2-1

Chapter 2 Thermoelectric Generators

2.1 Ideal 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. 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 [1]. 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 [2]. 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

∇⃑⃑ ∙ 푗 = 0 (2.1) 2-2

   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 [3]. The electric field is then expressed as

   E  j T (2.2)

  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 [2]. 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 (2.3) which is the most important equation in thermoelectrics (will be discussed later in detail). The general heat diffusion equation is given by

  T (2.4)    q  q  c p t

For steady state, we have

     q  q  0 (2.5) where q is expressed by [3]

2-3

    q  E  j  j 2   j T (2.6)

Substituting Equation (2.3) and (2.6) into (2.5) yields

  d   (2.7) kT  j 2  T j T  0 dT

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

d (2.8)   T dT

In Equation (2.7), 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  becomes zero and then the Thomson heat is absent. The two equations (2.3) and (2.7) 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

Figure 2.1 Cutaway of a thermoelectric generator module 2-4

(b)

Figure 2.2 p- and n-type unit thermocouple for a thermoelectric generator.

Consider a steady-state one-dimensional thermoelectric generator module as shown in Figure 2.1. The module consists of many p-type and n-type thermocouples, where one thermocouple (unicouple) with a circuit is shown in Figure 2.2. 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 (2.7) reduces to

d  dT  I 2  (2.9) kA    0 dx  dx  A 2-5

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

TxL  Tc ) in Figure 2.2 is derived as 2 (2.10) dT I L Th  Tc  2  dx x0 2A k L

Equation (2.3) is expressed in terms of p-type and n-type thermoelements.

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

 where Qh is the rate of heat absorbed at the hot junction in Figure 2.2 and n is the number of thermocouples. Substituting Equation (2.10) into (2.11) gives

 1   L  L   k A k A   (2.12) Q  n   T I  I 2  p p  n n    p p  n n T  T  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

(2.13)   1 2  Qh  n Th I  I R  KTh  Tc   2  where

   p n (2.14)

 L  L (2.15) R  p p  n n Ap An

k A k A (2.16) K  p p  n n Lp Ln where R is the electrical resistance and K is the thermal conductance. 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 2-6

and k = kp + kn. Equation (2.13) 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

(2.17)   1 2  Qc  n Tc I  I R  KTh  Tc   2 

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

 2 (2.18) Wn  nITh  Tc  I R

However, the total power output in Figure 2.2 can be defined by an external load resistance as

 2 (2.19) Wn  nI RL

 Equating Equations (2.18) and (2.19) with Wn  IVn gives the total voltage as

Vn  nIRL  nTh Tc  IR (2.20)

2.2 Performance Parameters of a Thermoelectric Module

From Equation (2.20), the electrical current for the module is obtained as

T T  (2.21) I  h c RL  R

Note that the current I is independent of the number of thermocouples. Inserting this into Equation (2.20) gives the voltage across the module by

2-7

nT  T  R  (2.22) V  h c  L  n R R L 1   R

Inserting Equation (2.21) in Equation (2.19) gives the power output as

RL (2.23) 2 2  n Th  Tc  R Wn  2 R  R  1 L   R 

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

W (2.24)   n th  Qh

Inserting Equations (2.13) and (2.23) into Equation (2.24) gives an expression for the conversion efficiency:

푇푐 푅퐿 (2.25) (1 − 푇 ) 푅 휂 = ℎ 푡ℎ 푅 1 푇 1 푅 2 푇 (1 − 퐿) − (1 − 푐 ) + (1 − 퐿) (1 + 푐 ) 푅 2 푇ℎ 2푍푇̅ 푅 푇ℎ

푇 +푇 Where the average temperature is defined as 푇̅ = ℎ 푐 . It is noted that the Carnot cycle 2 efficiency 휂푐 = (1 − 푇푐⁄푇ℎ).

2-8

2.3 Maximum Parameters for a thermoelectric Module

Since the maximum current inherently occurs at the short circuit where RL  0 in Equation (2.21), the maximum current for the module is

T  T  (2.26) I  h c max R

The maximum voltage inherently occurs at the open circuit where I = 0 in Equation (2.20). The maximum voltage is

Vmax  nTh  Tc  (2.27)

The maximum power output is attained by differentiating the power output W in Equation (2.23) 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 (2.28) W  h c max 4R

The maximum conversion efficiency can be obtained by differentiating the conversion efficiency in Equation (2.25) 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 (2.29)   1 c  max  T  T  h  1 ZT  c Th 2-9

 There are total four essential maximum parameters, which are I max , Vmax , Wmax , and max . However, there is also the maximum power efficiency. The maximum power efficiency is obtained by letting RL R  1in Equation (2.25). The maximum power efficiency mp is

푇푐 (2.30) (1 − 푇 ) 휂 = ℎ 푚푝 1 푇 2 푇 2 − (1 − 푐 ) + (1 + 푐 ) 2 푇ℎ 푍푇̅ 푇ℎ

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

2.4 Normalized Parameters

If we divide the actual values by the maximum values, we can normalize the characteristics of a thermoelectric generator. The normalized power output can be obtained by dividing Equation (2.23) by Equation (2.28), which leads to

R (2.31) 4 L W  R  2 W max  R   L 1  R 

Equations (2.21) and (2.26) give the normalized currents as

I 1 (2.32)  I R max L 1 R

Equations (2.22) and (2.27) give the normalized voltage as 2-10

RL (2.33) V n  R V R max L 1 R

Equations (2.25) and (2.29) give the normalized thermal efficiency as

푅퐿 √ ̅ 푇푐 (2.34) 휂 푅 ( 1 + 푍푇 + 푇 ) 푡ℎ = ℎ 휂 푅 1 푇 1 푅 2 푇 푚푎푥 [( 퐿 + 1) − (1 − 푐 ) + ( 퐿 + 1) (1 + 푐 )] (√1 + 푍푇̅ − 1) 푅 2 푇ℎ 2푍푇̅ 푅 푇ℎ

Note that the normalized values in Equations (2.31) – (2.33) are a function only of RL R , while ̅ Equation (2.34) is a function of three parameters, which are Tc Th , and 푍푇.

1 th 0.9  W   max 0.8 Wmax 0.7 V 0.6 Vmax 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 2.3 Normalized chart I for thermoelectric generators, where Tc/Th = 0.7 and 푍푇̅ = 1 are used.

It is noted, as shown in Figure 2.3, that the maximum power output and the maximum conversion efficiency appear close each other with respect to . The maximum power 푊̇푚푎푥 occurs at 2-11

RL R  1, while max occurs approximately at RL R 1.5. The various parameters are presented against the normalized current, which is shown in Figure 2.4. This plot is often used as a specification of the commercial module. Note that the current indicates half the maximum current for the maximum power output. The maximum conversion efficiency is presented in Figure

2.5 as a function of both the dimensionless figure of merit (푍푇̅) and Tc/Th. Considering the conventional combustion process (where the thermal efficiency is about 30%) where the high and low junction temperatures would be at 1500 K and 500 K, which leads to Tc/Th = 0.3. Therefore, in order to compete with the conventional way of the thermal efficiency (30%), the thermoelectric material should be at least 푍푇̅ = 3, which has been the goal in this field. Much development is needed when considering the current technology of thermoelectric material of 푍푇̅ = 1. However, it is thought that there is a strong potential that nanotechnology would contribute to 푍푇̅ = 3 in near future.

Figure 2.4 Normalized chart II for thermoelectric generators, where Tc/Th = 0.7 and 푍푇̅ = 1 are used.

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Figure 2.5 The maximum conversion efficiency versus 푍푇̅ as a function of the temperature ratio

Tc/Th for thermoelectric generators.

Example 2.1 Exhaust Waste Heat Recovery

We want to recover waste heat from the exhaust gas of a car using thermoelectric generator (TEG) modules as shown in Figure 12.6. An array of N = 24 TEG modules is installed on the exhaust of the car. Each module has n = 98 thermocouples, each of which 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

-3 -2 similar as p = −n = 168 V/K, p = n = 1.56 × 10 cm, and kp = kn = 1.18 × 10 W/cmK. 2 The cross-sectional area and leg length of the thermoelement are An = Ap = 12 mm and Ln = Lp = 4.6 mm, respectively, which are shown in the figure. (a) Per one TEG module, compute the electric current, the voltage, the maximum power output, and the maximum power efficiency. 2-13

(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 2.6 (a) a TEG device, (b) a thermocouple.

Solution:

푇ℎ = 503퐾 and 푇푐 = 323퐾 -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

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

2-14

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 (2.21), 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 (2.22), 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 (2.23), 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 (2.30), the maximum power efficiency is

푇푐 323퐾 (1 − 푇 ) 1 − 휂 = ℎ = 503퐾 푚푝 1 푇 2 푇 1 323퐾 2 323퐾 2 − (1 − 푐 ) + (1 + 푐 ) 2 − (1 − ) + (1 + ) 2 푇ℎ 푍푇̅ 푇ℎ 2 503퐾 0.633 503퐾 2-15

=0.051 (b) For the whole TEG device:

The maximum power output is

 Wn  247.493W 179.8W

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

mp  0.051 Using Equation (2.29), the maximum conversion efficiency is

 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

2.5 Effective Material Properties

Calculated performance curves using the ideal equations with intrinsic material properties usually show significant discrepancies compared with the measured performance curves of a commercial thermoelectric module because the ideal equations do not include the various losses such as the thermal and electrical contact resistances (manufacturability), the Thomson effect (temperature dependency), and heat losses (no perfect insulation). The problem is that the various losses are very difficult to be figured out, usually unknown. That is the reason why we 2-16

developed the effective material properties to include those losses for a system design rather than the intrinsic material properties. The following method shows how to determine the effective material properties. As mentioned before, the four maximum parameters are usually provided by manufacturers as specification for a commercial module. However, the thermoelectric material properties (, , and k) for the module are not usually provided by the manufacturers, which often causes a problem for system designers who wants to simulate the module operation using the ideal equations. 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) of thermocouple and two junction temperatures Th and Tc. Inversely, the three material properties can ideally be expressed from three out of the four maximum parameters. It turned out that, the two parameters ( and ) are essential and any one of and can be used. 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 with various energy losses. 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 of power output. The effective material properties are defined here as the realistic material properties that are extracted from the maximum parameters provided by the manufacturers. So that the calculated effective material properties include all kind of losses due to the thermal and electrical contact resistances, the temperature dependence of the properties, and heat losses to ambient in addition to the intrinsic material properties. This renders the dimensionless figure of merit for the effective material properties to be slightly lower than that of the intrinsic material properties. The effective electrical resistivity is obtained using Equations (2.26) and (2.28), which is

2-17

 (2.35)  4A LWmax   2 nI max 

The effective Seebeck coefficient can be obtained using Equation (2.26) and (2.34), which is

4W (2.36)    max nImax Th  Tc 

The effective figure of merit is obtained from Equation (2.29), which is

2 (2.37) 휂푚푎푥 푇푐 1 1 + 휂 푇 ∗ 푐 ℎ 푍 = ( 휂 ) − 1 푇̅ 1 − 푚푎푥 [ 휂푐 ] where c 1Tc Th which is the Carnot cycle efficiency. Alternatively, the effective figure of merit may be obtained from Equation (2.30) in terms of mp as

2 푇 (1 + 푐 ) (2.38) 푇̅ 푇ℎ 푍∗ = 1 1 휂푐 ( + ) − 2 휂푚푝 2

The effective thermal conductivity with Z * which is usually obtained from Equations (2.37) is now obtained

훼∗2 (2.39) 푘∗ = 휌∗푍∗

The effective material properties now include various effects such as the contact resistances, Thomson effect, and radiation and convection heat losses. Hence, the effective figure of merits in Table 2.1 might be slightly smaller than the intrinsic figure of merit (not shown in the table). 2-18

Note that these effective properties should be divided by two due to p-type and n-type thermoelements.

Comparison of calculations with a commercial product The effective material properties can be calculated from any commercial thermoelectric modules as long as the four maximum parameters are provided. Calculated effective material properties from the maximum parameters for four selected commercial thermoelectric modules are illustrated in Table 2.1. Then, we can simulate the performance curves of the module with these effective material properties using the ideal equations. This is very useful for a system design with thermoelectric modules. For example, we attained the effective material properties for TGM127- 1.4-2.5 module in Table 2.1 and compared the calculated performance curves with the performance curves provided by the manufacturer, which are shown in Figure 2.1 (a) – (f). It is found that the calculated results are in good agreement with the manufacturer’s performance curves (which are typically experimental values).

Table 2.1 Maximum parameters and effective material properties for some commercial modules.

Description TEG Module (Bismuth Telluride) Symbols TG12-4 HZ-2 TGM199-1.4-2.0 TGM127-1.4-2.5 o o o o Tc =50 C Tc =30 C Tc =30 C Tc =30 C o o o o Th =170 C Th =230 C Th =200 C Th =200 C

Number of n 127 97 199 127 thermocouples

Manufacturers’ 푊푚푎푥(W) 2.12 2.6 7.3 4.4 maximum parameters 퐼푚푎푥 (A) 1.32 1.6 2.65 2.37 η푚푝 (%) 4.08 4.5 5.3 5.4 푉푚푎푥 (V) 6.5 6.53 11 7.7 Rn () 6.32 4 3.7 3 A (mm2) 1 2.1 1.96 1.96 2-19

Measured geometry L (mm) 1.17 2.87 2 2.5 of thermoelements G = A/L (cm) 0.085 0.073 0.098 0.078 Dimension (W×L×H) mm 30 × 30 ×3.4 30 × 30 ×4.5 40 × 40 × 4.4 40 × 40 × 4.8 Effective material  V/K 210.7 167.5 162.8 171.9 properties (calculated cm 1.638 × 10-3 1.532 × 10-3 1.024 × 10-3 0.9672 × 10-3 using k(W/cmK) 0.015 0.016 0.015 0.017 commercial(Wmax , ZTavr 0.708 0.456 0.652 0.679 Imax, and ηmax) Maximum Wmax (W) 2.12 2.6 7.3 4.4 parameters using Imam (A) 1.32 1.6 2.65 2.37 effective material ηmax (%) 4.1 4.5 5.3 5.4 properties Vmax (V) 6.4 6.5 11.0 7.4 Rn () 4.86 3.90 3.67 3.13

(a)

(b) 2-20

(c)

(d) Figure 2.6 For module TGM127-1.4-2.5 of thermoelectric generator, comparison between calculations with effective material properties and commercial data. (a) matched output versus hot side temperature, (b) matched load voltage versus hotside temperature, (c) maximum efficiency versus load resistance, and (d) matched load voltage and output power versus output current.

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Problems

2.1 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 P2-1a. 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 fulfil 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 P2-1. (a) Curiosity rover on Mars, (b) p-type and n-type thermoelements.

2-22

2.2 We want to recover waste heat from the exhaust gas of a car using thermoelectric generator (TEG) modules as shown in Figure P2-2. 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 thermoelements 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.  2-23

(a) (b) Figure P2-2 (a) TEG device, (b) thermocouple.

2.3

References

1. Peltier, J.C., New experiments on the calorific electric currents. Annales de chimie de physique, 1834. 56(2): p. 371-386. 2. Thomson, W., Account of researches in thermo-electricity. Proceedings of the Royal Society of London, 1854. 7: p. 49-58. 3. Landau, L.D. and E.M. Lifshitz, Electrodynamics of continuous media. 1960: Pergamon Press, Oxford, UK. 4. Onsager, L., Reciprocal Relations in Irreversible Processes. I. Physical Review, 1931. 37(4): p. 405-426.