APPROXIMATE ANALYSIS OF THE OPERATION OF THERMOELECTRICGENERATORS WITH TEMPERATURE DEPENDENT PARAMETERS

Jose M. Borrego+(Received 1-22-6-3)

Summary

An approximate analysis of the operation ofthermoelectric generators with temperature depend-ent parameters is presented. Expressions for theoptimum current and optimum area to length ratioare obtained for the cases of maximum efficiencyand of maximum power output per unit volume.The principal assumption made in the analysis isthat the temperature distribution along the legs ofthe generator is determined, to the first order ofapproximation, by the thermal conductivity of thematerial. The equations derived in the analysisare applied to a particular solvable case, and theapproximate results obtained from them are com-pared with the exact results.

Introduction

The analysis of the performance of thermo-electric generators reported in the literature hasbeen carried on with the assumption of materialswith temperature independent parameters. 1 Thereason for making this assumption is that it is notpossible to solve, in closed form, the heat conduc-tion equation for a thermoelectric material withtemperature dependent parameters when an electriccurrent flows through the material. 2 Since theparameters of thermoelectric materials usuallyshow a strong temperature dependence, some authorshave taken either of the two following approaches inorder to account for such temperature dependence:

a) Use of the formulas derived for thetemperature independent parameter case withappropriate average values for the materialparameters. 1, 3.

b) Use of computers for an iterativesolution of the heat conduction equation. 4, 5.

Although approach a) can give results withinreasonable limits of accuracy, it has the drawbackthat no analytical justification has been given of theformulas used or of the "appropriate average values".Approach b) has the disadvantage of not giving anexplicit dependence of the performance of the deviceupon the relevant parameters. However, this approach

*This work was carried out at the Energy Conver-sion and Semiconductor Laboratory, Departmentof Electrical Engineering. M. I. T. This workwas supported by the Air Force Cambridge Re-search Laboratories (ARDC), U.S. Air Force,Contract AF-19(604)-4153 and reported in AFCRL-62-148, January 15, 1962.This work forms part of a thesis submitted to theDepartment of Electrical Engineering at theMassachusetts Institute of Technology in partialfulfillment of the requirements for the degree ofDoctor of Science, September 8, 1961.

+Associate Member, Centro de Investigacion yEstudios Avanzados del I. P. N. Mexico, D. F.MEXICO.

is necessary in order to obtain estimates of the orderof accuracy obtained in using approximate methodsin the analysis of the performance of the device.

It is the purpose of this paper to present a'middle of the way" approach developed by the author6, 7 and independently, by a Russian author. 8 Thisapproach consists of performing the analysis of theoperation of thermoelectric generators using anapproximate solution to the heat conduction equation.The principal assumption of the analysis is that thetemperature distribution in the material is deter-mined, to the first order of approximation, by thethermal conductivity of the material. This assump-tion proves to be valid for thermoelectric generatorsbut not for thermoelectric coolers. An importantfeature of this method of approach is that the derivedexpressions reduce to the exact expression in thecase of temperature independent parameters.

The paper starts with a detailed analysis of theperformance of thermoelectric generators with legsof similar materials, except for the sign of thethermoelectric power. A discussion is given inorder to explain why the approximate solution reducesto the exact expressions in the temperature inde-pendent parameter case. Next the case of thermo-electric generators with legs of dissimilar materialsis presented. Expressions are given for the optimumcurrent, maximum efficiency and maximum poweroutput per unit volume. In the last part of the paper,the equations derived in the analysis are applied toa particular solvable case and the approximate resultsare compared with the exact results.

Thermoelectric Generator with Legs of SimilarMaterials

The configuration pertinent to the analysis isshown in Fig. 1. For this particular case, whereboth legs are of similar materials except for thesign of the thermoelectric power, the efficiency ofthe device is the same as the efficiency of one of itslegs:

where

PO = power output = I f adT - I ., dx

Qi = power input = Ia(Th)Th + QhI = electric current.

(2)

(3)

Qh = heat conducted at the hot end by one of the legs.

h (A dx = oT = absolute temperature.

The quantity of heat Qh is determined by the heatconduction equation

dd-A + IT a

-

-Px - T d- A(I)

4

17= Po t1

with boundary conditions:

X = T = Th I x = T Te (5)

Double integration of equation 4 and use of boundaryconditions of equation 5 give the heat Qh theexpression:

3 h C + I a

v ( AJ a I iaA xThOTaJO (6)

(C1)0t0=A f?h TXI Te

An interpretation of this assemption is that the effectsof the Joule heat, Thompson heat and distributedPeltier heat upon the heat input are calculated usingthe temperature distribution under no-load conditions.This approximation is valid for thermoelectricgenerators but not for thermoelectric coolers wherethe Joule heat distorts to a large extent the no-loadtemperature distribution.

Substitution of equation 1 1 into equation 9 gives:

which can be written as follows:ITfh - 2 fh

I a IJ IdT f [ lT Tcc ..

1 Th Th T1h 1 Th [ hQO I + TAT 4- T - --- -C (12 )(7)Th-T IQh - T Ih dTTc where:

AT = Th-TC

where Q is defined as

Q & AdT

Substitution of equations 2, 3 and 7 in equation Iresults in the following expression for the effi-ciency of the device:

flh{Th hhT T dTXTJT

where use has been made of equation 8 and of theidentity:

(13)

Equation 12 is the first order approximation to the(R) efficiency of a thermoelectric generator with

temperature dependent parameters and has a formwhich makes it possible to find the optimumcurrent for maximum efficiency. The results ofsuch optimization are as follows:

rThT a dT

loI Tc (11

14~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1AT I__r__ 1 rTh r h (~~Th r,Th I (Thr/Th I

where:

|Td = aT - a(Th)Th + [ adTabT

(1o)

Equation 9 is the expression for the efficiency of athermoelectric generator with temperature dependentparameters. It is valid for the case of positiondependent parameters.

In order to evaluate the efficiency by means ofequation 9 it is necessary to know Q as a functionof T and I. This dependence may be found at leastin principle, from the solution of the heat conductionequation 4 and equation 8. Several authors 2 havestudied the solubility conditions of equation 4 andhave concluded that, in the most general case, thesolution cannot be represented in closed form. There-fore, in order to carry the analysis any further with-out restricting the temperature variation of theparameters, it is necessary to introduce an approxi-mation for the evaluation of the efficiency. Thesimplest approximation is to assume Q as a constant:

Tb rITh 1

rT

J PdT ic (16)

The following changes in the order of integration:

Th[ Th ] Th Te Th Th[J T 1] T dT 1 OdT = aTdT T OdTfTb T TTb (Tb lT,Thr(Th Th rT 1 Tb (Th1TU Jd T LdT dTJ~&k dT iT TJ9dT -- T J,oI dTC C T C (17)

transform equations 15 and 16 into:

5

TC

'M2 - I +

M - I-- Th ~

TadT 2 h oT,dTTc T

T - T (M-l)h f h hadT II J px dT

Tc Tc

JTh Th '0LX (Th23|at OQ dT, | i Q dT, IiT dT

These terms have the property that they vanish in(18) the temperature independent parameter case.

This property is shown as follows: Integration ofequation 8 gives:

M2 - I +

Th dT =4(24)(Io)

Equations 14, 18, and 19 are similar in form to theequations obtained with temperature independentparameters. The expression

htha dT)2Tc

T'h

&T J p9 dT (2n)Tc

play s the role of figur e of me rit and the expre ssion

Taking the derivative on both sides of the aboveequation with respect to I, we obtain:

T

| dT =

This last equation is valid for any dependence ofK independent of T we obtain:

{Th ,Th2 TadT T/AAdT

Tc Tc{Th rTh

a dT J pl(dTTC TC

plays the role of average temperature.interpretation may be given to equationplying both numerator and denominator

T'h ThadT)2

-- T ATfvET | VD dT J P/(dTT T

M dT-

0

(21 )

A simple20 by multi-by T:

(22)

where the indicated averages are averages withrespect to temperature. This figure of merit usingaverage parameters was suggested by loffe butwithout any justification. It should be pointed outthat there is not a priori justification to considerexpression 20 as the figure of merit for the tempera-ture dependent parameter case since the so-called"average temperature" depends also upon thematerial parameters. The only justification in thisanalysis is the similarity of the expression 20 tothe figure of merit for the case of temperatureindependent parameter. A more complete argumenthas been given in Reference 7.

It is a surprising result that equations 14, 18and 19, which are obtained by means of an approxi-mation, give the right expressions for the tempera-ture independent parameter case. This "anomaly"in our results can be explained as follows. One ofthe consequences of the approximation expressedby equation 11 is to neglect the dependence of Qupon the current I. If we take this dependence intoaccount in the evaluation of the derivative ofequation 9 with respect to the current I , we findthat the neglected terms contain either of thefollowing quantities as factors:

From the above equation it follows that the ex-pression in equation 23 vanish in the temperatureindependent parameter case.

THERMOELECTRIC GENERATOR WITH LEGS OFDISSIMILAR MATERIALS

The configuration pertinent to the analysis isshown in Figure 2. We choose, without any lossin generality, leg 1 an n-type semiconductor rodand leg 2 a p-type semiconductor rod. Many ofthe steps of the analysis presented here are omittedsince the development follows along the same linesas the one presented in the previous paragraph.

The efficiency of the device, that is, the ratioof the power output is given by:

2hTb (,P4)2 ~')IJ (a +a )dT -I2 [J O dT + 2 aErL2 f 2 fT 2 ]Th

1 TTh h rwT 1h7 * C X e IdTCQ1fdTC JTCEJdT dT dT'TJT~~~~T~~ [fTh JTha ~ ah dT a

AT c C +1 C C

where Q and Q2 are defined as:

(27)

(28)Q -- /( AT AdT1 x1 2 r x

6

Th?(M+l) fTI

Th(M iC

(25)

&ma;

c Lc

The numerator of equation 27 is the power outputof the device and the denominator is the powerinput. Introducing the simplifying approximations

A f hit! 010 = -FJ IdT

'r

2 O =P20=2 A/2dTrc ("I

and maximizing the efficiency with respect to thecurrent I, we obtain:

ThIrh (aI+ a2)d

I= , 'c II h fTh

fT ( 01"f2)dT2]Figure of Merit

AT fT JT

[(*al) + (YavV ][4Ftl;) av fw5;t ]

Equations 35, 37, 38,and 39 are the equations tothe first order of approximations for the optimumcurrent, maximum efficiency and optimum areasto lengths ratio of a generator with temperaturedependent parameters.

The electrical behavior of a thermoelectriggenerator is represented, under steady state con-ditions, by a voltage source in series with a resis-tance. The value VO of the voltage source is equalto the sum of the Seebeck voltage of the materials:

VO=b (al a2)dT

Tc

The maximum value of the above quantity is ob-tained when the areas fulfill the relation:

TTh {ThA12 I ) I dT f 2dAI1 T c Tcfl2 h h* d1(A, Sk2(IT 1 K IdT

c c

(45)

If the above relation is satisfied, the power out-put per unit volume has the maximum value of:

Mmlx

(141)

The value Rin of t1le resistance is given by theterm multiplying I in the numerator of equation27. Introducing the simplifying approximationsexpressed by equation 29, the value of this inter-nal resistance becomes:

Th Th

1d |2 fdTM '~~~~A2 ('42)

Te Tc

The maximum power ouput Pm delivered by thegenerator to an external load occurs when the re-sistance of the load is equal to the internal resis-tance of the generator. Under this matched con-dition, the value of Pm is:

Th 2

C C,) dTA Tc Tc

The above expression does not allow any furtheroptimization. Equations 36 and 45 indicate thatit is possible to choose the areas and lengths asto satisfy simultaneously the conditions of max-imum efficiency and maximum power output perunit volume. However this will make the lengthsof the segments different, which may be a dis-advantage for constructional simplicity.

The last point in our analysis of the efficiencyof the rmoelectric generators with temperaturedependent parameters is to apply our equationsto a particular solvable case. In this way, wemay obtain an estimate of the accuracy of our re-sults. Here, we take one of their cases and com-pare the exact results given by those authors withthe results obtained by our approximate analysis.The material parameters of the example to con-sider are given in Table I.

Tab] e IMaterlal Paraimeters

" p ~~~~~~~~~~A

leg SY/'C ) (S -cu) (witt/em0) (cm2) (cm)T2 = a T

p 200 1

T temperature In 0Telyimn.

More important than the power output is the poweroutput per unit volume. This quantity is given by:

-a= I(A11+2e2) (414)

The cold and hot temperature of the device are400K and 15000K respectively. The tabulationof the results obtained by means of equations 35,37, 38, 39 and 40 is given in Table II together withthe exact values reported in Reference 4.

Tuible IIComparison between #,xact and approximAte values

Quantity Our Resuilts Reference F,rror o/o('4)

A,,(cM2) 4.47 4.5o -n.7 o0oI0p amps) 52.5 55.0 -5 0/0'7 (o/O) 27.6 26.n +6 o/oLMax

8

Tb 2(al+a2)dTTc

The calculated values are in good agreement withthe reported exact values. The temperature dis-tribution along the n and p arms calculated fromequation 29 and the temperature distribution re-ported in Reference 4 are shown in figure 3. It isconcluded that the flow of current distorts in asmall measure the temperature distribution under'no load conditions.

CONCLUSIONS

A complete analysis of the performance of thermo-electric generators with temperature dependentparameters cannot be carried out without intro-ducing some approximations. A very simple anduseful approximation is to assume that the temp-erature distribution in the material is the sameas the temperature distribution under no loadconditions.

With this simplifying approximation, the optimumcurrent, maximum efficiency and maximum poweroutput per unit weight can be obtained for the caseof temperature dependent parameters. An impor-tant feature of this method of analysis is that theformulas obtained reduced to the exact express-ions in the temperature independent parametercase. Furthermore, the derived expressions seemto give results within the limits of accuracy towhich the parameters are known.

REFERENCES1. Ioffe, A. F., "Semiconductor Thermoelements

and Thermoelectric Cooling" (Infosearch Ltd.London), 1957.

2. Burshtein, A., "An investigation of the SteadyState Heat Flow through a Current CarryingConductor, " Soviet Physics - Tech. Phys. 2,1937, (1957).

3. Heikes, R. R. and Ure, R. W. Jr., Thermo-electricity: Science and Engineering (Inter-science Publishers, New York), 1961.

4. Sherman B., Heikes, R. R. and Ure, R. W. Jr.,"Computation of Efficiency of ThermoelectricDevices, " Scientific Paper 431FD410-P3,Westinghouse Research Laboratories, March1959.

5. Sherman, G., Heikes, R. R. and Ure, R. W. Jr."Calculation of Efficiency of ThermoelectricDevices, " J. of App. Physics, 31, (1960).

TTh//////I//////I L //////G MATERIAL PARAMETERS

a =ABSOLUTE VALUE OF THERMO-N P EL.ECTRIC POWERTYPE TYPE p ELECTRIC RESISTIVITY

/ K e THERMAL CO N DU C TIVIT YCROSS SECTIONAL AREA -A-

TC

Figure 1. Thermoelectric Generator withLegs of Similar Materials

////// M/// / MATERIAL PARAMETERS_ _ _ _7 ffi t -IABSOLUTE VALUE OF

THERMOELECTRIC POWERN X21 TPE K TERACNUTVTTYPE j YPE ELECTRIC RESISTIVITY/ Kt~~~~~X. K 2z THERMAL CO)NDUCT IV IT Y

LEG I LEG 2 t2

,/ /// ,// Tc Tc / // 1tCROSS SECTIONAL CROSS SECTIONAL

AREA AI AREA A2

Figure 2. Thermoelectric Generator withLegs of Dissimilar Materials

wwLLJcrQ)wa

L

1-

wa-

6. Borrego, J.M., Lyden, H.A., Blair, J., "TheEfficiency of Thermoelectric Generators, "WADC Technical Note 58-200, Project 6058,M. I. T. DSP 7672, September 1958.

7. Borrego, J. M., "Optimum Impurity Concen-tration in Semiconductor Thermoelements,Sc. D. Thesis, Massachusetts Institute ofTechnology, September 1961.

8. Moizhes, B. Ya., "The Influence of theTemperature Dependence of Physical Para-meters on the Efficiency of ThermoelectricGenerators and Refrigerators, " Soviet PhysicsSolid State, 4, 671, (1960).

NORMALIZED DISTANCE X

Figure 3. Temperature Distribution in theLegs of the Thermoelectric Generator

9