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Masters Theses
Student Research & Creative Works
1971
The design of a high temperature thermoelectric
generator element using silicon carbide
John Talmage Barrow Jr.
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THE DESIGN OF A HIGH TEMPERATURE THERMOELECTRIC
GENERATOR ELEMENT USING SILICON CARBIDE
BY
JOHN TALMAGE BARROW, JR., 1944-
A THESIS
Presented to the Faculty of the Graduate School of the
UNIVERSITY OF MISSOURI-ROLLA
In Partial Fulfillment of the Requirements for the Degree
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
1971
T2579
107 pages
~c.~
Approved by
ii
ABSTRACT
This report presents the design of an optimum, high
temperature silicon carbide thermoelectric generator element.
The analytical efforts have been divided into three
basic parts, the development of the theory, the accumulation
of the data, and the optimization of the design.
The first step in the theory development was the
derivation of accurate design equations.
With this done,
the design philosophy and computer program were constructed, the latter utilizing a subroutine to contain the
design equations.
The data was obtained from a survey of many references
and, for the most part, was found to be inexact, requiring
the consideration of ranges of loosely bounded values.
In evaluating the data and optimizing the designs,
both graphical and numerical methods were used.
The actual
calculations during the optimization process were performed
on the IBH 360/50 system, and entailed some twenty computer
runs, encompassing sixty designs.
The final result was an element that would produce
electrical power at a power density of 9.2 Megawatts/H 3 and
an efficiency of 9.17 Percent.
iii
TABLE OF CONTENTS
Page
ABSTRA.CT • ••••••••••••••••••••••••••••••••••••••••••••• i i
TABLE OF CONTENTS ••••••••••••..••••••..•.•••••••••••. i i i
LIST OF ILLUSTRATIONS •.••..•••....••.•..•.••.•••••....• v
LIST OF TABLES • .••••••••••.•••.•••••••.•••••••••••.•. vii
NOMENCLATURE • ••••••••••••••••••••••••••••••••••••••• viii
I.
THERMOELECTRIC DEVICES AND THE
CONSIDERATION OF SILICON CARBIDE ••.•••••••.....• 1
II •
III.
A.
INTRODUCTION •••••.•••••••••.•.••••.••..•... 1
B.
ACKNOWLEDGEMENT ••.••..••..•.•.•••••••...••• 5
DEVELOPMENT OF THE THEORY •••..•.••.•.••..•..•.•• 6
A.
DERIVING THE EQUATIONS •.•••••.••.••••.....• 6
B.
PHILOSOPHY OF THE DESIGN . • . . • . . • . . . . . . . . . . 16
C.
DEVELOPMENT OF THE COMPUTER PROGRAM ••..... 24
D.
SUMMARY AND CONCLUSIONS •..•.••••.••....•.. 30
EXAMINATION OF THE DATA .••.•....•..••.•••..•... 31
A.
CONSIDERATION OF THE MATERIALS
AND THEIR PROPERTIES .........•••.••......• 31
1.
INTRODUCTION OF THE MATERIALS ..•..... 31
2.
ANALYSIS OF THE EPOXY ADHESIVE .•..... 32
3.
ANALYSIS OF THE SILICON
CARBIDE AND TUNGSTEN •..••......••.... 4 3
IV.
B.
ESTIMATION OF THE TEMPERATURES EXPECTED ... 47
C.
COMPILATION OF THE REMAINING DATA •........ 49
D.
SUMMARY AND CONCLUSIONS .....•..•.•••••.•.. SO
ANALYSIS OF THE PRELIMINARY DESIGNS ••.•.•••.... 52
A.
REITERATION OF THEIR FUNCTION ..•.••••••..• 52
iv
TABLE OF CONTENTS
(CONT.)
Page
B.
EXAMINATION OF THE PRELIMINARY
OUTPUT DATA ••••••••••• •••.•••••••••••••.•• 53
C.
V.
SUMMARY AND CONCLUSIONS •....••.••.••.•.... 62
IMPLIMENTATION AND ANALYSIS
OF THE SECONDARY DESIGNS •.•••..•.•••••.••••...• 63
VI.
A.
REITERATION OF THEIR FUNCTION •..•••.••..•. 63
B.
CONSIDERATION OF PROPER DATA
AND ESTIMATION OF RESULTS .••••••••••••.•.. 64
C.
ANALYSIS OF THE SECONDARY DESIGNS •••...... 67
D.
SUMMARY AND CONCLUSIONS •..•..•.•.•....•••• 73
IMPLIMENTATION AND ANALYSIS
OF THE FINAL DESIGNS .•.•....•......•.•......•.. 7 4
A.
COMPILATION OF THE INPUT DATA AND
REITERATION OF THE FUNCTION OF THE
FINAL DESIGNS •.••••..••••...•••....•.••... 7 4
B.
ANALYSIS OF THE FINAL
DESIGNS VERSUS RATIO ........••..•.••.•.•.. 76
C.
VII.
SU~rnARY
AND CONCLUSIONS ......••.•.••...... 82
CONCLUSIONS AND RECOMMENDATIONS ...••...••...... 85
A.
CONCLUSIONS . • • . . . . • . . • . . . . . . . • . . . • . . . . . • . . 85
B.
RECOMMENDATIONS ••.•............•...••..... 86
APPENDIX A.
THE COMPUTER PROGRAM . . . . . . . . . . . . . . • • . . . . . 89
BIBLIOGRAPI-IY . . . • • . • . . . . . • • • • . . . • . • . . . . . . . . . • . . . . . . . . . . 9 4
VITA . • . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
v
LIST OF ILLUSTRATIONS
Figures
Page
1.
Proposed Design Layout of an Element . . . . . . . . . . . . . ?
2.
Example Power and Efficiency Curves . . . . • . . . . . . . . l9
3.
Flow Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6
4.
"Variation of Thermal Resistance with
Brinell Hardness Number of Adherend" ....•.•..... 35
5.
Cross Sections of (a) Copper Alloy and
(b) Tungsten Identically Bonded with
Epoxy, Showing the Values of BTR for Each . . . . . . . 38
6.
The Joining of Copper Alloy to Tungsten
with Epoxy, and the Resulting BTR'S •.......•.... 40
7.
"Results of Bond Thickness Test for
Epoxy Bonded Specimens" Expanded Upon . . . . . . . . . . . 41
8.
Load Voltage Versus Change Number
for the Preliminary Designs . . . . . . . . . . . . . . . . . . . . . 57
9.
10.
Load Current Versus Change Number
for the Preliminary Designs • . • . . . . . • . . . . . . . . . . . . 58
Output Power Versus Change Number
for the Preliminary Designs . . . . . . . . . . . . . . . . . . . . . 59
11.
Efficiency Versus Change Number
for the Preliminary Designs . . . . . . . . . . . . . . . . . . . . . 60
12.
Load Voltage Versus Change Number
(8) . . . . . . . . . . . 67
13.
Load Current Versus Change Number
(S) . . . . . . . . . . . 67
14.
Output Power Versus Change Number
(S) . . . . . . . . . . . 68
15.
Efficiency Versus Change Number (S) . . . . . . . . . . . . . 68
16.
Output Power Times Efficiency
Versus Change Number (S) . . . . . . . . . . . . . . . . . . . . . . . . 69
17 •
Load Vo 1 tage Versus Ratio
(S } . . . . • . . . . . • . . . . . . . . 7 0
vi
LIST OF ILLUSTRATIONS (CONT.)
Figures
Page
18.
Load Current Versus Ratio (S) •.•..••.••••..•...• 70
19.
Output Power Versus Ratio (S) ••..•.•.•..••••.... 70
20.
Efficiency Versus Ratio (S) •••••••..•.• o ••.. o ... 70
21.
Output Power Times Efficiency
Versus Ratio
(S) . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
22.
Load Voltage Versus Ratio (F).o •• oo•o•••o•••····78
23.
Load Current Versus Ratio (F) .•....•...•••••.... 78
24.
Output Power Versus Ratio (F) o .. o .•• o .••••••.... 79
25o
Efficiency Versus Ratio (F). o. o •..•••• o ••....•
26.
Output Power Times Efficiency
Versus Ratio
0.
79
(F) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 80
27o
Load Voltage Versus T 4 .... oooo•••••o•••·········81
28.
Load Current Versus T 4 o·······••ooo•o••·········81
29.
Output Power Versus T 4 •...........••.••....... o .81
30.
Efficiency Versus T 4 .• o·•······••o••············81
vii
LIST OF TABLES
Tables
Page
I.
Equation Order in the Computer ..•.•••••.•.••.... 24
II.
Divided Difference Table of the
Three Data Points of Figure 4 •.•••.•••••••.•.... 37
III.
Final Values of Bond Thermal
Resistance and Conductance for
the Copper Alloy-Epoxy-Tungsten
Bond as a Function of Epoxy Thickness ••.••.•..•. 43
IV.
Final Values of the Input Variables
for the Preliminary Designs ..•..•.••.••.•..•.... so
v.
Output Data of the Preliminary Designs ••...•.... 54
VI.
Percent Variations of the Four Basic
Output Quantities with the Change of
Six Input Variables .......•..•..•...•.•......... 64
VII.
Expected and Actual Results
of the Secondary Designs •..•..•.•.••.••.••.•.... 66
VIII.
Final Values of T 4 , T 5 , apn' and ~pn,
All in Relation to AT .•............•....••.•.... 75
IX.
Output Values of the First Final
Designs Versus Ratio and Temperature ..•..•.•.... 76
X.
Expected and Actual Final Design
Operating Output Variable Values .•.•.....•...... 77
XI.
Input and Output Values of the
Optimum Design of a Silicon Carbide
Thermoelectric Generator Element ...•..•....•.... 82
viii
NOMENCLATURE
Va~iable
Computer Name
Definition
Units
A
U(2)
Element Area
cm 2
An
U(2)
n Element Area
cm 2
Ap
U(2)
p Element Area
cm 2
A5
U(4)
Substrate Area
cm 2
General Seebeck Coeff.
n Element Seebeck Coeff.
p Element Seebeck Coeff.
B
DEl~OU
u ( 8)
pn Seebeck Coeff.
ALPH
n Thomson-Kelvin Ratio
u (2)
Element 'Vlidth
BFTA
Joule-Kelvin Ratio
Cl
Joule-Kelvin Factor
C2
n Thomson-Kelvin Factor
C3
p Thomson-Kelvin Factor
u ( 3)
Separation Between Elements
DENO
Efficiency Denominator
u ( 4)
Substrate Thickness
Cm
u ( 5)
Epoxy Thickness
ern
DJUN
Junction Thickness
em
DELT
p Thomson-Kelvin Ratio
ern
Cm
General Thomson Coeff.
VCrn/°K
u ( 11)
n Thomson Coeff.
VCm/°K
U(lO)
p Thomson Coeff.
VCrn/°K
Planck Constant
Joule-Sec
ix
NOHENCLATURE (CONT.)
·Variable
Computer Name
Definition
Units
n
EF
Actual Device Efficiency
%
ncarnot
CARN
Theoretical Carnot Eff.
%
Actual Carnot Efficiency
%
EFl
First Efficiency Check
%
EF2
Second Efficiency Check
%
EF3
Eff. Equation Coeff.
nsub CarnofARS
General Current Symbol
HI
K
U(l7)
K
K
n
p
L
m
m*A
Amps
Amps
Load Current
General Thermal Cond. or W/Cm°K
{Joule/°K
{Boltzmann Constant
Epoxy Thermal Cond.
W/Cm°K
u (13)
n Element Thermal Cond.
U(l2)
p Element Thermal Cond.
SK
Substrate Thermal Cond.
U(l)
Length of Active Device
ern
U(l)
n Element Length
em
U(l)
p Element Length
em
U(l6)
Resistance Ratio-RL/Rt
{Effective Mass of Charge
Carriers in p,n Elements
Rest Mass of Electron
Kg
{Concentration of Charge Part./Cm3
Carriers in p,n Elements
n
p
General Power Symbol
w
71'
General Peltier Coeff.
v
n Element Peltier Coeff.
v
p Element Peltier Coeff.
v
1T
1T
n
p
X
NOMENCLATURE (CONT.)
Variable
w
pn
Computer Name
Definition
Units
pn Peltier Coeff.
v
Electron Charge
Coulomb
HJ
Rate of Joule Heat
w
HK
Rate of Kelvin Heat
w
Rate of n Kelvin Heat
w
Rate of p Kelvin Heat
w
HP
Rate of Peltier Heat
w
HR
Rate of Resistive Heat
HTN
Rate of n Thomson Heat
w
HTP
Rate of p Thomson Heat
w
HO
Rate of Sink Heat
w
Hl
Rate of Source Heat
U(9)
q
.
QTp
ocm
ncm
ncm
U(6)
General Resistivity
{n Element-Substrate
Contact Resistivity
{P Element-Substrate
Contact Resistivity
n Element Resistivity
u (7)
p Element Resistivity
ncm
PCSN
PCSP
PS
r
RC
RCSN
RCSP
RL
Substrate Resistivity
{Carrier Scattering
Constant
Tungsten Contact Resis.
{n Element-Substrate
Contact Resistance
{p Element-Substrate
Contact Resistance
Load Resistance
ncm
ncm
xi
NOMENCLATURE (CONT.)
Variable
R
R
R
n
p
r
Computer Name
Definition
Units
RN
n Element Resistance
f2
RP
p Element Resistance
~
RR
Top Resistance
f2
RS
Substrate Resistance
f2
RSB
Bottom Substrate Resis.
f2
RST
Top Substrate Resistance
f2
RT
Total Internal Resis..
f2
General Temperature Symbol °K
TA
TA
Top Temperature Difference °K
TB
TB
Bottom Temperature Diff.
°K
To
TO
Sink Temperature
°K
Tl
Tl
Source Temperature
°K
T2
T2
Top Substrate Temperature
°K
T3,4
T4
Top Element Temperature
°K
T5,6
TS
Bottom Element Temperature °K
T7
T7
Bottom Substrate Temp.
U(I)
U (I)
Initial Data Vector
v
V (I)
General Voltage Symbol
v
V (I)
Second Data Vector
Vl
Load Voltage
v
WL
Output Power
w
General Length Expression
em
X
y (I)
°K
y (I)
Third Data Vector
1
I.
THEru·10ELECTRIC DEVICES AND THE
CONSIDERATION OF SILICON CARBIDE
A.
INTRODUCTION
The author's interest in the subject of direct energy
conversion, and especially in the study of thermoelectric
devices, was stimulated largely by a graduate course enti"Electrical Generation and Propulsion in Space",
tled,
taught at the University of Missouri-Rolla by Dr. James E.
Adair.
There it was revealed that thermoelectric generat-
ing apparatus are relatively in the infant stages of design
and construction, and suffer chiefly from an inability to
produce much voltage or power, as compared to other, more
colT'ITlon, forms of energy conversion devices.
This inability
stems from several main problems, the greatest being the
lack of semiconductor materials capable of operating at
high temperature with high output voltage {in millivolts).
In view of this, it seemed that the main difficulty
would be one of finding a material capable of generating
relatively high thermoelectric voltages at high temperature
levels.
Once a substance with this potential was found,
one could then derive a set of equations to design a generator using the material and develop a philosophy which
could be used to optimize such a design.
Finally, one
could write a digital computer program to help carry out
the optimization, since the calculations would undoubtedly
be too complex to manipulate by hand.
2
In examining different thermoelectric materials, S.L.
Soo (Ref. 1) listed such diverse substances as lead telluride, manganese telluride, and cerium sulfide.
All of
these have relatively high figures of merit (a desirable
design criteria), but none are capable of operating at very
high temperatures.
Their high temperature junctions are
limited to approximately 450, 900, and 1300° C, respectively.
None of these are really high enough to classify
them as high temperature materials.
In turn, this limits
their output voltage, as it depends on the Seebeck Coefficient and the temperature difference, both of which may be
increased by elevating the high temperature junction temperature.
Following this line of thinking, it was seen
that silicon carbide, an intrinsic semiconductor, is also
an excellent impurity semiconductor with quite a high melting temperature and consequently high voltage capability.
R.L. Weber (Ref. 2), in a plot of voltage versus temperature, showed that silicon carbide shows promise as the
material to solve the main problem.
Once a suitable material had perhaps been found, Soo
(Ref. 1) and Kettani
proper formulas.
(Ref. 3) were used as sources of
Both derived simplified equations which
would enable one to "design" a thermoelectric generator.
However, these formulas do not include important facets of
the design, such as;
resistances,
(1) complete consideration of internal
(2) inclusion of the Thomson Effect in the
material, and (3) optimization of the design with respect
3
to efficiency and power, independently and simultaneously.
Thus, their formulas, although being useful starting points
in the actual design process, needed to be expanded upon in
order to obtain a truly accurate design.
Some of the more
important details to be considered in this design are:
1.
The selection of materials to be used in the construction of the device.
2.
The properties of the component materials, including the boundary conditions between them.
3.
The change of the material properties with temperature.
4.
The consideration of important internal resistances, both electrical and thermal.
S.
The Thomson Effect.
6.
The preferred physical shape of a generator element.
Soo (Ref. l) discussed the temperature variation of
thermal and electrical resistances of metals and semiconductors, as did Kettani (Ref. 3), Holman (Ref. 4), Brown
and Marco (Ref. 5), and Austin (Ref. 6).
these
~orks
The study of
indicated that a hard, strong, high-melting-
point metal should be used as the substrate material on
which to bond the silicon carbide.
R.N. Hall (Ref. 7), in-
vestigating "Electrical Contacts to Silicon Carbide," discovered that tungsten (which fits the above criteria for a
substrate metal), when heated properly with silicon carbide, will form a mechanically strong, high temperature,
4
ohmic bond to the semiconductor.
He also discussed the
electrical resistances of this bond and other contacts to
silicon carbide, as well as other, important aspects of the
material.
These were also studied by vanDall, Greebe,
Knippenberg, and Vink (Ref. 8), Rutz (Ref. 9), and Farrell
(Ref. 10) .
The properties of tungsten and other component materials used were those given in the Handbook of Chemistry and
Physics
(Ref. 11) and in Sisler, VanderWerf, and Davidson
(Ref. 12) •
Eugene Charles Fadler (Ref. 13) studied the effects of
bonding agents between metals upon the overall thermal resistance.
M. Neuberger (Ref. 14) of Hughes Aircraft Corp.
supplied much valuable information about silicon carbide,
and Milton A. Levine (Ref. 15) of Melcor provided insight
into the physical shapes being used presently in the thermoelectric industry.
In addition, Wert and Thomson {Ref.
16) were helpful in the understanding of some of the Hughes
Aircraft data.
Finally, Soo (Ref. 1) and Kettani
(Ref. 3) were also
the sources of the Thomson Effect information.
Soo (Ref. 1)
showed that, for semiconductors, the Thomson Coefficient is
independent of the material and the temperature.
5
B.
ACKNOWLEDGEMENT
Dr. James E. Adair is to be thanked for suggesting and
encouraging this project.
In addition, his assistance and
helpful advice was greatly appreciated.
The author is also indebted to his wife, Anne W.
Barrow, and to Miss Paula Graves for overall support, and
especially for their assistance and encouragement in the
development of the computer program without which this work
would have been impossible.
6
II.
A.
DEVELOPMENT OF THE THEORY
DERIVING THE EQUATIONS
The first criteria used in this work was the finding
of a suitable semiconductor material, as was stated in the
introduction.
With this accomplished, upon the selection
of silicon carbide, however, very little further consideration of materials or actual data was then required until
the equations, philosophy, and computer program were developed.
At this point, the most fundamental part of the pro-
ject was started; that of deriving the equations.
It was at this stage that the geometry became important, as the equations depend largely upon areas, lengths,
widths, and separations of the individual generator element.
Levine's (Ref. 15) information showed that the shape
most commonly used in the industry is rectangular and the
individual elements resemble rather flat, rectangular
prisms.
An obvious advantage of this shape is that it
would easily fit into schemes for cascading and paralleling
many small elements for the production of a large scale
generator.
1.
Thus, the basic shape chosen is shown in Figure
Part (a) of this figure shows a front view of the de-
sign, while part (b) shows an orthogonal view of the same
element.
A list of the symbols, and their definitions,
used not only in this figure, but throughout this study, is
given in the Nomenclature, page viii.
7
Source
Sink
7'
Unit
(Source and sink
not shown)
(b)
(a)
FIG. 1 -
PROPOSED DESIGN LAYOUT OF AN
ELF~ENT
It was iP1rnediately obvious that the simplest, most
easily constructed and applied design would result if the
following constraints were made to the dimensions:
p
= An =
l\.
2.1
p
= L n = L.
2.2
A
L
Thickness
=
Unity
2.3
These would produce a design which is not only uniform
in shape and construction, but whose top and
botto~
sur-
faces are parallel, one of the oriqinal conditions of geornetry.
The unity thickness makes for an ease in calcu-
lation, which will be obvious as the discussion progresses.
However, the first two modifications above could not
si~ply
be made for the sake of convenience without
8
considering their possible effects upon the design.
(Ref.
1)
Soo
found that the relationship between the dimensions
and properties of the p and n type elements for optimum design were,
AnLp/ApLn I Optiroum
=
(pnKp/ppKn)
1/2
2.4
This indicated that the values of p ,
and K
n
needed to be researched further,
p
before proceeding with the
It was seen that the electrical re-
equation development.
sistivity and thermal conductivity of doped silicon carbide
do not vary appreciably with the kind or amount of dopant
used
(Ref. 14), thus it followed that;
K
n
"'
K
P
(p K /P K )l/ 2 "'
n p
2.5
•
p n
(1/l)l/ 2
=
1.
2.6
Therefore, the final result was
A L /A L
n p p n Optimum
= 1'
2.7
or, still considering the optimum case,
=
An /A p
L n /L p
2.8
is satisfied if,
A
p
=A,L
n
p
2.9
=L.
n
Thus, the constraints were found to be valid for this case,
and the derivation continued.
Consulting Figure 1
(a), it was seen that,
in view of
the above constraints, three of the variables could be
eliminated.
Since the thickness was assumed to be unity,
the following relationships could be established;
Ap
=
An
=
B,
2.10
9
and
2.11
These variables, with others, were then used to find
the expressions for the internal resistances of the device:
=
L/B.
2.12
Rn = pnL/B.
2.13
Rp
p
p
Rcsp
= Pcspo/B.
2.14
Rcsn
= Pcsno/B.
2.15
~
Rst
Rs
~
2.16
ps(D+B+d)/d.
2.17
ps(1.5D+2.0B+3.0d)/d.
The above, in turn, were seen to compose the major
resistances,
=
2.18
[Rst+O.S(Rcsp+Rcsn)],
= [Rs +(Rp +Rn )+(Rcsp +R csn )+2.0Rc ] ,
2.19
and
2.20
With the resistive part of the electrical circuit
completed, consideration shifted to the five Thermoelectric
Effects, which were found to be described by (Ref.
Seebeck Coefficient
=
Peltier Coefficient =
II
1) ,
a.
= 1 im
1T
= Q/I,
2.22
Ll VI Ll T
= 0'
2.21
Thomson Coefficient
=
y
= I (dT/dX) ,
2.23
Fourier Coefficient
=
K
= -Q/A (dT/dX) ,
2.24
=
p
=
PA/I 2 x.
2.25
and
Joule Coefficient
In addition to these expressions, Kelvin's Relations were
found to relate these coefficients as below;
10
TI
=
aT,
2.26
y = T(da/dT),
a pn
=
2.27
a p -a n'
2.28
= TI -TI •
2.29
pn
p
n
However, before the general expressions 2.21 to 2.29
TI
could be employed, an analysis of the temperatures in the
device had to be made.
Referring back to Figure 1
(a}, it
was found that some temperature drops would be significant.
The drop of temperature T 1 to T 2 , due to the organic
(epoxy) bond between the heat source and the substrate,
would probably be great
(Ref. 13).
This bond was to be
made organic so as to be electrically non-conducting.
Temperature T 2 would drop to T 4 due to the finite conductivity of the substrate.
It was noted that there would
probably be no appreciable drop in temperature at the substrate-silicon carbide junctions (Ref.
4), thus T 4 could be
assumed to be not only the bottom temperature of the top
substrate, but also the top temperature of the silicon carbide.
This then lead to the most important temperature drop
of the desion.
Temperature T 4 would drop to T 5 due to the
Thermoelectric Effects.
This would produce the temperature
gradiant which would, in turn, produce the electrical
power, and all the associated losses.
Following the T 4 to T 5 drop, temperature T 5 would drop
to T 7 ,
in the same manner as did T 2 to T 4 •
Likewise, the
T 7 to TO drop would resemble the T 1 to T 2 drop.
11
These considerations thus produced the following
results when analyzed further:
Tl-T2
T2-T4
T5-T7
T7-TO
=
=
=
=
.
2.30
llQ 1 /Ke(2.0B+D).
•
2.31
dQ 1 /Ks(2.0B+D).
.
dQ 0 /Ks(2.0B).
.
llQ 0 /Ke(2.0B).
2.32
2.33
These were then combined to give,
2.34
and
2.35
Upon obtaining these formulas, however, it was seen
that they could not be solved until the design was completed.
They could, however, show which temperatures to
use in analyzing the Thermoelectric Effects more thoroughly
and later be used to complete the design after the voltage,
current, and power calculations were made.
Thus, going
back to equations 2.21 to 2.29, the following considerations could be made.
Once the Seebeck Coefficient was cal-
culated using the expression (Ref. 1),
a p,n = ±K/q{ (r+2)+ln[(2/n) (27TmA*KT/h 2 ) 3 / 2 ]},
2.36
for both the p and n elements, the composite a pn was found.
This value and the temperatures T 4 and T 5 could then be
used to find the load current
2.37
The Peltier and Thomson Coefficients could be found by
7Tpn
and,
= apnT4'
2.38
12
y
p
=
T(da /dT)IT
T, y = T{da /dT)IT
p
=
4
n
n
=
T .*
4
2.39
With the above terms evaluated, the rates of heat exchange at the source, due to the Thermoelectric Effects,
could be calculated.
In the assessment of the effect of
Thomson Heat, it was assumed that, since the p and n type
elements are approximately homogeneous, and since they are
short with respect to their width and areas, then the ternperature gradients in the elements would be approximately
constant.
Thus,
2.40
With this kept in mind, the heat exchange rate equations
become:
. = IL.
pn
. = yniL(T4-T5)/L.
0 Tn
. = ypiL(T -T )/L.
Qp
4
QTp
.
2.41
TI
QK
=
OJ
=
2.42
2.43
5
(Kp+Kn)B(T 4 -T 5 )/L.
2
(Rp+Rn)IL.
2.44
2.45
These could be used in the simple equation for total
heat entering and leaving the top
.
01
. . . . .
= Qp+QK+QTp-QTn-QJ.
(source)
junction,
2.46
However, it was discovered that any heat produced or absorbed, simultaneously with and superimposed upon the ternperature gradient causing the Fourier Heat conduction,
would only partially be conducted to the hot junction.
*
y is actually independent ofT (Ref. 1), and showed
this property when calculated.
The
13
rest would be conducted away from it (Ref. 1).
Thus, not-
ing that there are three heats involved, the Joule Heat,
the n Thomson Heat, and the p Thomson Heat, it was necessary to analyze the behavior of each.
In the case of the rate of production of the Joule
Heat, QJ' Soo (Ref. 1) indicated that the amount of heat
flowing to the hot (source) junction from the element could
be given by the expression,
.
0 J(hot junction)
=
.
ClQJ'
2.47
where
c1 =
= 0 ;6 .
J
K
[(1/2)-(1/a>l Ia
2.48
Similarly, it was logical that this type of relationship would hold for the rate of production/absorption of
the Thomson Heat.
However, since the Thomson Heat is pro-
duced in the n element and absorbed in the p element, it
became clear that the relationship had to be evaluated for
each element separately.
The two ratios, l/a 1 , and 1/6 1 ,
respectively, were obtained and utilized by finding first
the n Thomson and Fourier Heat rates, and then the p Thornson and Fourier Heat rates, as below:
=
[1/(l+K p /K n )1QK.
2.49
(l/(l+Kn/Kp)1QK.
2.50
= [ (1/ 2 ) -
( 1/ a 1 ) 1 I
= r c112 > -
<116 1 > 1 1 6
al
=
2.51
2.52
1
Then, with this consideration, a more correct equation
for the source junction rate of heat flow became;
14
2.53
which was almost complete.
However, one more detail had to
be considered in the above expression.
not only would R
p
and R
n
It was found that
dump heat near the source, but so
, and (l/2)R
. As a matter of fact,
csp
csn
these resistances could be assumed to drop virtually all of
would Rst'
(l/2)R
their heat at the source, due to their extremely close
proximity to it.
Thus, the rate of heat production equa-
tion needed for this phenomena was found to be
QR = [Rs t+(l/2)R csp +(l/2)R csn ]IL2 .
2.54
Adding this to expression 2.53, the final equation for
source junction rate of heat flow became,
2.55
With this final heat rate equation determined, the expressions for load voltage, output power, rate of flow of
sink heat, source and sink temperatures, and efficiency
could be obtained, as shown below, respectively:
VL
=
wL =
Oo =
apn(T4-T5)-RtiL.
2
ILRL.
2.56
Ql-WL.
2.58
2.57
2.59
TO
=
2.60
TS-TB.
2.61
n = WL/Ql.
With the efficiency determined from expression 2.61,
it was seen that it could be checked by manipulating the
expressions for Carnot (temperature) efficiency.
The Car-
not efficiency for the entire device was found to be,
15
ncarnot =
(Tl-TO)/Tl'
2.62
and the Carnot efficiency from junction to junction (called
nsub Carnot> was evaluated as,
nsub Carnot
=
2.63
(T4-T5)/T4.
These two were found to be related by,
2.64
where
2.65
Furthermore, Soo (Ref. 1) found that the power efficiency,
n,
and the nsub Carnot were related by the expression;
DENOU =
<nsub Carnotm)/n,
where m is the resistance ratio, RL/Rt.
2.66
Therefore, using
expressions 2.62 through 2.66, a second term for efficiency
could be calculated;
n 2 = [(nC arno t-n 3 )m]/DENOU.
2.67
This would not only check efficiencies, but would also
check the temperatures, TA and TB' which are both important
design parameters.
16
B.
PHILOSOPHY OF THE DESIGN
With the derivation of the equations completed, the
philosophy of the design had to be constructed.
The list
of "independent" variables was assembled and, upon examining typical values, it was decided that seventeen could
truly be varied.
The other six should be held at the
values loosely estimated from the references (Ref. 11) (Ref.
14).
These six variables, DJUN, ps' pcsp' p csn , R,
c and Ks ,
were found to be relatively unimportant in the overall design and were, for the most part, estimated or extrapolated.
These methods of determination left little room for variation, since the limits could not be found.
Thus, seventeen
parameters were to be varied to achieve optimization.
The method chosen to do this was to construct a seventeen element vector of these variables,
Z(I) II= l-+l 7 =
[z(l), z(2),
... z(l7)],
2.68
such that;
Z (I) = [L,
B, D, d, 6. , p n, p p, apn,
TT pn,
Yp' Yn' Kp' Kn' T4' TS' m, Ke].
2.69
Each variable, L, B, D, etc., would be assigned three
possible values; its minimum value, its maximum value, and
the average of the two.
Following this procedure, the min-
imum values would be grouped together, as would the average
values and the maximum values.
Upon substitution of these
three sets of values into the variable vector, i(I), three
vectors would be produced; the minimum data vector, U(I),
17
the average data vector, V(I), and the maximum data vector,
y (I)
•
These would appear as below:
U (I)
= [Lmin.'
=
B
min.
••• K
e
min.
[u ( 1) , u (2), .•. u(l7)].
V (I) = [Lave.'
B
ave.
= [v (1) , v (2),
y (I) = [Lmax.'
=
I
B
max.
,
••• K
e
2.70
]
ave.
••. v(l7)].
I
••• K
e
2.71
]
max.
[y ( 1) , y (2), ... y(l7)].
2.72
It should be noted that the values to be used in each of
these three vectors, at this stage of the design, would not
be accurate values, but would only be typical.
As a matter
of fact, some of the variables were found to be slightly
dependent upon others, but this was ignored for this part
of the operation.
All that was desired here was to find
the absolute variation of power and efficiency with the
change of each variable, one at a time, and to check the
equations for correctness.
The method to be used to achieve this one by one variable testing was quite simple in conception.
The minimum
data vector was to be used first to calculate one design,
which would be known as the "original" design, and which
would be checked completely by hand, in order to verify the
equations.
In other words,
Design 1 (original)
= f[U(I)].
2.73
Following this design, the first element of the second data
vector, V(I), would be substituted for the first element of
18
U(I) and a second design would be run such that,
Design 2 = f{ [v(l), u(2), u(3), ••• u(l7)]}.
2.74
The third design would then be made by substituting the
first element of the third data vector, Y(I), for the first
element of the data vector obtained for the second design;
Design 3 = f { [y ( 1) , u ( 2) , u ( 3) , ••• u ( 17) ]} •
2. 7 5
These three designs would then each produce output
variable values.
Among these would be voltage, current,
power, and efficiency.
These four values could be plotted
as a function of the first data variable only.
As was
stated earlier, this type of functional determination was
desired for all seventeen variables, hence the above process was required to be repeated sixteen more times.*
How-
ever, before the fourth design could be started, the first
data variable, y(l), had to be replaced by the original
value of u(l).
The data vector would then be restored to
its original form, U(I).
The use of this vector again
would produce the first design, Design 1, as would be expected.
11
Hence, instead of performing the design again, the
0riginal 11 design would become the starting point of the
second variable's set of designs.
With this remembered, it
would then be a simple matter to substitute v(2) into U(I)
to produce a fourth design;
*
Actually, due to the later discovered invariability of
several variables, and to the further interdependence of
others, less than sixteen duplications were required.
19
Design 4 = f{[u(l), v(2), u(3), •.• u(l7)]}.
2.76
Likewise, a fifth design would be similarly produced using
y(2) such that,
Design 5
=
f{ [u(l), y(2), u(3),
••• u(l7)]}.
2.77
These two designs, coupled with the "original" design
would then produce functional plots similar to those of the
first set of three designs.
This is the process which,
when repeated the sixteen times previously discussed, would
produce, theoretically, the seventeen curves each of veltage, current, power, and efficiency.
Figure 2, parts (a) and (b), illustrates, respectively,
examples of power and efficiency curves which could be expected from the procedure described above.
n
0
1
CN
0
(a)
1
2
CN
(b)
FIG. 2 - EXAMPLE POWER AND EFFICIENCY CURVES
Several details of these plots should be noted:
1.
The ordinate of each curve was made independent of
the units of any variables.
Instead, it was marked
off in units of variable change number.
Since the
variables were to be changed linearly, the ordinate
was made a linear scale.
Thus, it was seen that,
if the abscissa was also made linear, then any non-
20
linearity in the functional relationships would be
indicated on the curves by their shapes.
2.
The functions were expected to vary both linearly
and non-linearly with the variables, as was indicated by the equations.
3.
The corresponding power and efficiency curves were
assumed to exhibit very little dependency.
As a
matter of fact, each could drop with change, rise
with change, or remain the same.
there were nine different
Consequently,
combination~
of behavior
that the power and efficiency curves could be expected to follow, respectively;
a.
Rise and rise,
b.
Rise and remain the same,
c.
Rise and drop,
d.
Drop and rise,
e.
Drop and remain the same,
f.
Drop and drop,
g.
Remain the same and rise,
h.
Remain the same and drop,
i.
Both remain the same.
Although there were actually seventeen variables,
and nine possible combinations of graphical behavior, there was no need to demonstrate the expected
plots that numerously.
It was decided that five
sample curves on each graph would suffice to show
the type of results expected.
It should be
21
remembered that these are not the actual predicted
variation for any variables, but only representations of what results could be expected.
From the seventeen curves each of voltage, current,
power, and efficiency, described above, it would then be
possible to determine which combinations of variable change
might serve to increase either power, or efficiency, or
both.
The way to do this would be to find the percentage
change of each output variable over its original value,
that was caused by each change of every data variable.
Any
data variable whose increase failed, simultaneously, to
cause power and efficiency increases could then be ruled
out for any increase over its minimum value.
Then, various
increases of the remaining variables could be superimposed
in different combinations, each of which would produce a
separate design.
An estimate of the output values of these designs
could be made by adding the percentage changes of the output variables caused by the corresponding input vari9ble
changes.
For example, one variable change which caused,
say, an eight percent increase in power could be coupled
with the change of another variable which caused a twentytwo percent increase.
The total increase in power, then,
due to the simultaneous change of both variables, could be
roughly expected to be thirty percent, neglecting any
interdependence of the variables.
22
The above procedure could easily be performed using
the three existing variable data vectors.
Instead of con-
taining the minimum, average, and maximum variable values
as before, they would be reconstructed to contain values
which were expected to produce promising designs.
These
new data vectors would be considered the secondary data veetors, and would be used, each independently, to produce
further designs.
Thus, when the first set of secondary
data vectors, U(I)A, V(I)A, and Y(I)A, were used, first
U(I)A would be applied to the equations, then it would be
replaced by V(I)A, and then by Y(I)A.
Since more than three
of these designs would probably be required, the variable
data vectors would have to be reconstructed several times,
using first the "A" subscript, then the "B" subscript, then
"C", etc.
It is seen, then, that the designs produced would follow the pattern;
Design Al
=
f[U(I)A],
Design ""2 = f[V(I)A],
Design A3
=
f[Y(I)A],
Des ism Bl = f[U(I)B],
Design B2 = f[V(I)B],
Design B3
=
f [Y (I) B] ,
=
f [Y (I)N]
and, finally,
Design N3
I
2.78
where N is the letter of the last secondary iteration of the
equations.
23
This process would produce information on how all the
variables should be varied to achieve the optimum design.
Once it is determined, then all variables would be given
their final, actual values.
At this time, a check for in-
terdependence would be made, and an optimum design would be
run.
It was decided to not try optimizing with temperature,
but to calculate the optimum design's performance versus
temperature.
This would give a good indication as to a
silicon carbide generator's temperature potential.
24
C.
DEVELOPMENT OF THE COMPUTER PROGRAM
Once the preceeding philosophy was completed, the com-
puter program, which would help carry it out, was to be
written.
equations.
The basis of the program would, of course, be the
In referring back to the theory, it was found
that the proper order of the equations would be as indicated in the table below:
TABLE I Place in
Program
EQUATION ORDER IN THE COMPUTER
Equation
Number
Place in
Program
Equation
Number
Place in
Program
Equation
Number
1
2.17
14
2.41
27
2.61
2
2.16
15
2.43
28
2.58
3
*
16
2.42
29
2.63
4
2.12
17
2.45
30
2.34
5
2.13
18
2.54
31
2.35
6
2.14
19
2.52 t
32
2.59
7
2.15
20
2.52 t
33
2.60
8
2.19
21
2.51 t
34
2.62
9
2.20
22
2.51 t
35
10
2.18
23
2.48 t
36
2.66
11
2.37
24
2.48 t
37
2.65
12
2.56
25
2.55
38
2.67
13
2.44
26
2.57
*
Special equations to simplify the program.
t
These equations were divided in the program.
*
25
Due to the nature of the logic (philosophy) to be used
in the program, it was found to be advantageous to place
the above listed equations into a computer subprogram subroutine.
This subprogram, designated as "CALCS", contains
no logic in itself, but merely proceeds step-by-step from
initiation to completion.
Thus, any time the main program
logic would dictate that a design were to be run, this subprogram would be called, the input data transferred into it,
the calculations made, and its output data returned to the
main program, for display via the write statements.
The logic statements were constructed from the portion
of the philosophy which dealt with the actual procedures
used in making the various design runs.
The optimization
decisions were not made in or by the computer program, but
were made by the author/operator upon examination of the
computer outputs.
This was deemed the most desirable way
to proceed, as the optimization of two (four, considering
voltage and current) variables with seventeen constraints
by computer programming/numerical analysis techniques would
have been excessively complicated and time consuming.
In
addition, the method used would give the author more exposure to the effects of the many changes as the designs
progressed.
Consequently, the computer program's main functions
were as follows:
26
1.
It would receive the data via the input data veetors and print it out.
2.
It would determine, from additional data, in what
mode it should operate.
There were three possible
modes from which to choose:
a.
Mode 1
-
The running of one design using
only the U (I) vector.
b.
Mode 2
-
The running of up to thirty-
three designs using the vectors, U (I) .
V (I), and Y(I).
c.
Mode 3 - The running of three secondary
(or final)
*
designs using the vectors,
U(I)X, V(I)X, and Y(I)x, where "X" represents whichever subscript 'i.vould be dietated at this stage of the process.
3.
It would, finally, run the designs in the mode
chosen and output the results.
This, of course, is highly simplified.
A more sophis-
ticated analysis of how the program logic would achieve the
design philosophy is shown by the flow chart of Figure 3.
Appropriate
Start
Comments
Subroutine
"CALCS"
Dimension
Data
Vectors
FIG. 3 - FLOW CHART
*
This mode would also be used in the final stage of
design.
27
"CALCS"
I=I+l
Call
"CALCS"
JJ=l
Calculate
Ml
Calculate
J=l
FIG.
3 -
FLON CHART
(CONT.)
Ml
28
J=J+l
Yes
L=J/2
X(L)=
U (L)
U(L)=
V(L)
Call
"CALCS"
Call
"CALCS"
WRITE All
Output Data
Variable
Stop
Values
FIG.
3 -
FLO\"l CHART
(CONT.)
29
Regarding Figure 3, it should be noted that, as indicated in the Nomenclature, the variable names were changed
from the original, so as to be applicable to the computer
language used, Fortran IV.
The actual program, with the
input and output data of the final design, may be seen in
Appendix A.
This actually shows the variable and data
arrays which are listed on the flow chart.
30
D.
SUMMARY AND CONCLUSIONS
The whole purpose of this work was to evaluate silicon
carbide as a thermoelectric material, as was stated in the
introduction.
Thus, since the material was chosen virtu-
ally before any work was started, the first major step in
the project became the development of the theory.
It was seen that Soo's (Ref. 1) basic design equations
were a good starting point in the design, but had to be
made more sophisticated.
Once modified, they could be in-
corporated into a computer program, to be used in the operation of an optimization philosophy.
This philosophy was
planned in such a way as to not only be relatively simple
in concept, but to provide output at every step of the process.
31
III.
A.
E~1INATION
OF THE DATA
CONSIDERATION OF THE MATERIALS AND THEIR PROPERTIES
1.
INTRODUCTION OF THE MATERIALS
Following the development of the theory, the next step
was to obtain, examine, and tabulate the necessary data to
effect a complete design.
However, the compilation of data
required not only a search, but a detailed examination of
the materials to be used.
Silicon carbide, composing the thermoelectric couple
elements, would be doped with either aluminum, boron, beryllium, or several other commonly used valence three elements to make it p type.
For the n type element, the
dopant most commonly used is nitrogen.
These dopants, how-
ever, whether p or n constituents, were all found to cause
approximately the same properties as impurities in silicon
carbide.
Therefore, the data for the p and n elements was
assumed to be accurate for any dopant, at the same concentrations.
The tungsten substrate would then be ohmically bonded
to the silicon carbide, top and bottom, by applying heat at
1800°
c, in a vacuum, to the two materials in close contact
(Ref. 7).
With the electrical-thermocouple stage of the de-
vice thus constructed, the heat source and sink material
would be adhesively bonded to the tungsten, top and bottom,
respectively, with an epoxy resin.
32
The
materia~
be a copper
a~loy,
to be used as the source and sink would
probably a bronze, and would not, for
this work, have any specific shape, except for the flat
polished adhesive bond surface.
This is due to the fact
that the heating and cooling of a single element would depend upon the overall configuration of the generator, of
which the element would be a small part.
Upon examination of the construction and materials, it
was decided to concentrate first on the epoxy adhesive
which was proposed to be the substrate to source/sink bonding agent, since information on this type of substance
appeared to be extremely scarce.
2.
ANALYSIS OF THE EPOXY ADHESIVE
Charles E. Fadler (Ref. 13) found that, when two
pieces of the same material are bonded by an adhesive agent,
the thermal resistance across that bond varies with (1) the
surface finish of the material,
material,
(2) the hardness of the
(3) the type of adhesive used, and (4) the thick-
ness of the adhesive.
In examining his treatment of these
four factors, it was discovered that
data for
Fad~er
(Ref. 13) had
(1) the type of polished surface that was needed
in this work,
(2) metals of varying hardness,
(3) an epoxy
adhesive, and (4) varying thicknesses of this adhesive.
Thus, it was deduced that, with some assumptions and manipulations, his results could be applied to this work.
Before any assumptions could made, some considerations had
33
to be brought forth.
In this work, the epoxy would have to stand temperatures perhaps as high as 2500° K, whereas Fadler's
(Ref.
13) data was taken at the considerably lower temperature of
364.5° K.
a.
Thus, three questions were raised:
Could an epoxy be found which could withstand such
high temperatures?
b.
If so, would its properties be retained at these
elevated temperatures?
c.
Could Fadler's
(Ref. 13) epoxy be considered
typical and therefore supply the data for this
work?
The first question was rather easily answered, as
there were found to be several epoxy adhesives which not
only were electrical insulators, but which could withstand
temperatures above 2300° K.
Unfortunately, no data could
be found on their thermal resistivities.
Therefore, in
answer to the second and third questions,
it had to be as-
sumed that not only was Fadler's
(Ref. 13) epoxy typical,
but that its properties were invariant with temperature.
In considering Fadler's (Ref. 13) work, it was seen
that he used copper, aluminum, and steel as adherends, and
concluded that, all other factors being equal,* the bond
thermal resistance of metals varied as a function of their
*
The same type of surface, adhesive, and adhesive thick-
ness being used in all cases.
34
Brinell Hardness Numbers.
Indeed, for the three above men-
tioned metals, he plotted a curve of bond thermal resistance (to be referred to hereafter as BTR) versus Brinell
Hardness Number (to be referred to hereafter as BHN) for
polished surfaces, with an epoxy adherent of 0.004 inches
thickness.
4.
A modified copy of this curve appears in Figure
It was observed that the curve drawn by Fadler (Ref. 13)
did not exceed a BHN of 250, the value for the steel that
he used.
Tungsten was the element in consideration for the substrate material, and a copper alloy was to be used as the
source and sink material.
Assuming the alloy to have
approximately the same hardness as copper, its BTR could
then be assumed to be that of copper, for which a value was
given.
However, the tungsten's BTR would have to be extra-
polated from the curve of Figure 4.
ficulty,
This presented a dif-
as the BHN of tungsten was seen to be 350 (Ref. 11)
which meant that tungsten's BTR-BHN value would be somewhat
beyond the last data point.
Thus, graphical extrapolation
could be expected to be somewhat innacurate.
However, the
alternative, numerical extrapolation, also had some disadvantages.
Noticing that, coincidentally, the step size
from the steel point to that of the tungsten was equal to
each previous step along the BHN scale, it became obvious
that a forward divided difference table cc,uld easily be
constructed from the three data points.
This
coul<.~
t_>en be
used to write an equation to approximate the curve beyond
2.0
Probable Extrapolation Area
;:J
E-4
Probable Extrapolation Path
al
'-....
~
0
N
.j..J
1.5
~
~
:r:
M
+
0
M
~
1.0
Q)
u
c
r.l
.j..J
en
en
·ri
0.5
Fadler's (Ref. 13)
Q)
!)::;
Last Point
M
rU
~
H
(!)
..c:
E-4
0
0
50
100
150
200
250
300
350
Brinell Hardness Number
FIG. 4 - "VARIATION OF THERMAL RESISTANCE WITH BRINELL HARDNESS NUMBER OF ADHEREND"
w
LT1
36
the three points.
The disadvantage, though, was found to
be the fact that only three data points would produce a
quadratic equation, and could, in turn, be expected to give
a rather low value when extrapolated another whole step.
In view of this, it was decided to use graphical extrapolation in conjunction with the numerical method, and to compare the results.
In order to assure some sort of accuracy, a probable
area of extrapolation was found.
This was done by extend-
ing the curve smoothly along two lines, the lower a reversed replica of the existing curve, and the upper a continuation of the existing curve's maximum estimated possible curvature.
These two lines, shown in Figure 4, were
considered to enclose the Probable Extrapolation Area, an
area in which the actual curve would probably lie.
Then, a
third line was drawn approximately equidistant between the
two previous lines, thus effectively bisecting the Probable
Extrapolation Area.
This line was considered the Probable
Extrapolation Path, and was assumed to be the correct extension of the curve, on which tungsten's BTR-BHN point
would be found.
The value thus found was 1.55 X 10- 3 in
the units of HrFt 2 °F/BTU, as compared to 0.45 X 10- 3 for
copper.
The above result was then checked by making a di-
vided difference table from the threE:' :1ata
This table,
po:i.nt~~;.
shown as Table II, rro·<uced an eqLwtion to
exactly fit the three points, and to approximately fit any
other point being considered.
37
TABLE II
BHN
=
-
DIVIDED DIFFERENCE TABLE OF THE
THREE DATA POINTS OF FIGURE 4
BTR = f(x)
X
0.45
50
>
150
0.54
250
0.83
~
P(x)
9.0 X 10- 4
> 29.0 X 10- 4
>
0.1 X 10- 4
This table produced the equation,
P(x)
=
(0.1 X l0- 4 )x 2 -(ll.O X 10- 4 )x+0.48,
3.1
which, when checked, proved to fit the three known data
points exactly.
When solved for the BTR of tungsten at a
BHN of 350, this equation performed as expected, producing
P(x) lx
=
350
=
1.325
(HrFt 2 °F/BTU),
3.2
which is only 85.5 percent of the value determined previously.
As a matter of fact, referring back to Figure 4,
it was seen that this value actually intersects the lower
line bounding the Probable Extrapolation Area.
Thus, it
was concluded that the actual value of BTR for tungsten
probably lies between the two points calculated, somewhere
in the lower half of the Probable Extrapolation Area.
Since this was the case, it was decided to use the larger
value, as it probably represented an upper limit and was
pessimistic.
Now that the values of BTR were found for two pieces
of tungsten bonded together, and for two pieces of copper
alloy bonded together, both with the same thicknesses of
the same epoxy, the BTR remained to be found for a piece of
38
tungsten identically bonded to a piece of copper alloy.
The first step in this procedure was to sketch the cross
sections of each case and to analyze their BTR's in
of component series thermal resistances.
ter~s
This is illus-
trated in Pigure 5.
"'- -'\vv~"V'
Copper
Alloy
BTP(a) = 0.45
Epoxy
0.004"
-~-
BTR(a)/2
Copper
Alloy
lA
--
-- -
Epoxy
~--A·
~._____
=
~
0.225~
____.,
(a)
o_;~:. .4--l--:-:-:-o_x_:_t_e_.n~~-BTl'(~
_o___
..
Epoxy
_: :_: s..:_ -
BTR(b)/2
=
-
-
L
0.778~
Tungsten
(b)
PH:;.
5
-
CP.OSS SECTIONS OP
(b)
(a)
COPPER ALLOY AND
TUNGSTEN IDENTICALLY BONDED T-TI'rH EPOXY,
SHOHING THE Vl\LUES OF BTR FOR EP.CH
39
It was noted that the BTR's were assumed to originate
and terminate just inside the metal, hence two basic regions were to be considered;
(l) the boundary regions be-
tween the epoxy and the metal, and (2) the region in the
epoxy itself.
Thus, it became obvious that, for any metal
being bonded to itself, the thermal resistances at both,
nearly identical, epoxy-metal boundaries would be approximately equal, differing only slightly due to the difference in temperatures at those boundaries.
was neglected for simplicity.
This difference
Furthermore, if the epoxy
was considered homogeneous, a valid assumption considering
its nature and thinness, then its intrinsic thermal resistance could be considered nearly uniform throughout.
Therefore, if the epoxy were divided into two 0.002 inch
thicknesses, each half would have approximately one half of
the whole's series thermal resistance.
Using the above deductions, it followed that, referring
to Figure 5, in general,
BTR
=
Rtop boundary+Repoxy+Rbottom boundary·
3. 3
But,
Rtop boundary
=
Rbottom boundary
3. 4
and,
Repoxy
=
2 (l/ 2 )Repoxy
3.5
thus,
BTR
=
2 (Rtop boundary)+ 2 (l/ 2 )Repoxy
=
2 [Rtop boundary+(l/ 2 )Repoxy 1 •
3.6
40
Therefore, it was concluded that,
BTR/2 =
[R top boundary +{l/2)R epoxy ]
3.7
= [Rbottom boundary+{l/ 2 )Repoxy]
This was all that was needed to find the half-cross
sectional value of BTR for each metal-epoxy case.
BTRcopper alloy/2
BTRt unqs t en /2
=
=
BTR{a)/2
BTR{b)/2
=
=
0.45/2
1.55/2
=
=
0.225.
0.778.
3.8
3.9
Referring to Figure 5 aqain, it was seen that, as the top
half of
(a) or
(a) or {b) was assumed to join the bottom half of
(b), respectively,
could join the top half of
to form BTR(ab).
BTR(ab)
=
to form BTR(a) or BTR(b), one
(a) with the bottom half of
(b)
Thus,
BTR(a)/2+BTR(b)/2,
3.10
which, with the proper values inserted, becomes,
3.11
BTR copper alloy-epoxy-tungsten = 1.003
in the units of HrFt 2 °F/BTU at a bond thickness of 0.004
inches.
This is demonstrated in Figure 6.
Copper
Alloy
J,
0.002"
Copper
Alloy
Epoxy
~
--- BTR(b)/2
0.002"
t
BTR(a)/2 =
0.225
Tungsten
'l'ungsten
0.778
BTR{ab) =
FIG. 6 -
THE JOINING OF COPPER ALLOY TO TUNGSTEN
\"JITH EPOXY, AND THE RESULTING BTR' S
41
This, however, still did not conclude the subject, as
the values for different thicknesses of epoxy were needed,
and the units had to be converted to those of the CGS systern.
First, making the expansion of the data with thick-
ness, i t was found that Fadler (Ref. 13) had plotted a
curve of BTR versus epoxy thickness, for an aluminum alloy
adherend.
This was modified to produce Figure 7.
It was
discovered that the BTR value of 0.88 at an epoxy thickness
of 0.004 inches did not agree with his previous value of
0.54, obtained from Figure 4.
This was supposedly also for
epoxy-bonded aluminum at the same thickness.
~
E-t
lXI
--- - - -
.........
Tungsten
/
/
/
~
0
N
~
1. 5
1-1
~
/
./
M
+0
....-!
Copper 1\lloyTungsten Bond
Orig. Aluminurr1
l!.lloy
Steel
1. 0
X
---- -- - --- -- ----------
<!)
{.)
s::
/
rei
+.J
/
Aluminur1
Copper
/
Cll
-~
Therefore, it
_..
0. 5
<!)
p:::
....-!
rei
E
1-1
<!)
..c:
E-t
0
0
0.002
0.004
Bond Thickness
FIG. 7 -
0.006
0.008
(Inches)
"RESULTS OF BOND THICKNESS TEST FOR
EPOXY BONDED SPECIMENS" EXPANDED UPON
42
was assumed that some other factor must have entered into
this later value and hence the curve.
This, however, did
not hinder the procedure which was used to effect the
thickness variation estimation.
It was clear that, whatever the other circumstances,
the original curve of Figure 7 still represented the BTR
variation with epoxy bond thickness only, of one test run.
It became logical, then, that whatever changes were made to
that test (changes in perhaps temperature, hardness, surface finish, etc.) would have no effect on the curve, other
than to shift it along the BTR scale, as these changes
would be independent of bond thickness.
Thus, if the value
for each material-epoxy bond previously discussed were entered on this curve, then the result would be a family of
data points, each at 0.004 inches, but each representing a
different hardness or combination of hardnesses.
If these
points were then expanded into curves similar in shape and
orientation to the original curve, they would represent a
family of curves of BTR versus epoxy bond thickness, which
varied from each other along the BTR axis as their construction points did.
These curves were added to Figure 7.
This being completed, the desired values were extracted from the copper alloy-epoxy-tungsten curve at the thicknesses of interest.
These values were then converted to
the CGS system of units, used in this work.
values appear in Table III.
Both sets
of
43
TABLE III - FINAL VALUES OF BOND THERMAL RESISTANCE
AND CONDUCTANCE FOR THE COPPER ALLOYEPOXY-TUNGSTEN BOND AS A FUNCTION OF
EPOXY THICKNESS
Bond Thickness
Inches
em
HrFt 2 °F/BTU
Seccm 2 °K/Cal
0.00100 0.00254
0.884
21.2
0.04720*
0.00300 0.00762
0.967
23.2
0.04310
0.00400 0.01016
1.003
24.1
0.04150
0.00425 0.01080
1.034
24.9
0.04015*
0.00600 0.01525
1.253
30.2
0.03310*
BTR
1/BTR
Cal/Seccm 2 °K
This table, therefore, established the preliminary
values for two of the seventeen previously discussed threevalue variables.
3.
ANALYSIS OF THE SILICON CARBIDE AND TUNGSTEN
Of the remaining fifteen three-value variables, eight
were properties of doped silicon carbide, and, in addition,
of the six single-value variables, previously discussed,
two were properties of tungsten.
These were then examined
in that order.
The properties of silicon carbide of interest could be
put into two categories;
a.
Properties approximately invariant with temperature,
b.
Properties whose temperature variation could not
be neglected.
*
Values actually used in the preliminary designs.
44
The first group included pp'
K
p
1
and
and the
second consisted of a pn , rr pn , yp' and Yn·
It was discovered that, as silicon carbide's melting point is near
3000° K, and since the temperature range of interest here
was below 2500° K, then the electrical resistivities and
thermal conductivities of the doped silicon carbide could
be assumed to be approximately in the flat or invariant
portion of their temperature curves.
In fact, data taken
at a much lower temperature could be used without serious
Thus, these values for both the p type and n type
error.
silicon carbide materials were extracted from the Hughes
Aircraft information (Ref. 14).
(SGCm)
3.12
0.01 > p
> 0.0001
p,n
3.13
0.335 > K
> 0.213
p,n
The second group of silicon carbide properties was
much more complicated to analyze.
The first of these, apn'
had to be calculated from the formulas of equations 2.36
and 2.28, given again as;
a
= ±K/q{ (r+2)+ln[ (2/n) (2rrmA*KT/h 2 ) 3/2 ] },
p,n
3.14
and,
=
a -a •
P
n
In equation 3.14, the symbols K, q, and h, represent
universally recognized constants.
However, r, n, and rnA_
were seen to be variables whose limits were,
r
=
1025
and,
2
(Dimensionless) (Ref. 1), 3.16
(Carriers/Cm 2 ) (Ref. 14), 3.17
45
(Dimensionless) (Ref. 14).t
3.18
Thus, the only variable of equation 3.14 remaining to be
determined was T, the temperature at which a.
was required.
pn
Since the design temperatures had not yet been firmly established, and a.
ture gradient,
actually had to be found over a temperapn
from T 4 to T 5 , it was decided to pick a max-
imum and minimum value for the average gradient temperature
and use this estimate for the preliminary work:
1623 > T > 1423
3.19
With this temperature range thus established, the minimum and maximum values of a.pn were calculated;
3.20
0.001368 > a.
> 0.000950
(V/°K)
pn
Next came the calculation of n pn , which proved to be
simply, from equation 2.38, a.
times its corresponding
pn
temperature, hence;
(V)
3.21
> 1.359
pnFollowing this yp and yn had to be expanded from equa-
1.790 > n
tion 2.39 to become,
Yp,n
=
2 3/2
Td/dT{±K/q[(r+2)+ln(2/n) (2nm_AKT/h )
] }. 3.22
'rhis produced,
y
P
=
y
n
=
3K/2q
=
Constant.
3. 2 3
Thus, the value for y p,n was calculated to be
yp
t
=
yn
=
0.00001292
3.24
These values for mA*/m
were actually only for the n
.
0
type silicon carbide, but were used for both the n and p
types at this stage.
Separate p values were found later.
46
and concluded the preliminary consideration of silicon carbide.
The two properties of tungsten were left to be found
in this part of the data accumulation.
The "Handbook of
Chemistry and Physics" (Ref. 11) produced,
p
s
= 0.0000325
(S1Cm)
3.25
and
=
0.413
3.26
These values, then, marked the end of the preliminary
consideration of the materials.
47
B.
ESTIMATION OF THE TEMPERATURES EXPECTED
The temperatures of the silicon carbide-tungsten junc-
tions,
although being the design parameters, were found not
to be the limiting temperatures.
Instead,
it was estiwated
that the series combination of the copper alloy-epoxy-tungsten BTR and the tungsten thermal resistance would produce
a temperature drop of up to 500° K from the source to the
top junction, and probably a similar drop from the
junction to the sink.
This could cause two serious prob-
lems, if the junction temperatures were ill chosen.
if the top junction temperature were set too high,
2000° K,
botto~
First,
say
then the temperature necessary at the source, to
produce it,
could soar to 2500° K,
bond to the source.
and could ruin the epoxy
On the other hand,
tion temperature were chosen too low,
if the bottoJTJ june-
say 1000° K,
then
the temperature at the sink could drop to perhaps 500° K,
which could be an inadequate temperature at which to dissipate the outgoing heat.
Therefore,
in view of the above,
the preliminary june-
tion temperature limits were chosen to be,
l 7 7 3 > Tt
-
.
.
> 15 7 3
op ]Unct1on -
3.27
and
3.28
.
.
> 1273.
147 3 > T
bottom JUnctlon Analyzing these further, it was seen that they would
produce a temperature difference range of
500 > 6T > 100
when used in the previously discussed philosophy.
3.29
They
48
could also lead to source and sink temperatures of perhaps,
2273 > T
>
source-
2073
{o
K)
3.30
and
( o K) •
973 > T . k > 773
3.31
- s~n These limits seemed to be reasonable and were judged to
still provide adequate margin for variation in the output.
49
C.
COHPILATION OF THE RElvlAINING DATA
The conclusion of the previous data consideration left
ten total variables to be evaluated.
The preliminary values for the six dimension variables
were rather arbitrarily chosen.
Since the dimensional unit
was chosen to be the centimeter and the thickness of the
design was to be of one unit, then the designs would all be
one centimeter thick.
Keeping in mind the previous discus-
sion of the shapes being presently used in industry, the
various dimensions were given values ranging from large
fractions to small multiples of one centimeter.
Following these determinations,
the quantities, m,
Pcsn'
Pcsp , and Rc became the remaining data variables to
be evaluated.
The resistance and resistivities were estimates resulting from a combination of the author's previous
experience and from discussions with associates.
The value
for m, the ratio of load resistance to internal resistance,
was merely started at one, the general maximum power transfer ratio, and increased slightly as it was suspected that
its optimum value would be near one.
50
D.
SUMMARY AND CONCLUSIONS
Upon examination of the materials, it was seen that
much calculation was necessary to produce viable data.
The
thermal properties of the epoxy, tungsten, and silicon carbide were evaluated.
In addition, the electrical proper-
ties of tungsten and silicon carbide were found, as were
the thermoelectric coefficients of silicon carbide.
After
logical estimates were made to determine the proper temperature limits, the dimensions, resistance ratio, and miscellaneous resistivities and resistance were found.
In most cases, the values found were only the limits
of the variable ranges.
Table IV was then constructed from
these limits, using the philosophy, to contain all of the
input data values used in the preliminary design.
TABLE IV - FINAL VALUES OF THE INPUT VARIABLES
FOR THE PRELIMINARY DESIGNS
Orig.
Name
Computer
Name
Initial
Value
Change 1
Value
Change 2
Value
Units
L
u (1)
1.5
2.0
2.5
em
B
u (2)
0.5
1.0
1.5
em
D
U(3)
0.2
0.4
0.6
em
d
u ( 4)
0.2
0.4
0.6
em
b.
u (5}
0.000254
0.01080
0.01524
em
Pn
u (6}
0.0001
0.001
0.01
pp
u ( 7)
0.0001
0.001
0.01
a.
u ( 8)
0.000950
0.001159
0.001368
pn
51
TABLE IV - FINAL VALUES OF THE INPUT VARIABLES
FOR THE PRELIMINARY DESIGNS (CONT. )
Orig.
Name
Computer * Initial
Name
Value
Change 1
Value
Change 2
Value
Units
1T
u (9)
1.359
1.790
2.220
v
yp
u (10)
0.00001292
0.00001292
0.00001292 VCm/°K
Yn
U(ll)
0.00001292
0.00001292
0.00001292 VCm/°K
K
p
U(l2)
0.213
0.255
0.335
W/Cm°K
n
u (13)
0.213
0.255
0.335
W/Cm°K
T4
u (14)
1573.0
1673.0
1773.0
OK
Ts
u (15)
1273.0
1373.0
1473.0
OK
m
u (16)
1.0
1.2
1.4
K
U(l7)
0.03310
0.04015
0
DJUN
0.005
0.04720 Cal/Seccm 2
OK
em
Ps
PS
0.0000325
ncm
Pcsn
PCSN
0.075
ncm
Pcsp
PCSP
0.075
nero
R
c
RC
0.000025
K
SK
0.413
K
*
pn
e
s
n
Cal/Seccm 2
OK
The vector U(I) II= 1417 was used in the computer as
the general input data vector.
The first column of data,
labeled "Initial Value," contains the actual vector 5(!)
that was discussed in the philosophy.
The column labeled
"Change 1 Value" is actually the V(I) data vector, and the
"Change 2 Value" column is the Y(I) data vector.
52
IV.
A.
ANALYSIS OF THE PRELIMINARY DESIGNS
REITERATION OF THEIR FUNCTION
As was stated before, in the philosophy, the purpose
of the preliminary designs was to establish the behavior of
the design equations with independent changes of each variable.
The first preliminary design was also to be hand
checked in order to verify the correctness of the equations
in the program subroutine, "CALCS".
53
B.
EXAMINATION OF THE PRELIMINARY OUTPUT DATA
Upon running the first design, it was seen that it did
give the same results as the hand calculations, within expected error.
Thus, the program formulas were shown to be
correct.
Following this check, the preliminary designs were
run; one original, and two designs for each variable as it
was changed.
If all the input data variables were varied
twice, as was originally proposed, the resulting number of
designs would be,
Number of Designs= ND = 1+2(17) = 35.
However,
4.1
as was seen previously, the variables yp and yn
were held to the same values throughout the preliminary
run.
Hence,
the use of their second and third values would
have merely reproduced the original design.
Therefore, de-
signs number 20 through 23 were not actually run, but were
numbered and set equal to the original design.
Similarly,
the thirty fourth and thirty fifth designs were not run;
but they were not numbered either.
iable they depended on,
ing
~
K
e
,
This is because the var-
was called when its correspond-
was encountered, in designs number 10 and 11.
i t was not needed later.
Therefore,
Thus
in the preliminary de-
sign run, the actual number of designs became,
ND'
=
35-2 = 33
4.2
and the number of independent designs that could be expected was seen to be,
54
ND"
=
33-4
=
29.
4.3
The results of these designs are tabulated below in
Table V.
It should be noted that, in addition to the val-
ues of load voltage, VL, load current, IL' output power,
WL, and efficiency,
n,
the computer output also contained
some twenty nine other variable values.
The first four
quantities, those included in Table V, are the only output
variables of interest in the optimization procedure.
TABLE V - OUTPUT DATA OF THE PRELIMINARY DESIGNS
Design Variable Change Variable Load
Load
Power Eff.
Number Changed Number Value t Voltage Current Out
(%)
(Volts) (Amps)
(Watts)
0.1425
58.76
8.37
4.96
2.0
0.1425
54.29
7.74
5.48
2
2.5
0.1425
50.44
7.19
5.85
u (2)
1
1.0
0.1425
93.44
13.32
4.39
5
u (2)
2
1.5
0.1425 107.55
15.33
3.75
6
u (3)
1
0.4
0.1425
57.69
8.22
4.91
7
u (3)
2
0.6
0.1425
56.66
8.07
4.87
8
u (4)
1
0.4
0.1425
61.22
8.72
5.06
9
u (4)
2
0.6
0.1425
62.09
8.85
5.09
10
U(5,17)
1
58.76
8.37
4.96
11
U(5,17)
2
58.76
8.37
4.96
12
u ( 6)
1
{0 · 01080 , 0.1425
0.04015
{0 · 01524 , 0.1425
0.04720
0.001
0.1425
27.81
3.96
3.18
13
u (6)
2
0.010
0.1425
4.44
0.63
0.69
14
u (7)
1
0.001
0.1425
27.81
3.96
3.18
15
u (7)
2
0.010
0.1425
4.44
0.63
0.69
1
None
0
2
U(l)
1
3
u (1)
4
55
TABLE V -
OUTPUT DATA OF THE PRELIMINARY DESIGNS
(CONT.)
Design Variable Change Variable Load
Number Changed
Number Value t
Load
Power Eff.
(%)
Voltage Current Out
(Volts) (Amps)
(Watts)
16
u ( 8)
1
0.001159 0.1740
71.75
12.49
5.94
17
u ( 8)
2
0.001368 0.2055
84.74
17.42
6.72
18
u ( 9)
1
1.790
0.1425
58.76
8.37
4.96
19
U(9)
2
2.220
0.1425
58.76
8.37
4.96
20
U(lO)
1
0.000013
*
*
*
*
21
U(10)
2
0.000013
*
*
*
*
22
U(11)
1
0.000013
*
*
*
*
23
u { 11)
2
0.000013
*
*
*
*
24
U(l2)
1
0.255
0.1425
58.76
8. 37
4.84
25
u ( 12)
2
0.335
0.1425
58.76
8.37
4.62
26
u ( 13)
1
0.255
0.1425
58.76
8.37
4.61
27
u ( 13)
2
0.335
0.1425
58.76
8.37
4.07
28
U(l4)
1
1673.0
0.1900
78.35
14.89
6.45
29
U(14)
2
1773.0
0.2375
97.94
23.26
7.87
30
ll(l5)
1
1373.0
0.0950
39.18
3.72
3.28
31
U(15)
2
1473.0
0.0475
19.59
0.93
l . 63
32
U(l6)
1
1.2
0.1555
53.42
8.31
5.14
33
u ( 16)
2
1.4
0.1663
48.97
8.14
5.23
*
1\ll o( these values are identical to the values of the
original design due to the invariability of U(lO)
t
and U(ll).
See Table IV for th0 units of each input data value.
56
Now that the output data was obtained for the preliminary designs, the first optimization step was taken.
From
the values of Table V, four curves were plotted, one for
each output variable versus the number of the variable
change.
The change number of zero indicates the "original"
design, which was the starting point of each curve.
Upon examining these curves, shown in Figures 8, 9,
10, and 11, it was discovered that eleven of the seventeen
variables either could or should be held at their minimum
values:
1.
Variables U(S), U(9), and U(l7) were found to be
unimportant in the designs, and could be held at
their minimum values.
This was rather ironic in
the case of U(S) and U(l7), since great effort was
made to calculate the epoxy BTR at the substratesource/sink bonds.
It was helpful, however, to
find that they could be neglected in further studies.
Considering U(9), it was remembered that,
although relatively unimportant, the Peltier Coefficient is a function of temperature and thus must
be re-evaluated later.
2.
Variables U(lO) and U(ll) were seen, as was stated
before, to be invariant and definitely could not
be increased.
3.
Variables U(3), U(6) and U(7), U(l2) and U(l3),
and U(lS) were all discovered to not be allowed to
increase without causing detrimental effects upon
57
0.25
Variable
14
Variable
8
Variable
16
0.20
O.l5
Variables 1,
[ 2, 3, 4, 5,
(I)
tTl
n:1
6,
+.J
7,
9,
12,
Variable
15
13.
r-1
0
:>
'd
n:1
0
...:!
0 .lO
0.05
0
0
l
2
Change Number
FIG.
8 - LOAD VOLTAGE VERSUS CHANGE
THE PRELIMINARY DESIGNS
NU~ffiER
FOR
58
Variable
2
Variable 14
Variable
80
8
Ul
n.
~
70
+l
60
~
s:::
Variable
4
(variables 5,
Q)
1-.t
1-.t
::J
u
~' 12, 13.
Variable
3
Variable 1
Variable 16
50
ro
ct:!
0
...:I
40
30
20
Variable 15
10
.variables 6,7.
0
0
1
2
Chanqe Number
FIG. 9 - LOAD CURRENT VEPSUS CHANGE NUMBER FOR
THE PRELIMINARY DESIGNS
59
25
Variable 14
20
[J)
+'
+'
cU
Variable
8
Variable
2
Variable
4
8:
lo-l
<lJ
15
~
0
A..
+'
:;:$
0..
.jJ
:;:$
10
0
(Var iab1es 5,
~9, 12, 13,
16.
Variable
Variable
5
Variable 15
Variables 6,7.
0
0
1
2
Change Number
FIG.
3
l
10 - OUTPUT POWER VERSUS aiANGE NUMBER FOR
THE PRELH1INARY DESIGNS
60
Variable 14
7
Variable
8
Variable
1
6
Variable 16
Variable
4
Variables 5,9.
Variable
3
Variable 12
5
dP
Variable 13
4
::>-!
{)
s::
Variable
(])
·.-i
2
{)
·.-i
4-.j
4-.j
li.l
3
2
Variable 15
1
Variables 6,7.
0
0
1
2
Change Number
FIG. 11 -
EFFICIENCY VERSUS CHANGE NUMBER FOR
THE PRELIMINARY DESIGNS
61
the design.
U(6) and U(7), then and p type re-
sistivities, respectively, and U(3), the distance
between the n and p type elements, were found to
decrease both WL and
n,
when increased.
U(l2) and
U(l3), then and p type thermal conductivities, respectively, were both found to, when increased,
cause
n
to decrease with no change in WL.
U(lS),
the bottom junction temperature, was found to
cause both WL and n to decrease almost as much as
they did with U(6) and U(7).
Thus, none of these
variables were allowed to be increased in future
designs as they were all seen to be detrimental.
Following this analysis of the negative changes, an
analysis of the other, possibly benificial, changes was
performed.
It was seen that the other six variables of the
original seventeen produced increases in either WL or
both, when increased.
n, or
However, some of the single in-
creases simultaneously accompanied decreases in the other
variable.
This meant that a more detailed analysis was
needed in order to determine which of the six should be increased to achieve optimization, and by how much.
62
C.
SU~1ARY
AND CONCLUSIONS
The preliminary designs, when plotted, showed very
well the independent functional relationships between the
four output variables and each input data variable.
Of the
seventeen original input data variables, eleven were found
to be either detrimental or non-beneficial when increased.
It was noted, however, that the Peltier Effect would still
have to be considered later when the final temperatures
were established.
An interesting conclusion was drawn.
The epoxy thick-
ness and its Bond Thermal Resistance were unimportant to
the design and could probably be neglected in future work.
63
V.
IMPLIMENTATION AND ANALYSIS OF
THE SECONDARY DESIGNS
A.
REITERATION OF THEIR FUNCTION
The preliminary designs indicated which variables were
to be further investigated.
It was now necessary to es-
tablish a method for estimating which quantities to vary
simultaneously, what their values should be, and what the
results of this variation would be.
With this accomplished,
the secondary designs could then be run and their analysis
would show which combinations of data variable changes would
further optimize the designs.
64
B.
CONSIDERATION OF PROPER DATA AND ESTIMATION OF RESULTS
In order to estimate what values to give the secondary
design vectors, and what results to expect from them, Table
VI was constructed from the data of Table
v.
This later
table shows the percentage changes from the "original" design values of VL,
IL' WL' and
n,
caused by the first and
second change of each of the six possibly beneficial variables.
TABLE VI - PERCENT VARIATIONS OF THE FOUR BASIC
OUTPUT QUANTITIES WITH THE CHANGE OF
SIX INPUT VARIABLES
Percent From "Original" Values
Variable
Name Symbol
n
u (1)
L
0.0
-7.3 -10.7 +10.0
0.0 -14.4
-14.0 +18.0
U(2)
B
0.0 +59.0 +58.2 -11.1
0.0 +83.0
+82.0 -24.4
U(4)
d
0.0
0.0
u ( 8)
a.
pn
U(l4)
T4
U(l6)
m
+0.6
+5.4
+2.0
+0.8
+2.8
+7.0
+21.7 +23.8 +49.0 +20.0 +43.0 +46.0 +108.0 +34.0
+32.8 +33.4 +77.0 +30.0 +66.0 +67.0 +176.0 +58. 0
+8.4
-9.0
-0.8
+3.6 +17.0 -18.0
+6.0
-2.8
It was immediately noticed that WL varied little with
the change in resistance ratio, m, as was expected.
It was
reasoned that, since the equations had previously been
proved correct, the WL versus m curve must peak somewhere
near an m value of one, and this peak must be rather flat.
65
In considering the percentage changes of Table VI, it
was assumed that appropriate combinations of input changes
would produce beneficial combinations of the output changes.
Thus, fifteen secondary runs were proposed to be made so as
to produce hopefully relevant data from combinations of the
input data changes.
These input change combinations were,
1.
Design Al - "Original Design" - No changes,
2.
Design A2 - All six variables change 1,
3.
Design A3 - All six variables change 2,
4.
Design Bl - Five variables change 1 with length
(L} its minimum value,
5.
Design B2 - Five change 1 with width (B) minimum,
6.
Design B3- Five change 1 with ratio (m) = 0.25,
7.
Design Cl
Five change 1 with ratio= 0.50,
8.
Design C2
Five change 1 with ratio= 2.00,
9.
Design C3
Five change 1 with ratio = 4.00,
10.
Design Dl - Five change 2 with length minimum,
11.
Design D2 - Five change 2 with width minimum,
12.
Design D3
Five change 2 with ratio
13.
Design El
Five change 2 with ratio = 0.50,
14.
Design E2 - Five change 2 with ratio= 2.00,
15.
Design E3 - Five change 2 with ratio= 4.00.
=
0.25,
These were combined, analyzed, and run as is shown in
Table VII.
The dashed places in the table indicate where,
due to limited preliminary data, there was not enough information to enable one to esti~ate an expected result with
reasonable accuracy.
TABLE VII Run
Design
Code r:UM.ber
EXPECTED fl-.ND ACTUAL RESCL'I'S OF THE SECONDARY DESIGNS
% Expected Change
vL
Over "original"
vL
IL
0
A
1
0
A
2
A
f,r
'L
1')
IL
(Volts) (Ar:lps)
rl1
(Watts)
Actual Values
ll
(%)
VL
IL
(Volts) (Amps)
WL
(Watts)
ll
(%)
0
0.1425
58.8
8.4
4.96
0.1425
58.8
8. 4
4.96
+63 +100 +178
+53
0.2320 117.0
23.3
7.60
0.2620 150.0
39.3
8.60
3
+126 +164 +356
+90
0.3230 155.0
38.2
9.40
0.4080 259.0 105.0 12.25
B
1
+63 +107 +189
+43
0.2320 121.0
24.2
7.10
0. 2620 161. 0
42.2
7.90
B
2
+63
+41 +120
+64
0.2320
83.0
18.4
8.10
0.2620
86.3
22.6
9.07
B
3
+31 +134 ----
+40
0.1870 137.0
----
6.90
0.0960 264.0
25.4
3.90
c
1
+47 +117 ----
+47
0.2100 127.0
----
7.30
0.1600 220.0
32.3
6.09
c
2
+97
+64 ----
+56
0.2810
97.0
----
7.70
0.3200 110.0
32.3
9.20
c
3
+184
-26 ----
+60
0.4050
44.0
----
7.90
0.3840
66.0
25.4
8.50
D
1
+126 +178 +370
+72
0.3220 163.0
39.4
8.50
0.4080 293.0 119.7 10.80
D
2
+126
+81 +274 +114
0.3220 106.0
31.3 10.60
0.4080 108.0
44.2 13.06
D
3
---- ---- ---- ---- ------ -----
---- -----
0.1400 496.0
64.5
5.12
E
1
0.2330 413.0
96.5
8.10
E
2
0.4670 207.0
96.5 12.93
E
3
0
Expected Values
---- ---- ---- ---- ------ ----- ---- -------- ---- ---- ---- ------ ----- ---- -------- ---- ---- ---- ------ ----- ---- -----
0.5600 124.0
69.5 12.51
0'1
0'1
67
C.
ANALYSIS OF THE SECONDARY DESIGNS
As can be seen from Table VII, the actual results ex-
ceeded the expected results in most cases.
This is mainly
due to the fact that the four output variables are all dependent.
all.
Any change in one actually causes some change in
In a few cases, this resulted in very significant and
n.
rather unexpected increases in WL and
The results of these runs, the "Actual Values" of
Table VII, were then plotted versus the change numbers in
Figures 12 through 15, and versus the ratio values in Figures 17 through 20.
300
(/)
250
0.5
[/)
+J
0..
,....j
0
:>
(l)
0"1
I'd
+J
-~
0.4
+J
s::(l)
0.3
1-1
1-1
::l
,....j
0
:>
"Original"
L-Orig.
B-Orig.
0.2
'"d
I'd
s
u
200
L-Orig.
150
100
'"d
I'd
0
H
0.1
50
0
0
1
0
2
Change Number
Change Number
FIG. 12
-
LOAD VOLTAGE
VERSUS CHANGE
NUMBER (S)
1
0
FIG. 13
-
LOAD CURRENT
VERSUS CHANGE
NUHBER (S}
2
68
150
12
Ul
+J
+J
rtl
cJP
8:
H
Q)
~
u
100
I:!
:3:
Q)
0
A.
·.-i
+J
::s
·r-i
4-1
4-1
+J
!:4
::s
8
u
0..
0
10
50
6
4
2
0
0
1
2
0
Change Number
1
2
Change Number
FIG. 14 - OUTPUT POWER
FIG. 15 - EFFICIENCY
VERSUS CHANGE
NUMBER (S)
VERSUS Cl:U~NGE
NUMBER (S)
Figures 12, 13, 14, and 15, displaying plots of
YvL, and
~
v~,
IL,
versus change number, do so first for all six
variables, then for five while keeping the length at its
minimum value, and finally for five while keeping the width
at its minimum value.
These curves show that leaving the
length minimum while increasing the other variables slightly increases WL and only slightly decreases
six-variable-change values.
n, over their
Ho\-Tever, doing the same with
the width causes a great decrease in WL, although causing
only a slight increase
timize both WL and
~
~n
n.
As the object here is to op-
together, a plot of WL times
n was
69
made from the data of Table VII and Figures 14 and 15.
This curve is shown in Figure 16 below.
150
dP
C/)
100
.jJ
.jJ
ro
-
~
6s::
QJ
·.-I
-~
500
~
4-1
ril
:X:
1-l
QJ
100
~
0
0..
0
1
2
Change Numl.>er
FIG. 16 - OUTPUT POWER TIMES EFFICIENCY VERSUS
CHANGE NUMBER (S)
The result of Figure 16 is that the WL X
n values ac-
tually become slightly higher if the length is kept minimum.
However, keeping the width minimum caused a marked decrease
in WL X
n,
result,
it was decided to leave the length at its original,
especially near the second change number.
As a
minimum value, and to use the change 2, maximum value of
width.
The length could have been decreased further, but
it was felt that a good balance of WL and n had been
reached at the value finally chosen.
70
Figures 17, 18, 19, and 20 were then considered, in
order to find the effects of ratio, as well as the other
variables, on the designs.
500
0.5
400
lfl
0...
0.4
3
300
0.3
+J
~
200
Q)
l-1
l-1
~
u
0.2
'tl
10
'd
Cti
0
...::!
100
0 • l -t-....__-+---t----+----1
0
1
2
4
3
s
0;-----r---~--~~--~
1
2
4
3
0
Resistance Ratio
Resistance Ratio
FIG. 18 - LOAD CURRENT
FIG. 17 - LOAD VOLTAGE
VERSUS IUI.TIO
(S)
VFRSUS RATIO
100
12
80
10
( S)
1
'Change 2
8
60
Peak l
20
+---t-J...-L---+--+--~
0
l
2
3
4
Resistance Ratio
FIG. 19 - OUTPUT POWER
VERSUS RATIO
11
I
6
1
40
(S)
I
44-L---4-----~-----~-~
0
4
l
2
3
Resistance Ratio
FIC. 20 - EFFICIENCY
VERSUS PATIO
As was decided earlier, the top junction temperature,
(S)
71
T 4 , was to be used as the final design variable, and was
thus eliminated at this point.
The only two remaining var-
iables to consider, besides m, were a
pn'
ficient, and d, the substrate thickness.
the Seebeck CoefAs neither of
these produced any ill effects on WL or n when increased,
it was decided to use the maximum values of both.
However,
it was remembered that a pn is dependent upon T 4 , thus still
remaining to be evaluated later.
The evaluation of the effects caused by changing the
resistance (power transfer) ratio, m, was next to be conIts variation, as can be seen from Figure 19 and
sidered.
20, caused the values of WL and
n
to peak, but it was noted
that they seemed to peak at different values of m.
the variable WL X
n
Hence,
was again constructed and plotted, in
Figure 21.
1200
eN>
Ul
+J
~
1000
=:::
Change 1
100~~~------~-----L~--~----------1---------~4
0
1
2
3
Resistance Ratio
FIG. 21- OUTPUT POWER TIMES EFFICIENCY VERSUS RATIO(S)
72
This plot showed that the optimum ratio, m
, shifted
opt
with the six-variable changes. The two values obtained
were;
mopt six-variable change 1 = 1 • 52 ,
5.1
and
mopt s1x-var1a
·
. bl e c h ange 2 = 1.60.
5.2
The shift was enough to indicate that the value for rnop t
should be recalculated for the final design.
In addition,
these results confirmed the earlier suspicion that rnop t
would be near a value of one, and that the peak in WL would
be rather flat.
73
D. SUMMARY AND CONCLUSIONS
Once the preliminary designs were made and some
thought was given to the possible data increase combinations, the secondary designs were performed to analyze the
changes in the six variables holding the key to optimization.
The determination of the final top junction tern-
perature, optimum ratio, and Seebeck Coefficient was deferred until later, as was the Peltier Coefficient earlier.
The values of length, width, and substrate thickness were
all set at their optimum.
It was concluded that the optimum ratio would fall
near a value of 1.5 for the final design, and that it might
vary with temperature.
74
VI.
IMPLEHENTATION 1\.ND ANALYSIS
OF THE FINAL DESIGNS
A.
COMPILATION OF THE INPUT DATA AND REITERATION OF 'I'HE
FUNCTION OF THE FINAL DESIGNS
Upon conclusion of the secondary designs, it was seen
that optimization of thirteen of the seventeen key variables had been achieved.
In considering the final values necessary to produce
optimization, i t was remembered that the T 4 values previously used were,
T4
=
1573, 1673, and 1773
6.1
On examining the results using these values, it was seen
that the source temperatures were approaching the value of
2000° K, which is close to the maximum epoxy temperature
previously found, of 2300° K.
Thus,
it was decided to use
the above temperatures as the final values of T 4 in evaluating the final design's performance.
With these temperatures,
the final values of a pn and
In evaluating a
earlier, the
could be calculated.
pn
pn
value of mA* /m was used for both the n and p type calculan o
However, it was found later that,
tions.
7T
0.30
.~mA.n/m 0
> 0.23
6.2
> 0.59.
6.3
as before, but
0.79 >rnA* /m
-
p
0
Therefore, the values used in the calculation of a pn Lecan:e,
6.4
rnA*
/m = 0.69, 0.265.
p,n o
75
Using the fact that a
and n
depend not only upon
pn
pn
T 4 , but on ~T as well, their corresponding values became as
given in Table VIII.
TABLE VIII - FINAL VALUES OF T 4 , T 5 , a
AND n
ALL
pn'
pn'
IN RELATION TO ~T
T
a
T4
T5
(OK)
(OK)
(OK)
1573
1273
300
0.001150
1.855
1673
1273
400
0.001158
1.993
1773
1273
500
0.001166
2.140
TI
pn
(V/°K)
pn
(V)
With these values, the only variable left to optimize
was the ratio, m.
1.60 -> mopt
Since the secondary designs produced,
>
1.52,
6.5
it was decided that this ratio should be the last variable
optimized, in order to assure its correct evaluation.
76
B.
ANALYSIS OF THE FINAL DESIGNS VERSUS RATIO
Five values of m were chosen so as to provide an ad-
equate range for investigation.
rn
=
These were,
0.25, 0.50, 1.00, 2.00, and 4.00.
6.6
The use of these values in conjunction with those of the
other sixteen variables produced the results shown in Table
IX.
TABLE IX - OUTPUT VALUES OF THE FIRST FINAL DESIGNS
VERSUS RATIO AND TEMPERATURE
rn
1573
1673
1773
n
WL X n
VL
(Volts)
IL
(A.mps)
WL
(Watts)
{%)
0.25
0.0690
277.39
19.14
2.65
50.6
0.50
0.1150
231. 16
26.58
4.10
109.0
1. 00
0.1725
173.37
29.91
5.38
161.0
2.00
0.2300
115.58
26.58
5.80
154.0
4.00
0.2760
69.35
19.14
5.05
96.7
0.25
0.0926
372.42
34.50
3.46
119.1
0.50
0.1544
310.35
47.92
5.34
255.5
1.00
0.2316
232.76
53.91
7.03
378.5
2.00
0.3088
155.18
47.92
7.60
364.0
4.00
0.3706
93.11
34.50
6.67
230.0
0.25
0.1166
468.74
54.66
4.23
231.5
0.50
0.1943
390.62
75.91
6.53
496.0
1.00
0.2915
292.97
85.40
8.61
735.0
2.00
0.3887
195.31
75.91
9.35
708.0
4.00
0.4664
117.19
54.66
8.26
451.0
(Watts%)
77
It was seen that the values of WL and n peaked separately, as in the secondary designs; thus the value of
WL X n was aqain calculated.
The data of Table IX was plot-
ted for all five variables, VL, IL, WL,
n, and WL X n.
results obtained are shown in Figures 22 through 26.
The
These
are families of curves in temperature, T 4 , and plotted versus m.
In examining the WL X n family of curves, Figure 26,
it was seen that each curve peaks at approximately
mpea k=l.37.
6. 7
It was thus concluded that this value of m is the optimum
value and that it does not vary with temperature.
With this
value of m
designated, the expected output values at
opt
optimum performance were interpolated from the curves of
Figures 22 through 25.
They were then compared with the
actual optimum values obtained in the final design computation,
using the optimum values of all seventeen variables.
These expected and actual values are shown in Table X.
TABLE X - EXPECTED AND ACTUAL FINAL DESIGN OPERATING
OUTPUT VARIABLE VALUES
Actual Values
Expected Values
T4
( o
K)
WL
IL
VL
(Volts) (Amps) (Watts)
n
(%)
IL
WL
VL
(Volts) (Amps) (Watts)
n
(%)
1573
0.1940
148.0
29.6
5.82
0.1994
146.3
29.18
5.72
1673
0.2570
196.0
53.7
7.50
0.2677
196.4
52.59
7.48
1773
0.3380
247.0
85.7
9.20
0.3370
247.0
83.32
9.17
0.4
{f}
400
0. 3
I
+J
........
,......j
{f}
0
Cl!
~
......
:>
.....
(l)
t:J'.
ro
1573° K
0.2
+J
s::
Q)
+J
,......j
0_
...
I
'U
I
....
ro
0
....:!
300
1-l
1-l
::l
u
'0
I
I
I
0 .l
200
Ill
0
....:!
100
I
I
1573° r:
I
I
0
0
0
l
2
3
4
Pesistance Ratio
riG. 22 - LOAD VOLTAGE VERSUS RATIO (F)
0
l
2
3
4
Resistance Ratio
FIG. 23 - LOAD CURRENT VERSUS RATIO
(F)
-...!
CX)
90
(I)
9
80
8
70
7
60
6
+J
+I
10
e.
50
-
5
rJP
---
H
Q)
:3:
~
0
c:...
cQ)
......
u
......
+I
~
Q.;
+I
~
0
4
u
3
4-1
lj.J
20
~
10
2
1
0
I
0
1
0
2
3
4
Resistance Ratio
FIG. 24 - OUTPUT POWER VERSUS RATIO (F)
0
1
2
3
4
Resistance Ratio
FIG. 25 - EFFICIENCY VERSUS RATIO (F)
-...I
\0
80
800
700
600
dP
Ul
+J
+J
ltj
:.s:
500
>t
u
r::
Q)
·r-i
u
·r-i
4-4
4-4
r:<:l
400
-
X
1-1
Q)
~
0
P..
300
200
100
1
0
2
3
4
Resistance Ratio
FIG.
26 -
OUTPUT POlvER TIMES EFFICIENCY VERSUS RATIO {F)
81
' These values compare within fractions of a percent;
the differences undoubtedly being due to the error in
graphical interpolation.
With the data thus obtained, the
curves of Figures 27 through 30 were constructed.
These
are the plots of performance versus T 4 of the optimum design.
0.35
400
Ul
Ul
-4..1
..--1
0
0. 30
:>
-~
Actual
7
'/
0.25
:/'
200
1673
T4
Expected
Expected
0.20~~-------+--------~
1573
300
1773
100~~-------+--------~
1673
1573
T4
(OK)
FIG. 27 - LOAD VOLTAGE
VERSUS T 4
1773
(OK)
FIG. 28 - LOAD CURRENT
VERSUS T 4
12
100
10
80
8
60
6
40
30~~------~--------~
1573
FIG. 29 -
1673
1773
OUTPUT POWER
VERSUS T 4
4+-----+--------1
1573
1673
FIG. 30 - EFFICIENCY
VERSUS T 4
1773
82
C.
SUMMARY AND CONCLUSIONS
The results of the secondary designs extended the
known optimum values to all variables except T 4 , a
, rr
,
pn
pn
Once T 4 was chosen, by examining the source and
and m.
sink temperatures, the values of a
and rr
were finalized.
pn
pn
The optimum ratio was found by using the first final design
runs.
With all of these quantities found, it was then a
matter of making one final design, the optimum one, and
analyzing its performance versus temperature.
The above final procedure produced a design which
would supply 247.0 amps and 83.32 watts at 0.3370 volts and
an efficiency of 9.17 percent, when operated at a top junetion temperature of 1773° K.
In conclusion, the complete final design actually produced by the computer, to operate at a temperature of 1773°
K, is given in Table XI.
TABLE XI -
INPUT AND OUTPUT VALUES OF THE OPTIMUM
DESIGN OF A SILICON CARBIDE THERMOELECTRIC
GENERATOR ELEMENT
Variable
Value
Input Variable Values
Variable Value
Units
Units
L
1.5
em
pp
0.0001
ncm
B
1.5
em
a pn
0.001166
V/°K
D
0.2
Cm
Tr
2.140
d
0.6
Cm
Yp
0.00001292 VCm/°K
/).
0.0025
em
Yn
0.00001292 VCm/°K
0.0001
ncrn
Kp
0.2130
Pn
pn
v
W/Cm°K
83
TABLE XI
-
INPUT AND OUTPUT VALUES OF THE OPTIHUH
DESIGN OF A SILICON CARBIDE THERMOELECTRIC
GENERATOR ELEHENT (CONT.}
Variable
K
n
Input Variable Values (Cont.)
Variable Value
Units
Value
W/Crn°K
0.2130
T4
1773
OK
T5
1273
OK
rn
1.37
-----
K
0.0331
e
0
Variable
2
Cal/SecCrn °K
ern
Ps
0.0000325
f2
P csn
0.075
ocrn
P csp
0.075
f2Crn
R
c
0.000025
K
0.413
s
0
Cal/Seccrn 2 °K
ern
0.005
Value
Units
Output Variable Values
Variable Values
Units
.
Units
1. 065
~l
1. 065
w
511.099
w
213.999
w
Rn
0.000100
Q
Rp
0.000100
Q
.QTp
Rst
0.000115
0
Qp
Rsb
0.000140
Q
.
.QK
R
s
0.000255
0
cl
-16.924
0.000250
Q
c2
-99.525
Rcsn
0.000250
Q
c3
-99.525
Rcsp
Rr
0.000365
Q
s
0.0574
Rt
0.000995
Q
al
0.0100
-------------
0.001363
Q
01
0.0100
-----
RL
Tl
2207.324
OK
QR
To
853.519
OK
ncarnot
w
TA
434.325
OK
Ql
908.676
825.358
w
TB
419.481
OK
12.224
w
ne~~not
.
.
Oo
.
QJ
QTn
.
---------
22.309
w
61.33
%
28.20
%
84
TABLE XI - INPUT AND OUTPUT VALUES OF THE OPTHlUM
DESIGN OF A SILICON CARBIDE THERMOELECTRIC
GENERATOR ELEMENT (CONT.)
Variable
Output Variable Values (Cont.)
Units
Variable Value
Value
-----
DENO
4.214
nl
9.17
%
N
n2
9.17
%
n
VL
0.3370
v
IL
L
247.0
83.32
9.17
Units
A
H
%
It is interesting to note several aspects of the data.
The source temperature, T 1 , carne out within less than one
hundred degrees of the maximum permissible epoxy temperature,
2300° K.
In addition, the sink temperature, T 0 , while
being low, was still high enough to remove the sink output
power,
.
o0 ,
with a moderate sized cooler of some type.
It is generally known that the Carnot efficiency of an
active thermal device cannot exceed approximately forty percent.
The nc arne t
calculated was 61.3 %, which is mislead-
ing, since the value of nc
arno t included insulation in the
form of temperature drops in series with the active part of
the device.
The actual value for the Carnot efficiency of
the element is given by nsub Carnot' which became 28.2 %,
and was indeed less than the maximum possible.
This effi-
ciency, then is the ideal efficiency possible for the de-
vice, but is reduced by thermoelectric heat generation,
giving a final efficiency of 9.17 %.
85
VII.
A.
CONCLUSIONS AND
RECOl~ENDATIONS
CONCLUSIONS
Three major conclusions can be made as a result of
this investigation.
1.
They are:
Silicon carbide is an excellent material to use in
constructing a high temperature thermoelectric
generator.
It would not only generate a respec-
table amount of power per unit volume at temperatures not previously attainable, but would produce
a large internal temperature drop, resulting in
fairly low output temperatures.
2.
A generator element constructed of silicon carbide
would, due to its high temperature capabilities,
generate a voltage ten to a hundred times as great
as elements composed of presently used thermoelectric materials.
3.
It is now technologically possible to build such a
silicon carbide thermoelectric generator.
The
fields of doped silicon carbide fabrication, electrical contacts to silicon carbide, and applications of adhesives are presently sufficiently advanced.
86
B.
RECm1NENDATIONS
In extending this work to future studies and applica-
tions, several observations and esti~ates may be offered
for consideration.
The optimum design obtained has a base of 3.2 cm 2 and
a height of 2.7 Cm.
These dimensions must be considered
when connecting generator elements either in series or parallel theriPally.
Consider a number of elements mounted in parallel
thermally, all operating at the same source and sink ternperatures.
These would effectively compose a "sheet" ther-
moelectric generator that could be shaped to conform to the
geometry of the heat source and sink available.
Since the
elements would all be operating nominally at the same peak
condition and output, each could be expected to produce
83.32 watts at 9.17 percent efficiency.
Therefore, if one
megawatt of power were required from the generator, twelve
thousand of these elements would be required.
Although
this seems a rather large number, it only represents a
2
"sheet" area of 3.8 X 10 4 em or 3.8 r12.
If this generator was placed around a cylindrical heat
source* 61.2 Cm in diameter, it would be 2.0
*
M
long and
This would be perhaps a conventional fuel furnace, a
liquid sodium heat exchanger, a nuclear pile, or a nuclear
fusion plasma.
87
2.7
Crn
thick.*
In other words, the whole active generator
would be a cylinder of 24.0 inches inside diameter, 78.8
inches long and 1.1 inches thick.
Its resulting power den-
sity would be 9.20 Mw/M 3 or 0.26 Mw/Ft 3 .
This is an im-
pressive figure, but it should be remembered that the efficiency is only 9.17 percent.
Therefore, in order to obtain
1.0 Mw of electrical power output, 10.92 Mw of thermal power
would have to be supplied at the source.
This difficulty could be partially resolved, however,
by one or more of the three following methods:
1.
Several different silicon carbide designs could be
produced, each operating at maximum WL X
certain fraction of the
~T
n
at a
of the original design.
Thus, for N designs,
~Tl+~T2+
. • • +~TN
=
~T or~g~na
. . 1"
7.1
Hence, the elements would be designed to operate
in series thermally, the whole combination performing between the temperature limits of the
original optimum design.
This type of construe-
tion, known as cascading, would probably reduce the
power output somewhat.
However, it would increase
the efficiency, as the total efficiency would be
given by,
n
=
N
1- [II (1-n.)
.
~
2.
*
~
1.
7. 2
Another "sheet" thermoelectric generator could be
This is excluding the necessary source and sink space.
88
constructed from a different, lower temperature
semiconductor, which would operate from the silicon carbide output temperature to a lower sink
temperature.
This additional, cascaded, generator
would increase the output and efficiency of the
overall device.
3.
Several distinct ther~oelectric generators could
be used either in series or parallel thermally, or
both.
These would all produce output power from
the heat source, and hence their efficiencies would
combine.
If these configurations were used in conjunction, they
would comprise a generator system which could possibly produce power at an overall efficiency approaching forty percent.
All of this leads to an important potential use for
the silicon carbide thermoelectric generator.
The waste
heat from any presently used powerplants, especially the
nuclear plants, must be dissipated through massive heat exchangers, in many cases causing severe l1eat pollution to
the environment.
This pollution could be decreased consid-
erably if some of the energy in this excess heat could be
salvaged, and converted into useful electrical power.
The
silicon carbide thermoelectric generator is the type of heat
converter that would be ineal for this application.
APPENDIX A.
/WI\1">
EEI4?4?2,TtM~=Ol,PlGES=010
I
I
c_
c
c
c
c
t:
c
c
c
c
c
c
c
c
c
c
c
c
2
3
4
5
6
7
~
q
10
11
12
13
14
15
16
17
1R
}0
20
21
22
23
24
Z5
26
27
2!L
zq
JO
~I
32
li~T
TS=I T=l ,P=lO,C=II
THE COMPUTER PROGRAl-1
BARROW JOHN T JR
71.088
CLlSS=W,PRlO~ITY=04,REGIO~=IOlOOK,OOOOKI 1 REAOER=REAOERZ
JOB 332
OEFAUL T
THIS·-is THE HNAL.Ru!\1·: THE.hPfiMUM DESIGN
·----~--
--------------
T~TS IS A PROG~AM FOR EE 4QO, TO DESIGN A THERMOELECTRIC
GE~ER~T~R ELEMENT USI~G SILICON CARBIDE AS THE SEMI-CONDUCTOR.
VARI~US VALUES OF "' AND K OETER"'IIIIE WHAT OPERATIONS THE
C:l"''PUTER t>EPFQR"''S. THE POSSIBLE COI18I~ATIONS ARE AS BELOW.
~=1,~=1
?
~
GIVES O~F OESIGN ONLY, FOR DEBUGGING PURPOSES.
M=2,K=2 TO 17 GIVES ZK+l PRELIMINARY DESIGNS.
~=3,~=17 GIVES 3 DESIGNS FOR FINAL CONSIDERATION.
··-
SU'3R'Jllll~E CALCS IQ,qSU,PSU,qSTU,RPU,RNU,RCSPU,PCSPU,DJUNUfRCSNU,P
lCS•.nr,RTU,RCU,RLU,RRU,HIU,VlU,IiKU 1 HPU,HTPU HTNU HJU HRU DEL U,C3U.tA
ZLPHU, C7U, ~ET AU 1 C1\1, HlU, WL U,EFU tHOU tCAR SU, f AU,JSU.rl {u,.TOU~CARNUaO~N
3ntl, H 1 U, FF3U, U 2U ,SKU 1 RSRU I
nl'oi(NSHI!Il 01171
RSU=IPSU*Il.5*0131+?.C*OIZI+3o0*0(4JII/0141
~STU=IPSU•IQI31+QI21+QI4111/QI4J
RSqU:RClJ-RSTU
RPlJ:(QI71*01111/0121
-··
·-
.. ·-.
-·------
~
·-
--------
R~U=IOI~I*OI1111QI21
PtSPU=IPCSPU*JJU~UIIQIZI
PCS~U= I PC S'lU*:"lJIJ~ II 0121
RTU= I~ SU+ RPU+RN1J+RCS PU+RC SNU+Z oO*RCUI
RLU=QI161*RTII
RRU=IRSTU+.5*1~CSPU+RCSNUII
HIU=IQI~I*IQI141-01151lli(RlU+RTUI
VIU:IQ(qi*IOI141-0il~II-RTU*~IUI
HKU=IIOI121+QI1~li*QI21*101141-QI1511110111
HPU:QI81*HIU*~Il41
HTPU=IQilOI*HlU*IQil41-0115111/Qill
HT '1111: I Q I 111 *'"H U*l Q114 I -0 I 1511 I /Q( 1 I
HJU:(RPU+~NUI*IHIU**ZI
HRU=RRU* Cil U** 21
r)EL TU= I liT PIJ*Il. 0+1 Q1131/01 121 lIt IHKU
---- -------- .. ---
C~U:(.~-11.Df~ELTUII
---
ALPHU=IHT~U*(l.O+IQI1211Cil31tii/HKU
C2U=C.5-Clo0/~LPHUII
~ETAU=HJUIHKU
C111= I • ~-I 1. vI '3 H 41J II
H1U:HPU+HKlJ+C3U*HTPU-C2U*HTNU-ClU*HJU-HRU
WLU=RLU*CH1U**21
EFlJ:WL Ulli 1U
HOU=!-11 U-WLU
CARSU=IOI141-0I151110114t
TI\U=IH1U/112.0*01211+013111*11QI5t/QI1711+0141/SKUI
.
--
·-
---------· --------·-·
(X)
\0
33
34
TBU=IiOU*
I (Q(Sii(Oiln• I I 2 .0*01211 +QI31111 +CO (41/( SKU*l.O$Q(211 )J
T1U=QI
l41+TAU
TOU=Qil51-TBU
Ui~~H31 ~ TOUJ /IllL
UPlflU=I CARSU*Oil6 I 1/Et-lU
35
·---1t
38
)q
EF3U=!tTAU*Qil5li+ITBU*~Il4111/IQtl41**2+(TAU*Q(l4111
'tO
47
46
4<:J
50
'H
52
53
54
c;s
5b
1)7
58
r;q
60
bl
62
&1
h4
6')
611
67
68
6Q
7C
11
7i'
73
74
75
76
77
78
79
1!0
Rl
82
!!3
94
85
86
- 81
RB
A9
_QO
91
·-- -·--- ...... --- ··-"'-··-
EF2U=ICA~NU-EF3UI*O!l6l*llo0/0ENOUI
ENO
43
44
-·
q,ETUR~
.. --" 1
42
45
46
~
DI~F.NSION U!171,VI171,VIl7) 1 XIl71
K.,pot
( lt4001
READ !1,4101
GO TO !4(11 ,K
REAO 11,4101
REo\0 .(1 1 4101
40 READ 11,4101
GO TO I651 1 K
~EAD
IUIII,I=l,l71
IVIII,t=l,l71
1Vfii,I=l 1 171
DJUN,PS,PCSN,PCSP,RC,SK
WRITE 13,4201
WRITE 13,4301
GO Tn 75
()') WRITE 13,4401
WRlTE !3,4301
7 5 WRITE 1.3, 460) I U.l I I , I= 1 ,17 I
GI'J Tf1 I 9 c; I , K
WR IT E I 3 , 460 I IV I I I , I= 1 , 1 7)
WRITE (3,4601 IVIJI,I=l 171
0 5 WRITE 13 1 4701 K,~,OJUN,PS,PCSN,PCSP,RC,SK
GO TO llo5 105,1101,~
toe; IF fi<.E0.1ft GO TO 106
1141=12*K+ll
GO TO 115 ..
106 "'l=I?.*K-11
GO T!l 115
ll'l .. 1=1
111)
310 J=1 Ml
IJ.FQ.261 GO TO" 1g5
IJ.F0.211 GO TO 1R5
IJ.F0.221 GO TO 195
IJ.EQ.?.31 GO TO \85
GO TO 1190 I , r.fl ··-- ..
Gn TO 1130), J
GO T!l 135
130 Co\LL CALCS !U,~S,PS 1_RST,RD,R~,RCSP,PCSP,DJUN 1 RCSN,PCSN,RT,RC,RL 1 RR
- l,~I,Vl,H~,HP,HTP,HTN,HJ,HR,OELT,Cl 1 ALPH,C2,8ETA,Cl,Hl,Wl,EF,HO,cAR
2S,TA,TB,Tl,TO,CARN,DENO,FF1,EF3,EFz,SK,RSBt
GO TO 11'15
13c; IF IJ,EQ.JOI GO.TO l36
IF tJ.EQ.lll GO TO 38
r,o Tn 140
136 XI 17l=lH 171
u1 111 "'v 11 71
r,o rn 140
l3'l V!l71=Y!l71
140 Ni:!o!OOIJ,21
!F INI 150,1<;0,170
150 l=J/2
'lQ
I~'
II'
IF
!I'
X Ill =U Ill
Ul L I =VI Ll
CUl rAL cc; llJ, RS, P S ,R STt RP, RN t RC SP_,_PCSP, OJU~E ~l~NfPCS~.t.!lrRC, R.ktRR
l 1 HI , V1, HK, HP, HTP, HTN1HJ, HR ,DEL T, C3, ll1Jlr1 C7f
,C I RI, Wlo!::F,HO ,\.IR--
2~,TA,TR,Tl,TO,C4RN,OtNO,EFl,EF3,EF2,SK~~SBt
Grl TIJ 1R5
\0
0
92
170 t=IJ-U/2
Flll"'Htls IU,RS,PS,RST,RP,RN,RCSP,PtsP,DJUN 1 RtsN PcsN,RT 1 Rc,Rt 1 RR
1 ,HI tY1tHK,HP, HTPlHTNtHJ,HR,OEl T tC3tAlPH,C2jBH AtC 1,H1 ,WL;tF,HO ,~AR
2S;TA,TH,T1,TO,CAKN,DtNO,EFl,EF3,EFz,SK,RS8
qo;
185 .. =1
96
G TO 195
"'2=M280 JJ=l,M2
'
91
190
98
195 00
99
_
1~ 1 J.Eg·~YJ GO
~o8
100
I
(J,E,
GO 0 10
·
101
IF (J,EQ,22l GO TO 310
IF (J,EQ.23l GO TO 310
102
103
_ ________1F__11U._f.O_.U GO TO 250
104
- GOIOf"205,210,230ftJJ
105
205 CALL CALCS IU,RS,PS,RST,RP,RN{RCSP,PCSP,OJUN 1 RCSN{PCSN,RTtRC,RltRR
1tHI,Vl,HKiHP,HTP,HTN,HJ,HRlDE T,C3~ALPH,C2JBtTA,C ,H1,WL,tF,HO,~AR
2S TA,TB~T ,TQ,CARN,OENQ,EF ,EF3,EF •SK,RSB
G6 TO 2 0
106
101
210 00 220 1=1.17
108
Ulll=VIIl
lQCL_____ 220 CONTINUE
flO-CALL CALCS (U ,RS, PS, RST, RP, RN, RCSP, PCSP, OJUN 1 RC SN, PCSN, RT .tRC,RL 1 RR
1tHitV1,HK,HP,HTP,HTN,HJ,HR,OELT,C3,ALPH,C2,8tTA,t1,HltWL,tF,HO,~AR
2S,TA,TB,Tl,TO,CARN,DENO,EF1,EF3,EFZ,SK,RSBI
111
GO TO 250
112
230 DO 240 1:1,11
113
Ulll=YIII
}14
240 CONTI~UE
ts __
,., , c•• c u
ps
~. ,,~~
·e~"H~~~s~ 1 P~fNorT~'cABltfR
l,Hl,V1,HK,HP,HTP,HTN 1 HJ,H t 8 L, t LPH, t
t
t
t t t t
2S,TA,TB,Tl,TO,CARN,OtNO,EFl,EF3,EF2,SK,RS81
116
250 GO TO 1265,265,2551,M
ll1_
255 Jl=JJ
118
GO TO 270
119
265 Jl=J
120
270 WRITE 13,490)J1 1 RN 1 RP:RST 1 RSB,RS,RCSNtRCSP:RR~R.T 1 RLtTllTO,Hl:HOtHJ
l.HTN.HJP,HP.HK~1~2~~3.B~TA,llPH.OFL_,HB•GAR~,l&.L•~Ta~•ut~_uR~Su·~O~Eu!uO~,~~F~1L-__--------------------------------2,EFZ
121
WRITE 13,5001 Vl,Hl,WL,EF
122
280 CONTINUE
123
GO TO !315,290,315l,M
12~
290 GOT~ I305),J
125
IF IN.EQ.OI GO TO 305
126
UILI=XILI
127
30i_1f~oEO.lll GO TO 3QI.
----------------------------------------------------------------128
GO TO 310
129
107 Ull71=X(171
GO TO 310
130
Lll__ __10B WRITE 13,5101
132
310 CONTINUE
133
31 'i STOP
13~
400 FORMAT 121101
135
~fr_f~AI l'iEl~.Bl
• ~'
42~010'1Af 1f
L>TiftOtYli"IT'CfW ARE TWCl -~v~-ron!F~Y"TA"~~"AriAr18111lne!'""""'INnl.-.M~E--~----------------------------1S. THE FIRST IS THE APRAY OF INPUT VARIABLES AND THE SECOND IS'//
2'
THE ARRAY OF OUTPUT SECONDARY VARIABLES. THE SUBSEQUENT NU
3!1ERICAL INPUT AND OUTPUT DATA ARRAYS COR~n~~..,D...r:,.TOIL.rtJT~HE~S&.E'~/ttJl'r.,.,I'IITir-------------4EL£~ENT BY ELE~ENT, RE~~ECTIVELY.
Jij lD
-ro TRESE, TRE FOUR
5PRIMARY OUTPUT VARIABLES ARE SHOWN FOR EACH 1 / / 1
DESIGN.'////'
6 FIRST SET
LENGTH
WIDTH
SEPARATE SUB.THKN EPOX.THKN N-R
11-m
P-RHO
__SEEBECJl __ _p£L_Il£R
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94
BIBLIOGRAPHY
1.
Soo, S.L.
Direct Energy Conversion.
New Jersey:
2.
Webe!, R.L.
Prentice Hall, Inc., 1968.
"Temperature Measurement and Control,"
Philadelphia, Penn:
Ref.
Englewood Cliffs,
Blakiston Co., 1941.
Reference Data for Radio Engineers.
Edition]
New York:
Telegraph Corp.,
Also
[Fourth
International Telephone and
P~erican
Book-Strafford Press Inc.,
1964.
3.
Kettani, M. Ali.
Direct Energy Conversion.
London:
Addison-Wesley Publishing Co., 1970.
4.
Holman, J.P.
Heat Transfer.
New York:
McGraw-Hill
Book Company, 1968.
5.
Brown, Aubrey I.
and Marco, Salvatore M.
to Heat Transfer.
Company,
6.
7.
HcGraw-Hill Book
19 68.
Austin, J.B.
Ohio:
New York:
Introduction
The Flow of Heat in Metals.
Cleveland,
American Society for r1etals, 1942.
Hall, R.N.
"Electrical Contacts to Silicon Carbide,"
Journal of Applied Physics, 1958, Vol. 29, 914-917.
a.
vanDall, H.J., Greebe, C.A.A.J., Knippenberg, W.F., and
Vink, H.J.
"Investigations on Silicon Carbide,"
Journal of Applied Physics, 1961, Vol. 32, 2225-2233.
9.
Rutz, R.F.
"Negative Resistance Tunnel Diodes in
Silicon carbide," IBM Journal, Nov. 1964, 539-542.
10.
Fa rrell, Richard.
"A SiC Backward Diode," IEEE Pro-
ceedings Letter, Feb. 1969, 221-222.
95
BIBLIOGRAPHY (CONT.)
11.
"Hodgman, Charles P., Weast, Robert
Clarence W."
Handbook of Chemistry and Physics.
Cleveland, Ohio:
12.
c., and Wallace,
Chemical Rubber Company, 1953.
Sisler, Harry H., VanderWerf, Calvin A., and Davidson,
Arthur W.
College Chemistry.
New York:
The
Macmillan Company, 1963.
13.
Fadler, Eugene Charles.
"Heat Transfer Coefficients
Across Bonds and Contacts," Thesis No. 1816,
University of Missouri-Rolla, Rolla, Missouri, 1965.
14.
"Neuberger, M."
"Silicon Carbide Data Table and
Supplimentary Bibliography," Hughes Aircraft Co.,
Electronic Properties Information Center,
Report No.
15.
Interium
62, Aug. 1968.
"Levine, Milton A."
"Electronic Cooling with
Thermoelectrics," Catalog from Materials Electronics
Products Corp.
[MELCOR]
Trenton, New Jersey
(Received Oct. 1970}.
16.
Wert, Charles A. and Thomson, Robb W.
Solids.
New York:
Physics of
McGraw-Hill Book Company, 1964.
96
VITA
John Talmage Barrow, Jr. was born on November 21,
1944, in Washington D.C.
He received his primary and
secondary education in Webster Groves, Missouri.
He has
received his college education from the University of
Missouri-Rolla, in Rolla, Missouri.
He received a
Bachelor of Science degree in Electrical Engineering from
the University of Missouri-Rolla, in Rolla, Missouri, in
August 1967.
After graduation he spent two years with Westinghouse
Electric Corporation, Power Transformer Division, as a
design engineer.
During this time he also attended and
completed the Westinghouse Design School, held at the company's Educational Center.
He has been enrolled in the Graduate School of the
University of Missouri-Rolla since September 1969.