Analysis for relating changes in heat input to power output of thermoelectric devic

Raag, V.

English

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MEMORANDUM #2 
RELATIONSHIP OF CHANGES IN THE HEAT 
INPUT TO THE POWER OUTPUT OF A 
THERMOELECTRIC DEVICE 
1 
1 
I 
J E T  P R O P U L S I O N  L A B O R A T O R Y  
C A L I F O R N I A  I N S T I T U T E  O F  T E C H N O L O G Y  
P A S A D E N A ,  C A L I F O R N I A  
https://ntrs.nasa.gov/search.jsp?R=19700016379 2020-03-12T03:05:32+00:00Z
MEMORANDUM #2 
RELATIONSHIP OF CHANGES IN THE HEAT 
INPUT TO THE POWER OUTPUT OF A 
THERMOEUCTRIG DEVICE 
July 30, 1969 
Prepared For: 
Jet Propulsion Laboratory 
Pasadena, California 
bY 
Ualvo Raag 
RESALAB SCIENTIFIC 
Menlo Park, California 
This work was performed for the J 
California Institute of Techno1 
National Aeronautics and Space 
Contract N-7-100. 
INTRODUCTION 
It is well known that the electrical power output of a thermoelectric 
device is proportional to the square of the temperature difference across the 
thermoelements . It is sometimes assumed that the heat traversing the device 
is proportional to the first power of the temperature difference across the 
thermoelements and therefore that the electrical power output is doubly 
changed for a given change in the heat input . This assumption is only valid 
in the l imi t  of a vanishing current inasmuch as it neglects all current dependent 
effects . Other equally unacceptable assumptions are frequently made with 
the result that considerable confusion exists on the heat input dependence 
of the power output of a thermoelectric device. The purpose of the present 
memorandum is to investigate this problem in somewhat more detail than is 
done usually and to derive a relationship that correctly relates the heat in- 
put to the power output of a thermoelectric device. 
TYPICAL ANALYSES 
Before proceeding with the analysis that correctly relates the heat 
input to the power output of a thermoelectric device, it may be worthwhile to 
consider two typical, albeit incorrect, approaches to the problem. For con- 
venience, all calculations will be performed on the basis of single thermo- 
couples. The generalization of the results to multi-thermocouple devices is 
straightforward. 
Probably the mos t  common method used for relating the heat input to 
the power output of a thermoelectric device assumes that all of the heat 
traversing the thermocouples is transported by con+uction. Thus, it is 
assumed that 
Q = KAT, 
1. 
where Q is the total heat input to the thermocouples, K is the thermal con- 
ductance of the thermocouples and AT is the temperature difference between 
the hot and cold junctions of the thermocouples. The power output P of a 
thermocouple is given in terms of current I and load resistance RL as 
2 P = I  RL. 
The current I may be defined as 
where S is the combined average Seebeck coefficient of the n- and p-type 
thermoelements , R is the total internal electrical resistance of the thermo- 
couple and m = R /R is the ratio of load to internal electrical resistance. 
The cQmbination of Eqs  . (1) 8 (2) and (3) and their differentials results in the 
following relationship between changes in the power output and heat input to 
L 
a thermocouple: 
- dP 
P 
= 2  do_ (4) 
Equation (4) shows that changes in the heat input result in a doubly changed 
power output of a thermocouple. This result of course is incorrect because 
the heat balance, Eq. (l), neglects all current dependent effects such as 
Peltier, Joule and Thomson heating. Only in the l imi t  of vanishing current, 
infinite load resistance, does it apply. 
Another analysis that yields an incorrect relationship between the heat 
input and power output of a thermoelectric device makes use of the efficiency 
expression as the starting point. The efficiency $bf ,  a thermoelectric device 
may be defined as 
2. 
Differentiation of Eq. (5) and division by q shows that 
It is now assumed that efficiency may be expressed as the product of a Carnot 
efficiency q and a material efficiency qM, with the latter independent of 
temperature. Although not rigorous , this assumption is reasonable and is 
based on the traditional analysis of the fixed operating temperature performance 
of a thermoelectric device (see, for example, A.F. Ioffe, Sem iconductqh 
Thermoelements and Thermoelectric Coolinu, Infosearch, London 1957). As d 
result of this assumption dq/q may now be written 
C 
where T and T represent thermocouple hot and cold junction.temperatures 
respectively and use has been made of the definition of Carnot efficiency, 
H C 
-T T Assuming that the cold junction temperature is constant 
%=(TH C)/ H' - l-qc, Eq. (7) becomes 
CITH and rewriting T 
where the abbreviated notation AT = TH-Tc has been used m With the assumption 
of a fixed cold junction temperature, an inspection of Eq. (1) shows that dTH/OT 
may be replaced by dQ/Qm Making this replacement in Eq. (8) and substituting 
the resultant equation in Eq. (6), it is possible to write the final result as 
k.., 
3 .  
1 
It should be noted that although Eq.  (9) in some cases yields more accurate 
answers than the corresponding Eq. (4), it itself, in making use of Eq. (l), 
neglects all current dependent effects in the thermocouple and therefore is 
subject to the same shortcomings as E q .  (4) e Moreover, the assumption of a 
fixed cold junction temperature is highly questionable and therefore if anything, 
the second derivation is even more unacceptable than the first. In any case, 
both derivations are grossly inaccurate and their results should consequently 
never be used in serious engineering work in thermoelectrics. 
I 
CORRECT ANALYSIS 
The main problem with the typical analyses of the relationship between 
the heat input and the power output of a thermocouple as given above is the 
over-simplification of the heat input equation, Eq. (1). As pointed out, the 
neglect of all current dependent effects in Eq.  (1) is unjustified and unaccept- 
able. A more meaningful representation of the heat input to a thermocouple is 
given by (see V. Raag, Eng. Conv. 8, 173 (1968) ) 
Q = KAT f ISHTH -11/2)12R - [l/dIpAT, 
where SH pertains to the combined Seebeck coefficient of the n- and p=type 
thermoelements at the hot junction temperature of the thermocouple and p is 
the average combined Thomson coefficient of the n- and p-type thermoelements . 
The rest of the symbols have previously been defined. The use of Eq. (3) in 
Eq. (10) enables the elimination of the AT's8 and the resultant quadratic 
equation in current may be solved to yield 
4. 
where 
The substitution of Eq e (11) in Eq e (21, differentiation and simplification enables 
the relationship between changes in the heat input and the power output of a 
thermocouple to be expressed a s  
where QL is defined as 
The derivative of A with respect to heat input to the thermocouple may be written 
where it has implicitly been assumed that the thermoelectric properties are in= 
dependent of changes in the heat input and consequently independent of changes 
in the operating temperatures 
valid for reasonably s m a l l  changes in heat input and operating temperatures e 
If temperature independent thermoelectric propeqies were to be used in the 
analysis, then of course E q s .  (13) and (15) would be exact. In this case, how- 
This same assumption underlies Eq a (13) and is 
ever, the Thomson coefficient would vanish and S and S 
In the treatment as given, it is assumed that all q toe lec t r ic  properties 
represent temperature integrated averages . 
would become identical. H 
5. 
The use of the expression for the power output of the thermocouple 
enables the derivative of the hot junction temperature of the thermocouple 
with respect t o  the heat input to  be expressed as 
The substitution of E q s  . (16) and (15) in Eq. (13) permits the relationship 
between changes in the heat input and the power output of a thermocouple 
to  be finally written as 
where f l  and I? are defined by 
s = +  aA 8 
It is noted that the terms entering Eq.  (17) are functions of thermocouple operat- 
ing temperatures 8 dimensions 8 thermoelectric properties 8 heat input and load 
characteristics. The use of Eq.  (17) thus enables the determination of the ef- 
fect of changes in heat input on the power output of a thermocouple, The 
derivative of the cold junction temperature with respect to  heat input in Eq . (17) 
depends on system heat rejection characteristics and thus in a general treat- 
ment of the present type cannot be precisely specified. In case of heat rejection 
by radiation, such as in space thermoelectric systems, the term may be written as 
%.: 
dT T -c=& 
dQ 4Q 8 (1 9) 
where the sink temperature has been assumed to be negligible in comparison to 
the thermocouple cold junction temperature 
Attention is finally called to  the fact that the present treatment is not 
only farily rigorous within its presently intended scope, but also reasonably 
represents the behavior of actual thermoelectric devices when typically 
experienced thermal and electrical losses are included in the thermal con- 
ductance K and the electrical resistance R of the thermocouple. Thermoelectric 
generator thermal end losses can approximately be taken into account by 
including them in the total heat input . 
ILLUSTRATION 
A silicon-germanium Air-Vac thermocouple* has been selected for the 
purpose of numerically illustrating the results of the preceding discussion, 
The thermocouple component dimensions are given as follows: 
Thermoelement length 2.54 c m  
Cross-sectional area 
* 2  
2 
n-type thermoelement 0.2101 cm 
p-type thermoelement 0.1168 cm 
2 
2 
Hot shoe area 4.00 c m  
Radiator area per couple 15,OO c m  
The heat input into the thermocouple is 5.40 watts, this resulting in the ap- 
proximate operating temperatures of 1000° C and 260° C for the hot and cold 
junctions respectively at the maximum power output point. Using thermo- 
electric property data that pertain to  those extant initially in s ilicon-germanium 
alloys, the performance of the thermocouple as a function of load current may 
be shown to be given by the solid curve in Figure 1. If the heat input into the 
thermocouple is decreased to  4 . 93 watts 0 the opeiating temperatures of the 
thermocouple decrease and the performance is given by the dashed curve in 
Figure 1 a The reduction of heat input to the thermocouple from 5.40 watts to 
4 e 93 watts represents the decay experienced with a Pu-2 38 heat source after 
* The same thermoco configuration was derived a 
RESAZAB SCIENTIFIC, f a  #l, prepared by V o  
some 11.4 years of operation e It is noted from Figure 1 that for the given 
decrease in heat input, the decrease in power output is load dependent, as 
predicted by Eq . (17) of the preceding section . Using Eq a (17) 8 it may be 
calculated that for the thermocouple in question, the dependence of power 
output on heat input as a function of the ratio of load to internal electrical 
resistance m is given as follows: 
m 0 0.5 1.0 1.5 3.0 10.0 e3 
- d P / a  1.58 1.67 1.72 1.74 1.83 1.93 2.00 
P Q  
The agreement of the calculated values of (dP/P)/(dQ/Q) with those obtainable 
from' Figure 1 is'good . It should be noted that the internal resistance of the 
thermocouple in this example is 85.9 milliohms . From the above table it is 
seen that dependence of changes in power output on heat input near the 
maximum power point is less than a factor of two, being of the order of 1.70 
to 1.75 . Only when m becomes very large, the l imi t  being the open circuit 
operation of the thermocouple, does the power output have a square dependence 
on heat input. The reason for this is that under open circuit operating condi- 
tions, the heat balance, as given by Eq. (lo), only possesses the conduction 
term and therefore changes in the temperature differential across the thermo- 
elements are directly proportional to changes in heat input. Power output, 
however, changes as the second power of the temperature differential across 
the thermoelements and therefore also as the second power of heat input. 
A comparison of the results obtained with the present analysis and 
those calculable by means of more approximate methods such as  those 
discussed in the Section entitled "Typical Analyses'' # shows the inaccuracy 
of the results commonly obtained . For example, in the illustration just discussed, 
it is found that the present analysis yields a value of 1.73 for (dP/P)/(dQ/Q) at 
the maximum power output point (m = 1.3) The two common methods of analysis 
discussed under "Typical Analyses" give the respective values of 2 .OO and 1.42 
for the same case. The fallacy is obvious . 
< .  "k 
8. 
SUMMARY 
A new analysis has h e n  developed for relating changes in the heat 
input to the power output of a thermoelectric device because most  existing 
treatments of this problem are completely inadequate. A numerical example 
has been used to illustrate the results obtainable with the new method and 
also to indicate the gross difference between the present results and those 
begotten by the methods previously used. 
FIGURE 1 
O.5Q 
0.45 
0.40 
0.35 
0.30 
0.25 
0.20  
0.15 
0.10 
0.05 
0 
1.5 
I 
0.50 
0.45 
0.40 
0 . 35, 
0.30 
0.25 
0 . 2 0  
0.15 
0.10. 
0 . 05. 


Relationship of changes in the heat input to the power output of a thermoelectric device

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