1

Thermoelectric Generators

HoSung Lee,

Nomenclature A cross-sectional area of thermoelement (m2)

COP the coefficient of performance, dimensionless

I electric current (A)

maxI maximum current (A)

j

electric current density vector (A/m2)

K thermal conductance (W/K)

L length of thermoelement (m)

k thermal conductivity (W/mK)

n the number of thermocouples

q

heat flux vector (W/m2)

cQ cooling power, heat absorbed at cold junction (W)

hQ heat liberated at hot junction (W)

maxcQ maximum cooling power (W)

R internal electrical resistance ()

nR total internal electrical resistance for a module ()

T temperature (°C)

cT low junction temperature (°C)

hT high junction temperature (°C)

T average temperature 2ch TT (°C)

V Voltage of a module (V)

maxV maximum voltage (V)

nW module power output (W)

x distance of thermoelement leg (m)

Z the figure of merit (K-1), kZ 2

T temperature difference ch TT (°C),

maxT maximum temperature difference (°C)

Greek symbols

Seebeck coefficient (V/K)

electrical resistivity (cm)

Subscript

2

p p-type element

n n-type element

Superscript

* effective quantity

1. Introduction

2. Formulation of Basic Equations

2.1 Basic Equations

In 1821, Thomas J. Seebeck discovered that an electromotive force or potential

difference could be produced by a circuit made from two dissimilar wires when one

junction was heated [1]. This is called the Seebeck effect. In 1834, Jean Peltier discovered

the reverse process that the passage of an electric current through a thermocouple

produces heating or cooling depended on its direction [2]. This is called the Peltier effect

(or Peltier cooling). In 1854, William Thomson discovered that if a temperature

difference exists between any two points of a current-carrying conductor, heat is either

absorbed or liberated depending on the direction of current and material [3]. This is called

the Thomson effect (or Thomson heat). These three effects are called the thermoelectric

effects.

Let us consider a non-uniformly heated thermoelectric material. For an isotropic

substance, the continuity equation for a constant current gives

0 j

(1)

The electric field E

is affected by the current density j

and the temperature

gradient T

. The coefficients are known according to the Ohm’s law and the Seebeck

effect [5]. The field is then expressed as

TjE

(2)

The heat flux q

is also affected by both the field E

and the temperature gradient T

.

However, the coefficients were not readily attainable at that time. Thomson in 1854

arrived at the relationship assuming that thermoelectric phenomena and thermal

conduction are independent [3]. Later, Onsager [4] supported that relationship by

presenting the reciprocal principle which was experimentally proved. The Thomson

relationship and the Onsager’s principle yielded the heat flow density vector (heat flux),

which is expressed as

TkjTq

(3)

3

The general heat diffusion equation is given by

t

Tcqq p

(4)

For steady state, we have

0 qq

(5)

where q is expressed by [5]

TjjjEq

2 (6)

Substituting Equations (3) and (6) in Equation (5) yields

02 TjdT

dTjTk

(7)

The Thomson coefficient , originally obtained from the Thomson relations, is written

dT

dT

(8)

In Equation (7), the first term is the thermal conduction, the second term is the Joule

heating, and the third term is the Thomson heat. Note that if the Seebeck coefficient is

independent of temperature, the Thomson coefficient is zero and then the Thomson heat

is absent. The above two equation governs the thermoelectric phenomena.

(a)

p

n

p

n

np

p

pn

Positive (+)

Negative (-)

Heat Absorbed

Heat Rejected

Electrical Conductor (copper)Electrical Insulator (Ceramic)

p-type Semiconcuctor

n-type Semiconductor

4

(b)

Figure 1. (a) Cutaway of a thermoelectric generator module, and (b) a p-type and n-type

thermocouple.

Consider a steady-state one-dimensional thermoelectric generator module in Figure

1a. The module consists of many p-type and n-type thermocouples as shown in Figure 1b.

We assume that the electrical and thermal contact resistances are negligible, the Seebeck

coefficient is independent of temperature, and the radiation and convection at the surfaces

of the elements are negligible. Then Equation (7) reduces to

02

A

I

dx

dTkA

dx

d (9)

The solution for the temperature gradient with two boundary conditions ( hx TT 0

and cLx TT ) is

L

TT

kA

LI

dx

dT ch

x

2

2

0 2

(10)

Equation (3) is expressed in terms of p-type and n-type thermoelements.

nxpx

cnphdx

dTkA

dx

dTkAITnQ

00

(11)

5

where hQ is the rate of heat absorbed at the hot junction. Substituting Equation (10) in

(11) gives

ch

n

nn

p

pp

n

nn

p

pp

hnph TTL

Ak

L

Ak

A

L

A

LIITnQ

2

2

1 (12)

Finally, the heat absorbed at the hot junction of temperature Th is expressed as

chhh TTKRIITnQ 2

2

1 (13)

where

np (14)

n

nn

p

pp

A

L

A

LR

(15)

n

nn

p

pp

L

Ak

L

AkK (16)

If we assume that p-type and n-type thermocouples are similar, we have that R =

L/A and K = kA/L, where = p + n and k = kp + kn. Equation (13) is called the ideal

equation which has been widely used in science and industry. The rate of heat liberated at

the cold junction is given by

chcc TTKRIITnQ 2

2

1 (17)

From the 1st law of thermodynamics across the thermoelectric module, which is

chn QQW . The power output is then expressed in terms of the internal properties as

RITTInW chn

2 (18)

However, the power output in Figure 1b can be defined by an external load

resistance as

Ln RnIW 2 (19)

Equating Equations (18) and (19) with nn IVW gives the voltage as

6

IRTTnnIRV chLn (20)

2.2 Performance Parameters of a Thermoelectric Module

From Equation (20), the electrical current for the module is obtained as

RR

TTI

L

ch

(21)

Note that the current I is independent of the number of thermocouples. Inserting this

into Equation (20) gives the voltage across the module by

R

R

R

R

TTnV L

L

chn

1

(22)

Inserting Equation (21) in Equation (19) gives the power output as

2

22

1

R

R

R

R

R

TTnW

L

L

chn

(23)

The thermal (or conversion) efficiency is defined as the ratio of the power output to

the heat absorbed at the hot junction:

h

nth

Q

W

(24)

Inserting Equations (13) and (23) into Equation (24) gives an expression for the

thermal efficiency:

c

h

cL

h

cL

L

h

c

th

ZT

T

T

R

R

T

T

R

R

R

R

T

T

2

1

12

11

1

(25)

7

where k

Z

2

or, equivalently, RK

Z2

.

2.3 Maximum Parameters for a Thermoelectric Generator Module

Since the maximum current inherently occurs at the short circuit where 0LR in

Equation (21), the maximum current for the module is

R

TTI ch

max (26)

The maximum voltage inherently occurs at the open circuit, where I = 0 in Equation

(20). The maximum voltage is

ch TTnV max (27)

The maximum power output is attained by differentiating the power output W in

Equation (23) with respect to the ratio of the load resistance to the internal resistance and

setting it to zero. The result yields a relationship of 1RRL , which leads to the

maximum power output as

R

TTnW ch

4

22

max

(28)

The maximum conversion efficiency can be obtained by differentiating the thermal

efficiency in Equation (25) with respect to the ratio of the load resistance to the internal

resistance and setting it to zero. The result yields a relationship of TZRRL 1 .

Then, the maximum conversion efficiency max is

h

ch

c

T

TTZ

TZ

T

T

1

111max (29)

where kZ 2 and T is the average temperature of cT and hT . On the basis of cT ,

TZ is expressed by

1

12 h

cc

T

TZTTZ (30)

8

There are so far four essential maximum parameters, which are maxI , maxV , maxW , and

max . However, there is also the maximum power efficiency. Most manufacturers have

been using the maximum power efficiency as a specification for their products. The

maximum power efficiency is obtained by letting 1RRL in Equation (25). The

maximum power efficiency mp is

c

h

c

h

c

h

c

mp

ZT

T

T

T

T

T

T

4

12

12

1

(31)

Note there are two thermal efficiencies: the maximum power efficiency mp and the

maximum conversion efficiency max .

2.4 Normalized Parameters

If we divide the active values by the maximum values, we can normalize the

characteristics of a thermoelectric generator. The normalized power output can be

obtained by dividing Equation (23) by Equation (28), which is

2

max1

4

R

R

R

R

W

W

L

L

(32)

Equations (21) and (26) give the normalized currents as

1

1

max

R

RI

I

L

(33)

Equations (22) and (27) give the normalized voltage as

1max

R

RR

R

V

V

L

L

n (34)

Equations (25) and (29) give the normalized thermal efficiency as

9

112

1

1

12

11

12

1

1

2

1

max

h

cc

c

h

cL

h

cL

h

c

h

ccL

th

T

TZT

ZT

T

T

R

R

T

T

R

R

T

T

T

TZT

R

R

(35)

Note that the above normalized values in Equations (32) – (34) are a function only of

RRL , while Equation (35) is a function of three parameters, which are hc TT , RRL and

cZT . Also, note that the present analysis is on the basis of cT .

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

max

th

maxW

W

maxV

V

maxI

I

R

RL

Figure 2. Normalized chart I, where Tc/Th = 0.7 and ZTc = 1 are used.

It is first noted, as shown in Figures 2 and 3, that the maximum power output and the

maximum conversion efficiency appear close each other with respect to RRL . mp

occurs at 1RRL , while max occurs approximately at 5.1RRL . The maximum

conversion efficiency max is presented in Figure 4 as a function of both the

dimensionless figure of merit (ZTc) and Tc/Th. Considering the conventional combustion

process (where the thermal efficiency is about 30%) where the high and low junction

temperatures would be typically at 1000 K and 400 K, which leads to Tc/Th = 0.4.

Therefore, in order to compete with the conventional way of the thermal conversion

(30%), the thermoelectric material should be at least ZTc = 3, which has been the goal.

10

Much development is needed when considering the current technology of thermoelectric

material of ZTc = 1. However, there is a strong potential that the nanotechnology would

provide a solution toward ZTc = 3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

1

2

3

4

5

maxW

W

max

th

maxW

W

R

RL

max

th

maxV

V

R

RL

maxV

V

maxI

I

Figure 3. Normalized chart II, where Tc/Th = 0.7 and ZTc = 1 are used.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Tc

Th

0.1

0.2

0.3

max 0.4

0.5

0.6

0.7

0.8

0.9

ZTc

Figure 4. The maximum conversion efficiency versus ZTc as a function of the

temperature ratio Tc/Th.

11

2.5 Effective Material Properties

As mentioned in Section 2.3, we have four maximum parameters ( maxI , maxV , maxW ,

and mp ), which are ideally a function of three material properties (, and k) with given

geometry (A/L) and two junction temperatures Th and Tc. Inversely, the three material

properties can ideally be expressed in terms of three among the four maximum

parameters. It turned out that, the two parameters ( maxI and mp ) are essential and any

one of maxV and maxW can be valid. In real world, the three material properties are difficult

to attain as the manufactures’ proprietary. Instead, the manufactures usually provide the

four measured maximum parameters which naturally include the thermal and electrical

contact resistances, the Thomson effect, and the radiation and convection losses. We wish

to deduce the three material properties from the four manufactures’ maximum parameters.

It is of interest to find that there would be no convergence of the three material properties

from the four measured maximum parameters because of the contradiction of the ideal

formulation and real measurements. This enforces us to choose one of the two parameters

( maxV and maxW ) and the essential two parameters ( maxI and mp ). We choose the

maximum power output instead of the maximum voltage because of the practical

importance. The effective material properties are defined here as the material properties

that are extracted from the maximum parameters provided by the manufacturers. The

effective electrical resistivity is obtained using Equations (26) and (28), which is

2max

max4

In

WLA (35)

The effective Seebeck coefficient can be obtained using Equation (26) and (35),

which is

ch TTnI

W

max

max4 (36)

Note that both the effective resistivity obtained and the maximum current equally

affect the Seebeck coefficient. The effective figure of merit is obtained from Equation

(29), which is

1

1

1

1

2

2

max

max

1

c

h

c

c

h

cc

T

T

T

TT

Z

(37)

where hcc TT1 which is the Carnot efficiency. Alternatively, the effective figure of

merit may be obtained from Equation (31) in terms of mp as

12

22

11

4

mp

c

h

c

c T

T

TZ

(38)

The effective thermal conductivity with *Z whichever is available from Equations

(37) or (38) is now obtained

Zk

2

(39)

The effective material properties include various effects such as the contact

resistances, Thomson effect, and radiation and convection. Hence, the effective figure of

merit appears slightly smaller than the intrinsic figure of merit as shown in Table 1. Note

that these effective properties should be divided by two for the single p-type or n-type

thermocouple.

13

Table.1

Description TEG Module (Bismuth Telluride)

Symbol Hi-Z

HZ-9

Tc = 50°C

Th = 230°C

Crystal

G-127-10-

05

Tc = 50°C

Th = 150°C

Kryotherm

TGM-199-

1.4-1.2

Tc = 50°C

Th = 150°C

Kryotherm

TGM-31-2.8-3.5

Tc = 50°C

Th = 280°C

# of thermocouples n 98 127 199 31

Intrinsic material

properties (provided

by manufacturer at

average

temperature)

V/K 189 - - -

cm 1.26 × 10-3 - - -

k (W/cmK) 1.13 × 10-2 - - -

ZTc 0.811 - - -

Effective material

properties

(calculated using

commercial Wmax,

Imax, and mp)

V/K 168.2 225.1 121.3 141.7

cm 1.563 × 10-3 0.677 × 10-3 0.854 × 10-3 0.946 × 10-3

k (W/cmK) 1.18 × 10-2 3.0 × 10-2 1.3 × 10-2 1.8 × 10-2

ZTc 0.497 0.806 0.428 0.381

Measured geometry

of thermoelement

A (mm2) 12 (1.0) 1.96 7.84

L (mm) 4.62 (1.17) 1.2 3.5

G=A/L (cm) 0.26 (0.085) 0.163 0.224

Dimension

(W×L×H)

mm 62.7 × 62.7 ×

6.5

30× 30 × 2.8 40 × 40 × 3.7 40 × 40 × 6.5

Manufacturers’

maximum

parameters

maxW (W) 7.46 4.06 2.8 3.9

Imax (A)a 5.03 2.84 2.32 7.72

Vmax (V)b 5.94 5.74 4.8 2.0

mp (%) 5.12 4.2 2.6 5.2

nR () -

module

1.18 2.03 2.1 0.26

Effective maximum

parameters

(calculated using

andk

maxW (W) 7.46 4.06 2.8 3.9

Imax (A)a 5.03 2.84 2.32 7.72

Vmax (V)b 5.93 5.72 4.83 2.0

mp (%) 5.1 4.2 2.6 5.2

nR () -

module

1.18 2.01 2.08 0.26

a Short circuit current b Open circuit voltage

14

Example E-1

We want to recover waste heat from the exhaust gas of a car using thermoelectric

generator (TEG) modules as shown in Figure E-1a. An array of N = 24 TEG modules is

installed on the exhaust of the car. Each module has n = 98 thermocouples that consist of

p-type and n-type thermoelements. Exhaust gases flow through the TEG device, wherein

one side of the modules experiences the exhaust gases while the other side of the modules

experiences coolant flows. These cause the hot and cold junction temperatures of the

modules to be at 230 °C and 50 °C, respectively. To maintain the junction temperatures,

the significant amount of heat should be absorbed at the hot junction and liberated at the

cold junction, which usually achieved by heat sinks. The material properties for the p-

type and n-type thermoelements are assumed to be similar as p = −n = 168 V/K, p =

n = 1.56 × 10-3 cm, and kp = kn = 1.18 × 10-2 W/cmK. The cross-sectional area and leg

length of the thermoelement are An = Ap = 12 mm2 and Ln = Lp = 4.6 mm, respectively,

which are shown in Figure E-1b.

(a) Per one TEG module, compute the electric current, the voltage, the maximum

power output, and the maximum power efficiency.

(b) For the whole TEG device, compute the maximum power output, the maximum

power efficiency, the maximum conversion efficiency and the total heat absorbed

at the hot junction.

(a) (b)

Figure E-1 (a) TEG device, (b) thermocouple.

Solution:

Material properties: =p − n = 336 × 10-6 V/K, =p + n = 3.12 × 10-5 m,

and k= kp + kn = 2.36 W/mK

The figure of merit is

13

5

262

10533.136.21012.3

10336

K

mKWm

KV

kZ

and

15

495.032310533.1 13 KKZTc

For the maximum power output, we use the condition of 1RRL . The internal

resistance R is

012.01012

106.41012.326

35

m

mm

A

LR

(a) For one TEG module:

Using Equation (21), the electric current per module is

A

KKKV

RR

TTI

L

ch 528.2012.0012.0

)27350(27323010336 6

Using Equation (22), the voltage per module is

V

KKKV

R

R

R

R

TTnV L

L

chn 964.2

11

)27350(2732301033698

1

6

Using Equation (23), the maximum power output is

W

KKKV

R

R

R

R

R

TTnW

L

L

chn 493.7

2012.0

3235031033698

1

2

226

2

22

Using Equation (31), the maximum power efficiency is

051.0

495.0

503

3234

503

3231

2

12

503

3231

4

12

12

1

K

K

K

K

K

K

ZT

T

T

T

T

T

T

c

h

c

h

c

h

c

mp

(b) For the whole TEG device:

The maximum power output is

16

WWWn 8.179493.724

The maximum power efficiency is same as the one for the module, so

051.0mp

Using Equation (29), the maximum conversion efficiency is

633.02

50332310533.1

2

13

KK

KTT

ZTZ hc

052.0

503

323633.01

1633.01

503

3231

1

111max

K

KK

K

T

TTZ

TZ

T

T

h

ch

c

The total heat absorbed is

WWW

Qmp

nh 525,3

051.0

8.179

References

[1] Seebeck T.J., Magnetic polarization of metals and minerals, Abhandlungen der

Deutschen Akademie der Wiessenschaften zu Berlin, 265-373, 1822

[2] Peltier J.C., Nouvelle experiences sur la caloricite des courans electrique, Ann.

Chim.LV1 371, 1834

[3] W . Thomson, Account of researchers in thermo-electricity, Philos. Mag. [5], 8, 62,

1854.

[4] Onsager L., Phys. Rev., 37, 405-526, 1931.

[5] Landau L.D., Lifshitz E.M., Elecrodynamics of continuous media, Pergamon Press,

Oxford, UK, 1960.

[6] Ioffe A.F., Semiconductor thermoelements and thermoelectric cooling, Infoserch

Limited, London, UK, 1957.

[7] D.M. Rowe, CRC Handbook of Thermoelectrics, CRC Press, Boca Raton, FL, USA,

1995.

[8] Lee H.S., Thermal design: heat sinks, thermoelectrics, heat pipes, compact heat

exchangers, and solar cells, John Wiley & Sons, Inc., Hoboken, New Jersey, USA, 2010.

[9] Goldsmid H.J., Introduction to thermoelectricity, Spriner, Heidelberg, Germany, 2010.

[10] Nolas G.S., Sharp J., Goldsmid H.J., Thermoelectrics, Springer, Heidelberg,

Germany, 2001.

17

Problem P-1

NASA’s Curiosity rover is working (February, 2013) on the Mars surface to collect a

sample of bedrock that might offer evidence of a long-gone wet environment, as shown

in Figure P-1a. In order to provide the electric power for the work, a radioisotope

thermoelectric generator (RTG) wherein Plutonium fuel pellets provide thermal energy is

used. The p-type and n-type thermoelements are assumed to be similar and to have the

dimensions as the cross-sectional area A = 0.196 cm2 and the leg length L = 1 cm. The

thermoelectric material used is lead telluride (PbTe) having p = −n = 187 V/K, p =

n = 1.64 × 10-3 cm, and kp = kn = 1.46 × 10-2 W/cmK. The hot and cold junction

temperatures are at 815 K and 483 K, respectively. If the power output of 123 W is

required to fulfill the work, estimate the number of thermocouples, the maximum power

efficiency and the rate of heat liberated at the cold junction of the RTG.

(a) (b)

Figure P-1. (a) Curiosity rover on Mars, (b) p-type and n-type thermoelements.

18

Problem P-1-2

We want to recover waste heat from the exhaust gas of a car using thermoelectric

generator (TEG) modules as shown in Figure P-1-2a. An array of N = 36 TEG modules is

installed on the exhaust of the car. Each module has n = 127 thermocouples that consist

of p-type and n-type thermoelements. Exhaust gases flow through the TEG device,

wherein one side of the modules experiences the exhaust gases while the other side of the

modules experiences coolant flows. These cause the hot and cold junction temperatures

of the modules to be at 230 °C and 50 °C, respectively. To maintain the junction

temperatures, the significant amount of heat should be absorbed at the hot junction and

liberated at the cold junction, which usually achieved by heat sinks. The material

properties for the p-type and n-type thermoelements are assumed to be similar. The most

appropriate module of TG12-4 for this work found in the commercial products shows the

maximum parameters rather than the material properties as the number of couples of 127,

the maximum power of 4.05 W, the short circuit current of 1.71 A, the maximum

efficiency of 4.97 %, and the open circuit voltage of 9.45 V. The cross-sectional area and

leg length of the thermoelement are An = Ap = 1.0 mm2 and Ln = Lp = 1.17 mm,

respectively, which are shown in Figure P-1b.

(a) Estimate the effective material properties: the Seebeck coefficient, the electrical

resistivity, and the thermal conductivity.

(b) Per one TEG module, compute the electric current, the voltage, the maximum

power output, and the maximum power efficiency.

(c) For the whole TEG device, compute the maximum power output, the maximum

power efficiency, the maximum conversion efficiency and the total heat absorbed

at the hot junction.

(a) (b)

Figure P-1-2 (a) TEG device, (b) thermocouple.