1

Thermoelectric Generators

HoSung Lee,

Nomenclature A cross-sectional area of thermoelement (m2) COP the coefficient of performance, dimensionless I electric current (A) I maximum current (A) max 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)  Qc cooling power, heat absorbed at cold junction (W)  Qh heat liberated at hot junction (W)  Qc max maximum cooling power (W) R internal electrical resistance ()

Rn total internal electrical resistance for a module () T temperature (°C)

Tc low junction temperature (°C)

Th high junction temperature (°C)

T average temperature Th  Tc  2 (°C) V Voltage of a module (V)

Vmax maximum voltage (V)  Wn module power output (W) x distance of thermoelement leg (m) Z the figure of merit (K-1), Z   2 k

T temperature difference Th  Tc (°C),

Tmax 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

    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 . 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)

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The general heat diffusion equation is given by

  T    q  q  c (4) p t

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 (7) dT

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

d   T (8) dT

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 n p Positive (+) p n p-type Semiconcuctor p

Negative (-) Electrical Conductor (copper) Electrical Insulator (Ceramic) Heat Rejected

(a) 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  kA    0 (9) dx  dx  A

The solution for the temperature gradient with two boundary conditions (Tx0  Th and TxL  Tc ) is

2 dT I L Th  Tc  2  (10) dx x0 2A k L

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

  dT   dT   Q  n   T I   kA    kA   (11) h p n c  dx   dx    x0  p  x0 n  5

 where Qh is the rate of heat absorbed at the hot junction. Substituting Equation (10) in (11) gives

 1   L  L   k A k A   Q  n   T I  I 2  p p  n n    p p  n n T  T  (12) h p n h 2  A A   L L  h c   p n   p n  

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

  1 2  Qh  n Th I  I R  KTh  Tc  (13)  2  where

   p n (14)

 L  L R  p p  n n (15) Ap An

k A k A K  p p  n n (16) Lp Ln

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 2  Qc  n Tc I  I R  KTh  Tc  (17)  2 

From the 1st law of thermodynamics across the thermoelectric module, which is    Wn  Qh  Qc . The power output is then expressed in terms of the internal properties as

 2 Wn  nITh  Tc  I R (18)

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

 2 Wn  nI RL (19)

 Equating Equations (18) and (19) with Wn  IVn gives the voltage as 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

T T  I  h c (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

nT  T  R  V  h c  L  (22) n R R L 1   R

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

RL 2 2  n Th  Tc  R Wn  2 (23) R  R  1 L   R 

The thermal (or conversion) efficiency is defined as the ratio of the power output to the heat absorbed at the hot junction:

W   n (24) th  Qh

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

 Tc  RL 1   Th  R th  2 (25)  R  T 1 L  c  RL  1  Tc   R  Th 1   1    R  2  Th  ZTc

7

 2  2 where Z  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

T  T  I  h c (26) max 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 T T 2 W  h c (28) max 4R

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  1 ZT 1   1 c  (29) max  T  T  h  1 ZT  c Th

2 where Z   k and T is the average temperature of Tc and Th . On the basis of Tc , ZT is expressed by

1 ZT   T   ZT  c 1  c   (30) 2   T     h  

8

 There are so far four essential maximum parameters, which are I max , Vmax , Wmax , 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 RL R  1 in Equation (25). The maximum power efficiency mp is

T 1 c T   h mp T (31) 4 c 1  Tc  Th 2  1   2  Th  ZTc

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

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

R 4 L W  R (32)  2 W max  R   L 1  R 

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

I 1  (33) I R max L 1 R Equations (22) and (27) give the normalized voltage as

RL V n  R (34) V R max L 1 R

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

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 1  R  ZT   T   T  L 1 c 1  c    c R  2   T   T      h   h  th    (35)   2  max  RL  Tc   1  1   R  1  T   R  T  ZT   T     L 1  1 c   h  1 c 1  c   1           R  2  Th  ZTc  2  Th         

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 , and

ZTc . Also, note that the present analysis is on the basis of Tc .

1 th 0.9 W   max 0.8 Wmax

0.7 V

0.6 Vmax

0.5

0.4

0.3 I I 0.2 max

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 . mp occurs at RL R  1 , while max occurs approximately at RL R 1.5 . The maximum conversion efficiency 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 W   Wmax 0.8 th 4 W max  Wmax 0.7 0.6 3 R  L th 0.5 V R max V 0.4 max 2

0.3 V RL 0.2 R 1 Vmax 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 I I max Figure 3. Normalized chart II, where Tc/Th = 0.7 and ZTc = 1 are used.

0.7 Tc 0.1 0.6 Th

0.5 0.2

0.4 0.3  max 0.4 0.3 0.5 0.2 0.6

0.1 0.7 0.8 0.9 0 0 0.5 1 1.5 2 2.5 3 ZT c Figure 4. The maximum conversion efficiency versus ZTc as a function of the temperature ratio Tc/Th.

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2.5 Effective Material Properties  As mentioned in Section 2.3, we have four maximum parameters ( I max , Vmax , Wmax , 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 ( and ) are essential and any one of and 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 ( and ) and the essential two parameters ( and ). 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Wmax   2 (35) nI max 

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

4W    max (36) nImax Th  Tc 

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

2   T   1 max c   2  T Z    c h  1 (37) 1       T   max c  1   Tc 1      T   c     h  

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

4  T   c  T  T  Z   c  h  (38)  1 1       2 c   mp 2 

The effective thermal conductivity with Z * whichever is available from Equations (37) or (38) is now obtained

2   k   (39)  Z 

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 Crystal Kryotherm Kryotherm HZ-9 G-127-10- TGM-199- TGM-31-2.8-3.5 Tc = 50°C 05 1.4-1.2 Tc = 50°C Th = 230°C Tc = 50°C Tc = 50°C Th = 280°C Th = 150°C Th = 150°C # of thermocouples n 98 127 199 31 Intrinsic material V/K 189 - - - properties (provided cm 1.26 × 10-3 - - - by manufacturer at k (W/cmK) 1.13 × 10-2 - - - average ZTc 0.811 - - - temperature) Effective material V/K 168.2 225.1 121.3 141.7 properties cm 1.563 × 10-3 0.677 × 10-3 0.854 × 10-3 0.946 × 10-3 (calculated using k (W/cmK) 1.18 × 10-2 3.0 × 10-2 1.3 × 10-2 1.8 × 10-2 commercial Wmax, ZTc 0.497 0.806 0.428 0.381 Imax, and mp) Measured geometry A (mm2) 12 (1.0) 1.96 7.84 of thermoelement L (mm) 4.62 (1.17) 1.2 3.5 G=A/L (cm) 0.26 (0.085) 0.163 0.224 Dimension mm 62.7 × 62.7 × 30× 30 × 2.8 40 × 40 × 3.7 40 × 40 × 6.5 (W×L×H) 6.5 Manufacturers’ W (W) 7.46 4.06 2.8 3.9 maximum max a parameters Imax (A) 5.03 2.84 2.32 7.72 b Vmax (V) 5.94 5.74 4.8 2.0

mp (%) 5.12 4.2 2.6 5.2 nR () - 1.18 2.03 2.1 0.26 module Effective maximum (W) 7.46 4.06 2.8 3.9 parameters a (calculated using Imax (A) 5.03 2.84 2.32 7.72 b andk Vmax (V) 5.93 5.72 4.83 2.0

mp (%) 5.1 4.2 2.6 5.2 nR () - 1.18 2.01 2.08 0.26 module 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 = -3 -2 n = 1.56 × 10 cm, and kp = kn = 1.18 × 10 W/cmK. The cross-sectional area and leg 2 length of the thermoelement are An = Ap = 12 mm 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: -6 -5 Material properties:  =p − n = 336 × 10 V/K,  =p + n = 3.12 × 10 m, and k= kp + kn = 2.36 W/mK

The figure of merit is

2 6 2  33610 V K 3 1 Z    1.53310 K k 3.12105 m2.36W mK  and 15

3 1 ZTc  1.53310 K 323K  0.495

For the maximum power output, we use the condition of RL R  1 . The internal resistance R is

L 3.12105 m4.6103 m R    0.012 A 12106 m2

(a) For one TEG module:

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

T  T  336106 V K230 273K  (50  273)K I  h c   2.528A RL  R 0.012  0.012

Using Equation (22), the voltage per module is

nT  T  R  98 336106 V K 230 273K  (50  273)K V  h c  L    2.964V n R R 11 L 1   R

Using Equation (23), the maximum power output is

RL 2 2 6 2 2  n Th  Tc  R 98 33610 V K  503K  323K Wn  2  2  7.493W R  R  0.012  2 1 L   R  Using Equation (31), the maximum power efficiency is

Tc 323K 1 1 T   h  503K  0.051 mp T 323K 4 c 4 1  T  T 1  323K  503K 2  1 c   h 2  1     2  503K  0.495 2  Th  ZTc

(b) For the whole TEG device:

The maximum power output is 16

 Wn  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  T  3 1  323K  503K  ZT  Z c h   1.53310 K    0.633  2   2 

 T  1 ZT 1  323K  1 0.633 1   1 c   1   0.052 max  T  T 503K 323K  h  1 ZT  c   1 0.633  Th 503K The total heat absorbed is

  Wn 179.8W Qh    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 = -3 -2 n = 1.64 × 10 cm, and kp = kn = 1.46 × 10 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 2 leg length of the thermoelement are An = Ap = 1.0 mm 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.