SERI/TP-35-253 

UC CATEGORY: UC-64 

ANALYSIS OF A HEAT EXCHANGERTHERMOELECTRIC GENERATOR SYSTEM 

JON HENDERSON 

To BE PRESENTED AT tHE 14TH 

INTERSOCIETY ENERGY CONVERSION 

ENGINEERING CONFERENCE 

BOSTON, MASSACHUSETTS 

AUGUST 5-10, 1979 

Solar Energy Research Institute 

1536 Cole Boulevard 

Golden, Colorado 80401 

A Division of Midwest Research Institute 

Prepared for the 

U.S. Department of Energy 

Contract No. EG '7TC'Ol4042 
ANALYSIS OF A HEAT EXCHANGER-'THERMOELECTRIC 

GENERATOR SYSTEM 

Jon Henderson 

Solar Energy Research Institute 

Golden, Colorado 80401 

~ Division of Midwest Research Institute 

ABSTRACT 

Analysis of a thermoelectric generator (TEG) in an 

ocean thermal energy conversion (OTEC) application is 

presented. An analytic model is developed for describing the heat exchanger-TEG interactions. This 

model is used to illustrate limitations of applying 

conventional fixed junction temperature assumptions to 

systems experiencing significant temperature drops 

across the heat exchanger surfaces. Design methods 

are developed for determining the thermoelectric element geometry that produces maximum output power. Results show that a heat exchanger-TEG system may deli-

'ver about 100 W/m2 of heat exchanger surface. This 

compares favorably with conventional OTEG schemes. 

RECENT WO~~ performed at the Solar Energy Research Institute has considered thermoelectric generators 

(TEGs) for electric power generation from low temperature solar thermal sources, such as ocean thermal layers or salinity gradient ponds (1*). These applications consider a TEG sandwiched between the warm and 

cold flow channels of a plate-fin or parallel-plane 

heat exchanger. Mathematical analysis of TEG systems 

is relatively straightforward; however, no previous 

investigations addressing this type of heat exchangerTEG system were uncovered by the author. Most conventional design methods assume known, fixed temperatures 

at the warm and cold junctions. This approach fails 

to address the temperature dependence between ~,e energy flux passing through the TEG and the associated 

temperature drops across the heat exchanger surfaces. 

Since these temperature drops strongly affect power 

generation and are sensitive to TEG design, fixed temperature techniques are inadequate for this analysis. 

Fixed heat input TEG analysis considers variations in 

junction temperatures ~ut is intended for systems significantly different from ocean thermal energy conversion (OTEC) or other fixed-temperature-source applications. 

This paper describes an analytic !!Iodel suitable 

for predicting performance of a heat exchanger-TEG 

system. The model is used to illustrate the limitation of applying conventional fixed junction temperature analysis when significant temperature drops occur 

across the heat exchanger surfaces. An OTEC performance analysis is included that shows the relationship 

between the ratio of thel"llloelectric element area to 

length and the output power generated by a TEG using 

standard maxilJum efficiency and maxh~um power fixed 

junction temperature design constraints. 

For OTEC applications, the thermal energy source 

is free. Thus, the cost of power Is controlled by the 

cost of the conversion equipment. The cost of electrical energy for a given heat exchanger is minimized 

*Numbers in brackets des ignate References at end of 

paper. 

1 

by maximizing the TEGpower generation. The method of 

Lagrange multipliers is used to solve numerically the 

heat exchanger-TEG model for the maximum power generation condition. This technique is rather tedious; 

therefore, a simplified approximate technique to design a TEG that delivers maximum power within a heat 

exchanger-TEG system is also developed. 

NOMENCLATURE 

A,B,C,D,E,F heat exchanger-TEG constants defined in 

text 

COY 

I 

K 

n 

p 

R 

R1. 

Rt 

TWj 

TCj 

tJ.Tc 

fraction of heat exchanger surface covered with thermoelectric material 

heat transfer coefficient between cold 

flow and TEG cold junction 

heat transfer coefficient between warm 

flow and TEG warm junction 

current 

conduction heat 

through the TEG 

transfer coefficient 

Lagrange multipliers 

Yn/Yp 

R1./R 

TEG electrical power output 

heat fluxes at the TEG warm and cold 

junctions 

internal resi~tance of TEG 

electric load resistance 

total. circuit resistance 

TEG warm junction temperature 

TEG cold junction temperature 

temperature difference between the TEG 

cold junction and the cold flow 

temperature difference across the TEG 

junctions 

temperature difference between the warm 

flow and the warm TEG junction 

z ATteg/Rt 

v 

Seebeck coefficient of the TEG 

Seebeck coefficients of the n-type and ptype thermoelectric 'materials 

Yp x COY 

ratios of the thermoelectric element area 

to length for the n-type and p-type materials 

TEG conversiorr efficiency 

thermal conductivities of the nand p 

thermoelectric materials 

resistivities of the nand p ther:noelectric matei"ials 

gradient operator 

nEAT ZXCHANGER-TEG DISCUSSION 

A typical thermoelectric element configuration is 

assumed and illustrated in Fig. 1. The analysis is 

developed !Jy incorpor.ating ther:o!oelectric theory tdth 

the heat transfer coefficients of the heat exchanger. 

Assumptions used i~ this paper are: 

• The heat exchanger heat transfer coefficients include the thermal resistance of the TEG electrical 

insulation in addition to the thermal resistances 
• 

• 

• 

associated with the flow boundary layer and the 

heat exchanger material. 

The thermal resistance between thermoelectric materials is assumed to introduce negligible conduction losses. 

Contact electric resistance is insignificant relative to the electric resistance of the thermoelectric elements. 

The analysis is performed on a single thermoelectric element. Power is investigated, but output 

voltage and current remain dependent on the connection scheme. 

• The device operates under small temperature differences so that thermoelectric material properties remain constant. Thus, Thomson effects are 

neglected. 

Warm Flow 

Heat Exchanger Surface / / 

Electric Insulation ./ 

"-

Thermal 

Cold Flow 

,;/; 

iN 

//': 

Fig. 1 - Thermocouple detail 

/ 

The generator performance is determined by equating the heat fluxes across the heat exchanger surfaces 

to the heat fluxes at the warm and cold junctions of 

the thermoelectric generator. 

qw -trrwhy (1 ) 

qw .. Kitrteg + a 1Twj - t 12R (2) 

qc = f.Tchc (3) 

qc .. Kf.Tteg + a 1Tcj + .! r2R 

2 (4) 

The heat fluxes through the warm and cold sides of 

the heat exchanger are determined from the associated 

thermal conduct ion coef ficients. The heat fluxes at 

the thermoelectric junctions are slightly more complicated. At these junctions the heat flux is due to the 

sum of thermal conduction, the Peltier effect, and 

joule heating (2). The joule heating within the generator is assumed to disperse evenly between the warm 

and cold junctions. 

The lumped-parameters used to describe the thermal and electrical properties of the generator are determined from: 

K ,. (Y A + Y A )COV n n p p (5) 

a_ fa I + I a I (6) n p 

p /y + p /Y 

R .. nnE e (7) COY 

By using the relationships between the temperatures: 

(8) 

2 

Tcj - Tw - f.Tw - f.T teg (9) 

l!r - T - l!r - trr - T (10) c w w teg c 

and the following expressions for the total resistance 

and TEG current: 

R

t .. R + R,t (11) 

~ 

I -~ R

t 

(12) 

Eqs. (1) through (4) can be solved for f.Tw' Eqs. (1) 

and (2) yield 

KATt + 2R iT f.T - Ra2 !!I:2 

eg t w teg teg (13) 

2R2 h + 2R J f.T 

t w t teg 

and Eqs. (3) and (4) produce the expression 

AT [2R2h (T -T ) +. 2R2 (h +K) AT + 2R a

2T f.T w tee 101 t c teg t w teg 

(14) 

By equating these two 

braic manipulation, 

ATteg/Rt results: 

expressions and performing algea third-order polynomial in 

ARl:tegY + B(2R,t+R>(:tegY 

t t 

with 

(16) 

(17 ) 

(18) 

(19) 

Although Eq. (15) is rather complicated, a numeric 

solution for f.Tteg can be calculated when the heat exchanger and TEG Parameters are known. Once the temperature difference across the TEG is known, the electrical power output is determined from the product of 

the current squared and the load resistance. 

(20) 

FIXED JUNCTION TEMPERATURE DESIGN PERFORMANCE -

The design of TEGs often neglects the temperature drop 
across the heat exchanger surface. For many higher 

temperature applications this assumption is reasonable. However, the temperature drop across the heat 

exchanger is determined by the TEG design, and illustration of this interaction is informative. 

For convenience, Eq. (15), which describes the 

TEG and heat exchanger interaction, is modified to express the TEG design by: 

(21) 

(22) 

(23) 

These expressions and Eqs. (5) and (7), are used 

to replace Rt ' R, and K. Defining 

(24 ) 

the resulting polynomial is 

nAZ3 + B(2n+l)Z2 

+ [2 Q (n+l)(h +h ) + 2a(n+l)(h +h )(m>' +>' ) w c w c n p 

+ P I;; + P ] Z + P 7:+ P .. 0 n p n p 

(25) 

The power output of a heat exchanger-TEG system is 

affected by numerous parameters. Since this paper has 

evo1yed from investigating TEG performance in an OTEC 

application, heat transfer coefficients and flow temperatures representative of OTEC systems are considered. Also, the values of the p and n materials approach current limits of technology. These parameter 

values are given in Table 1. 

a 

p 

Figure 

of merit 

Table 1 - Constants for OTEe 

Heat Exchanger-TEG Example 

Flow temperatures 

Heat Transfer Coefficients 

h - 0.3 w/cm2 ·C c 

Thermoelectric Properties 

n-type material 

0.143 x 10-3 v/·C 

0.79 x 10-3 1"1 cm 

8.1 x 10-3 1~/cm ·c 

p-type material 

0.173 x 10-3 v/·e 

0.87 x 10-3 1"1 cm 

8.7 x 10-3 W/cm °c 

-3 -1 3.4 x 10 !{ 

Now only the design parameters Y, n, ~nd m need to 

be investtg~ted. Two standard fixed junction temperature design methods are examined: maximum efficiency 

and maximum power constraints. They are: 

3 

for maximum efficiency 

(26) 

(27 ) 

and for maximum power 

(28) 

n .. 1 (29) 

Since no constraint is applied to Y, performance 

of the heat exchanger-TEG system is evaluated for a 

wide range of y. Results are illustrated in Figs. 2 

and 3. 

12 

-.10

1 

____ _ 

E ATleg 

~ 8 

--b 6 ~ 

:I( 

~ 4 

o == Il. 2 

20 

0-

° 

--

~~~'------------~-------------1~0~'-------------1~0' 

o<cm) 

Fig. 2 - Heat exchanger-TEG performance using fixedjunction-temperature maximum-efficiency 

constraints 

12 

.... E 10 20 

u 

~ 8 

.. ..... , (J 0 

~ 6 ~ 

:I( 

.. 10~ III 4 !q == 0 

Il. 

2 

0 

10-' 

o<cm) 

10' 

Fig. 3 - Heat exchanger-TEG performance using fixedjunction-temperature maximum-power 

constraints 

Inspection of Figs. 2 and 3 shows little variation in performance between maximum-efficiency and 

maximum-power fixed-june t ion-temperature constraints. 

However, for this calculation the largest value of 

electrical output power occurs when maximurn efficiency, not maximum power, constraints are used. This 
is an obvious error and illustrates one limitation of 

applying fixed temperature analysis on heat exchangerTEG systems with significant temperature drops across 

the heat exchanger surfaces_ Also, fixed junction 

temperature techniques do not address the important 

relationship between Y and AT teg - In practice, an acceptable value for Y that results in a satisfactory 

trade-off between heat exchanger losses and r.educed 

conversion efficiency may be found by trial and error. 

MAXIMm-! POWER CONDITIONS - To design a TEG imbedded within a heat exchanger that will deliver maximum 

power, appropriate values of Y, n, and m need to be 

identified _ The method of Lagrange multipliers is 

well-suited to this optimization problem. The problem 

restated is to maximize the power function 

(30) 

subject to the constraints that describe the heat exchanger-TEG system. 

An analytic solution for m can be readily found 

by using Eqs. (5), (7), and (15) as constraints on the 

power function. These equations are restated in terms 

of Y, m, R, R l' K, and Z. 

F 1 (Y,m,K) = (A m + A )Y - K .. 0 n p (31) 

= (:n + pp)/Y _ R '" 0 (32) 

+ D - 0 (33) 

The maximum occurs when 

Looking only at the partial derivatives of Eq. (34) 

with respect to m and Y yields 

. L2 P

n -L A Y + --- 0 (35) 1 n· rm2 

These two equations may be equated to yield 

m - (37) 

With the constraint on m satisfying the maximum 

power condition, a single constraint, G(Z,n,Y), on the 

power function is developed from Eq. (2?): 

G(Z,n,Y) ~ nAZ3 + B(2n + 1)Z2 

+ [en + 1)(2h h + YE)+ ~FC]z + Y ~ = 0 (38) w c F 

with the additional constantS;., 

4 

E so 2(h wen + h )(mA +- A ) p (39) 

(40) 

Eq. (38), the constraint on the power function, is dependent on Y, n, and Z. Power expressed in these variables is 

(41) 

Again, the maximum power condition must satisfy 

the Lagrange multiplier expression 

Vp(Z,n, Y) - Ll VG(Z,n, Y) '" 0 (42) 

This gradient expression and Eq. (38) yield the 

four nonlinear simultaneous equations: 

2a2

nFZ [ 2 hI (Z,n, Y,L) = -Y-- - Ll 3AnZ + 2B(2n+l)Z 

+ 2nh h + n'IE + Y(E + -Fe) + 2h h ] w ewe 

- 0 (43) 

(44 ) 

(45) 

(46) 

These expressions provide sufficient information 

to determine the values of Z, n, Y, and Ll that satisfy the maximum power condition. However, solution of 

these four nonlinear equations is extremely difficult 

analytically. Consequently, a numeric iterative approach is better suited. 

Newton's method has been used successfully. When 

an approximate solution of Z, n, Y, and Ll is known, 

an improved estimate is found by adding a correction 

to the current approximate solution. With 

(47) 

and 

(48) 

the correction y satisfies the linear simultaneous 

equations: 

[HJ' y s [HJ x (49) 

In Eq. (49), [H]' is the Jacobian matrix of [H] (3). 

Using the constraints applied previously on the 

heat exchanger, flow temperatures, and thermoelectric 

material, the maximum power solution has been calculated and is presented in Table 2. 

APPROXHlATE MAXIMUM POI."ER MODEL - Analysis of several heat exchanger-TEG systems has shown that maximum 

power generation occurs. when lITteg is approximately 
Table 2 - Maximum Power Operating Conditions 

for the Heat Exchanger-TEG Example 

TWj 

Y 

• 21.24"C Tcj • 11.23"C 

D • 1.399 

Power" 1.078 x 10-2 W/cm2 • 0.57367% 

equal to one-half of the total available temperature 

difference between the warm and cold flows and maximum 

efficiency fixed temperature constraints are applied 

to the TEG. This result, which is almost intuitively 

obvious, yields a quite simple approximate approach 

for designing a TEG to yield maxinum power. 

With n, the conversion efficiency, the TEG power 

output can be calculated from the heat flux passing 

through the warm Side of the heat exchanger 

(50) 

Applying conservation of energy through the TEG, the 

warm and cold heat fluxes are related by 

(51 ) 

This approximate maximum power solution uses the 

assumptions that 

(52 ) 

The additional assumption is that the TEG is designed 

to maximize efficiency at its operating temperature. 

Applying this constraint, the ratio of the load resistance to internal resistance is calculated from Eq. 

(27). The resulting conversion efficiency is known to 

be 

(53) 

which is the maximum efficiency possible for a TEG 

operating under fixed junction temperatures (2) •. 

Using Eqs. (10), (51), and (52) the resulting expression for the temperature drop across the warm flow 

heat exchange surface is 

lrr w 

h lJ'r c teg 

(54) 

In Eqs. (27) and (53), the junction temperatures 

are assumed to be known. This is not the case, but 

the solution can be rapidly found by iterating through 

Eqs. (27), (53), and (54) several times. The convergence progresses by setting the junction temperatures 

at T1o1 and Tc and then refining their estimates after 

I::.T has been calculated. Accurate values for n, 11, 

an~ I::.T are obtained in three or so iterations. After 

this c't;nvergence is completed, the necessary value of 

Y is calculated from . 

h NT l1(n+l)2 

y = ~w-=~w~ ____ _ 

iar2 n teg 

(55 ) 

This expression for l' is found by equating the power 

expressions (41) and (SO) and replacing Z by 

6T l' 

Z .. ~ F(n+1) (56 ) 

5 

Using this approximate design technique on -- the 

previously used example system yields results similar 

to 'the actual maximum power solution developed previously. These results are tabulated in Table 3. The 

operating point varies slightly, but power generation 

is reduced by only 0.0001%. Agreement 1s well within 

experimental tolerances, and little if any improvement 

is possible by using the exact maximum power solution. 

Table 3 - Comparison of Actual and 

Approximate Maximum Power Solutions 

Actual Maximum Power Approximate Maximum Power 

P • 10.78 X 10-3 W/cm2 P ~ 10.78 X 10-3 W/cm2 

l' • 8.29 cm y ,. 8.31 

n .. 1.40 n ,. 1.40 

lJ'rteg " 10.cloC &... • 10.cOoe "teg 

ADDITIONAL HEAT EXCHANGER COMMENTS - The preceding 

discussion considers TEG design when the heat transfer 

coefficients of the heat exchanger surfaces are known. 

To actually design a TEG for maximum power generation, 

the additional system losses need to be subtracted 

from the TEG output power. The major loss in an OTEC 

application is the pumping power. 

For heat exchangers, the heat transfer coefficient increases roughly linearly with Reynolds number, 

while pumping losses are more closely related to the 

cube of the Reynolds number. Figure 4 illustrates a 

typical relationship between gross TEG power and net 

system power for a range of Reynolds numbers. As is 

obvious from the graph, the net power generation of a 

heat exchanger-TEG system achieves a maximum at a specific Reynolds number. 

5 

...... 

!! 

C 4 ;, 

Q) 

~ 

iii 3 

~ 

--.... 

Q) 2 

== 0 

D. 

0 

10' 10' .10' 

Reynolds Number 

Fig. 4 - TEG power output and heat exchanger-TEG 

system power output 

Normally the Reynolds number for the warm and 

cold flows will differ at the point of maximum power 

generation. Since heat exchanger analysis is not well 

suited to analytic description, numeric trial and error of various Reynolds numbers for the warm and cold 

flows is a direct method for locating a desirable 

operating point. 

CONCLUSION 

Conventional fixed temperature TEG analysis techniques are inadequate for investigating heat exchanger-TEG s),stem performance. Analysis of such sys-
tems illustrates the relationship between the thermoelectric element area-to-length ratio and both the 

available temperature difference across the TEG and 

the resulting electric power generation. For applications, such as OTEC, where maximum power generation is 

desired, detailed and simplified numeric techniques 

for designing TEGs within heat exchanger-TEG systems 

are presented. 

6 

REFERENCES 

1. T. S. Jayadev, et al., "Thermoelectric OTEC," 6th 

OTEC Conference, Washington, D.C., June 1979. 

2. S. W. Angrist, "Direct Energy Conversion," Boston, 

Allyn and Bacon, Inc., 1971. 

3. J. M. Ortega, "Numerical Analysis: A Second 

Course," New York, Academic Press., 1972.