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1
Thermoelectric Generators
HoSung Lee,
Nomenclature
cross-sectional area of thermoelement (m2)
A
the coefficient of performance, dimensionless
COP
I
electric current (A)
I max
maximum current (A)

electric current density vector (A/m2)
j
K
thermal conductance (W/K)
L
length of thermoelement (m)
thermal conductivity (W/mK)
k
the number of thermocouples
n

q
heat flux vector (W/m2)
cooling power, heat absorbed at cold junction (W)
Q
c
Q h
Q
c max
R
Rn
T
Tc
Th
T
V
Vmax
W
n
x
Z
T
Tmax
heat liberated at hot junction (W)
maximum cooling power (W)
internal electrical resistance ()
total internal electrical resistance for a module ()
temperature (°C)
low junction temperature (°C)
high junction temperature (°C)
average temperature Th  Tc  2 (°C)
Voltage of a module (V)
maximum voltage (V)
module power output (W)
distance of thermoelement leg (m)
the figure of merit (K-1), Z   2  k
temperature difference Th  Tc (°C),
maximum temperature difference (°C)
Greek symbols

Seebeck coefficient (V/K)

electrical resistivity (cm)
Subscript
2
p
n
p-type element
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
 
 j  0
(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
 

E  j   T
(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



q  Tj  kT
(3)
3
The general heat diffusion equation is given by
 
T
   q  q  c p
t
(4)
For steady state, we have
 
   q  q  0
(5)
where q is expressed by [5]
 
 
q  E  j  j 2   j  T
(6)
Substituting Equations (3) and (6) in Equation (5) yields


 
d  
  kT  j 2   T
j  T  0
dT
(7)
The Thomson coefficient , originally obtained from the Thomson relations, is written
 T
d
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.
Heat Absorbed
p
n
p
n-type Semiconductor
n
p
p-type Semiconcuctor
p
n
p
Negative (-)
Electrical Insulator (Ceramic)
(a)
Heat Rejected
n
Positive (+)
Electrical Conductor (copper)
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
d  dT  I 2 
0
 kA  
dx  dx 
A
(9)
The solution for the temperature gradient with two boundary conditions ( Tx 0  Th
and Tx  L  Tc ) is
dT
dx

x 0
I 2 L Th  Tc

2 A2 k
L
(10)
Equation (3) is expressed in terms of p-type and n-type thermoelements.


dT
Q h  n  p   n Tc I    kA
dx




dT
    kA
dx
x 0  p

 
 
x 0  n 

(11)
5
where Q h is the rate of heat absorbed at the hot junction. Substituting Equation (10) in
(11) gives


1   p L p  n Ln   k p Ap k n An 

Q h  n  p   n Th I  I 2 



Th  Tc  (12)
2  Ap
An   L p
Ln 


Finally, the heat absorbed at the hot junction of temperature Th is expressed as
1


Q h  n Th I  I 2 R  K Th  Tc 
2


(13)
   p  n
(14)
where
R
K
 p Lp
Ap
k p Ap
Lp


 n Ln
(15)
An
k n An
Ln
(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
1


Q c  n Tc I  I 2 R  K Th  Tc 
2


(17)
From the 1st law of thermodynamics across the thermoelectric module, which is
W n  Q h  Q c . The power output is then expressed in terms of the internal properties as

W n  n I Th  Tc   I 2 R

(18)
However, the power output in Figure 1b can be defined by an external load
resistance as
W n  nI 2 RL
Equating Equations (18) and (19) with W n  IVn gives the voltage as
(19)
6
Vn  nIRL  n Th  Tc   IR 
(20)
2.2 Performance Parameters of a Thermoelectric Module
From Equation (20), the electrical current for the module is obtained as
I
 Th  Tc 
(21)
RL  R
Note that the current I is independent of the number of thermocouples. Inserting this
into Equation (20) gives the voltage across the module by
Vn 
n Th  Tc   RL 
 
RL
R
1  
R
(22)
Inserting Equation (21) in Equation (19) gives the power output as
n Th  Tc 
W n 
R
2
2
RL
R
2
 RL 
1 

R 

(23)
The thermal (or conversion) efficiency is defined as the ratio of the power output to
the heat absorbed at the hot junction:
W
th   n
Qh
(24)
Inserting Equations (13) and (23) into Equation (24) gives an expression for the
thermal efficiency:
 Tc  RL
1  
 Th  R
 th 
2
 RL  Tc
1 

R  Th
 RL  1  Tc  
1 
  1   
R  2  Th 
ZTc

(25)
7
where Z 
2
2
or, equivalently, Z 
.
k
RK
2.3 Maximum Parameters for a Thermoelectric Generator Module
Since the maximum current inherently occurs at the short circuit where RL  0 in
Equation (21), the maximum current for the module is
I max 
 Th  Tc 
(26)
R
The maximum voltage inherently occurs at the open circuit, where I = 0 in Equation
(20). The maximum voltage is
Vmax  n Th  Tc 
(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 RL R  1 , which leads to the
maximum power output as
n 2 Th  Tc 
W max 
4R
2
(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 RL R  1  ZT .
Then, the maximum conversion efficiency  max is

T 

h
1  ZT  1
 1  ZT  Tc
Th
 max  1  c 
T
(29)
where Z   2  k and T is the average temperature of Tc and Th . On the basis of Tc ,
ZT is expressed by
ZT
ZT  c
2
  T  1 
1   c  
  Th  


(30)
8
There are so far four essential maximum parameters, which are I max , Vmax , W max , 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 R L R  1 in Equation (25). The
maximum power efficiency  mp is
1
mp 
Tc
Th
T
4 c
1 T 
T
2  1  c   h
2  Th  ZTc
(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
W
W

max
4
RL
R
 RL

 1

 R

2
(32)
Equations (21) and (26) give the normalized currents as
I

1
RL
1
R
Equations (22) and (27) give the normalized voltage as
I max
RL
Vn
 R
Vmax RL
1
R
Equations (25) and (29) give the normalized thermal efficiency as
(33)
(34)
9
 th
 max
1


ZTc   Tc   Tc 
RL 
1
1   
R 
2   Th   Th 





2


 RL
 T
 1 c 


ZTc   Tc
 Th 
 RL  1  1 1  Tc    R
1

1 


 R

ZTc
2   Th
 2  Th 





(35)



1
 
 1
 
 
Note that the above normalized values in Equations (32) – (34) are a function only of
RL R , while Equation (35) is a function of three parameters, which are Tc Th , RL R and
ZTc . Also, note that the present analysis is on the basis of Tc .
1
th
max
W

Wmax
0.9
0.8
0.7
V
Vmax
0.6
0.5
0.4
I
0.3
I max
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
RL
R
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 RL R .  mp
occurs at R L R  1 , while  max occurs approximately at RL R  1.5 . 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.
1
5
0.9
th
max
0.8
W

Wmax
0.7
W

Wmax
4
0.6
th
max
3
0.5
V
Vmax
0.4
0.3
V
Vmax
2
RL
R
0.2
RL
R
1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
I
I max
Figure 3. Normalized chart II, where Tc/Th = 0.7 and ZTc = 1 are used.
0.7
Tc
Th
0.6
0.1
0.2
0.5
0.3
0.4
max
0.4
0.3
0.5
0.2
0.6
0.7
0.8
0.9
0.1
0
0
0.5
1
1.5
2
2.5
3
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 ( I max , Vmax , W max ,
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 ( I max and  mp ) are essential and any
one of Vmax and W max 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
( Vmax and W max ) and the essential two parameters ( I max 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
 
4 A L W max
2
n I max 
(35)
The effective Seebeck coefficient can be obtained using Equation (26) and (35),
which is
 
4W max
nI max Th  Tc 
(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
  max Tc
 1 
 c Th
2


Z 
1 
 max
 T  
Tc 1   c    1  
c
  Th   
2




  1






(37)
where  c  1  Tc Th which is the Carnot efficiency. Alternatively, the effective figure of
merit may be obtained from Equation (31) in terms of  mp as
12
 Tc 
 
 Th 
Z 
 1
1
 c 
 2

  mp 2 
4
Tc
(38)
The effective thermal conductivity with Z * whichever is available from Equations
(37) or (38) is now obtained
2

k   
 Z

(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
# of thermocouples
Intrinsic material
properties (provided
by manufacturer at
average
temperature)
Effective material
properties
(calculated using
commercial Wmax,
Imax, and mp)
Measured geometry
of thermoelement
Dimension
(W×L×H)
Manufacturers’
maximum
parameters
Effective maximum
parameters
(calculated using
andk
a
b
TEG Module (Bismuth Telluride)
Symbol
Hi-Z
Crystal
HZ-9
G-127-10Tc = 50°C
05
Th = 230°C
Tc = 50°C
Th = 150°C
n
98
127
189
V/K
-3
1.26
×
10
cm
-2
k (W/cmK)
1.13 × 10
ZTc
0.811
-
V/K
cm
Kryotherm
TGM-1991.4-1.2
Tc = 50°C
Th = 150°C
199
-
Kryotherm
TGM-31-2.8-3.5
Tc = 50°C
Th = 280°C
31
-
168.2
1.563 × 10-3
1.18 × 10-2
0.497
225.1
0.677 × 10-3
3.0 × 10-2
0.806
121.3
0.854 × 10-3
1.3 × 10-2
0.428
141.7
0.946 × 10-3
1.8 × 10-2
0.381
12
4.62
0.26
62.7 × 62.7 ×
6.5
7.46
(1.0)
(1.17)
(0.085)
30× 30 × 2.8
1.96
1.2
0.163
40 × 40 × 3.7
7.84
3.5
0.224
40 × 40 × 6.5
4.06
2.8
3.9
5.03
5.94
5.12
1.18
2.84
5.74
4.2
2.03
2.32
4.8
2.6
2.1
7.72
2.0
5.2
0.26
W max (W)
7.46
4.06
2.8
3.9
Imax (A)a
Vmax (V)b
mp (%)
nR ()
module
5.03
5.93
5.1
1.18
2.84
5.72
4.2
2.01
2.32
4.83
2.6
2.08
7.72
2.0
5.2
0.26
k (W/cmK)
ZTc
A (mm2)
L (mm)
G=A/L (cm)
mm
W max (W)
Imax (A)a
Vmax (V)b
mp (%)
nR ()
module
Short circuit current
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 ptype 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)
Figure E-1 (a) TEG device, (b) thermocouple.
(b)
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
336 106 V K 
2
Z

 1.533  103 K 1
5
k 3.12  10 m 2.36W mK 
2
and
15
ZTc  1.533  103 K 1 323K   0.495
For the maximum power output, we use the condition of RL R  1 . The internal
resistance R is
R
L
A

3.12 10
m 4.6  10 3 m 
 0.012
12  10 6 m 2
5
(a) For one TEG module:
Using Equation (21), the electric current per module is
I
 Th  Tc 
RL  R

336  10
6
V K 230  273K  (50  273) K 
 2.528 A
0.012  0.012
Using Equation (22), the voltage per module is
Vn 
n Th  Tc   RL  98  336  10 6 V K 230  273K  (50  273) K 
 2.964V
 
RL
R 
11

1
R
Using Equation (23), the maximum power output is
RL
2
2
2
2


n

T

T
98  336  10 6 V K  503K  323K 
h
c
R

Wn 

 7.493W
2
R
0.012  2 2
 RL 
1 

R 

Using Equation (31), the maximum power efficiency is
1
 mp 
Tc
Th
T
4 c
T
1 T 
2  1  c   h
2  Th  ZTc

1
323K
503K
323K
4
1  323K 
503K
2  1 

2  503K  0.495
(b) For the whole TEG device:
The maximum power output is
 0.051
16
W n  24  7.493W  179.8W
The maximum power efficiency is same as the one for the module, so
mp  0.051
Using Equation (29), the maximum conversion efficiency is
 T  Th 
3
1  323K  503K 
ZT  Z  c
  1.533  10 K 
  0.633
2


 2 

T 
1  ZT  1  323K 
1  0.633  1
 1 
 0.052

T
323
K
503
K


c
h  1  ZT 

1  0.633 
503K
Th
The total heat absorbed is
 max  1  c 
T
W
179.8W
Q h  n 
 3,525W
 mp
0.051
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)
Figure P-1-2 (a) TEG device, (b) thermocouple.
(b)
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