
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00026392/00001
Material Information
 Title:
 Dynamic heat transfer in composite miniature structures
 Creator:
 Ariet Antiga, Mario, 1939
 Publication Date:
 1965
 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 ) nonfiction ( marcgt )
Notes
 Thesis:
 ThesisUniversity of Florida, 1965.
 Bibliography:
 Bibliography: leaves 127129.
 Additional Physical Form:
 Also available online.
 General Note:
 Manuscript copy.
 General Note:
 Vita.
Record Information
 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 nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 13409735 ( OCLC )
0021888922 ( ALEPH )

Downloads 
This item has the following downloads:

Full Text 
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 IntraCell 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 DifferentialDifference Equatiohs for Mathematical
Model I.............................................. 28
2 Coefficients for Programmed DifferentialDifference
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 DifferentialDifference Equations for Mathematical
Model II............................................. 69
6 Coefficients for Programmed DifferentialDifference
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
A1 Summary of Runs........................................... 132
A2 Summary of Potentiometer Settings......................... 134
B1 Summary of Runs........................................... 143
B2 Summary of Potentiometer Settings......................... 144
Cl Data Used in Simulation of Standard Thermal Battery....... 150
C2 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 IntraCell Chemical Heat Generation...... 57
16 Change of Initial Rate of IntraCell Chemical Heat
Generation Term................................. 59
17 Change of Decay Rate of IntraCell 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 1CaSeatuc~H a Eistories in Simulatea Theeal Eattery
wi~th vayin Insulation C z,"nI; n ....... .
3 Yristories in Simualated he la Lterzey with.
V ying insulation Arranngeme t. ................ 30
Im Terertura 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
A1 Analog Circuit of Heat of Reaction Term.............. 137
Bl Details of the Shape of Insulation Elements.......... 139
Cl 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 wellmixed 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 differentialdifference equations
which in turn were solved by the AdamsMoultonShell 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 allThermoflex 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 intracell 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 cutoff temperature imposed by a highmelting
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 cutoff 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 cutoff 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/LiClKClK2Cr204/Ni and
Mg/LiClKCl/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 cellgenerator 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 timetemperature 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 unsteadystate 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 (II1)
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 (II1) 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 (II1) 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 (112)
Equation (112) 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 (113)
have been used since Stefan (11) found this relation and Boltzmann
(12) proved it theoretically for a perfectly black surface. Equation
(113) 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 blackbody
10
radiation. For surfaces not absolutely black, 0 must be modified if
the StefanBoltzmann law is to be applicable.
The basic equation of heat accumulation for small linear
changes of temperature is
Q = pC VAT (114)
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 flowtemperature field and an electrical flow
voltage field of the same geometrical configuration.
Beuken (34), and Paschkis (35) developed largescale, permanent
analog devices whose principal elements were resistors and condensers,
and they were able to solve unsteadystate 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 onetoone 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
tionresponse 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 highspeed 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 threedimensional steady
state heat transfer processes. Although the relaxation technique can
be used for unsteadystate 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 finitedifferences 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 finitedifference
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 finitedifference formulation, and solved on a digital
computer the problem of heating homogeneous cylindrical briquettes.
Dickert (47) used the explicit finitedifference approach for the
solution on an IBM 650 digital computer of the unsteadystate 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 wellmixed 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 (115)
while the volume is given by
V = iR 2h (116)
If the volume is considered to be fixed, the area can be expressed by
17
A= 271R2 + 2nR V (117)
which can be differentiated with respect to R and equated to zero
to give
dA 2V
S= 47nR  = 0 (118)
"dR 2
R
Equation (118) may be solved for the volume to obtain
V = 2nR3 (119)
This would be the value of the volume corresponding to a minimum area,
but
V 7R2h (II10)
hence
h = 2R (II11)
Equation (II11) 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 (1112)
while the area is given by
A = 2xy + 2xz + 2yz (1113)
If the volume is considered to be fixed the area can be expressed as
2xV 2yV
A = 2xy + 2xV + 2V (1114)
xy xy
which can be differentiated partially with respect to x and y to give
aA 2V
S= 2x 2 =0 (1115)
x
18
aA 2V
= 2y 2 0 (1116)
y
These equations may be solved for x and y,respectively, to give
x = V1/3 and y= Vl/3 (1117)
1/3
which results in a value of z = V when substituted into equation
(1112). 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 (1118)
and the area is given by 1/3
A = 47T( (1119)
For the cylinder, the radius is given by
( V\ 1/3
c = IV1/ (1120)
and the area 2/3
A = ,47r (1I21)
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 (II22)
A 4
c
For a cube the side 1 is given by
1 = VI/3 (1123)
and the area
A = 6V2/3 (II24)
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 (1125)
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 suggeststhat the temperature with
in each element would be rather uniform, hence it was assumed that each
element was "wellstirred", 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 intracell 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
*r4
Top Generator
Cell H
 Middle Generator 
e1
Fiure 3 Schematic Diaram of Battery Described by Model
Figure 3 Schematic Diagram of Battery Described by Model I
24
wellmixed 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 intracell 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 reactiontime 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 wellmixed 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 pseudoheat 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 pseudoheat 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 (III1)
Resistance of Cell of Generator
h1 1 hc_ 1 ht (1112)
2 2
H k~ 2 k 2
gc g
H = 1 (III3)
gc
1 h
k 2 k\ 2
c g
where H is the heat transfer coefficient between the heat generator
gc
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 onehalf
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 (1114)
gli
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) (III5)
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 StefanBoltzmann constant
(4.92 x 10 kgcal/m hr k ), and T is the temperature in degrees Kelvin.
F =1 1 (1116)
__. + 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 (1117)
gcR AT + gc
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 = 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)
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
RR2Htia(Tti Ta) 2RhtiHtili(Tti li) R2Htig(Tti Ttg) PiCpiR 2h ti (III8)
dt
Initial Condition For Equation III8 at t = 0 Tti = T (III9)
Top Generator
2 2 2 dT 00
R2H ig(T 27Rh Htg g (litg Tl) RHg (T Tc) = pgC R2hdt (III10)
tig tg ti tg gli tg li gc tg c g t
Initial Condition For Equation III10 at t 0 Tt = 22000C (III11)
Cell
2 2 2 dT
RH (T T ) 2Rh H li(T T i 7R H (T Tr) = pc PR2h dT (11112)
gc tg c li c li gc ng c c pc e
Initial Condition For Equation III12 at t = 0 T = T (11113)
Middle Generator
R2H (T T) 2Rh H (Tmg Tli)= pC R2h dTm (III14)
gc mg c g gli mg li gpg t dt
2 2
Initial Condition For Equation 11114 at t = 0 T = 22000C (11115)
mg
TABLE 1 (Continued)
Lateral Insulation
t27Rhi tili li Tti 2 tg gli (Ti Ttg 27RhcHcli Tli c 2R glili mg
2
27Ro(hti + htg + h + h )Ha(Ti Ta) = Cp2(RoR)(ht + + h + h dTi ( 16)
Sti tg t lia i a i i o ti tg c dt
2 2
Initial Condition For Equation III16 at t = 0 Tli = T (III17)
i a (1 7
49V \ 944
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 c11 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 III7)
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 firstorder 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 tig 0.123
pC h
g pg tg
3 1 H + 2ht H + H 0.744
p tig R g1 c
g Cpghtg
4 Htig 0.286
PiCpihti
5 c 0.143
pC h
c pc c
6 htg gli 0.042
p.C .( o l)(ht + 3/2 h + h )
Spi R ti tg c
7 ti Htili 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 cli c
9 2 H +t H 0.126
gCpghtg LC R
AH h
11 c 0.0035
100
2H
12 2c 0.123
PgCpg tg
33
TABLE 2 (Continued)
Pot. No. Mathematical Expression Pot. Setting
h H
14 c cli 0.1395
R
Pp. ( /2)(h11 + 3/2 h + h )
15 (htg/2)Hgli 0.021
p.C .( o l)(hti + 3/2 h + h )
R hi. tg c
17 H Ta 0.0037
17 tia a
Pipih ti
18 R/R(Hla T) 0.0063
18 o lia a
lO1piC (O 1)
R
19 cli + 3/2 H l tg + Htiliti
.C (~o 1)(hti + 3/2 ht + h )
i tg c
Ro/R(ht + 3/2 htg + h)Hlia 0.8644
PiC i(o l)(hti + 3/2 ht + h )
20 2tili 0.0107
Rp.C P
RPiCpi
21 2Hli 0.006
Rp C
g pg
2H 0.0046
22 cli 0.0046
Rp cC
c pc
34
TABLE 2 (Continued)
Pot.No. Mathematical Expression Pot. Setting
23 Hli 0.003
Rp C
g pg
24 H c 0.615
pC h
g pg tg
25 gc 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
gcR g11
pC h R
g pg tg
43 2HcR 0.615
pC h
g pg tg
44 1 H + tg H l + Hg R 0.321"
44hI Htig g11 gcR
pCh R
g pg g
45 gcR 0.715
p C h
c pc c
2h 1
49 1 H + c Hc + H 0.143
50 gcR 0.308
p C h
g pg tg
53 gcR 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 intracell 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 pseudoconvection 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 ii
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 25Std.
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. IntraCell 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 intracell 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 intracell
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 onehalf 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) intracell 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 intracell chemical heat generation built in),
the peak temperatures decrease as the cell thickness increases; however,
the intracell 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 intracell 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 intracell 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 intracell 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 IntraCell 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 intracell 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 IntraCell 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 onehalf to 1.3 times that of the standard. It
is clear from examination of timetemperature 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 intracell 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 IntraCell 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 cellgenerator 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 wellmixed 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 cellgenerator 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 hightemperature 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 differentialdifference 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 intracell 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 Blb, the volume of the
top insulation element is given by
66
VTi = (R +RR + R) (III1)
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 ) (1112)
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 (1113)
L1 3 i+l i i+1 R+1+ R
where S1 is the height of the element as shown in Figure Blb.
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 differentialdifference 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 differentialdifference equations which describe
/
the system take the form shown in Table 5.
F. (R2 + RR. + (1114)
1 3 i+1 i 1+1 i
and
G C ih(2R RR R ) + 2Cp 2 (R (II5)
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
1T
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
L2L
A 32 T
LL
C 1 I.C. = 40.0 Volts
Figure 19
1Analog Comuter Diagram of Model II
TA Cmt \ o \
69
TABLE 5
DIFFERENTIALDIFFERENCE 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 + (LT
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
DIFFERENTIALDIFFERENCE 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 differentialdifference
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 indicatedthat 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 wellmixed 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 steptype 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 coreinsulation 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
S4000 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 multilayered 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 IntraCell 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 DifferentialDifference Equations for Mathematical Model I... 28 2 Coefficients for Programmed DifferentialDifference 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 DifferentialDifference Equations for Mathematical Model II 69 6 Coefficients for Programmed DifferentialDifference 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 Al Summary of Runs 132 A2 Summary of Potentiometer Settings ....". 134 B1 Summary of Runs 143 B2 Summary of Potentiometer Settings 144 Cl Data Used in Simulation of Standard Thermal Ba'ttery 150 C2 Summary of Runs 151 vi
PAGE 7
LIST OF FIGURES Figure Â— 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 Â„ 55 15 Effect on Cell Temperature of Changing the Magnitude of the IntraCell Chemical Heat Generation 57 Change of Initial Rate of IntraCell Chemical Heat Generation Term 59 17 Change of Decay Rate of IntraCell Chemical Heat Generation Term. 6^ Vll
PAGE 8
JIGw 25 ^Continued) Sattsry 79 =rature Histories in Simulated Thermal Battery rrature Histories in Siauiaaad "hernial lattery srature Histories in Sir^nluaed Thermal Lattery with Varying Insulation Arrangement C5 ilated Thermal Battery neura riistorias in Sim Â„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 Â•:; 107 29 Effect en Cell Temperature of Idealized Chaages in Jy ^j"aC. en msulcit ion i iiTipsr&iurt: oz idea i.12 sd Cn&ng&s jL.,1 iUciuui .luJ^^iCu w j_ Cue Xiiaua.SL.lOZl Ill iiirxsct on ^eii IS:inp6XS.cu2rG cz Dxf Â£Â£3TÂ£iit v fir A, jv, w L. ij 'aÂ„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 Al Analog Circuit of Heat of Reaction Term 137 Bl Details of the Shape of Insulation Elements 139 Cl 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 wellmixed 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 differentialdifference equations which in turn were solved by the AdamsMoultonShell 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 allThermof 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 intracell 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 cutoff temperature imposed by a highmelting 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 cutoff 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 cutoff 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/LiClKCl^Cr^/Ni and Mg/LiClKCl/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 cellgenerator 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 unsteadystate 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 (iiD 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 (IIl) 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 (IIl) 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 (H2) Equation (II2) 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 (H3) have been used since Stefan (11) found this relation and Boltzmann (12) proved it theoretically for a perfectly black surface. Equation (II3) 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 blackbody
PAGE 22
10 radiation. For surfaces not absolutely black, O must be modified if the StefanBoltzmann law is to be applicable. The basic equation of heat accumulation for small linear changes of temperature is Q = pC VAT (II4) 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 flowtemperature field and an electrical flowvoltage field of the same geometrical configuration. Beuken (34), and Paschkis (35) developed largescale, permanent analog devices whose principal elements were resistors and condensers, and they were able to solve unsteadystate 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 onetoone 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 excitationresponse 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 highspeed 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 threedimensional steady state heat transfer processes. Although the relaxation technique can be used for unsteadystate 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 finitedifferences 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 finitedifference 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 finitedifference formulation, and solved on a digital computer the problem of heating homogeneous cylindrical briquettes. Dickert (47) used the explicit finitedifference approach for the solution on an IBM 650 digital computer of the unsteadystate 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 wellmixed 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 = 2nR 2 + 2nRh (II5) while the volume is given by V = 7lR 2 h (II6) If the volume is considered to be fixed, the area can be expressed by
PAGE 29
17 A = 27lR 2 + 271R V (II7) 7IR 7 which can be differentiated with respect to R and equated to zero to give Â•di = 47TR ^1 = (II_8) R Equation (II8) may be solved for the volume to obtain V = 27lR 3 (H9) This would be the value of the volume corresponding to a minimum area, but V = 7iR 2 h (1110) hence h = 2R (IIll) Equation (IIll) 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 (1112) while the area is given by A = 2xy + 2xz + 2yz (1113) If the volume is considered to be fixed the area can be expressed as A = 2xy+2 ^+^ (1114) xy xy which can be differentiated partially with respect to x and y to give ^ 2x *! (1115) X
PAGE 30
18 Ufr.if.O (1116) 1 y r These equations may be solved for x and y, respectively, to give x V l/3 and y = V 1/3 (1117) 1/3 which results in a value of z = V when substituted into equation (1112). 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 ,Â„ / f 3v S I 47T R = Â— VI (1118) A 47T rH (1119) For the cylinder, the radius is given by 1/3 C \ 7T and the area R = Â— V ) (1120) A 4TT \zr\ (1121) 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 (1122) For a cube the side 1 is given by and the area 1 = V 1 3 (1123) A = 6V 2/3 (1124) 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 (1125) 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 Â•H rH o Â•H O 42 H (4 co o C o> 50 O g o CO 0) >H u o 4) CO Â•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 "wellstirred", 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 intracell 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 ti 3 en (3 M 1 h t. Top Generator Â• Cell i >l M Â• H Figure 3 Schematic Diagram of Battery Described by Model I
PAGE 36
24 wellmixed 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 intracell 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 reactiontime 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 wellmixed 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 pseudoheat 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 pseudoheat 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 (IIIl) Resistance of Cell of Generator H 1 (IH3) where H is the heat transfer coefficient between the heat generator gc 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 onehalf 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 (III4) 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 ) (III5) 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 StefanBoltzmann constant (4.92 x 10 kgcal/m hr k ), and T is the temperature in degrees Kelvin. / F 12 = 1 (III6) 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 (IH7) gcR ^ + gc 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 TÂ„, 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 Â§ < CJ M H eg o En 22 O H H 3 cy w w u W 11 < H H as w Pi w Â§ H T3 Â•t4 CM CJ oo H i Â•H 4J 60 Â•r4 Jl EC H i i Â•H Â•U SB c o H 3 CO c 11 a o H i CO I CM H II O II CO 3 O cfl 3 cr W M o 3 O Â•H T3 C o CJ !i O 4J 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 41 o 71 I c o Â•rl 41 CO 3 W W o 3 Â•H 4J Â•H 3 O U CO Â•H 3 Ol 41 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 4J M H M 3 O cd 3 a" M H O 3 13 3 O Â•H 3 M O 4J CO M 01 3 QJ o ID 13 Â•H S oc E C CM 00 3u 60 oo e o I oo u o o o CN CN op o II 41 CO i
PAGE 41
vO 29 M f I H 1 Â•H 11 Â•H H IH 41 v^ H "O Â•H *o il /n Â• 60 JCM 03 ^ x: a w + J2 1 CM H + II 60 /Â— \ u r O Â£1 H H i + Â•H H 41 il J3 H v' V' e"> H ftf iH 1 1 y P3 Â•s W N^ 13 o Â£ O 0) ^3 CM 3 I Â•H II Â•H CM o 41 4J Â•11 B i Q. 4J o ca 60 II U /v Il H H* 3 Â•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) Ml O I u 60 efl Â•H Q H CD 3 a g o u 60 O .Â—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 III7) t = refers to top i.e. ti refers to the top insulation Example H refers to the heat transfer coefficient between the tili 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 firstorder 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 Â£i=g p C h. *g pg tg p C h *g Pg tg H tig p.C .h r i pi ti H gcp C h c pc c H + 2h ti H + H tia Â— Â— tili tig K H + 2h tg H + H tig Â£B gli gR gc h H tg gli ,R p.C .(o l)(h + 3/2 h + h ) r x pi Â— ti tg c R ti tili ,R p.C .( o l)(h,.. + 3/2 h + h ) H i pi Â— 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 gc Â— cli gc H + tg H gc Â—^ gli 2H gc 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 cli 0.1395 15 ( V 2) Vli 0.021 P^pA" fr tl + 3/2 h + h c ) R 17 H tia T a 0037 Pi C pi h ti 18 R o /R(H lia T a ) 00063 lOOp.C .( R o 1) 19 ViA + 3/2 Vii h t g + H tiii 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^ Â• 1)(h ti + 3/2 h tg + v : R 20 2H tili 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 gli Rp C *g PS 24 H gc p C h r g Pg tg 25 H KC 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 gcR p C h r g Pg tg 44 1 p C h /g Pg g 45 H gcR p C h c pc c 49 1 p C h c pc c 50 H gcR p C h. g Pg tg 53 H gcR p C h c pc c Mathematical Expression H D +S& H ,gcR jp gli H 2h. + te H + H tig jS* gli gcR. 2h H + c H , + H gcR rcli gcR 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 intracell 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 Â„ 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 pseudoconvection 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 iH 3 CO cu D &, B o u w 3 CD Â• h o 0) o 01 > Â•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 Â•65F Sec. A. Insulation 2X* 40 Heat Generators 1. Thickness 2X 2. Enthalpy p = 2X, h Â• 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. IntraCell 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. !Â• 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 onehalf 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
PAGE 63
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) intracell 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 intracell chemical heat generation built in), the peak temperatures decrease as the cell thickness increases; however, the intracell 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 intracell 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 ~Â£ 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 intracell 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 intracell chemical heat generation 10 20 30 Time, sec. Figure 15 Effect on Cell Temperature of Changing the Magnitude of the IntraCell Chemical Heat Generation 550C 60
PAGE 70
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 intracell chemical heat generation rate more than twice as large as the standard should be studied, a
PAGE 71
60 r 59 50 40 o > 30 20 10 30 Time, sec. Figure 16 Change of Initial Rate of IntraCell Chemical Heat Generation Term 50 60
PAGE 72
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 onehalf to 1.3 times that of the standard. It is clear from examination of timetemperature 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 intracell 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.
PAGE 73
60 r 61 50 40 30 2010 Slope = 0.5X 40 30 Time, sec. Figure 17 Change of Decay Rate of IntraCell Chemical Heat Generation Term 50 60
PAGE 74
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 cellgenerator 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 wellmixed 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
PAGE 75
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 cellgenerator 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 hightemperature 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
PAGE 76
64 .Figure 18 Schematic Diagram of Battery Described by Model II /
PAGE 77
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 differentialdifference 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 intracell 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 Blb, the volume of the top insulation element is given by
PAGE 78
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 ) (IH2) 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 Blb. 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 differentialdifference 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 differentialdifference equations which describe / the system take the form shown in Table 5. p.C .h. 9 V iEii (a J .. + R.R ... +RT) (III4) i 3 i+l i i+l i and G. = 1HL1(2r r.r R 2 ) + p.C ,S.(R?..R?>, (IH5) i 3 i+l i i+l i r i pi i i+l i
PAGE 79
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
PAGE 80
68 > o o St II (5 H I (S I 4) U 3 60 il En 0) O s 141 o B 60 n5 Si 3 u 60 O
PAGE 81
69 TABLE 5 DIFFERENTIALDIFFERENCE EQUATIONS FOR MATHEMATICAL MODEL II Top Element 4 2Â„ 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 Â• ^f 1 ^ v Â• 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) Â— 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
PAGE 82
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 Â— 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 Â— Â— (L i c) 5; Â— W L i V dt
PAGE 83
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 Â• Pot. Setting 26.35 24.44 1.824 0.086 34.31 32.94 1.371 0.0014 0.2314 0.1762
PAGE 84
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 Rp 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
PAGE 85
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 (RÂ„ + R a )hBLÂ„v 3 2 2 TL2 0.00038 40 2RS H, _^ 3 3 L32 GÂ„ 0.1762 41 2R 2 S 2 H L21 0544
PAGE 86
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
PAGE 87
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
PAGE 88
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 differentialdifference 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
PAGE 89
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
PAGE 90
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
PAGE 91
79 750 Configuration 4 = Metal 3 = Mica 2 = Asbestos 1 = Thermoflex 625 550C Core 500 u u a u
PAGE 92
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 wellmixed 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 steptype temperature gradient in the simulation of an actual layer of homogeneous material
PAGE 93
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
PAGE 94
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 coreinsulation 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
PAGE 95
83 750 r o o u 41 SJ I (U H 625500 375 250 125 Figure 22 60 Time, sec. Temperature Histories in Simulated Thermal Battery with Varying Insulation Arrangement 120
PAGE 96
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'
PAGE 97
85 750 625 500 o o
PAGE 98
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
PAGE 99
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
PAGE 100
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
PAGE 101
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
PAGE 102
90 be expected. The foregoing considerations suggested that the core could be considered to consist of two sections, an inner wellmixed 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 wellmixed 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 intracell heat of reaction, and the impulsetype 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 (Vl) at r r Thermal energy out at 47r(r + Ar) .qj L ; (V2) r + Ar I r + Ar
PAGE 103
91 Thermal energy generated Accumulation 47TT ArAH 2 4Tir Ar G C oT Pp o7 (V3) (V4) The thermal energy balance equation states that rate of rate of rate of rate of thermal thermal + thermal energy = accumula(V5) energy in energy out generation tion If equation (Vl) to (V4) are substituted into equation (V5) and one takes the limit as Ar approaches zero, the following relation results: ST lim 4Tr(r + Ar) q + 47Tr ql + 47rr ArAH (V6) ST = ArÂ— ^ 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(V7 )becomes (V7) (V8) oT o7 = a a T 2 oT \ 2 r or AH (V9) where T is the temperature, t is the time, r is the radius, and a is the thermal diffusivity, Â— Â— (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
PAGE 105
93 Figure 26 Schematic Diagram of Battery Described by Â• Model III
PAGE 106
94 C. FiniteDifferences Approximation The partial derivatives which appear in equations (V10) to (V15) 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(rAr s t) T Ar ^ + ^~ "j Â— J + or or (V16) (V17) If equations (V16) and (V17),., 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) (V18) Sr 2Ar If equations (V16) and (V17) 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) (V19) 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 (V10), (V12), etc., would take the following form: dt n T r*Ar 2T r + T rAr 2 T r4Ar T rAr (Ar) 2 2(Ar) (V20)
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 Â£ r ~ Ar i rJ&r r Â„Â„. k n Â— ~ = 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 j1 are functions of time of the form giver in equation (V20) The heat generation term, the last term of equation (V9) 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 wellmixed 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 AÂ— k B A Ar~~ + AHV A < v "23) B and T l = T 2 (V24) 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 wellmixed 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 intracell
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 AdamsMoultonShell (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 firstorder 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 mÂ£(x i> %) CV25)
PAGE 110
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 predictorconvector 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 AdamsMoulton. Because of stability considerations and the possibility of error control, the AdamsMoulton 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 RungeKuttaGill 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 AdamsMoultonShell method is a divided difference technique based on Newton's interpolation formula and is an extension by Shell (55) of the AdamsMoulton constant increment method to a nonconstant 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
PAGE 111
99 generate new starting points after each change in interval size, whereas a method which can utilize nonconstant 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 Gl. 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
PAGE 112
100 model because it was developed to simulate a heterogeneous sphere having a wellmixed 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 wellmixed 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 Cl 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
PAGE 113
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 wellmixed 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
PAGE 114
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 intracell 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 Cl. This simulation was effected using 27 grid points. The temperature profiles in the battery are shown in Figure 27, and the
PAGE 115
103 o o o o o o CO o o <>4 o o in aan^Basduiax o
PAGE 116
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 allThermof 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 lexasbestos structure. On the other hand, the use of an allasbestos 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,
PAGE 117
105 TABLE 9 EFFECT OF PARAMETERS AND CHANGES ON LIFE TO 400C OF A SIMULATEDTHERMAL 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 AllThermof lex 32.8 1.09 28 All Asbestos 16.2 0.54 29 AsbestosThermofL 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
PAGE 118
106 TABLE 9 (Continued) Life at Heat Sink Temperature of 25Life Run No Description 25C, sec 25Std. 36 Radius of Core = 1.40 AllThermoflex 32.0 1.06 37 Radius of Core = 1.60 30.6 1.02 AllThermoflex 38 No IntraCell 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 AllThermoflex 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
PAGE 119
107 a CD i 60 C CO u U < (3 o 3 en C en 3 O Â•H M > o. cu u 3 41 cfl }i
PAGE 120
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 allThermof 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
PAGE 121
109 thermal conductivity. This result can be explained in the following manner. When a high cutoff 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 cutoff temperature is chosen low enough, the life above the cutoff 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 cutoff 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
PAGE 122
110 '.* v / c / * /td 3 CO /Â•o erf / r C cfl 3 u / erf ai '/ in / CO / a / 3 / erf/ { y J / Â• i "I 1 i o CO o m in o o m o 5 o o o .in /co o Q CO I CD co cu 00 Ctfl Â£. c_> C T3 o CD H N 4J H ttj iH 1Â—1 cd 3 CD 01 13 C H ii O CM Ul cu I o J3 ,' 41 CD >J <H CD 3 o CO 41 / cd en Â•> u 0) U CD Â•h a 41 E U CD CD H O* o iÂ—l SJ rÂ— 1 Â£ CD O 1Â— i en a s o u a) O 41 J3 ii o H CD IW Hl W ON CM CD H 3 60 Pn /q ajtn^Basduraj; /'
PAGE 123
Ill c Â•ll co CD tsO C cd JS U "0 0) N Â•H rl C n) O m ii T3 U M cd .Â—I O 01 c 0) M !i 3 01 M .Â£ n) u Si CD Ul D, O 6 CD CO H cu Â•H c o cu o u 3 eu ca co c lH e o o cu "M w o CO
PAGE 124
112 U o 3 CO C T3 N cd c 0) Jl 01 <4< <4( Â•H Q Ml O
PAGE 125
113 o CO 3 Â•H 13 CO Pi CD H o a c il CO S 5 o o a) M 3 41 cd 0) a) H (3 O o 0) CM w CM CO 0) Ml Â•H fa
PAGE 126
114 core both greater and smaller than the standard core radius. 5. Effect of IntraCell Heat Generation The effect of the intracell heat generation on the life of the battery was shown to be a significant one. Figure 33 indicates that the side benefit of intracell 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
PAGE 127
115 01 CO d) Ml a nj X U B O U4 >H O 4J cO 01 S4 M 0> 3 C Â• 41 0) o cO O 0) u rn 0> 41 O, cd n g cu <1> 5 S3 a H Â•H ii H rl r( rl CD (U o U 1 C !i 4> C 4J M O d) IW Hl Ed sr> CO a) u 3 60 aan^Baaduiaj, o
PAGE 128
116 9jn^Ba9dui9j, o
PAGE 129
117 0> O M 3 03 cd u 01 &, cu e s CD jj H o 01 4) c o o ca o. w ca o o cu yi ca w 0) CO 01 u o oo Â•H Pn o o o o o in o to O O o 9anaBJ9draax
PAGE 130
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 allThermof lex layer could be used, that an electrochemical system could be developed which would yield twice the intracell 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 Â— Â— 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
PAGE 131
119 o C vÂ£> a cu > o 51 1 c Â•H W a) CD a) 60 O (VI
PAGE 132
120 c o 3 C 0) J3 C H ,d 4J Â•il is c o C8 U =s Â•rl o in O o o o o o LA o o <3" 3 9 an 3 Baa dura x
PAGE 133
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).
PAGE 134
122 u 41 u P3 01 <4l o 01 01 M c u O a) 4J tfl Vi
PAGE 135
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 allThermof lex insulation were used instead of the standard Thermof lexasbestos 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 intracell 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
PAGE 136
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 cutoff 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 cutoff temperature is high and high if the cutoff 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 $
PAGE 137
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 III4, dimensionless factor to allow for interchange between gray surfaces F dimensionless factor to allow for interchange between black surfaces G defined by equation III5 h height of element, cm H pseudoconvection 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 C5 125
PAGE 138
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
PAGE 139
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., McGrawHill, 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
PAGE 140
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", McGrawHill 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", McGrawHill, New York (1954). 40. Dusinberre, G. M., Trans. Am. Soc. Mech. Engrs. 67, p. 703 (1945).
PAGE 141
129 41. Mickley, H. S., Sherwood, J. K. and Reed, C.E., "Applied Mathematics in Chemical Engineering", Second Ed. McGrawHill, 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, McGrawHill, New York (1950). .. 52. Brown, G. G., "Unit Operations", John Wiley & Sons, Inc., New York, (1956). 53. Johnson, C. L., "Analog Computer Techniques", McGrawHill, 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", McGrawHill, 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).
PAGE 142
APPENDICES
PAGE 143
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 :ia [0.0002 = 0.00256 1 H tirli H tig 0.46 L_ 0.156 0.0002 0.0002 2 = 0.000372 1 1 0.156 1 T H gli 0.1 0.0002 2 0.0005 2 i 1 1 0.156 0.46 + = 0.002 H gc 0. 0002 2 0.0005. Â„ 1 0.1 1 0.1 "P = 0.000763 H cli 0.0005 2 0.10 2 1 0.156 .46 0.0002 2 .10 1 = 0.01 0.00254 H. = H lia tia Note: The value of 0.46 is the radius of the circle having an area equal to onehalf of the circle whose radius is 1 56 131
PAGE 144
132 TABLE Al 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
PAGE 145
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 Intracell Heat Generation Tabulated in Table 3; Shown in Figure 16 14 Change Decay Rate of Intracell Heat Generation Tabulated in Table 3; Shown in Figure 17 1
PAGE 146
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
PAGE 147
135 TABLE A2 (Continued) Run No, 8 10 Pot No. 1 Â• '. 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 Â•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
PAGE 148
Run No. Â• 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)
PAGE 149
137 e u 0) H a o Â•H U o
PAGE 150
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 Bla 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 2nRdy J 2 2?tRh (B2) 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 (B4) which becomes 3 3 ,RÂ„ R V, ^L 2 A ^(R? + R,R, + R?) (B5) 3(R 2 R x ) 3 v 2 1 2 138
PAGE 151
139 (a) (R r P) (R 2 ,h) (b) >* *2 *> t Â„ D i 1 ; *> c I ..* (c) Top Insulation Lateral Insulation Figure Bl Details of the Shape of Insulation Elements
PAGE 152
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 Blb. 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(B7)it becomes v 7T(R 2 R L ) (2R 2 R R R 2) + ^^2 R 2 } (B8) or = ^2 V (2R* R X R 2 R*) + R X (R 2 + V (B9) L 3 which yields upon simplification Y<*2 +R l*2 + *l> (B10) Equation(BlO) 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 Blc 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.
PAGE 153
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
PAGE 154
142 **TL1 .0002 (.470 + .820) .05 + 1 .465 = .00015 .0002 (.005) 1 = .00292
PAGE 155
143 TABLE Bl 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
PAGE 156
144 TABLE B2 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
PAGE 157
145 TABLE B2 (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
PAGE 158
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 (Bll) dJ J dt (B12) InJ = t+c, J = ce at t = J = 40 t J 40e t volts Since 1 volt =25 J = 1025e" t / (B13) (B14) (B15) (B16) Since Pot. 29 = 0.5, AH_ / 512e (12.6cc) = 12.6(512) (B17) PC and hence the total heat given off by the core is given by AH = (.432)12.6(512) = 2785 cal (B18)
PAGE 159
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) (220025) = 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)
PAGE 160
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 Cl. 148 1
PAGE 161
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>
PAGE 162
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(t15.0) for t > 15 The wellmixed portion of the core has a radius of 1.323 cm. >:v ,:m : ;'..k
PAGE 163
151 TABLE C2 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 lexasbestos) 10 grid points 30 grid points 30 grid points, AM0S 4 40 grid points AllThermof lex Allasbestos Asbestos Thermof lex Mixinsulation AllThermof lex k 0.0001 Allasbestos k 0.0001 Allasbestos 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
PAGE 164
152 TABLE C2 (Continued) Run No. 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Description Allasbestos 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 AllThermof lex Generator outside Thermoflex (10 Sec, delay) Generator outside Thermoflex (2 Sec. delay) Generator (p,C 4X) >& Pg outside Thermoflex (2 Sec. delay) Generator (PÂ„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
PAGE 165
153 MATHEMATICAL ILLUSTRATION OF THE SIMILARITY BETWEEN THE FINITE DIFFERENCE APPROXIMATION AND THE APPROXIMATION BASED ON WELLMIXED (LUMPED PARAMETER) ELEMENTS The equation which describes the unsteadystate heat transfer by conduction in a homogeneous body in onedimensional Cartesian coordinates is given by St St (Cl) where a k/pC The threepoint central difference approximation of this partial derivative is given by ii V 2T ) + Yi dx 2 <*o* ( C2) hence equation Cl takes the form dT. dt a L j+1 2T. + T. J Jl < C3) (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 Â•,.' i ,'i'. i ,i_ ; ;_ 3
PAGE 166
154 A heat balance on element j gives the following relation AH(T. T.) AH(T. T. .,) = pC AAx dT. Jl J J J+l P 1 dt (C4) and this equation can be rewritten in the following form A. (Ax) 2 L T.^ n 2T. + T. J+l J Jl (C5) dT. dt where ]3 = HAx If equations (C3)and (C5)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 ii 2 Vi AR 2 R M 111 (C6) If R = jAR equation ( C6 ) reduces to dT. dt a jAR (j+l)T j+1 2jT. + ODTj.! (C7) The wellmixed (lumped resistance) approximation takes the following form
PAGE 167
155 47TR^_ 1 H(T._ 1 T.) 4nR 2 H(T. T 4J1 ) = pC,47iR
PAGE 168
APPENDIX D DETAILS OF THE COMPUTER PROGRAM TABLE Dl 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
PAGE 169
157 TABLE Dl (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 WellMixed Section of the Core RHO Density T Time TBREAK Time at Which the IntraCell 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 IntraCell Heat Generation Term
PAGE 170
158 TABLE Dl (Continued) SL0PE2 Decay Slope of IntraCell Heat Generator Term SUM Sum of Grid Point Temperature Within an Element i e : i
PAGE 171
159 Figure Dl.. 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
PAGE 172
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
PAGE 173
161 c5 oj iÂ— i ,' rs! i_ ' 0. 00 o 00 tÂ— x 2: z 3 O UJ D vtil o uj o_ < OS oS Z 3 O < tt o or o. z _i hZ LU DW 3QO WW 3 Â— tLUVI~>tcÂ£ < z O Z a. <3(< a o o o 1* uj o a. wa: Â— i < a. a. a: 3 <'3ii3(Jm>3iH < UJ O 1 Â— I Â— Â— i Â— s Â— a! X X *X 00 orz 03 a o33tujtu_^: h u. a 3 3 3 3 3 3 3 Â— uj Â— 1 '3 o X> > a.1Â— K2as Z 3 ~ < OS Z 3 ~ 3 StOOOOaOOtU!i:ZZ{SOZU")eÂ£H< a es3HZiLÂ£OHZu.a.HZÂ£uj^Q.2:s:zxÂ£Q:oooua:ouujoz.> os 3ll u. DOH^ia.j:<. Â• > Â• Ol ex JJ i Â— ^3 1 Â— 2S 3 MOi'Nfft'f K.l i~ > Â— 1 Â— 1 Â— 1 Â— Â— ( Â— 1 Â— m Z < Â•> Â• UJ oo > N, s DZ HHU 1 r1 X tÂ— ^Â— liNJrsjlÂ— 00_J333O333333ii33a;p<iiMfOilÂ— X X co QSoSnSoS3 Z QMHWHQHMMM^iioi/inja 1Â— a. a; % Â— 1 ~m m ~ i^r > Z z as a. 3 < i/1 00 L0 00 00 co x ll z Â— m < 3 HtIÂ— 3 3 3 3 3 3 OS Â—i 3 OS < 3 3 oo Jj 00 X Z X X XX UJ X OS UJ X Â— Â— 3 00 OS < < < I < < !Â— 00 Â— Q XX X XX XXXXXXX X XX X XX X XX XXXXXXX X XX X XX X XX XXXXXXX X, XX X Â— 1 oj com oo^rvro^'o rv co o x rt ~ Â— HiÂ— i ~* Â—t y* .Â— I r^ (\J UOOUUU.UO O 3 3 3 3 3 3 3 O U O O U 13 OO U U O O O O O O O 3
PAGE 174
162 UJ 1Â— X h*~\ lÂ— X LJ u Â— Â• U) *~* 3 (Â—( LU o ~ X X < X p <; LU UJ m SI CO LU UJ < a? _J >" 1Â— _j :*: CÂ£ i Â— UJ 03 uj 3 2 UJ 3 O 1Â— 2 3 < j O kn4 _l < O 3 LL) CO iÂ— CD 3 a 13 1X1 < a. < :Â— Q. < QO _' Â• Â— UJ lÂ— a: <Â— '3 &. < .3 Cl < 3 a. O 3 3 ^_ UJ 3 UJ ;Â— Z jj o s: Â•Â— aÂ£ 3 < 3 to i~4 3 < lH LU 3 3 Z 3 JJ ~ 3 __, Â— . tO ZZ < 5 < aC < < O O < f e* m c* !> p H* Â£ ~H C\J Â— i fM ,Â— I M W: Â•H CM 11 II II II l II to UJ II II 3 IS) rÂ— < 1 Â— 1 CM (\1 f<1 rn HÂ— ( Â—t to > > > > > > 3 a. Q. LU a 3 3 Q a 3 2. tO to a; Â—> "5 ~) ~> > ~> 3 O ^ 3 O Q < to X X X x uuououuuuu
PAGE 175
163 Z X Q LU CO a LU LU iÂ— 1 X hMl CO co + 1 Â— i a (Â— z < < 44 1Â— H < o LU LU 1 XT 3S 31 X 3! co cc m CÂ£ cc X a i Â— o o rÂ— H1 Â— 1Â— cc rÂ— LU < LU z LU X ^ Z 2 00 2T LU iÂ— _j \/ (Â— 1 Â— 1 Q 1Â—4 Â™i >Â— 1Â— X Â•^ Z Z o u X CD Â— < a Â— 1 a Â— i a cl X ^< Â— ^ a.CC X 0. x CO co i Â— i a + Â•fc9a. a. + a > Q. hco X JJ 00 LU u a O 1 X 1 X 1 X CD a X X O ha H~ (Â— >Â— IÂ— iÂ— 1LU h>QC < :~ X Â£ LU X X (Â— hhi_ a < 53 < i Â— i 2 X LU LU *ft X r4 X rj X ro LU UJ ce. X a: at +A M4 h3Â£ Â£ t **"* 1 1 Â— 1 1 iÂ—i 1 QC Z cc CO X O CO CO 1i Â— X LU LU a X X O Â—1 CO < LL LL > ~^ Z o JJ rS ~^ tÂ— C_ U_ rv 1 Â— Cl *Â— X Â£ 1> Jj O X 1 1 Â— 1 1Â— t "< X u O z Z X X Â— ci X < z i2 jS > a CC < o o Â£ Â— t 2J X Â• a K~< a >Â—4 O tÂ— 4 LU Â— CC a a 1Â•V *s Otf a. 3. 14 1 Â— 1 a h a 1Â— O 1 Â— O 'U IÂ— O 5S 1 Â— 1 CL Â— 1Â—1 CO < JJ CO X 0. a. 3l 0. s: LU CL < !Â— CO CO LU 7" X ** i.n CO X UJ LU r( M m 1Â—1 a < X X ro lu LU Ci Â— < LU Q2 X cc Â£ 2 1 2; 1 2 1 Â£ IÂ— 2; OS Â£ _l _J Â£ J J 1. O rÂ— LU LU UJ ^ CO 1 Â— 1 o >< a <* a * UJ X ^ s; iJJ X X OS 1/5 X uu u_ o 1Â— X < I u.i o LU O LU O LU X C3 lu a: X X X X X 1Â— (Â— J O X a Â— CO 2j a s 2: a. Â£ a. 2Z a. X Â•< 2 x 1^ Â—I Â£J 2J ^_ LU > >Â• > S^ Â£ OJ X CC 5Â£ cC X CC CO LU > pÂ— iÂ— Â— X z 1Â— X J_ 1Â— 1 ^ iÂ— < LU a M ca o ca o o O a O O LU LU '.J X X X z z X i~* kH 2 CO rX h1Â— LU 1Â— h~5  Â— X OS "S* a: ;r* ^ X 1Â— 3^ X CO X L_i LU UJ Â— 1Â— iÂ— LU X 3 X X X ~< u. Â— 1 LL *Â— 1 LL h1 CO ^1 1Â— t z x> 2: 2: X X X x a lu 'jj CO Â£3 O *""* o a > q 1O LU 1Â— v: htf Â£ LU U LU >X Q UJ Z Â£ LU O QÂ£ tc X cÂ£ N M i_ < H UJ LU LU LU LU LU LU O z LU CO X Â—1 CL cL CÂ£ Si.cC C^ LL O X CC *Â— cq CO U CO LU > ^ u O O Â£ Â£ Â£ X Â£ Hjj" X L5 3j 3 X Cu Cl X LJ Â£ ca 00 X co Z 3 X < X X Q UJ < 1 < tÂ—i tÂ— i i Â— 1 >Â— Â— ; Â— 1 tX LLI 0Â£ IÂ— 1 1Â—4 _j' <( X z z X X ^J i~j LU at LU a \A <0 Â£ co u_ ~7 Q LL u. u_ l_ i_ w ~~ 1Â—  Â— r"" .1Â— CO X IÂ— 1u_ X Â— ) a X > SÂ— /' y ~3 rl Q X "3 ?1Â— > X ^Â•J X i%J Â— X Â— *" MM Â— i pj r\j co **" v* M a > LU 0Â£ LU 1Â— ~^1 Â— 1 J _) Q X > CO CO co Â— o LU iÂ—i ~r u as: cO '.jj Â•^t X ^L :j.. rÂ£ CC X CC "< X a o id w < Â— a: 5 X o *4 "Z A 3f +A ."5 UJ X UJ < 0/0 O O < a Â•W X Â£ X _J J <5 LU X iÂ— < CO A ^ yC _i _J z Â£ O O 3Z a X SI ^ 2: X X < < < t u X Q Q Li. LU U_ X X X X X X X X X X " 1Â—1 Â—i >H 1Â— 1Â— 1 1Â— < >< Â— Â• 355 a u u u u a u a o u a 0.0 u u u u u lj a u u u o UJJ o u .uuuiUJ. Qu u o o
PAGE 176
164 z CO OS OS to a: uj oi sXJ r; 3.. a a O LU u. > (. ol h* OC Z 3 UJ o t Â— CC ca huj z m tu z o 3 LU H3S " Q X Â— to uj a 3 ar iÂ— uj tÂ— o Z a. < 3 o 3 as a. O iÂ— X < iÂ— oiooca: t~ v> 14 3 OH 03 OS O OS O IÂ— H > CO 03 hii 3 UJ CO UJ U. < Z ;? a, < 'JJ Â•> o > Â• tco a. o idz co x D x iÂ— a m ^ z s: as s: a: 3 h 3a: Â£^a3 tto hh wwh hto Â— m huÂ— c< 3 2: iÂ— 3 O O 3 1Â— :JJ YSJJ Z3h>> m LL JLL Q. D OS < < OS c Â—ifto to co 1Â— Â—to < a a a tÂ— Â• n Â•Â— uua: t; UJ Â•Â— X LUaCQ. 3hLU>'3cO;z;i3 3 Z uj uj z hi uj 1Â— jju.ij.za: a to lo to a < to _i z >h a. 3 Â— z z: f* hz 2 < z o o > < o z 2 z 3 i < Â— uj ho hÂ— 1 1Â— 1 o d m m o. 1Â— 1 i_ _i o a o uj 3 > Â— 1 tÂ— tÂ— 3 z < o 3 tÂ— 1Â— a. a 1Â— H 1Â— z z 3 q: . m tfl _j ua: ru. d a Â— 1 1 a 3 3 j r33to DOOOOsJositz ta jjmmcl a: z ci icy: o 3 o 3 o 3 3 o Â• t oi u. 3 < < o < z wa z is ca 1 uj coaa^o^uzocraso OS" IÂ— tco u. f~ m m rj < o 3 1 co o 3 < h3 3 ca C3 3 Â•Â— I CO 03 Â— i a<<2hOO" h hw I i/) j; i/) o rH i/3 fÂ— Â•Â— i O 2T co 3 3 fX 3 3 > 3 Z Z CO C UJ _J _J<_I l< oS cC Z OS O to to JJ ai o CO CO JJ tO 3 3 1 LU CS ~ Â— i 3 Â— 3 Z 3 3 3 3 (Â— 2 3 3 u a NH CO Q 3 CO c.O > U_ CO LL u_ U_ co Z IOZZZZIIZhS, CO hIÂ— 1Â— CO CO IÂ— (Â— OS IÂ— 1Â—3 COOZ333Z Â— 3 iÂ— Â—t Â— ihh Â— Â— i hÂ— I Z Z Z Z J JZZll Z J JZHS J 3 QiZZ^hi(lil 3 f> t3 3 3Ml3 3liJZ3HM3oÂ£wlCO'0333333UJM 0< LU Q^ < 3 u_ 'Â— Z 3C UJ i Â— i > Â— 3 < to 3 a. 3 UJ *z 0. 0. j%Â£ uj o O oS 3 OS /+Â— !Â— iÂ— a. CrT OS a 3 w MlOlVl cO 1Â— H h f(Â— z 3 u Q UJ G. CO CO UJ X 3 UJ UJ u3 ~5 "7 o 3 3 3 ~3 3 3 7 3 5 D Â—> *Â£, iÂ£ 3 z Z z Z O a. os co CO to Â— 1Â— tO 3 U (J U> O U U U O U 3 3 3 3 3 3 3 3 :J '^ '^> ^ U U U ^J 'J> U U U iJ U L> ^S ^
PAGE 177
165 (Â— LU as _j <4 aa th< hQ lO ..'** 1 < Z LU nS < ess UJ J < LU > _J :aJ a; as CO CQ 03 Q < 1Â— < < or 1 M Z Â— t~ as OS OS H /*Â• UJ 0; OS o UJ X 1< O < < os os a z > z: > > H ts* (z iÂ— a: i~ 4Â— 1 UJ cs; UJ Z < Z UJ x < a (/> a, z 0h(O LU Z z *Â— ?Â• tÂ— iÂ— 1Â— Â•Â— z os LU Z O LU OJ 00 x Â—r to < to X srS X Â— Q X Q X LU Â•Â—i UJ z z: Â—t a a < Z >Â— 0: z Z OS hX > z a 1Â— u. u "LU 1Â— X LU Â•*Â• LU < UJ UJ iX Q Â•Â— 1 >Â— 1 u_ OS LU it 3 1Â— UJ !Â— ^ i Â— (Â— si tO tÂ— i Â— X LU X iÂ— LU XXX O X UJ Qu 2" OS + X UJ zoo. Â— X < OS X x a u_ X 3 UJ U. 3 ~s OS X gr X U_ OS ~ hu_ _j J LU < < >Â— < z OJ O OS UJ O th LU as lu z O Z O < z < oj 1z. ~ lH X (Â— a. X 0V ;_(_(_ UJ Â•Â— LU > i Z > Â•~* O X u_ nSLX z 0' UUJ U Â•Â— z LU X X LU O !JJ * LL 2; < uOS OS a ;Â— a z ;r OTSh X 3: Z > X to Â— 1 > >i Z 2 < tÂ— a hf"H Â— tH *4 O Hl >Â— LU Â— 1 LU 1 Â— i Â— uu mio J >" X UJ 1Â— ?Â— tH ttÂ— _J (Â— UJ rÂ— a. 1Â— >Â— 1 Â— 1 Z < 1Â— it h Z UJ Z O Z X u + X y~ O < CO < UJ < CJ < z < U L. os z ST _l M J LU (J) UJ t/i iÂ—* cC O > UJ Â— !___ X X UJ >~i Â— 1 < X a OS O 2* O O OS X X LU UJ UJ LLi <Â—> cs X to S Q Z Â•Â— 1 X Cl. 3E O X 1Â— u to OS X 5 X as ca Â—> uj a. x x X LU ciS OS Z OS LU. Â•Â— sC X O UJ J2J a < UJ O OS .^ os < X a CSHw hÂ— >Â— 1 X LU X a X LU CC O LU OS O _j ca UJ iÂ— 5; a: a r~ H iÂ— _i to .. OS Â• Z Â— J ~Â— iii J Â— qSLUJ; Â— ) ^3 UlOQCBMO 0. O X LJ i*C LU X Â— m 11 jujzoctqsioouzzzz a a. Â— 3 Â— 11 z Â— Â— o a uj a. uj uj x x x x x x x a. hi a Â— ***Â•sVÂ— j x x i. a. as os os os cs as os tf3sosiAu ^ x c. 0. 3 I Â— 1 Â— 1 Â— Â— I Â— Â— J Â— I Â— Â— Â— JÂ— ~ tÂ— htSSXXX XXX XX o o u a u x a u> o a o u u b a u u u u a u o u o o u o x x
PAGE 178
166 LU a X ~7 s 0Â£ 2J 1Â— UJ ?~i < 3E X o O X p~ _j u. fÂ— 1Â— !Â— X ^, CC cÂ£ UJ UJ UJ Q CL rÂ— *Â— i a. Z) ^ O tÂ— _J m >Â— i o Â• CÂ£ Â— W) o LU '.Li Â— t Â£ 5 a: lu 00 3 *^ Â— 14. Â— Q 2 < rvj a. r/ < 3 ** *. h< X o UJ o 1/5 t \ a ^ 1^ x > O _j x UJ >Â— ** J < < < m hJÂ— X oÂ£ Â• e* < u. >Â£ n :s c* U> ^ a Q o O o 1Â—1 o tzz iÂ— i >Â— c/l X 0Â£ tM z> ^ 3 ^ Â— TZ 3 Â— 1 >Â— i i Â— i _i >Â— (Â— m> i Â— a Â•> <Â• Â• < o < o X jr LLi SI < Su < a s: oi hÂ—  QÂ£ Â£\l ,> a Â— a a J *~ 1 Â— j u_ IÂ— : _j_ Â—j LL rvi ro Q Q a oC DC 3/ < < < O U u DC < UUUOUtJUOUUUO
PAGE 179
167 > < < o cc Z tu 3B O x fti X o i Â— t < LL OO < I LU CD UJ < o UJ 3 Z cr> kÂ— a < w* ^ t UJ Q tÂ— LL X Q 3S ft ft o (N Â£ UJ a: JJ ^ X a 1Â— Â—> *? X n^ < < :Â— 1Â— O ^ z tÂ— Â— 1 i Â— i o iÂ— 'JJ m ft ft ft Ki a < 1Â— r* oo X ft 1Â— X 1Â— to x a o X UJ O cc a. a X < IÂ— lÂ— Â™. ~~. rd ^ LU < 0Â£ Â£ X !>Â• X LL. tLU X Q m Â— i X O Â£ LU < ci JÂ— 00 a O0 hUJ ft CM H ro X x: 1Â— > iÂ— o LU Â— < LU m Z Â— X rft O I* ha: fcC : Â— t m o rn ft qÂ£ to UJ O0 LU t. Â•> rn o SÂ£ 3 $Â£. a "j _J ; 3 o 00 O X X x O ft. rn x x Ha H Â£ tÂ— X rÂ— O Â— i rn o X Â•Â• 1/5 Â•* ^r IÂ— < co < O0 LU Â— < O w Â— I x X x in "Â• OS LU 1 Â— I z X m X 0. rn ~" f a. ^ 2 Q N. X Q o o X Â•Â• X ~^z *Â• Lfi h Â—i LU hLU oo X X a. c *Â— o CL ft cÂ£ 'Â— a* tÂ— 1Â— ft< 0 _j oÂ£. CL aj a Â— t _, m m Â• O :JJ ex a. a. \A o CO "S CO CM e. O Â— < w** e hCNJ O Â£ 3 < LU < X X O Â— I Â— o O C\i ") a. z ^~ Â— __ _J X ft a: o i Â— o ft* X _l Â— < CO z Â— \ a. Â— i Â—Â• am Â— 'X ft Â•> ~ > ^ <^ > UJ w iÂ— 1 MOX 0ft hÂ— H ft ^ Q_ L_ U Â— 1 (Â— 00 Â— X X X x X LU r^ a. <> 2^ rs X cC X X CC & "< UJ z 1fc X Â—* J j.i (K ^^ UJ LU UJ XJ X 1Â— D =>. Â•t * ft 6 J 3 ft c ijH (. 31 o :U _J a o 1Â— "X Â— CC o ~*  ro ft en X 3. Q O ,Â— X CC C3 Â•x X I 1Â— X rO cO ^ LU ~5 UJ .7! UJ O UJ < a m ^ tÂ— 4 Â• hft X iÂ— L.0 .U ~ a O O rn o 0S~ I ^3! Â• LU Z iij 1Â— rpJ < U1 Â£ Q CM X m m m O X 0. X ca 3 X X 3 t/1 X o?Â• C3 I* Q. r( =5 *. <:'.. 1 Â— ft < X oi a r i Â— i X 1 X ft4 >_ X M ^ Â•Â• O > M ft i0 O 'X Â— 1Â— ft cÂ£ X ciX ^ ftH lÂ— O O i o LL 3 a LU Â— i Â•JCO lO 1_ ca Â— 1 a: "" ^ ^ z ft1 .Â— 1 Â— Â— 1 Si ^ 2 n ft ^u 3 1 Â— X > LU *Â• Â— ft1 Â•JCL UJ m Â•~~ Â•Â— o Mi ^^ ft a v/* X L0 LU oo LU tÂ— Tvi Â— 1 Â— oo X Â•' q Â— Â—< <Â• rs o UJ k Â— 4 UJ '.' o 1/5 Â•Â— Q ~s Â— ft" X ft" ~ < 2 XXX X Â£Z m X > &Â• 7* Â• I! f^ < X ci 00 00 Â£ e LU o m *:t ^ ft o 3 rn <5 X ?' n < < X a. o < > u ~T O i51 Q ."S ~j a: ii C7 ri Â— i Â•30 o X Â• U3 X X > cÂ£ X 5 i Â— i i Â— j LL X Â— ..." X : ^Â— X Â— Â• X OS Â— < X oo 01 a a _i LU X 1! X X t/i z X uLU LU IÂ— a c^ X 00 1 Â— 1 Â— z a o < rvl rÂ— < iÂ— i. < 00 UJ Â£ s X ft j"^Â— < o _J X _l a X x x jg a: i/) a / a_i Â—I _J Â£ _J 1 Â— 1 a a a 1* Â— X LU LU < < < <5 < < o u a LiÂ— Â• LL *u CC 1 Â— X X o X u Â—I Â•Â— i
PAGE 180
168 > Q GO > o c* < r\i 3 > a Â• j 0* Â•Â• o N rft p> d r o o X rft ri r\j ro iH Q. > > 2: O LU o a rÂ— IT, > ST > j 5 iÂ— ro > a Â— 1 rvl Â—> CM > aa 9 pn .< w m' pi rft Q rO i\l 3. > > X O 3 *Â— t,n rft a. O ~ i/i M O f\l "5 jx o o Â— << O ("ft jcÂ— 1Â— 1Â— 1Â— o > a: in o o Â— 1 Â— 1 eg ro tÂ£ in oc o^ Â— Â• Â— < 3Ha.y.rtwQi>fs:a:oowoi/ioioow2 Â— < a << o so z o o 00 Â•Â— a. 1 1 fsj sC "Z. N C\J < oÂ£ CiflQHOriO^a^D UJ si Z 0") tÂ— !fl "ft 3 O > O LU IÂ— GO X ~LU w*OU**Z( i/i o w a 11 <> iÂ— < < < < < 1Â— Q 11 od iÂ— < < o do oaoooaoiÂ— a a o a o a _! _J _J J UJ r=. Â— Â— I I ? Â— Â— I 1 Â— i Â— I ? Â— I t Â— Â— ) l_LJ _l > IÂ— talH IÂ— J'fÂ— lJ >Â— J I 1 1 J _J oÂ£ J_l J J I J J fl! II J tf 'J J Â—1 I <<< o u j o ooooiAOifl^^ooino*ftii*i v +in>oN03a>Oinoi;, moÂ— imomo Â— 1 cm ro < lfl>0OO^HN(<^iniflOOOOOOOOOO'l^NNrt^'f't^ifliflOiflJ _i _i ,< v \j f\i cm e\i cm fNj rs) CM cm rft eft rn pO rn ft eft ("ft en rH ft <ft ro romrc, m^ fn fiTimfOr'i
PAGE 181
169 re 3. o 0 a. a Â— t ~5 ~5 "J Z "M O t/1 a 3 < o 1/5 r a o < CJ3 U + oc o < Â— ^ O h(Si CÂ£ ""? fl iTi o ^ o a. Â£1 CL < IÂ— _Ol Oii CD OlflZZOIOIÂ£UZO ns: ii cr: Â— OvjooÂ— iOri
PAGE 182
170 X < s. > X LU ?~ t/1 X X, 0LU iÂ— X 00 4> a 3: o M Â•' fv) >_l Â• Â•%. H LU r/ 00 X CO z X X LU = X r\J 2J O 1Â— a P 13 X Â—1 00 LU X. o X al X z Â• X X X a 1Â— a 1Â— r< O Â— i X T~5 o 1 Oi ~ X 5 u^ H < i i.O Â— 1 Â— 1 a I z !Â— H X 2T rÂ— CM s >Â— 1 Z X X co I. ^ LU < X o 5C CÂ£ X x r>. X a. Â• 1Â— CO 1Â— e> X X lU X Lf\ co a: CD O tÂ— cr X Â• crn a Â—i CO X E o < X o LU O Z Â— 1 w 1Â— in *. < (Â— Â• Â•> ,Â—. fO St. LU X < aÂ£ X e h> H X <<>Â• > X Â•O < X x O x po Â— X O * o as rai 1 Â— CO LU LU a. CNJ X X 2! X z Oi s> ft m o X 3 "3 ^Â£ Â— ] _J 3T iÂ— 1 CI X X 1 Â— 2" s: ...u LU X X X Q O X a !N o t> m X >r iÂ— a m 01 la jf X O 03 iU X Â£ Â•> O E ra JO IB 7* 1 iU n SÂ— 4 'X ci J UJ Â— X '/ _l Â— 1 X ro CD Â—j Â•>. M Q Â—  Â—y o cc LU < m 2: Â— I X X X X Q o Â—4 o <Â•_Â• X Z ~3 l__ o < iÂ— X o CM (~l VI hx M X X a. O Â—4 Â— r~4 X 3 M X Â— i in ea: rÂ— 4 H 1Â— X 00 z 1 Cl cc X 00 X Â—1 X m X a. m w c^ S! Tg lU X < a h^~ CO c LU X X X X < CO Â•Â• o X f*. X x e M J^ tÂ— 1 Â•at X X 27 ^i LU X X X z i Â— < 3 tO ^ (\J X tf Â— o oi a p QÂ£ H ^57 C17 1 X IU j Â— X X ^Â— ( X a Â— 1 X Z X cr ^ X > 3E C X Â• Â— m CM X <^ 3 u^_ a. 1c O "S. X M 1 Â— i LU X X ro 1 j ^ ~ 1 Â— f<*l Â•> m  1 Â— J hla) X < 1Â— X 1Â— < o ro s (* iÂ— rn Â•. cc f Â— i 1 Â— ( < O X z S X p, c* qi; X Q >Â— 1 r4 X a O rn o fH J fNJ iÂ— a. ca UJ P* X iÂ— >v 3 O0 1 pÂ— 4 HZ X , fH hfM 4X o fh fft X X _J 00 oi CL CT l>r? h,n^ i Â— i X O Â— CM c^ tr X X O VCO n o Q. LU Â•> a. CL Â• O LU Â—? l_J tÂ— < X Â— 1 rÂ— 1 Â•3 2 LU Â— 1 (VI a. o Â—j Â•ft *Â•> cu Â— fSJ >_ X O X QC >* i Â— i > z .. LU .Â— ^ O 14 X B O <t ~^ o o Cvl X "3 CS(_1 Â— rsj o X a 7^* 3 a: in LU CM *. f> X O Â— ^^ a St o Â—j Â— a 1 X J >~ i ijj vf X X X i Â— i IU X *4" p* *. a. LL ~( 1 Â— , rH Â— Â— 1 Â— Mi 00 LL) SI X I Â— z > z iH _J jy LU C^* I Â— ( 9 S. ^s 'X 1 Â— Â•<Â— uj IK LU OJ Â— V* 3 M Â—1 Â— < Â• X Â•ft Â•a Â— i L~D _l 0. D M Â£ 1Â— X o "N > Â• ;"[ 00 Â— t Â— i Â— Â— Â— < O CO LL < o a: ". 'rn Â— oo X CJ CL Q lÂ— X ^1 Â£Â• _) X LU < 3 < 2* >> X Tr' Â— j a X N. rÂ— i X yn m c<*i LL ^^ 5 LU LU Â•0 < >Â— 1 jv; O *y < CM IM z> 1Â— X X ~~, C" ^>Â— < ryf a< (Â— Â•ro < X iÂ— X z a ih X II II X < a 00 re ""J Â• m I*. o ^ r* o a. (JJ o LC ."M Â•H Â— i X jv; (/) iÂ— hX LU St X sr X a Z X OO 7^ a CM m rOi pr CJ X ^ 0. CO X X o Di O0 LU 1 Â— C3 'X 5 X Â— 1 m H 00 k^a O Â— O M X x <^ a a. X Â— 1 o .O V Â•^ 3E 00 c? i Â— !Â— Â— i X rsl X X oÂ£ LU LU 1Â— Â— ( kÂ—4 00 y 1 Â— LJ CM y \Â— Â•Â— o O rt o J3: Â— i a LU *Â• vf ro X Â— J hX X^ 7* X UJ LU HH X < Hx Â— ( x t> ii !NI PO _l X X 'JU d X Â•Â— X O LU o Â• Q, LU Â— 1 ) Â— n X r^ a: SX X Na ro ki X X 00 O Â— 1 00 Â• 'Â— St ClS Â•* pÂ—4 Â— v^ r\ Â—4 Â—* X X X X 2 <* X > ^ 2T ~ i 21 51" Q X .X iÂ£ ~4 Â— 1 r?\ LU Â•SC X J X < j x Â— LU CM ai jX Â—t iÂ—* X a LL. x X > Tg VÂ— Â— X Â— _> Â— i z Â• _' Q J\ a. X Â— fSl Â— fNl Â— X Â— Â•*Â• X ^_ Â— *Â— i 4 X X 1 Â— X Â—. Â— 1 X O Xi c!! 00 2 hhc* rX rÂ— 1 x Â•> a hCO h00 1 Â— 00 LU rJ 1Â— a a z 3 Â•"! T* 2^ ea s: 21 >" Oi rv' LU a; X X CO X > CrCX Â•Â• x Oi < X X mO X X O Â— i Q O a > m CO LT LU r\l ^^ X rj^ O a* O X rÂ— 1 fcÂ— X 00 o Q U u. X X X T LU m St LU X X LL UJ X X =Â— Â—4 Â• CO Â— 1 Â— < jÂ— \ CM lO St O o o c m O o o rÂ— 1 o Â— 1 OvJ fSl CM CM
PAGE 183
171 x J < 3 ac <n ~5 UJ : HX *CC *Â• CO a < 7 2 X < to Â•> UJ X CC UJ CM 0Â£ z CO s: CM X 0CC CC QÂ£ 1Â—1 II a. Â— i II O 3 11 tÂ— < 1Â—* 3 0Â£ O a* X X 1Â—4 K* 5 V Â— < UJ Â—4 \*m m <Â• CC _, UJ 2 3 tÂ— r11 rÂ— 1 O ^ ^ Â• UJ Â— 2: CC 3 1Â— 1 1 2 O *< Â•Â— 1 O Â— Â— "O _J O Â— t 2 0 tZ Â—1 Â— 1 Â— a O f<"! 1 Â— Â— < 3 p1 i/> 1 rn ~^ a Â— H 1Â— ~ LU rÂ—> IÂ— IN 3 ^ !Â— a 1 Â— 1 tt/> a. UJ 0Â£ fH 3 LU UJ U~l a. jj Â— > z LU UJ O l/J X rÂ— iÂ— a: >?O. LJ 01 CÂ£ a. ~^ m 3 X a Ml O CT> 3 1 Â— c> Â— X 3 z Â— 4Â—* 7 UJ Â— 1 r'J> ~> X O *Â— Â— w X rÂ— < p. ^ r^Â•1 Â•r*. O a: ** O 1 Â— Â—> m rÂ—  a: Â•* 3 en JO u 3 O CM fÂ— Â— *Â— 1 Â— 1 CM 1 Â— 1 fv ^ IÂ— m ;jj CM jE 0, CM CM O 3 1Â— 1 3; <Â— _J tÂ— 1 r * c 3 1Â— 1 C3 , rl Â— 1 _j 5 X _i Â— 1 iC^ _J 3 Â— Â— Â— 1 O Â—1 iÂ— a < 0. 3 00 P* CM I s 3 w rÂ— CL >t .A !>Â• Â— O Â— > Â£ sC ht/> UJ sO Â— O IU ^H CL "O 3 Â•0 LA 03 n c:5 a. 11 UJ CC a. (M Cl *Â• 3C 3 a. Â•N Â•1 ClÂ— 1 LT\ cc m tÂ— UJ cc fÂ— ^~ hÂ— Uj Â— Â•Â• 1tc ^ LU UJ UJ 01 Â— o CL LU 3 ^ tÂ— 7 IK Â—1 3l iÂ— c> Â• Â—) LU O 3l vCO 3 CL a Â•rÂ— Q. < 1 1 2S 1Â— < or. Â— Q. < i_ 1 1 s: 3 J~ O "J Cl uÂ— 1 X LJ + UJ 1Â— UJ tÂ— JS 1 Â— O UJ Q1 1 i_1 1 Â— ~~ LL Â— 1 O 3 a < 0. > >Â— a. a. Â— 1 fV) Q. 3 a J_ CL hUJ H > x < x 13 Â• < p. < 4OT < tÂ— _, Â•Â• 1 "Si < s, Â— 1 a. > H ;jj a 1Q. HQ. hO Â— ~^j f_ O O CO 1 Â— to X X a CÂ£ Â•s^ +* (Â— (/> 7" A Â— O a s Â— Â— w. \Â— vÂ— 3 ~ (Â— c/5 Â— ^ Â— v^ *U0 CL Â— b O Â— j *Â—< 3 1 "5 Â— 3 3 Â— 1 O 3 a .H LU *J Â£. 3 Â— < O l Â£ X a. iÂ— u> I Â— 3S a. > O CL > cC Â— > Q ^ 21 > > 21 ^L 1 Â— 1 Cl O 3 < X <; h1 1 3 CO iÂ— ci 4" hf^ jÂ£ rÂ— I kO jÂ— ^.x X i'jJ cc ro 1Â— 1 Â— 1 iÂ— Tt a. ~5 2 3 0. L'J rÂ— 3 3 a NÂ— 3 Q Â— *Â• II _i  Â— _j th1 Q Â•Â— Zi Â— Â— < cÂ£ _l ._! *cC u cc: Â— Cl 3 UJ < Sh LU CZ ^ 3 L_ _J Â— 1 _i *Â— 1 _J 1Â— 1 Â— 1 SI 'JU _J _J CL ^ LU ~^ hiD xT 3 LU SÂ£ 1Â— UJ cC (Â• UJ Li. U. 3C <Â£ agj r1>r >^ cc CO CC CO cc c> c> CC. c0>
PAGE 184
172 a so n ii n o sos O X ^ t< x ,Ti Â• ^ r*X Â•> H" lO Â•> IÂ— Â•Â• X Â— rl 1/1 O X 2C O Â•" q m ioÂ— *' X, Â— re) Â•> O jÂ— iÂ— ihfrt om~o~~ **'CÂ£ o <~ ?o o^3:)i<;aJ3 a. w q n O Â•Â• r<*i X ST. rÂ—O IE Â— i O <** Â• O X t/} ''^ L,J X m O Â— i >. t. Q ^ "UJ Ll Z CÂ£ Â„ Â—i ~o *joi o. io: Â£ Â—Â• o oa^l*o^ hw ix C3 o i a Â• cÂ£ 1 OrÂ— ^XdXiLOQi^ 2 S x Â— m x a. i^ Â•Â•a.xzstÂ— l 1 O X Â• X X Â•> H < L*> X Â— O aÂ£ 0. QÂ£ x *. ~ .^ Niznao.!x rry (Ci I r~ iÂ— Â„ r\ O Â• X Â• i m Offi s iw ci Â• pr> o cl uj Â•> cl a. Â•< ,0 a C\4 Or^OI'"MXXX IÂ— fX 0H~0 0(M2")ClHH rOÂ— ( Q Â— IXÂ— I 1 1 Â— 1 Â— a m Â— x o a a SÂ£ ?!K 000 o o o rm o x 1 <> a a o '** n Â£< Â™ x o'xx xuj*oo.z:3Z * ~ c_ o o, o o 2? h j z UJ (K Â— UJ Â• r^ < ^ 3 0 JO3 Â£hE Â— t>1 (3 ,Â•., ,~ *o Â— * ~XO~ X hm m PO LL Z ") UJ n O Â• O Â•> 0. O O L"\ sO o oofioa: J iz^Zid uj co zj a. ~i x < ts a: x hooiuo uj ri uj < jj >x. .0 .zuiaiH .0 1 1 t a. x m a, Â• o. Â• ^5? Q. rhOOtOU.3SMOU) ^ ^ uj c^ < < f +< f^ + 2 ^ ,_ fc z M ^ ,h ^ ~5Xv: ^J'^^ *~ x .'~" x o ^ _h x x >< X Z X > 3 Z X X II Q 3 I O 3 103 X U 3 X O KinGr^ *<^OD^"i!!.'* \T a. 3Â£ a. > ^a.> ^ Â— >> ^: f . S^ssOJai^ * uj m Â— jj Â— H oi + Â— Â— cc + Â— Â— oc + O^"C0 X _#i_o0 IÂ—l^ IÂ— I~ ?: Â•. o < < Â— o uj O u Â— w Â— uj a S ^ .'r. y x 3C h'I *: l_i 11 oi h1 11 as lJ i! nfi Â— s =: 9: v .,0 rÂ£ x "1 o Â— i o Â— Â• i o 1 J o ^ g 5 5 o<*o*:azof$azoÂ£t3zo ^ ^ 000000 oooooooooo ,i m .a rft n> ^ ro ,Â• .ri ro r^ 1*1 ~"\ ,0 f^ n rn r*l (njÂ—J'O O N Orsj Â£ N II Cl Cl. 0Cl. m. a. LI I Â— Â• I I" j q "J Q ""} a
PAGE 185
173 o o X X IN M o o OL a: o u z 2J a o cm cm o o m ro ** Â— ^. Ml vO IN o C\J a a. tÂ— i Â— *.. 5 O Â—> o 1Â— (~ LU r4 UJ a. Q. *. < c + 3t IX > LÂ— ^ + ^ Â— 0Â£ 3 b. 13 Q a o Q II o O tÂ— ( h*Â— : 2 LU C2 m ':i >Â— _J II a; i Â— Â— 1 3 Â— _l O SÂ— J 1 1 U_ C5 cr < a 3? eÂ£
PAGE 186
174 a a '..5 Â•> n sjUJ < cÂ£ UJ Q. ii x za:2:ox ' r\l O O O *~> 3 iÂ£ X "5 O CÂ£ Q X Z I r^i 'iC ^ t. Â•Â• >Â— />Â•.(Â— Â•Â• X fa Â— H l/l O X 2 O o *m io :*Â£ H t~ O x Â— m *Â• o Â• fÂ— q o Â• mix iÂ— a Â— < O fl O X Â• oo Â• x m o Â—Â• *Â• Â• q o uj iu Â•^iw.o^" Â• o; o. iCi z Â• O Â— QÂ—tX^OH*Â— u 1 a o i a in o; x o m OHjwha.tauJ om .filljn o;n. ~ m .o c uj Â•Â• a. a. (\ OH UhNS3 X Â„ O I Â— OONZTOih O Â— Â— O Â— IX J ^ > _ Â— Qflw X .. ft ft ft Â—i n c x a Â• ftÂ— Â— Â• o a, civ)iunQ^ x x _i x !Â— .0 r*i fl U. 2 "" UJ to 3 .. Â• Â— i OS H "5 ^ ii o Â• Â•Â• c a. c o m E 5 o ~oomoo:jiÂ— ^uj^Nift *Â• ~. 2 Q N (f on m O X X a. Q3X *Â— 3q.h t. z < tÂ•> se as o XIÂ— IÂ— OOrIOO.3Â— Olil vT^ iÂ— ft^^''^Â— < i^ "7 Z Â•':*Â£ ^ .'' ^ uj < 3. UJ Â— Â— Q Â— ft X Â•> cC '*!. *< >w k x a i 3 Z O UVO ruj uj o m Â•Â• Z H II II' 'Â•r c3 a 5" o 'X 1L.' : u ^ 0; i tk a: 5u x ^'^ *o z r > jj .n in n .n in u*> ; n u^*m tf\ >"
PAGE 187
175 4tm hLU Â„ Â•> Â•> *> 00 I Q ;Â£ (.0 ,, 4. ;\j LU 0. X ZiS:Â£OX OX Â•< NO 2 3 m 3 UiO q X Z tÂ—t "> ro > iii O < X Â• tÂ— 00 Â• IÂ— X IÂ— X ^ Hlfl Q X X O  x rvj ,. ,m x~3^^t0 Pi "1 Â— < x Â— m Â•> o Â— iu a o Â• co x x iÂ—3 ujwliii *. ~, Â— o ^" Â• a: atÂ— a: s: *Â— oo >t oi ^ C O h Q >t x X iX Â— 5 H O X O Z X ori Â— rtxox>iLr\'a:Q;<Â— i Â•Â— x Â— wx s.!*i .a z3 aou *Â• j^2 ox*xx ""Ln'< X .^ aÂ£ a x Â•Â•Â— ooiLt'c;'' i/)in"u '_) X ^(fl sail ^3: IÂ— X Â—< x Â— .CO .0JXZ5O.0.HX "^ UJ LL Â— <*J "' Â— rt mi h H m X < > <*S Â• o Â• y? Â• ~* Â— *Â• x on OHjNH4,aiy <Â•< o 3 iÂ— o m ffixx_jooa:a0'xx rl cfj o eu uj Â• a. a. jo oÂ— i as N S>S) OH Ut(MS3 00 Â• x Â— I vO X = T IÂ— X OHOON2T0.HI<03 Q O ** Â— 3HIJh X H Oi Â•> "5 Â•. i o fl Â— X *Â•Â•>Â• lj i Â• o o o .< Â— CM O X 0. Â•> IÂ— Â— O C X IÂ— O P"Â— < O x u x x x 'u << o a. z d z sr t Â— 'Â•*Â— Â• Â™ r < 5; > ;jj iÂ— 4  3g UJ rt h ifl Â• LU Â• O 0 Â•Â— ^0 5 Â• Â— Â• O J 0 3 X *X iU <.o Â— cm O ci Â— ~ m ~ to X O QQ tÂ— X 4* X IÂ— Â•> j. <Â• Q X i_rCica COLL JfJUJ Â• >* X o aouiA( Â£ Â• Â— ko < iÂ— L) OOrtOttl Â•tZUlZSiWllUlU'f MD 311 5 4 2 DM (^ m ff> O X X 3. CO XX < Qi X Â• Cl'S: 35 X iÂ— 5 0. Â—i X < h* SI oÂ£ O Â—. O * lu j: 3: huj t 3^0UJ ^u v uu o Â—<:Â•+ ^<^^i_ 4 l_ ^s;^^,^,, ^ 5 Z i^ 2C < h" QlÂ— C '30. i/> a. oj o Â— Â• x X. a < Â— Â— x Â•> or" x x Â•Â— xx x + + CÂ£ IÂ— X Q Â— I f N U UJ >!Â•Â•Â• ;JJ LJ H "i O O "' Â— Â•Â— Â— X Â— Â— I 3 ^XXXXZ* X>3SZ*0'JL1 LU LU Dfl <(203I3! Â— I I '3 1 > 1' O 3E > a. > O Ci z o u x o^2ZQ Jtii:>c p. r* u a w h ct 3 >c oo ta; z Q Â— t4 X O LL X X 3 X IIÂ— X Â— X 0 Â— *J c^s: Â£cc x a oi x Â— t Â— Â• Â— i' i Â— Â— i _i Â— i o a; 3 o ~( o a < uj o st> o u. a 0 O O 0 O O <5 O mD C O O O
PAGE 188
176 X X X >i < < < c5 Â£ II O O O O II II II I il Tl lO ") i Â— SHT >D O O til UJ hCC UJ II sO O ^0 > T. Z i/) O jr w o O U IO tÂ— x Q a o x 'Ja. 3 eg uj OOÂ— IMflOtfiO NO f\l rr\ t 4" Jj\ m m in O J)O>0Ov0'CO^^N vO O O 0 td m3 O 'O O O
PAGE 189
177 q S J Â• N jld < cÂ£ oj a. ii .,' *Z X Z it nÂ£ X O X "\ x M o a o >< 3 _T a x z t x Â• Â•Â• tÂ— rt Â• IÂ— X Â• Â— ro ^ Hi Â£ < o: D O rn Â— lO, a. z 3 "s ox xiin h t1 h x Â— o ac o. aÂ£ *Â• i^> ^ ( .Â£Uh O w a. O U '.ri M 2: r> Â— Â— r\i X X x o ^ X O OOmOKlJ "HZLIJZ^ > j.Q.rOv. : J<< S Cw (**  Jt Z X H. + +III + ^MI II uw o m Â• < a: a 3 m + a. Â— z a u z o ^ 3 2 q j r<: ^ z 1 :.~ 3: o Â— o Z 00^ a z: 11 11 11 11 11 11 n 11 o o o se. az v \jj x 'u 11 H i! n / Â— Â— Â— Â— 3 rn Â£ Z. Z. Z 3B 3E 3 >(VI fl O H3E^JZN(<1^Jf^iiX^O3 0.S U Z rt u a u axxxxc.a.Q,xxxxxoaax ec u Â—1 ~i rvj ro nT Â— 1 cm
PAGE 190
178 a 3 'j> Â•Â• Â•Â• O Â• IÂ— cO O CO Q Â• Â• OS Â• CO O 5Â£ 3C "5 5*S Q Â— 12 Q O mit IÂ— O Â— 1 Of(\ o X Â•" t/> Â•Â• X CO O .I Â• Â•Â• QO~ULU Â— 1 Â— o Â•$ Â•Â• cs a. 1Â— a: s S3 O t S3 Â•Â£ ~ X X "3 H x fix cs.ro a. x .5 ox x 1 in Â— Â• x ~ o oc a. oS J'(fl Â•SOhtlU^ X Â• Â— PI ~lNIZ"5Q.0rÂ— co Â• ro X t~ h" "" ""* en o Â•. ii is ro ow 3 K1 *a! Â• Â•> x oft Oh. irgHct sQuj ~5 "^ O j*l CIIJt/l tic *Â• ro O Cl LU Q. 0. < Â—Â• cm 0 W U W (M t 3 Â— **" x oÂ— i~oofMxx>a. riÂ— a a O^OhIJh O Gt Â— < Â— am1 Â• Â• x x Â— Â— cm o x a. Â• tÂ— >* Q a. x u x X X 'JJ Â•Â• ri o cl z X Z >< O 'C Z Â•> z Â—1 _j Z tii ri Â— 1 x x x Â•> Â•> Â— x _j a. x Z h x < Â— O OS Â— Â— r^ Â— ifllOHQH X ~3 + 4Â—t X 1Â— co co rnu_ at "5 UJ Â• _i Â• Â•> ~i OS H Â• Â— Â™ < Â•> ~ o Â•> o a. O !_5 .0 Â— X X X ii X X X as 7 o Â— o o co o os j izwzi:o<<< <<< Â— ZCNCMfl flUIXCL C0XXO51XS7 X X X ^> 3 Cl Â—1 M j < H XX ^ i; ) Â— Â•Â•> i^~5Â— 'Â• "ia!X 0 Â•IWO'ht" h 11 f4 (jou. 1Â— o O >Â— < O U3 Â— 1 O UJ "4* i Â• Â— *Â•<*Â•. Â— O ^_iÂ— 1^,Â— 1 ^ ~5 X Â• iti Â— 1 Â— ( rJ O 1Â— *Â•Â• 14 IÂ— I O &; Cl Uj Â— Â— X Â— X OS *Â£ X X "' X < !Â— X S3 rt .? c\ O UJ w LiJ tj O II H I' 11 !l ^ X Â— X X X X X X > 3C Z "> O H II Â• LU LU X CO m < OS X X CO N ii "5 Â— >Â£Â—)<Â— O ZCOZ U^SZO J0!^H Â— ^ m X Q UX X 3 Z h H X Â— I O O O iÂ£ O O O i<_ I! LÂ— I w* }" sj" xj~ P >J* sj" Vj" ** X X LO 2 '""I iN ^J ~5 v ^ CO CO T ^ Z 3 X X O O 1 ' Â— t& <* Â• Â— 1 Â•Â• O OS K E LU X IÂ— 1 Â— 1 Â— 1 Â— > X Â— 1 Â— 1 Â— Â— 1 X/X aa Â£ x z Â— *Â— *" x' iÂ— x x x hh o ox a a 0. u x a x o as uu z (/) o x o x aaax as sa a a x jÂ— os uj Â—I Â— tCMrOsjH M fl < Jl i > O OOHMOOOtNOOO O O O O X J" 10
PAGE 191
179 o SO Â•> Â•> fSI uj < a; lu a ^ X 2 ii QÂ£ X O X NCaDMD si I J O IÂ— a: n q x ar i Â— i ro Â• jjj X Â•.(Â—(/)Â•. tÂ— Â•Â• X Â— ro y:ai x< ^ D a Â•> m Â— r o<Â£ujq:icm .. ~ rOX5i;t(o Â— x Â— po o tÂ— :*: Â— rooro.o"Q: Â•Â• Â— o o *Â• m x x to Â•> Â•. Â— I O rO O X "* IS* *> Â— "5 xroo'' > OvO ,> 'LUiiJ Â— COÂ— I O <Â• Â— Â• Z 2T Â• h x a c hwhxq.im ino; o x . m x a. <*> Â•a.zs cm x OX XI Mfl H 0. Â•> X ""Â• O Ot! fii fiÂ£ + X~r< s .~CMX2"JCLa.hÂ— Â— n m x ~ hÂ— :*: <*v t> o Â• Â•> Â•< 54 X On Oh JNhÂ£l.a UJ 5 r<") a a.jj~a.Q_ Â— Â• Â— cv om~om;vjs:o ~ Q X 0 r*w OOf\iX~5CL(Â— hQ Q. O Â— Â— OÂ— I X J "t O xo Â— i Â— M X ,*.. X Â— Si m MOXO. Â• Â• I 1 Q 0. # X X o X X KU HOU.23J:h Â—i / _i ^ ^! X Â— i _1 2 UJ (3Â£ Â— CM ii ft. O > 3 ~ "O jao zm x < x x ~o oÂ£i(n"(<1l l '.( IÂ— > Â•Â— ."O < > Â• Â• Â• o Â•> o a. ts uiA"> x x x Â— x x x **Â£ "i ~jo oomoaiJ *lZlUZi;o < < ro 3 a. Â— ( m X X OC fi Â— i ii Â— 3 Â— Â— 3 Â•.Â£ j _i Â•Â• m .njoxÂ— >o Â—Â•Â• xtooftÂ— rÂ— ~(\jx ~ .n i/>iÂ— MOOriOU_3:>Â— 'OUJ ^r^s .>*< ,o O 2 ~ < it w. Â— i ^ ) Z Â• ^ O Â—4Â—i Â— I Â— .'' h H rH O rfi x a. '.j a Â— x oÂ£ *c x Â• x cm o""' x 36 2 O O II ll Â• UJ UJ Of <0Â£O3flNI +i!TM x ^ > Â— a 2 Q O 2! U^SXSJKiCN Â— Â— Â— i Â— i x o u.iiizt"t"Xiiftoo o o *: '.n o o o x :i tÂ— Â•Â— i ^X '2 O MNNTZOClCimTjiZ O a 2 O D II N CM N flJ M (M CM O PC a: x _u x L_i jiimoh^h^mtj C Z 2 X Â—i Â— IÂ— w ID 1Â— Q o o i a ON^ocaamiooouKiii? f o a o a a. x a 2 a x cc a a a a < Â— ^ uj rH Â— i rg rrj ^> Â— ( f\ fO >JIT\ O I s O OOÂ— lOHMOOOOÂ— tfMOOO o ooo*iiÂ— ij ro ^O n ro r^i r 'S\ CM CM CM r M 'J rv 1 CM CM r \l CM CM <\J rsj CM CM N
PAGE 192
180 3 36 O Â•Â•Â• fSj $ LU < Of UJ CL iÂ£ ojoo o^ %Â£ X "> U IOS a X 35 tÂ— <~" ,0 Â•Â•Â•*&Â£ >< lI/) lÂ•. X Â•Â• HiflOXSU X ~ Ct Â•> O Â• IÂ— ~> "5 a *o mis ho Â•Â— Â— _i O fft O X '"> Q Q xr r >o<i'QO>LUUj iÂ— i o ri Â— ocro:c.Ha:s: x x 0 Â— Qr^X^O^IÂ— Â— X O Â— Â• Q vf Â•Â— JE 31 ~~i m tÂ— fSi r\i X ~ m X a. m 0. X 3: X X OX XI Â• vO 1 xÂ— Â• o aÂ£ a. ac Â• i I x ~n ~(Njizoa.cL^ Â— Â— Â— ci .mi m IÂ— Â— ... ^ sÂ£ Â•Â•r0 fc O Â•*'**Â•* Â— ** m O fn Â•> 3 Â• tÂ— eft Â•> 0!! "C ~~> O ~~> X"OolOri_irvJIÂ— G.CQIJJ X X Â•> n o cl iu Â•> a. a. h Â— I Â— CNJ oi Â• Â• IJI^SD X X Q XOÂ— I Â— OOOJX~3Q(Â— Â— Â— O Â• C_ 0Â—<~Q'. ^ x pH *X Â— liQlflw X Â•Â•Â•Â•Â•Â• Â— XO I* Â— I Â— fNl O X a. !Â— ~ 4 G SrO Â—IÂ— xZ i* X O X X X UJ r4 O 07!" 3 X r~* 0. Â— Â— "iCnC Â—CX i^ ,r gr Â— i _j 2! iu oJ Â—Â• fft + rsi Â—' x ox* x x Â— o_ia.xxiÂ— 3C < I ^ ~7 sÂ£ Â— >J Â— O oÂ£Â— Â— fO Â— i/>XOCOhX3 ~ # X +Â•> X X + iHXH<*Vf0WU.Z"9UJ o Â— < Â— t =>f 14 Â— ^ ro ** aÂ£. IÂ— Â•> ci ^INXXX< or<~i^xxx Â— < n > o <Â• Â• o a. a ow n x Â— x < < < ^ q x a. Â— < < < aÂ£ 5<_; Â— OoroOo:_l<"l^UJX^'i ci +SZSZ *N Â— + rf X x X Â— 2* C (\l f"l fO fOOXXa. S3XO P"> j Â—. \vthTX ^ 0^5>~' 3 X 0. Â•< Â•> 'Z. < h x oi ^o m x tt\ Â—Â• Â• Â— Â• Â•+ rtjjXw o **! 'Si/lOl^ fl fi o. m h m .~oo m m + *m is) Â— hOO"^OU.3Sw3lU 4" <~i + a. n 4 Â— Â— ^ O. X Â— * <> 4 Â— Q g; _ r Â— <* ^ 5 X <* 3X"0 un I I ^ i ii X <Â— Â— sÂ£ Â—Â• II uj uj Ofl Â—4 O OOÂ— i cnj Â— irxjrOOÂ— irsJO OlfMflOHNO OmOOOOOONNN^ tn O O O S r\j CM
PAGE 193
181 o Â£ IE 1* o Â•J+ o CNJ Â— o X ~1 IÂ— < I X + o i*o iÂ£ \ / X Â— ^a o X o :Â£ CC UJ Ia: a 0
PAGE 194
182 Q 2(J M s}LU < CC LU Q. Si x 2Ma:su x n o a a o "n ^ I TJ h oÂ£ Â— x x iÂ— >* rn Â•> si Â£X Â• m ^iflOXSU *"Â• Q m iQ hX > f rrj o r^ o *Â•*Â•Â• *Â• oÂ£ + '"* > Â•i o if* s i/i Â•> x Â— Â— < xmo* j *>ciO'ijjLu + x Â— oi cm x .2 ; a. a. IÂ— Â— Nf" m Â— ox Â— m m i m iÂ— r* n, X X w x Â— roiO n Â• Â• _i Â— Â— Â•Â— Â—> '2 Â— m ^o^xxsiÂ— pn Â• cc x .4*^ ^ ro *Â• m x oi*i ouNHao'jj ex Â— Â— Â— i x Â— ex ro~Q o, m a a si 0.0. o Â— X .. _i Q (fl x *Â•Â•> _J X >Â— Â• <Â—< Â— < I Â—i* ri Â— r\joxa. fÂ— >i a a. x + _j si ii ^ X O X X XU .HOaZDZJ Â— i <i_ X xÂ— XX 7t 2c Â— i ii + jsi + i D '3JD.D Â£tZ< X X i << + Â•> Â• H Â— o a: Â— ^ m ~ t/) x o a. a tÂ— x j + Â— xx ~5 x :*; j x *Â— (ftm silt ztuj Â• iÂ— i Â— Â•Â— ~ Â— x o 0(J ftft ft ^T fO X X X ii Â— x < p p pO o Â•> a. o o tf* Â•* x a. rh en Â— CM +Â• Â• I 3 ai p z < iÂ™ 3E GC rn r"* x f1 ~i Â— i X Â— * t 1*1 Â— 'JJH!,Â•Â—Â• h i; h t/> p iÂ— ooÂ— loUii* an iÂ—i mifl *iijy:HH^om si X si O X Â— I Â— I Â— Â•Â— < Si ~S Z Â• Si CM pO CM'* X X XSi _^*x .Â£ 0. i:j Â— Â• a Â— X CÂ£ Si X ft Â• CO Ql "Â£* <* l tl II X ft Â—* I* < Â— x 2 Â— 3:2T0 mid I I I I ^ T JÂ— X LU UJ ft 3 PO Â•Â• < Ci. O X CO rH O O O Â• Â• + * ST a u z <_d si :x s: a _j ci si  Â— X a U. X X "9 3K IÂ— tÂ— XÂ— ***#*OXQQ.aOOOSi XXX _ hi O l\ rO m M I Â— Â—. Â— ^..JsJvJ.. Â— Â— Â— 3 ,~ i/i ^r x a. x o. x. n m n "? z xox o a ii it ji ii J" ft oc h ii C S uj x iÂ— ii ii iiiim 7^ Â— < Â— Â• i~ 3 jjS e s Â— irsims}Â— tÂ— .I r\j 3 Q ftx 3 3 M r*\ J" W N si Si Si si O 3 O Q. UJ si ^ to u Q '3 O C X. X X X X X X X X X X X aÂ£ X x o ooÂ— i (\i fi
PAGE 195
183 *Â£. > "5 o X rÂ— 4 LI a CM Â•Â• o O o Â— X o a *Â£ O O X Â—1 CM + v 1_ id .Â— Â• CM X :*. o 4 X t c Â• X i^ 1 Â— i m x I X e% \ X Â— rÂ—l Â— 1 Â— 1 Â— c\ id Â— pÂ— i X Â•w "T* Â•. XI id # ~ >_Â• Â•~ CM X o X "" X :Â—j Â— s ~Â— # Â•Â— IT* id po X *S X Â— 1 1 X CM a. n < <5I < ii Â— ;":_ + Q. 2 SI 5C Â• a Â— H Â•^ a ic ~ Â— i ~> rÂ— X u rH m. X K + X ^ Â—1 c* m >H in Cd X X Â— H Â•Â—I <" m Â• in 1 Â— o ^ a Â— N, o c II 1! II X X Â• Â— f 'J T~ ~P + # o X ii ~> H :I o A ci CM r\i Â£ ff, rÂ— Q. C X a 1 I o O o i^ II Â•w j"3" ^ *Â•
PAGE 196
184 Q 35 O p 0~ CM *! 3 X <* x x "3 rÂ— X Q x S! 1Â— H m r> Â•Â• ^ X fc Â•H Â— i 00 a X Â— X ... n. ~ ro v^ LU jg < ,vX 0C a m Â— X O ^ X LU oi (Â— CM a* .. > ro X "3 X hH p X m o ro f* o f Â— 0: K Â• rn o s> ~Â§ 5 X X X 3* o X ro o rl o Â•* 0. CX a. 1Â— LU X LU X m o Â— Q Â— i HI H hJ1 Â— 1 .Â— o ~> Q t a: X o en X X cs a. X f0 X Â„ ^ uo Q X 3* X n Â— ,> X CX a. rÂ£ s}" Â• *Â•* ro Â• Â•k X tÂ— LU "J; X ^ iO ro .. oi CM Â£ 1 Â— t Â— ) X Hr* ro o n\ Â•> O Q. OJ O X X CM Â• o Â— i <Â• *1 ^ U \Â— CM Â£ X X Q O Â— 1 Â— o O o X CM ^ 5 X iÂ— >Â— Â• Â•) ^^ ^ ro Â— Â•> Â—i Â— CM O X Q. Â• hrl c ,\ X o X X & :< LU 52 Â— 1 O 0. LU X L0 / *Â• 5 a Â— *Â• O _J X X Â• Â£ h^r < Â— o X Â— Â— i <Â£ r> *Â• m o X Q i* X f ^ 'O ji ~i o Â— o o m o X _J ~ hX LU 2 X *Â• 2! Q fM m ro m u JX X CQ X X 0 5 a. f* ~ 2T Â• Â• o n h~4 ** 2C O0 Of H ; Â— n JT\ CM CO H1 Â— o o ^t O X X? 1 Â— i UJ vT r X 3 "Z Â—J Â— 1 Â— r~i ^ ~ > is. H e> X + X ex UJ Â— Â— Q Â— .. zm n O^ X X t^ r\j ',. f~4 rl Â—1 .Â— ; ~i ~! Â—4 Â•H < Â— X a Â— H *~ f\l o u HH ? ^ 'JLJ O m rl X Â— 1 j .~j Â— 1 Â— 1 _J X 5C CM X Z Â—4 X X X X 2 *> X > 3E ^ Â•> O iM in X X X X. X X X r X X a. X dj UJ G* O rn c% : X aÂ£ X b ro r ^ O O Â• Â• I + + + X X X Yi + 2 Â— u j O : S X O X iX ^ in x X j_ O + Â—J n Â— t .Â— < f4 O >Â— M X a U, X ~J3 Â—> 2; (Â— ,Â— X H ^ # & X ) N, & rl Â—1 Â•M r~* Â—1 PMl rj X \/ J x i Â— a O CM f*1 >^" in >J" r^ CM Â— 1 Â—4 ^j ^ CM CM "J CM CM CM CM .0 i/i a Â— 1 a rt iM ."0 Â•aa Â— M rsi m m rO X a> a, Â— X X X X Cu X X X CL ^ Cl a. a X X o O O rl oj ,0 < uo O r1 fM ',"0 Â—i c\i .0, nT m o m O C3 ** O ,J H Â—4 rJ CM r\: M CM CM CM CM 0 o a in LTi in .ri in n m n i^, in 'T 1 in m in L^ in LO n n LOi n Â—> mM
PAGE 197
185 CM j Â— Â— i Â—I CM Â— o J f} _j X XL O ^_ X V X X T_J x X x CM V CM CM o ^ X Â— < SÂ£ J * iiC H _J ^ X : X 31 ii X ^ X ^ X Ki >!< X o. Â— Â— + J + Â— + Â— CM Â—4 Â— Â— i iÂ— i CM X ~~^ ^^ Â— Â— CM in a. 1 Â—i X _i i _^ r~ X X ii O ft. X X X X x + 'Â•0 + a. + + k "ii + + + 1 Bi CM Â•H + o o 1 ^ X CM o o J Â— 1 Â—) rO a X <4 Â— 1 fi i vO CC CO 1 Â— Â•> Â—i w T 'SI Js Â— Â• Â— i o a. a. c_ CÂ— oft. # . Â— Â— ^ x + o Â— Â— Â— X ~* m 1)1 X Â— nT X a >*Â• ft. o J4nT Â—J cm a. i^^ ft, + ft. + + + Â•> X <Â— ft. l a, + a. + 4=. x + Â— in Â— nT x Â— Â— Â— O O v0 X ^ + N, Q. Q_ ft. a. p* a. a. x o_ a. a. o r4 Â— + + + r + 5 + + Â— + + r + 3 m n ur\ rr, pH O Q M ^ m Â—J. o o Cl 0. ft. CM CM CM CM. X X X Â— ^ tÂ— vC ra. O o sC X X X Â— rl X< + + CL a. Q_ Â£L Â— 1 ). Â— J 5 CM A O ^Â— v^ 5 1 Â— 1 ~5 Â—i ~0. Â—j P0 m rn m Â•> X m rÂ— o m ^i n > X Â— Â— i a. Q_ a. a e. <~ iÂ—i m a. a. CO a. a. a. c Â• p. >" rl Â• a. + + + *. Â—1 ^ .Â—4 Â— ro o Â•itQ + + Â— Â—* i Â— i rl Â— rn Â— i CM + O.ii Q_ o + o X X Â— 1 j Â— i X o o CM > Â— i o II II II X X rÂ— 1 Â—j Â— Â—< Â— ; Â— i Xi o ro o i~\j Â— i o II II II X X X V X 21 H m ,, re. ro + Â— i v' _i Â— t Â— .ii _J _J Â•J "T O, Â— 1 >_ r^** *> + + X Â•r 31 a. p a. a. 2 ii ; Â— 1 II X X X ^ o X X X X X Â— a. a. a. a. ixi Â—> II o & # # + M & A X a. >!> ^i + ~Â— >> X o Â—i m (N CM CM CM Â— /^ o c3 + O o r4 Q CM rM CM eg CM Â— Â— >N r\l IN O >t c_ ft. c_ ft, o O r X 0. c_ o O o X Cl o_ a. X a. t** Â— Â— Â— CM CM 3 II II II il II 11 II II II in Â— i in m i Â— i X ii II II II ii II II ii II II n II II II in in in rl ** X u rl cm ffi Â— 1 CM pH vt in Â— ho Â—1 CM f) *^" o Â— i (M o Â—i CM rfl din Â— X rn rO rt*l x y* X X ^ o CZ3 a. uu v0 vO D o o s. r^rD x ^: X ^ ^ ^Â™1 a 3 o X x r Â— G X Â— X X X X o a a X Â—4 c& a. a. a. ru a. ; Â— a. *Â• a. X X Â— 1 X X X 5 ^ X X Â— 1 H CM m iÂ—i CM m tf Lft o Â— i r\j O o Q rÂ— I vN ro < o Â— i CM o rl CM n m m i.n m vn ,/> m in s.n Â•j~. m J^i xt ~ n in in m Â— 1
PAGE 198
o o a. 186 a Â• Â•n sO CM Q. a + IT, <\J a o rl a m + c o o :d O ho o ur\ <Â— ii y. z o a: O 2: 3 r> 3 1Â— q ci ci lu a: Â— tQÂ£ UU ( rsj o o o o si m m
PAGE 199
187 < ixi~ X o uu SiXTOtQj Â• rt J Q X Z H* Â•* UJ O ro Â•> ^ en O X to i_ t/> Â„ _ .. X UJ vO <Â• HViDXEO O I s o co >io XJiii*Â— rÂ— OUJX X po ~ o ^ H UJ O O rnOrni0'>e:o O q o m x x Va uj rÂ— i a pn o x v> >Â— X ro o Â— I Â• Q vO Â•Â• LU UJ UJ CC ,i Â— Ocrara.Ka'.X X UJ Â• Â•0 Â— Oh I uj! Â— i Â— m *< > O o o Â— t O <) Â— x z "? h Â•Â— 'xi m OriÂ— iXQX"Â— V\ CC rO X ^ co x Q_ ro. 0. Z 3 UJ UJ I s O X X X VTi Â—i i/l oo Â—I x .Â— ec a. cÂ£ X 3 ,jÂ•Â•^rO^^STOIÂ— <'IJJ3<i o o ra ea >X en *o<^3>Â— ro~: Â— ox x z x O ro OHj^haam w> o > o c< Â— o ro ro X x Â—i to c^ 0. lu uj cm uj i x ro ~0 Q. UJ O. Q. (Â— IÂ— hI s !Â— x X Oh Â— OONZ">MI2 z uj Â• Â•Â— > oi Â— o Â— ix_j< Â— <Â— x o ^ o Â— i Â— X ro Â— X Â•> Â• "> TÂ£ Â£ iJJ o Ifl ^c ri Â— CM O X QÂ•* Â• I 1 Q Cu or. QC Oi O O ~i o X O X X X XI itOfl. Z D ZLU UJ U CM I s '' I s z z Â— J z uj or. Â•Â— iÂ— !Â— z < Â— i > h 3 QJ1D S Â— X i Â— < iÂ— i Â— o sfli/)ioaQix z z Â• Â— x (Â— ro m c\ u. z s uj Â•> O o^Oifi x o h h q; i_ ..,_,_ r>o c a. X Â• O Â• Â•> O O. ',0 O m ~IÂ— Q en tÂ— X X X I Â— x O ^OOfl OciJ lZ'iJjZii;< < X r3 <<< 3 TZQNWffl rOOXXCO 3 X irr( X X X Â— Â• Â— 3 a i Z < tÂ• X fli 3 X Z X' ^thv wU cv;xÂ— ^oÂ— z wan iÂ— c a.Â— 4Â—.a_ .. q, siÂ— ro o Â— i o x. 3 ^ a uj ! ci ^: X ;_> u>, O 1 1Â— (Â— X O t 4" <\ O UJ Â—Â• 3 z X X rX) O X 1 uj uj a m < 0Â£ O x ^ in ou Â• rx v: 5 Â— < x 0.0 Z X U Z O ih* s^ Ps s xh P"Nr*Â— f*I s
PAGE 200
188 Q ~g U> Â• (N O IÂ— flfi *Â• a x 3 tÂ— Â—t ro Â•> iC X iÂ— go Â• tÂ— Â•Â• X ~ Â— Â— iOOOXSO Â— m :*: iu Â£: < a: x Â•Â• f^ km .. rn X J X IÂ— hO x Â— > rA O Â•Â• fÂ— q Â• o roxs: >O Â—i Dfl 0 X Â• *" CO *Â• x en O Â— t O sO LU LU Â•!Â— o >* "Q^CLiÂ— c^s: O Â— Q f X i LJ *Â— iÂ— <Â— Q OrÂ— O ^ 'Â— 51 ^T "^ Â• HO Â— Â— Â— < X O Â•Â• X Â•Â— ur\ m. ci XÂ— roxCi.ro C_ Z 3: ox x X Â• Â• ifi m x Â— o c' i q: X Â•Â• Â— rO i"\IIZ5C.a.iÂ— m x Â— tx O ro OHjNha.cau om mxxJ^ oi o. .. ro O Q. LU Â• 0. 0_ x Â• o r< oorjz5CL.iÂ— (Â— O < Â— O >< X Â—I 1 " Â— i Â— a i*\ Â— x <Â• Â— < Â— f\ O X O. Â•> I i O Q. xoxxxujii ,. Â„ Â— qji3 x HX O of Â— * fO .Â— < CO X O 3Q IÂ— X .^ ^ x hm rn rfl ll 2: 7 lu ii ^ ^ ; Â„ o Â• o a. & o to xxxxxxxx"">"3~?~? O 'COrOO^J hZU~^<<< ... z C3 cm m m ro o x x a. aX'X^:2:2;55? 3)'<'''''Â— ^ a .< Z < H* X 0Â£ x s Â• ....~<._~~ Â•Â— oÂ£ X Â—Â• *Â• O Â•Â• Â— 1 I W3IIÂ• Â— 1Â— ifi*>OC5QQC5 LL, hiOOiiOILShOUI .3...Â•.Â— Â—..Â— ^wror\r40X iH ZHp^wH X ""5 Z Â•> X Â— I r 1 ^ M 1 OjL) X X X X x a. lu Â— Â• Â— Â• a Â— x & x x x x x x x + t/5 Â— X Q ri ,r\j O LU '>~i Â• LU O II II H l I! ; l II 2! > X X X X Z Â• X > 3t Z II II II II II ~i utu am Â•> <^r x ^ 3 Z t/i ^ r i^'^3) 5^ 3llXXXXX a a z q cots<>f">iHMM 11 11 5C X : LI Â£ Â—I < Â—1 * >~t * Â—4 i~< Â— Â— Â— ^ LU tt II II II II C3 X X 2: Â— Â— Â— ~ Â— QQQOX m r\i OOÂ—i O O a Q ',*^N*0'tflN"'H>j'fl(\j 1 iOii^ wu a o ci O O X X X X X X X X X 1Â— X X X X X X X Â— I r< :\j m Â•$Â• rH ^J ,"0 4* ^\ O P"" O O^fMOOOOOOOOOOOOOOOOO O COOOi'M^^OH^rOOi(MfltWi,'\j SO TO CO CO X) 03 03 CO CO CO CO CO 33 CO =0 CO CO CO CO CO CO
PAGE 201
189 cm en X X + Â•+ o O ^ O O O O 3T y. o J Jii2 JX X X X j_ I X II I! II il II !l H II Z X X X X X X X aC UJ oooooooo.o fCl njÂ— J ny ft\ _i f\ r^ Q LT\ L", vO sO vO hPI s 63 33 00 CO CO 00 SO 33 CO CO
PAGE 202
190 Q Â„ i. > HI o :s o !\i hx z >: s o x oj Â— Â— !M o o o Â— '3 ii 5. II X X "5 U HQÂ£ o L.J X X q x ~z. yÂ— Â— < uj : Â— i :jj Â— i X n IÂ— t/) tÂ— p. X >Â— l V) H i (/) _. Â— I 00 Q X X 'X (0 *"" O VI Â— X ~ ro ^uj 2 I*" 'LI s I*" ; u Q O nfllS tÂ— o Â•" *Â• Â— I O t*l O X Â•> /) "Â• vO O H '<0 x II X rO O Â—< Â•> G O >lU uj iÂ— i Â• Â— < Â• ..,i~.oJ~a:3>a:x (muj (M in O ** X *Â— 'XHOiÂ— ) Â— i Â— i ni si w N Â— O O Â— < Q ^ Â— X X O ~ rt:j Â— i iJJ Â— Â• Â— q _,_,_ixax in oc iÂ— ooiÂ— oo x X w m x 0. <*> iz3S Z 2 Â• Â•* Â— 3 X XI lfl "" HI O'il 3 "*>"" Â„ x Â— o cd a. ai Â• X II jÂ— >_ h Â— Â•> Â— oo v c Â•,, Â— i a tOh JUJ3: uj lu u ^oo x .. Â—. m r>jiz)iiii.rk lu i i lj i w ir> m Â— no Â•> po x 14 iÂ— on i o ^j Â—< o~>cnc^ Â•> i^l o "Â• "" *Â• x iÂ— Â— X Â— Â— Â• x Â• Â—* * CO Oco...3:iÂ— mX "Â— oo Â— oo "1 X O rO O Â— IÂ— I f\J tÂ— Cl CO UJ Â•> Â•> I *Â— q f0 ,c, X X _! tO cC CL O O O O O Â•. fi O 2. iu d a. X X X X Â— Â• X o o ; nj >o^ > *U>MÂ£3 <\i o* <\i <\i C> r\i ^ X uo UJ > Â— Â— Â— fsj Â• Â— i Â— x ro Â— X Â•Â• *Â• Â•* o Â• v0 Â— rC <> Â— CM O X Q. *Â• IÂ— Â— X &, 0 ih fc o Â— Â• u. Â— X U X X X !.! "CN > BJ c_ CJ Â— i o 5 ~ .QjaD 'Zhz1 ara ix < ;m Â— C7^ C^ _i X V~ PO ro fA U. Z ~? UJ x TO X X H ii c ; taiÂ£Â£^>UJ'Jji U.J UJ Â•. iÂ£ '*Â— ~ n o o a. o o l;o, x x x x Â— x x x x x Â— Â— x x x jo oooafJ wzujz^ujUiZWLJZo << XXX r>Qi' x >Â• XXXÂ— nO h Â•EuiOIllÂ— ZOH X'XO" \Â— 'rXX ^1_ k00H0 123LU x x x .'5 <^ X Z XX o ^ <^ uj uj Â• am <(iODfn? v rfi Â• c^ ""< Â•> ' o x 5 Â— x x x o o x ~> *Z X O X X X 3! X X _J OS X h X Â— X CT 1 "5 UJ UJ X *_, x O JX X o Z iÂ— iÂ— X Â— O Â— O H l o O O II 3 a. Il N> r^ N. Â— i z t.o z iÂ— x n, Â— o v, fÂ— Â— t Â—i Â—i Â— iÂ— _i x yMflfl o q 2 o < Â• < Â• ouuiff^auHiis^oai K S UJ X X N Q S N IÂ— X Â— Â— Â— I X X Â— Â— oo Lii Â— ( Â— I Â— I : 5 s z s s:h zaJH x xx lu x ) O M X O UJ < O u X X O OO^sJU.U.I0 O O X ._/) XX U. JX HOOH.*'jX ^ _i Â— i f\j oo <Â— t .\j ro >t ur> o rÂ— < : "j '' t\i O O OOO<(Mr0OOOO'OiCN q J t\ JO" I iH r1 Â— '(MOÂ— 'OOÂ— <Â— (Â— I o o o Â— i Â— < Â— Â— Â— i^MM^nmflfi O :T> 0> C7> 1 7> J>i 'T 1 L> ^ OV ?> O1> CT" C ^ ^
PAGE 203
191 U) UJ Z . Z a a c/0 o (Â• Â• o o x x ft CM 1 J a. a. o o z z Z Z v: uj. uj uj uj Â£ Z IÂ— Â£ 2 !Â— 5 hQIhOI1Â— X co hX oo iÂ—i UJ UJ Â— tÂ— Â•> (Â— o o Â—> o s Z Â• rl C\J 3 'JO Â— 'J3 O w '' Ci < 0* c < C Â— k>h iÂ— i 5 Â• tn Â— > a. a. h uj Â• O > O Z UJ C ~5 O UJ O > Q ~i ^ O X O Â— < flZ O X sr t uj Â•#< aj tss Â•< u. "< x x uj si u. uj jr M ij i_ ^ ij.j iÂ— x 2 "J o Â•Â— o iÂ— 3 H 5 O Z X> PsJ Z O CÂ£ + + O O Z X CM CC O 'J o i/i JÂ— i a. uj o o u. o iÂ— c w ^ u. rn ii Â— i Â— < n x Â— i ,j_ o> ii cm z o xz ) Â—i ii 3 Â— < ^ ii o n w q Â— ii Â— Â• ii o x x x ii Â— o Â— ii Â— ii ^ 5 ii z Â—J iÂ— II X X cm IÂ— e; :s ojm xi u a uj iÂ— n n n cc :s cj _i uj a uj x u4 iÂ— h 4 D :.JJ u i p H 2 >Z  