Optimal design of a multi-couple thermoelectric generator

Semicond. Sci. Technol. 15 (2000) 184 188. Printed in the UK PII: S0268-1242(00)06772-9 Optimal design of a multi-couple thermoelectric generator Jincan Chen, Bihong Lin, Hongjie Wang Guoxing Lin Department of Physics, Xiamen University, Xiamen 361005, People s Republic of China Department of Physics, Quanzhou Normal College, Quanzhou, Fujian 362000, People s Republic of China E-mail: jcchen@xmu.edu.cn Received 5 August 1999, in final form 16 November 1999, accepted for publication 14 December 1999 Abstract. The performance of a multi-couple thermoelectric device as a generator is investigated. The general expressions of two important performance parameters, the efficiency power output, are given. The η K characteristic curves of a thermoelectric generator are presented for some differently constrained conditions, where η is the efficiency of the thermoelectric system K is the total thermal conductance of the multi-couple thermoelectric device. The maximum efficiency of the system is calculated. The structure parameters of the device are optimized. The effect of the thermal conductances between the thermoelectric device the external heat reservoirs on the performance of the system is expounded by using some representative numerical examples. The results obtained here will be useful for a more detailed investigation for the optimal design of real thermoelectric generators. Nomenclature A I K l L C L H n P Q C Q H R S T C T Cj T H T Hj Z α κ ρ η total Seebeck coefficient of the device electric current total thermal conductance of the device length of arms thermal conductance between the device the heat sink thermal conductance between the device the heat source number of couples power output heat flow from the device to the heat sink heat flow from the heat source to the device total electrical resistance of the device cross-sectional area of arms temperature of the heat sink temperature of the cold junction of the thermoelectric device temperature of the heat source temperature of the hot junction of the thermoelectric device figure of merit of the device Seebeck coefficient thermal conductivity electrical resistivity efficiency 1. Introduction In the optimal design of thermoelectric devices, there are two interesting problems: one is to find the important relation between the structure parameters l n /S n l p /S p of semiconductor arms the other is to determine the optimal value of l n /S n or l p /S p, where l S are the length the cross-sectional area of semiconductor arms, the subscripts n p designate the n- p-type arms, respectively. As described in the literature, to maximize the figure of merit Z may easily solve the first problem [1 4]. It is well known that the efficiency of a thermoelectric device as a generator depends on material parameters through the figure of merit Z. The larger the figure of merit is, the better the performance of a thermoelectric device is the higher the efficiency of the thermoelectric system is. In the case of a thermoelectric device with n- p-type arms, the figure of merit Z = A 2 /(RK), where A, R K are, respectively, the total Seebeck coefficient, electrical resistance thermal conductance of the thermoelectric device. For given semiconductor materials, to maximize Z requires that the structural parameters of semiconductor arms satisfy the following equation l n = l p κn ρ p (1) S n S p κ p ρ n where ρ is the electrical resistivity κ is the thermal conductivity. Equation (1) gives the important relation between l n /S n l p /S p. In this case, the geometric configuration of the thermoelectric device is optimal [5]. To solve the second problem will be more troublesome, because the optimal value of l n /S n or l p /S p depends on both the parameters of semiconductor materials the operative conditions of thermoelectric devices, specially when the 0268-1242/00/020184+05$30.00 2000 IOP Publishing Ltd

Multi-couple thermoelectric generator finite thermal conductances between the thermoelectric device the external heat reservoirs are considered [6, 7]. Thus, this problem is rarely discussed. However, for real thermoelectric generators, the operative conditions are different from each other the thermal conductances between the thermoelectric device the external heat reservoirs are always finite. These questions are necessarily considered in the optimal design of thermoelectric devices. This has come to notice in recent years [8 10]. In the present paper, we will analyse how the different operative conditions the finite-rate heat transfer between the thermoelectric device the external heat reservoirs affect the performance of a multi-couple thermoelectric generator optimize the value of (S p /l p )n, where n is the number of couples. 2. A multi-couple thermoelectric generator For the sake of generality, figure 1 shows a schematic diagram of a multi-couple thermoelectric generator, which involves a large number of semiconductor elements connected electrically in series thermally in parallel. It is assumed that the Seebeck coefficient α, the electrical resistivity ρ, the thermal conductivity κ of the semiconductor materials are independent of temperature. The total electrical resistance R, Seebeck coefficient A thermal conductance K of a multi-couple thermoelectric device may be given by ( ρp l p R = S p + ρ nl n S n ) n (2) A = (α p α n )n (3) ( κp S p K = l p + κ ) ns n n (4) l n respectively. It may be easily proven that equation (1) still holds for a multi-couple thermoelectric device. Substituting equation (1) into equations (2) (4) gives R = (ρ p + ρ p ρ n κ n /κ p ) l p S p n (5) K = (κ p + κ p κ n ρ n /ρ p ) S p l p n. (6) From figure 1, one obtains the heat balance equations as Q H = (T H T Hj )L H (7) Q H = AIT Hj 1 2 I 2 R + K(T Hj T Cj ) (8) Q C = AIT Cj + 1 2 I 2 R + K(T Hj T Cj ) (9) Q C = (T Cj T C )L C (10) where Q H Q C are the heat flows from the heat source to the generator from the generator to the heat sink, T Hj T Cj are the temperatures of the hot cold junctions of the thermoelectric device, T H T C are the temperatures of the heat source sink, L H L C are the thermal Figure 1. A schematic diagram of a multi-couple thermoelectric generator. conductances between the generator the heat source sink I is the electric current. The model mentioned above is general useful. It may be conveniently used to discuss the optimal performance of an arbitrary-couple thermoelectric generator. For example, the optimal performance of a single-couple thermoelectric generator which is often discussed in the literature may be directly obtained from the results derived in the present paper as long as we choose 1. Eliminating the temperatures T Hj T Cj of the hot cold junctions from equations (7) (10), one can obtain q H = j(1 λj ) 1 2 j 2 (1/ )(1 λj +2λ) + (1 θ C ) (1+λ λj )(1+βj ) + (1 λj )β (11) q C = j(1+βj ) + 1 2 j 2 (1/ZT C )(1+βj +2β) + (θ H 1) (1+λ λj )(1+βj ) + (1 λj )β (12) where the dimensionless quantities j = AI/K, β = K/L H, λ = K/L C, q H = Q H /(KT H ), q C = Q C /(KT C ) θ H = 1/θ C = T H /T C. Using equations (11) (12), we obtain the efficiency power output of the system as η = Q H Q C = θ H q H q C Q H θ H q { H = j[1 λj θ C (1+βj )] 1 } 2 j 2 1 [2(1+λ+β) + (β λ)j] [ j(1 λj ) 1 ] 2 j 2 1 1 (1 λj +2λ) + (1 θ C ) p = P KT H = Q { H Q C = j[1 λj θ C (1+βj )] KT H (13) 1 2 j 2 1 [2(1+λ+β) + (β λ)j] [(1+λ λj )(1+βj ) + (1 λj )β] 1 (14) respectively, where p = P /(KT H ) is the dimensionless power output. } 185

J Chen et al Table 1. The optimal values of K at maximum efficiency for given L C = L H = 1WK 1,θ H =2 = 1.6. Q H /T H (W K 1 ) η max K opt j(η max ) 0.005 0.120 0.006 46 0.316 0.01 0.118 0.0132 0.310 0.02 0.113 0.0275 0.299 0.04 0.104 0.0603 0.273 0.08 0.0862 0.149 0.224 (a) P/T H (W K 1 ) η max K opt j(η max ) 0.0005 0.120 0.005 36 0.318 0.001 0.118 0.0111 0.311 0.002 0.114 0.0238 0.302 0.004 0.105 0.0567 0.276 0.008 0.0704 0.264 0.182 quantities. It is seen from equation (11) or (14) that the dimensionless current j is a function of the thermal conductance K Q H /T H or P/T H, so the efficiency may be written as η = η[j(q H /T H, K), K] (15) Figure 2. The efficiency η as a function of K for θ H = 2, = 1.6, (a) Q H /T H = 0.01 W K 1, (b) Q H /T H = 0.02 W K 1 (c) Q H /T H = 0.04 W K 1. Curves I II correspond to the cases of L H = L C = 1 10 W K 1, respectively. (b) (c) 3. Optimal design of the device For given semiconductor materials specified operating conditions, the parameters, L C, L H θ C are known or η = η[j(p/t H, K), K]. (16) If Q H /T H is given, the efficiency is only a function of K. From equations (11) (13), we can generate the η K curves of a thermoelectric generator for some given values of Q H /T H, as shown in figure 2. It is seen from figure 2 that for a thermoelectric device with the maximum figure of merit Z max, the optimum value of K still depends on the choice of the parameters L H, L C, T H T C even when Q H /T H is given. Obviously, the optimum value of (S p /l p )n may be obtained from these curves. For example, when L C = L H = 1WK 1, = 1.6, θ H = 2 Q H /T H = 0.02 W K 1, the maximum efficiency occurs at K = K opt = 0.0275 W K 1. Then, the optimum value of (S p /l p )n is 0.0275 l p opt κ p +. (17) κ p κ n ρ n /ρ p According to equation (17) the technological requirements, one may determine the values of S p /l p n. Thus, equation (17) may provide theoretical guidance for an engineer to design thermoelectric devices. For the different choices of Q H /T H, K will have different optimum values, which are listed in table 1. It is clearly seen from table 1 that the optimum values of K are closely dependent on Q H /T H. This shows that for the thermoelectric generators with the different operative conditions which are required practically, the optimum values of the structure parameters (S p /l p )n of the thermoelectric device must be different from each other. This is worthwhile to notice in the design of thermoelectric generators. In the general case, the concrete expression of equation (15) is complicated. The maximum efficiency can only be calculated numerically. Table 2 gives the optimal values of K at maximum efficiency for some different values 186

Table 2. The optimal values of K at maximum efficiency for given Q H /T H = 0.02 W K 1, θ H = 2 = 1.6. L H (W K 1 ) L C (W K 1 ) η max K opt j(η max ) 0.122 0.0253 0.323 1 0.118 0.0265 0.311 1 0.117 0.0262 0.311 1 10 0.117 0.0266 0.310 10 1 0.117 0.0264 0.308 1 1 0.113 0.0275 0.299 Multi-couple thermoelectric generator of L H L C. It is seen from table 2 that when the thermal conductances between the device the external heat reservoirs become small, the optimal value of K (i.e. (S p /l p )n) must be increased in the design of the device so that the device has an optimum structure. It is thus clear that when β = λ = 0, [(S p /l p )n] opt gives the lower bound for optimal values of the structure parameters (S p /l p )n of the device. For this case, equation (15) may be written as η = 2bb 1 (1+b 1 b 2 )K + (1+b 2 )K 2 2bK bb 1 (18) where b = Q H /(ZTH 2 ), b 1 = 1/(1 +θ C ) b 2 = 2(1 θ C )/( ). From equation (18), we can find that when the efficiency of the system is maximum for a given Q H /T H, the optimum value of (S p /l p )n is b/(1+b 2 ) l p κ p + κ p κ n ρ n /ρ p opt [ 1+ 1+ 1+b 2 (1+b 1 b 2 ) 2 (1+b 2 ) = n. (19) l p min Obviously, equation (19) may provide theoretically an instruction for the optimal design of the thermoelectric device. For real thermoelectric systems with the finite thermal conductances between the device the external heat reservoirs, the optimum value of (S p /l p )n should be chosen to be larger than [(S p /l p )n] min. If P/T H is given, equations (13) (14) may be used to plot the η K curves of a thermoelectric generator, as shown in figure 3, determine the optimal value of (S p /l p )n at maximum efficiency. Some relevant data have been listed in tables 1 3, from which some important results can be obtained. For example, when L C = L H = 1WK 1, = 1.6, θ H = 2 P/T H = 0.002 W K 1, the maximum efficiency occurs at K = K opt = 0.0238 W K 1. Then, the optimum value of (S p /l p )n is 0.0238 l p opt κ p +. (20) κ p κ n ρ n /ρ p For the special case of β = λ = 0, one can determine the maximum efficiency of a thermoelectric generator with the given value of P/T H the optimum value of the structure parameters (S p /l p )n by using the similar method mentioned above. ] Figure 3. The efficiency η as a function of K for P/T H = 0.002 W K 1. The values of the parameters θ H,, L H L C are the same as those used in figure 2. Table 3. The optimal values of K at maximum efficiency for given P/T H = 0.002 W K 1, θ H = 2 = 1.6. L H (W K 1 ) L C (W K 1 ) η max K opt j(η max ) 0.122 0.0208 0.322 1 0.118 0.0223 0.310 1 0.118 0.0221 0.312 1 10 0.118 0.0224 0.311 10 1 0.118 0.0223 0.310 1 1 0.114 0.0238 0.302 4. Conclusions A new cycle model consisting of a multi-couple thermoelectric device involving several key irreversibilities of real thermoelectric generators is established used to optimize the performance of a multi-couple thermoelectric generator. Some fundamental relations such as equations (11) (14) are derived. On the basis of these relations, the problem relative to the optimal structure of the thermoelectric device is discussed in detail. The influence of the rate of heat supplying the thermal conductances between the generator the external heat reservoirs on the performance of the system are analysed quantitatively. The results obtained here may reveal some general characteristics of real multi-couple thermoelectric generators be used to instruct the optimal design of real thermoelectric generators. Acknowledgments This work has been supported by National Natural Science Foundation No 59976033 Trans-Century Training Programme Foundation for the Talents by the State Education Commission, People s Republic of China. References [1] Ioffe A F 1957 Semiconductor Thermoelement Thermoelectric Cooling (London: Infosearch) [2] Bollmeier E W 1960 Direct Conversion of Heat to Electricity ed J Kaye JAWelsh (New York: Wiley) [3] Mikloš P Formánek B 1995 J. Electr. Eng. 46 380 187

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