area of contact between the top and lateral elements is given by the expression A TU = *(R. + R. +1 ) (IH-2) and the volume of the lateral insulation element is given by \i l^+i + R i R i+ i R i> + (R ? + i R 5 )s i (III 3) where S 1 is the height of the element as shown in Figure B-lb. The heat transfer coefficients were calculated in a manner identical to the one utilized in Model I, that is, by assuming the thermal resistance between two adjacent elements to be lumped at the interface. This resistance was evaluated by consideration of the mean heat path between two adjacent elements and their respective thermal conductivities. The numerical values of the heat transfer coefficients for Run 15 are shown in Appendix B. The differential-difference equations were developed by heat balance considerations of each of the elements making up the simulated battery. If the following substitutions are utilized for the purpose of simplification, the differential-difference equations which describe / the system take the form shown in Table 5. p.C .h. 9 V i-Ei-i (a J .. + R.R ... +RT) (III-4) i 3 i+l i i+l i and G. = -1-HL1(2r r.r R 2 ) + p.C ,S.(R?..-R?>, (IH-5) i 3 i+l i i+l i r i pi i i+l i

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67 C. Analog Computer Solution l The equations shown in Table 5 were programmed on the analog computer and the circuit is shown in Figure 19. The potentiometer settings corresponding to Run 15 are listed in Table 6 together with the corresponding mathematical expression for each potentiometer. The values of the physical parameters used in these simulations are shown in Table 7. It should be noted that Amplifier 17 in the computer diagram represents the exponential decay type of heat generation attributed to the core. This exponential function was designed so that the integral over time of this relation, namely the total amount of heat generated by unit volume of core, resulted in a core peak temperature near the maximum chosen. The value which produced the most realistic core peak temperature (525 C) corresponded to 353 ,cal./g of heat generator in the core (see Appendix B) Because of the high initial rate of change of temperature in the 4 core and the nature of the simplifying assumptions of this model, it was realized that the model could not predict these temperatures accurately during the first few seconds of operation. In any case, the primary concern in this study was with the cooling period, and with the effect of changes in the thermal properties of the materials and the geometric arrangement of the assembly on the cooling rate. D. Discussion of Results This model is a variation of Model I and it is based on essentially the same assumptions as the latter; therefore, many of the considerations discussed in the preceding chapter are applicable

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68 > o o St II (5 H I (S I 4) U 3 60 i-l En 0) O s 14-1 o B 60 n5 S-i 3 u 60 O

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69 TABLE 5 DIFFERENTIALDIFFERENCE EQUATIONS FOR MATHEMATICAL MODEL II Top Element 4 2„ 2, 5 4 L h R 5 \hk R A \l% < R s + R / > h / dT / F, U 4 A; P. CT 4 V F, H TL4 (T 4 V ^T Top Element 3 R 4 2r T43 R 3 2r t32 (R 4 + VN dT "Sf< T 3 V -Sf(*3 V ^7^ H TL3< T 3 V = 1? Top Element 2 2 2 ^32 R 2 l F 2 (T 2 V ~T 2 R 3 ^32 R 2 H T21 ( S + R 2 )h ? dT 9 Top Element 1 R 2 H T21 R 1 2r t1C (R 2 + V h i dT • ^f 1 ^ v • -Sf"**! ',/ x w^ v ii Core *r flic 2H L1C AH dC p c s/ c V r iP c (c V p c dt c pc 1 Fc pc H c pc Lateral Element 4 2R 5 S 5 H LAG n A 2R 4 S 4 H L43 T T < R 5 + W dL 4 ~ < L 4 A) — Oj (L 4 L 3> 5J "W L 4 ~ V dT Lateral Element 3 2R 4 S 4 H 143 a y 2R 3 S 3 H L32 T T < R 4 + R 3 )h 3 ~ (L 3 V :.-.-6 ? (L 3 V G 3 H TL3 (L 3 T 3 ) -fa dt

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70 TABLE 5 (Continued) Lateral Element 2 2R 3 S 3 H L32 L T 2R 2 S 2 H L21 /T \\ (R 3 + R 2 )h 2 % (L 2 V — T 2 < L 2 V ^ H TL2< L 2 V dL 2 = dt~ Lateral Element 1 %l fl T 2R l S l H LlC fT r < R 2 + V h l 5J (L i V — — (L i c) 5; — W L i V dt

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71 TABLE 6 COEFFICIENTS FOR PROGRAMMED DIFFERENTIALDIFFERENCE EQUATIONS: MODEL II Pot. No. 1 Mathematical Expression R 5 2r T4A + R 4 2H T43 + (R 5 + V h 4*W F 4 4 5 T4A F 4 5 4 T43 F 4 6 (R 5 + Vty^ F 4 2 R 4 2H T43 + R 3 2H T32 + < R 4 + V h 3 H TL. F 3 11 R 4 H T43 F 3 10 R 3 H T32 F 3 12 (R 4 + R 3 )h 3 H TL3 F 3 3 R 3 2H T32 + R 2 2H T21 + < R 3 + V h 2 H TT.2 F 2 15 R 3 H T32 F -' : ^ 2 • Pot. Setting 26.35 24.44 1.824 0.086 34.31 32.94 1.371 0.0014 0.2314 0.1762

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Pot. No. 16 72 TABLE 6 (Continued) Mathematical Expression 2 Il T21 R."^ Pot. Setti ins 0.0544 17 (R 3 + R 2 )h 2HTL2 0.00075 R 2 2H T21 + R 1 2H T1C + (R 2 + R l) h l H TLl 1.155 22 23 24 R 2 H T21 27 25 28 29 19 F l R i "tic F l (R 2 + VVtli F l "tic 2H L1C p G s 1 r c pc 1 Vc C pc AH ..; "tic p C S. *c pc 1 2H L1C R-p C r c pc AH p c *c pc (R 5 + R 4 )h 4HTU + 2R 5 S 5Hla4 f 2R 4 S aHl43 G, 0.635 0.516 0.0040 0.0121 0.1 0.00357 0.0085 0.500 :. 26.31

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73 TABLE 6 (Continued) Pot. No. 31 Mathematical Expression (r 5 + y y^ G, Pot. Setting 0.043 32 2R 5 S 5 H LA4 24.44 33 2R 4 S 4 H L43 1.824 20 (R 4 + R 3 )h 3 H TL3 H2R 4 S 4 H U3 + 2R 3 S 3 H L32 34.31 35 (R 4 + R 3 )h 3 H TL3 0.0007 36 2R 4 S 4 H U3 32.94 37 2R 3 S 3 H L32 1.371 21 (R 3 + Vh 2 H TL2 + 2R 3 S 3 G L32 + 2R^S 2 H L21 0.2310 39 (R„ + R a )h-BL„v 3 2 2 TL2 0.00038 40 2R-S -H, _^ 3 3 L32 G„ 0.1762 41 2R 2 S 2 H L21 0544

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74 TABLE 6 (Continued) Pot. No. 26 Mathematical Expression (R 2 + R 1 )h 1 H TU + 2R 2 S 2 H L21 + 2R lSl H Llc Pot. Setting 1.153 43 (R 2 + ^H^ 0.0020 44 2R 2 S 2 H L21 0.635 46 a i 8 Aic 0.516

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75 TABLE 7 AVERAGE PHYSICAL PROPERTIES AND DIMENSIONS OF THERMAL BATTERY COMPONENTS: MODEL II Core p 2.36 gms/cc C 0.183 cal/gmC k 0.05 cal cm U cm C sec. Mica p 2.7 c p 0.206 k 0.001 Asbestos P 1.5 C P 0.308 k 0.0004 Thermo flex P 0.193 C P 0.232 k 0.0002 Metal Steel P 7.9 C P 0.12 k 0.11

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76 here. The size of the available analog computer was again an important limitation which restricted the complexity that could be built into the model. In spite of this. limitation, it was decided to continue with the analog computer study because it was believed that it could lead to qualitative information which would be of value in more precise studies utilizing a digital computer. Thus some valuable qualitative information was obtained which considerably facilitated the development of the digital computer model. The results of the analog simulation of Model II are summarized in Table 81. Geometrical Shape of Insulation Elements The geometrical shape assumed for the insulation elements in Model II is really no more arbitrary than. that assumed for Model I but it has two advantages. 1. Compared to the shape assumed for the insulation elements in Model I, the shape used for the insulation elements in Model II reduced by one the number of elements -with which each insulation element was in contact, thus considerably simplifying the differential-difference equations of the system with the obvious consequence of a simplification in the analog computer circuit. 2. The shape assumed for the insulation elements in Model II yields expressions for the area and volume of the corresponding top and lateral insulation elements which illustrate the similarity of the radial and axial modes of heat transfer in a cylindrical structure described by this approximate model (see Appendix B) The similarity lies in the fact that the assumed shape yields values for the area and

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77 TABLE 8 EFFECT OF CHANGES IN THE INSULATION STRUCTURE ON LIFE TO 400C OF SIMULATED THERMAL BATTERIES Structure Run No Element No. Material 15 4 3 2 1 Metal (Standard) Mica Asbestos Thermof lex 16 4 3 2 1 Thermo flex > Mica Asbestos Metal 17 4 3 2 1 Asbestos (New St Asbestos dard) Thermof lex Thermo flex 18 4 3 2 1 Thermoflex Thermof lex Asbestos Asbestos 19 4 3 2 1 Asbestos Thermoflex Thermoflex Thermoflex 20 4 3 2 1 Thermoflex Thermoflex Thermoflex Thermoflex Life at Heat Sink Temp, of 25sec. 52 25 95 90 97 130 25Life 25Std. 1.00 0.50 1.00 0.95 1.02 1.37

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78 volume of a lateral insulation element which are twice the values obtained for these quantities in the corresponding top insulation element. Thus these elements have the same ratio of area to volume and the equations describing the differential heat balance in these elements have very similar coefficients, which is equivalent to very similar potentiometer settings for corresponding potentiometers as shown in Table 6 for the case of potentiometers 1 and 19, 4 and 32, and many others. Figure 23 illustrates the almost identical temperature histories of the corresponding top and lateral insulation elements in Model II as a result of the above discussed similarity in their descriptive equations. These results provided some justification for the assumption of spherical symmetry utilized in the digital computer model. 2. Simulation of Standard Thermal Battery The standard thermal battery which, in the framework of this model, consisted of a cylindrical core surrounded in order by Thermoflex insulation, asbestos insulation, a mica layer and a metal can, was simulated and the results are shown in Figure 20. A life above 400 C of 52.5 seconds was indicated. This figure is in reasonable agreement with experimental results. One significant aspect of this run was that the mica and metal layers did not rise appreciably above the temperature of the heat sink. In view of these results, the mica and metal layers could be considered extensions of the heat sink. This assumption liberated a section of the analog computer which could be used to simulate in more

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79 750 Configuration 4 = Metal 3 = Mica 2 = Asbestos 1 = Thermoflex 625 550C Core 500 u u a u
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80 detail the heat transfer process in other sections of the battery. 3 Effect of a Metal Layer Next to the Core on Core Temperature Some considerations indicated that it might be advantageous from the standpoint of maximization of battery life, to place the metal can next to the core instead of on the outside of the assembly. The intuitive justification of this arrangement was based on the high heat capacity of the metal. It appeared that the metal would absorb the heat given off by the core and act as a buffer between the core and the insulation. The results of a run testing this idea clearly revealed the inefficiency of this arrangement which clearly yielded a lower battery life than the standard as shown in Figure 21. The reason for this result is that the metal layer is essentially an addition of inert material to the core (inert in the sense that there is no heat generation associated with it)which only serves to increase its heat transfer area and hence the rate of heat loss from the core to the insulation. 4. Effect of Varying Insulation Arrangements on Core Temperature The negligible temperature rise of the outer mica and metal layers discussed before for the case of the standard battery allowed a more precise simulation of the effect of different insulation arrangements on the life of the battery. The number of well-mixed elements used to simulate the insulation space, which in the case of the standard battery is composed of a layer of Thermoflex and a layer of asbestos, was doubled. This allowed for the existence of a step-type temperature gradient in the simulation of an actual layer of homogeneous material

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750 625 500 o o o u ttf a H 375 250 125 30 Time, sec. Figure 21 Temperature Histories in Simulated Thermal Battery with Metal Next to Core

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82 such as Thermo flex. Figure 22 illustrates this effect on the simulation of the standard thermal battery. The results shown in this figure indicate that the mathematical model was very sensitive to this type of change. The standard battery life was almost doubled as a result of inaccuracies introduced by the new mode of simulation. It should be noted that battery life is a very stringent test for agreement between two models because of the low rate of cooling of the battery which makes small changes in the rate of cooling have a large effect on battery life. The significant discrepancy between the lives of the two simulations of the standard battery is believed to be a consequence of the assumptions of the model. The heat transfer coefficients utilized in the model are calculated on the basis of the thermal resistance of the two adjacent elements. The introduction of extra insulation elements reduces the thickness of the element in contact with the core and makes the value of the core-insulation coefficient more dependent on the core. Therefore, in a sense it may be stated that increasing the number of insulation elements decreases the relative approximation of the core results. Since in this case, the core was being represented by only one element already having by far the greatest volume of all the elements present, the intended improvement in the simulation of the standard battery, actually resulted in a poorer :.'.' i approximation. In spite of this disagreement between the two simulations of the standard battery, a valid qualitative comparison between different

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83 750 -r o o u 4-1 S-J I (U H 625500 375 250 125 Figure 22 60 Time, sec. Temperature Histories in Simulated Thermal Battery with Varying Insulation Arrangement 120

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84 insulation arrangements should exist using, of course, the new standard as the basis for the comparison. Figure 23 shows the effect of interchanging the asbestos and Thermo flex layers to be detrimental to the life of the battery. Figure 24 shows that a slight increase in battery life is obtained when three layers of Thermoflex and only one outer layer of asbestos are utilized. This result indicates the desirability of having as much Thermoflex as possible adjacent to the core. Figure 25 illustrates the obvious arrangement which results as a consequence of the data in Figure 24, this is, an all Thermoflex insulation. In connection with this figure it should be noted that, although it predicts the longest life of all the arrangements tested, the magnitude of the predicted life is clearly in disagreement with the trend established in the preceding figures. The reason for this is probably experimental error in setting the numerous potentiometers in the analog circuit, and the fact that this particular experiment was carried out on a different occasion which means that all the potentiometers had to be set, as opposed to the other three arrangements which were run on the same day. For these runs only a few potentiometers had to be changed to affect the simulation of a particular insulation arrangement :VM 11'

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85 750 625 500 o o
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86 750 Configuration 4 as Asbestos 3 = Thermoflex 2 = Thermoflex 1 Thermoflex 625" 550C 500 375 250125 20 Figure 24 40 60 Time, sec. Temperature Histories in Simulated Thermal Battery with Varying Insulation Arrangement 120

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87 750 625 Configuration 4 ~ Thermoflex 3 = Thermoflex 2 = Thermoflex 1 a Thermoflex 550C 500 Core o u 3 m U a) H 400C 375 250 125 20 Figure 25 T^ 60 40 60 80 100 Time, sec. Temperature Histories in Simulated Thermal Battery with Varying Insulation Arrangement 120

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CHAPTER V DIGITAL COMPUTER MODEL III A. Description of the Model The digital computer model which was programmed on the IBM 709 computer was that of a multilayered sphere consisting of a wellmixed spherical core surrounded by a core film in which heat conduction is occurring; this, in turn, is surrounded by spherical shells of insulation, and the whole is immersed in an infinite heat sink. This particular physical model was arrived at after careful consideration of the results of the two analog computer models and the requirements of the problem. One important factor in the use of the digital computer for solution of problems of this type is the matter of the convergence of the approximate (digital) solution to the exact one. If this factor is considered together with the dimensionality of the problem, that is, the number of dimensions involved iti the partial differential equation, it may be seen that the amount of effort (time and/or money) necessary to achieve an accurate solution, i.e., assymptotic approach of the digital computer solution to the exact one, becomes prohibitive. Therefore, simplifying assumptions must ordinarily be made which will lead to a solution on the computer, at reasonable expense, and at the same time conserve enough characteristics of the physical model so that the results may be meaningful. The result of the analog computer study on Model II (a right 88

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89 circular cylinder with a height equal to the diameter) indicated that the temperature difference between similar points in top and lateral insulation is essentially negligible. .In view of these results it appears, then, that the dimensionality can be reduced assuming spherical symmetry without seriously reducing its practicality. The choice of a spherical model reduces the partial differential equation to one containing a single spatial variable and time. The assumption of a homogeneous core was retained because, it was desired to place the greater emphasis in the study of the insulation materials and configuration as these affect battery life. It should be noted that the period during which the battery is hot enough for operation requires that the electrolyte contained in the cells be in a molten state. Furthermore, there exists during this period a continuous electrochemical reaction with its inherent ionic migration which contributes to convection heat transfer inside the cell. These two facts, together with the very high thermal conductivity of the molten electrolyte, suggest that the temperature gradients in any direction within the cells must be very small. Moreover, since the core is made of alternate disks of heat generators and cells, and the heat generators and the cells are at very nearly the same temperature after the first few seconds of operation, the temperature of most of the core should be essentially independent of position and be dependent on time only. On the other hand, the heat generators in the core provide a significant resistance to heat transfer and within the cells, in the region near the coreinsulation interface, a significant temperature gradient would

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90 be expected. The foregoing considerations suggested that the core could be considered to consist of two sections, an inner well-mixed section where the temperature was a function of time only, and an outer section consisting of a film where temperature gradients existed. This film concept is implied in all the computations of convection heat transfer which make use of relations such as Newton's cooling law. The preceding discussion indicates that it would be desirable to simulate heat transfer from the core to the insulation by transfer from a well-mixed element through an imaginary film where heat conduction would be the heat transfer mechanism. The problem of estimating the thickness of this film is a very difficult one, specially from the analytical viewpoint. On the other hand, if the film thickness is used as the simulation parameter, it could be adjusted in such a manner that the performance of a standard battery would be approximated very closely by the mathematical simulation, provided that the same values for all other parameters were used. This model also includes the effect of intra-cell heat of reaction, and the impulse-type heat input of the generators in the core. B. Development of the Mathematical Model On making a thermal energy balance over a spherical shell of thickness Ar within the sphere, the following relations can be obtained: 2 Thermal energy in Uttt ql (V-l) at r r Thermal energy out at 47r(r + Ar) .qj L ; (V-2) r + Ar I r + Ar

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91 Thermal energy generated Accumulation 47TT ArAH 2 4Tir Ar G C oT Pp o7 (V-3) (V-4) The thermal energy balance equation states that rate of rate of rate of rate of thermal thermal + thermal energy = accumula(V-5) energy in energy out generation tion If equation (V-l) to (V-4) are substituted into equation (V-5) and one takes the limit as Ar approaches zero, the following relation results: ST lim 4Tr(r + Ar) q| + 47Tr ql + 47rr ArAH (V-6) ST = Ar— ^ r Ar -IS 47TT ArpC hence gj m -S r q at ^r" + AH P C r and since q = -k 9T r -3or equation(V-7 )becomes (V-7) (V-8) oT o7 = a a T 2 oT -\ 2 r or AH (V-9) where T is the temperature, t is the time, r is the radius, and a is the thermal diffusivity, — — (54). P p The mathematical model describing the temperature distribution in a composite sphere consisting of a spherical central core having internal heat generation surrounded by concentric spherical layers of \

PAGE 104

92 different materials, takes the following form: for 0
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93 Figure 26 Schematic Diagram of Battery Described by • Model III

PAGE 106

94 C. Finite-Differences Approximation The partial derivatives which appear in equations (V-10) to (V-15) can be approximated by finite difference equations using Taylor's series. In the present case, T = f(r,t), the f(r,t) may be expanded about r for a fixed value of t as follows (41): 2 -s2. 3 a 3 i or The function f(r,t) can also be expanded as follows: a ST (Ar) 2 o 2 T Ar 3 o 3 T T(r-Ar s t) T Ar ^ + ^~ -"j — -J + or or (V-16) (V-17) If equations (V-16) and (V-17),., are subtracted and terms of the order of (Ar) are neglected, an approximation of oT/or is given by oT T(r + Ar,t) T(r Ar,t) (V-18) Sr 2Ar If equations (V-16) and (V-17) are added and terms of the order of i 2 2 (Ar) are neglected an approximation of S T/or is found to be O T T(r + Ar,t) 2T(r,t) + T(r Ar.t) (V-19) or^ (Ar)' D. Development of the Computer Program The elements of the idealized spherical thermal battery can be divided into a finite number of sections by dividing the radius into finite increments. Within one of the insulation elements, equations (V-10), (V-12), etc., would take the following form: dt n T r-*Ar 2T r + T r-Ar 2 T r4Ar T r-Ar (Ar) 2 2(Ar) (V-20)

PAGE 107

95 where n is the element number and R n and the equation on the right can be expressed similarily. Hence, the equation at an interface point takes the following form T T T T u £ r ~ Ar i r-J&r r „„. k n — ~ = k n + l -5TT / (V 22) n n+1 This equation can be solved for T. as a function of T and T which J J+1 j-1 are functions of time of the form giver in equation (V-20) The heat generation term, the last term of equation (V-9) is not affected by the differencing technique since it does not involve

PAGE 108

96 any derivatives. Therefore, it remains unchanged in the finite difference form of the equation. The form of the AH term is an exponential decay function of time. Another important assumption which is common to all three models is the consideration of the battery as being in contact with a perfect heat sink. In this model, this assumption affects the form of the boundary condition at the surface of the external element of the battery. Since that interface is in contact with a material whose temperature does not change with time, and its thermal conductivity is essentially infinite, the temperature at that interface must remain constant and hence the time derivative at that point is equal to zero. The heat transfer process in the well-mixed portion of the core requires a particular analysis. Since the temperature within this sphere is a function of time only, a heat balance in the sphere can be expressed in finite difference form as dT 2 T T P A C P A V A— k B A -Ar~~ + AHV A < v "23) B and T l = T 2 (V-24) where V A is the volume of element A. These equations with their corresponding constants were programmed as the DERIV subroutine of the AM0S program. The model programmed described the heat transfer in the core which included a well-mixed section where the temperature was a function of time only, and an outer 'section or film where heat conduction resistance existed. In both sections of the core, the heat generation term which simulates the heat given off by the generators and the intra-cell

PAGE 109

97 chemical reaction was included. Six other elements were included which can be assumed to possess different values of thickness, density, thermal conductivity and heat capacity: The program was modified during the investigation to include the feature that any one of the "insulation" elements (all the elements except the core) may be considered to possess heat generation capabilities. This modification permitted the simulation of a battery where heat generators might be used in place of normal insulating materials in order to provide extra heat to the battery, and thus improve its performance. This point will be discussed more fully in the Discussion of Results. E. The AM0S Program The AM0S program is an extension of the Adams-Moulton-Shell (55) routine programmed at the University of Florida by Fairchild, Wengrow and May as a package of F0RTRAN subprograms. AM0S has the unique advantage of automatic truncation error control of increment size without requiring any special procedure for restart due to change in increment size. AM0S requires that the problem be stated as a set of first-order ordinary differential equations with initial conditions specified. In this case, the system of partial differential equations was approximated by ordinary differential equations using appropriate differencing techniques. The general problem for which the AM0S program was designed consists of solving sets of equations of the form dx. ir m£(x i> %) CV-25)

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98 Immediately, techniques such as Simpson's Rule or the trapozoidal rule must be discarded because x. appears on the right hand side of the equation. The simplest constant increment method is Euler's method which can be extended by introducing the concept of predictor and corrector. This is known as the Modified Euler method. A number of predictor-convector methods have been devised based on Newton's interpolation formula. Those which predict from point n to point n+1 based on derivatives at point n and preceding points, and which correct the same interval adding the derivative at point n+1, carry the name Adams-Moulton. Because of stability considerations and the possibility of error control, the Adams-Moulton method is the best available constant increment procedure. Milne (56) has developed similar equations which have lower error for comparable effort but the possibility of instability makes it unsuitable for machine computation. The Runge-Kutta-Gill method (57) has the advantage of requiring no history points, but error control is a significant problem in this method. The constant increment equations are more thoroughly discussed in standard numerical analysis tests (58,59). The Adams-Moulton-Shell method is a divided difference technique based on Newton's interpolation formula and is an extension by Shell (55) of the Adams-Moulton constant increment method to a non-constant increment method. The advantage of the method can be appreciated where truncation error control is included with the result of frequent changes in interval size. A method requiring constant interval sizes must

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99 generate new starting points after each change in interval size, whereas a method which can utilize non-constant increments can continue after a change of interval size without .interruption. The AM0S program consists of a main program and sixteen subprograms, fourteen of which remain unchanged for all problems of the general type discussed. The other two subprograms are called DERIV and I0DRV, and they contain the information which is specific to each problem such as the particular form of the differential equations involved and the specification of the input and output requirements of the problem. The program listings used in this study are shown in Appendix D (60) F. Discussion of Results The digital computer program was subjected to its first test by studying the simulation of a physical model for which an analytical solution can be evaluated readily. This model consisted of a homogeneous sphere having a uniform initial temperature of 500 C everywhere except at its surface, where the initial temperature was C. The analytical and computed temperature profiles are shown in Figure G-l. It may be seen that, with 27 simultaneous equations describing the system, the computed and analytical results agreed within 5% up to values of time of 20 seconds. It was observed that the computed results always predicted a lower temperature than the exact solution, and that the difference between the two solutions increased with time. On the other hand, it should be pointed out that the computer program was not ideally suited to simulate the homogeneous sphere

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100 model because it was developed to simulate a heterogeneous sphere having a well-mixed section in the core where no radial temperature gradients exist. Therefore, a number of changes had to be made in the input data in order to adapt the program to the homogeneous sphere model, such as assuming the radius of the well-mixed section of the core to be zero and assuming all of the insulation layers to be composed of the same material. This assumption produces an inherent loss of precision due to the fact that at each interface point in the model, the equation describing the heat transfer process is the continuity boundary condition between adjacent but supposedly different materials, whereas in the case of a homogeneous sphere the more precise equation involving the second partial derivative could be used. In view of the above factors, the agreement between the approximate and exact solutions can be considered quite satisfactory. Figure C-l also illustrates the effect of geometry on the behavior of the homogeneous system. The figure shows a temperature profile of the homogeneous sphere after 50 seconds, and the center point temperatures after 50 seconds of homogeneous right circular cylinders having equal height and diameter, and having (1) a radius and (2) a volume equal to those of the sphere. The temperature of the cylinders was calculated from the tables due to Olson and Schultz (23). It should be noted that the sphere is a more efficient heat reservoir than the cylinder having equal volume owing to the smaller area to volume ratio of the former, and that the center temperatures of these two solids differ by only approximately ten per cent after 50 seconds. This may be consi-

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101 dered another justification for the assumption of spherical symmetry in the simulation of thermal batteries especially because these contain a core where smaller temperature gradients than those encountered in pure heat conduction exist, owing to the molten state of the electrochemical cells and to the ionic migration within them. 1Simulation of the Standard Thermal Battery In this model a different approach was used to the battery than had been used in the two analog models. In the analog models, the thermal properties of the materials comprising the battery were incorporated into the model and the value predicted for the life of the thermal battery was accepted. In the case of the spherical model studied on the digital computer, as a consequence of the assumption of a well-mixed section in the core where no temperature gradients and hence no resistance to heat transfer exist, and of a film in the core where the thermal resistance of the core is concentrated, an unknown parameter, namely the thermal resistance of the film, must be considered. It has been pointed out by Jakob (7) that this is the proper approach to analysis of convection heat transfer. *""""* The amount of effort involved in obtaining a very precise solution of the mathematical model under consideration would be prohibitive because of the numerous different alternatives tttat had to be considered for the purpose of design optimization. In any case, a high degree of accuracy would not be justified because the choice of spherical symmetry was only an approximation of a battery. In view of the above considerations it was decided to choose the

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102 arbitrary parameter in such a manner that a feasible model would yield results very similar to those given by a standard battery. Specifically, the life of the simulated thermal battery at heat sink temperature of 25C was arbitrarily made to be 30 seconds. At the beginning of the investigation, it appeared that there were two arbitrary parameters which resulted from the assumption of a finite film in the core, the thickness of the film and the thermal conductivity of the film. A number of simulations involving different values of these two parameters indicated that within the range of values tested, the same response was obtained for different values of thickness and thermal conductivity of the film if their ratio was the same. This result indicated that essentially only a single thermal resistance parameter had to be chosen arbitrarily. The form of the heat generation term was chosen to be an exponential decay which would make the core reach 400C in approximately 0.2 seconds and of such a magnitude that the total heat generated would be equal to the heat given off by the equivalent number of heat generators in the core as determined from calorimetric measurements. The value of the intra-cell heat generation was established to be 10% of the total heat generated by the heat generators. It was again given a triangular form with the maximum heat generation being reached after 15 seconds. The values of the parameters used in the simulation of the standard battery are listed in Table C-l. This simulation was effected using 27 grid points. The temperature profiles in the battery are shown in Figure 27, and the

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104 temperature history for the core is illustrated in Figure 28. The numerical results are summarized in Table 9. The life of the battery was dependent on the number of grid points used in the simulation, but since the deviation from the exact solution was estimated to be of about 107, the number of 27 grid points was chosen in order to keep the problem within the limits of economic feasibility. The film resistance was chosen so that with this number of points the life of the standard battery would coincide with the arbitrarily selected life, i.e., 30 seconds. As will be discussed later, the model proved to be very satisfactory when used in simulating battery performance under other conditions because it predicted life above 400 C which agreed reasonably well with experimental results under these new conditions. 2. Effect of Varying Insulation Arrangements on Core Temperature The experience gained from the analog study greatly reduced the range of possible insulation arrangements which might have yielded an improved battery performance. The all-Thermof lex insulation structure proved to be the optimum available insulating material. The results of this and other insulation arrangements are shown in Figure 28, Table 9 shows that a slight increase in battery life may be obtained if Thermoflex only is used instead of the Thermof lex-asbestos structure. On the other hand, the use of an all-asbestos insulation is seen to be considerably worse than both the standard and the optimum. Similarily an idealized material having the average properties of equal volumes of Thermoflex and asbestos, which was denominated "mixinsulation", also showed a decrease in battery life compared to the standard. In short,

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105 TABLE 9 EFFECT OF PARAMETERS AND CHANGES ON LIFE TO 400C OF A SIMULATED-THERMAL BATTERY Life at Heat Sink Run No. Description temperature of 25C sec. 25Life 25Std. 22 Standard (Thermof lexAsbestos Radius of Core=1.50cm) 30.0 ; 1.00 27 All-Thermof lex 32.8 1.09 28 All Asbestos 16.2 0.54 29 Asbestos-ThermofL 2X 17.0 0.57 30 All Mixinsulation (See Page 104 ) 20.6 0.68 31 p = 0.193, c = o.: 232, k 0.0001 51.0 1.70 32 p = 1.50, C 0.308, P k = 0.0001 24.0 0.80 33 p 15.0, C 0.308, r p k = 0.0001 16.2 0.54 34 p 150, C 0.308, P k = 0,0001 15.0 0.50 35 p 0.015, C = 0.308, p k = 0.0001 60 2.00

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106 TABLE 9 (Continued) Life at Heat Sink Temperature of 25Life Run No Description 25C, sec 25Std. 36 Radius of Core = 1.40 All-Thermoflex 32.0 1.06 37 Radius of Core = 1.60 30.6 1.02 All-Thermoflex 38 No Intra-Cell Heat Generation 25.0 0.83 39 AH = 2X* (Double Intra Cell Heat Generation) 37.0 1.23 42 p C = 2X r g Pg 36.0 1.20 43 p C = 4X r s pg 47.0 1.56 44 p C = 2X, AH = 2X ^8 Pg ( 47.0 1.56 All-Thermoflex 45 Generator Outside Thermoflex (10 sec. de lay) 55.5 1.85 46 Generator Outside > Peak Temp. Thermoflex (2 sec. delay) Too High 47 Generator (p C = 4X) ^g Pg Outside Thermoflex (2 delay) sec. 62.0 2.06 48 Generator (p C 4X) 40.0 / 1.33 g Pg / Outside Asbestos (2 sec. delay) 49 3 inch diameter battery 122.0 4.05 X times standard value of parameter being varied

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108 from the standpoint of heat transfer, mixed fiber insulation is less effective than a layer of Thermoflex surrounded by layers of less effective insulation. Reversing the order of the Thermoflex and asbestos layers in the standard battery yielded a lower battery life than the standard. All of the above findings are consistent among themselves, and they substantiate the finding that an all-Thermof lex insulation should give the optimum performance among the available insulating materials. 3. Effect of Idealized Insulating Materials on the Core Temperature The investigation of the response of battery performance to insulating materials having arbitrary idealized properties was probably the most interesting aspect of the work and probably the most instructive development from an academic standpoint. A number of materials having some thermal properties equal to real materials and having other unrealistic properties were assumed in order to test the effect on battery life of the parameters of the system. It was found that the arbitrary manner in which the optimization criterion was defined, namely life above 400 C, greatly affected the choice of an optimum idealized insulator. It is believed, and the results shown in Figure 31 seem to confirm it, that the insulator with the lowest thermal diffusivity will yield the longest life above the heat sink temperature, however, if the life above 400 C is considered the criterion, then it is seen that for the values tested, an insulation having low thermal diffusivity will produce a shorter battery life than an insulation having a high thermal diffusivity if both have the same

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109 thermal conductivity. This result can be explained in the following manner. When a high cut-off temperature is chosen as the criterion for battery life, the initial mechanism of .heat transfer becomes very significant. This is illustrated in Figure 30 where the temperature at a point in the insulation close to the core is seen to be greater for insulation with a low heat capacity than for insulation with a high heat capacity if both materials have the same thermal conductivity, and the same initial temperature difference exists between the point in the insulation and the core. The higher temperature of the insulation with the low heat capacity produces a smaller temperature gradient between this point and the core, thus reducing the rate of heat loss from the core. It should be noted that as time proceeds, the insulation with the lower heat capacity cools more rapidly, and therefore, if the cut-off temperature is chosen low enough, the life above the cut-off temperature will eventually be longer for the material having the higher heat capacity. It was noted that, if the thermal diffusivity is reduced by reducing the thermal conductivity while the density and heat capacity are kept constant, then the' battery life is improved regardless of the cut-off temperature. 4, Effect of Core Radius on Core Temperature The effect of changing the radius of the core on the life of the battery was tested and the results are shown in Figure 32. These results indicate that, though arrived at empirically, the radius of the core had already been optimized in the standard thermal battery. Battery lives lower than that of the standard were obtained for radii of the

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114 core both greater and smaller than the standard core radius. 5. Effect of Intra-Cell Heat Generation The effect of the intra-cell heat generation on the life of the battery was shown to be a significant one. Figure 33 indicates that the side benefit of intra-cell heat generation provides almost a 207 increase in battery life. This figure also shows that another 207= increase could be gained if reacting systems yielding twice the heat of reaction of the standard system could be utilized. 6. Effect of Heat Sink Temperature on Core Temperature Figure 34 illustrates the effect on the core temperature of a change in the temperature of the external heat sink. The low temperature of -65 F reduces the battery life by 507, while the high temperao ture of 160 F produces an increase in battery life of about 307 relative, of course, to the life of the standard battery at a heat sink temperature of 25 C. These results including the values obtained for the peak temperatures, are in very close agreement with experimental values obtained under these conditions. 7. Effect of Changes in the Heat Capacity of the Heat Generators on the Core Temperature The effect of increasing the heat capacity of the heat generators was considered a feasible and beneficial possibility because it permitted a larger quantity of heat in the core without increasing the peak temperature. It was assumed that this change in the heat capacity of the generators did not affect the value for the thermal conductivity of the core. The results are shown in Figure 35. They indicate that

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118 an increase of 20% above the life of the standard battery could be achieved by doubling the heat capacity (or the density) of the heat generators, and that an increase of 50% could be accomplished if it could be quadrupled. 8. Core Temperature of Improved Thermal Battery Figure 36 illustrates the effect that some feasible design improvements could have on the life of the battery. It was assumed that an all-Thermof lex layer could be used, that an electrochemical system could be developed which would yield twice the intra-cell heat generation of the standard battery, and that the density or heat capacity of the heat generators could be doubled: The results indicate that an increase in battery life of about 56% could be obtained. 9. Effect on Core Temperature of Delayed Heat Generation Within the Insulation L — — The effect of delayed heat generation within the insulation was found to be very beneficial to battery life. Figure 37 indicates that in some arrangements of this type, the life of the battery could be doubled. This change appears to be more difficult to implement from a design standpoint than those suggested before. The main difficulty appears to be in the manner in which the delayed heat generation is to be achieved. It was found that commonly used heat generators could not be placed in intimate contact with the core because their ignition would increase the temperature of the core above the upper limit. Placing a layer: of Thermoflex of about 1 mm in thickness between the core and the heat generator seemed to prevent the core temperature from rising too high and at the same time provided good insulation for the

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121 core, thus increasing the life of battery considerably. 10. Effect of Change in the Volume of the Battery on Core Temperature The effect of a change in the volume of the battery on the core temperature constituted an important test of the mathematical model. Figure 38 illustrates the results which again are in reasonable agreement with experimental observations. It should be noted that an increase in the volume of the battery does not change the peak temperature appreciably but it reduces the rate of temperature decay very considerably. This is a consequence of the volume increasing as I the third power of the radius while the area increases only as the square of the radius. Therefore, the total amount of heat generated will increase more rapidly (since it is constant per unit volume) than the increase in the rate of heat loss (which is proportional to the area).

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CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS The objectives of this work were to provide temperature histories for specific locations within a simulated thermal battery, and to study the effect of changing the construction configuration and properties of the materials as to optimize battery life and to understand the effects that these changes produce. It was concluded that the analog computer models provided an adequate representation of thermal batteries for drawing qualitative conclusions. They also provided very useful insight into the heat transfer processes that occur in a battery, and this was quite valuable for the development of the digital computer model. The digital model was a more complete representation of a thermal battery than were the analog models, and it permitted more complex configurations to be simulated. The results of this investigation indicate that some gain in battery life could be effected if an all-Thermof lex insulation were used instead of the standard Thermof lex-asbestos configuration. It was concluded that the ratio of core volume to insulation volume in the standard battery studied was essentially optimum. The results also indicated the desirability of developing systems which would have higher intra-cell chemical heat generation than the one used in the standard battery. Heat generators having a higher thermal conductivity or density than the standard were shown to be beneficial to battery life. 123

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124 An examination of the cooling portion of the temperature histories shows that battery life could be extended appreciably if electrochemical systems could be developed which would remain in a molten state below 400 C. It was also shown that placement of heat generators in the insulation section surrounding the core could provide a delayed heat input to the assembly and substantially increased life, but this improvement may not be feasible at this time because of difficulty in controlling the time delay of the heat generator ignition. It was further concluded that the choice of a cut-off temperature of 400 C as the criterion for battery life strongly influenced the choice of an optimum idealized insulator. For an arbitrary cutoff temperature there exists an ideal material which should have the lowest possible thermal conductivity and value of density and heat capacity which should be low if the cut-off temperature is high and high if the cut-off temperature is low. It should be noted that the lowest value of thermal diffusivity for the insulation does not yield o the maximum life above 400 C. w $

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LIST OF SYMBOLS 2 A area cm o A heat sink temperature, C C temperature of the core, C C heat capacity, cal P gmC F defined by equation III-4, dimensionless factor to allow for interchange between gray surfaces F dimensionless factor to allow for interchange between black surfaces G defined by equation III-5 h height of element, cm H pseudo-convection heat transfer coefficient, cal cm z, secC k thermal conductivity, cal. cm. cm z secC o L temperature of lateral insulation, C q rate of heat transfer, cal sec Q rate of heat transfer R radius of element, cm S height of lateral insulation, cm t time, sec. / T temperature of element, C 3 V volume, cm Greek Letters a thermal diffusivity j3 defined by equation C-5 125

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126 Greek Letters (continued) e / emissivity, dimensionless / 2 p density, g/cm / a StefanBo It zmann Constant Subscripts a refers to the heat sink A refers to the heat sink c refers to the cell, or to the core g refers to the generator th i refers to the insulation or to the i element j refers to the j element 1 refers to lateral L refers to lateral m refers to middle, i.e., mg refers to middle generators r refers to radius R refers to radiation t refers to top T refers to top 1 refers to element number 1 2 refers to element number 2 / 3 refers to element number 3 4 refers to element number 4

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LITERATURE CITED 1. Goodrich, R. B., and Evans, R. C, J. Electrochem. Soc, 99, p. 207c (1952). 2. Vinal, G. W. "Primary Batteries," John Wiley & Sons, Inc., New York (1950). 3. Selis, S. M., et al, J. Electrochem. Soc, 110, p. 469 (1963). 4. McKee, E. S., Proc. 10th Annual Battery Res. and Dev. Conf., p. 26, New Jersey (1956). 5. Johnson, L. B., Ind. Eng. Chem. 52, p. 868 (1960). 6. Hill, R. A. W., Trans. Farad. Soc, 53, p. 1136 (1957). 7. Jakob, M., "Heat Transfer", vol. I, John Wiley and Sons, Inc., New York, p. 393 (1949). 8. Biot, J. B., Biblioteque Britannique, 27, p. 310 (1804). 9. Fourier, J. B. J., "Theorie Analitique de la Chaleur", Paris (1822). 10. Newton, I., Phil. Trans. Roy. Soc London, 22, p. 824 (1701). 11. Stefan, J., Sitzungsber, D. Kais Akad D. Wiess, Wien. Math.Naturwiss. Klasse, 79, p. 391 (1879). 12. Boltzmann, L. Wiedemanns Annalen, 22, p. 291 (1884). 13. Carslaw, H. S., and Jaeger, J. C., "Heat Conduction in Solids", Second Ed. Oxford (1959). 14. McAdams, W. H., "Heat Transmission", Third Ed., McGraw-Hill, New York (1954). 15. Anthony, M.L., General Discussion on Heat Transfer, Inst. Mech. Engrs. (London) and Am. Soc. of Mech. Engrs. (1951). 16. Newcomb, T. B., British J. Appl. Phys 9, p. 370 (1958). 17. Wasserman, B. a J. Aero. Science, 24, p. 924 (1957). 18. Siede, P., J. Aero Science, 25, p. 523 (1958). 19. Friedman, N. E., Trans. Am. Inst. Mech. Engr 80, p. 633 (1958). 127

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128 20. Yang, K. T., J. Appl. Mech., 25, p. 146 (1958). 21. Gurney, H. P., and Lurie, J. Ind. Eng Chem., 15, p. 1170 (1923). 22. Groeber, J., Zeitschr d. Ver deutsch. Ing., 69, p. 705 (1925). 23. Olson, F.C.W. and Schultz, O.T., Ind. Eng. Chem., 34, p. 874 (1942). 24. Newman, A. B., Ind. Eng. Chem., 38, p. 545 (1936). 25. Heisler, M. P., Trans. Am. Soc Mech. Engrs 69, p. 227 (1947). 26. Rosenhow, W. M., Trans. Am. Inst. Mech. Engrs., 68, p. 195 (1946). 27. Paschkis, V., and Hlinka, J. W., Trans. Am. Inst. Mech. Engrs., 79, p. 1742 (1957). 28. Binder, L. Dissertation, Techn. Hochschule Muenchen, Munich (1911) 29. Schmidt, E., Forch. Gebiete Ing., 13, p. 177 (1942). 30. Nessi, A., and Nissolle, L., "Methodes graphiques pour l'etude des installations de chauffage", Dunod, Paris (1929). 31. Patton, T. C, Trans. Am. Inst. Mech. Engrs., 66, p. 990 (1944). 32. Longwell, P. A., A.I.Ch.E. Journal, 4, p. 53 (1958). 33. Langmuir, I., Adams, E. Q., and Meikle, G. S., Trans. Am. Electro. Chem. Soc, 24, p. 53 (1913). 34. Beuken, C. L., Dissertation, Saechs Bergakademie Frieberg, Triltsch and Huther, Berlin (1936). 35. Paschkis, V., and Baker, H. D. Trans. Am. Inst. Mech. Engrs., 64, p. 105 (1942). 36. Karplus, W. J., and Soroka, W. W. "Analog Methods", McGraw-Hill New York (1959). 7 37. Emmons, H. W. Trans. Am. Soc. Mech. Engrs., 65, p. 607 (1943). 38. Southwell, R. V., "Relaxation Methods in Engineering Science", Oxford University Press, New York (1940). 39. Allen, D. N. de G., "Relaxation Methods", McGraw-Hill, New York (1954). 40. Dusinberre, G. M., Trans. Am. Soc. Mech. Engrs. 67, p. 703 (1945).

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129 41. Mickley, H. S., Sherwood, J. K. and Reed, C.E., "Applied Mathematics in Chemical Engineering", Second Ed. McGraw-Hill, New York (1957). 42. Hildebrand, F. B., J. Math. Phys., 31, p. 35 (1952). 43. Evans, G. W., et al, J. Math. Phys!, 34, p. 267 (1956). 44. Brian, P. L. T., A.I.Ch.E. Journal 7, p. 367 (1961). 45. Douglas, J., Jr., and Peaceman, D.W., A.I.Ch.E. Journal 1, p. 505 (1955). 46. Yavorsky, P. M., et al, Ind. Eng. Chem., 51, p. 833 (1959). 47. Dickert, B. F., M. S. Thesis, U. of Fla., Gainesville, Fla. (1960). 48. Home, R. A., and Richardson, D.L., Proc. 18th Annual Battery Res. and Dev. Conf., p. 75, New Jersey (1964). 49. Forsyth, A.R., "Calculus of Variations", Dover Publications, Inc., New York (1960). 50. Walker, R. D., Jr., and Chipley, E. L., private communication, (Dec, 1964). 51. Perry, J. H., "Chemical Engineers' Handbook", Third Ed., 0. 488, McGraw-Hill, New York (1950). .. 52. Brown, G. G., "Unit Operations", John Wiley & Sons, Inc., New York, (1956). 53. Johnson, C. L., "Analog Computer Techniques", McGraw-Hill, New York, (1956). 54. Bird, R. B., Steward, W. E., and Lightfoot, E. N., "Transport Phenomena", John Wiley and Sons, Inc., New York (1960). 55. Shell, D. L. General Electric Company Technical Information Series No. DF 58AGT679, G.E. Co., Cincinnati, Ohio. z 56. Milne, W. E., "Numerical Calculus',' Princeton University Press, New Jersey (1949), 57. Gill, F., Proc. Cambr. Phil. Soc. 47, Part 1,(1951). 58. Lapidus, L., "Digital Computation for Chemical Engineers", McGrawHill, New York (1962). 59. Hamming, R. W. "Numerical Methods for Scientists and Engineers", McGraw-Hill, New York (1962). 60. Fairchild, B. T., Wengrow, H. R., and May, F. P., "AM0S: Numerical Integration of Differential Equations with the Adams -Moul tonShe 11 Method", Chem. Eng. Dept U. of Fla. Gainesville, Fla. (1965).

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APPENDICES

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APPENDIX A DETAILS OF ANALOG MODEL I COMPUTATION OF HEAT TRANSFER COEFFICIENTS The following heat transfer coefficients yield the values for the potentiometer settings listed in Table 1 (based on the data given in Table 2). -1 1 n 156 H :i-a [0.0002 = 0.00256 -1 H tirli H ti-g 0.46 L_ 0.156 0.0002 0.0002 2 = 0.000372 -1 1 0.156 1 T H g-li 0.1 0.0002 2 0.0005 2 -i -1 1 0.156 0.46 + = 0.002 H g-c 0. 0002 2 0.0005. „ 1 0.1 1 0.1 "P = 0.000763 H c-li 0.0005 2 0.10 2 1 0.156 .46 0.0002 2 .10 -1 = 0.01 0.00254 H. = H li-a ti-a Note: The value of 0.46 is the radius of the circle having an area equal to one-half of the circle whose radius is 1 56 131

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132 TABLE A-l SUMMARY OF RUNS Run No, Description Standard Results Tabulated in Table 3; Shown in Figures 5, 6 and 7 k = 1/2 X Tabulated in Table 3; Shown in Figure 8 k = 1.5X h = 2X ti Tabulated in Table 3; Shown in Figure 9 Tabulated in Table 3; Shown in Figure 10 h = 2X tg Tabulated in Table 3; Shown in Figure 11 p = 2X, h = 1/2 X 'g tg Tabulated in Table 3; Shown in Figure 12 h = 1.2X c Tabulated in Table 3; Shown in Figure 13 h = 1.5X c Tabulated in Table 3; Shown in Figure 13 h = 2X c Tabulated in Table 3; Shown in Figure 13 10 p C = 0.8X 'c pc Tabulated in Table 3; Shown in Figure 14

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133 Run No. Description Results 11 p C = 1.2X *c pc Tabulated in Table 3; Shown in Figure 14 12 Change Magnitude of Intracell Heat Generation Tabulated in Table 3; Shown in Figure 15 13 Change Initial Rate of Intra-cell Heat Generation Tabulated in Table 3; Shown in Figure 16 14 Change Decay Rate of Intra-cell Heat Generation Tabulated in Table 3; Shown in Figure 17 1

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Run No, 134 TABLE A2 SUMMARY OF POTENTIOMETER SETTINGS Pot. No, 1 2 3 4. 5 6 7 8 9 11 12 14 15 17 18 19 20 21 22 23 24 25 31 32 34 42 43 44 45 49 50 53 0.663 0.3583 0.974 0.328 0.700 0.123 0.075 0.184 0.132 0.076 0.744 0.692 0.804 0.748 0.235 0,286 0,163 0,398 0.143 0.324 0.143 0.143 0.143 0.143 0.0715 0.042 0.0378 0.0565 0.022 0.067 0.0319 0.0152 0.015 0.015 0.0256 0.2907 0.288 0.292 0.292 0.148 0.126 0.130 0.132 0.138 0.064 0.00346 0.0034 0.0034 0.0034 0.00346 0.123 0.123 0.123 0.123 0.0615 0.1395 0.070 0.208 0.069 0.1195 0.021 0.0176 0.0254 0.0097 0.017 0.00366 0.0019 0.0056 0.0018 0.00366 0.0063 0.0031 0.0095 0,0063 0.0063 0.8644 0.451 1.280 0.745 0.8591 0.0107 0.0053 0.016 0.005 0.0107 0.006 0.0056 0.008 0.0062 0.003 0.00465 0.00234 0.0069 0.0046 0.0058 0.003 0.0030 0.00435 0.0034 0.0015 0.615 0.615 0.615 0.615 0.153 0.143 0.143 0.143 0.143 0.0715 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.0071 0.0071 0.0071 0.0071 0.0071 0.615 0.615 0.615 0.615 0.615 0.615 0.615 0.615 0.615 0.615 0.321 0.321 0.321 0.321 / 0.312 0.715 0.714 0.714 0.714 0.715 0.143 0.145 0.145 0.145 0.143 0.308 0.308 0.308 0.308 0.308 0.715 0.714 0.714 0.714 0.715

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135 TABLE A-2 (Continued) Run No, 8 10 Pot No. 1 • '. 0.621 0.700 0.700 0.663 0.663 2 0.105 0.076 0.076 0.123 0.123 3 1.408 0.235 0.235 0.744 0.744 4 0.243 0.324 0.324 0.286 0.286 5 0.286 0.0948 0.119 0.0715 0.179 6 0.024 0.0374 0.040 0.0337 0.042 7 0.0364 0.0284 0.0304 0.0256 0.0319 8 0.577 0.194 0.243 0.148 0.364 9 0.247 0.064 0.064 0.126 0.126 11 0.00346 0.0052 0.00415 0.007 0.00346 12 0.246 0.0615 0.0615 0.123 0.123 14 0.159 0.186 0.159 0.224 0.1395 15 0.012 0.018 0.020 0.0168 0.021 17 0.00366 0.00366 0.00366 0.00366 0.00366 18 0.0063 0.0063 0.0063 0.0063 0.0063 19 0.8614 0.900 0.879 0.9301 0.8644 20 0.0107 0.0107 0.0107 0.0107 0.0107 21 0.003 0.003 0.003 .006 0.006 22 0.0058 0.0058 0.0058 0.0058 0.0058 23 0.0015 0.0015 0.0015 0.003 0.003 24 1.30 0.153 0.153 0.615 0.615 25 0.286 0.948 0.119 0.0715 0.179 31 0.050 0.050 0.050 0.050 0.050 32 0.050 0.050 0.050 0.050 0.050 34 •0.0071 0.0071 0.0071 : 0.0071 0.0071 42 0.615 0.615 0.615 0.615 0.615 43 0.615 0.615 0.615 0.615 0.615 44 0.321 0.321 0.321 0.321 0.321 45 0.715 0.472 0.594 0.357 0.895 49 0.143 0.0948 0.119 0.0719 0.179 50 0.308 0.308 0.308 0.308 / 0.308 53 0.715 0.472 0.594 0.357 0.895

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Run No. • 11 : Pot. No. 1 0.663 2 0.123 3 0.744 4 0.286 5 0.119 6 0.042 7 0.0319 8 0.242 9 0.126 11 0.00346 12 0.123 14 0.1395 15 0.021 17 0.00366 18 0.0063 19 0.8644 20 0.0107 21 0.006 22 0.0039 23 0.003 24 0.615 25 0.119 31 0.050 32 0.050 34 0.0071 42 0.615 43 0.615 44 0.321 45 0.595 49 0.119 50 0.308 53 0.595 136 TABLE A2 (Continued)

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137 e u 0) H a o •H U o
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APPENDIX B DETAILS OF ANALOG MODEL II AREA AND VOLUME OF THE INSULATION ELEMENTS OF MODEL II The lateral area and volume of the top insulation elements assumed in Model II can be evaluated by the methods of elementary calculus. The equation of the straight line shown in Figure B-la is "=7v^v< E -V < B 1 > This line can be rotated around the y axis to yield the shape of the insulation element. The area of the element is given by A T / 2 dA = J 2 2-nRdy J 2 2?tRh (B-2) Va J J -a ^O ^1 k i y 1 R l 2 l Therefore, the area is R 2 R 2 *r 17?^ -S1 2 + V (B 3 > The volume of a top element is given by V T / 2 dV = / 2 7lR 2 dy -/ 2 2 TiR h dR % H"*l (B-4) which becomes 3 3 ,R„ R V, ^L 2 A ^(R? + R,R, + R?) (B-5) 3(R 2 R x ) 3 v 2 1 2 138

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139 (a) (R r P) (R 2 ,h) (b) >* *2 *> t „ D i 1 ; *> c I ..* (c) Top Insulation Lateral Insulation Figure B-l Details of the Shape of Insulation Elements

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140 The volume of a lateral insulation element can now be expressed as V L = TlR^h 2|(R* + R^ + R*> + 77(3*2 R i )S l ( fi 6) where S x is defined in Figure B-lb. This expression simplifies to V L = ^(2R^ R X R 2 R*) + tt(R* R^)S 1 < B 7 > It should be noted that the dimensions of the model are such that S = R and h = R 9 R, When these expressions are substituted into equation(B-7)it becomes v 7T(R 2 R L ) (2R 2 R R R 2) + ^^2 R 2 } (B-8) or = ^2 V (2R* R X R 2 R*) + R X (R 2 + V (B-9) L 3 which yields upon simplification Y-<*2 +R l*2 + *l> (B-10) Equation(B-lO) illustrates the fact that for these dimensions and this model the volume of a lateral insulation element is twice the volume of the corresponding top insulation element. This circumstance thus offers a simplification in the calculation of the potentiometer settings; furthermore, it reduces the number of elements with which a given element of lateral insulation may be contact. This is illustrated in Figure B-lc where it is seen that the element of lateral insulation in an alternate structure has two faces in contact with the heat sink, thus giving a more complicated system of equations.

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141 COMPUTATION OF HEAT TRANSFER COEFFICIENTS The following equations for heat transfer coefficients yield the values for the potentiometer settings listed in Table 6 based on the data given in Table 7. Element 4 Metal Element 3 Mica Element 2 Asbestos R 5 = 1.90 R. = 1.80 4 R 3 = 1.79 R 2 = 1.69 Element 1 Thermoflex R. = 1.59 I W = H L4A ^43 \kZ ^32 == ^32 ^21 = H L21 .05 ,110 2.20 "I -1 .110 1 .05 1 .005 ,001 = .183 ^lC ^1,2 J_ .005 + ,05 .001 1 .0004 0004 ,05 1 .05 .0002 .05 .795 .0002 1 .110 _1 .001 JL .0004 (.005) (.530 + .925) (.527 + .897) (.500 + .870) -1 = .0077 --I = .00267 = .00245 = .0756 -1 .0007 .0003

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142 **TL1 .0002 (.470 + .820) .05 + -1 .465 = .00015 .0002 (.005) -1 = .00292

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143 TABLE B-l SUMMARY OF RUNS Run No, Insulation Structure Results 15 4 Metal 3 Mica 2 Asbestos (Standard) 1 Thermo flex Tabulated in Table 8 Shown in Figure 20 16 4 Thermo flex 3 Mica 2 Asbestos 1 Metal Tabulated in Table 8 Shown in Figure 21 17 4 Asbestos 3 Asbestos 2 Thermoflex (Standard) 1 Thermoflex Tabulated in Table 8 Shown in Figure 22 18 4 Thermoflex 3 Thermoflex 2 Asbestos 1 Asbestos Tabulated in Table 8 Shown in Figure 23 19 4 Asbestos 3 Thermo felx 2 Thermoflex 1 Thermoflex .Tabulated in. Table 8 Shown in Figure 24 20 4 Thermoflex 3 Thermoflex 2 Thermoflex 1 Thermoflex Tabulated in Table 8 Shown in Figure 25

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144 TABLE B-2 SUMMARY OF POTENTIOMETER SETTINGS -; Run No. 15 16 17 18 19 Pot. No. 1 26.35 2.635 1.066 5.40 0.934 4 24.44 1.40 0.710 3.665 0.710 5 1.824 1.23 0.355 1.73 0.223 6 0.086 0.0054 0.00068 0.002 0.00068 2 34.31 2.077 0.587 4.20 4.24 11 32.94 0.705 0.360 1.86 2.48 10 1.371 1.371 0.226 2.34 1.76 12 0.0014 0.0014 0.00076 0.0037 0.0039 3 0.2314 0.340 4.244 0.571 3.624 15 0.1762 0.1762 2.48 0.240 1.86 16 0544 0.163 1.76 0.331 1.76 17 0.00075 0.00075 0.0041 0.00077 0.0041 8 1.155 0.249 3.384 0.547 3.384 22 0.635 0.0895 1.85 0.357 1.85 23 0.516 0.0620 1.53 0:190 1.53 24 0.0040 0.0972 0044 0.00083 0.0044 9 0.0121 0.0246 0.0118 0.01608 0.0118 25 0.00357 0.00912 0.00513 0.00658 0.00513 28 0.0085 0.0155 0.00667 0.0095 0.00667 29 0.500 0.500 0.500 0.500 0.500 19 26.31 2.633 1.065 5.40 0.933 31 0.043 0.0027 0.0003 : 0.001 0.00034 32 24.44 1.40 0.710 3.665 0.710 33 1.824 1.23 0.355 1.73 0.223 20 34.31 2.077 0.586 4.20 4.24 35 0.0007 0.0007 0004 0.0001 0.002 36 32.94 0.705 0.360 1.86 2.48 37 1.371 1.371 0.226 2.34 / 1.76 21 0.2310 0.339 4.242 0.571 3.62 39 0.00038 0.00038 0.002 0004 0.002 40 0.1762 0.1762 2.48 0.240 1.86 41 0.0544 0.163 1.76 0.331 1.76 26 1.153 0.198 3.842 0.628 3.842 43 0.0020 0.0436 0.0022 0004 0.0022 44 0.635 ,0.0895 1.85 ,0.357 1.85 46 0.516 0.0620 1.99 0.271 1.99

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145 TABLE B-2 (Continued) Run No. 20 Pot. No. 1 5.40 4 3.665 5 1.73 6 0.002 2 3.61 11 1.86 10 1.75 12 0.0037 3 3.624 15 1.86 16 1.76 17 0.0041 8 3.384 22 1.85 23 1.53 24 0.0044 9 0.0118 25 0.00513 28 0.00667 29 0.5 .19 5.40 31 0.001 32 3.665 33 1.73 20 3.61 35 0.002 36 1.86 37 1.75 21 3.624 39 0.002 40 1.86 41 1.76 26 3.84 43 0.002 44 1.85 46 1.99

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146 COMPUTATION OF FIRST ORDER TRANSFER FUNCTION USED TO SIMULATE HEAT GENERATORS Pot. 27 = 0.1 and Pot. 29 = 0,5 The amplifier circuit represents the following function J = dj dt (B-ll) dJ J dt (B-12) InJ = -t+c, J = ce at t = J = -40 -t J -40e -t volts Since 1 volt =25 J = -1025e" t / (B-13) (B-14) (B-15) (B-16) Since Pot. 29 -= 0.5, AH_ / 512e (12.6cc) = 12.6(512) (B-17) PC and hence the total heat given off by the core is given by AH = (.432)12.6(512) = 2785 cal (B-18)

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147 COMPUTATION OF HEAT GIVEN OFF BY CORE Model I Volume of Gen. = 7r(1.56) 2 ( 15) = 1.15 cc Heat Given Off by Gen. = 1.25 (. 130) (1. 15) (2200-25) = 406 cal. Volume of Cell = 7T(1.56) 2 ( 1) .765 cc Weight of Cell 3.48(.765) =2.67 gms Heat Per Unit Volume of Core in the First 5 sec. = 213 cal/cc Heat Per Unit Volume of Core From 5 to 80 sec. = 120 cal/cc Heat Per Unit Volume of Gen. in first 5 sec. = 353 cal/cc 283 cal/gm Model II Volume of Core = 7T(1.59) 2 (1.59) = 12.6 cc Heat Given Off by Core = 2785 cal Heat Per Unit Volume of Core in the First 5 sec. = 221 cal/cc Heat Per Unit Weight of Gen. in First 5 sec. = 442 cal/cc 353 cal/gm This result is a consequence of assuming equal generators and cell volumes in Model II whereas there is 50% more generators than cell volumes in Model I. It should be noted that on the basis of core volume, / the amount of heat given off is essentially the same (213 and 221 cal/cc)

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APPENDIX C DETAILS OF DIGITAL MODEL III SIMULATION OF CONDUCTION HEAT TRANSFER IN A HOMOGENEOUS SPHERE In order to test the validity of the finite difference approximation to the partial differential equations describing heat conduction in a composite sphere, the computer program was used to simulate a system for which the analytical solution was known. The model consisted of a homogeneous sphere having an initial temperature equal to 500 C everywhere except at the surface where the temperature was constant at zero degrees. The analytical solution for this model is presented by Carslaw and Jaeger (13). The results for both the approximate and exact solutions are shown in Figure C-l. 148 1

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149 a o en 3 cd CD M cu Cu C/3 CO O 01 p CD to o i o n o 60 C O O o CU o 3 CO 0) oi CD 4J 3 g o O en 3 en U CU > cd C o o o c>
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150 TABLE C1 DATA USED IN SIMULATION OF STANDARD THERMAL BATTERY This computation simultaneously solves 27 ordinary differential equations. Material Radius Density Thermal Cond. Heat Capacity Core 1.50 2.36 0.00054 0.183 Thermo flex 1.60 0.193 0.0002 0.232 Thermo flex 1.70 0.193 0.0002 0.232 Asbestos 1.75 1.50 0.0004 0.308 Asbestos 1.79 1.50 0.0004 0.308 Mica 1.80 2.70 0.001 0.206 Metal 1.90 7.90 0.110 0.120 AH is represented by AH = 2185.0e" 10 0t + 0.050t for t<15.0 and AH 2185.0e" 10, ^ 15,0) + 0.050(15.0) 0.017(t-15.0) for t > 15 The well-mixed portion of the core has a radius of 1.323 cm. >:v ,:m : ;'..k

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151 TABLE C-2 SUMMARY OF RUNS Run No. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Description Car slaw and Jaeger p. 234 (13) Standard (20 points) (Thermof lex-asbestos) 10 grid points 30 grid points 30 grid points, AM0S 4 40 grid points AllThermof lex All-asbestos Asbestos -Thermof lex Mixinsulation AllThermof lex k 0.0001 All-asbestos k 0.0001 All-asbestos k = 0.0001 3 p Results Shown in Figure Tabulated in Table 9 Shown in Figure 27 = 15 All"asbestos k 0.0001, p 150 Tabulated in Table 9 Shown in Figure 28 Tabulated in Table 9 Shown in Figure 28 Tabulated in Table 9 Shown in Figure 28 Tabulated in Table 9 Shown in Figure 28 Tabulated in Table 9 Shown in Figures 29 and 31 Tabulated in Table 9 Shown in Figures 29 and 31 Tabulated in Table 9 Shown in Figure 29 Tabulated in Table 9 Shown in Figure 29

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152 TABLE C-2 (Continued) Run No. 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Description All-asbestos k = 0.0001, p = 0.015 Radius of core = 1.40 Radius of core = 1.60 No heat of reaction ZH = 2X (double intracell chemical heat generation) Heat sink temperature = -65F Heat sink temperature = 160F s c Pg = 2X ^g C pg 4X VW = 2X H = 2X All-Thermof lex Generator outside Thermoflex (10 Sec, delay) Generator outside Thermoflex (2 Sec. delay) Generator (p,C 4X) >& Pg outside Thermoflex (2 Sec. delay) Generator (P„C: = 4X) g Pg outside, asbestos (2 Sec. delay) 3 -inch battery diameter Results Tabulated in Table 9 Shown in Figure 29 Tabulated in Table 9 Shown in Figure 32 Tabulated in Table 9 Shown in Figure 32 Tabulated in Table 9 Shown in Figure 33 Tabulated in Table 9 Shown in Figure 33 Shown in Figure 34 Shown in Figure 34 Tabulated in Table 9 Shown in Figure 35 Tabulated in Table 9 Shown in Figure 35 Tabulated in Table 9 Shown in Figure 36 Tabulated in Table 9 Shown in Figure 37 Tabulated in Table 9 Shown in Figure 37 Tabulated in Table 9 Shown in Figure 37 Tabulated: inTable 9 Shown in Figure 37 Tabulated in Table 9 Shown in ^Figure 38 y

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153 MATHEMATICAL ILLUSTRATION OF THE SIMILARITY BETWEEN THE FINITE DIFFERENCE APPROXIMATION AND THE APPROXIMATION BASED ON WELL-MIXED (LUMPED PARAMETER) ELEMENTS The equation which describes the unsteady-state heat transfer by conduction in a homogeneous body in one-dimensional Cartesian coordinates is given by St St (C-l) where a k/pC The threepoint central difference approximation of this partial derivative is given by ii V 2T ) + Yi dx 2 <*o* ( C-2) hence equation C-l takes the form dT. dt a L j+1 2T. + T. J J-l < C-3) (Axr L The heat transfer may be assumed to occur by transfer from one wellmixed element to an adjacent one having the thermal resistance lumped at the interface as shown in the following diagram: Yi j+i •,.' i ,'-i'. i ,i_ ; ;-_ 3

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154 A heat balance on element j gives the following relation AH(T. T.) AH(T. T. .,) = pC AAx dT. J-l J J J+l P 1 dt (C-4) and this equation can be rewritten in the following form A. (Ax) 2 L T.^ n 2T. + T. J+l J J-l (C-5) dT. dt where ]3 = HAx If equations (C-3)and (C-5)are compared, it is seen that if H is calculated by the relation H = k/Ax the two equations are identical. In the case of spherical coordinates, the finite difference form of the heat transfer equation is given by dT. dt = a Vi : 2T i + T i-i 2 Vi AR 2 R M 111 (C-6) If R = jAR equation ( C-6 ) reduces to dT. dt a jAR (j+l)T j+1 2jT. + O-DTj.! (C-7) The well-mixed (lumped resistance) approximation takes the following form

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155 47TR^_ 1 H(T._ 1 T.) 4-nR 2 H(T. T 4J-1 ) = pC,47iR
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APPENDIX D DETAILS OF THE COMPUTER PROGRAM TABLE D-l MNEMONICS OF I0DRV AND DERIV ALPHA a, Thermal Diffusivity AVGT Average Temperature of Element B2 Auxiliary Constant B3 Auxiliary Constant C Constants of Differential Equations CP Heat Capacity DELH Heat Generation of the Core DELHG Heat Generation of Generator Surrounding the Core FJ Floating Point Values of Jl, J2, etc. HBEG Constant in Heat Generation of Core INDEX Parameter Indicating Whether a Heat Generator Exists in the Insulation Jl Highest Grid Point Number of Element 1 J2 Highest Grid Point Number of Element 2 J3 Highest Grid Point Number of Element ,3 J4 Highest Grid Point Number of Element 4 J5 Highest Grid Point Number of Element 5 J6 Highest Grid Point Number of Element 6 J7 Highest Grid Point Number of Element 7 JDCORE Parameter Indicating Whether an Artificial Profile Will be Used to Initiate Run Instead of the Core Heat Generation 156

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157 TABLE D-l (Continued) N Number of Grid Points in Each Element NAME Material Making up Each Element NELG Element Number Which Becomes a Heat Generator NFREQ1 Frequency of Output in Element 1 NFREQ2 Frequency of Output in Element 2 NFREQ3 Frequency of Output in Element 3 NFREQ4 Frequency of Output in Element 4 NFREQ5 Frequency of Output in Element 5 NFREQ6 Frequency of Output in Element 6 NFREQ7 Frequency of Output in Element 7 NPOINT Highest Grid Point Number in the Assembly R Radius RC Radius of the Well-Mixed Section of the Core RHO Density T Time TBREAK Time at Which the Intra-Cell Heat Generation Term Begins to Decay TEMP Temperature TEMPD Derivative of the Temperature 7 / THRCON Thermal Conductivity TKON Coefficient of Exponent in Core Heat Generation Term TST Coefficient of Exponent in Generator Heat Generation Term SL0PE1 Initial Slope of Intra-Cell Heat Generation Term

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158 TABLE D-l (Continued) SL0PE2 Decay Slope of Intra-Cell Heat Generator Term SUM Sum of Grid Point Temperature Within an Element i e : i

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159 Figure D-l.. Flow Diagram for the AM0S Program Start 100 h CL0CK Note Beginning Time 120 I0AM0S Read and Write Initial AM0S Inf. 130 ; I0DRV Read and Write Initial DERN Inf, 150 # STPTS Compute Starting Points 160 g 0STPTS Write Starting Points 200 g SHIFT 420 \7 I0AM0S Write Final Output 430 {/ CL0CK Note Final Time 100 START 400 Diagnostic Statement V 420 g I0AM0S

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160 410 Jl. Diagnostic Statement 420 "I0AM0S 340 \f RSTART Calculate STPTS 150 y STPTS 365 ^7 AM0S3 Calculate XC 380 7 TRUNK Test Truncation Error 200 fr Shift 240 I0AM0S Write AM0S Information 250 TERM Check For Termination 305 AM0S3 Calculate XP 320 DERN Calculate XPD 350 % TRUNK Call Trunk with Truct Too Large J 305 g AM0S3 Repeat XP

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BIOGRAPHICAL SKETCH Mario Ariet Antiga was born July 9, 1939, in Havana, Cuba. In June, 1956, he was graduated from Colegio de Belen, Havana, Cuba. He entered the University of Florida in September, 1956; he received the degree of Bachelor of Science in Chemical Engineering with Honors in June, 1960, and the degree of Master of Science in Engineering in December, 1962. Since that time he has been employed at the University by the Department of Chemical Engineering as a full-time engineering assistant He is married to the former Laudelina Rosa Sust. They have three children, Ana Maria, Maria Elena and Mario Alberto. He is a member of Sigma Tau, Tau Beta Pi, Phi Kappa Phi, and Sigma Xi. 209

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This dissertation was prepared under the direction of the Chairman of the Candidate's Supervisory Committee, and has been approved by all members of that committee. It. was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. April 24, 1965 ./ Dean, College of Engineering 5 Dean, Graduate School SUPERVISORY COMMITTEE: Chairman ~7~V (I .C o, \rm,0i^cl—^ VA ) \ T^/S f. 7tl U // K. Xy M/'^A


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