Efficiency and output power of thermoelectric module by taking into account corrected Joule and Thomson heat Hee Seok Kim, Weishu Liu, and Zhifeng Ren Citation: Journal of Applied Physics 118, 115103 (2015); View online: https://doi.org/10.1063/1.4930869 View Table of Contents: http://aip.scitation.org/toc/jap/118/11 Published by the American Institute of Physics Articles you may be interested in Calculation of Efficiency of Thermoelectric Devices Journal of Applied Physics 31, 1 (2004); 10.1063/1.1735380 Modeling of concentrating solar thermoelectric generators Journal of Applied Physics 110, 074502 (2011); 10.1063/1.3642988 Enhanced thermoelectric figure of merit in nanostructured -type silicon germanium bulk alloy Applied Physics Letters 93, 193121 (2008); 10.1063/1.3027060 Numerical modeling and optimization of the segmented PbTe–BiTe-based thermoelectric leg Journal of Applied Physics 120, 124503 (2016); 10.1063/1.4962317 Improved mechanical properties of thermoelectric (Bi0.2Sb0.8)2Te3 by nanostructuring APL Materials 4, 104807 (2016); 10.1063/1.4953173 Acoustic phonon softening and reduced thermal conductivity in Mg2Si1-xSnx solid solutions Applied Physics Letters 110, 143903 (2017); 10.1063/1.4979871 JOURNAL OF APPLIED PHYSICS 118, 115103 (2015) Efficiency and output power of thermoelectric module by taking into account corrected Joule and Thomson heat Hee Seok Kim, Weishu Liu,a) and Zhifeng Renb) Department of Physics and Texas Center for Superconductivity, University of Houston, Houston, Texas 77204, USA (Received 5 July 2015; accepted 31 August 2015; published online 16 September 2015) The maximum conversion efficiency of a thermoelectric module composed of p- and n-type materials has been widely calculated using a constant property model since the 1950s, but this conventional model is only valid in limited conditions and no Thomson heat is accounted for. Since Thomson heat causes the efficiency under- or over-rated depending on the temperature dependence of Seebeck coefficient, it cannot be ignored especially in large temperature difference between the hot and cold sides. In addition, incorrect Joule heat is taken into consideration for heat flux evaluation of a thermoelectric module at thermal boundaries due to the assumption of constant properties in the conventional model. For this reason, more practical predictions for efficiency and output power and its corresponding optimum conditions of p- and n-type materials need to be revisited. In this study, generic formulae are derived based on a cumulative temperature dependence model including Thomson effect. The formulae reliably predict the maximum efficiency and output power C 2015 AIP Publishing LLC. of a thermoelectric module at a large temperature. V [http://dx.doi.org/10.1063/1.4930869] I. INTRODUCTION Thermoelectric modules are environmentally benign, quiet, and direct energy converters from thermal to electric power.1–5 The energy conversion efficiency of a typical thermoelectric module consisting of p-type and n-type materials connected thermally in parallel and electrically in series is a function of the figure of merit [Z] of a p-n pair, defined as6 ½Z ¼ ðSp Sn Þ2 ; ðRp þ Rn ÞðKp þ Kn Þ (1) where S, R, and K are the Seebeck coefficient, electrical resistance, and thermal conductance, respectively, and the subscript p, n, and square brackets denote p-type, n-type materials, and p-n module property, respectively. The efficiency of a thermoelectric module has been analytically calculated based on an assumption that S, R, and K are temperature independent as6 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ DT 1 þ ½Z Tavg 1 g¼ (2) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ T ; Th c 1 þ ½Z Tavg þ Th where Th, Tc, DT, and Tavg are temperatures of hot end, cold end, their difference, and average, respectively. This is reasonably accurate when its operating temperature difference DT is small, or when [Z] is temperature independent over the operating temperature range, but most thermoelectric materials have strongly temperature dependent Seebeck coefficient S, electrical resistivity q, and thermal conductivity j. In a) Electronic mail: firstname.lastname@example.org Electronic mail: email@example.com b) 0021-8979/2015/118(11)/115103/9/$30.00 addition, thermoelectric modules as electric power generators are gaining attention in various applications such as unmanned aerial vehicle (UAV),7,8 solar energy harvesting,9,10 and heat recovery of gasoline/diesel engine,11,12 which requires larger temperature gradient on thermoelectric modules. Hence, the conventional model by Eq. (2) is not reliable on prediction for practical conversion efficiency of a module operating at large temperature difference between hot and cold sides. In addition, Thomson effect is neglected in the conventional model, which leads to shifting the efficiency and output power depending on the degree of Thomson heat at a given temperature gradient. Sunderland and Burak evaluated how Thomson heat affects output power and conversion efficiency,13 and Min et al. developed a modified ZT considering Thomson effect,14 but both assumed that Thomson coefficient is constant. The conversion efficiency accounting for Thomson effect was reported based on linear behavior of S, q, and j and temperature dependent Seebeck coefficient with constant q and j.15–17 Kim et al. recently established new formulae for maximum efficiency of a homogeneous thermoelectric material to analytically calculate efficiency including Thomson effect at a large temperature gradient.18 This model evaluated the maximum efficiency of a single thermoelectric material, which is good to first examine the characteristics of individual thermoelectric materials. However, usually thermoelectric modules operate with pairing of p- and n-type materials, which requires a balance of thermal flow and electric current of dissimilar p-type and n-type materials. Here, we report formulae for maximum efficiency of thermoelectric modules based on cumulative temperature dependence model leading to more reliable predictions than the conventional model, and define the maximum engineering figure of merit and engineering power factor density as direct indicators of a 118, 115103-1 C 2015 AIP Publishing LLC V 115103-2 Kim, Liu, and Ren J. Appl. Phys. 118, 115103 (2015) ð Th ð Th thermoelectric module at practical temperature differences. The thermoelectric module performance is demonstrated in three different p-n pairings: (1) skutterudites (SKU): Ce0.45Nd0.45Fe3.5Co0.5Sb12 (p-type)19 and Ba0.08 La0.05Yb0.04Co4Sb12 (n-type),20 (2) chalcogenides: K0.02 Pb0.98Te0.75Se0.25 (p-type)21 and Pb0.995SeCr0.005 (n-type),22 and (3) chalcogenides: SnSe (p-type)23 and PbSe (n-type).22 Tc WJ;p or n ¼ Tc ð Th : (9) sp or n ðT ÞdT Tc where x is the distance from the hot side surface of the thermoelectric leg, and J is the current density. The temperature dependent Thomson coefficient is defined as s(T) ¼ TdS(T)/ dT. Integrating Eq. (3) twice with respect to x with applying the boundary conditions, Tjx¼0 ¼ Th and Tjx¼Lp ¼ Tc , yields the conduction heat at Th of p-type leg as ð ð ð dT Ap Th I 2 Lp x Ap jp ðT Þ ¼ jp ðT ÞdT q ðT Þdxdx Lp Tc Ap Lp 0 0 p dx Th ð ð I Lp T þ sp ðT ÞdTdx; (4) Lp 0 Th where Lp, Ap and I are the leg length, cross sectional area of p-type leg and electric current, respectively. The conduction heat of n-type leg is also obtained in the same way. From the constitutive relation for heat Q and electric current I (5) the heat transferred into each thermoelectric leg at T ¼ Th becomes ð Ap Th jp ðT ÞdT WJ;p Ip2 Rp Qh;p ¼ Ip Th Sp ðTh Þ þ Lp Tc ð Th WT;p Ip sp ðT ÞdT; (6) Tc Qh;n (8) sp or n ðT ÞdTdT T DT The one-dimensional governing equation for a steadystate heat flow in thermoelectric module is24 d dT dT ð Þ jT þ J 2 qðT Þ JsðT Þ ¼ 0; (3) dx dx dx ð An Th ¼ I n T h S n ðT h Þ þ jn ðT ÞdT WJ;n In2 Rn Ln Tc ð Th WT;n In sn ðT ÞdT; ; qp or n ðT ÞdT Tc ð Th ð Th II. CUMULATIVE TEMPERATURE DEPENDENCE MODEL OF P-N MODULE dT ; dx ð Th DT WT;p or n ¼ Q ¼ ITS Aj qp or n ðT ÞdTdT T By Eqs. (6) and (7), total input heat Qh (¼Qh,p þ Qh,n) at Th becomes ð Ap Th jp ðT ÞdT Qh ¼ ITh Sp ðTh Þ Sn ðTh Þ þ Lp Tc ð An Th þ jn ðT ÞdT ðWJ;p Rp þ WJ;n Rn ÞI2 Ln Tc ! ð ð Th Th sp ðT ÞdT WT;n WT;p sn ðT ÞdT I; Tc where I ¼ Ip ¼ In. The output power accounting for Thomson effect is (see Appendix A) Pout ¼ 2 Voc m ; Rint ð1 þ mÞ2 Tc where S(Th) is the Seebeck coefficient at Th, and Rp and Rn are DT-dependent electrical resistance at Ða given thermal T boundary, defined as R ¼ ðLA1 DT 1 Þ Tch qðTÞdT. The terms on the right-hand side in Eqs. (6) and (7) represent Peltier, conduction, Joule, and Thomson heat, respectively. WJ and WT are DT-dependent weight factors for practical fraction to the hot side of the total Joule and Thomson heat associated with the temperature dependence of q(T) and s(T), defined as18 (12) Tc m is the ratio of load (RL) to internal (Rint ¼ Rp þ Rn) electrical resistance, m ¼ RL/Rint. The conversion efficiency of the module is expressed as the ratio of the output power to input heat g¼ Pout : Qh (13) Eqs. (10)–(13) yield 2 Th Sp ðTh Þ Sn ðTh Þ m 1þm g¼ 6 þ ð Th ð Th 1 þ m 4½ZT eng Sp ðT ÞdT Sn ðT ÞdT Tc (11) where Voc is the open circuit voltage expressed as ð Th ð Th Sp ðTÞdT Sn ðTÞdT: Voc ¼ Tc (7) (10) Tc WJ;p R0p WJ;n R0n þ 1þm Tc WT;p s0p WT;n s0n #1 ; (14) where R0p ¼ Rp =Rint and R0n ¼ Rn =Rint . s0p and s0p are expressed as ð Th sp or n ðT ÞdT Tc : (15) s0p or n ¼ ð Th ð Th Sp ðT ÞdT Sn ðT ÞdT Tc Tc Here, the engineering dimensionless figure of merit of p-n module [ZT]eng is defined as 115103-3 Kim, Liu, and Ren J. Appl. Phys. 118, 115103 (2015) ð Th ½ZT eng ¼ Sp ðT ÞdT Tc Lp Ap ð Th Tc Ln qp ðT ÞdT þ An ð Th ! qn ðT ÞdT Tc The engineering power factor of p-n module [PF]eng in W K1 is defined as !2 ð ð Th Th Sp ðT ÞdT ½PFeng ¼ Lp Ap Sn ðT ÞdT Tc Tc Tc Ln qp ðT ÞdT þ An ð Th ð Th : ð Th Tc An jp ðT ÞdT þ Ln ð Th ! DT: Tc Tc jn ðT ÞdT Substituting Eq. (21) into Eqs. (18) and (19) leads to the efficiency formula without Thomson effect on heat flux based on the cumulative temperature dependence model. If S, q, and j are assumed to be temperature-independent, Eq. (20) is further reduced to gc i: 2 (22) Due to [ZT]eng ¼ [Z]DT from Eq. (16) for temperature independent properties, Eq. (19) becomes mopt ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ ½ZTavg ; Tc qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ ½ZT eng a1 g1 c 1 ¼ gc qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : þ a a0 1 þ ½ZT eng a1 g1 2 c When S, q, and j are temperature independent, (25) is reduced to ½Zeng (26) in Eq. (23) and its corresponding efficiency is identical to Eq. (2), which indicates Eqs. (18)–(20) are generic formulae that can be converted to the expression based on conventional as well as the cumulative temperature dependence model. The optimized efficiency by Eq. (18) can be maximized by matching the leg size of p- and n-type materials for balancing thermal flow and electric current. Taking derivative of the denominator of Eq. (16) and setting it equal to zero yields be for the maximum [ZT]eng as vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð Th uð Th u u qn ðT ÞdT jp ðT ÞdT An Lp u Tc Tc be ¼ ¼u ; ð ð Th Ap Ln u t Th qp ðT ÞdT jn ðT ÞdT Tc (24) Tc and the corresponding maximum engineering figure of merit of the module ½ZTeng and the maximum efficiency become Sp ðT ÞdT ð Th !2 Sn ðT ÞdT Tc ¼ 0sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ12 DT; ð Th ð Th ð Th ð Th @ jp ðT ÞdT qp ðT ÞdT þ jn ðT ÞdT qn ðT ÞdT A Tc (16) Tc ai ¼ 1 ð Th gmax Ap Lp qn ðT ÞdT where i ¼ 0, 1, and 2. When Thomson effect and weight factors are not taken into consideration, i.e., sp or n ¼ 0 and WJ,p or n ¼ WT,p or n ¼ 1/2, Eq. (20) is reduced to Sp ðTh Þ Sn ðTh Þ DT g (21) c i: ai ¼ ð Th ð Th 2 ð Þ ð Þ Sp T dT Sn T dT ½Z eng DT Sn ðT ÞdT Tc Tc where gc is the Carnot efficiency (DT/Th), and ai is defined as Sp ðTh Þ Sn ðTh Þ DT ai ¼ ð Th ð Th Sp ðT ÞdT Sn ðT ÞdT Tc Tc WT;p s0p WT;n s0n gc i WJ;p R0p þ WJ;n R0n gc ; (20) ½ZT eng ¼ !2 (17) By optimizing the ratio m satisfying dg/dm ¼ 0 regarding Eq. (14), the optimized efficiency and its corresponding mopt are obtained as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ ½ZT eng a1 g1 c 1 ; (18) gopt ¼ gc qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a0 1 þ ½ZT eng a1 g1 þ a 2 c qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ mopt ¼ 1 þ ½ZTeng a1 g1 (19) c ; Tc ð Th Tc (25) Tc ðS S Þ2 ½Z eng ¼ ½Z ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃ pnﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 ; jp qp þ jn qn (27) where [Z]* is the optimal figure of merit for the maximum efficiency based on conventional model, and its maximum efficiency is6 115103-4 Kim, Liu, and Ren gmax J. Appl. Phys. 118, 115103 (2015) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ DT 1 þ ½ZT Tavg 1 ¼ : qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Tc Th 1 þ ½ZT Tavg þ Th (28) III. RESULTS AND DISCUSSION The output power density Pd in W m2 by Eqs. (11) and (17) is expressed as Pd ¼ ¼ Pout Ap þ An 2 Voc m DT ðAp þ An ÞRint ð1 þ mÞ2 !2 ð Th ð Th Sp ðT ÞdT Sn ðT ÞdT m DT ð1 þ mÞ2 ! ¼ ð ð Lp Th Ln Th q ðT ÞdT þ q ðT ÞdT ðAp þ An Þ Ap Tc p An Tc n Tc ¼ Tc ½PFeng m DT ðAp þ An Þ ð1 þ mÞ2 m DT: ¼ ½PFeng;d ð1 þ m Þ2 (29) When the optimal value m ¼ 1 by taking derivative m/(1 þ m)2 with respect to m and setting it to zero, the output power density Pd is proportional to the engineering power factor density [PF]eng,d at a given DT, so the maximum Pd and its corresponding condition bp are obtained by taking derivative of the denominator of [PF]eng,d with respect to An/Ap Pd;max ¼ ½PFeng;d DT 4 ð Th ð Th !2 DT 4 Tc Tc ¼ 0sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ12 ; (30) ð Th ð Th @ L q ðT ÞdT þ L q ðT ÞdT A Sp ðT ÞdT p Sn ðT ÞdT n p Tc n Tc vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u ð Th u uLn qn ðT ÞdT An u Tc u bp ¼ ¼ ; ð Th Ap u t ð Þ qp T dT Lp Tc where ½PFeng;d is the maximum engineering power factor density of a module. (31) The DT-dependent maximum efficiency for p-SKU/nSKU is shown in Fig. 1(a), where Tc is fixed at 100 C and Th is ramped up to 500 C. The efficiency predictions by the cumulative temperature dependence model and conventional model have good agreement in whole temperature range and show 2.6% and 2.5% of relative difference as compared with results by the numerical simulation25,26 at DT ¼ 400 C [Fig. 1(b)], respectively. The conventional model can show the reliable efficiency prediction for p-SKU/n-SKU because S, q, and j of p-type and n-type SKUs have weak temperature dependence,19,20 i.e., linear behavior of ZT. Thus, the conventional model can be simply utilized in such limited material systems. For p-PbTe/n-PbSe, however, the conventional model underestimates the maximum efficiency over whole temperature range [Fig. 1(c)] and gives rise to 27% of relative difference as compared with the simulation result at DT ¼ 400 C [Fig. 1(d)], whereas the cumulative temperature dependence model without considering Thomson effect on the input heat flux shows more accurate prediction with the relative difference by 10.3% mainly because of the cumulative effect of temperature dependent S, q, and j taken into account. By accounting for Thomson effect, the efficiency prediction becomes further accurate over the whole temperature range and leads to only 2.8% of underestimation as compared to the simulation result. For p-SnSe/n-PbSe paired module shown in Figs. 1(e) and 1(f), the cumulative temperature dependence model without Thomson effect overrates the efficiency by 4.6% at DT ¼ 400 C, and the prediction is improved by accounting for Thomson effect leading to the relative difference by 3% while the conventional model overrates it by 17% as compared with the simulation. In some cases, however, the conventional model shows more accurate prediction. For example, in p-SnSe/n-PbSe if DT ¼ 500 C, the conventional model only overrates by 2.5% as shown in the inset of Fig. 1(e), while it predicts mostly inaccurate efficiency in the rest of temperature range [Fig. 1(e)]. This is because the averaged constant values of S, q, and j at DT ¼ 500 C are by chance very close to each FIG. 1. The maximum efficiency as a function of DT, and its relative difference as compared to the numerical simulation at DT ¼ 400 C for (a) and (b) p-SKU/n-SKU, (c) and (d) p-PbTe/ n-PbSe, and (e) and (f) p-SnSe/n-PbSe. The efficiency of p-SnSe/n-PbSe up to DT ¼ 500 C is shown as the inset in (e). The x-axis in (b), (d), and (e) represents A: numerical simulation, B: CTD model with Thomson effect, C: CTD model without Thomson effect, and D: conventional model. CTD stands for the cumulative temperature dependence model. 115103-5 Kim, Liu, and Ren J. Appl. Phys. 118, 115103 (2015) equivalent value which includes the influence of temperature dependent properties as well as Thomson effect taken into account by the numerical simulation. Thus, the conventional model is not reliable, since it sometimes has inadvertent accuracy with lack of analogy. To examine the effect of Thomson heat on the thermoelectric module based on the cumulative temperature dependence model, Figs. 2(a) and 2(b) show the cumulative effect of practical Joule and Thomson heat on the input heat flux at hot side, respectively. The conventional model assumes a half of the Joule heat returning to the hot and cold side by the temperature-independent q. The concept of the cumulative temperature dependence provides a practical contribution of Joule heat on each thermal boundary of a single thermoelectric leg as WJ, and its combination effect in p-n pair module is defined by referring to Eq. (14) WJ;pn ¼ WJ;p R0p þ WJ;n R0n : (32) In Fig. 2(a), WJ,pn of p-SKU/n-SKU and p-PbTe/n-PbSe is over 1/2 indicating more Joule heat on the hot side than that based on the conventional model, and it has an increasing trend as DT increases due to dq/dT > 0 of each material. In p-SnSe/n-PbSe, WJ,p–n fluctuates and becomes below 1/2 at DT ¼ 300 C due to a large reduction of q of SnSe with higher temperature.23 Figure 2(b) shows the combined effect of Thomson heat for input heat flux, which is also defined as WT;pn ¼ WT;p s0p WT;n s0n : (33) WT,p–n for p-SKU/n-SKU and p-PbTe/n-PbSe gradually decreases with higher temperature due to the decreasing FIG. 2. The cumulative effect of (a) Joule heat and (b) Thomson heat on the input heat flux at the hot side. tendency of djSj=dT indicating less Thomson effect is associated. For p-SnSe/n-PbSe, WT,p–n shows a parabolic behavior up to DT ¼ 350 C and decreases below zero. The negative WT,p–n corresponds to the cumulative effect of heat absorbed by Thomson effect and exerts an influence on decreasing the conversion efficiency at the given thermal boundary condition as compared with the model without Thomson effect, as shown in Fig. 1(e). In contrast, the positive WT,pn for p-SKU/n-SKU and p-PbTe/n-PbSe has an effect on increasing the efficiency compared with that in absence of Thomson heat as shown in Figs. 1(a) and 1(c). Figure 3 shows the maximum Pd with respect to DT for three p-n module configurations in which the temperature gradient is assumed to be 200 C mm1 for the leg length of 2 mm where Lp ¼ Ln in a typical p-shape module, and the temperature profiles of each thermoelectric leg at Th ¼ 500 C by the numerical computation. The analytical prediction for p-SKU/n-SKU by the conventional model has good agreement with the simulation results by 2.7% at DT ¼ 400 C [Fig. 3(a)] due to the high linearity of ZT curves of each SKU material, while the maximum Pd predictions for p-PbTe/n-PbSe [Fig. 3(c)] and p-SnSe/n-PbSe [Fig. 3(e)] lead to 23% and 42% of relative difference at DT ¼ 400 C as compared to the simulation results, respectively. As the conventional model is reliable only when thermoelectric properties are not much varied with temperature such as p-type and n-type SKUs, the cumulative temperature dependence model computes more accurate prediction through the whole temperature range [Figs. 3(c) and 3(e)] and gives rise to 0.5% and 4.9% of relative difference at DT ¼ 400 C for p-PbTe/n-PbSe and p-SnSe/n-PbSe, respectively, which shows Eq. (30) is more reliable by taking Thomson effect into consideration. The differences between the cumulative temperature dependence model and numerical simulations are mainly caused by the assumption of the linear approximation for temperature gradient in evaluating Rint for integrating q(T) with respect to x. In p-SKU/n-SKU, the temperature profiles by the simulations are over the linear approximation in terms of x (solid line) in Fig. 3(b), for which lower Rint based on the linear approximation is obtained due to q / T in SKUs.19,20 This results in the overestimated output power density by this model compared by the numerical simulation as shown in Fig. 3(a). In contrast, since the temperature distributions of p-PbTe/n-PbSe by the simulation are lower than the linear profile [Fig. 3(d)], the overestimated Rint by this model leads to the lower output power density [Fig. 3(c)] as q of p-PbTe and n-PbSe is proportional to T.21,22 For p-SnSe/n-PbSe, the temperature profiles of both p-type and n-type leg are below the linear distribution with respect to x [Fig. 3(f)], but q of the p-type SnSe is higher than that of n-type PbSe by two order of magnitude, and inversely proportional to T.23 This causes the overestimated output power density by this model [Fig. 3(e)] resulting from the underrated Rint that is dominant by Rp of SnSe. Therefore, the cumulative temperature dependence model including Thomson effect more correctly predicts the output power generation than the conventional model, and the main difference between the cumulative temperature dependence model and the numerical analysis is caused by 115103-6 Kim, Liu, and Ren J. Appl. Phys. 118, 115103 (2015) FIG. 3. The maximum output power density as a function of DT up to 400 C and the temperature profile through each leg at Th ¼ 500 C for (a) and (b) pSKU/n-SKU, (c) and (d) p-PbTe/n-PbSe, and (e) and (f) p-SnSe/n-PbSe. The length of legs was determined based on the temperature gradient DTmax/ L ¼ 200 C mm1. the linear approximation for the temperature distribution in this model. This over- or under-estimate of output power gives rise to the tendency of over- or under-rated maximum efficiency prediction by this model shown in Fig. 1 as compared with the numerical results. Another possible reason for the difference is that this model accounts for the cumulative effect of the temperature dependence of each property at thermal boundaries, which is path-independent according to the temperature while the numerical model incorporates the instantaneous temperature dependence that is path dependent associated with the temperature. Figures 4(a)–4(c) show the maximum efficiency with respect to be for 1-pair module of p-SKU/n-SKU, p-PbTe/nPbSe, and p-SnSe/n-PbSe, respectively, at DT ¼ 400 C and Tc ¼ 100 C with Lp ¼ Ln. The be by the model in this study (open square and open diamond) has better agreement with that by numerical simulation (solid line and open triangle in inset) than the prediction by the conventional model (open circle). The efficiency of the p-SKU/n-SKU is maximized when An is about 20% smaller than Ap (be ¼ 0.802), while p-PbTe/n-PbSe requires about 16% larger An (be ¼ 1.164) for the maximum efficiency. For p-SnSe/n-PbSe, be ¼ 0.103 FIG. 4. The efficiency with respect to the maximum efficiency condition be of (a) p-SKU/n-SKU, (b) p-PbTe/nPbSe, and (c) p-SnSe/n-PbSe, where numerical simulation (solid line and open triangle in insets), conventional model (open circle), the model in this study without Thomson effect (open square), and with Thomson effect (open diamond). The area near peak efficiencies is magnified as inset. (d) Comparison of be (open symbols) and bp (solid symbols) corresponding to the maximum efficiency and maximum output power density, respectively. 115103-7 Kim, Liu, and Ren should be met for the maximum efficiency, which requires the large difference between An and Ap. In Fig. 4(d), bp (solid symbols) for the maximum output power density is obtained by Eq. (31), and compared with be (open symbols). For small variations of S, q, and j between p- and n-type materials like p-SKU/n-SKU and p-PbTe/n-PbSe, the maximum efficiency is not much different from those at Ap ¼ An (be ¼ 1), but for p-SnSe/n-PbSe showing large property variations between pand n-type materials, the maximum conditions should be examined instead of assuming be or bp ¼ 1. The relative differences between bp and be for p-SKU/n-SKU, p-PbTe/ n-PbSe, and p-SnSe/n-PbSe are 1.5%, 14%, and 65%, respectively, where the difference between them is caused by the ratio of thermal conductivity of p-type and n-type materials [Eqs. (24) and (31)]. However, the efficiencies led by be and bp are close each other within 1% in p-SKU/ n-SKU and p-SnSe/n-PbSe, and within 4% in p-SnSe/nPbSe since they are on the plateau curve near the peak efficiency of each p-n configuration as shown in Figs. 4(a)–4(c). Thus, finding the range of be or bp is crucial for a thermal and electric balance of p-n configuration, but it is not a critical design parameter of a module to select either be or bp unless a fine tuning of a module is necessary to see what the module’s primary concern is, e.g., its efficiency or output power. J. Appl. Phys. 118, 115103 (2015) Figure 5(a) shows ½ZTeng as a function of DT for pSKU/n-SKU, p-PbTe/n-PbSe, and p-SnSe/n-PbSe, and the maximum efficiency as inset. Note that ½ZTeng has a close correspondence to the maximum efficiency (inset), which means that ½ZTeng is the practical indicator to judge which p-n configuration is superior in terms of the efficiency. In Fig. 5(b), ½PFeng;d is the intrinsic output power density per DT as a material property, so it directly represents the power generation at a given temperature boundaries and the ratio m by showing similar tendency to Pd at matched load condition (inset). Thus, ½PFeng;d is also the practical indicator for the output power generation of each p-n module. Figures 5(c)–5(e) show ½ZTeng ; ðZTÞeng of p- and n-type materials, and their averaged (ZT)eng for each p-n configuration, where parentheses denotes a single material’s engineering figure of merit. For p-SKU/n-SKU and p-PbTe/n-PbSe, the simply averaged (ZT)eng of each p- and n-type materials has a good agreement with ½ZTeng of their corresponding p-n module within 1% as shown in Figs. 5(c) and 5(d), while the averaged (ZT)eng of p-SnSe/n-PbSe is overrated by 47% as compared to ½ZTeng of the module at DT ¼ 400 C shown in Fig. 5(e). The differences are caused by the correspondence of the cumulative temperature-dependent S, q, and j for p- and n-type materials in the p-n module (see Appendix B), so it should be careful to predict a module efficiency by simply FIG. 5. (a) The maximum engineering figure of merit of p-n module ½ZTeng and the efficiency (inset) at Tc ¼ 100 C, and (b) the maximum engineering power factor density ½PFeng;d and output power density (inset) as a function of DT of the p-n configurations, where Tc ¼ 100 C and Comparison of Lp ¼ Ln ¼ 2 mm. ½ZTeng (larger symbols), (ZT)eng of individual p- (solid lines) and n-type (dashed lines) materials, and their averaged (ZT)eng (smaller symbols) in (c) p-SKU/n-SKU, (d) p-PbTe/n-PbSe, and (e) p-SnSe/n-PbSe. 115103-8 Kim, Liu, and Ren J. Appl. Phys. 118, 115103 (2015) averaging ZTs of p- and n-type materials when using Eq. (28). IV. CONCLUSION We have established the maximum efficiency formulae of thermoelectric module consisting of p-n pair based on the cumulative temperature dependence model including Thomson heat showing how Thomson heat affects the output power and conversion efficiency. The effect of the cumulative Joule and Thomson heat enables the formulae to predict the efficiency and output power more reliably than the conventional model. The optimized engineering figure of merit ½ZTeng and engineering power factor density ½PFeng;d of modules are direct indicators of practical efficiency and output power. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy under Contract No. DOE DE-FG02-13ER46917/DESC0010831. The second term on the right-hand side for p-type becomes ðL ðL dT dSp ðT Þ dT I sp ðT Þ dx ¼ I T dx dx dT dx 0 0 ð Tc dSp ðT Þ dT T ¼I dT Th ð Th ¼ I Tc Sp ðTc Þ Th Sp ðTh Þ þ I Sp ðT ÞdT: Tc (A5) In the same manner for n-type ð Th ðL dT Sn ðT ÞdT: I sn ðT Þ dx ¼ I ðTc Sn ðTc Þ Th Sn ðTh ÞÞ þ I dx Tc 0 (A6) By substituting Eqs. (A5) and (A6) into Eq. (A4), and applyÐT ÐT oc and Voc ¼ Tch Sp ðTÞdT Tch Sn ðTÞdT ing I ¼ Rint Vð1þmÞ ! ð ð Th Tc APPENDIX A: OUTPUT POWER PREDICTION ACCOUNTING FOR THOMSON EFFECT Assuming no parasitic electric and/or thermal losses through thermoelectric configuration, the output power generation is expressed based on an energy balance as Pout ¼ Qh Qc : (A1) Qh and Qc are the heat flux at hot and cold side, shown by Eqs. (3)–(7) ð A Th jp ðT ÞdT Qh ¼ ITh Sp ðTh Þ Sn ðTh Þ þ L Tc ð A Th jn ðT ÞdT I2 ðWJ;p Rp þ WJ;n Rn Þ þ L Tc ! ð ð Th Th sp ðT ÞdT WT;n I WT;p Tc sn ðT ÞdT ; (A2) Tc ð A Th Qc ¼ ITc Sp ðTc Þ Sn ðTc Þ þ jp ðT ÞdT L Tc ð A Th þ jn ðT ÞdT I2 ðWJ;p Rp þ WJ;n Rn Þ L Tc ! ð Th ð Th sp ðT ÞdT WT;n sn ðT ÞdT I WT;p Tc I ðL 0 dT sp ðT Þ dx dx Tc ðL 0 dT sn ðT Þ dx dx Th Sp ðT ÞdT Pout ¼ I ! þ I2 Rint ; (A3) where Rint is electric resistance of thermoelectric legs, and the same geometry for p-type and n-type legs is considered. Substituting Eqs. (A2) and (A3) into Eq. (A1) yields Pout ¼ I Th Sp ðTh Þ Th Sn ðTh Þ Tc Sp ðTc Þ þ Tc Sn ðTc Þ ! ðL ðL dT dT þI sp ðT Þ dx sn ðT Þ dx I2 Rint : (A4) dx dx 0 0 Sn ðT ÞdT I2 Rint Tc ¼ 2 2 Voc Voc Rint ð1 þ mÞ Rint ð1 þ mÞ2 ¼ 2 Voc m : Rint ð1 þ mÞ2 (A7) Eq. (A7) for the output power generation taking Thomson effect into account is analytically exact expression based on Rint with linear approximation of dx (L/DT)dT, and it is proved that integrating the Seebeck coefficient with respect to temperature gives rise to the open circuit voltage influenced by Thomson effect. APPENDIX B: ½ZT eng VS. AVERAGED (ZT)eng To examine the relative difference of averaged (ZT)eng and ½ZTeng i 1h ðZT Þeng;p þ ðZT Þeng;n ½ZT eng 2 ! 2 2 S^n 1 S^p ðSp Sn Þ2 ¼ DT þ DT pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 DT; ^ pj ^p ^ nj ^n q 2 q qp jp þ qn jn <¼ (B1) where symbol with a hat indicates the integral with respect to Th and Tc. Eq. (B1) is reduced to 2 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ12 !2 0 2 2 ^ ^ ^ pj ^p S p DT 6 S q ^ nj ^ nA 7 S^n @1 þ q 41 þ p2 5; 2 1 <¼ ^ ^p ^ pj ^p q 2^ qpj Sp ^ j ^ S^ q n n n (B2) which is a function of the ratios of p-type to n-type properties. < can be positive or negative value depending on the ratio of ^ n =^ ^ p , e.g., < > 0 for p-SnSe/n-PbSe as ^ nj qpj S^n =S^p and q shown in Fig. 5(e), and (ZT)eng and ½ZTeng become compara^ n =^ ^ p 1. ^ nj ble, i.e., < 0 when S^p =S^n 1 and q qpj 115103-9 1 Kim, Liu, and Ren D. M. Rowe, CRC Handbook of Thermoelectrics (CRC Press, 1995). L. E. Bell, Science 321, 1457 (2008). 3 G. J. Snyder and E. S. Toberer, Nat. Mater. 7, 105 (2008). 4 J. Yang, H. L. Yip, and A. Jen, Adv. Energy Mater. 3, 549 (2013). 5 W. Liu, Q. Jie, H. S. Kim, and Z. Ren, Acta Mater. 87, 357 (2015). 6 S. W. Angrist, Direct Energy Conversion (Allyn and Bacon, Boston, 1965). 7 J. P. Thomas, M. A. Qidwai, and J. C. Kellogg, J. Power Sources 159, 1494 (2006). 8 H. S. Kim, T. Itoh, T. Iida, M. 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